Microwave control of superconducting cavity and qubit

Group members

LI Yifan e0653565@u.nus.edu

Zhao Luheng e0647245@u.nus.edu

Introduction

Circuit QED is the study of the interaction between light confined in a cavity or resonator and artificial atoms. Usually, the artificial atom is denoted as the qubit which is an essential element in the superconducting circuit. For the qubits, we expect to control its ground state and the first excited state, which provides a well-defined two-level system. In the past decades, we have witnessed enormous progress in technology and control over the quantum system. With these state-of-art engineering technologies, the superconducting circuit architecture is a powerful platform to explore quantum physics and can serve as a testbed for quantum information,

In our setup, we have two aluminum 3D superconducting cavity samples A and B, both embedded with transmon qubit chips inside. We have deposited these two samples on the bracket of MXC stage in the Bluefors dilution refrigerator, which provides an extremely cold environment with a temperature down to 10mK. In this case, the environmental thermal noise can be suppressed, while the quantum effects of mesoscopic objects, e.g. transmon qubit, non-classical photon state in the cavity, emerge from the measurement. At the same time, quantum technologies enable the manipulation and engineering of these quantum states.

This project is intended to characterize the properties of superconducting cavities and the transmon qubits and perform the measurement of qubits. In this characterization project, we will obtain the coupling strength between transmon qubit and cavity $\chi$, the anharmonicity of qubits $K_{tt}$, coherent time of the qubit $T_1,T_2$, the quality factor of superconducting cavity $Q$. We will benefit from this characterization when we try to accurately engineer the qubit state by the microwave pulse.

Setup

Setup

RF system

A schematic description of the connectivity between the OPX and the experimental system mounted in the low dilution refrigerator. The OPX is a quantum device integrated with FPGA, DAC, and ADC. The analog outputs send the I and Q signals at the range of MHz, which will be modulated by the IQ mixer. to upconvert to the appropriate microwave signal

The microwave signals are modulated by the I signal component and the Q signal which are both at the range of MHz.

A schematic description of the connectivity insides the fridge. The BlueFors fridge provides an extremely low temperature (around 10mK) at the MXC flange where the experimental sample is mounted.

A schematic description of the connectivity insides the fridge. The BlueFors fridge provides an extremely low temperature (around 10mK) at the MXC flange where the experimental sample is mounted.

The quantum device is a superconducting circuit composed of a single transmon qubit and a readout resonator, with the following Hamiltonian $H=\hslash {\omega }_r{a}^{†}a+\frac{\hslash }{2}{\omega }_q{\sigma }_z+\hslash g(a^{†}{\sigma }^{-}+a{\sigma }^{+}).$ In the dispersive regime that $|{\omega }_r-{\omega }_q|\ll g$, the dispersive Hamiltonian is given as $H=\frac{\hslash }{2}({\omega }_q+\frac{g^2}{\Delta }){\sigma }_z+\hslash ({\omega }_r+\frac{g^2}{\Delta }{\sigma }_z)a^{†}a$

Hamiltonian

The quantum system is composed of a transmon qubit and a readout resonator, with the following Hamiltonian

$H=\frac{\hslash }{2}{\omega }_q{\sigma }_z+\hslash {\omega }_r{a}^{†}a+\hslash g(a^{†}{\sigma }^{-}+a{\sigma }^{+}).$

The quantum system works in the dispersive regime where the qubit is strongly detuned from the oscillator with $|\Delta ={\omega }_q-{\omega }_r|\gg g$. There, the Hamiltonian is approximated as

$H_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{p}}\approx \hslash {\omega }_r{\hat{a}}^{†}\hat{a}+\frac{\hslash }{2}({\omega }_q+\chi ){\hat{\sigma }}_z+\hslash \chi {\hat{a}}^{†}\hat{a}{\hat{\sigma }}_z,$

The third term, i.e., dispersive coupling term, represents the qubit-state dependent shift of oscillator frequency, or equivalently, a photon-number dependent frequency shift in the qubit spectroscopy, which is also denoted as the ac-Stark shift. Due to the dispersive coupling, the qubit frequency is dressed by a Lamb shift corresponding to the second term in this equation. There, the cross-Kerr nonlinearity strength $\chi$ is derived from the coupling strength $g$ as

$\chi =\frac{g^2}{\Delta }.$

When we send the resonator driving pulse and the qubit driving pulse to the system, the Hamiltonian is given as

$H=H_0+\hslash s(t){\sigma }_x+\frac{m(t)}{2}(a^{†}{e}^{-\omega t}+a{e}^{-\omega t}),$

where the second term rotates the Bloch vector of the qubit around the axis which has an angle $\varphi$ from the x-axis on the x-y plane, namely Rabi oscillation of the qubit. The details are shown in the experiment section.

Experiments

IQ mixer calibration

The IQ mixer suffers from two major drawbacks: one is the IQ imbalance, and the other is DC offset. Hence the calibration and the control of imbalance are essential to limit the signal modulation error. We should pass the proper amplitude and phase correction and DC-offsets to the OPX to calibrate them in our setup. A more detailed discussion of IQ imbalance refers to the reference.

Theoretical analysis

For an ideal mixer with a local oscillator (LO) with the frequency of $\Omega$, the LO signal is described as

$A_{LO}(t)=\mathrm{R}\mathrm{e}[A_0{e}^{-\Omega t}],$

while the RF signal emitted from the IQ mixer is modulated by the I Q signals denoted by $z(t)=z_I(t)+i{z}_Q(t)$. Hence, the RF signal is represented as

$A_{RF}(t)=Re[z(t)A_0{e}^{i\Omega t}]=A_0(z_I(t)\cos (\Omega t)-z_Q(t)\sin (\Omega t))$

In the frequency domain, the RF port generates two sidebands at the two sides of $\Omega$. When we regard the lower sideband is the signal, the upper sideband becomes the image component that needs to be suppressed by a proper choice of $z(t)$. IQ imbalances occur due to the mismatches between the parallel in-phase (I) and quadrature (Q) signal paths.

Another feature that needs to be calibrated is the LO leakage. For a non-ideal mixer, the LO signal leaks into the RF path, resulting in unwanted components at LO frequency. We can handle this issue by setting the proper DC-offsets for the I Q ports of the OPX.

Measurement setup

As seen in the schematic diagram of the RF system, the RF signal is connected to a power splitter whose one branch sends the signal to the fridge input port. At the same time, the other is attached to the spectrum analyzer, detecting the RF output signal. The I Q ports of the IQ mixer receive the I and Q signals from the OPX correspondingly, attenuated by a 10dB attenuator to fulfill the power limitation. In the calibration process, we implement the optimization algorithm, which tries different IQ DC-offset and IQ imbalance correction and minimizes them using

scipy.optimize.minimize

function to lower down LO signal and the image sideband signal.

One tone spectroscopy

From the analysis of the interaction between the cavity mode and the qubit mode, the cavity frequency can be shifted based on the qubit state, moreover, the frequency shift is determined by the coupling strength between the cavity and qubit. For a new sample, we don't know the qubit frequency to carefully manipulate it. Fortunately, even we have no prerequisite knowledge about the qubit. we can observe this frequency shift caused by the coupling of cavity and qubit.

Spectroscopy measured by VNA

By using the Vector Network Analyzer (VNA), we can observe the cavity resonance at the scattering parameters $S_{21}$. We can compare cavity frequency with different sweep signal power and see if the frequency shifts. In the high-power mode, we send a vast number of photons into the cavity, which essentially overwhelms the effect of the qubit, resulting in the measurement of the bare frequency of the readout resonator. In contrast, the qubit is in its ground state $|g⟩$ in the low-power regime, the cavity frequency is dispersively shifted due to the ground state of the qubit $|g⟩$. Since the $\Delta$ between the cavity frequency and the qubit frequency is large, we can ensure that when the cavity is driving (the VNA signal sweeps around the cavity frequency), the qubit can be maintained at the ground state in the low-power regime.

A bigger frequency shift means a stronger coupling between the cavity and the qubit exists. The frequency shift of the cavity is a rough estimation of the coupling $\chi =g^2/\Delta$ rather than the exact value.

According to Ref, if the qubit frequency is close to the cavity frequency $\Delta <g$, the qubit and cavity may enter the polariton regime, and there will be clearly two peaks (separated by ~2g) in the cavity at low-power transmission. Inversely, the qubit of our system is far detuned from the cavity frequency $\Delta \ll g$.

Rabi sweeps

One- and two-dimensional Rabi sweeps are critical qubit characterization protocols. In this section, we perform the power Rabi pulse sequence and time Rabi pulse sequence to find the right amplitude and time of pulse to execute a particular single-qubit gate, such as $\pi$-pulse around the x-axis which rotates the ground state $|g⟩$ to the excited state $|e⟩$ and vice versa.

The control pulse follows

$s(t)=A(t)\cos ({\omega }_{d}t+\varphi ),$

where $A(t)$ is the time-dependent amplitude of the pulse. When the drive frequency is detuned from the qubit as $\Delta ={\omega }_d-{\omega }_q\ne 0$, it allows the rotations around the z-axis in the Bloch sphere. When the drive frequency is exactly same as the qubit frequency, the qubit rotations around an axis in the x-y plane, which is defined accordingly to the phase $\varphi$. For the rotation around the x-axis, the phase is set as 0, and the rotation angle $\theta =\int A(t)dt$.

Power-Rabi experiment

In the Power-Rabi experiment, we fix the drive pulse duration and sweep the power of the drive pulse. The sequence - Prepare the qubit to the ground state $|g⟩$, which is realized by a long wait time allowing the qubit is relaxed to its ground state. - Execute a gaussian shaped pulse with a fixed duration $\tau$ and varying peak amplitude $a$ which rotates the qubits by $\theta \propto a\times \tau e^{-{\tau }^2/2{\sigma }^2}.-Executeaweakreadoutpulsetothereadoutresonator,whichiscoupledtothequbit.Fromthephaseofthereflectedpulse,wecandeducethestateofthequbit.Underthedrivepulse,thestateofthequbitevolvesas<math>\cos (\theta )|0⟩+\sim (\theta )e^{i\varphi }|1⟩$, hence the probability of measuring the state $|1⟩$ is $P_{|1⟩}=|{\sin }^2\theta |$. To obtain an obvious result, the above sequence is repeated large times, therefore, the measurement sample is averaged.