QT5201U wiki - User contributions [en]

Optical control of TMDCs valley pseudospin qubits

2021-04-30T16:30:08Z

Chu WenHao: /* Numerical calculation results by Quantum Espresso */

==Linked project==
* This project is conducted by the same people as 👉 [[Control over the atomic spins within certain molecules by NMR technique]]

==Group members==
*'''XU ZIZHOU (Bensen)'''
**'''Matric Number:''' ''A0229645W''
**'''Email:''' ''[mailto:info@example.org zizhou_xu@u.nus.edu]''
*'''CHU WENHAO'''
**'''Matric Number:''' ''A0228747R''
**'''Email:''' ''[mailto:info@example.org e0675585@u.nus.edu]''
==Description==

* The strategies and materials for building universal quantum computers are being actively researched, and various platforms for holding qubits have been proposed and tested<ref>Experimental perfect state transfer of an entangled photonic qubit[https://www.nature.com/articles/ncomms11339]</ref><ref>Superconducting Qubits: Current State of Play[https://www.annualreviews.org/doi/abs/10.1146/annurev-conmatphys-031119-050605]</ref><ref>Two-qubit entangling gates within arbitrarily long chains of trapped ions[https://journals.aps.org/pra/abstract/10.1103/PhysRevA.100.022332]</ref><ref>Digital Coherent Control of a Superconducting Qubit[https://journals.aps.org/prapplied/abstract/10.1103/PhysRevApplied.11.014009]</ref><ref>Fast quantum logic gates with trapped-ion qubits[https://www.nature.com/articles/nature25737]</ref>. Within all the options such as superconducting qubits and ion trap qubits, solid-state qubits have illustrated great benefits of the compatibility with the existing semiconductor technology, which attracts lots of attention to this platform. Moreover, because of the direct band gap and strong spin-orbit coupling discovered in the monolayer transition metal dichalcogenides (TMDCs), especially some semiconductor monolayers possess a sizeable direct bandgap of ≈1.5–2 eV in the optical range allowing electrostatic confinement and optical manipulation of carriers<ref>Atomically Thin MoS2: A New Direct-Gap Semiconductor[https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.105.136805]</ref><ref>Emerging Photoluminescence in Monolayer MoS2[https://pubs.acs.org/doi/10.1021/nl903868w]</ref>, chances are TMDCs materials have the potential to become the promising platform for fabricating qubits. There are four main types of TMDCs that have been suggested as advantageous in acting qubits, <math>\text{WS}_2</math> , <math>\text{WSe}_2</math> , <math>\text{MoS}_2</math> and <math>\text{MoSe}_2</math> . Kormányos et al used DFT calculations to confirm that <math>\text{WS}_2</math> and <math>\text{WSe}_2</math> , are better than <math>\text{MoS}_2</math> and <math>\text{MoSe}_2</math> in terms of the spin-valley coupling<ref>Spin-Orbit Coupling, Quantum Dots, and Qubits in Monolayer Transition Metal Dichalcogenides[https://journals.aps.org/prx/abstract/10.1103/PhysRevX.4.011034]</ref>, which is really important since spin-valley coupling is regarded as a key factor that can improve the coherence lifetime of spin-valley states.

* After several decades of active research on the properties of TMDCs, some interesting features have been discovered. Such as the extra degree of freedom offered by valley magnetic moment, which can realise bit 0 and bit 1 to form a qubit in momentum space. In addition, these valleys are independently addressable by an optical signal, and optical controlling technique is currently under research by many groups both theoretically and experimentally. As a result, as inspired by the idea from one of the theoretical work<ref>Spin-valley qubit in nanostructures of monolayer semiconductors: Optical control and hyperfine interaction[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.93.045313]</ref>, we intend to build a quantum simulator and investigate optical control of the spin-valley pseudospin on 2D heterobilayer TMDCs.

==Q & A==

===What are K & K' points?===
[[File:K_&_K'_points.png|thumb|400px|K & K' points]]
* K-points are sampling points of Brillouin zone in reciprocal lattice
* The monolayer TMDs have a unique band structure with the conduction and valence band edges both at the degenerate K and K‘ valleys at the corners of the hexagonal Brillouin zone. The direct-gap optical transitions have a selection rule: left- (right-) handed circular polarized photons couple to the interband transitions in the K (K’) valley only.
===The Tight-Binding Model===
* By introducing the tight-binding model for TMDs, we will give a Hamiltonian of TMDs that can be further deduced and calculated. As is showned in figure (b):under the tight-binding ansatz,the atom(e.g.the red one)can only jump to the nearest position, whose interaction processing is called "hopping" in physics,If we use the lattice represetation and the corresponding creation operators commonly used in condensed matter physics,then the nearest-neighbor tight-binding Hamiltonian has the simple form:

[[File:graphene.png|thumb|400px|Graphene]]

<math>
\hat{H}=-t\sum_{l} \sum_{j=1}^{3}a_{l+\tau_j}^{\dagger}b_{l}+h.c.
</math>

By Fourier Transformation,<math>a_l</math> and <math>b_{l}</math> can be written as:

<math>
a_l=\frac{1}{\sqrt{N}}\sum_{k} e^{ik\cdot R_l}\alpha_k
</math>

<math>
b_l=\frac{1}{\sqrt{N}}\sum_{k} e^{ik\cdot R_l}\beta_k
</math>

<math>
\tau_1=(0,\frac{1}{\sqrt{3}})a,\tau_2=(-\frac{1}{2},-\frac{1}{2\sqrt{3}})a,\tau_3=(\frac{1}{2},-\frac{1}{2\sqrt{3}})a
</math>

Then we can rewrite the Hamiltonian in reciprocal space:

<math>
\hat{H}=\sum_{k}c^{\dagger}(k)\begin{bmatrix}
0&f^{*}\\
f&0
\end{bmatrix}c(k),\qquad c(k)=\begin{bmatrix}
\alpha(k)\\
\beta(k)
\end{bmatrix},\qquad f=-t\sum_{j=1}^{3}e^{ik\cdot\tau_{j}}
</math>
===Choice of K values===
*We define the first Brillouin zone of the reciprocal lattice in the standard way.It is clear that the six points at the corners of the FBZ fall into two groups of three which are equivalent, so we need consider only two equivalent corners that we label K and K0 as in the figure. Explicitly, their positions in momentum space are given by:

[[File:K values.png|thumb|400px|K values]]

<math>
K=\frac{2\pi}{a}(\frac{1}{3},\frac{1}{\sqrt{3}}),\qquad K'=\frac{2\pi}{a}(\frac{2}{3},0)
</math>

These K and K' point are also called "Dirac points".
===Solution of TB model Hamiltonian===
<ref>Neto, AH Castro, et al. "The electronic properties of graphene." Reviews of modern physics 81.1 (2009): 109.[https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.81.109]</ref>
*We can rewrite the Schrödinger equation in reciprocal space

<math>
H(k)\Psi_k=E\Psi_k
</math>

From the explicit expressions for the nearest neighbor vectors we obtain:

<math>
f=-t\exp(ik_xa(1+2e^{i\cdot\frac{3k_xa}{2}}cos\frac{\sqrt{3}}{2}k_ya))
</math>

And the eigenvalues are given by

<math>
\epsilon_{s,k}=s|f|=\pm t(1+4cos(\frac{3}{2}k_xa)cos(\frac{\sqrt{3}}{2}k_ya)+4cos^2\frac{\sqrt{3}}{2}k_ya)^{\frac{1}{2}}
</math>

It is interesting to enquire whether there are any values of k for which f is zero? Any such value must satisfy the conditions? Luckily, the answer is yes and the points "at the corner" which satisfied these conditions are exactly the "Dirac points".

===Dirac model near K points--Not spins but like spins===
*With the Dirac model in Quantum Fireld Theory , We have the Dirac's original:

<math>(i\gamma^\mu\partial_\mu-m)\Psi=0</math>,
<math>\gamma^\mu\gamma^v+\gamma^v\gamma^\mu=2g^{v\mu}</math>,
<math>v,\mu=0,1,2,3</math>

It is very suggestive to write the Hamiltonian in the form:

[[File:Dirac points.png|thumb|400px|Dirac points]]

<math>
\hat{H}=\hbar v_F\begin{bmatrix}
0&k_x+ik_y\\
k_x-ik_y&0
\end{bmatrix}=\hbar v_F\hat{\sigma}\cdot q,\qquad \epsilon(k)=\pm v_F|k|
</math>

where the components of the operator <math>\sigma</math> are the usual Pauli matrices,and <math>v_F</math> is the Fermi velocity. It is clear that the eigenvalues are a function only of the magnitude of k, not its direction in the 2D space,therefore, although the Hamiltonian expression contains spin operators, the Dirac point derived from this is not the spin of particle or quasiparticle, (which can be compared with the quasiparticle ansatz in BCS theory).However, it still inspires us to regard these K and K' points of TMDs as some kinds of qubit resources and control them on this theoretical basis.

===Numerical calculation results by Quantum Espresso===
<ref>Ghosal, Supriya, Arka Bandyopadhyay, and Debnarayan Jana. "Electric field induced band tuning, optical and thermoelectric responses in tetragonal germanene: a theoretical approach." Physical Chemistry Chemical Physics 22.35 (2020): 19957-19968.[https://pubs.rsc.org/en/content/articlelanding/cp/2020/d0cp03892j#!divAbstract]</ref>
[[File:Dirac points in Band structure.png|thumb|400px|Band structure]]

The Dirac points in the band structure of graphene are shown in the Figure, In the next step, we scheduled to add the effect of external filed (like magnetic field or optical field), which will change the hamiltonian and choose the appropriate pseudopotential to calculate the change of K and K' point. We plan to use Quantum Espresso as our calculation programme.

[[File:QE.png|thumb|400px|QE code]]

===Why choose heterobilayer TMDCs instead of monolayer===
* There are four main carrier properties that optimal Opto-valleytronics should possess.
*# long carrier lifetime
*# long valley lifetime
*# high valley polarization
*# long valley coherence time
* By adopting 2D heterostructure TMDCs materials, we can create these conditions for building promising quantum platform. (eg. Due to the type II band alignment and weak hybridization of van der Waals heterostructure, the electron–hole layer separation, the electron–hole exchange interaction is greatly reduced, resulting in a long cryogenic lifetime (ns to <math>\mu</math>s)and valley lifetime (~ 10 ns) of the interlayer exciton<ref>Opto-valleytronics in the 2D van der Waals heterostructure[https://doi.org/10.1007/s12274-020-3036-x]</ref>)

==Method==
===Principle===
* It was shown that inversion symmetry breaking can lead to opposite circular dichroism in different momentum space regions, which induces an interesting phenomena of optical selection rules at the symmetry corner points of the Brillouin zone. This enables a valley-dependent control of electrons with optical signal of different circular polarizations.
* However, valley optical selection rule in the heterobilayer TMDCs is a bit different than in the monolayer, which depends on the emission location and the emission energy. Additionally, in most cases, owing to the type II band alignment and weak hybridization, the conduction band minimum and valence band maximum of the van der Waals heterostructure are dominated by orbitals from different layers. As a result, electrons and holes tend to reside in different layers of the structure.
* It was found that the g-factor value of the interlayer exciton depends on the stacking sequence of the layers. For the AA-stacked <math>\text{MoSe}_2</math>/<math>\text{WSe}_2</math>, the value of the singlet g-factor between 6 to 7 has been reported; while, for the AB-stacked <math>\text{WSe}_2</math>/<math>\text{MoSe}_2</math>, the value of g-factor ~ −15 has been reported

===Experimental setup===

====Setup====
[[File:Experimental Setup.jpg|thumb|400px|Experimental Setup]]
* As shown in figure

====Quantum simulator====
* The <math>\text{MoSe}_2</math>/<math>\text{WSe}_2</math> heterostructure is fabricated via mechanical exfoliation and aligned transfer method. The stacking sequence of the sample is checked using second harmonic generation (SHG)-based measurement and it is found to be AA stacking. The sample is loaded into a magnetocryostat to control the magnetic ﬁeld and sample temperature.

====Qubit initialisation====
* Use circularly polarised light to create valley exciton polarization
* Use linearly polarised light to pump the qubits to rotate the valley pseudospin in the superposition plane of K and K′ valley polarizations
* The polarization state of the excitation and collection is controlled using a combination of polarizer, quarter-wave plate, and half-wave plate.

====Qubit control====
* Most popular control techniques these days
*# 1st approach: The coupling between the optical (electromagnetic) signal and the valley information can also be used to control the valley information. For the intralayer exciton, the exciton can be moved from one valley to another by applying intense THz radiation. This is shown by the observation of linearly polarized emission from higher-order sidebands when the sample is excited with a 100-fs near-resonant circularly polarized light pulse together with a 33-fs 40-THz pulse. Theoretically, it is shown that this is caused by a THz pulse driven intervalley transition. It is also predicted that a complete transfer from K (K’) to K’ (K) valley can be done within a period less than 5 fs if a more intense and shorter THz pulse is used.
*# 2nd approach: On the other hand, by utilizing the optical Stark effect, the phase control has been demonstrated. In this case, the valley state is first prepared in the <math>|K\rang+|K'\rang</math> state by using a near-resonant linearly polarized excitation. A red-detuned 100-fs circularly polarized light pulse is then used to break the valley degeneracy, which allows the coherent rotation of the valley state to the <math>|K\rang+e^{i\Delta\phi}|K'\rang</math> state with <math>\Delta\phi</math> depends on the pulse intensity and duration. This rotation is reflected in the linear polarization state of the exciton emission.

====Qubit readout====
* The PL emission is directed by a multimode optical ﬁber into a spectrometer (Andor Shamrock) with a CCD detector for spectroscopic recording

==Results==

==References==
<references />

Optical control of TMDCs valley pseudospin qubits

2021-04-30T16:29:10Z

Chu WenHao: /* Numerical calculation results by Quantum Espresso */

==Linked project==
* This project is conducted by the same people as 👉 [[Control over the atomic spins within certain molecules by NMR technique]]

==Group members==
*'''XU ZIZHOU (Bensen)'''
**'''Matric Number:''' ''A0229645W''
**'''Email:''' ''[mailto:info@example.org zizhou_xu@u.nus.edu]''
*'''CHU WENHAO'''
**'''Matric Number:''' ''A0228747R''
**'''Email:''' ''[mailto:info@example.org e0675585@u.nus.edu]''
==Description==

* The strategies and materials for building universal quantum computers are being actively researched, and various platforms for holding qubits have been proposed and tested<ref>Experimental perfect state transfer of an entangled photonic qubit[https://www.nature.com/articles/ncomms11339]</ref><ref>Superconducting Qubits: Current State of Play[https://www.annualreviews.org/doi/abs/10.1146/annurev-conmatphys-031119-050605]</ref><ref>Two-qubit entangling gates within arbitrarily long chains of trapped ions[https://journals.aps.org/pra/abstract/10.1103/PhysRevA.100.022332]</ref><ref>Digital Coherent Control of a Superconducting Qubit[https://journals.aps.org/prapplied/abstract/10.1103/PhysRevApplied.11.014009]</ref><ref>Fast quantum logic gates with trapped-ion qubits[https://www.nature.com/articles/nature25737]</ref>. Within all the options such as superconducting qubits and ion trap qubits, solid-state qubits have illustrated great benefits of the compatibility with the existing semiconductor technology, which attracts lots of attention to this platform. Moreover, because of the direct band gap and strong spin-orbit coupling discovered in the monolayer transition metal dichalcogenides (TMDCs), especially some semiconductor monolayers possess a sizeable direct bandgap of ≈1.5–2 eV in the optical range allowing electrostatic confinement and optical manipulation of carriers<ref>Atomically Thin MoS2: A New Direct-Gap Semiconductor[https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.105.136805]</ref><ref>Emerging Photoluminescence in Monolayer MoS2[https://pubs.acs.org/doi/10.1021/nl903868w]</ref>, chances are TMDCs materials have the potential to become the promising platform for fabricating qubits. There are four main types of TMDCs that have been suggested as advantageous in acting qubits, <math>\text{WS}_2</math> , <math>\text{WSe}_2</math> , <math>\text{MoS}_2</math> and <math>\text{MoSe}_2</math> . Kormányos et al used DFT calculations to confirm that <math>\text{WS}_2</math> and <math>\text{WSe}_2</math> , are better than <math>\text{MoS}_2</math> and <math>\text{MoSe}_2</math> in terms of the spin-valley coupling<ref>Spin-Orbit Coupling, Quantum Dots, and Qubits in Monolayer Transition Metal Dichalcogenides[https://journals.aps.org/prx/abstract/10.1103/PhysRevX.4.011034]</ref>, which is really important since spin-valley coupling is regarded as a key factor that can improve the coherence lifetime of spin-valley states.

* After several decades of active research on the properties of TMDCs, some interesting features have been discovered. Such as the extra degree of freedom offered by valley magnetic moment, which can realise bit 0 and bit 1 to form a qubit in momentum space. In addition, these valleys are independently addressable by an optical signal, and optical controlling technique is currently under research by many groups both theoretically and experimentally. As a result, as inspired by the idea from one of the theoretical work<ref>Spin-valley qubit in nanostructures of monolayer semiconductors: Optical control and hyperfine interaction[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.93.045313]</ref>, we intend to build a quantum simulator and investigate optical control of the spin-valley pseudospin on 2D heterobilayer TMDCs.

==Q & A==

===What are K & K' points?===
[[File:K_&_K'_points.png|thumb|400px|K & K' points]]
* K-points are sampling points of Brillouin zone in reciprocal lattice
* The monolayer TMDs have a unique band structure with the conduction and valence band edges both at the degenerate K and K‘ valleys at the corners of the hexagonal Brillouin zone. The direct-gap optical transitions have a selection rule: left- (right-) handed circular polarized photons couple to the interband transitions in the K (K’) valley only.
===The Tight-Binding Model===
* By introducing the tight-binding model for TMDs, we will give a Hamiltonian of TMDs that can be further deduced and calculated. As is showned in figure (b):under the tight-binding ansatz,the atom(e.g.the red one)can only jump to the nearest position, whose interaction processing is called "hopping" in physics,If we use the lattice represetation and the corresponding creation operators commonly used in condensed matter physics,then the nearest-neighbor tight-binding Hamiltonian has the simple form:

[[File:graphene.png|thumb|400px|Graphene]]

<math>
\hat{H}=-t\sum_{l} \sum_{j=1}^{3}a_{l+\tau_j}^{\dagger}b_{l}+h.c.
</math>

By Fourier Transformation,<math>a_l</math> and <math>b_{l}</math> can be written as:

<math>
a_l=\frac{1}{\sqrt{N}}\sum_{k} e^{ik\cdot R_l}\alpha_k
</math>

<math>
b_l=\frac{1}{\sqrt{N}}\sum_{k} e^{ik\cdot R_l}\beta_k
</math>

<math>
\tau_1=(0,\frac{1}{\sqrt{3}})a,\tau_2=(-\frac{1}{2},-\frac{1}{2\sqrt{3}})a,\tau_3=(\frac{1}{2},-\frac{1}{2\sqrt{3}})a
</math>

Then we can rewrite the Hamiltonian in reciprocal space:

<math>
\hat{H}=\sum_{k}c^{\dagger}(k)\begin{bmatrix}
0&f^{*}\\
f&0
\end{bmatrix}c(k),\qquad c(k)=\begin{bmatrix}
\alpha(k)\\
\beta(k)
\end{bmatrix},\qquad f=-t\sum_{j=1}^{3}e^{ik\cdot\tau_{j}}
</math>
===Choice of K values===
*We define the first Brillouin zone of the reciprocal lattice in the standard way.It is clear that the six points at the corners of the FBZ fall into two groups of three which are equivalent, so we need consider only two equivalent corners that we label K and K0 as in the figure. Explicitly, their positions in momentum space are given by:

[[File:K values.png|thumb|400px|K values]]

<math>
K=\frac{2\pi}{a}(\frac{1}{3},\frac{1}{\sqrt{3}}),\qquad K'=\frac{2\pi}{a}(\frac{2}{3},0)
</math>

These K and K' point are also called "Dirac points".
===Solution of TB model Hamiltonian===
<ref>Neto, AH Castro, et al. "The electronic properties of graphene." Reviews of modern physics 81.1 (2009): 109.[https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.81.109]</ref>
*We can rewrite the Schrödinger equation in reciprocal space

<math>
H(k)\Psi_k=E\Psi_k
</math>

From the explicit expressions for the nearest neighbor vectors we obtain:

<math>
f=-t\exp(ik_xa(1+2e^{i\cdot\frac{3k_xa}{2}}cos\frac{\sqrt{3}}{2}k_ya))
</math>

And the eigenvalues are given by

<math>
\epsilon_{s,k}=s|f|=\pm t(1+4cos(\frac{3}{2}k_xa)cos(\frac{\sqrt{3}}{2}k_ya)+4cos^2\frac{\sqrt{3}}{2}k_ya)^{\frac{1}{2}}
</math>

It is interesting to enquire whether there are any values of k for which f is zero? Any such value must satisfy the conditions? Luckily, the answer is yes and the points "at the corner" which satisfied these conditions are exactly the "Dirac points".

===Dirac model near K points--Not spins but like spins===
*With the Dirac model in Quantum Fireld Theory , We have the Dirac's original:

<math>(i\gamma^\mu\partial_\mu-m)\Psi=0</math>,
<math>\gamma^\mu\gamma^v+\gamma^v\gamma^\mu=2g^{v\mu}</math>,
<math>v,\mu=0,1,2,3</math>

It is very suggestive to write the Hamiltonian in the form:

[[File:Dirac points.png|thumb|400px|Dirac points]]

<math>
\hat{H}=\hbar v_F\begin{bmatrix}
0&k_x+ik_y\\
k_x-ik_y&0
\end{bmatrix}=\hbar v_F\hat{\sigma}\cdot q,\qquad \epsilon(k)=\pm v_F|k|
</math>

where the components of the operator <math>\sigma</math> are the usual Pauli matrices,and <math>v_F</math> is the Fermi velocity. It is clear that the eigenvalues are a function only of the magnitude of k, not its direction in the 2D space,therefore, although the Hamiltonian expression contains spin operators, the Dirac point derived from this is not the spin of particle or quasiparticle, (which can be compared with the quasiparticle ansatz in BCS theory).However, it still inspires us to regard these K and K' points of TMDs as some kinds of qubit resources and control them on this theoretical basis.

===Numerical calculation results by Quantum Espresso===
<ref>Ghosal, Supriya, Arka Bandyopadhyay, and Debnarayan Jana. "Electric field induced band tuning, optical and thermoelectric responses in tetragonal germanene: a theoretical approach." Physical Chemistry Chemical Physics 22.35 (2020): 19957-19968.</ref>
[[File:Dirac points in Band structure.png|thumb|400px|Band structure]]

The Dirac points in the band structure of graphene are shown in the Figure, In the next step, we scheduled to add the effect of external filed (like magnetic field or optical field), which will change the hamiltonian and choose the appropriate pseudopotential to calculate the change of K and K' point. We plan to use Quantum Espresso as our calculation programme.

[[File:QE.png|thumb|400px|QE code]]

===Why choose heterobilayer TMDCs instead of monolayer===
* There are four main carrier properties that optimal Opto-valleytronics should possess.
*# long carrier lifetime
*# long valley lifetime
*# high valley polarization
*# long valley coherence time
* By adopting 2D heterostructure TMDCs materials, we can create these conditions for building promising quantum platform. (eg. Due to the type II band alignment and weak hybridization of van der Waals heterostructure, the electron–hole layer separation, the electron–hole exchange interaction is greatly reduced, resulting in a long cryogenic lifetime (ns to <math>\mu</math>s)and valley lifetime (~ 10 ns) of the interlayer exciton<ref>Opto-valleytronics in the 2D van der Waals heterostructure[https://doi.org/10.1007/s12274-020-3036-x]</ref>)

==Method==
===Principle===
* It was shown that inversion symmetry breaking can lead to opposite circular dichroism in different momentum space regions, which induces an interesting phenomena of optical selection rules at the symmetry corner points of the Brillouin zone. This enables a valley-dependent control of electrons with optical signal of different circular polarizations.
* However, valley optical selection rule in the heterobilayer TMDCs is a bit different than in the monolayer, which depends on the emission location and the emission energy. Additionally, in most cases, owing to the type II band alignment and weak hybridization, the conduction band minimum and valence band maximum of the van der Waals heterostructure are dominated by orbitals from different layers. As a result, electrons and holes tend to reside in different layers of the structure.
* It was found that the g-factor value of the interlayer exciton depends on the stacking sequence of the layers. For the AA-stacked <math>\text{MoSe}_2</math>/<math>\text{WSe}_2</math>, the value of the singlet g-factor between 6 to 7 has been reported; while, for the AB-stacked <math>\text{WSe}_2</math>/<math>\text{MoSe}_2</math>, the value of g-factor ~ −15 has been reported

===Experimental setup===

====Setup====
[[File:Experimental Setup.jpg|thumb|400px|Experimental Setup]]
* As shown in figure

====Quantum simulator====
* The <math>\text{MoSe}_2</math>/<math>\text{WSe}_2</math> heterostructure is fabricated via mechanical exfoliation and aligned transfer method. The stacking sequence of the sample is checked using second harmonic generation (SHG)-based measurement and it is found to be AA stacking. The sample is loaded into a magnetocryostat to control the magnetic ﬁeld and sample temperature.

====Qubit initialisation====
* Use circularly polarised light to create valley exciton polarization
* Use linearly polarised light to pump the qubits to rotate the valley pseudospin in the superposition plane of K and K′ valley polarizations
* The polarization state of the excitation and collection is controlled using a combination of polarizer, quarter-wave plate, and half-wave plate.

====Qubit control====
* Most popular control techniques these days
*# 1st approach: The coupling between the optical (electromagnetic) signal and the valley information can also be used to control the valley information. For the intralayer exciton, the exciton can be moved from one valley to another by applying intense THz radiation. This is shown by the observation of linearly polarized emission from higher-order sidebands when the sample is excited with a 100-fs near-resonant circularly polarized light pulse together with a 33-fs 40-THz pulse. Theoretically, it is shown that this is caused by a THz pulse driven intervalley transition. It is also predicted that a complete transfer from K (K’) to K’ (K) valley can be done within a period less than 5 fs if a more intense and shorter THz pulse is used.
*# 2nd approach: On the other hand, by utilizing the optical Stark effect, the phase control has been demonstrated. In this case, the valley state is first prepared in the <math>|K\rang+|K'\rang</math> state by using a near-resonant linearly polarized excitation. A red-detuned 100-fs circularly polarized light pulse is then used to break the valley degeneracy, which allows the coherent rotation of the valley state to the <math>|K\rang+e^{i\Delta\phi}|K'\rang</math> state with <math>\Delta\phi</math> depends on the pulse intensity and duration. This rotation is reflected in the linear polarization state of the exciton emission.

====Qubit readout====
* The PL emission is directed by a multimode optical ﬁber into a spectrometer (Andor Shamrock) with a CCD detector for spectroscopic recording

==Results==

==References==
<references />

Optical control of TMDCs valley pseudospin qubits

2021-04-30T16:28:06Z

Chu WenHao: /* Solution of TB model Hamiltonian */

==Linked project==
* This project is conducted by the same people as 👉 [[Control over the atomic spins within certain molecules by NMR technique]]

==Group members==
*'''XU ZIZHOU (Bensen)'''
**'''Matric Number:''' ''A0229645W''
**'''Email:''' ''[mailto:info@example.org zizhou_xu@u.nus.edu]''
*'''CHU WENHAO'''
**'''Matric Number:''' ''A0228747R''
**'''Email:''' ''[mailto:info@example.org e0675585@u.nus.edu]''
==Description==

* The strategies and materials for building universal quantum computers are being actively researched, and various platforms for holding qubits have been proposed and tested<ref>Experimental perfect state transfer of an entangled photonic qubit[https://www.nature.com/articles/ncomms11339]</ref><ref>Superconducting Qubits: Current State of Play[https://www.annualreviews.org/doi/abs/10.1146/annurev-conmatphys-031119-050605]</ref><ref>Two-qubit entangling gates within arbitrarily long chains of trapped ions[https://journals.aps.org/pra/abstract/10.1103/PhysRevA.100.022332]</ref><ref>Digital Coherent Control of a Superconducting Qubit[https://journals.aps.org/prapplied/abstract/10.1103/PhysRevApplied.11.014009]</ref><ref>Fast quantum logic gates with trapped-ion qubits[https://www.nature.com/articles/nature25737]</ref>. Within all the options such as superconducting qubits and ion trap qubits, solid-state qubits have illustrated great benefits of the compatibility with the existing semiconductor technology, which attracts lots of attention to this platform. Moreover, because of the direct band gap and strong spin-orbit coupling discovered in the monolayer transition metal dichalcogenides (TMDCs), especially some semiconductor monolayers possess a sizeable direct bandgap of ≈1.5–2 eV in the optical range allowing electrostatic confinement and optical manipulation of carriers<ref>Atomically Thin MoS2: A New Direct-Gap Semiconductor[https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.105.136805]</ref><ref>Emerging Photoluminescence in Monolayer MoS2[https://pubs.acs.org/doi/10.1021/nl903868w]</ref>, chances are TMDCs materials have the potential to become the promising platform for fabricating qubits. There are four main types of TMDCs that have been suggested as advantageous in acting qubits, <math>\text{WS}_2</math> , <math>\text{WSe}_2</math> , <math>\text{MoS}_2</math> and <math>\text{MoSe}_2</math> . Kormányos et al used DFT calculations to confirm that <math>\text{WS}_2</math> and <math>\text{WSe}_2</math> , are better than <math>\text{MoS}_2</math> and <math>\text{MoSe}_2</math> in terms of the spin-valley coupling<ref>Spin-Orbit Coupling, Quantum Dots, and Qubits in Monolayer Transition Metal Dichalcogenides[https://journals.aps.org/prx/abstract/10.1103/PhysRevX.4.011034]</ref>, which is really important since spin-valley coupling is regarded as a key factor that can improve the coherence lifetime of spin-valley states.

* After several decades of active research on the properties of TMDCs, some interesting features have been discovered. Such as the extra degree of freedom offered by valley magnetic moment, which can realise bit 0 and bit 1 to form a qubit in momentum space. In addition, these valleys are independently addressable by an optical signal, and optical controlling technique is currently under research by many groups both theoretically and experimentally. As a result, as inspired by the idea from one of the theoretical work<ref>Spin-valley qubit in nanostructures of monolayer semiconductors: Optical control and hyperfine interaction[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.93.045313]</ref>, we intend to build a quantum simulator and investigate optical control of the spin-valley pseudospin on 2D heterobilayer TMDCs.

==Q & A==

===What are K & K' points?===
[[File:K_&_K'_points.png|thumb|400px|K & K' points]]
* K-points are sampling points of Brillouin zone in reciprocal lattice
* The monolayer TMDs have a unique band structure with the conduction and valence band edges both at the degenerate K and K‘ valleys at the corners of the hexagonal Brillouin zone. The direct-gap optical transitions have a selection rule: left- (right-) handed circular polarized photons couple to the interband transitions in the K (K’) valley only.
===The Tight-Binding Model===
* By introducing the tight-binding model for TMDs, we will give a Hamiltonian of TMDs that can be further deduced and calculated. As is showned in figure (b):under the tight-binding ansatz,the atom(e.g.the red one)can only jump to the nearest position, whose interaction processing is called "hopping" in physics,If we use the lattice represetation and the corresponding creation operators commonly used in condensed matter physics,then the nearest-neighbor tight-binding Hamiltonian has the simple form:

[[File:graphene.png|thumb|400px|Graphene]]

<math>
\hat{H}=-t\sum_{l} \sum_{j=1}^{3}a_{l+\tau_j}^{\dagger}b_{l}+h.c.
</math>

By Fourier Transformation,<math>a_l</math> and <math>b_{l}</math> can be written as:

<math>
a_l=\frac{1}{\sqrt{N}}\sum_{k} e^{ik\cdot R_l}\alpha_k
</math>

<math>
b_l=\frac{1}{\sqrt{N}}\sum_{k} e^{ik\cdot R_l}\beta_k
</math>

<math>
\tau_1=(0,\frac{1}{\sqrt{3}})a,\tau_2=(-\frac{1}{2},-\frac{1}{2\sqrt{3}})a,\tau_3=(\frac{1}{2},-\frac{1}{2\sqrt{3}})a
</math>

Then we can rewrite the Hamiltonian in reciprocal space:

<math>
\hat{H}=\sum_{k}c^{\dagger}(k)\begin{bmatrix}
0&f^{*}\\
f&0
\end{bmatrix}c(k),\qquad c(k)=\begin{bmatrix}
\alpha(k)\\
\beta(k)
\end{bmatrix},\qquad f=-t\sum_{j=1}^{3}e^{ik\cdot\tau_{j}}
</math>
===Choice of K values===
*We define the first Brillouin zone of the reciprocal lattice in the standard way.It is clear that the six points at the corners of the FBZ fall into two groups of three which are equivalent, so we need consider only two equivalent corners that we label K and K0 as in the figure. Explicitly, their positions in momentum space are given by:

[[File:K values.png|thumb|400px|K values]]

<math>
K=\frac{2\pi}{a}(\frac{1}{3},\frac{1}{\sqrt{3}}),\qquad K'=\frac{2\pi}{a}(\frac{2}{3},0)
</math>

These K and K' point are also called "Dirac points".
===Solution of TB model Hamiltonian===
<ref>Neto, AH Castro, et al. "The electronic properties of graphene." Reviews of modern physics 81.1 (2009): 109.[https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.81.109]</ref>
*We can rewrite the Schrödinger equation in reciprocal space

<math>
H(k)\Psi_k=E\Psi_k
</math>

From the explicit expressions for the nearest neighbor vectors we obtain:

<math>
f=-t\exp(ik_xa(1+2e^{i\cdot\frac{3k_xa}{2}}cos\frac{\sqrt{3}}{2}k_ya))
</math>

And the eigenvalues are given by

<math>
\epsilon_{s,k}=s|f|=\pm t(1+4cos(\frac{3}{2}k_xa)cos(\frac{\sqrt{3}}{2}k_ya)+4cos^2\frac{\sqrt{3}}{2}k_ya)^{\frac{1}{2}}
</math>

It is interesting to enquire whether there are any values of k for which f is zero? Any such value must satisfy the conditions? Luckily, the answer is yes and the points "at the corner" which satisfied these conditions are exactly the "Dirac points".

===Dirac model near K points--Not spins but like spins===
*With the Dirac model in Quantum Fireld Theory , We have the Dirac's original:

<math>(i\gamma^\mu\partial_\mu-m)\Psi=0</math>,
<math>\gamma^\mu\gamma^v+\gamma^v\gamma^\mu=2g^{v\mu}</math>,
<math>v,\mu=0,1,2,3</math>

It is very suggestive to write the Hamiltonian in the form:

[[File:Dirac points.png|thumb|400px|Dirac points]]

<math>
\hat{H}=\hbar v_F\begin{bmatrix}
0&k_x+ik_y\\
k_x-ik_y&0
\end{bmatrix}=\hbar v_F\hat{\sigma}\cdot q,\qquad \epsilon(k)=\pm v_F|k|
</math>

where the components of the operator <math>\sigma</math> are the usual Pauli matrices,and <math>v_F</math> is the Fermi velocity. It is clear that the eigenvalues are a function only of the magnitude of k, not its direction in the 2D space,therefore, although the Hamiltonian expression contains spin operators, the Dirac point derived from this is not the spin of particle or quasiparticle, (which can be compared with the quasiparticle ansatz in BCS theory).However, it still inspires us to regard these K and K' points of TMDs as some kinds of qubit resources and control them on this theoretical basis.

===Numerical calculation results by Quantum Espresso===
[[File:Dirac points in Band structure.png|thumb|400px|Band structure]]

The Dirac points in the band structure of graphene are shown in the Figure, In the next step, we scheduled to add the effect of external filed (like magnetic field or optical field), which will change the hamiltonian and choose the appropriate pseudopotential to calculate the change of K and K' point. We plan to use Quantum Espresso as our calculation programme.

[[File:QE.png|thumb|400px|QE code]]

===Why choose heterobilayer TMDCs instead of monolayer===
* There are four main carrier properties that optimal Opto-valleytronics should possess.
*# long carrier lifetime
*# long valley lifetime
*# high valley polarization
*# long valley coherence time
* By adopting 2D heterostructure TMDCs materials, we can create these conditions for building promising quantum platform. (eg. Due to the type II band alignment and weak hybridization of van der Waals heterostructure, the electron–hole layer separation, the electron–hole exchange interaction is greatly reduced, resulting in a long cryogenic lifetime (ns to <math>\mu</math>s)and valley lifetime (~ 10 ns) of the interlayer exciton<ref>Opto-valleytronics in the 2D van der Waals heterostructure[https://doi.org/10.1007/s12274-020-3036-x]</ref>)

==Method==
===Principle===
* It was shown that inversion symmetry breaking can lead to opposite circular dichroism in different momentum space regions, which induces an interesting phenomena of optical selection rules at the symmetry corner points of the Brillouin zone. This enables a valley-dependent control of electrons with optical signal of different circular polarizations.
* However, valley optical selection rule in the heterobilayer TMDCs is a bit different than in the monolayer, which depends on the emission location and the emission energy. Additionally, in most cases, owing to the type II band alignment and weak hybridization, the conduction band minimum and valence band maximum of the van der Waals heterostructure are dominated by orbitals from different layers. As a result, electrons and holes tend to reside in different layers of the structure.
* It was found that the g-factor value of the interlayer exciton depends on the stacking sequence of the layers. For the AA-stacked <math>\text{MoSe}_2</math>/<math>\text{WSe}_2</math>, the value of the singlet g-factor between 6 to 7 has been reported; while, for the AB-stacked <math>\text{WSe}_2</math>/<math>\text{MoSe}_2</math>, the value of g-factor ~ −15 has been reported

===Experimental setup===

====Setup====
[[File:Experimental Setup.jpg|thumb|400px|Experimental Setup]]
* As shown in figure

====Quantum simulator====
* The <math>\text{MoSe}_2</math>/<math>\text{WSe}_2</math> heterostructure is fabricated via mechanical exfoliation and aligned transfer method. The stacking sequence of the sample is checked using second harmonic generation (SHG)-based measurement and it is found to be AA stacking. The sample is loaded into a magnetocryostat to control the magnetic ﬁeld and sample temperature.

====Qubit initialisation====
* Use circularly polarised light to create valley exciton polarization
* Use linearly polarised light to pump the qubits to rotate the valley pseudospin in the superposition plane of K and K′ valley polarizations
* The polarization state of the excitation and collection is controlled using a combination of polarizer, quarter-wave plate, and half-wave plate.

====Qubit control====
* Most popular control techniques these days
*# 1st approach: The coupling between the optical (electromagnetic) signal and the valley information can also be used to control the valley information. For the intralayer exciton, the exciton can be moved from one valley to another by applying intense THz radiation. This is shown by the observation of linearly polarized emission from higher-order sidebands when the sample is excited with a 100-fs near-resonant circularly polarized light pulse together with a 33-fs 40-THz pulse. Theoretically, it is shown that this is caused by a THz pulse driven intervalley transition. It is also predicted that a complete transfer from K (K’) to K’ (K) valley can be done within a period less than 5 fs if a more intense and shorter THz pulse is used.
*# 2nd approach: On the other hand, by utilizing the optical Stark effect, the phase control has been demonstrated. In this case, the valley state is first prepared in the <math>|K\rang+|K'\rang</math> state by using a near-resonant linearly polarized excitation. A red-detuned 100-fs circularly polarized light pulse is then used to break the valley degeneracy, which allows the coherent rotation of the valley state to the <math>|K\rang+e^{i\Delta\phi}|K'\rang</math> state with <math>\Delta\phi</math> depends on the pulse intensity and duration. This rotation is reflected in the linear polarization state of the exciton emission.

====Qubit readout====
* The PL emission is directed by a multimode optical ﬁber into a spectrometer (Andor Shamrock) with a CCD detector for spectroscopic recording

==Results==

==References==
<references />

File:QE.png

2021-04-30T16:22:38Z

Chu WenHao:

Optical control of TMDCs valley pseudospin qubits

2021-04-30T16:22:15Z

Chu WenHao: /* Numerical calculation results by Quantum Espresso */

==Linked project==
* This project is conducted by the same people as 👉 [[Control over the atomic spins within certain molecules by NMR technique]]

==Group members==
*'''XU ZIZHOU (Bensen)'''
**'''Matric Number:''' ''A0229645W''
**'''Email:''' ''[mailto:info@example.org zizhou_xu@u.nus.edu]''
*'''CHU WENHAO'''
**'''Matric Number:''' ''A0228747R''
**'''Email:''' ''[mailto:info@example.org e0675585@u.nus.edu]''
==Description==

* The strategies and materials for building universal quantum computers are being actively researched, and various platforms for holding qubits have been proposed and tested<ref>Experimental perfect state transfer of an entangled photonic qubit[https://www.nature.com/articles/ncomms11339]</ref><ref>Superconducting Qubits: Current State of Play[https://www.annualreviews.org/doi/abs/10.1146/annurev-conmatphys-031119-050605]</ref><ref>Two-qubit entangling gates within arbitrarily long chains of trapped ions[https://journals.aps.org/pra/abstract/10.1103/PhysRevA.100.022332]</ref><ref>Digital Coherent Control of a Superconducting Qubit[https://journals.aps.org/prapplied/abstract/10.1103/PhysRevApplied.11.014009]</ref><ref>Fast quantum logic gates with trapped-ion qubits[https://www.nature.com/articles/nature25737]</ref>. Within all the options such as superconducting qubits and ion trap qubits, solid-state qubits have illustrated great benefits of the compatibility with the existing semiconductor technology, which attracts lots of attention to this platform. Moreover, because of the direct band gap and strong spin-orbit coupling discovered in the monolayer transition metal dichalcogenides (TMDCs), especially some semiconductor monolayers possess a sizeable direct bandgap of ≈1.5–2 eV in the optical range allowing electrostatic confinement and optical manipulation of carriers<ref>Atomically Thin MoS2: A New Direct-Gap Semiconductor[https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.105.136805]</ref><ref>Emerging Photoluminescence in Monolayer MoS2[https://pubs.acs.org/doi/10.1021/nl903868w]</ref>, chances are TMDCs materials have the potential to become the promising platform for fabricating qubits. There are four main types of TMDCs that have been suggested as advantageous in acting qubits, <math>\text{WS}_2</math> , <math>\text{WSe}_2</math> , <math>\text{MoS}_2</math> and <math>\text{MoSe}_2</math> . Kormányos et al used DFT calculations to confirm that <math>\text{WS}_2</math> and <math>\text{WSe}_2</math> , are better than <math>\text{MoS}_2</math> and <math>\text{MoSe}_2</math> in terms of the spin-valley coupling<ref>Spin-Orbit Coupling, Quantum Dots, and Qubits in Monolayer Transition Metal Dichalcogenides[https://journals.aps.org/prx/abstract/10.1103/PhysRevX.4.011034]</ref>, which is really important since spin-valley coupling is regarded as a key factor that can improve the coherence lifetime of spin-valley states.

* After several decades of active research on the properties of TMDCs, some interesting features have been discovered. Such as the extra degree of freedom offered by valley magnetic moment, which can realise bit 0 and bit 1 to form a qubit in momentum space. In addition, these valleys are independently addressable by an optical signal, and optical controlling technique is currently under research by many groups both theoretically and experimentally. As a result, as inspired by the idea from one of the theoretical work<ref>Spin-valley qubit in nanostructures of monolayer semiconductors: Optical control and hyperfine interaction[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.93.045313]</ref>, we intend to build a quantum simulator and investigate optical control of the spin-valley pseudospin on 2D heterobilayer TMDCs.

==Q & A==

===What are K & K' points?===
[[File:K_&_K'_points.png|thumb|400px|K & K' points]]
* K-points are sampling points of Brillouin zone in reciprocal lattice
* The monolayer TMDs have a unique band structure with the conduction and valence band edges both at the degenerate K and K‘ valleys at the corners of the hexagonal Brillouin zone. The direct-gap optical transitions have a selection rule: left- (right-) handed circular polarized photons couple to the interband transitions in the K (K’) valley only.
===The Tight-Binding Model===
* By introducing the tight-binding model for TMDs, we will give a Hamiltonian of TMDs that can be further deduced and calculated. As is showned in figure (b):under the tight-binding ansatz,the atom(e.g.the red one)can only jump to the nearest position, whose interaction processing is called "hopping" in physics,If we use the lattice represetation and the corresponding creation operators commonly used in condensed matter physics,then the nearest-neighbor tight-binding Hamiltonian has the simple form:

[[File:graphene.png|thumb|400px|Graphene]]

<math>
\hat{H}=-t\sum_{l} \sum_{j=1}^{3}a_{l+\tau_j}^{\dagger}b_{l}+h.c.
</math>

By Fourier Transformation,<math>a_l</math> and <math>b_{l}</math> can be written as:

<math>
a_l=\frac{1}{\sqrt{N}}\sum_{k} e^{ik\cdot R_l}\alpha_k
</math>

<math>
b_l=\frac{1}{\sqrt{N}}\sum_{k} e^{ik\cdot R_l}\beta_k
</math>

<math>
\tau_1=(0,\frac{1}{\sqrt{3}})a,\tau_2=(-\frac{1}{2},-\frac{1}{2\sqrt{3}})a,\tau_3=(\frac{1}{2},-\frac{1}{2\sqrt{3}})a
</math>

Then we can rewrite the Hamiltonian in reciprocal space:

<math>
\hat{H}=\sum_{k}c^{\dagger}(k)\begin{bmatrix}
0&f^{*}\\
f&0
\end{bmatrix}c(k),\qquad c(k)=\begin{bmatrix}
\alpha(k)\\
\beta(k)
\end{bmatrix},\qquad f=-t\sum_{j=1}^{3}e^{ik\cdot\tau_{j}}
</math>
===Choice of K values===
*We define the first Brillouin zone of the reciprocal lattice in the standard way.It is clear that the six points at the corners of the FBZ fall into two groups of three which are equivalent, so we need consider only two equivalent corners that we label K and K0 as in the figure. Explicitly, their positions in momentum space are given by:

[[File:K values.png|thumb|400px|K values]]

<math>
K=\frac{2\pi}{a}(\frac{1}{3},\frac{1}{\sqrt{3}}),\qquad K'=\frac{2\pi}{a}(\frac{2}{3},0)
</math>

These K and K' point are also called "Dirac points".
===Solution of TB model Hamiltonian===
*We can rewrite the Schrödinger equation in reciprocal space

<math>
H(k)\Psi_k=E\Psi_k
</math>

From the explicit expressions for the nearest neighbor vectors we obtain:

<math>
f=-t\exp(ik_xa(1+2e^{i\cdot\frac{3k_xa}{2}}cos\frac{\sqrt{3}}{2}k_ya))
</math>

And the eigenvalues are given by

<math>
\epsilon_{s,k}=s|f|=\pm t(1+4cos(\frac{3}{2}k_xa)cos(\frac{\sqrt{3}}{2}k_ya)+4cos^2\frac{\sqrt{3}}{2}k_ya)^{\frac{1}{2}}
</math>

It is interesting to enquire whether there are any values of k for which f is zero? Any such value must satisfy the conditions? Luckily, the answer is yes and the points "at the corner" which satisfied these conditions are exactly the "Dirac points".
===Dirac model near K points--Not spins but like spins===
*With the Dirac model in Quantum Fireld Theory , We have the Dirac's original:

<math>(i\gamma^\mu\partial_\mu-m)\Psi=0</math>,
<math>\gamma^\mu\gamma^v+\gamma^v\gamma^\mu=2g^{v\mu}</math>,
<math>v,\mu=0,1,2,3</math>

It is very suggestive to write the Hamiltonian in the form:

[[File:Dirac points.png|thumb|400px|Dirac points]]

<math>
\hat{H}=\hbar v_F\begin{bmatrix}
0&k_x+ik_y\\
k_x-ik_y&0
\end{bmatrix}=\hbar v_F\hat{\sigma}\cdot q,\qquad \epsilon(k)=\pm v_F|k|
</math>

where the components of the operator <math>\sigma</math> are the usual Pauli matrices,and <math>v_F</math> is the Fermi velocity. It is clear that the eigenvalues are a function only of the magnitude of k, not its direction in the 2D space,therefore, although the Hamiltonian expression contains spin operators, the Dirac point derived from this is not the spin of particle or quasiparticle, (which can be compared with the quasiparticle ansatz in BCS theory).However, it still inspires us to regard these K and K' points of TMDs as some kinds of qubit resources and control them on this theoretical basis.

===Numerical calculation results by Quantum Espresso===
[[File:Dirac points in Band structure.png|thumb|400px|Band structure]]

The Dirac points in the band structure of graphene are shown in the Figure, In the next step, we scheduled to add the effect of external filed (like magnetic field or optical field), which will change the hamiltonian and choose the appropriate pseudopotential to calculate the change of K and K' point. We plan to use Quantum Espresso as our calculation programme.

[[File:QE.png|thumb|400px|QE code]]

===Why choose heterobilayer TMDCs instead of monolayer===
* There are four main carrier properties that optimal Opto-valleytronics should possess.
*# long carrier lifetime
*# long valley lifetime
*# high valley polarization
*# long valley coherence time
* By adopting 2D heterostructure TMDCs materials, we can create these conditions for building promising quantum platform. (eg. Due to the type II band alignment and weak hybridization of van der Waals heterostructure, the electron–hole layer separation, the electron–hole exchange interaction is greatly reduced, resulting in a long cryogenic lifetime (ns to <math>\mu</math>s)and valley lifetime (~ 10 ns) of the interlayer exciton<ref>Opto-valleytronics in the 2D van der Waals heterostructure[https://doi.org/10.1007/s12274-020-3036-x]</ref>)

==Method==
===Principle===
* It was shown that inversion symmetry breaking can lead to opposite circular dichroism in different momentum space regions, which induces an interesting phenomena of optical selection rules at the symmetry corner points of the Brillouin zone. This enables a valley-dependent control of electrons with optical signal of different circular polarizations.
* However, valley optical selection rule in the heterobilayer TMDCs is a bit different than in the monolayer, which depends on the emission location and the emission energy. Additionally, in most cases, owing to the type II band alignment and weak hybridization, the conduction band minimum and valence band maximum of the van der Waals heterostructure are dominated by orbitals from different layers. As a result, electrons and holes tend to reside in different layers of the structure.
* It was found that the g-factor value of the interlayer exciton depends on the stacking sequence of the layers. For the AA-stacked <math>\text{MoSe}_2</math>/<math>\text{WSe}_2</math>, the value of the singlet g-factor between 6 to 7 has been reported; while, for the AB-stacked <math>\text{WSe}_2</math>/<math>\text{MoSe}_2</math>, the value of g-factor ~ −15 has been reported

===Experimental setup===

====Setup====
[[File:Experimental Setup.jpg|thumb|400px|Experimental Setup]]
* As shown in figure

====Quantum simulator====
* The <math>\text{MoSe}_2</math>/<math>\text{WSe}_2</math> heterostructure is fabricated via mechanical exfoliation and aligned transfer method. The stacking sequence of the sample is checked using second harmonic generation (SHG)-based measurement and it is found to be AA stacking. The sample is loaded into a magnetocryostat to control the magnetic ﬁeld and sample temperature.

====Qubit initialisation====
* Use circularly polarised light to create valley exciton polarization
* Use linearly polarised light to pump the qubits to rotate the valley pseudospin in the superposition plane of K and K′ valley polarizations
* The polarization state of the excitation and collection is controlled using a combination of polarizer, quarter-wave plate, and half-wave plate.

====Qubit control====
* Most popular control techniques these days
*# 1st approach: The coupling between the optical (electromagnetic) signal and the valley information can also be used to control the valley information. For the intralayer exciton, the exciton can be moved from one valley to another by applying intense THz radiation. This is shown by the observation of linearly polarized emission from higher-order sidebands when the sample is excited with a 100-fs near-resonant circularly polarized light pulse together with a 33-fs 40-THz pulse. Theoretically, it is shown that this is caused by a THz pulse driven intervalley transition. It is also predicted that a complete transfer from K (K’) to K’ (K) valley can be done within a period less than 5 fs if a more intense and shorter THz pulse is used.
*# 2nd approach: On the other hand, by utilizing the optical Stark effect, the phase control has been demonstrated. In this case, the valley state is first prepared in the <math>|K\rang+|K'\rang</math> state by using a near-resonant linearly polarized excitation. A red-detuned 100-fs circularly polarized light pulse is then used to break the valley degeneracy, which allows the coherent rotation of the valley state to the <math>|K\rang+e^{i\Delta\phi}|K'\rang</math> state with <math>\Delta\phi</math> depends on the pulse intensity and duration. This rotation is reflected in the linear polarization state of the exciton emission.

====Qubit readout====
* The PL emission is directed by a multimode optical ﬁber into a spectrometer (Andor Shamrock) with a CCD detector for spectroscopic recording

==Results==

==References==
<references />

Optical control of TMDCs valley pseudospin qubits

2021-04-30T16:14:17Z

Chu WenHao: /* Numerical calculation results by Quantum Espresso */

==Linked project==
* This project is conducted by the same people as 👉 [[Control over the atomic spins within certain molecules by NMR technique]]

==Group members==
*'''XU ZIZHOU (Bensen)'''
**'''Matric Number:''' ''A0229645W''
**'''Email:''' ''[mailto:info@example.org zizhou_xu@u.nus.edu]''
*'''CHU WENHAO'''
**'''Matric Number:''' ''A0228747R''
**'''Email:''' ''[mailto:info@example.org e0675585@u.nus.edu]''
==Description==

* The strategies and materials for building universal quantum computers are being actively researched, and various platforms for holding qubits have been proposed and tested<ref>Experimental perfect state transfer of an entangled photonic qubit[https://www.nature.com/articles/ncomms11339]</ref><ref>Superconducting Qubits: Current State of Play[https://www.annualreviews.org/doi/abs/10.1146/annurev-conmatphys-031119-050605]</ref><ref>Two-qubit entangling gates within arbitrarily long chains of trapped ions[https://journals.aps.org/pra/abstract/10.1103/PhysRevA.100.022332]</ref><ref>Digital Coherent Control of a Superconducting Qubit[https://journals.aps.org/prapplied/abstract/10.1103/PhysRevApplied.11.014009]</ref><ref>Fast quantum logic gates with trapped-ion qubits[https://www.nature.com/articles/nature25737]</ref>. Within all the options such as superconducting qubits and ion trap qubits, solid-state qubits have illustrated great benefits of the compatibility with the existing semiconductor technology, which attracts lots of attention to this platform. Moreover, because of the direct band gap and strong spin-orbit coupling discovered in the monolayer transition metal dichalcogenides (TMDCs), especially some semiconductor monolayers possess a sizeable direct bandgap of ≈1.5–2 eV in the optical range allowing electrostatic confinement and optical manipulation of carriers<ref>Atomically Thin MoS2: A New Direct-Gap Semiconductor[https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.105.136805]</ref><ref>Emerging Photoluminescence in Monolayer MoS2[https://pubs.acs.org/doi/10.1021/nl903868w]</ref>, chances are TMDCs materials have the potential to become the promising platform for fabricating qubits. There are four main types of TMDCs that have been suggested as advantageous in acting qubits, <math>\text{WS}_2</math> , <math>\text{WSe}_2</math> , <math>\text{MoS}_2</math> and <math>\text{MoSe}_2</math> . Kormányos et al used DFT calculations to confirm that <math>\text{WS}_2</math> and <math>\text{WSe}_2</math> , are better than <math>\text{MoS}_2</math> and <math>\text{MoSe}_2</math> in terms of the spin-valley coupling<ref>Spin-Orbit Coupling, Quantum Dots, and Qubits in Monolayer Transition Metal Dichalcogenides[https://journals.aps.org/prx/abstract/10.1103/PhysRevX.4.011034]</ref>, which is really important since spin-valley coupling is regarded as a key factor that can improve the coherence lifetime of spin-valley states.

* After several decades of active research on the properties of TMDCs, some interesting features have been discovered. Such as the extra degree of freedom offered by valley magnetic moment, which can realise bit 0 and bit 1 to form a qubit in momentum space. In addition, these valleys are independently addressable by an optical signal, and optical controlling technique is currently under research by many groups both theoretically and experimentally. As a result, as inspired by the idea from one of the theoretical work<ref>Spin-valley qubit in nanostructures of monolayer semiconductors: Optical control and hyperfine interaction[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.93.045313]</ref>, we intend to build a quantum simulator and investigate optical control of the spin-valley pseudospin on 2D heterobilayer TMDCs.

==Q & A==

===What are K & K' points?===
[[File:K_&_K'_points.png|thumb|400px|K & K' points]]
* K-points are sampling points of Brillouin zone in reciprocal lattice
* The monolayer TMDs have a unique band structure with the conduction and valence band edges both at the degenerate K and K‘ valleys at the corners of the hexagonal Brillouin zone. The direct-gap optical transitions have a selection rule: left- (right-) handed circular polarized photons couple to the interband transitions in the K (K’) valley only.
===The Tight-Binding Model===
* By introducing the tight-binding model for TMDs, we will give a Hamiltonian of TMDs that can be further deduced and calculated. As is showned in figure (b):under the tight-binding ansatz,the atom(e.g.the red one)can only jump to the nearest position, whose interaction processing is called "hopping" in physics,If we use the lattice represetation and the corresponding creation operators commonly used in condensed matter physics,then the nearest-neighbor tight-binding Hamiltonian has the simple form:

[[File:graphene.png|thumb|400px|Graphene]]

<math>
\hat{H}=-t\sum_{l} \sum_{j=1}^{3}a_{l+\tau_j}^{\dagger}b_{l}+h.c.
</math>

By Fourier Transformation,<math>a_l</math> and <math>b_{l}</math> can be written as:

<math>
a_l=\frac{1}{\sqrt{N}}\sum_{k} e^{ik\cdot R_l}\alpha_k
</math>

<math>
b_l=\frac{1}{\sqrt{N}}\sum_{k} e^{ik\cdot R_l}\beta_k
</math>

<math>
\tau_1=(0,\frac{1}{\sqrt{3}})a,\tau_2=(-\frac{1}{2},-\frac{1}{2\sqrt{3}})a,\tau_3=(\frac{1}{2},-\frac{1}{2\sqrt{3}})a
</math>

Then we can rewrite the Hamiltonian in reciprocal space:

<math>
\hat{H}=\sum_{k}c^{\dagger}(k)\begin{bmatrix}
0&f^{*}\\
f&0
\end{bmatrix}c(k),\qquad c(k)=\begin{bmatrix}
\alpha(k)\\
\beta(k)
\end{bmatrix},\qquad f=-t\sum_{j=1}^{3}e^{ik\cdot\tau_{j}}
</math>
===Choice of K values===
*We define the first Brillouin zone of the reciprocal lattice in the standard way.It is clear that the six points at the corners of the FBZ fall into two groups of three which are equivalent, so we need consider only two equivalent corners that we label K and K0 as in the figure. Explicitly, their positions in momentum space are given by:

[[File:K values.png|thumb|400px|K values]]

<math>
K=\frac{2\pi}{a}(\frac{1}{3},\frac{1}{\sqrt{3}}),\qquad K'=\frac{2\pi}{a}(\frac{2}{3},0)
</math>

These K and K' point are also called "Dirac points".
===Solution of TB model Hamiltonian===
*We can rewrite the Schrödinger equation in reciprocal space

<math>
H(k)\Psi_k=E\Psi_k
</math>

From the explicit expressions for the nearest neighbor vectors we obtain:

<math>
f=-t\exp(ik_xa(1+2e^{i\cdot\frac{3k_xa}{2}}cos\frac{\sqrt{3}}{2}k_ya))
</math>

And the eigenvalues are given by

<math>
\epsilon_{s,k}=s|f|=\pm t(1+4cos(\frac{3}{2}k_xa)cos(\frac{\sqrt{3}}{2}k_ya)+4cos^2\frac{\sqrt{3}}{2}k_ya)^{\frac{1}{2}}
</math>

It is interesting to enquire whether there are any values of k for which f is zero? Any such value must satisfy the conditions? Luckily, the answer is yes and the points "at the corner" which satisfied these conditions are exactly the "Dirac points".
===Dirac model near K points--Not spins but like spins===
*With the Dirac model in Quantum Fireld Theory , We have the Dirac's original:

<math>(i\gamma^\mu\partial_\mu-m)\Psi=0</math>,
<math>\gamma^\mu\gamma^v+\gamma^v\gamma^\mu=2g^{v\mu}</math>,
<math>v,\mu=0,1,2,3</math>

It is very suggestive to write the Hamiltonian in the form:

[[File:Dirac points.png|thumb|400px|Dirac points]]

<math>
\hat{H}=\hbar v_F\begin{bmatrix}
0&k_x+ik_y\\
k_x-ik_y&0
\end{bmatrix}=\hbar v_F\hat{\sigma}\cdot q,\qquad \epsilon(k)=\pm v_F|k|
</math>

where the components of the operator <math>\sigma</math> are the usual Pauli matrices,and <math>v_F</math> is the Fermi velocity. It is clear that the eigenvalues are a function only of the magnitude of k, not its direction in the 2D space,therefore, although the Hamiltonian expression contains spin operators, the Dirac point derived from this is not the spin of particle or quasiparticle, (which can be compared with the quasiparticle ansatz in BCS theory).However, it still inspires us to regard these K and K' points of TMDs as some kinds of qubit resources and control them on this theoretical basis.

===Numerical calculation results by Quantum Espresso===
[[File:Dirac points in Band structure.png|thumb|400px|Band structure]]

The Dirac points in the band structure of graphene are shown in the Figure, In the next step, we scheduled to add the effect of external filed (like magnetic field or optical field), which will change the hamiltonian and choose the appropriate pseudopotential to calculate the change of K and K' point. We plan to use Quantum Espresso as our calculation programme.

===Why choose heterobilayer TMDCs instead of monolayer===
* There are four main carrier properties that optimal Opto-valleytronics should possess.
*# long carrier lifetime
*# long valley lifetime
*# high valley polarization
*# long valley coherence time
* By adopting 2D heterostructure TMDCs materials, we can create these conditions for building promising quantum platform. (eg. Due to the type II band alignment and weak hybridization of van der Waals heterostructure, the electron–hole layer separation, the electron–hole exchange interaction is greatly reduced, resulting in a long cryogenic lifetime (ns to <math>\mu</math>s)and valley lifetime (~ 10 ns) of the interlayer exciton<ref>Opto-valleytronics in the 2D van der Waals heterostructure[https://doi.org/10.1007/s12274-020-3036-x]</ref>)

==Method==
===Principle===
* It was shown that inversion symmetry breaking can lead to opposite circular dichroism in different momentum space regions, which induces an interesting phenomena of optical selection rules at the symmetry corner points of the Brillouin zone. This enables a valley-dependent control of electrons with optical signal of different circular polarizations.
* However, valley optical selection rule in the heterobilayer TMDCs is a bit different than in the monolayer, which depends on the emission location and the emission energy. Additionally, in most cases, owing to the type II band alignment and weak hybridization, the conduction band minimum and valence band maximum of the van der Waals heterostructure are dominated by orbitals from different layers. As a result, electrons and holes tend to reside in different layers of the structure.
* It was found that the g-factor value of the interlayer exciton depends on the stacking sequence of the layers. For the AA-stacked <math>\text{MoSe}_2</math>/<math>\text{WSe}_2</math>, the value of the singlet g-factor between 6 to 7 has been reported; while, for the AB-stacked <math>\text{WSe}_2</math>/<math>\text{MoSe}_2</math>, the value of g-factor ~ −15 has been reported

===Experimental setup===

====Setup====
[[File:Experimental Setup.jpg|thumb|400px|Experimental Setup]]
* As shown in figure

====Quantum simulator====
* The <math>\text{MoSe}_2</math>/<math>\text{WSe}_2</math> heterostructure is fabricated via mechanical exfoliation and aligned transfer method. The stacking sequence of the sample is checked using second harmonic generation (SHG)-based measurement and it is found to be AA stacking. The sample is loaded into a magnetocryostat to control the magnetic ﬁeld and sample temperature.

====Qubit initialisation====
* Use circularly polarised light to create valley exciton polarization
* Use linearly polarised light to pump the qubits to rotate the valley pseudospin in the superposition plane of K and K′ valley polarizations
* The polarization state of the excitation and collection is controlled using a combination of polarizer, quarter-wave plate, and half-wave plate.

====Qubit control====
* Most popular control techniques these days
*# 1st approach: The coupling between the optical (electromagnetic) signal and the valley information can also be used to control the valley information. For the intralayer exciton, the exciton can be moved from one valley to another by applying intense THz radiation. This is shown by the observation of linearly polarized emission from higher-order sidebands when the sample is excited with a 100-fs near-resonant circularly polarized light pulse together with a 33-fs 40-THz pulse. Theoretically, it is shown that this is caused by a THz pulse driven intervalley transition. It is also predicted that a complete transfer from K (K’) to K’ (K) valley can be done within a period less than 5 fs if a more intense and shorter THz pulse is used.
*# 2nd approach: On the other hand, by utilizing the optical Stark effect, the phase control has been demonstrated. In this case, the valley state is first prepared in the <math>|K\rang+|K'\rang</math> state by using a near-resonant linearly polarized excitation. A red-detuned 100-fs circularly polarized light pulse is then used to break the valley degeneracy, which allows the coherent rotation of the valley state to the <math>|K\rang+e^{i\Delta\phi}|K'\rang</math> state with <math>\Delta\phi</math> depends on the pulse intensity and duration. This rotation is reflected in the linear polarization state of the exciton emission.

====Qubit readout====
* The PL emission is directed by a multimode optical ﬁber into a spectrometer (Andor Shamrock) with a CCD detector for spectroscopic recording

==Results==

==References==
<references />

Optical control of TMDCs valley pseudospin qubits

2021-04-30T16:13:37Z

Chu WenHao: /* Numerical calculation results by Quantum Espresso */

==Linked project==
* This project is conducted by the same people as 👉 [[Control over the atomic spins within certain molecules by NMR technique]]

==Group members==
*'''XU ZIZHOU (Bensen)'''
**'''Matric Number:''' ''A0229645W''
**'''Email:''' ''[mailto:info@example.org zizhou_xu@u.nus.edu]''
*'''CHU WENHAO'''
**'''Matric Number:''' ''A0228747R''
**'''Email:''' ''[mailto:info@example.org e0675585@u.nus.edu]''
==Description==

* The strategies and materials for building universal quantum computers are being actively researched, and various platforms for holding qubits have been proposed and tested<ref>Experimental perfect state transfer of an entangled photonic qubit[https://www.nature.com/articles/ncomms11339]</ref><ref>Superconducting Qubits: Current State of Play[https://www.annualreviews.org/doi/abs/10.1146/annurev-conmatphys-031119-050605]</ref><ref>Two-qubit entangling gates within arbitrarily long chains of trapped ions[https://journals.aps.org/pra/abstract/10.1103/PhysRevA.100.022332]</ref><ref>Digital Coherent Control of a Superconducting Qubit[https://journals.aps.org/prapplied/abstract/10.1103/PhysRevApplied.11.014009]</ref><ref>Fast quantum logic gates with trapped-ion qubits[https://www.nature.com/articles/nature25737]</ref>. Within all the options such as superconducting qubits and ion trap qubits, solid-state qubits have illustrated great benefits of the compatibility with the existing semiconductor technology, which attracts lots of attention to this platform. Moreover, because of the direct band gap and strong spin-orbit coupling discovered in the monolayer transition metal dichalcogenides (TMDCs), especially some semiconductor monolayers possess a sizeable direct bandgap of ≈1.5–2 eV in the optical range allowing electrostatic confinement and optical manipulation of carriers<ref>Atomically Thin MoS2: A New Direct-Gap Semiconductor[https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.105.136805]</ref><ref>Emerging Photoluminescence in Monolayer MoS2[https://pubs.acs.org/doi/10.1021/nl903868w]</ref>, chances are TMDCs materials have the potential to become the promising platform for fabricating qubits. There are four main types of TMDCs that have been suggested as advantageous in acting qubits, <math>\text{WS}_2</math> , <math>\text{WSe}_2</math> , <math>\text{MoS}_2</math> and <math>\text{MoSe}_2</math> . Kormányos et al used DFT calculations to confirm that <math>\text{WS}_2</math> and <math>\text{WSe}_2</math> , are better than <math>\text{MoS}_2</math> and <math>\text{MoSe}_2</math> in terms of the spin-valley coupling<ref>Spin-Orbit Coupling, Quantum Dots, and Qubits in Monolayer Transition Metal Dichalcogenides[https://journals.aps.org/prx/abstract/10.1103/PhysRevX.4.011034]</ref>, which is really important since spin-valley coupling is regarded as a key factor that can improve the coherence lifetime of spin-valley states.

* After several decades of active research on the properties of TMDCs, some interesting features have been discovered. Such as the extra degree of freedom offered by valley magnetic moment, which can realise bit 0 and bit 1 to form a qubit in momentum space. In addition, these valleys are independently addressable by an optical signal, and optical controlling technique is currently under research by many groups both theoretically and experimentally. As a result, as inspired by the idea from one of the theoretical work<ref>Spin-valley qubit in nanostructures of monolayer semiconductors: Optical control and hyperfine interaction[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.93.045313]</ref>, we intend to build a quantum simulator and investigate optical control of the spin-valley pseudospin on 2D heterobilayer TMDCs.

==Q & A==

===What are K & K' points?===
[[File:K_&_K'_points.png|thumb|400px|K & K' points]]
* K-points are sampling points of Brillouin zone in reciprocal lattice
* The monolayer TMDs have a unique band structure with the conduction and valence band edges both at the degenerate K and K‘ valleys at the corners of the hexagonal Brillouin zone. The direct-gap optical transitions have a selection rule: left- (right-) handed circular polarized photons couple to the interband transitions in the K (K’) valley only.
===The Tight-Binding Model===
* By introducing the tight-binding model for TMDs, we will give a Hamiltonian of TMDs that can be further deduced and calculated. As is showned in figure (b):under the tight-binding ansatz,the atom(e.g.the red one)can only jump to the nearest position, whose interaction processing is called "hopping" in physics,If we use the lattice represetation and the corresponding creation operators commonly used in condensed matter physics,then the nearest-neighbor tight-binding Hamiltonian has the simple form:

[[File:graphene.png|thumb|400px|Graphene]]

<math>
\hat{H}=-t\sum_{l} \sum_{j=1}^{3}a_{l+\tau_j}^{\dagger}b_{l}+h.c.
</math>

By Fourier Transformation,<math>a_l</math> and <math>b_{l}</math> can be written as:

<math>
a_l=\frac{1}{\sqrt{N}}\sum_{k} e^{ik\cdot R_l}\alpha_k
</math>

<math>
b_l=\frac{1}{\sqrt{N}}\sum_{k} e^{ik\cdot R_l}\beta_k
</math>

<math>
\tau_1=(0,\frac{1}{\sqrt{3}})a,\tau_2=(-\frac{1}{2},-\frac{1}{2\sqrt{3}})a,\tau_3=(\frac{1}{2},-\frac{1}{2\sqrt{3}})a
</math>

Then we can rewrite the Hamiltonian in reciprocal space:

<math>
\hat{H}=\sum_{k}c^{\dagger}(k)\begin{bmatrix}
0&f^{*}\\
f&0
\end{bmatrix}c(k),\qquad c(k)=\begin{bmatrix}
\alpha(k)\\
\beta(k)
\end{bmatrix},\qquad f=-t\sum_{j=1}^{3}e^{ik\cdot\tau_{j}}
</math>
===Choice of K values===
*We define the first Brillouin zone of the reciprocal lattice in the standard way.It is clear that the six points at the corners of the FBZ fall into two groups of three which are equivalent, so we need consider only two equivalent corners that we label K and K0 as in the figure. Explicitly, their positions in momentum space are given by:

[[File:K values.png|thumb|400px|K values]]

<math>
K=\frac{2\pi}{a}(\frac{1}{3},\frac{1}{\sqrt{3}}),\qquad K'=\frac{2\pi}{a}(\frac{2}{3},0)
</math>

These K and K' point are also called "Dirac points".
===Solution of TB model Hamiltonian===
*We can rewrite the Schrödinger equation in reciprocal space

<math>
H(k)\Psi_k=E\Psi_k
</math>

From the explicit expressions for the nearest neighbor vectors we obtain:

<math>
f=-t\exp(ik_xa(1+2e^{i\cdot\frac{3k_xa}{2}}cos\frac{\sqrt{3}}{2}k_ya))
</math>

And the eigenvalues are given by

<math>
\epsilon_{s,k}=s|f|=\pm t(1+4cos(\frac{3}{2}k_xa)cos(\frac{\sqrt{3}}{2}k_ya)+4cos^2\frac{\sqrt{3}}{2}k_ya)^{\frac{1}{2}}
</math>

It is interesting to enquire whether there are any values of k for which f is zero? Any such value must satisfy the conditions? Luckily, the answer is yes and the points "at the corner" which satisfied these conditions are exactly the "Dirac points".
===Dirac model near K points--Not spins but like spins===
*With the Dirac model in Quantum Fireld Theory , We have the Dirac's original:

<math>(i\gamma^\mu\partial_\mu-m)\Psi=0</math>,
<math>\gamma^\mu\gamma^v+\gamma^v\gamma^\mu=2g^{v\mu}</math>,
<math>v,\mu=0,1,2,3</math>

It is very suggestive to write the Hamiltonian in the form:

[[File:Dirac points.png|thumb|400px|Dirac points]]

<math>
\hat{H}=\hbar v_F\begin{bmatrix}
0&k_x+ik_y\\
k_x-ik_y&0
\end{bmatrix}=\hbar v_F\hat{\sigma}\cdot q,\qquad \epsilon(k)=\pm v_F|k|
</math>

where the components of the operator <math>\sigma</math> are the usual Pauli matrices,and <math>v_F</math> is the Fermi velocity. It is clear that the eigenvalues are a function only of the magnitude of k, not its direction in the 2D space,therefore, although the Hamiltonian expression contains spin operators, the Dirac point derived from this is not the spin of particle or quasiparticle, (which can be compared with the quasiparticle ansatz in BCS theory).However, it still inspires us to regard these K and K' points of TMDs as some kinds of qubit resources and control them on this theoretical basis.

===Numerical calculation results by Quantum Espresso===
[[File:Dirac points in Band structure.png|thumb|400px|Band structure]]

The Dirac points in the band structure of graphene are shown in the Figure, In the next step, we scheduled to add the effect of external filed (like magnetic field or opticcal field), which will change the hamiltonian and choose the appropriate pseudopotential to calculate the change of K and K' point. We plan to use Quantum Espresso as our calculation programme.

===Why choose heterobilayer TMDCs instead of monolayer===
* There are four main carrier properties that optimal Opto-valleytronics should possess.
*# long carrier lifetime
*# long valley lifetime
*# high valley polarization
*# long valley coherence time
* By adopting 2D heterostructure TMDCs materials, we can create these conditions for building promising quantum platform. (eg. Due to the type II band alignment and weak hybridization of van der Waals heterostructure, the electron–hole layer separation, the electron–hole exchange interaction is greatly reduced, resulting in a long cryogenic lifetime (ns to <math>\mu</math>s)and valley lifetime (~ 10 ns) of the interlayer exciton<ref>Opto-valleytronics in the 2D van der Waals heterostructure[https://doi.org/10.1007/s12274-020-3036-x]</ref>)

==Method==
===Principle===
* It was shown that inversion symmetry breaking can lead to opposite circular dichroism in different momentum space regions, which induces an interesting phenomena of optical selection rules at the symmetry corner points of the Brillouin zone. This enables a valley-dependent control of electrons with optical signal of different circular polarizations.
* However, valley optical selection rule in the heterobilayer TMDCs is a bit different than in the monolayer, which depends on the emission location and the emission energy. Additionally, in most cases, owing to the type II band alignment and weak hybridization, the conduction band minimum and valence band maximum of the van der Waals heterostructure are dominated by orbitals from different layers. As a result, electrons and holes tend to reside in different layers of the structure.
* It was found that the g-factor value of the interlayer exciton depends on the stacking sequence of the layers. For the AA-stacked <math>\text{MoSe}_2</math>/<math>\text{WSe}_2</math>, the value of the singlet g-factor between 6 to 7 has been reported; while, for the AB-stacked <math>\text{WSe}_2</math>/<math>\text{MoSe}_2</math>, the value of g-factor ~ −15 has been reported

===Experimental setup===

====Setup====
[[File:Experimental Setup.jpg|thumb|400px|Experimental Setup]]
* As shown in figure

====Quantum simulator====
* The <math>\text{MoSe}_2</math>/<math>\text{WSe}_2</math> heterostructure is fabricated via mechanical exfoliation and aligned transfer method. The stacking sequence of the sample is checked using second harmonic generation (SHG)-based measurement and it is found to be AA stacking. The sample is loaded into a magnetocryostat to control the magnetic ﬁeld and sample temperature.

====Qubit initialisation====
* Use circularly polarised light to create valley exciton polarization
* Use linearly polarised light to pump the qubits to rotate the valley pseudospin in the superposition plane of K and K′ valley polarizations
* The polarization state of the excitation and collection is controlled using a combination of polarizer, quarter-wave plate, and half-wave plate.

====Qubit control====
* Most popular control techniques these days
*# 1st approach: The coupling between the optical (electromagnetic) signal and the valley information can also be used to control the valley information. For the intralayer exciton, the exciton can be moved from one valley to another by applying intense THz radiation. This is shown by the observation of linearly polarized emission from higher-order sidebands when the sample is excited with a 100-fs near-resonant circularly polarized light pulse together with a 33-fs 40-THz pulse. Theoretically, it is shown that this is caused by a THz pulse driven intervalley transition. It is also predicted that a complete transfer from K (K’) to K’ (K) valley can be done within a period less than 5 fs if a more intense and shorter THz pulse is used.
*# 2nd approach: On the other hand, by utilizing the optical Stark effect, the phase control has been demonstrated. In this case, the valley state is first prepared in the <math>|K\rang+|K'\rang</math> state by using a near-resonant linearly polarized excitation. A red-detuned 100-fs circularly polarized light pulse is then used to break the valley degeneracy, which allows the coherent rotation of the valley state to the <math>|K\rang+e^{i\Delta\phi}|K'\rang</math> state with <math>\Delta\phi</math> depends on the pulse intensity and duration. This rotation is reflected in the linear polarization state of the exciton emission.

====Qubit readout====
* The PL emission is directed by a multimode optical ﬁber into a spectrometer (Andor Shamrock) with a CCD detector for spectroscopic recording

==Results==

==References==
<references />

Optical control of TMDCs valley pseudospin qubits

2021-04-30T16:05:37Z

Chu WenHao: /* Dirac model near K points--Not spins but like spins */

==Linked project==
* This project is conducted by the same people as 👉 [[Control over the atomic spins within certain molecules by NMR technique]]

==Group members==
*'''XU ZIZHOU (Bensen)'''
**'''Matric Number:''' ''A0229645W''
**'''Email:''' ''[mailto:info@example.org zizhou_xu@u.nus.edu]''
*'''CHU WENHAO'''
**'''Matric Number:''' ''A0228747R''
**'''Email:''' ''[mailto:info@example.org e0675585@u.nus.edu]''
==Description==

* The strategies and materials for building universal quantum computers are being actively researched, and various platforms for holding qubits have been proposed and tested<ref>Experimental perfect state transfer of an entangled photonic qubit[https://www.nature.com/articles/ncomms11339]</ref><ref>Superconducting Qubits: Current State of Play[https://www.annualreviews.org/doi/abs/10.1146/annurev-conmatphys-031119-050605]</ref><ref>Two-qubit entangling gates within arbitrarily long chains of trapped ions[https://journals.aps.org/pra/abstract/10.1103/PhysRevA.100.022332]</ref><ref>Digital Coherent Control of a Superconducting Qubit[https://journals.aps.org/prapplied/abstract/10.1103/PhysRevApplied.11.014009]</ref><ref>Fast quantum logic gates with trapped-ion qubits[https://www.nature.com/articles/nature25737]</ref>. Within all the options such as superconducting qubits and ion trap qubits, solid-state qubits have illustrated great benefits of the compatibility with the existing semiconductor technology, which attracts lots of attention to this platform. Moreover, because of the direct band gap and strong spin-orbit coupling discovered in the monolayer transition metal dichalcogenides (TMDCs), especially some semiconductor monolayers possess a sizeable direct bandgap of ≈1.5–2 eV in the optical range allowing electrostatic confinement and optical manipulation of carriers<ref>Atomically Thin MoS2: A New Direct-Gap Semiconductor[https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.105.136805]</ref><ref>Emerging Photoluminescence in Monolayer MoS2[https://pubs.acs.org/doi/10.1021/nl903868w]</ref>, chances are TMDCs materials have the potential to become the promising platform for fabricating qubits. There are four main types of TMDCs that have been suggested as advantageous in acting qubits, <math>\text{WS}_2</math> , <math>\text{WSe}_2</math> , <math>\text{MoS}_2</math> and <math>\text{MoSe}_2</math> . Kormányos et al used DFT calculations to confirm that <math>\text{WS}_2</math> and <math>\text{WSe}_2</math> , are better than <math>\text{MoS}_2</math> and <math>\text{MoSe}_2</math> in terms of the spin-valley coupling<ref>Spin-Orbit Coupling, Quantum Dots, and Qubits in Monolayer Transition Metal Dichalcogenides[https://journals.aps.org/prx/abstract/10.1103/PhysRevX.4.011034]</ref>, which is really important since spin-valley coupling is regarded as a key factor that can improve the coherence lifetime of spin-valley states.

* After several decades of active research on the properties of TMDCs, some interesting features have been discovered. Such as the extra degree of freedom offered by valley magnetic moment, which can realise bit 0 and bit 1 to form a qubit in momentum space. In addition, these valleys are independently addressable by an optical signal, and optical controlling technique is currently under research by many groups both theoretically and experimentally. As a result, as inspired by the idea from one of the theoretical work<ref>Spin-valley qubit in nanostructures of monolayer semiconductors: Optical control and hyperfine interaction[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.93.045313]</ref>, we intend to build a quantum simulator and investigate optical control of the spin-valley pseudospin on 2D heterobilayer TMDCs.

==Q & A==

===What are K & K' points?===
[[File:K_&_K'_points.png|thumb|400px|K & K' points]]
* K-points are sampling points of Brillouin zone in reciprocal lattice
* The monolayer TMDs have a unique band structure with the conduction and valence band edges both at the degenerate K and K‘ valleys at the corners of the hexagonal Brillouin zone. The direct-gap optical transitions have a selection rule: left- (right-) handed circular polarized photons couple to the interband transitions in the K (K’) valley only.
===The Tight-Binding Model===
* By introducing the tight-binding model for TMDs, we will give a Hamiltonian of TMDs that can be further deduced and calculated. As is showned in figure (b):under the tight-binding ansatz,the atom(e.g.the red one)can only jump to the nearest position, whose interaction processing is called "hopping" in physics,If we use the lattice represetation and the corresponding creation operators commonly used in condensed matter physics,then the nearest-neighbor tight-binding Hamiltonian has the simple form:

[[File:graphene.png|thumb|400px|Graphene]]

<math>
\hat{H}=-t\sum_{l} \sum_{j=1}^{3}a_{l+\tau_j}^{\dagger}b_{l}+h.c.
</math>

By Fourier Transformation,<math>a_l</math> and <math>b_{l}</math> can be written as:

<math>
a_l=\frac{1}{\sqrt{N}}\sum_{k} e^{ik\cdot R_l}\alpha_k
</math>

<math>
b_l=\frac{1}{\sqrt{N}}\sum_{k} e^{ik\cdot R_l}\beta_k
</math>

<math>
\tau_1=(0,\frac{1}{\sqrt{3}})a,\tau_2=(-\frac{1}{2},-\frac{1}{2\sqrt{3}})a,\tau_3=(\frac{1}{2},-\frac{1}{2\sqrt{3}})a
</math>

Then we can rewrite the Hamiltonian in reciprocal space:

<math>
\hat{H}=\sum_{k}c^{\dagger}(k)\begin{bmatrix}
0&f^{*}\\
f&0
\end{bmatrix}c(k),\qquad c(k)=\begin{bmatrix}
\alpha(k)\\
\beta(k)
\end{bmatrix},\qquad f=-t\sum_{j=1}^{3}e^{ik\cdot\tau_{j}}
</math>
===Choice of K values===
*We define the first Brillouin zone of the reciprocal lattice in the standard way.It is clear that the six points at the corners of the FBZ fall into two groups of three which are equivalent, so we need consider only two equivalent corners that we label K and K0 as in the figure. Explicitly, their positions in momentum space are given by:

[[File:K values.png|thumb|400px|K values]]

<math>
K=\frac{2\pi}{a}(\frac{1}{3},\frac{1}{\sqrt{3}}),\qquad K'=\frac{2\pi}{a}(\frac{2}{3},0)
</math>

These K and K' point are also called "Dirac points".
===Solution of TB model Hamiltonian===
*We can rewrite the Schrödinger equation in reciprocal space

<math>
H(k)\Psi_k=E\Psi_k
</math>

From the explicit expressions for the nearest neighbor vectors we obtain:

<math>
f=-t\exp(ik_xa(1+2e^{i\cdot\frac{3k_xa}{2}}cos\frac{\sqrt{3}}{2}k_ya))
</math>

And the eigenvalues are given by

<math>
\epsilon_{s,k}=s|f|=\pm t(1+4cos(\frac{3}{2}k_xa)cos(\frac{\sqrt{3}}{2}k_ya)+4cos^2\frac{\sqrt{3}}{2}k_ya)^{\frac{1}{2}}
</math>

It is interesting to enquire whether there are any values of k for which f is zero? Any such value must satisfy the conditions? Luckily, the answer is yes and the points "at the corner" which satisfied these conditions are exactly the "Dirac points".
===Dirac model near K points--Not spins but like spins===
*With the Dirac model in Quantum Fireld Theory , We have the Dirac's original:

<math>(i\gamma^\mu\partial_\mu-m)\Psi=0</math>,
<math>\gamma^\mu\gamma^v+\gamma^v\gamma^\mu=2g^{v\mu}</math>,
<math>v,\mu=0,1,2,3</math>

It is very suggestive to write the Hamiltonian in the form:

[[File:Dirac points.png|thumb|400px|Dirac points]]

<math>
\hat{H}=\hbar v_F\begin{bmatrix}
0&k_x+ik_y\\
k_x-ik_y&0
\end{bmatrix}=\hbar v_F\hat{\sigma}\cdot q,\qquad \epsilon(k)=\pm v_F|k|
</math>

where the components of the operator <math>\sigma</math> are the usual Pauli matrices,and <math>v_F</math> is the Fermi velocity. It is clear that the eigenvalues are a function only of the magnitude of k, not its direction in the 2D space,therefore, although the Hamiltonian expression contains spin operators, the Dirac point derived from this is not the spin of particle or quasiparticle, (which can be compared with the quasiparticle ansatz in BCS theory).However, it still inspires us to regard these K and K' points of TMDs as some kinds of qubit resources and control them on this theoretical basis.

===Numerical calculation results by Quantum Espresso===
[[File:Dirac points in Band structure.png|thumb|400px|Band structure]]
===Why choose heterobilayer TMDCs instead of monolayer===
* There are four main carrier properties that optimal Opto-valleytronics should possess.
*# long carrier lifetime
*# long valley lifetime
*# high valley polarization
*# long valley coherence time
* By adopting 2D heterostructure TMDCs materials, we can create these conditions for building promising quantum platform. (eg. Due to the type II band alignment and weak hybridization of van der Waals heterostructure, the electron–hole layer separation, the electron–hole exchange interaction is greatly reduced, resulting in a long cryogenic lifetime (ns to <math>\mu</math>s)and valley lifetime (~ 10 ns) of the interlayer exciton<ref>Opto-valleytronics in the 2D van der Waals heterostructure[https://doi.org/10.1007/s12274-020-3036-x]</ref>)

==Method==
===Principle===
* It was shown that inversion symmetry breaking can lead to opposite circular dichroism in different momentum space regions, which induces an interesting phenomena of optical selection rules at the symmetry corner points of the Brillouin zone. This enables a valley-dependent control of electrons with optical signal of different circular polarizations.
* However, valley optical selection rule in the heterobilayer TMDCs is a bit different than in the monolayer, which depends on the emission location and the emission energy. Additionally, in most cases, owing to the type II band alignment and weak hybridization, the conduction band minimum and valence band maximum of the van der Waals heterostructure are dominated by orbitals from different layers. As a result, electrons and holes tend to reside in different layers of the structure.
* It was found that the g-factor value of the interlayer exciton depends on the stacking sequence of the layers. For the AA-stacked <math>\text{MoSe}_2</math>/<math>\text{WSe}_2</math>, the value of the singlet g-factor between 6 to 7 has been reported; while, for the AB-stacked <math>\text{WSe}_2</math>/<math>\text{MoSe}_2</math>, the value of g-factor ~ −15 has been reported

===Experimental setup===

====Setup====
[[File:Experimental Setup.jpg|thumb|400px|Experimental Setup]]
* As shown in figure

====Quantum simulator====
* The <math>\text{MoSe}_2</math>/<math>\text{WSe}_2</math> heterostructure is fabricated via mechanical exfoliation and aligned transfer method. The stacking sequence of the sample is checked using second harmonic generation (SHG)-based measurement and it is found to be AA stacking. The sample is loaded into a magnetocryostat to control the magnetic ﬁeld and sample temperature.

====Qubit initialisation====
* Use circularly polarised light to create valley exciton polarization
* Use linearly polarised light to pump the qubits to rotate the valley pseudospin in the superposition plane of K and K′ valley polarizations
* The polarization state of the excitation and collection is controlled using a combination of polarizer, quarter-wave plate, and half-wave plate.

====Qubit control====
* Most popular control techniques these days
*# 1st approach: The coupling between the optical (electromagnetic) signal and the valley information can also be used to control the valley information. For the intralayer exciton, the exciton can be moved from one valley to another by applying intense THz radiation. This is shown by the observation of linearly polarized emission from higher-order sidebands when the sample is excited with a 100-fs near-resonant circularly polarized light pulse together with a 33-fs 40-THz pulse. Theoretically, it is shown that this is caused by a THz pulse driven intervalley transition. It is also predicted that a complete transfer from K (K’) to K’ (K) valley can be done within a period less than 5 fs if a more intense and shorter THz pulse is used.
*# 2nd approach: On the other hand, by utilizing the optical Stark effect, the phase control has been demonstrated. In this case, the valley state is first prepared in the <math>|K\rang+|K'\rang</math> state by using a near-resonant linearly polarized excitation. A red-detuned 100-fs circularly polarized light pulse is then used to break the valley degeneracy, which allows the coherent rotation of the valley state to the <math>|K\rang+e^{i\Delta\phi}|K'\rang</math> state with <math>\Delta\phi</math> depends on the pulse intensity and duration. This rotation is reflected in the linear polarization state of the exciton emission.

====Qubit readout====
* The PL emission is directed by a multimode optical ﬁber into a spectrometer (Andor Shamrock) with a CCD detector for spectroscopic recording

==Results==

==References==
<references />

File:Dirac points in Band structure.png

2021-04-30T16:03:23Z

Chu WenHao:

Optical control of TMDCs valley pseudospin qubits

2021-04-30T16:02:35Z

Chu WenHao: /* Q & A */

==Linked project==
* This project is conducted by the same people as 👉 [[Control over the atomic spins within certain molecules by NMR technique]]

==Group members==
*'''XU ZIZHOU (Bensen)'''
**'''Matric Number:''' ''A0229645W''
**'''Email:''' ''[mailto:info@example.org zizhou_xu@u.nus.edu]''
*'''CHU WENHAO'''
**'''Matric Number:''' ''A0228747R''
**'''Email:''' ''[mailto:info@example.org e0675585@u.nus.edu]''
==Description==

* The strategies and materials for building universal quantum computers are being actively researched, and various platforms for holding qubits have been proposed and tested<ref>Experimental perfect state transfer of an entangled photonic qubit[https://www.nature.com/articles/ncomms11339]</ref><ref>Superconducting Qubits: Current State of Play[https://www.annualreviews.org/doi/abs/10.1146/annurev-conmatphys-031119-050605]</ref><ref>Two-qubit entangling gates within arbitrarily long chains of trapped ions[https://journals.aps.org/pra/abstract/10.1103/PhysRevA.100.022332]</ref><ref>Digital Coherent Control of a Superconducting Qubit[https://journals.aps.org/prapplied/abstract/10.1103/PhysRevApplied.11.014009]</ref><ref>Fast quantum logic gates with trapped-ion qubits[https://www.nature.com/articles/nature25737]</ref>. Within all the options such as superconducting qubits and ion trap qubits, solid-state qubits have illustrated great benefits of the compatibility with the existing semiconductor technology, which attracts lots of attention to this platform. Moreover, because of the direct band gap and strong spin-orbit coupling discovered in the monolayer transition metal dichalcogenides (TMDCs), especially some semiconductor monolayers possess a sizeable direct bandgap of ≈1.5–2 eV in the optical range allowing electrostatic confinement and optical manipulation of carriers<ref>Atomically Thin MoS2: A New Direct-Gap Semiconductor[https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.105.136805]</ref><ref>Emerging Photoluminescence in Monolayer MoS2[https://pubs.acs.org/doi/10.1021/nl903868w]</ref>, chances are TMDCs materials have the potential to become the promising platform for fabricating qubits. There are four main types of TMDCs that have been suggested as advantageous in acting qubits, <math>\text{WS}_2</math> , <math>\text{WSe}_2</math> , <math>\text{MoS}_2</math> and <math>\text{MoSe}_2</math> . Kormányos et al used DFT calculations to confirm that <math>\text{WS}_2</math> and <math>\text{WSe}_2</math> , are better than <math>\text{MoS}_2</math> and <math>\text{MoSe}_2</math> in terms of the spin-valley coupling<ref>Spin-Orbit Coupling, Quantum Dots, and Qubits in Monolayer Transition Metal Dichalcogenides[https://journals.aps.org/prx/abstract/10.1103/PhysRevX.4.011034]</ref>, which is really important since spin-valley coupling is regarded as a key factor that can improve the coherence lifetime of spin-valley states.

* After several decades of active research on the properties of TMDCs, some interesting features have been discovered. Such as the extra degree of freedom offered by valley magnetic moment, which can realise bit 0 and bit 1 to form a qubit in momentum space. In addition, these valleys are independently addressable by an optical signal, and optical controlling technique is currently under research by many groups both theoretically and experimentally. As a result, as inspired by the idea from one of the theoretical work<ref>Spin-valley qubit in nanostructures of monolayer semiconductors: Optical control and hyperfine interaction[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.93.045313]</ref>, we intend to build a quantum simulator and investigate optical control of the spin-valley pseudospin on 2D heterobilayer TMDCs.

==Q & A==

===What are K & K' points?===
[[File:K_&_K'_points.png|thumb|400px|K & K' points]]
* K-points are sampling points of Brillouin zone in reciprocal lattice
* The monolayer TMDs have a unique band structure with the conduction and valence band edges both at the degenerate K and K‘ valleys at the corners of the hexagonal Brillouin zone. The direct-gap optical transitions have a selection rule: left- (right-) handed circular polarized photons couple to the interband transitions in the K (K’) valley only.
===The Tight-Binding Model===
* By introducing the tight-binding model for TMDs, we will give a Hamiltonian of TMDs that can be further deduced and calculated. As is showned in figure (b):under the tight-binding ansatz,the atom(e.g.the red one)can only jump to the nearest position, whose interaction processing is called "hopping" in physics,If we use the lattice represetation and the corresponding creation operators commonly used in condensed matter physics,then the nearest-neighbor tight-binding Hamiltonian has the simple form:

[[File:graphene.png|thumb|400px|Graphene]]

<math>
\hat{H}=-t\sum_{l} \sum_{j=1}^{3}a_{l+\tau_j}^{\dagger}b_{l}+h.c.
</math>

By Fourier Transformation,<math>a_l</math> and <math>b_{l}</math> can be written as:

<math>
a_l=\frac{1}{\sqrt{N}}\sum_{k} e^{ik\cdot R_l}\alpha_k
</math>

<math>
b_l=\frac{1}{\sqrt{N}}\sum_{k} e^{ik\cdot R_l}\beta_k
</math>

<math>
\tau_1=(0,\frac{1}{\sqrt{3}})a,\tau_2=(-\frac{1}{2},-\frac{1}{2\sqrt{3}})a,\tau_3=(\frac{1}{2},-\frac{1}{2\sqrt{3}})a
</math>

Then we can rewrite the Hamiltonian in reciprocal space:

<math>
\hat{H}=\sum_{k}c^{\dagger}(k)\begin{bmatrix}
0&f^{*}\\
f&0
\end{bmatrix}c(k),\qquad c(k)=\begin{bmatrix}
\alpha(k)\\
\beta(k)
\end{bmatrix},\qquad f=-t\sum_{j=1}^{3}e^{ik\cdot\tau_{j}}
</math>
===Choice of K values===
*We define the first Brillouin zone of the reciprocal lattice in the standard way.It is clear that the six points at the corners of the FBZ fall into two groups of three which are equivalent, so we need consider only two equivalent corners that we label K and K0 as in the figure. Explicitly, their positions in momentum space are given by:

[[File:K values.png|thumb|400px|K values]]

<math>
K=\frac{2\pi}{a}(\frac{1}{3},\frac{1}{\sqrt{3}}),\qquad K'=\frac{2\pi}{a}(\frac{2}{3},0)
</math>

These K and K' point are also called "Dirac points".
===Solution of TB model Hamiltonian===
*We can rewrite the Schrödinger equation in reciprocal space

<math>
H(k)\Psi_k=E\Psi_k
</math>

From the explicit expressions for the nearest neighbor vectors we obtain:

<math>
f=-t\exp(ik_xa(1+2e^{i\cdot\frac{3k_xa}{2}}cos\frac{\sqrt{3}}{2}k_ya))
</math>

And the eigenvalues are given by

<math>
\epsilon_{s,k}=s|f|=\pm t(1+4cos(\frac{3}{2}k_xa)cos(\frac{\sqrt{3}}{2}k_ya)+4cos^2\frac{\sqrt{3}}{2}k_ya)^{\frac{1}{2}}
</math>

It is interesting to enquire whether there are any values of k for which f is zero? Any such value must satisfy the conditions? Luckily, the answer is yes and the points "at the corner" which satisfied these conditions are exactly the "Dirac points".
===Dirac model near K points--Not spins but like spins===
*With the Dirac model in Quantum Fireld Theory, it is very suggestive to write the Hamiltonian in the form:

[[File:Dirac points.png|thumb|400px|Dirac points]]

<math>
\hat{H}=\hbar v_F\begin{bmatrix}
0&k_x+ik_y\\
k_x-ik_y&0
\end{bmatrix}=\hbar v_F\hat{\sigma}\cdot q,\qquad \epsilon(k)=\pm v_F|k|
</math>

where the components of the operator <math>\sigma</math> are the usual Pauli matrices,and <math>v_F</math> is the Fermi velocity. It is clear that the eigenvalues are a function only of the magnitude of k, not its direction in the 2D space,therefore, although the Hamiltonian expression contains spin operators, the Dirac point derived from this is not the spin of particle or quasiparticle, (which can be compared with the quasiparticle ansatz in BCS theory).However, it still inspires us to regard these K and K' points of TMDs as some kinds of qubit resources and control them on this theoretical basis.

===Numerical calculation results by Quantum Espresso===
[[File:Dirac points in Band structure.png|thumb|400px|Band structure]]
===Why choose heterobilayer TMDCs instead of monolayer===
* There are four main carrier properties that optimal Opto-valleytronics should possess.
*# long carrier lifetime
*# long valley lifetime
*# high valley polarization
*# long valley coherence time
* By adopting 2D heterostructure TMDCs materials, we can create these conditions for building promising quantum platform. (eg. Due to the type II band alignment and weak hybridization of van der Waals heterostructure, the electron–hole layer separation, the electron–hole exchange interaction is greatly reduced, resulting in a long cryogenic lifetime (ns to <math>\mu</math>s)and valley lifetime (~ 10 ns) of the interlayer exciton<ref>Opto-valleytronics in the 2D van der Waals heterostructure[https://doi.org/10.1007/s12274-020-3036-x]</ref>)

==Method==
===Principle===
* It was shown that inversion symmetry breaking can lead to opposite circular dichroism in different momentum space regions, which induces an interesting phenomena of optical selection rules at the symmetry corner points of the Brillouin zone. This enables a valley-dependent control of electrons with optical signal of different circular polarizations.
* However, valley optical selection rule in the heterobilayer TMDCs is a bit different than in the monolayer, which depends on the emission location and the emission energy. Additionally, in most cases, owing to the type II band alignment and weak hybridization, the conduction band minimum and valence band maximum of the van der Waals heterostructure are dominated by orbitals from different layers. As a result, electrons and holes tend to reside in different layers of the structure.
* It was found that the g-factor value of the interlayer exciton depends on the stacking sequence of the layers. For the AA-stacked <math>\text{MoSe}_2</math>/<math>\text{WSe}_2</math>, the value of the singlet g-factor between 6 to 7 has been reported; while, for the AB-stacked <math>\text{WSe}_2</math>/<math>\text{MoSe}_2</math>, the value of g-factor ~ −15 has been reported

===Experimental setup===

====Setup====
[[File:Experimental Setup.jpg|thumb|400px|Experimental Setup]]
* As shown in figure

====Quantum simulator====
* The <math>\text{MoSe}_2</math>/<math>\text{WSe}_2</math> heterostructure is fabricated via mechanical exfoliation and aligned transfer method. The stacking sequence of the sample is checked using second harmonic generation (SHG)-based measurement and it is found to be AA stacking. The sample is loaded into a magnetocryostat to control the magnetic ﬁeld and sample temperature.

====Qubit initialisation====
* Use circularly polarised light to create valley exciton polarization
* Use linearly polarised light to pump the qubits to rotate the valley pseudospin in the superposition plane of K and K′ valley polarizations
* The polarization state of the excitation and collection is controlled using a combination of polarizer, quarter-wave plate, and half-wave plate.

====Qubit control====
* Most popular control techniques these days
*# 1st approach: The coupling between the optical (electromagnetic) signal and the valley information can also be used to control the valley information. For the intralayer exciton, the exciton can be moved from one valley to another by applying intense THz radiation. This is shown by the observation of linearly polarized emission from higher-order sidebands when the sample is excited with a 100-fs near-resonant circularly polarized light pulse together with a 33-fs 40-THz pulse. Theoretically, it is shown that this is caused by a THz pulse driven intervalley transition. It is also predicted that a complete transfer from K (K’) to K’ (K) valley can be done within a period less than 5 fs if a more intense and shorter THz pulse is used.
*# 2nd approach: On the other hand, by utilizing the optical Stark effect, the phase control has been demonstrated. In this case, the valley state is first prepared in the <math>|K\rang+|K'\rang</math> state by using a near-resonant linearly polarized excitation. A red-detuned 100-fs circularly polarized light pulse is then used to break the valley degeneracy, which allows the coherent rotation of the valley state to the <math>|K\rang+e^{i\Delta\phi}|K'\rang</math> state with <math>\Delta\phi</math> depends on the pulse intensity and duration. This rotation is reflected in the linear polarization state of the exciton emission.

====Qubit readout====
* The PL emission is directed by a multimode optical ﬁber into a spectrometer (Andor Shamrock) with a CCD detector for spectroscopic recording

==Results==

==References==
<references />

File:Dirac points.png

2021-04-30T15:41:00Z

Chu WenHao:

File:K values.png

2021-04-30T15:40:34Z

Chu WenHao:

Optical control of TMDCs valley pseudospin qubits

2021-04-30T15:40:05Z

Chu WenHao: /* What are K & K' points? */

==Linked project==
* This project is conducted by the same people as 👉 [[Control over the atomic spins within certain molecules by NMR technique]]

==Group members==
*'''XU ZIZHOU (Bensen)'''
**'''Matric Number:''' ''A0229645W''
**'''Email:''' ''[mailto:info@example.org zizhou_xu@u.nus.edu]''
*'''CHU WENHAO'''
**'''Matric Number:''' ''A0228747R''
**'''Email:''' ''[mailto:info@example.org e0675585@u.nus.edu]''
==Description==

* The strategies and materials for building universal quantum computers are being actively researched, and various platforms for holding qubits have been proposed and tested<ref>Experimental perfect state transfer of an entangled photonic qubit[https://www.nature.com/articles/ncomms11339]</ref><ref>Superconducting Qubits: Current State of Play[https://www.annualreviews.org/doi/abs/10.1146/annurev-conmatphys-031119-050605]</ref><ref>Two-qubit entangling gates within arbitrarily long chains of trapped ions[https://journals.aps.org/pra/abstract/10.1103/PhysRevA.100.022332]</ref><ref>Digital Coherent Control of a Superconducting Qubit[https://journals.aps.org/prapplied/abstract/10.1103/PhysRevApplied.11.014009]</ref><ref>Fast quantum logic gates with trapped-ion qubits[https://www.nature.com/articles/nature25737]</ref>. Within all the options such as superconducting qubits and ion trap qubits, solid-state qubits have illustrated great benefits of the compatibility with the existing semiconductor technology, which attracts lots of attention to this platform. Moreover, because of the direct band gap and strong spin-orbit coupling discovered in the monolayer transition metal dichalcogenides (TMDCs), especially some semiconductor monolayers possess a sizeable direct bandgap of ≈1.5–2 eV in the optical range allowing electrostatic confinement and optical manipulation of carriers<ref>Atomically Thin MoS2: A New Direct-Gap Semiconductor[https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.105.136805]</ref><ref>Emerging Photoluminescence in Monolayer MoS2[https://pubs.acs.org/doi/10.1021/nl903868w]</ref>, chances are TMDCs materials have the potential to become the promising platform for fabricating qubits. There are four main types of TMDCs that have been suggested as advantageous in acting qubits, <math>\text{WS}_2</math> , <math>\text{WSe}_2</math> , <math>\text{MoS}_2</math> and <math>\text{MoSe}_2</math> . Kormányos et al used DFT calculations to confirm that <math>\text{WS}_2</math> and <math>\text{WSe}_2</math> , are better than <math>\text{MoS}_2</math> and <math>\text{MoSe}_2</math> in terms of the spin-valley coupling<ref>Spin-Orbit Coupling, Quantum Dots, and Qubits in Monolayer Transition Metal Dichalcogenides[https://journals.aps.org/prx/abstract/10.1103/PhysRevX.4.011034]</ref>, which is really important since spin-valley coupling is regarded as a key factor that can improve the coherence lifetime of spin-valley states.

* After several decades of active research on the properties of TMDCs, some interesting features have been discovered. Such as the extra degree of freedom offered by valley magnetic moment, which can realise bit 0 and bit 1 to form a qubit in momentum space. In addition, these valleys are independently addressable by an optical signal, and optical controlling technique is currently under research by many groups both theoretically and experimentally. As a result, as inspired by the idea from one of the theoretical work<ref>Spin-valley qubit in nanostructures of monolayer semiconductors: Optical control and hyperfine interaction[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.93.045313]</ref>, we intend to build a quantum simulator and investigate optical control of the spin-valley pseudospin on 2D heterobilayer TMDCs.

==Q & A==

===What are K & K' points?===
[[File:K_&_K'_points.png|thumb|400px|K & K' points]]
* K-points are sampling points of Brillouin zone in reciprocal lattice
* The monolayer TMDs have a unique band structure with the conduction and valence band edges both at the degenerate K and K‘ valleys at the corners of the hexagonal Brillouin zone. The direct-gap optical transitions have a selection rule: left- (right-) handed circular polarized photons couple to the interband transitions in the K (K’) valley only.
* The Tight-Binding Model:By introducing the tight-binding model for TMDs, we will give a Hamiltonian of TMDs that can be further deduced and calculated. As is showned in figure (b):under the tight-binding ansatz,the atom(e.g.the red one)can only jump to the nearest position, whose interaction processing is called "hopping" in physics,If we use the lattice represetation and the corresponding creation operators commonly used in condensed matter physics,then the nearest-neighbor tight-binding Hamiltonian has the simple form:

[[File:graphene.png|thumb|400px|Graphene]]

<math>
\hat{H}=-t\sum_{l} \sum_{j=1}^{3}a_{l+\tau_j}^{\dagger}b_{l}+h.c.
</math>

By Fourier Transformation,<math>a_l</math> and <math>b_{l}</math> can be written as:

<math>
a_l=\frac{1}{\sqrt{N}}\sum_{k} e^{ik\cdot R_l}\alpha_k
</math>

<math>
b_l=\frac{1}{\sqrt{N}}\sum_{k} e^{ik\cdot R_l}\beta_k
</math>

<math>
\tau_1=(0,\frac{1}{\sqrt{3}})a,\tau_2=(-\frac{1}{2},-\frac{1}{2\sqrt{3}})a,\tau_3=(\frac{1}{2},-\frac{1}{2\sqrt{3}})a
</math>

Then we can rewrite the Hamiltonian in reciprocal space:

<math>
\hat{H}=\sum_{k}c^{\dagger}(k)\begin{bmatrix}
0&f^{*}\\
f&0
\end{bmatrix}c(k),\qquad c(k)=\begin{bmatrix}
\alpha(k)\\
\beta(k)
\end{bmatrix},\qquad f=-t\sum_{j=1}^{3}e^{ik\cdot\tau_{j}}
</math>

*Choice of K values:We define the first Brillouin zone of the reciprocal lattice in the standard way.It is clear that the six points at the corners of the FBZ fall into two groups of three which are equivalent, so we need consider only two equivalent corners that we label K and K0 as in the figure. Explicitly, their positions in momentum space are given by:

[[File:K values.png|thumb|400px|K values]]

<math>
K=\frac{2\pi}{a}(\frac{1}{3},\frac{1}{\sqrt{3}}),\qquad K'=\frac{2\pi}{a}(\frac{2}{3},0)
</math>

These K and K' point are also called "Dirac points".

*Solution of TB model Hamiltonian:

We can rewrite the Schrödinger equation in reciprocal space

<math>
H(k)\Psi_k=E\Psi_k
</math>

From the explicit expressions for the nearest neighbor vectors we obtain:

<math>
f=-t\exp(ik_xa(1+2e^{i\cdot\frac{3k_xa}{2}}cos\frac{\sqrt{3}}{2}k_ya))
</math>

And the eigenvalues are given by

<math>
\epsilon_{s,k}=s|f|=\pm t(1+4cos(\frac{3}{2}k_xa)cos(\frac{\sqrt{3}}{2}k_ya)+4cos^2\frac{\sqrt{3}}{2}k_ya)^{\frac{1}{2}}
</math>

It is interesting to enquire whether there are any values of k for which f is zero? Any such value must satisfy the conditions? Luckily, the answer is yes and the points "at the corner" which satisfied these conditions are exactly the "Dirac points".

*Not spins but like spins (Dirac model near K points):With the Dirac model in Quantum Fireld Theory, it is very suggestive to write the Hamiltonian in the form:

[[File:Dirac points.png|thumb|400px|Dirac points]]

<math>
\hat{H}=\hbar v_F\begin{bmatrix}
0&k_x+ik_y\\
k_x-ik_y&0
\end{bmatrix}=\hbar v_F\hat{\sigma}\cdot q,\qquad \epsilon(k)=\pm v_F|k|
</math>

where the components of the operator <math>\sigma</math> are the usual Pauli matrices,and <math>v_F</math> is the Fermi velocity. It is clear that the eigenvalues are a function only of the magnitude of k, not its direction in the 2D space,therefore, although the Hamiltonian expression contains spin operators, the Dirac point derived from this is not the spin of particle or quasiparticle, (which can be compared with the quasiparticle ansatz in BCS theory).However, it still inspires us to regard these K and K' points of TMDs as some kinds of qubit resources and control them on this theoretical basis.

===Why choose heterobilayer TMDCs instead of monolayer===
* There are four main carrier properties that optimal Opto-valleytronics should possess.
*# long carrier lifetime
*# long valley lifetime
*# high valley polarization
*# long valley coherence time
* By adopting 2D heterostructure TMDCs materials, we can create these conditions for building promising quantum platform. (eg. Due to the type II band alignment and weak hybridization of van der Waals heterostructure, the electron–hole layer separation, the electron–hole exchange interaction is greatly reduced, resulting in a long cryogenic lifetime (ns to <math>\mu</math>s)and valley lifetime (~ 10 ns) of the interlayer exciton<ref>Opto-valleytronics in the 2D van der Waals heterostructure[https://doi.org/10.1007/s12274-020-3036-x]</ref>)

==Method==
===Principle===
* It was shown that inversion symmetry breaking can lead to opposite circular dichroism in different momentum space regions, which induces an interesting phenomena of optical selection rules at the symmetry corner points of the Brillouin zone. This enables a valley-dependent control of electrons with optical signal of different circular polarizations.
* However, valley optical selection rule in the heterobilayer TMDCs is a bit different than in the monolayer, which depends on the emission location and the emission energy. Additionally, in most cases, owing to the type II band alignment and weak hybridization, the conduction band minimum and valence band maximum of the van der Waals heterostructure are dominated by orbitals from different layers. As a result, electrons and holes tend to reside in different layers of the structure.
* It was found that the g-factor value of the interlayer exciton depends on the stacking sequence of the layers. For the AA-stacked <math>\text{MoSe}_2</math>/<math>\text{WSe}_2</math>, the value of the singlet g-factor between 6 to 7 has been reported; while, for the AB-stacked <math>\text{WSe}_2</math>/<math>\text{MoSe}_2</math>, the value of g-factor ~ −15 has been reported

===Experimental setup===

====Setup====
[[File:Experimental Setup.jpg|thumb|400px|Experimental Setup]]
* As shown in figure

====Quantum simulator====
* The <math>\text{MoSe}_2</math>/<math>\text{WSe}_2</math> heterostructure is fabricated via mechanical exfoliation and aligned transfer method. The stacking sequence of the sample is checked using second harmonic generation (SHG)-based measurement and it is found to be AA stacking. The sample is loaded into a magnetocryostat to control the magnetic ﬁeld and sample temperature.

====Qubit initialisation====
* Use circularly polarised light to create valley exciton polarization
* Use linearly polarised light to pump the qubits to rotate the valley pseudospin in the superposition plane of K and K′ valley polarizations
* The polarization state of the excitation and collection is controlled using a combination of polarizer, quarter-wave plate, and half-wave plate.

====Qubit control====
* Most popular control techniques these days
*# 1st approach: The coupling between the optical (electromagnetic) signal and the valley information can also be used to control the valley information. For the intralayer exciton, the exciton can be moved from one valley to another by applying intense THz radiation. This is shown by the observation of linearly polarized emission from higher-order sidebands when the sample is excited with a 100-fs near-resonant circularly polarized light pulse together with a 33-fs 40-THz pulse. Theoretically, it is shown that this is caused by a THz pulse driven intervalley transition. It is also predicted that a complete transfer from K (K’) to K’ (K) valley can be done within a period less than 5 fs if a more intense and shorter THz pulse is used.
*# 2nd approach: On the other hand, by utilizing the optical Stark effect, the phase control has been demonstrated. In this case, the valley state is first prepared in the <math>|K\rang+|K'\rang</math> state by using a near-resonant linearly polarized excitation. A red-detuned 100-fs circularly polarized light pulse is then used to break the valley degeneracy, which allows the coherent rotation of the valley state to the <math>|K\rang+e^{i\Delta\phi}|K'\rang</math> state with <math>\Delta\phi</math> depends on the pulse intensity and duration. This rotation is reflected in the linear polarization state of the exciton emission.

====Qubit readout====
* The PL emission is directed by a multimode optical ﬁber into a spectrometer (Andor Shamrock) with a CCD detector for spectroscopic recording

==Results==

==References==
<references />

Optical control of TMDCs valley pseudospin qubits

2021-04-30T15:37:33Z

Chu WenHao: /* What are K & K' points? */

==Linked project==
* This project is conducted by the same people as 👉 [[Control over the atomic spins within certain molecules by NMR technique]]

==Group members==
*'''XU ZIZHOU (Bensen)'''
**'''Matric Number:''' ''A0229645W''
**'''Email:''' ''[mailto:info@example.org zizhou_xu@u.nus.edu]''
*'''CHU WENHAO'''
**'''Matric Number:''' ''A0228747R''
**'''Email:''' ''[mailto:info@example.org e0675585@u.nus.edu]''
==Description==

* The strategies and materials for building universal quantum computers are being actively researched, and various platforms for holding qubits have been proposed and tested<ref>Experimental perfect state transfer of an entangled photonic qubit[https://www.nature.com/articles/ncomms11339]</ref><ref>Superconducting Qubits: Current State of Play[https://www.annualreviews.org/doi/abs/10.1146/annurev-conmatphys-031119-050605]</ref><ref>Two-qubit entangling gates within arbitrarily long chains of trapped ions[https://journals.aps.org/pra/abstract/10.1103/PhysRevA.100.022332]</ref><ref>Digital Coherent Control of a Superconducting Qubit[https://journals.aps.org/prapplied/abstract/10.1103/PhysRevApplied.11.014009]</ref><ref>Fast quantum logic gates with trapped-ion qubits[https://www.nature.com/articles/nature25737]</ref>. Within all the options such as superconducting qubits and ion trap qubits, solid-state qubits have illustrated great benefits of the compatibility with the existing semiconductor technology, which attracts lots of attention to this platform. Moreover, because of the direct band gap and strong spin-orbit coupling discovered in the monolayer transition metal dichalcogenides (TMDCs), especially some semiconductor monolayers possess a sizeable direct bandgap of ≈1.5–2 eV in the optical range allowing electrostatic confinement and optical manipulation of carriers<ref>Atomically Thin MoS2: A New Direct-Gap Semiconductor[https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.105.136805]</ref><ref>Emerging Photoluminescence in Monolayer MoS2[https://pubs.acs.org/doi/10.1021/nl903868w]</ref>, chances are TMDCs materials have the potential to become the promising platform for fabricating qubits. There are four main types of TMDCs that have been suggested as advantageous in acting qubits, <math>\text{WS}_2</math> , <math>\text{WSe}_2</math> , <math>\text{MoS}_2</math> and <math>\text{MoSe}_2</math> . Kormányos et al used DFT calculations to confirm that <math>\text{WS}_2</math> and <math>\text{WSe}_2</math> , are better than <math>\text{MoS}_2</math> and <math>\text{MoSe}_2</math> in terms of the spin-valley coupling<ref>Spin-Orbit Coupling, Quantum Dots, and Qubits in Monolayer Transition Metal Dichalcogenides[https://journals.aps.org/prx/abstract/10.1103/PhysRevX.4.011034]</ref>, which is really important since spin-valley coupling is regarded as a key factor that can improve the coherence lifetime of spin-valley states.

* After several decades of active research on the properties of TMDCs, some interesting features have been discovered. Such as the extra degree of freedom offered by valley magnetic moment, which can realise bit 0 and bit 1 to form a qubit in momentum space. In addition, these valleys are independently addressable by an optical signal, and optical controlling technique is currently under research by many groups both theoretically and experimentally. As a result, as inspired by the idea from one of the theoretical work<ref>Spin-valley qubit in nanostructures of monolayer semiconductors: Optical control and hyperfine interaction[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.93.045313]</ref>, we intend to build a quantum simulator and investigate optical control of the spin-valley pseudospin on 2D heterobilayer TMDCs.

==Q & A==

===What are K & K' points?===
[[File:K_&_K'_points.png|thumb|400px|K & K' points]]
* K-points are sampling points of Brillouin zone in reciprocal lattice
* The monolayer TMDs have a unique band structure with the conduction and valence band edges both at the degenerate K and K‘ valleys at the corners of the hexagonal Brillouin zone. The direct-gap optical transitions have a selection rule: left- (right-) handed circular polarized photons couple to the interband transitions in the K (K’) valley only.
* The Tight-Binding Model:By introducing the tight-binding model for TMDs, we will give a Hamiltonian of TMDs that can be further deduced and calculated. As is showned in figure (b):under the tight-binding ansatz,the atom(e.g.the red one)can only jump to the nearest position, whose interaction processing is called "hopping" in physics,If we use the lattice represetation and the corresponding creation operators commonly used in condensed matter physics,then the nearest-neighbor tight-binding Hamiltonian has the simple form:

[[File:graphene.png|thumb|400px|Graphene]]

<math>
\hat{H}=-t\sum_{l} \sum_{j=1}^{3}a_{l+\tau_j}^{\dagger}b_{l}+h.c.
</math>

By Fourier Transformation,<math>a_l</math> and <math>b_{l}</math> can be written as:

<math>
a_l=\frac{1}{\sqrt{N}}\sum_{k} e^{ik\cdot R_l}\alpha_k
</math>

<math>
b_l=\frac{1}{\sqrt{N}}\sum_{k} e^{ik\cdot R_l}\beta_k
</math>

<math>
\tau_1=(0,\frac{1}{\sqrt{3}})a,\tau_2=(-\frac{1}{2},-\frac{1}{2\sqrt{3}})a,\tau_3=(\frac{1}{2},-\frac{1}{2\sqrt{3}})a
</math>

Then we can rewrite the Hamiltonian in reciprocal space:

<math>
\hat{H}=\sum_{k}c^{\dagger}(k)\begin{bmatrix}
0&f^{*}\\
f&0
\end{bmatrix}c(k),\qquad c(k)=\begin{bmatrix}
\alpha(k)\\
\beta(k)
\end{bmatrix},\qquad f=-t\sum_{j=1}^{3}e^{ik\cdot\tau_{j}}
</math>

*Choice of K values:We define the first Brillouin zone of the reciprocal lattice in the standard way.It is clear that the six points at the corners of the FBZ fall into two groups of three which are equivalent, so we need consider only two equivalent corners that we label K and K0 as in the figure. Explicitly, their positions in momentum space are given by:

<math>
K=\frac{2\pi}{a}(\frac{1}{3},\frac{1}{\sqrt{3}}),\qquad K'=\frac{2\pi}{a}(\frac{2}{3},0)
</math>

These K and K' point are also called "Dirac points".

*Solution of TB model Hamiltonian:

We can rewrite the Schrödinger equation in reciprocal space

<math>
H(k)\Psi_k=E\Psi_k
</math>

From the explicit expressions for the nearest neighbor vectors we obtain:

<math>
f=-t\exp(ik_xa(1+2e^{i\cdot\frac{3k_xa}{2}}cos\frac{\sqrt{3}}{2}k_ya))
</math>

And the eigenvalues are given by

<math>
\epsilon_{s,k}=s|f|=\pm t(1+4cos(\frac{3}{2}k_xa)cos(\frac{\sqrt{3}}{2}k_ya)+4cos^2\frac{\sqrt{3}}{2}k_ya)^{\frac{1}{2}}
</math>

It is interesting to enquire whether there are any values of k for which f is zero? Any such value must satisfy the conditions? Luckily, the answer is yes and the points "at the corner" which satisfied these conditions are exactly the "Dirac points".

*Not spins but like spins (Dirac model near K points):With the Dirac model in Quantum Fireld Theory, it is very suggestive to write the Hamiltonian in the form:

<math>
\hat{H}=\hbar v_F\begin{bmatrix}
0&k_x+ik_y\\
k_x-ik_y&0
\end{bmatrix}=\hbar v_F\hat{\sigma}\cdot q,\qquad \epsilon(k)=\pm v_F|k|
</math>

where the components of the operator <math>\sigma</math> are the usual Pauli matrices,and <math>v_F</math> is the Fermi velocity. It is clear that the eigenvalues are a function only of the magnitude of k, not its direction in the 2D space,therefore, although the Hamiltonian expression contains spin operators, the Dirac point derived from this is not the spin of particle or quasiparticle, (which can be compared with the quasiparticle ansatz in BCS theory).However, it still inspires us to regard these K and K' points of TMDs as some kinds of qubit resources and control them on this theoretical basis.

===Why choose heterobilayer TMDCs instead of monolayer===
* There are four main carrier properties that optimal Opto-valleytronics should possess.
*# long carrier lifetime
*# long valley lifetime
*# high valley polarization
*# long valley coherence time
* By adopting 2D heterostructure TMDCs materials, we can create these conditions for building promising quantum platform. (eg. Due to the type II band alignment and weak hybridization of van der Waals heterostructure, the electron–hole layer separation, the electron–hole exchange interaction is greatly reduced, resulting in a long cryogenic lifetime (ns to <math>\mu</math>s)and valley lifetime (~ 10 ns) of the interlayer exciton<ref>Opto-valleytronics in the 2D van der Waals heterostructure[https://doi.org/10.1007/s12274-020-3036-x]</ref>)

==Method==
===Principle===
* It was shown that inversion symmetry breaking can lead to opposite circular dichroism in different momentum space regions, which induces an interesting phenomena of optical selection rules at the symmetry corner points of the Brillouin zone. This enables a valley-dependent control of electrons with optical signal of different circular polarizations.
* However, valley optical selection rule in the heterobilayer TMDCs is a bit different than in the monolayer, which depends on the emission location and the emission energy. Additionally, in most cases, owing to the type II band alignment and weak hybridization, the conduction band minimum and valence band maximum of the van der Waals heterostructure are dominated by orbitals from different layers. As a result, electrons and holes tend to reside in different layers of the structure.
* It was found that the g-factor value of the interlayer exciton depends on the stacking sequence of the layers. For the AA-stacked <math>\text{MoSe}_2</math>/<math>\text{WSe}_2</math>, the value of the singlet g-factor between 6 to 7 has been reported; while, for the AB-stacked <math>\text{WSe}_2</math>/<math>\text{MoSe}_2</math>, the value of g-factor ~ −15 has been reported

===Experimental setup===

====Setup====
[[File:Experimental Setup.jpg|thumb|400px|Experimental Setup]]
* As shown in figure

====Quantum simulator====
* The <math>\text{MoSe}_2</math>/<math>\text{WSe}_2</math> heterostructure is fabricated via mechanical exfoliation and aligned transfer method. The stacking sequence of the sample is checked using second harmonic generation (SHG)-based measurement and it is found to be AA stacking. The sample is loaded into a magnetocryostat to control the magnetic ﬁeld and sample temperature.

====Qubit initialisation====
* Use circularly polarised light to create valley exciton polarization
* Use linearly polarised light to pump the qubits to rotate the valley pseudospin in the superposition plane of K and K′ valley polarizations
* The polarization state of the excitation and collection is controlled using a combination of polarizer, quarter-wave plate, and half-wave plate.

====Qubit control====
* Most popular control techniques these days
*# 1st approach: The coupling between the optical (electromagnetic) signal and the valley information can also be used to control the valley information. For the intralayer exciton, the exciton can be moved from one valley to another by applying intense THz radiation. This is shown by the observation of linearly polarized emission from higher-order sidebands when the sample is excited with a 100-fs near-resonant circularly polarized light pulse together with a 33-fs 40-THz pulse. Theoretically, it is shown that this is caused by a THz pulse driven intervalley transition. It is also predicted that a complete transfer from K (K’) to K’ (K) valley can be done within a period less than 5 fs if a more intense and shorter THz pulse is used.
*# 2nd approach: On the other hand, by utilizing the optical Stark effect, the phase control has been demonstrated. In this case, the valley state is first prepared in the <math>|K\rang+|K'\rang</math> state by using a near-resonant linearly polarized excitation. A red-detuned 100-fs circularly polarized light pulse is then used to break the valley degeneracy, which allows the coherent rotation of the valley state to the <math>|K\rang+e^{i\Delta\phi}|K'\rang</math> state with <math>\Delta\phi</math> depends on the pulse intensity and duration. This rotation is reflected in the linear polarization state of the exciton emission.

====Qubit readout====
* The PL emission is directed by a multimode optical ﬁber into a spectrometer (Andor Shamrock) with a CCD detector for spectroscopic recording

==Results==

==References==
<references />

Optical control of TMDCs valley pseudospin qubits

2021-04-30T15:36:42Z

Chu WenHao: /* What are K & K' points? */

==Linked project==
* This project is conducted by the same people as 👉 [[Control over the atomic spins within certain molecules by NMR technique]]

==Group members==
*'''XU ZIZHOU (Bensen)'''
**'''Matric Number:''' ''A0229645W''
**'''Email:''' ''[mailto:info@example.org zizhou_xu@u.nus.edu]''
*'''CHU WENHAO'''
**'''Matric Number:''' ''A0228747R''
**'''Email:''' ''[mailto:info@example.org e0675585@u.nus.edu]''
==Description==

* The strategies and materials for building universal quantum computers are being actively researched, and various platforms for holding qubits have been proposed and tested<ref>Experimental perfect state transfer of an entangled photonic qubit[https://www.nature.com/articles/ncomms11339]</ref><ref>Superconducting Qubits: Current State of Play[https://www.annualreviews.org/doi/abs/10.1146/annurev-conmatphys-031119-050605]</ref><ref>Two-qubit entangling gates within arbitrarily long chains of trapped ions[https://journals.aps.org/pra/abstract/10.1103/PhysRevA.100.022332]</ref><ref>Digital Coherent Control of a Superconducting Qubit[https://journals.aps.org/prapplied/abstract/10.1103/PhysRevApplied.11.014009]</ref><ref>Fast quantum logic gates with trapped-ion qubits[https://www.nature.com/articles/nature25737]</ref>. Within all the options such as superconducting qubits and ion trap qubits, solid-state qubits have illustrated great benefits of the compatibility with the existing semiconductor technology, which attracts lots of attention to this platform. Moreover, because of the direct band gap and strong spin-orbit coupling discovered in the monolayer transition metal dichalcogenides (TMDCs), especially some semiconductor monolayers possess a sizeable direct bandgap of ≈1.5–2 eV in the optical range allowing electrostatic confinement and optical manipulation of carriers<ref>Atomically Thin MoS2: A New Direct-Gap Semiconductor[https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.105.136805]</ref><ref>Emerging Photoluminescence in Monolayer MoS2[https://pubs.acs.org/doi/10.1021/nl903868w]</ref>, chances are TMDCs materials have the potential to become the promising platform for fabricating qubits. There are four main types of TMDCs that have been suggested as advantageous in acting qubits, <math>\text{WS}_2</math> , <math>\text{WSe}_2</math> , <math>\text{MoS}_2</math> and <math>\text{MoSe}_2</math> . Kormányos et al used DFT calculations to confirm that <math>\text{WS}_2</math> and <math>\text{WSe}_2</math> , are better than <math>\text{MoS}_2</math> and <math>\text{MoSe}_2</math> in terms of the spin-valley coupling<ref>Spin-Orbit Coupling, Quantum Dots, and Qubits in Monolayer Transition Metal Dichalcogenides[https://journals.aps.org/prx/abstract/10.1103/PhysRevX.4.011034]</ref>, which is really important since spin-valley coupling is regarded as a key factor that can improve the coherence lifetime of spin-valley states.

* After several decades of active research on the properties of TMDCs, some interesting features have been discovered. Such as the extra degree of freedom offered by valley magnetic moment, which can realise bit 0 and bit 1 to form a qubit in momentum space. In addition, these valleys are independently addressable by an optical signal, and optical controlling technique is currently under research by many groups both theoretically and experimentally. As a result, as inspired by the idea from one of the theoretical work<ref>Spin-valley qubit in nanostructures of monolayer semiconductors: Optical control and hyperfine interaction[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.93.045313]</ref>, we intend to build a quantum simulator and investigate optical control of the spin-valley pseudospin on 2D heterobilayer TMDCs.

==Q & A==

===What are K & K' points?===
[[File:K_&_K'_points.png|thumb|400px|K & K' points]]
* K-points are sampling points of Brillouin zone in reciprocal lattice
* The monolayer TMDs have a unique band structure with the conduction and valence band edges both at the degenerate K and K‘ valleys at the corners of the hexagonal Brillouin zone. The direct-gap optical transitions have a selection rule: left- (right-) handed circular polarized photons couple to the interband transitions in the K (K’) valley only.
* The Tight-Binding Model:By introducing the tight-binding model for TMDs, we will give a Hamiltonian of TMDs that can be further deduced and calculated. As is showned in figure (b):under the tight-binding ansatz,the atom(e.g.the red one)can only jump to the nearest position, whose interaction processing is called "hopping" in physics,If we use the lattice represetation and the corresponding creation operators commonly used in condensed matter physics,then the nearest-neighbor tight-binding Hamiltonian has the simple form:

[[File:graphene.png|thumb|400px|center]]

<math>
\hat{H}=-t\sum_{l} \sum_{j=1}^{3}a_{l+\tau_j}^{\dagger}b_{l}+h.c.
</math>

By Fourier Transformation,<math>a_l</math> and <math>b_{l}</math> can be written as:

<math>
a_l=\frac{1}{\sqrt{N}}\sum_{k} e^{ik\cdot R_l}\alpha_k
</math>

<math>
b_l=\frac{1}{\sqrt{N}}\sum_{k} e^{ik\cdot R_l}\beta_k
</math>

<math>
\tau_1=(0,\frac{1}{\sqrt{3}})a,\tau_2=(-\frac{1}{2},-\frac{1}{2\sqrt{3}})a,\tau_3=(\frac{1}{2},-\frac{1}{2\sqrt{3}})a
</math>

Then we can rewrite the Hamiltonian in reciprocal space:

<math>
\hat{H}=\sum_{k}c^{\dagger}(k)\begin{bmatrix}
0&f^{*}\\
f&0
\end{bmatrix}c(k),\qquad c(k)=\begin{bmatrix}
\alpha(k)\\
\beta(k)
\end{bmatrix},\qquad f=-t\sum_{j=1}^{3}e^{ik\cdot\tau_{j}}
</math>

*Choice of K values:We define the first Brillouin zone of the reciprocal lattice in the standard way.It is clear that the six points at the corners of the FBZ fall into two groups of three which are equivalent, so we need consider only two equivalent corners that we label K and K0 as in the figure. Explicitly, their positions in momentum space are given by:

<math>
K=\frac{2\pi}{a}(\frac{1}{3},\frac{1}{\sqrt{3}}),\qquad K'=\frac{2\pi}{a}(\frac{2}{3},0)
</math>

These K and K' point are also called "Dirac points".

*Solution of TB model Hamiltonian:

We can rewrite the Schrödinger equation in reciprocal space

<math>
H(k)\Psi_k=E\Psi_k
</math>

From the explicit expressions for the nearest neighbor vectors we obtain:

<math>
f=-t\exp(ik_xa(1+2e^{i\cdot\frac{3k_xa}{2}}cos\frac{\sqrt{3}}{2}k_ya))
</math>

And the eigenvalues are given by

<math>
\epsilon_{s,k}=s|f|=\pm t(1+4cos(\frac{3}{2}k_xa)cos(\frac{\sqrt{3}}{2}k_ya)+4cos^2\frac{\sqrt{3}}{2}k_ya)^{\frac{1}{2}}
</math>

It is interesting to enquire whether there are any values of k for which f is zero? Any such value must satisfy the conditions? Luckily, the answer is yes and the points "at the corner" which satisfied these conditions are exactly the "Dirac points".

*Not spins but like spins (Dirac model near K points):With the Dirac model in Quantum Fireld Theory, it is very suggestive to write the Hamiltonian in the form:

<math>
\hat{H}=\hbar v_F\begin{bmatrix}
0&k_x+ik_y\\
k_x-ik_y&0
\end{bmatrix}=\hbar v_F\hat{\sigma}\cdot q,\qquad \epsilon(k)=\pm v_F|k|
</math>

where the components of the operator <math>\sigma</math> are the usual Pauli matrices,and <math>v_F</math> is the Fermi velocity. It is clear that the eigenvalues are a function only of the magnitude of k, not its direction in the 2D space,therefore, although the Hamiltonian expression contains spin operators, the Dirac point derived from this is not the spin of particle or quasiparticle, (which can be compared with the quasiparticle ansatz in BCS theory).However, it still inspires us to regard these K and K' points of TMDs as some kinds of qubit resources and control them on this theoretical basis.

===Why choose heterobilayer TMDCs instead of monolayer===
* There are four main carrier properties that optimal Opto-valleytronics should possess.
*# long carrier lifetime
*# long valley lifetime
*# high valley polarization
*# long valley coherence time
* By adopting 2D heterostructure TMDCs materials, we can create these conditions for building promising quantum platform. (eg. Due to the type II band alignment and weak hybridization of van der Waals heterostructure, the electron–hole layer separation, the electron–hole exchange interaction is greatly reduced, resulting in a long cryogenic lifetime (ns to <math>\mu</math>s)and valley lifetime (~ 10 ns) of the interlayer exciton<ref>Opto-valleytronics in the 2D van der Waals heterostructure[https://doi.org/10.1007/s12274-020-3036-x]</ref>)

==Method==
===Principle===
* It was shown that inversion symmetry breaking can lead to opposite circular dichroism in different momentum space regions, which induces an interesting phenomena of optical selection rules at the symmetry corner points of the Brillouin zone. This enables a valley-dependent control of electrons with optical signal of different circular polarizations.
* However, valley optical selection rule in the heterobilayer TMDCs is a bit different than in the monolayer, which depends on the emission location and the emission energy. Additionally, in most cases, owing to the type II band alignment and weak hybridization, the conduction band minimum and valence band maximum of the van der Waals heterostructure are dominated by orbitals from different layers. As a result, electrons and holes tend to reside in different layers of the structure.
* It was found that the g-factor value of the interlayer exciton depends on the stacking sequence of the layers. For the AA-stacked <math>\text{MoSe}_2</math>/<math>\text{WSe}_2</math>, the value of the singlet g-factor between 6 to 7 has been reported; while, for the AB-stacked <math>\text{WSe}_2</math>/<math>\text{MoSe}_2</math>, the value of g-factor ~ −15 has been reported

===Experimental setup===

====Setup====
[[File:Experimental Setup.jpg|thumb|400px|Experimental Setup]]
* As shown in figure

====Quantum simulator====
* The <math>\text{MoSe}_2</math>/<math>\text{WSe}_2</math> heterostructure is fabricated via mechanical exfoliation and aligned transfer method. The stacking sequence of the sample is checked using second harmonic generation (SHG)-based measurement and it is found to be AA stacking. The sample is loaded into a magnetocryostat to control the magnetic ﬁeld and sample temperature.

====Qubit initialisation====
* Use circularly polarised light to create valley exciton polarization
* Use linearly polarised light to pump the qubits to rotate the valley pseudospin in the superposition plane of K and K′ valley polarizations
* The polarization state of the excitation and collection is controlled using a combination of polarizer, quarter-wave plate, and half-wave plate.

====Qubit control====
* Most popular control techniques these days
*# 1st approach: The coupling between the optical (electromagnetic) signal and the valley information can also be used to control the valley information. For the intralayer exciton, the exciton can be moved from one valley to another by applying intense THz radiation. This is shown by the observation of linearly polarized emission from higher-order sidebands when the sample is excited with a 100-fs near-resonant circularly polarized light pulse together with a 33-fs 40-THz pulse. Theoretically, it is shown that this is caused by a THz pulse driven intervalley transition. It is also predicted that a complete transfer from K (K’) to K’ (K) valley can be done within a period less than 5 fs if a more intense and shorter THz pulse is used.
*# 2nd approach: On the other hand, by utilizing the optical Stark effect, the phase control has been demonstrated. In this case, the valley state is first prepared in the <math>|K\rang+|K'\rang</math> state by using a near-resonant linearly polarized excitation. A red-detuned 100-fs circularly polarized light pulse is then used to break the valley degeneracy, which allows the coherent rotation of the valley state to the <math>|K\rang+e^{i\Delta\phi}|K'\rang</math> state with <math>\Delta\phi</math> depends on the pulse intensity and duration. This rotation is reflected in the linear polarization state of the exciton emission.

====Qubit readout====
* The PL emission is directed by a multimode optical ﬁber into a spectrometer (Andor Shamrock) with a CCD detector for spectroscopic recording

==Results==

==References==
<references />

Optical control of TMDCs valley pseudospin qubits

2021-04-30T15:34:27Z

Chu WenHao: /* What are K & K' points? */

==Linked project==
* This project is conducted by the same people as 👉 [[Control over the atomic spins within certain molecules by NMR technique]]

==Group members==
*'''XU ZIZHOU (Bensen)'''
**'''Matric Number:''' ''A0229645W''
**'''Email:''' ''[mailto:info@example.org zizhou_xu@u.nus.edu]''
*'''CHU WENHAO'''
**'''Matric Number:''' ''A0228747R''
**'''Email:''' ''[mailto:info@example.org e0675585@u.nus.edu]''
==Description==

* The strategies and materials for building universal quantum computers are being actively researched, and various platforms for holding qubits have been proposed and tested<ref>Experimental perfect state transfer of an entangled photonic qubit[https://www.nature.com/articles/ncomms11339]</ref><ref>Superconducting Qubits: Current State of Play[https://www.annualreviews.org/doi/abs/10.1146/annurev-conmatphys-031119-050605]</ref><ref>Two-qubit entangling gates within arbitrarily long chains of trapped ions[https://journals.aps.org/pra/abstract/10.1103/PhysRevA.100.022332]</ref><ref>Digital Coherent Control of a Superconducting Qubit[https://journals.aps.org/prapplied/abstract/10.1103/PhysRevApplied.11.014009]</ref><ref>Fast quantum logic gates with trapped-ion qubits[https://www.nature.com/articles/nature25737]</ref>. Within all the options such as superconducting qubits and ion trap qubits, solid-state qubits have illustrated great benefits of the compatibility with the existing semiconductor technology, which attracts lots of attention to this platform. Moreover, because of the direct band gap and strong spin-orbit coupling discovered in the monolayer transition metal dichalcogenides (TMDCs), especially some semiconductor monolayers possess a sizeable direct bandgap of ≈1.5–2 eV in the optical range allowing electrostatic confinement and optical manipulation of carriers<ref>Atomically Thin MoS2: A New Direct-Gap Semiconductor[https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.105.136805]</ref><ref>Emerging Photoluminescence in Monolayer MoS2[https://pubs.acs.org/doi/10.1021/nl903868w]</ref>, chances are TMDCs materials have the potential to become the promising platform for fabricating qubits. There are four main types of TMDCs that have been suggested as advantageous in acting qubits, <math>\text{WS}_2</math> , <math>\text{WSe}_2</math> , <math>\text{MoS}_2</math> and <math>\text{MoSe}_2</math> . Kormányos et al used DFT calculations to confirm that <math>\text{WS}_2</math> and <math>\text{WSe}_2</math> , are better than <math>\text{MoS}_2</math> and <math>\text{MoSe}_2</math> in terms of the spin-valley coupling<ref>Spin-Orbit Coupling, Quantum Dots, and Qubits in Monolayer Transition Metal Dichalcogenides[https://journals.aps.org/prx/abstract/10.1103/PhysRevX.4.011034]</ref>, which is really important since spin-valley coupling is regarded as a key factor that can improve the coherence lifetime of spin-valley states.

* After several decades of active research on the properties of TMDCs, some interesting features have been discovered. Such as the extra degree of freedom offered by valley magnetic moment, which can realise bit 0 and bit 1 to form a qubit in momentum space. In addition, these valleys are independently addressable by an optical signal, and optical controlling technique is currently under research by many groups both theoretically and experimentally. As a result, as inspired by the idea from one of the theoretical work<ref>Spin-valley qubit in nanostructures of monolayer semiconductors: Optical control and hyperfine interaction[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.93.045313]</ref>, we intend to build a quantum simulator and investigate optical control of the spin-valley pseudospin on 2D heterobilayer TMDCs.

==Q & A==

===What are K & K' points?===
[[File:K_&_K'_points.png|thumb|400px|K & K' points]]
* K-points are sampling points of Brillouin zone in reciprocal lattice
* The monolayer TMDs have a unique band structure with the conduction and valence band edges both at the degenerate K and K‘ valleys at the corners of the hexagonal Brillouin zone. The direct-gap optical transitions have a selection rule: left- (right-) handed circular polarized photons couple to the interband transitions in the K (K’) valley only.
* The Tight-Binding Model:By introducing the tight-binding model for TMDs, we will give a Hamiltonian of TMDs that can be further deduced and calculated. As is showned in figure (b):under the tight-binding ansatz,the atom(e.g.the red one)can only jump to the nearest position, whose interaction processing is called "hopping" in physics,If we use the lattice represetation and the corresponding creation operators commonly used in condensed matter physics,then the nearest-neighbor tight-binding Hamiltonian has the simple form:

[[File:graphene.png|500px|center]]

<math>
\hat{H}=-t\sum_{l} \sum_{j=1}^{3}a_{l+\tau_j}^{\dagger}b_{l}+h.c.
</math>

By Fourier Transformation,<math>a_l</math> and <math>b_{l}</math> can be written as:

<math>
a_l=\frac{1}{\sqrt{N}}\sum_{k} e^{ik\cdot R_l}\alpha_k
</math>

<math>
b_l=\frac{1}{\sqrt{N}}\sum_{k} e^{ik\cdot R_l}\beta_k
</math>

<math>
\tau_1=(0,\frac{1}{\sqrt{3}})a,\tau_2=(-\frac{1}{2},-\frac{1}{2\sqrt{3}})a,\tau_3=(\frac{1}{2},-\frac{1}{2\sqrt{3}})a
</math>

Then we can rewrite the Hamiltonian in reciprocal space:

<math>
\hat{H}=\sum_{k}c^{\dagger}(k)\begin{bmatrix}
0&f^{*}\\
f&0
\end{bmatrix}c(k),\qquad c(k)=\begin{bmatrix}
\alpha(k)\\
\beta(k)
\end{bmatrix},\qquad f=-t\sum_{j=1}^{3}e^{ik\cdot\tau_{j}}
</math>

*Choice of K values:We define the first Brillouin zone of the reciprocal lattice in the standard way.It is clear that the six points at the corners of the FBZ fall into two groups of three which are equivalent, so we need consider only two equivalent corners that we label K and K0 as in the figure. Explicitly, their positions in momentum space are given by:

<math>
K=\frac{2\pi}{a}(\frac{1}{3},\frac{1}{\sqrt{3}}),\qquad K'=\frac{2\pi}{a}(\frac{2}{3},0)
</math>

These K and K' point are also called "Dirac points".

*Solution of TB model Hamiltonian:

We can rewrite the Schrödinger equation in reciprocal space

<math>
H(k)\Psi_k=E\Psi_k
</math>

From the explicit expressions for the nearest neighbor vectors we obtain:

<math>
f=-t\exp(ik_xa(1+2e^{i\cdot\frac{3k_xa}{2}}cos\frac{\sqrt{3}}{2}k_ya))
</math>

And the eigenvalues are given by

<math>
\epsilon_{s,k}=s|f|=\pm t(1+4cos(\frac{3}{2}k_xa)cos(\frac{\sqrt{3}}{2}k_ya)+4cos^2\frac{\sqrt{3}}{2}k_ya)^{\frac{1}{2}}
</math>

It is interesting to enquire whether there are any values of k for which f is zero? Any such value must satisfy the conditions? Luckily, the answer is yes and the points "at the corner" which satisfied these conditions are exactly the "Dirac points".

*Not spins but like spins (Dirac model near K points):With the Dirac model in Quantum Fireld Theory, it is very suggestive to write the Hamiltonian in the form:

<math>
\hat{H}=\hbar v_F\begin{bmatrix}
0&k_x+ik_y\\
k_x-ik_y&0
\end{bmatrix}=\hbar v_F\hat{\sigma}\cdot q,\qquad \epsilon(k)=\pm v_F|k|
</math>

where the components of the operator <math>\sigma</math> are the usual Pauli matrices,and <math>v_F</math> is the Fermi velocity. It is clear that the eigenvalues are a function only of the magnitude of k, not its direction in the 2D space,therefore, although the Hamiltonian expression contains spin operators, the Dirac point derived from this is not the spin of particle or quasiparticle, (which can be compared with the quasiparticle ansatz in BCS theory).However, it still inspires us to regard these K and K' points of TMDs as some kinds of qubit resources and control them on this theoretical basis.

===Why choose heterobilayer TMDCs instead of monolayer===
* There are four main carrier properties that optimal Opto-valleytronics should possess.
*# long carrier lifetime
*# long valley lifetime
*# high valley polarization
*# long valley coherence time
* By adopting 2D heterostructure TMDCs materials, we can create these conditions for building promising quantum platform. (eg. Due to the type II band alignment and weak hybridization of van der Waals heterostructure, the electron–hole layer separation, the electron–hole exchange interaction is greatly reduced, resulting in a long cryogenic lifetime (ns to <math>\mu</math>s)and valley lifetime (~ 10 ns) of the interlayer exciton<ref>Opto-valleytronics in the 2D van der Waals heterostructure[https://doi.org/10.1007/s12274-020-3036-x]</ref>)

==Method==
===Principle===
* It was shown that inversion symmetry breaking can lead to opposite circular dichroism in different momentum space regions, which induces an interesting phenomena of optical selection rules at the symmetry corner points of the Brillouin zone. This enables a valley-dependent control of electrons with optical signal of different circular polarizations.
* However, valley optical selection rule in the heterobilayer TMDCs is a bit different than in the monolayer, which depends on the emission location and the emission energy. Additionally, in most cases, owing to the type II band alignment and weak hybridization, the conduction band minimum and valence band maximum of the van der Waals heterostructure are dominated by orbitals from different layers. As a result, electrons and holes tend to reside in different layers of the structure.
* It was found that the g-factor value of the interlayer exciton depends on the stacking sequence of the layers. For the AA-stacked <math>\text{MoSe}_2</math>/<math>\text{WSe}_2</math>, the value of the singlet g-factor between 6 to 7 has been reported; while, for the AB-stacked <math>\text{WSe}_2</math>/<math>\text{MoSe}_2</math>, the value of g-factor ~ −15 has been reported

===Experimental setup===

====Setup====
[[File:Experimental Setup.jpg|thumb|400px|Experimental Setup]]
* As shown in figure

====Quantum simulator====
* The <math>\text{MoSe}_2</math>/<math>\text{WSe}_2</math> heterostructure is fabricated via mechanical exfoliation and aligned transfer method. The stacking sequence of the sample is checked using second harmonic generation (SHG)-based measurement and it is found to be AA stacking. The sample is loaded into a magnetocryostat to control the magnetic ﬁeld and sample temperature.

====Qubit initialisation====
* Use circularly polarised light to create valley exciton polarization
* Use linearly polarised light to pump the qubits to rotate the valley pseudospin in the superposition plane of K and K′ valley polarizations
* The polarization state of the excitation and collection is controlled using a combination of polarizer, quarter-wave plate, and half-wave plate.

====Qubit control====
* Most popular control techniques these days
*# 1st approach: The coupling between the optical (electromagnetic) signal and the valley information can also be used to control the valley information. For the intralayer exciton, the exciton can be moved from one valley to another by applying intense THz radiation. This is shown by the observation of linearly polarized emission from higher-order sidebands when the sample is excited with a 100-fs near-resonant circularly polarized light pulse together with a 33-fs 40-THz pulse. Theoretically, it is shown that this is caused by a THz pulse driven intervalley transition. It is also predicted that a complete transfer from K (K’) to K’ (K) valley can be done within a period less than 5 fs if a more intense and shorter THz pulse is used.
*# 2nd approach: On the other hand, by utilizing the optical Stark effect, the phase control has been demonstrated. In this case, the valley state is first prepared in the <math>|K\rang+|K'\rang</math> state by using a near-resonant linearly polarized excitation. A red-detuned 100-fs circularly polarized light pulse is then used to break the valley degeneracy, which allows the coherent rotation of the valley state to the <math>|K\rang+e^{i\Delta\phi}|K'\rang</math> state with <math>\Delta\phi</math> depends on the pulse intensity and duration. This rotation is reflected in the linear polarization state of the exciton emission.

====Qubit readout====
* The PL emission is directed by a multimode optical ﬁber into a spectrometer (Andor Shamrock) with a CCD detector for spectroscopic recording

==Results==

==References==
<references />

Optical control of TMDCs valley pseudospin qubits

2021-04-30T15:05:00Z

Chu WenHao: /* What are K & K' points? */

==Linked project==
* This project is conducted by the same people as 👉 [[Control over the atomic spins within certain molecules by NMR technique]]

==Group members==
*'''XU ZIZHOU (Bensen)'''
**'''Matric Number:''' ''A0229645W''
**'''Email:''' ''[mailto:info@example.org zizhou_xu@u.nus.edu]''
*'''CHU WENHAO'''
**'''Matric Number:''' ''A0228747R''
**'''Email:''' ''[mailto:info@example.org e0675585@u.nus.edu]''
==Description==

* The strategies and materials for building universal quantum computers are being actively researched, and various platforms for holding qubits have been proposed and tested<ref>Experimental perfect state transfer of an entangled photonic qubit[https://www.nature.com/articles/ncomms11339]</ref><ref>Superconducting Qubits: Current State of Play[https://www.annualreviews.org/doi/abs/10.1146/annurev-conmatphys-031119-050605]</ref><ref>Two-qubit entangling gates within arbitrarily long chains of trapped ions[https://journals.aps.org/pra/abstract/10.1103/PhysRevA.100.022332]</ref><ref>Digital Coherent Control of a Superconducting Qubit[https://journals.aps.org/prapplied/abstract/10.1103/PhysRevApplied.11.014009]</ref><ref>Fast quantum logic gates with trapped-ion qubits[https://www.nature.com/articles/nature25737]</ref>. Within all the options such as superconducting qubits and ion trap qubits, solid-state qubits have illustrated great benefits of the compatibility with the existing semiconductor technology, which attracts lots of attention to this platform. Moreover, because of the direct band gap and strong spin-orbit coupling discovered in the monolayer transition metal dichalcogenides (TMDCs), especially some semiconductor monolayers possess a sizeable direct bandgap of ≈1.5–2 eV in the optical range allowing electrostatic confinement and optical manipulation of carriers<ref>Atomically Thin MoS2: A New Direct-Gap Semiconductor[https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.105.136805]</ref><ref>Emerging Photoluminescence in Monolayer MoS2[https://pubs.acs.org/doi/10.1021/nl903868w]</ref>, chances are TMDCs materials have the potential to become the promising platform for fabricating qubits. There are four main types of TMDCs that have been suggested as advantageous in acting qubits, <math>\text{WS}_2</math> , <math>\text{WSe}_2</math> , <math>\text{MoS}_2</math> and <math>\text{MoSe}_2</math> . Kormányos et al used DFT calculations to confirm that <math>\text{WS}_2</math> and <math>\text{WSe}_2</math> , are better than <math>\text{MoS}_2</math> and <math>\text{MoSe}_2</math> in terms of the spin-valley coupling<ref>Spin-Orbit Coupling, Quantum Dots, and Qubits in Monolayer Transition Metal Dichalcogenides[https://journals.aps.org/prx/abstract/10.1103/PhysRevX.4.011034]</ref>, which is really important since spin-valley coupling is regarded as a key factor that can improve the coherence lifetime of spin-valley states.

* After several decades of active research on the properties of TMDCs, some interesting features have been discovered. Such as the extra degree of freedom offered by valley magnetic moment, which can realise bit 0 and bit 1 to form a qubit in momentum space. In addition, these valleys are independently addressable by an optical signal, and optical controlling technique is currently under research by many groups both theoretically and experimentally. As a result, as inspired by the idea from one of the theoretical work<ref>Spin-valley qubit in nanostructures of monolayer semiconductors: Optical control and hyperfine interaction[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.93.045313]</ref>, we intend to build a quantum simulator and investigate optical control of the spin-valley pseudospin on 2D heterobilayer TMDCs.

==Q & A==

===What are K & K' points?===
[[File:K_&_K'_points.png|thumb|400px|K & K' points]]
* K-points are sampling points of Brillouin zone in reciprocal lattice
* The monolayer TMDs have a unique band structure with the conduction and valence band edges both at the degenerate K and K‘ valleys at the corners of the hexagonal Brillouin zone. The direct-gap optical transitions have a selection rule: left- (right-) handed circular polarized photons couple to the interband transitions in the K (K’) valley only.
* The Tight-Binding Model:By introducing the tight-binding model for TMDs, we will give a Hamiltonian of TMDs that can be further deduced and calculated. As is showned in figure (b):under the tight-binding ansatz,the atom(e.g.the red one)can only jump to the nearest position, whose interaction processing is called "hopping" in physics,If we use the lattice represetation and the corresponding creation operators commonly used in condensed matter physics,then the nearest-neighbor tight-binding Hamiltonian has the simple form:

[[File:graphene.png|500px|center]]

<math>
\hat{H}=-t\sum_{l} \sum_{j=1}^{3}a_{l+\tau_j}^{\dagger}b_{l}+h.c.
</math>

By Fourier Transformation,<math>a_l</math> and <math>b_{l}</math> can be written as:

<math>
a_l=\frac{1}{\sqrt{N}}\sum_{k} e^{ik\cdot R_l}\alpha_k
</math>

<math>
b_l=\frac{1}{\sqrt{N}}\sum_{k} e^{ik\cdot R_l}\beta_k
</math>

<math>
\tau_1=(0,\frac{1}{\sqrt{3}})a,\tau_2=(-\frac{1}{2},-\frac{1}{2\sqrt{3}})a,\tau_3=(\frac{1}{2},-\frac{1}{2\sqrt{3}})a
</math>

Then we can rewrite the Hamiltonian in reciprocal space:

<math>
\hat{H}=\sum_{k}c^{\dagger}(k)\begin{bmatrix}
0&f^{*}\\
f&0
\end{bmatrix}c(k),\qquad c(k)=\begin{bmatrix}
\alpha(k)\\
\beta(k)
\end{bmatrix},\qquad f=-t\sum_{j=1}^{3}e^{ik\cdot\tau_{j}}
</math>

*Choice of K values:We define the first Brillouin zone of the reciprocal lattice in the standard way.It is clear that the six points at the corners of the FBZ fall into two groups of three which are equivalent, so we need consider only two equivalent corners that we label K and K0 as in the figure. Explicitly, their positions in momentum space are given by:

<math>
K=\frac{2\pi}{a}(\frac{1}{3},\frac{1}{\sqrt{3}}),\qquad K'=\frac{2\pi}{a}(\frac{2}{3},0)
</math>

These K and K' point are also called "Dirac points".

*Solution of TB model Hamiltonian:

We can rewrite the Schrödinger equation in reciprocal space

<math>
H(k)\Psi_k=E\Psi_k
</math>

From the explicit expressions for the nearest neighbor vectors we obtain:

<math>
f=-t\exp(ik_xa(1+2e^{i\cdot\frac{3k_xa}{2}}cos\frac{\sqrt{3}}{2}k_ya))
</math>

And the eigenvalues are given by

<math>
\epsilon_{s,k}=s|f|=\pm t(1+4cos(\frac{3}{2}k_xa)cos(\frac{\sqrt{3}}{2}k_ya)+4cos^2\frac{\sqrt{3}}{2}k_ya)^{\frac{1}{2}}
</math>

It is interesting to enquire whether there are any values of k for which f is zero? Any such value must satisfy the conditions? Luckily, the answer is yes and the points "at the corner" which satisfied these conditions are exactly the "Dirac points".

===Why choose heterobilayer TMDCs instead of monolayer===
* There are four main carrier properties that optimal Opto-valleytronics should possess.
*# long carrier lifetime
*# long valley lifetime
*# high valley polarization
*# long valley coherence time
* By adopting 2D heterostructure TMDCs materials, we can create these conditions for building promising quantum platform. (eg. Due to the type II band alignment and weak hybridization of van der Waals heterostructure, the electron–hole layer separation, the electron–hole exchange interaction is greatly reduced, resulting in a long cryogenic lifetime (ns to <math>\mu</math>s)and valley lifetime (~ 10 ns) of the interlayer exciton<ref>Opto-valleytronics in the 2D van der Waals heterostructure[https://doi.org/10.1007/s12274-020-3036-x]</ref>)

==Method==
===Principle===
* It was shown that inversion symmetry breaking can lead to opposite circular dichroism in different momentum space regions, which induces an interesting phenomena of optical selection rules at the symmetry corner points of the Brillouin zone. This enables a valley-dependent control of electrons with optical signal of different circular polarizations.
* However, valley optical selection rule in the heterobilayer TMDCs is a bit different than in the monolayer, which depends on the emission location and the emission energy. Additionally, in most cases, owing to the type II band alignment and weak hybridization, the conduction band minimum and valence band maximum of the van der Waals heterostructure are dominated by orbitals from different layers. As a result, electrons and holes tend to reside in different layers of the structure.
* It was found that the g-factor value of the interlayer exciton depends on the stacking sequence of the layers. For the AA-stacked <math>\text{MoSe}_2</math>/<math>\text{WSe}_2</math>, the value of the singlet g-factor between 6 to 7 has been reported; while, for the AB-stacked <math>\text{WSe}_2</math>/<math>\text{MoSe}_2</math>, the value of g-factor ~ −15 has been reported

===Experimental setup===

====Setup====
[[File:Experimental Setup.jpg|thumb|400px|Experimental Setup]]
* As shown in figure

====Quantum simulator====
* The <math>\text{MoSe}_2</math>/<math>\text{WSe}_2</math> heterostructure is fabricated via mechanical exfoliation and aligned transfer method. The stacking sequence of the sample is checked using second harmonic generation (SHG)-based measurement and it is found to be AA stacking. The sample is loaded into a magnetocryostat to control the magnetic ﬁeld and sample temperature.

====Qubit initialisation====
* Use circularly polarised light to create valley exciton polarization
* Use linearly polarised light to pump the qubits to rotate the valley pseudospin in the superposition plane of K and K′ valley polarizations
* The polarization state of the excitation and collection is controlled using a combination of polarizer, quarter-wave plate, and half-wave plate.

====Qubit control====
* Most popular control techniques these days
*# 1st approach: The coupling between the optical (electromagnetic) signal and the valley information can also be used to control the valley information. For the intralayer exciton, the exciton can be moved from one valley to another by applying intense THz radiation. This is shown by the observation of linearly polarized emission from higher-order sidebands when the sample is excited with a 100-fs near-resonant circularly polarized light pulse together with a 33-fs 40-THz pulse. Theoretically, it is shown that this is caused by a THz pulse driven intervalley transition. It is also predicted that a complete transfer from K (K’) to K’ (K) valley can be done within a period less than 5 fs if a more intense and shorter THz pulse is used.
*# 2nd approach: On the other hand, by utilizing the optical Stark effect, the phase control has been demonstrated. In this case, the valley state is first prepared in the <math>|K\rang+|K'\rang</math> state by using a near-resonant linearly polarized excitation. A red-detuned 100-fs circularly polarized light pulse is then used to break the valley degeneracy, which allows the coherent rotation of the valley state to the <math>|K\rang+e^{i\Delta\phi}|K'\rang</math> state with <math>\Delta\phi</math> depends on the pulse intensity and duration. This rotation is reflected in the linear polarization state of the exciton emission.

====Qubit readout====
* The PL emission is directed by a multimode optical ﬁber into a spectrometer (Andor Shamrock) with a CCD detector for spectroscopic recording

==Results==

==References==
<references />

Optical control of TMDCs valley pseudospin qubits

2021-04-30T14:26:46Z

Chu WenHao: /* What are K & K' points? */

==Linked project==
* This project is conducted by the same people as 👉 [[Control over the atomic spins within certain molecules by NMR technique]]

==Group members==
*'''XU ZIZHOU (Bensen)'''
**'''Matric Number:''' ''A0229645W''
**'''Email:''' ''[mailto:info@example.org zizhou_xu@u.nus.edu]''
*'''CHU WENHAO'''
**'''Matric Number:''' ''A0228747R''
**'''Email:''' ''[mailto:info@example.org e0675585@u.nus.edu]''
==Description==

* The strategies and materials for building universal quantum computers are being actively researched, and various platforms for holding qubits have been proposed and tested<ref>Experimental perfect state transfer of an entangled photonic qubit[https://www.nature.com/articles/ncomms11339]</ref><ref>Superconducting Qubits: Current State of Play[https://www.annualreviews.org/doi/abs/10.1146/annurev-conmatphys-031119-050605]</ref><ref>Two-qubit entangling gates within arbitrarily long chains of trapped ions[https://journals.aps.org/pra/abstract/10.1103/PhysRevA.100.022332]</ref><ref>Digital Coherent Control of a Superconducting Qubit[https://journals.aps.org/prapplied/abstract/10.1103/PhysRevApplied.11.014009]</ref><ref>Fast quantum logic gates with trapped-ion qubits[https://www.nature.com/articles/nature25737]</ref>. Within all the options such as superconducting qubits and ion trap qubits, solid-state qubits have illustrated great benefits of the compatibility with the existing semiconductor technology, which attracts lots of attention to this platform. Moreover, because of the direct band gap and strong spin-orbit coupling discovered in the monolayer transition metal dichalcogenides (TMDCs), especially some semiconductor monolayers possess a sizeable direct bandgap of ≈1.5–2 eV in the optical range allowing electrostatic confinement and optical manipulation of carriers<ref>Atomically Thin MoS2: A New Direct-Gap Semiconductor[https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.105.136805]</ref><ref>Emerging Photoluminescence in Monolayer MoS2[https://pubs.acs.org/doi/10.1021/nl903868w]</ref>, chances are TMDCs materials have the potential to become the promising platform for fabricating qubits. There are four main types of TMDCs that have been suggested as advantageous in acting qubits, <math>\text{WS}_2</math> , <math>\text{WSe}_2</math> , <math>\text{MoS}_2</math> and <math>\text{MoSe}_2</math> . Kormányos et al used DFT calculations to confirm that <math>\text{WS}_2</math> and <math>\text{WSe}_2</math> , are better than <math>\text{MoS}_2</math> and <math>\text{MoSe}_2</math> in terms of the spin-valley coupling<ref>Spin-Orbit Coupling, Quantum Dots, and Qubits in Monolayer Transition Metal Dichalcogenides[https://journals.aps.org/prx/abstract/10.1103/PhysRevX.4.011034]</ref>, which is really important since spin-valley coupling is regarded as a key factor that can improve the coherence lifetime of spin-valley states.

* After several decades of active research on the properties of TMDCs, some interesting features have been discovered. Such as the extra degree of freedom offered by valley magnetic moment, which can realise bit 0 and bit 1 to form a qubit in momentum space. In addition, these valleys are independently addressable by an optical signal, and optical controlling technique is currently under research by many groups both theoretically and experimentally. As a result, as inspired by the idea from one of the theoretical work<ref>Spin-valley qubit in nanostructures of monolayer semiconductors: Optical control and hyperfine interaction[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.93.045313]</ref>, we intend to build a quantum simulator and investigate optical control of the spin-valley pseudospin on 2D heterobilayer TMDCs.

==Q & A==

===What are K & K' points?===
[[File:K_&_K'_points.png|thumb|400px|K & K' points]]
* K-points are sampling points of Brillouin zone in reciprocal lattice
* The monolayer TMDs have a unique band structure with the conduction and valence band edges both at the degenerate K and K‘ valleys at the corners of the hexagonal Brillouin zone. The direct-gap optical transitions have a selection rule: left- (right-) handed circular polarized photons couple to the interband transitions in the K (K’) valley only.
* The Tight-Binding Model:By introducing the tight-binding model for TMDs, we will give a Hamiltonian of TMDs that can be further deduced and calculated. As is showned in figure (b):under the tight-binding ansatz,the atom(e.g.the red one)can only jump to the nearest position, whose interaction processing is called "hopping" in physics,If we use the lattice represetation and the corresponding creation operators commonly used in condensed matter physics,then the nearest-neighbor tight-binding Hamiltonian has the simple form:

[[File:graphene.png|500px|center]]

<math>
\hat{H}=-t\sum_{l} \sum_{j=1}^{3}a_{l+\tau_j}^{\dagger}b_{l}+h.c.
</math>

By Fourier Transformation,<math>a_l</math> and <math>b_{l}</math> can be written as:

<math>
a_l=\frac{1}{\sqrt{N}}\sum_{k} e^{ik\cdot R_l}\alpha_k
</math>

<math>
b_l=\frac{1}{\sqrt{N}}\sum_{k} e^{ik\cdot R_l}\beta_k
</math>

<math>
\tau_1=(0,\frac{1}{\sqrt{3}})a,\tau_2=(-\frac{1}{2},-\frac{1}{2\sqrt{3}})a,\tau_3=(\frac{1}{2},-\frac{1}{2\sqrt{3}})a
</math>

Then we can rewrite the Hamiltonian in reciprocal space:

<math>
\hat{H}=\sum_{k}c^{\dagger}(k)\begin{bmatrix}
0&f^{*}\\
f&0
\end{bmatrix}c(k),\qquad c(k)=\begin{bmatrix}
\alpha(k)\\
\beta(k)
\end{bmatrix},\qquad f=-t\sum_{j=1}^{3}e^{ik\cdot\tau_{j}}
</math>

*Choice of K values:We define the first Brillouin zone of the reciprocal lattice in the standard way.It is clear that the six points at the corners of the FBZ fall into two groups of three which are equivalent, so we need consider only two equivalent corners that we label K and K0 as in the figure. Explicitly, their positions in momentum space are given by:

<math>
K=\frac{2\pi}{a}(\frac{1}{3},\frac{1}{\sqrt{3}}),\qquad K'=\frac{2\pi}{a}(\frac{2}{3},0)
</math>

===Why choose heterobilayer TMDCs instead of monolayer===
* There are four main carrier properties that optimal Opto-valleytronics should possess.
*# long carrier lifetime
*# long valley lifetime
*# high valley polarization
*# long valley coherence time
* By adopting 2D heterostructure TMDCs materials, we can create these conditions for building promising quantum platform. (eg. Due to the type II band alignment and weak hybridization of van der Waals heterostructure, the electron–hole layer separation, the electron–hole exchange interaction is greatly reduced, resulting in a long cryogenic lifetime (ns to <math>\mu</math>s)and valley lifetime (~ 10 ns) of the interlayer exciton<ref>Opto-valleytronics in the 2D van der Waals heterostructure[https://doi.org/10.1007/s12274-020-3036-x]</ref>)

==Method==
===Principle===
* It was shown that inversion symmetry breaking can lead to opposite circular dichroism in different momentum space regions, which induces an interesting phenomena of optical selection rules at the symmetry corner points of the Brillouin zone. This enables a valley-dependent control of electrons with optical signal of different circular polarizations.
* However, valley optical selection rule in the heterobilayer TMDCs is a bit different than in the monolayer, which depends on the emission location and the emission energy. Additionally, in most cases, owing to the type II band alignment and weak hybridization, the conduction band minimum and valence band maximum of the van der Waals heterostructure are dominated by orbitals from different layers. As a result, electrons and holes tend to reside in different layers of the structure.
* It was found that the g-factor value of the interlayer exciton depends on the stacking sequence of the layers. For the AA-stacked <math>\text{MoSe}_2</math>/<math>\text{WSe}_2</math>, the value of the singlet g-factor between 6 to 7 has been reported; while, for the AB-stacked <math>\text{WSe}_2</math>/<math>\text{MoSe}_2</math>, the value of g-factor ~ −15 has been reported

===Experimental setup===

====Setup====
[[File:Experimental Setup.jpg|thumb|400px|Experimental Setup]]
* As shown in figure

====Quantum simulator====
* The <math>\text{MoSe}_2</math>/<math>\text{WSe}_2</math> heterostructure is fabricated via mechanical exfoliation and aligned transfer method. The stacking sequence of the sample is checked using second harmonic generation (SHG)-based measurement and it is found to be AA stacking. The sample is loaded into a magnetocryostat to control the magnetic ﬁeld and sample temperature.

====Qubit initialisation====
* Use circularly polarised light to create valley exciton polarization
* Use linearly polarised light to pump the qubits to rotate the valley pseudospin in the superposition plane of K and K′ valley polarizations
* The polarization state of the excitation and collection is controlled using a combination of polarizer, quarter-wave plate, and half-wave plate.

====Qubit control====
* Most popular control techniques these days
*# 1st approach: The coupling between the optical (electromagnetic) signal and the valley information can also be used to control the valley information. For the intralayer exciton, the exciton can be moved from one valley to another by applying intense THz radiation. This is shown by the observation of linearly polarized emission from higher-order sidebands when the sample is excited with a 100-fs near-resonant circularly polarized light pulse together with a 33-fs 40-THz pulse. Theoretically, it is shown that this is caused by a THz pulse driven intervalley transition. It is also predicted that a complete transfer from K (K’) to K’ (K) valley can be done within a period less than 5 fs if a more intense and shorter THz pulse is used.
*# 2nd approach: On the other hand, by utilizing the optical Stark effect, the phase control has been demonstrated. In this case, the valley state is first prepared in the <math>|K\rang+|K'\rang</math> state by using a near-resonant linearly polarized excitation. A red-detuned 100-fs circularly polarized light pulse is then used to break the valley degeneracy, which allows the coherent rotation of the valley state to the <math>|K\rang+e^{i\Delta\phi}|K'\rang</math> state with <math>\Delta\phi</math> depends on the pulse intensity and duration. This rotation is reflected in the linear polarization state of the exciton emission.

====Qubit readout====
* The PL emission is directed by a multimode optical ﬁber into a spectrometer (Andor Shamrock) with a CCD detector for spectroscopic recording

==Results==

==References==
<references />

Optical control of TMDCs valley pseudospin qubits

2021-04-30T14:26:14Z

Chu WenHao: /* What are K & K' points? */

==Linked project==
* This project is conducted by the same people as 👉 [[Control over the atomic spins within certain molecules by NMR technique]]

==Group members==
*'''XU ZIZHOU (Bensen)'''
**'''Matric Number:''' ''A0229645W''
**'''Email:''' ''[mailto:info@example.org zizhou_xu@u.nus.edu]''
*'''CHU WENHAO'''
**'''Matric Number:''' ''A0228747R''
**'''Email:''' ''[mailto:info@example.org e0675585@u.nus.edu]''
==Description==

* The strategies and materials for building universal quantum computers are being actively researched, and various platforms for holding qubits have been proposed and tested<ref>Experimental perfect state transfer of an entangled photonic qubit[https://www.nature.com/articles/ncomms11339]</ref><ref>Superconducting Qubits: Current State of Play[https://www.annualreviews.org/doi/abs/10.1146/annurev-conmatphys-031119-050605]</ref><ref>Two-qubit entangling gates within arbitrarily long chains of trapped ions[https://journals.aps.org/pra/abstract/10.1103/PhysRevA.100.022332]</ref><ref>Digital Coherent Control of a Superconducting Qubit[https://journals.aps.org/prapplied/abstract/10.1103/PhysRevApplied.11.014009]</ref><ref>Fast quantum logic gates with trapped-ion qubits[https://www.nature.com/articles/nature25737]</ref>. Within all the options such as superconducting qubits and ion trap qubits, solid-state qubits have illustrated great benefits of the compatibility with the existing semiconductor technology, which attracts lots of attention to this platform. Moreover, because of the direct band gap and strong spin-orbit coupling discovered in the monolayer transition metal dichalcogenides (TMDCs), especially some semiconductor monolayers possess a sizeable direct bandgap of ≈1.5–2 eV in the optical range allowing electrostatic confinement and optical manipulation of carriers<ref>Atomically Thin MoS2: A New Direct-Gap Semiconductor[https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.105.136805]</ref><ref>Emerging Photoluminescence in Monolayer MoS2[https://pubs.acs.org/doi/10.1021/nl903868w]</ref>, chances are TMDCs materials have the potential to become the promising platform for fabricating qubits. There are four main types of TMDCs that have been suggested as advantageous in acting qubits, <math>\text{WS}_2</math> , <math>\text{WSe}_2</math> , <math>\text{MoS}_2</math> and <math>\text{MoSe}_2</math> . Kormányos et al used DFT calculations to confirm that <math>\text{WS}_2</math> and <math>\text{WSe}_2</math> , are better than <math>\text{MoS}_2</math> and <math>\text{MoSe}_2</math> in terms of the spin-valley coupling<ref>Spin-Orbit Coupling, Quantum Dots, and Qubits in Monolayer Transition Metal Dichalcogenides[https://journals.aps.org/prx/abstract/10.1103/PhysRevX.4.011034]</ref>, which is really important since spin-valley coupling is regarded as a key factor that can improve the coherence lifetime of spin-valley states.

* After several decades of active research on the properties of TMDCs, some interesting features have been discovered. Such as the extra degree of freedom offered by valley magnetic moment, which can realise bit 0 and bit 1 to form a qubit in momentum space. In addition, these valleys are independently addressable by an optical signal, and optical controlling technique is currently under research by many groups both theoretically and experimentally. As a result, as inspired by the idea from one of the theoretical work<ref>Spin-valley qubit in nanostructures of monolayer semiconductors: Optical control and hyperfine interaction[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.93.045313]</ref>, we intend to build a quantum simulator and investigate optical control of the spin-valley pseudospin on 2D heterobilayer TMDCs.

==Q & A==

===What are K & K' points?===
[[File:K_&_K'_points.png|thumb|400px|K & K' points]]
* K-points are sampling points of Brillouin zone in reciprocal lattice
* The monolayer TMDs have a unique band structure with the conduction and valence band edges both at the degenerate K and K‘ valleys at the corners of the hexagonal Brillouin zone. The direct-gap optical transitions have a selection rule: left- (right-) handed circular polarized photons couple to the interband transitions in the K (K’) valley only.
* The Tight-Binding Model:By introducing the tight-binding model for TMDs, we will give a Hamiltonian of TMDs that can be further deduced and calculated. As is showned in figure (b):under the tight-binding ansatz,the atom(e.g.the red one)can only jump to the nearest position, whose interaction processing is called "hopping" in physics,If we use the lattice represetation and the corresponding creation operators commonly used in condensed matter physics,then the nearest-neighbor tight-binding Hamiltonian has the simple form:

[[File:graphene.png|500px|center]]

<math>
\hat{H}=-t\sum_{l} \sum_{j=1}^{3}a_{l+\tau_j}^{\dagger}b_{l}+h.c.
</math>

By Fourier Transformation,<math>a_l</math> and <math>b_{l}</math> can be written as:

<math>
a_l=\frac{1}{\sqrt{N}}\sum_{k} e^{ik\cdot R_l}\alpha_k
</math>

<math>
b_l=\frac{1}{\sqrt{N}}\sum_{k} e^{ik\cdot R_l}\beta_k
</math>

<math>
\tau_1=(0,\frac{1}{\sqrt{3}})a,\tau_2=(-\frac{1}{2},-\frac{1}{2\sqrt{3}})a,\tau_3=(\frac{1}{2},-\frac{1}{2\sqrt{3}})a
</math>

Then we can rewrite the Hamiltonian in reciprocal space:

<math>
\hat{H}=\sum_{k}c^{\dagger}(k)\begin{bmatrix}
0&f^{*}\\
f&0
\end{bmatrix}c(k),\qquad c(k)=\begin{bmatrix}
\alpha(k)\\
\beta(k)
\end{bmatrix},\qquad f=-t\sum_{j=1}^{3}e^{ik\cdot\tau_{j}}
</math>

*Choice of K values:We define the first Brillouin zone of the reciprocal lattice in the standard way.It is clear that the six points at the corners of the FBZ fall into two groups of three which are equivalent, so we need consider only two equivalent corners that we label K and K0 as in the figure. Explicitly, their positions in momentum space are given by:

<math>
K=\frac{2\pi}{a}(\frac{1}{3},\frac{1}{\sqrt{3}}),\qquad K'=\frac{2\pi}{a}(\frac{2}{3},0)
<\math>

===Why choose heterobilayer TMDCs instead of monolayer===
* There are four main carrier properties that optimal Opto-valleytronics should possess.
*# long carrier lifetime
*# long valley lifetime
*# high valley polarization
*# long valley coherence time
* By adopting 2D heterostructure TMDCs materials, we can create these conditions for building promising quantum platform. (eg. Due to the type II band alignment and weak hybridization of van der Waals heterostructure, the electron–hole layer separation, the electron–hole exchange interaction is greatly reduced, resulting in a long cryogenic lifetime (ns to <math>\mu</math>s)and valley lifetime (~ 10 ns) of the interlayer exciton<ref>Opto-valleytronics in the 2D van der Waals heterostructure[https://doi.org/10.1007/s12274-020-3036-x]</ref>)

==Method==
===Principle===
* It was shown that inversion symmetry breaking can lead to opposite circular dichroism in different momentum space regions, which induces an interesting phenomena of optical selection rules at the symmetry corner points of the Brillouin zone. This enables a valley-dependent control of electrons with optical signal of different circular polarizations.
* However, valley optical selection rule in the heterobilayer TMDCs is a bit different than in the monolayer, which depends on the emission location and the emission energy. Additionally, in most cases, owing to the type II band alignment and weak hybridization, the conduction band minimum and valence band maximum of the van der Waals heterostructure are dominated by orbitals from different layers. As a result, electrons and holes tend to reside in different layers of the structure.
* It was found that the g-factor value of the interlayer exciton depends on the stacking sequence of the layers. For the AA-stacked <math>\text{MoSe}_2</math>/<math>\text{WSe}_2</math>, the value of the singlet g-factor between 6 to 7 has been reported; while, for the AB-stacked <math>\text{WSe}_2</math>/<math>\text{MoSe}_2</math>, the value of g-factor ~ −15 has been reported

===Experimental setup===

====Setup====
[[File:Experimental Setup.jpg|thumb|400px|Experimental Setup]]
* As shown in figure

====Quantum simulator====
* The <math>\text{MoSe}_2</math>/<math>\text{WSe}_2</math> heterostructure is fabricated via mechanical exfoliation and aligned transfer method. The stacking sequence of the sample is checked using second harmonic generation (SHG)-based measurement and it is found to be AA stacking. The sample is loaded into a magnetocryostat to control the magnetic ﬁeld and sample temperature.

====Qubit initialisation====
* Use circularly polarised light to create valley exciton polarization
* Use linearly polarised light to pump the qubits to rotate the valley pseudospin in the superposition plane of K and K′ valley polarizations
* The polarization state of the excitation and collection is controlled using a combination of polarizer, quarter-wave plate, and half-wave plate.

====Qubit control====
* Most popular control techniques these days
*# 1st approach: The coupling between the optical (electromagnetic) signal and the valley information can also be used to control the valley information. For the intralayer exciton, the exciton can be moved from one valley to another by applying intense THz radiation. This is shown by the observation of linearly polarized emission from higher-order sidebands when the sample is excited with a 100-fs near-resonant circularly polarized light pulse together with a 33-fs 40-THz pulse. Theoretically, it is shown that this is caused by a THz pulse driven intervalley transition. It is also predicted that a complete transfer from K (K’) to K’ (K) valley can be done within a period less than 5 fs if a more intense and shorter THz pulse is used.
*# 2nd approach: On the other hand, by utilizing the optical Stark effect, the phase control has been demonstrated. In this case, the valley state is first prepared in the <math>|K\rang+|K'\rang</math> state by using a near-resonant linearly polarized excitation. A red-detuned 100-fs circularly polarized light pulse is then used to break the valley degeneracy, which allows the coherent rotation of the valley state to the <math>|K\rang+e^{i\Delta\phi}|K'\rang</math> state with <math>\Delta\phi</math> depends on the pulse intensity and duration. This rotation is reflected in the linear polarization state of the exciton emission.

====Qubit readout====
* The PL emission is directed by a multimode optical ﬁber into a spectrometer (Andor Shamrock) with a CCD detector for spectroscopic recording

==Results==

==References==
<references />

Optical control of TMDCs valley pseudospin qubits

2021-04-30T14:25:07Z

Chu WenHao: /* What are K & K' points? */

==Linked project==
* This project is conducted by the same people as 👉 [[Control over the atomic spins within certain molecules by NMR technique]]

==Group members==
*'''XU ZIZHOU (Bensen)'''
**'''Matric Number:''' ''A0229645W''
**'''Email:''' ''[mailto:info@example.org zizhou_xu@u.nus.edu]''
*'''CHU WENHAO'''
**'''Matric Number:''' ''A0228747R''
**'''Email:''' ''[mailto:info@example.org e0675585@u.nus.edu]''
==Description==

* The strategies and materials for building universal quantum computers are being actively researched, and various platforms for holding qubits have been proposed and tested<ref>Experimental perfect state transfer of an entangled photonic qubit[https://www.nature.com/articles/ncomms11339]</ref><ref>Superconducting Qubits: Current State of Play[https://www.annualreviews.org/doi/abs/10.1146/annurev-conmatphys-031119-050605]</ref><ref>Two-qubit entangling gates within arbitrarily long chains of trapped ions[https://journals.aps.org/pra/abstract/10.1103/PhysRevA.100.022332]</ref><ref>Digital Coherent Control of a Superconducting Qubit[https://journals.aps.org/prapplied/abstract/10.1103/PhysRevApplied.11.014009]</ref><ref>Fast quantum logic gates with trapped-ion qubits[https://www.nature.com/articles/nature25737]</ref>. Within all the options such as superconducting qubits and ion trap qubits, solid-state qubits have illustrated great benefits of the compatibility with the existing semiconductor technology, which attracts lots of attention to this platform. Moreover, because of the direct band gap and strong spin-orbit coupling discovered in the monolayer transition metal dichalcogenides (TMDCs), especially some semiconductor monolayers possess a sizeable direct bandgap of ≈1.5–2 eV in the optical range allowing electrostatic confinement and optical manipulation of carriers<ref>Atomically Thin MoS2: A New Direct-Gap Semiconductor[https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.105.136805]</ref><ref>Emerging Photoluminescence in Monolayer MoS2[https://pubs.acs.org/doi/10.1021/nl903868w]</ref>, chances are TMDCs materials have the potential to become the promising platform for fabricating qubits. There are four main types of TMDCs that have been suggested as advantageous in acting qubits, <math>\text{WS}_2</math> , <math>\text{WSe}_2</math> , <math>\text{MoS}_2</math> and <math>\text{MoSe}_2</math> . Kormányos et al used DFT calculations to confirm that <math>\text{WS}_2</math> and <math>\text{WSe}_2</math> , are better than <math>\text{MoS}_2</math> and <math>\text{MoSe}_2</math> in terms of the spin-valley coupling<ref>Spin-Orbit Coupling, Quantum Dots, and Qubits in Monolayer Transition Metal Dichalcogenides[https://journals.aps.org/prx/abstract/10.1103/PhysRevX.4.011034]</ref>, which is really important since spin-valley coupling is regarded as a key factor that can improve the coherence lifetime of spin-valley states.

* After several decades of active research on the properties of TMDCs, some interesting features have been discovered. Such as the extra degree of freedom offered by valley magnetic moment, which can realise bit 0 and bit 1 to form a qubit in momentum space. In addition, these valleys are independently addressable by an optical signal, and optical controlling technique is currently under research by many groups both theoretically and experimentally. As a result, as inspired by the idea from one of the theoretical work<ref>Spin-valley qubit in nanostructures of monolayer semiconductors: Optical control and hyperfine interaction[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.93.045313]</ref>, we intend to build a quantum simulator and investigate optical control of the spin-valley pseudospin on 2D heterobilayer TMDCs.

==Q & A==

===What are K & K' points?===
[[File:K_&_K'_points.png|thumb|400px|K & K' points]]
* K-points are sampling points of Brillouin zone in reciprocal lattice
* The monolayer TMDs have a unique band structure with the conduction and valence band edges both at the degenerate K and K‘ valleys at the corners of the hexagonal Brillouin zone. The direct-gap optical transitions have a selection rule: left- (right-) handed circular polarized photons couple to the interband transitions in the K (K’) valley only.
* The Tight-Binding Model:By introducing the tight-binding model for TMDs, we will give a Hamiltonian of TMDs that can be further deduced and calculated. As is showned in figure (b):under the tight-binding ansatz,the atom(e.g.the red one)can only jump to the nearest position, whose interaction processing is called "hopping" in physics,If we use the lattice represetation and the corresponding creation operators commonly used in condensed matter physics,then the nearest-neighbor tight-binding Hamiltonian has the simple form:

[[File:graphene.png|500px|center]]

<math>
\hat{H}=-t\sum_{l} \sum_{j=1}^{3}a_{l+\tau_j}^{\dagger}b_{l}+h.c.
</math>

By Fourier Transformation,<math>a_l</math> and <math>b_{l}</math> can be written as:

<math>
a_l=\frac{1}{\sqrt{N}}\sum_{k} e^{ik\cdot R_l}\alpha_k
</math>

<math>
b_l=\frac{1}{\sqrt{N}}\sum_{k} e^{ik\cdot R_l}\beta_k
</math>

<math>
\tau_1=(0,\frac{1}{\sqrt{3}})a,\tau_2=(-\frac{1}{2},-\frac{1}{2\sqrt{3}})a,\tau_3=(\frac{1}{2},-\frac{1}{2\sqrt{3}})a
</math>

Then we can rewrite the Hamiltonian in reciprocal space:

<math>
\hat{H}=\sum_{k}c^{\dagger}(k)\begin{bmatrix}
0&f^{*}\\
f&0
\end{bmatrix}c(k),\qquad c(k)=\begin{bmatrix}
\alpha(k)\\
\beta(k)
\end{bmatrix},\qquad f=-t\sum_{j=1}^{3}e^{ik\cdot\tau_{j}}
</math>

*Choice of K values:We define the first Brillouin zone of the reciprocal lattice in the standard way.It is clear that the six points at the corners of the FBZ fall into two groups of three which are equivalent, so we need consider only two equivalent corners that we label K and K0 as in the figure. Explicitly, their positions in momentum space are given by:

<math>
K=\frac{2\pi}{a}(\frac{1}{3},\frac{1}{\sqrt{3}}),\qquad K'=\frac{2\pi}{a}(\frac{2}{3},0)
<\math>

===Why choose heterobilayer TMDCs instead of monolayer===
* There are four main carrier properties that optimal Opto-valleytronics should possess.
*# long carrier lifetime
*# long valley lifetime
*# high valley polarization
*# long valley coherence time
* By adopting 2D heterostructure TMDCs materials, we can create these conditions for building promising quantum platform. (eg. Due to the type II band alignment and weak hybridization of van der Waals heterostructure, the electron–hole layer separation, the electron–hole exchange interaction is greatly reduced, resulting in a long cryogenic lifetime (ns to <math>\mu</math>s)and valley lifetime (~ 10 ns) of the interlayer exciton<ref>Opto-valleytronics in the 2D van der Waals heterostructure[https://doi.org/10.1007/s12274-020-3036-x]</ref>)

==Method==
===Principle===
* It was shown that inversion symmetry breaking can lead to opposite circular dichroism in different momentum space regions, which induces an interesting phenomena of optical selection rules at the symmetry corner points of the Brillouin zone. This enables a valley-dependent control of electrons with optical signal of different circular polarizations.
* However, valley optical selection rule in the heterobilayer TMDCs is a bit different than in the monolayer, which depends on the emission location and the emission energy. Additionally, in most cases, owing to the type II band alignment and weak hybridization, the conduction band minimum and valence band maximum of the van der Waals heterostructure are dominated by orbitals from different layers. As a result, electrons and holes tend to reside in different layers of the structure.
* It was found that the g-factor value of the interlayer exciton depends on the stacking sequence of the layers. For the AA-stacked <math>\text{MoSe}_2</math>/<math>\text{WSe}_2</math>, the value of the singlet g-factor between 6 to 7 has been reported; while, for the AB-stacked <math>\text{WSe}_2</math>/<math>\text{MoSe}_2</math>, the value of g-factor ~ −15 has been reported

===Experimental setup===

====Setup====
[[File:Experimental Setup.jpg|thumb|400px|Experimental Setup]]
* As shown in figure

====Quantum simulator====
* The <math>\text{MoSe}_2</math>/<math>\text{WSe}_2</math> heterostructure is fabricated via mechanical exfoliation and aligned transfer method. The stacking sequence of the sample is checked using second harmonic generation (SHG)-based measurement and it is found to be AA stacking. The sample is loaded into a magnetocryostat to control the magnetic ﬁeld and sample temperature.

====Qubit initialisation====
* Use circularly polarised light to create valley exciton polarization
* Use linearly polarised light to pump the qubits to rotate the valley pseudospin in the superposition plane of K and K′ valley polarizations
* The polarization state of the excitation and collection is controlled using a combination of polarizer, quarter-wave plate, and half-wave plate.

====Qubit control====
* Most popular control techniques these days
*# 1st approach: The coupling between the optical (electromagnetic) signal and the valley information can also be used to control the valley information. For the intralayer exciton, the exciton can be moved from one valley to another by applying intense THz radiation. This is shown by the observation of linearly polarized emission from higher-order sidebands when the sample is excited with a 100-fs near-resonant circularly polarized light pulse together with a 33-fs 40-THz pulse. Theoretically, it is shown that this is caused by a THz pulse driven intervalley transition. It is also predicted that a complete transfer from K (K’) to K’ (K) valley can be done within a period less than 5 fs if a more intense and shorter THz pulse is used.
*# 2nd approach: On the other hand, by utilizing the optical Stark effect, the phase control has been demonstrated. In this case, the valley state is first prepared in the <math>|K\rang+|K'\rang</math> state by using a near-resonant linearly polarized excitation. A red-detuned 100-fs circularly polarized light pulse is then used to break the valley degeneracy, which allows the coherent rotation of the valley state to the <math>|K\rang+e^{i\Delta\phi}|K'\rang</math> state with <math>\Delta\phi</math> depends on the pulse intensity and duration. This rotation is reflected in the linear polarization state of the exciton emission.

====Qubit readout====
* The PL emission is directed by a multimode optical ﬁber into a spectrometer (Andor Shamrock) with a CCD detector for spectroscopic recording

==Results==

==References==
<references />

File:Graphene.png

2021-04-30T14:13:23Z

Chu WenHao: Take Graphenes as an example, and the notation of derivations below are marked.

== Summary ==
Take Graphenes as an example, and the notation of derivations below are marked.

Optical control of TMDCs valley pseudospin qubits

2021-04-30T14:10:02Z

Chu WenHao: /* What are K & K' points? */

==Linked project==
* This project is conducted by the same people as 👉 [[Control over the atomic spins within certain molecules by NMR technique]]

==Group members==
*'''XU ZIZHOU (Bensen)'''
**'''Matric Number:''' ''A0229645W''
**'''Email:''' ''[mailto:info@example.org zizhou_xu@u.nus.edu]''
*'''CHU WENHAO'''
**'''Matric Number:''' ''A0228747R''
**'''Email:''' ''[mailto:info@example.org e0675585@u.nus.edu]''
==Description==

* The strategies and materials for building universal quantum computers are being actively researched, and various platforms for holding qubits have been proposed and tested<ref>Experimental perfect state transfer of an entangled photonic qubit[https://www.nature.com/articles/ncomms11339]</ref><ref>Superconducting Qubits: Current State of Play[https://www.annualreviews.org/doi/abs/10.1146/annurev-conmatphys-031119-050605]</ref><ref>Two-qubit entangling gates within arbitrarily long chains of trapped ions[https://journals.aps.org/pra/abstract/10.1103/PhysRevA.100.022332]</ref><ref>Digital Coherent Control of a Superconducting Qubit[https://journals.aps.org/prapplied/abstract/10.1103/PhysRevApplied.11.014009]</ref><ref>Fast quantum logic gates with trapped-ion qubits[https://www.nature.com/articles/nature25737]</ref>. Within all the options such as superconducting qubits and ion trap qubits, solid-state qubits have illustrated great benefits of the compatibility with the existing semiconductor technology, which attracts lots of attention to this platform. Moreover, because of the direct band gap and strong spin-orbit coupling discovered in the monolayer transition metal dichalcogenides (TMDCs), especially some semiconductor monolayers possess a sizeable direct bandgap of ≈1.5–2 eV in the optical range allowing electrostatic confinement and optical manipulation of carriers<ref>Atomically Thin MoS2: A New Direct-Gap Semiconductor[https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.105.136805]</ref><ref>Emerging Photoluminescence in Monolayer MoS2[https://pubs.acs.org/doi/10.1021/nl903868w]</ref>, chances are TMDCs materials have the potential to become the promising platform for fabricating qubits. There are four main types of TMDCs that have been suggested as advantageous in acting qubits, <math>\text{WS}_2</math> , <math>\text{WSe}_2</math> , <math>\text{MoS}_2</math> and <math>\text{MoSe}_2</math> . Kormányos et al used DFT calculations to confirm that <math>\text{WS}_2</math> and <math>\text{WSe}_2</math> , are better than <math>\text{MoS}_2</math> and <math>\text{MoSe}_2</math> in terms of the spin-valley coupling<ref>Spin-Orbit Coupling, Quantum Dots, and Qubits in Monolayer Transition Metal Dichalcogenides[https://journals.aps.org/prx/abstract/10.1103/PhysRevX.4.011034]</ref>, which is really important since spin-valley coupling is regarded as a key factor that can improve the coherence lifetime of spin-valley states.

* After several decades of active research on the properties of TMDCs, some interesting features have been discovered. Such as the extra degree of freedom offered by valley magnetic moment, which can realise bit 0 and bit 1 to form a qubit in momentum space. In addition, these valleys are independently addressable by an optical signal, and optical controlling technique is currently under research by many groups both theoretically and experimentally. As a result, as inspired by the idea from one of the theoretical work<ref>Spin-valley qubit in nanostructures of monolayer semiconductors: Optical control and hyperfine interaction[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.93.045313]</ref>, we intend to build a quantum simulator and investigate optical control of the spin-valley pseudospin on 2D heterobilayer TMDCs.

==Q & A==

===What are K & K' points?===
[[File:K_&_K'_points.png|thumb|400px|K & K' points]]
* K-points are sampling points of Brillouin zone in reciprocal lattice
* The monolayer TMDs have a unique band structure with the conduction and valence band edges both at the degenerate K and K‘ valleys at the corners of the hexagonal Brillouin zone. The direct-gap optical transitions have a selection rule: left- (right-) handed circular polarized photons couple to the interband transitions in the K (K’) valley only.
* The Tight-Binding Model:By introducing the tight-binding model for TMDs, we will give a Hamiltonian of TMDs that can be further deduced and calculated. As is showned in figure (b):under the tight-binding ansatz,the atom(e.g.the red one)can only jump to the nearest position, whose interaction processing is called "hopping" in physics,If we use the lattice represetation and the corresponding creation operators commonly used in condensed matter physics,then the nearest-neighbor tight-binding Hamiltonian has the simple form:

[[File:graphene.png|500px|center]]

<math>
\hat{H}=-t\sum_{l} \sum_{j=1}^{3}a_{l+\tau_j}^{\dagger}b_{l}+h.c.
</math>

By Fourier Transformation,<math>a_l</math> and <math>b_{l}</math> can be written as:

<math>
a_l=\frac{1}{\sqrt{N}}\sum_{k} e^{ik\cdot R_l}\alpha_k
</math>

<math>
b_l=\frac{1}{\sqrt{N}}\sum_{k} e^{ik\cdot R_l}\beta_k
</math>

<math>
\tau_1=(0,\frac{1}{\sqrt{3}})a,\tau_2=(-\frac{1}{2},-\frac{1}{2\sqrt{3}})a,\tau_3=(\frac{1}{2},-\frac{1}{2\sqrt{3}})a
</math>

Then we can rewrite the Hamiltonian in reciprocal space:

<math>
\hat{H}=\sum_{k}c^{\dagger}(k)\begin{bmatrix}
0&f^{*}\\
f&0
\end{bmatrix}c(k),\qquad c(k)=\begin{bmatrix}
\alpha(k)\\
\beta(k)
\end{bmatrix},\qquad f=-t\sum_{j=1}^{3}e^{ik\cdot\tau_{j}}
</math>

===Why choose heterobilayer TMDCs instead of monolayer===
* There are four main carrier properties that optimal Opto-valleytronics should possess.
*# long carrier lifetime
*# long valley lifetime
*# high valley polarization
*# long valley coherence time
* By adopting 2D heterostructure TMDCs materials, we can create these conditions for building promising quantum platform. (eg. Due to the type II band alignment and weak hybridization of van der Waals heterostructure, the electron–hole layer separation, the electron–hole exchange interaction is greatly reduced, resulting in a long cryogenic lifetime (ns to <math>\mu</math>s)and valley lifetime (~ 10 ns) of the interlayer exciton<ref>Opto-valleytronics in the 2D van der Waals heterostructure[https://doi.org/10.1007/s12274-020-3036-x]</ref>)

==Method==
===Principle===
* It was shown that inversion symmetry breaking can lead to opposite circular dichroism in different momentum space regions, which induces an interesting phenomena of optical selection rules at the symmetry corner points of the Brillouin zone. This enables a valley-dependent control of electrons with optical signal of different circular polarizations.
* However, valley optical selection rule in the heterobilayer TMDCs is a bit different than in the monolayer, which depends on the emission location and the emission energy. Additionally, in most cases, owing to the type II band alignment and weak hybridization, the conduction band minimum and valence band maximum of the van der Waals heterostructure are dominated by orbitals from different layers. As a result, electrons and holes tend to reside in different layers of the structure.
* It was found that the g-factor value of the interlayer exciton depends on the stacking sequence of the layers. For the AA-stacked <math>\text{MoSe}_2</math>/<math>\text{WSe}_2</math>, the value of the singlet g-factor between 6 to 7 has been reported; while, for the AB-stacked <math>\text{WSe}_2</math>/<math>\text{MoSe}_2</math>, the value of g-factor ~ −15 has been reported

===Experimental setup===

====Setup====
[[File:Experimental Setup.jpg|thumb|400px|Experimental Setup]]
* As shown in figure

====Quantum simulator====
* The <math>\text{MoSe}_2</math>/<math>\text{WSe}_2</math> heterostructure is fabricated via mechanical exfoliation and aligned transfer method. The stacking sequence of the sample is checked using second harmonic generation (SHG)-based measurement and it is found to be AA stacking. The sample is loaded into a magnetocryostat to control the magnetic ﬁeld and sample temperature.

====Qubit initialisation====
* Use circularly polarised light to create valley exciton polarization
* Use linearly polarised light to pump the qubits to rotate the valley pseudospin in the superposition plane of K and K′ valley polarizations
* The polarization state of the excitation and collection is controlled using a combination of polarizer, quarter-wave plate, and half-wave plate.

====Qubit control====
* Most popular control techniques these days
*# 1st approach: The coupling between the optical (electromagnetic) signal and the valley information can also be used to control the valley information. For the intralayer exciton, the exciton can be moved from one valley to another by applying intense THz radiation. This is shown by the observation of linearly polarized emission from higher-order sidebands when the sample is excited with a 100-fs near-resonant circularly polarized light pulse together with a 33-fs 40-THz pulse. Theoretically, it is shown that this is caused by a THz pulse driven intervalley transition. It is also predicted that a complete transfer from K (K’) to K’ (K) valley can be done within a period less than 5 fs if a more intense and shorter THz pulse is used.
*# 2nd approach: On the other hand, by utilizing the optical Stark effect, the phase control has been demonstrated. In this case, the valley state is first prepared in the <math>|K\rang+|K'\rang</math> state by using a near-resonant linearly polarized excitation. A red-detuned 100-fs circularly polarized light pulse is then used to break the valley degeneracy, which allows the coherent rotation of the valley state to the <math>|K\rang+e^{i\Delta\phi}|K'\rang</math> state with <math>\Delta\phi</math> depends on the pulse intensity and duration. This rotation is reflected in the linear polarization state of the exciton emission.

====Qubit readout====
* The PL emission is directed by a multimode optical ﬁber into a spectrometer (Andor Shamrock) with a CCD detector for spectroscopic recording

==Results==

==References==
<references />

Optical control of TMDCs valley pseudospin qubits

2021-04-30T13:49:50Z

Chu WenHao: /* What are K & K' points? */

==Linked project==
* This project is conducted by the same people as 👉 [[Control over the atomic spins within certain molecules by NMR technique]]

==Group members==
*'''XU ZIZHOU (Bensen)'''
**'''Matric Number:''' ''A0229645W''
**'''Email:''' ''[mailto:info@example.org zizhou_xu@u.nus.edu]''
*'''CHU WENHAO'''
**'''Matric Number:''' ''A0228747R''
**'''Email:''' ''[mailto:info@example.org e0675585@u.nus.edu]''
==Description==

* The strategies and materials for building universal quantum computers are being actively researched, and various platforms for holding qubits have been proposed and tested<ref>Experimental perfect state transfer of an entangled photonic qubit[https://www.nature.com/articles/ncomms11339]</ref><ref>Superconducting Qubits: Current State of Play[https://www.annualreviews.org/doi/abs/10.1146/annurev-conmatphys-031119-050605]</ref><ref>Two-qubit entangling gates within arbitrarily long chains of trapped ions[https://journals.aps.org/pra/abstract/10.1103/PhysRevA.100.022332]</ref><ref>Digital Coherent Control of a Superconducting Qubit[https://journals.aps.org/prapplied/abstract/10.1103/PhysRevApplied.11.014009]</ref><ref>Fast quantum logic gates with trapped-ion qubits[https://www.nature.com/articles/nature25737]</ref>. Within all the options such as superconducting qubits and ion trap qubits, solid-state qubits have illustrated great benefits of the compatibility with the existing semiconductor technology, which attracts lots of attention to this platform. Moreover, because of the direct band gap and strong spin-orbit coupling discovered in the monolayer transition metal dichalcogenides (TMDCs), especially some semiconductor monolayers possess a sizeable direct bandgap of ≈1.5–2 eV in the optical range allowing electrostatic confinement and optical manipulation of carriers<ref>Atomically Thin MoS2: A New Direct-Gap Semiconductor[https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.105.136805]</ref><ref>Emerging Photoluminescence in Monolayer MoS2[https://pubs.acs.org/doi/10.1021/nl903868w]</ref>, chances are TMDCs materials have the potential to become the promising platform for fabricating qubits. There are four main types of TMDCs that have been suggested as advantageous in acting qubits, <math>\text{WS}_2</math> , <math>\text{WSe}_2</math> , <math>\text{MoS}_2</math> and <math>\text{MoSe}_2</math> . Kormányos et al used DFT calculations to confirm that <math>\text{WS}_2</math> and <math>\text{WSe}_2</math> , are better than <math>\text{MoS}_2</math> and <math>\text{MoSe}_2</math> in terms of the spin-valley coupling<ref>Spin-Orbit Coupling, Quantum Dots, and Qubits in Monolayer Transition Metal Dichalcogenides[https://journals.aps.org/prx/abstract/10.1103/PhysRevX.4.011034]</ref>, which is really important since spin-valley coupling is regarded as a key factor that can improve the coherence lifetime of spin-valley states.

* After several decades of active research on the properties of TMDCs, some interesting features have been discovered. Such as the extra degree of freedom offered by valley magnetic moment, which can realise bit 0 and bit 1 to form a qubit in momentum space. In addition, these valleys are independently addressable by an optical signal, and optical controlling technique is currently under research by many groups both theoretically and experimentally. As a result, as inspired by the idea from one of the theoretical work<ref>Spin-valley qubit in nanostructures of monolayer semiconductors: Optical control and hyperfine interaction[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.93.045313]</ref>, we intend to build a quantum simulator and investigate optical control of the spin-valley pseudospin on 2D heterobilayer TMDCs.

==Q & A==

===What are K & K' points?===
[[File:K_&_K'_points.png|thumb|400px|K & K' points]]
* K-points are sampling points of Brillouin zone in reciprocal lattice
* The monolayer TMDs have a unique band structure with the conduction and valence band edges both at the degenerate K and K‘ valleys at the corners of the hexagonal Brillouin zone. The direct-gap optical transitions have a selection rule: left- (right-) handed circular polarized photons couple to the interband transitions in the K (K’) valley only.
* The Tight-Binding Model:By introducing the tight-binding model for garphene, we will give a Hamiltonian of garphene that can be further deduced and calculated. As is showned in figure (b):under the tight-binding ansatz,the atom(e.g.the red one)can only jump to the nearest position, whose interaction processing is called "hopping" in physics,If we use the lattice represetation and the corresponding creation operators commonly used in condensed matter physics,then the nearest-neighbor tight-binding Hamiltonian has the simple form:
<math>
\hat{H}=-t\sum_{l} \sum_{j=1}^{3}a_{l+\tau_j}^{\dagger}b_{l}+h.c.
</math>

By Fourier Transformation,<math>a_l</math> and <math>b_{l}</math> can be written as:

<math>
a_l=\frac{1}{\sqrt{N}}\sum_{k} e^{ik\cdot R_l}a_k
</math>

<math>
b_l=\frac{1}{\sqrt{N}}\sum_{k} e^{ik\cdot R_l}b_k
</math>

<math>
\tau_1=(0,\frac{1}{\sqrt{3}})a,\tau_2=(-\frac{1}{2},-\frac{1}{2\sqrt{3}})a,\tau_3=(\frac{1}{2},-\frac{1}{2\sqrt{3}})a
</math>

===Why choose heterobilayer TMDCs instead of monolayer===
* There are four main carrier properties that optimal Opto-valleytronics should possess.
*# long carrier lifetime
*# long valley lifetime
*# high valley polarization
*# long valley coherence time
* By adopting 2D heterostructure TMDCs materials, we can create these conditions for building promising quantum platform. (eg. Due to the type II band alignment and weak hybridization of van der Waals heterostructure, the electron–hole layer separation, the electron–hole exchange interaction is greatly reduced, resulting in a long cryogenic lifetime (ns to <math>\mu</math>s)and valley lifetime (~ 10 ns) of the interlayer exciton<ref>Opto-valleytronics in the 2D van der Waals heterostructure[https://doi.org/10.1007/s12274-020-3036-x]</ref>)

==Method==
===Principle===
* It was shown that inversion symmetry breaking can lead to opposite circular dichroism in different momentum space regions, which induces an interesting phenomena of optical selection rules at the symmetry corner points of the Brillouin zone. This enables a valley-dependent control of electrons with optical signal of different circular polarizations.
* However, valley optical selection rule in the heterobilayer TMDCs is a bit different than in the monolayer, which depends on the emission location and the emission energy. Additionally, in most cases, owing to the type II band alignment and weak hybridization, the conduction band minimum and valence band maximum of the van der Waals heterostructure are dominated by orbitals from different layers. As a result, electrons and holes tend to reside in different layers of the structure.
* It was found that the g-factor value of the interlayer exciton depends on the stacking sequence of the layers. For the AA-stacked <math>\text{MoSe}_2</math>/<math>\text{WSe}_2</math>, the value of the singlet g-factor between 6 to 7 has been reported; while, for the AB-stacked <math>\text{WSe}_2</math>/<math>\text{MoSe}_2</math>, the value of g-factor ~ −15 has been reported

===Experimental setup===

====Setup====
[[File:Experimental Setup.jpg|thumb|400px|Experimental Setup]]
* As shown in figure

====Quantum simulator====
* The <math>\text{MoSe}_2</math>/<math>\text{WSe}_2</math> heterostructure is fabricated via mechanical exfoliation and aligned transfer method. The stacking sequence of the sample is checked using second harmonic generation (SHG)-based measurement and it is found to be AA stacking. The sample is loaded into a magnetocryostat to control the magnetic ﬁeld and sample temperature.

====Qubit initialisation====
* Use circularly polarised light to create valley exciton polarization
* Use linearly polarised light to pump the qubits to rotate the valley pseudospin in the superposition plane of K and K′ valley polarizations
* The polarization state of the excitation and collection is controlled using a combination of polarizer, quarter-wave plate, and half-wave plate.

====Qubit control====
* Most popular control techniques these days
*# 1st approach: The coupling between the optical (electromagnetic) signal and the valley information can also be used to control the valley information. For the intralayer exciton, the exciton can be moved from one valley to another by applying intense THz radiation. This is shown by the observation of linearly polarized emission from higher-order sidebands when the sample is excited with a 100-fs near-resonant circularly polarized light pulse together with a 33-fs 40-THz pulse. Theoretically, it is shown that this is caused by a THz pulse driven intervalley transition. It is also predicted that a complete transfer from K (K’) to K’ (K) valley can be done within a period less than 5 fs if a more intense and shorter THz pulse is used.
*# 2nd approach: On the other hand, by utilizing the optical Stark effect, the phase control has been demonstrated. In this case, the valley state is first prepared in the <math>|K\rang+|K'\rang</math> state by using a near-resonant linearly polarized excitation. A red-detuned 100-fs circularly polarized light pulse is then used to break the valley degeneracy, which allows the coherent rotation of the valley state to the <math>|K\rang+e^{i\Delta\phi}|K'\rang</math> state with <math>\Delta\phi</math> depends on the pulse intensity and duration. This rotation is reflected in the linear polarization state of the exciton emission.

====Qubit readout====
* The PL emission is directed by a multimode optical ﬁber into a spectrometer (Andor Shamrock) with a CCD detector for spectroscopic recording

==Results==

==References==
<references />

Optical control of TMDCs valley pseudospin qubits

2021-04-30T13:40:37Z

Chu WenHao: /* What are K & K' points? */

Optical control of TMDCs valley pseudospin qubits

2021-04-30T13:32:41Z

Chu WenHao: /* What are K & K' points? */

Control over the atomic spins within certain molecules by NMR technique

2021-04-30T13:03:40Z

Chu WenHao: /* Description */

==Linked project==
* This project is conducted by the same people as 👉 [[Optical control of TMDCs valley pseudospin qubits]]

==Group members==
*'''XU ZIZHOU (Bensen)'''
**'''Matric Number:''' ''A0229645W''
**'''Email:''' ''[mailto:info@example.org zizhou_xu@u.nus.edu]''
*'''CHU WENHAO'''
**'''Matric Number:''' ''A0228747R''
**'''Email:''' ''[mailto:info@example.org e0675585@u.nus.edu]''

==Description==
* NMR has been the workhorse for the experimental implementation of quantum protocols, allowing exquisite control of systems up to seven qubits in size. However, there exists some experimental limitations in terms of the cross-talk, coupled evolution, instrumental errors and so on. Thanks to the current advanced pulse techniques, we can reduce these influences and extend this technique to a new stage that the experimental limits can be neglected. In this experiment, we try to use composite pulses to compensate RF field strength variations and frequency offsets.
* Introduction of NMR:The phenomenon of nuclear magnetic resonance is based on the fact that nuclei of certain elements possess a spin angular momentum and an associated magnetic moment.some well-known magnetic nuclei, such as <math>^{1}H</math> and <math>^{31}P</math> were able to absorb radio frequency energy when placed in a magnetic field of a strength that was specific to the nucleus. Upon absorption, the nuclei begin to resonate and different atoms within a molecule resonated at different frequencies. For example, when such neclei are placed in a magnetic field, they can adopt one of a number of quantized orientations,each orientation corresponding to a particular energy level.Especially, The orientation with the lowest energy is the one in which the nuclear magnetic moment is most closely aligned with the external magnetic field while the orientation with the highest energy is the one in which the nuclear magnetic moment is least closely aligned with the magnetic field.
*Quantized orientations:It is a fundamental fact of physics that a spinning charged body produces a magnetic moment. Since a nucleus is positively charged, if it then has a spin angular momentum, <math>P</math>, its spinning will result in the rotation of the positive charge which may be compared to a current flowing in a circle. This would produce a magnetic field parallel to the spin axis, and the nucleus would have a magnetic moment,In quantum mechanics, the angular momentum P is given by the relationship

==Method==

===Principle===
We aim to use the C-shaped electromagnet that we build, to send magnetic pulse to the solid sample, which is between the poles of the magnet. Then we will observe the reaction of the sample to see if we can control the electron spin direction within it.

===Experimental setup===

====Fabrication of C-shaped electromagnet====
[[File:C-shaped Electromagnet.jpg|thumb|400px|C-shaped Electromagnet]]
# The main components of the electromagnet we are building are non-grain oriented electrical steel, by using laser beam, we cut the C-shaped piece out of the steel plate. After that, we accumulate all the pieces that we cut, forming ... cm thick. (''Note that the gap between two poles should be ... mm, as long as suitable enough to put sample in'')
# Then we use copper wiles to entwine each pole of the steel, ..., number of coils required for generating ... radio frequency pulse is according to the formula below:
#* <math>B=I\cdot\mu_0\frac{\mu_r}{L+d\cdot\mu_r}</math>
#** <math>\mu_0</math> is the permeability of free space
#** <math>\mu_r</math> is the relative permeability
#** <math>L</math> is the length of the magnet
#** <math>d</math> is the gap between two poles
#** <math>B</math> is the magnetic flux density (Tesla or <math>Wb/m^2</math>)
#* <math>B=2\mu_0nI</math>
#** <math>\mu_0=4\pi\times10^{-7}(\text{T}\cdot\text{m}/A)</math>
#** n is the number of loops per unit length of the solenoid (<math>n=\frac{N}{L}</math>, with N being the number of loops and L the length)
# Next, connect the electromagnet we made to the power source
====Procedure====
# Put our sample within the gap of the C-shaped electromagnet<ref>Solid-State NMR Spectroscopy Principles and Applications[https://onlinelibrary.wiley.com/doi/book/10.1002/9780470999394]</ref>
# Change the input radio frequency pulse to control the spin of the electrons within sample
# Use hall probe connected to sample to readout the result

==Results==

==References==

Control over the atomic spins within certain molecules by NMR technique

2021-04-30T12:36:14Z

Chu WenHao: /* Description */

==Linked project==
* This project is conducted by the same people as 👉 [[Optical control of TMDCs valley pseudospin qubits]]

==Group members==
*'''XU ZIZHOU (Bensen)'''
**'''Matric Number:''' ''A0229645W''
**'''Email:''' ''[mailto:info@example.org zizhou_xu@u.nus.edu]''
*'''CHU WENHAO'''
**'''Matric Number:''' ''A0228747R''
**'''Email:''' ''[mailto:info@example.org e0675585@u.nus.edu]''

==Description==
* NMR has been the workhorse for the experimental implementation of quantum protocols, allowing exquisite control of systems up to seven qubits in size. However, there exists some experimental limitations in terms of the cross-talk, coupled evolution, instrumental errors and so on. Thanks to the current advanced pulse techniques, we can reduce these influences and extend this technique to a new stage that the experimental limits can be neglected. In this experiment, we try to use composite pulses to compensate RF field strength variations and frequency offsets.
* Introduction of NMR:The phenomenon of nuclear magnetic resonance is based on the fact that nuclei of certain elements possess a spin angular momentum and an associated magnetic moment.some well-known magnetic nuclei, such as <math>^{1}H</math> and <math>^{31}P</math> were able to absorb radio frequency energy when placed in a magnetic field of a strength that was specific to the nucleus. Upon absorption, the nuclei begin to resonate and different atoms within a molecule resonated at different frequencies. For example, when such neclei are placed in a magnetic field, they can adopt one of a number of quantized orientations,each orientation corresponding to a particular energy level.Especially, The orientation with the lowest energy is the one in which the nuclear magnetic moment is most closely aligned with the external magnetic field while the orientation with the highest energy is the one in which the nuclear magnetic moment is least closely aligned with the magnetic field.

==Method==

===Principle===
We aim to use the C-shaped electromagnet that we build, to send magnetic pulse to the solid sample, which is between the poles of the magnet. Then we will observe the reaction of the sample to see if we can control the electron spin direction within it.

===Experimental setup===

====Fabrication of C-shaped electromagnet====
[[File:C-shaped Electromagnet.jpg|thumb|400px|C-shaped Electromagnet]]
# The main components of the electromagnet we are building are non-grain oriented electrical steel, by using laser beam, we cut the C-shaped piece out of the steel plate. After that, we accumulate all the pieces that we cut, forming ... cm thick. (''Note that the gap between two poles should be ... mm, as long as suitable enough to put sample in'')
# Then we use copper wiles to entwine each pole of the steel, ..., number of coils required for generating ... radio frequency pulse is according to the formula below:
#* <math>B=I\cdot\mu_0\frac{\mu_r}{L+d\cdot\mu_r}</math>
#** <math>\mu_0</math> is the permeability of free space
#** <math>\mu_r</math> is the relative permeability
#** <math>L</math> is the length of the magnet
#** <math>d</math> is the gap between two poles
#** <math>B</math> is the magnetic flux density (Tesla or <math>Wb/m^2</math>)
#* <math>B=2\mu_0nI</math>
#** <math>\mu_0=4\pi\times10^{-7}(\text{T}\cdot\text{m}/A)</math>
#** n is the number of loops per unit length of the solenoid (<math>n=\frac{N}{L}</math>, with N being the number of loops and L the length)
# Next, connect the electromagnet we made to the power source
====Procedure====
# Put our sample within the gap of the C-shaped electromagnet...
# Change the input radio frequency pulse to control the spin of the electrons within sample
# Use hall probe connected to sample to readout the result

==Results==

==References==

Control over the atomic spins within certain molecules by NMR technique

2021-04-30T12:30:10Z

Chu WenHao: /* Description */

==Linked project==
* This project is conducted by the same people as 👉 [[Optical control of TMDCs valley pseudospin qubits]]

==Group members==
*'''XU ZIZHOU (Bensen)'''
**'''Matric Number:''' ''A0229645W''
**'''Email:''' ''[mailto:info@example.org zizhou_xu@u.nus.edu]''
*'''CHU WENHAO'''
**'''Matric Number:''' ''A0228747R''
**'''Email:''' ''[mailto:info@example.org e0675585@u.nus.edu]''

==Description==
* NMR has been the workhorse for the experimental implementation of quantum protocols, allowing exquisite control of systems up to seven qubits in size. However, there exists some experimental limitations in terms of the cross-talk, coupled evolution, instrumental errors and so on. Thanks to the current advanced pulse techniques, we can reduce these influences and extend this technique to a new stage that the experimental limits can be neglected. In this experiment, we try to use composite pulses to compensate RF field strength variations and frequency offsets.
* Introduction of NMR:The phenomenon of nuclear magnetic resonance is based on the fact that
nuclei of certain elements possess a spin angular momentum and an associated
magnetic moment.some well-known magnetic nuclei, such as <math>^{1}H</math> and <math>^{31}P</math> were able to absorb radio frequency energy when placed in a magnetic field of a strength that was specific to the nucleus. Upon absorption, the nuclei begin to resonate and different atoms within a molecule resonated at different frequencies. For example, when such neclei are placed in a magnetic field, they can adopt one of a number of quantized orientations,each orientation corresponding to a particular energy level

==Method==

===Principle===
We aim to use the C-shaped electromagnet that we build, to send magnetic pulse to the solid sample, which is between the poles of the magnet. Then we will observe the reaction of the sample to see if we can control the electron spin direction within it.

===Experimental setup===

====Fabrication of C-shaped electromagnet====
[[File:C-shaped Electromagnet.jpg|thumb|400px|C-shaped Electromagnet]]
# The main components of the electromagnet we are building are non-grain oriented electrical steel, by using laser beam, we cut the C-shaped piece out of the steel plate. After that, we accumulate all the pieces that we cut, forming ... cm thick. (''Note that the gap between two poles should be ... mm, as long as suitable enough to put sample in'')
# Then we use copper wiles to entwine each pole of the steel, ..., number of coils required for generating ... radio frequency pulse is according to the formula below:
#* <math>B=I\cdot\mu_0\frac{\mu_r}{L+d\cdot\mu_r}</math>
#** <math>\mu_0</math> is the permeability of free space
#** <math>\mu_r</math> is the relative permeability
#** <math>L</math> is the length of the magnet
#** <math>d</math> is the gap between two poles
#** <math>B</math> is the magnetic flux density (Tesla or <math>Wb/m^2</math>)
#* <math>B=2\mu_0nI</math>
#** <math>\mu_0=4\pi\times10^{-7}(\text{T}\cdot\text{m}/A)</math>
#** n is the number of loops per unit length of the solenoid (<math>n=\frac{N}{L}</math>, with N being the number of loops and L the length)
# Next, connect the electromagnet we made to the power source
====Procedure====
# Put our sample within the gap of the C-shaped electromagnet...
# Change the input radio frequency pulse to control the spin of the electrons within sample
# Use hall probe connected to sample to readout the result

==Results==

==References==

Control over the atomic spins within certain molecules by NMR technique

2021-03-21T17:15:17Z

Chu WenHao: /* Group members */

Optical control of TMDCs valley pseudospin qubits

2021-03-21T17:00:12Z

Chu WenHao: