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Physical review. B./Physical review. B

Coupled superconducting qudit-resonator system: Energy spectrum, state population, and state transition under microwave drive

Weiyuan Liu, Huikai Xu, Feifan Su, Zhiyuan Li, Ye Tian, Siyuan Han, S. P. Zhao

Published Mar 26, 2018• 5 citations
Publisher

Abstract

Superconducting quantum multilevel systems coupled to resonators have recently been considered in some applications such as microwave lasing and high-fidelity quantum logical gates. In this work, using an rf-SQUID type phase qudit coupled to a microwave coplanar waveguide resonator, we study both theoretically and experimentally the energy spectrum of the system when the qudit level spacings are varied around the resonator frequency by changing the magnetic flux applied to the qudit loop. We show that the experimental result can be well described by a theoretical model that extends from the usual two-level Jaynes-Cummings system to the present four-level system. It is also shown that due to the small anharmonicity of the phase device a simplified model capturing the leading state interactions fits the experimental spectra very well. Furthermore we use the Lindblad master equation containing various relaxation and dephasing processes to calculate the level populations in the simpler qutrit-resonator system, which allows a clear understanding of the dynamics of the system under the microwave drive. Our results help to better understand and perform the experiments of coupled multilevel and resonator systems and can be applied in the case of transmon or Xmon qudits having similar anharmonicity to the present phase device.

DOI
10.1103/physrevb.97.094513

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Superconducting quantum multilevel systems coupled to resonators have recently been considered in some applications such as microwave lasing and high-fidelity quantum logical gates. In this work, using an rf-SQUID type phase qudit coupled to a microwave coplanar waveguide resonator, we study both theoretically and experimentally the energy spectrum of the system when the qudit level spacings are varied around the resonator frequency by changing the magnetic flux applied to the qudit loop. We show that the experimental result can be well described by a theoretical model that extends from the usual two-level Jaynes-Cummings system to the present four-level system. It is also shown that due to the small anharmonicity of the phase device a simplified model capturing the leading state interactions fits the experimental spectra very well. Furthermore we use the Lindblad master equation containing various relaxation and dephasing processes to calculate the level populations in the simpler qutrit-resonator system, which allows a clear understanding of the dynamics of the system under the microwave drive. Our results help to better understand and perform the experiments of coupled multilevel and resonator systems and can be applied in the case of transmon or Xmon qudits having similar anharmonicity to the present phase device.
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By coupling a superconducting weak link to a microwave resonator, recent experiments probed the spectrum and achieved the quantum manipulation of Andreev states in various systems.However, the quantitative understanding of the response of the resonator to changes in the occupancy of the Andreev levels, which are of fermionic nature, is missing.Here, using Bogoliubov-de Gennes formalism to describe the weak link and a general formulation of the coupling to the resonator, we calculate the shift of the resonator frequency as a function of the levels occupancy and describe how transitions are induced by phase or electric field microwave drives.We apply this formalism to analyze recent experimental results obtained using circuit-QED techniques on superconducting atomic contacts and semiconducting nanowire Josephson junctions.
Achieving fast gates and high-fidelity readout in superconducting quantum circuits often benefits from the use of large amplitude microwave drives. As the strength increases, unwanted effects such as excitation out of the logical states also become more prevalent. In transmon qubits, coherent controls relying on sub-resonant drives are fundamentally limited by the eventual appearance of strong hybridization in the spectrum. Understanding how this threshold depends on intrinsic device parameters is vital for the design and operation of circuits utilizing the transmon as a source of nonlinearity. In this work, we present an experimental study of 14 transmons with a wide range of anharmonicities, and use comparison with Floquet branch analysis and semiclassical simulations to establish a straightforward relationship between the critical value of the effective drive amplitude ηmax⁡η_{\max}ηmax​ and the ratio γ=ωq/αγ=ω_q/αγ=ωq​/α, where ωqω_qωq​ is the transmon frequency and −α-α−α is the anharmonicity. For the case where the transmon is used as a parametrically driven coupler, these results further allow us to estimate maximum parametric interaction rates, which informs the circuit design process for optimizing gate rates and fidelities.
By coupling a superconducting weak link to a microwave resonator, recent experiments probed the spectrum and achieved the quantum manipulation of Andreev states in various systems. However, the quantitative understanding of the response of the resonator to changes in the occupancy of the Andreev levels, which are of fermionic nature, is missing. Here, using Bogoliubov-de Gennes formalism to describe the weak link and a general formulation of the coupling to the resonator, we calculate the shift of the resonator frequency as a function of the levels occupancy and describe how transitions are induced by phase or electric field microwave drives. We apply this formalism to analyze recent experimental results obtained using circuit-QED techniques on superconducting atomic contacts and semiconducting nanowire Josephson junctions.
At low temperature and low damping, the magnetic flux of a rf Superconducting QUantum Interference Device (SQUID) acts as a macroscopic quantum variable whose dynamic is described by a double well potential. A rf SQUID consists of a superconducting loop interrupted by a thin insulating barrier, known as a Josephson junction. The modified rf SQUID design discussed in this thesis has independent, in situ, controls that allow varying the barrier height between the potential wells and their relative tilt. This high level of control of the potential together with the capability to implement an experimental setup optimized to reduce coupling to external sources of noise, made this system ideal not only for tests of quantum transition at a macroscopic level, but also as a scalable solid state device for quantum computation. A variety of measurements of quantum phenomena are reported in this thesis, including microwave level spectroscopy, coherent oscillations, and macroscopic resonant tunneling (MRT). These measurements are compared with theoretical models to study the possible sources of decoherence in a rf SQUID qubit. The analysis confirmed that the main source of decoherence is not due to coupling to the external environment, but it can be linked to intrinsic low frequency flux noise. Work from many groups over the last several years has shown that the performance of a superconductive qubit is limited by coupling to spurious defects in the materials used for the device fabrication. In this work we analyze the possibility of using the MRT as a tool for future studies of effects of materials and fabrication process in the realization of improved qubits, and we also explore the feasibility for measurements of quantum phenomena that involve fast pulsing of the potential barrier height.
We demonstrate amplification of a microwave signal by a strongly driven two-level system in a coplanar waveguide resonator. The effect, similar to the dressed-state lasing known from quantum optics, is observed with a single quantum system formed by a persistent current (flux) qubit. The transmission through the resonator is enhanced when the Rabi frequency of the driven qubit is tuned into resonance with one of the resonator modes. Amplification as well as linewidth narrowing of a weak probe signal has been observed. The stimulated emission in the resonator has been studied by measuring the emission spectrum. We analyzed our system and found an excellent agreement between the experimental results and the theoretical predictions obtained in the dressed-state model.