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.

The experimental setup inside fridge. The input line will send the qubit manipulation signals and cavity probe signals into the cavity sample. The output line will transmit the signal from the output port to the outside. At the 4K stage, there is HEMT amplifier which provides amplification of the signal.

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

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 $|\Delta |\gg g$. There, the Hamiltonian is approximated as

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

where 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

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

Experiments

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 $\Delta \ll g$.