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Microwave Bistability in Circuit QED

Andrew C. Doherty

Published Jan 1, 2011• 0 citations
Publisher

Abstract

Recent experiments have realized a familiar model of quantum optics, dispersive optical bistability, in the context of microwave coplanar resonators with internal Josephson junctions that are used to achieve large effective Kerr nonlinearities with very low dissipation. A Josephson junction qubit constitutes an effective atom that may be coupled to such a nonlinear resonator to provide a very useful probe that is sensitive to the quantum noise in the nonlinear cavity. In particular, I will explain how sidebands on the qubit excitation spectrum are sensitive to the effective temperature, oscillation frequency and damping rate of the intracavity field fluctuations.

DOI
10.1364/iqec.2011.i124

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Recent experiments have realized a familiar model of quantum optics, dispersive optical bistability, in the context of microwave coplanar resonators with internal Josephson junctions that are used to achieve large effective Kerr nonlinearities with very low dissipation. A Josephson junction qubit constitutes an effective atom that may be coupled to such a nonlinear resonator to provide a very useful probe that is sensitive to the quantum noise in the nonlinear cavity. In particular, I will explain how sidebands on the qubit excitation spectrum are sensitive to the effective temperature, oscillation frequency and damping rate of the intracavity field fluctuations.
Recent experiments have realized a familiar model of quantum optics, dispersive optical bistability, in the context of microwave coplanar resonators with internal Josephson junctions that are used to achieve large effective Kerr nonlinearities with very low dissipation. A Josephson junction qubit constitutes an effective atom that may be coupled to such a nonlinear resonator to provide a very useful probe that is sensitive to the quantum noise in the nonlinear cavity. In particular, I will explain how sidebands on the qubit excitation spectrum are sensitive to the effective temperature, oscillation frequency and damping rate of the intracavity field fluctuations.
One of the most studied model systems in quantum optics is a two-level atom strongly coupled to a single mode of the electromagnetic field stored in a cavity, a research field named cavity quantum electrodynamics or CQED. CQED has recently received renewed attention due to its implementation with superconducting artificial atoms and coplanar resonators in the so-called circuit quantum electrodynamics (cQED) architecture. In cQED, the couplings can be much stronger than in CQED due to the design flexibility of superconducting circuits and to the enhanced field confinement in one-dimensional cavities. This enabled the realization of fundamental quantum physics and quantum information processing experiments with a degree of control comparable to that obtained in CQED. The purpose of this chapter is to investigate the situation where the resonator to which the atom is coupled is made nonlinear with a Kerr-type nonlinearity, causing its energy levels to be nonequidistant. The system is then described by a nonlinear Jaynes-Cummings Hamiltonian. This considerably enriches the physics since a pumped nonlinear resonator displays bistability, parametric amplification, and squeezing. The interplay of strong coupling and these nonlinear effects constitutes a novel model system for quantum optics that can be implemented experimentally with superconducting circuits. This chapter is organized as follows. In a first section we present the system consisting of a superconducting Kerr nonlinear resonator strongly coupled to a transmon qubit. In the second section, we describe the response of the sole nonlinear resonator to an external drive. In the third section, we show how the resonator bistability can be used to perform a high-fidelity readout of the transmon qubit. In the last section, we investigate the quantum backaction exerted by the intracavity field on the qubit.
In this thesis, we study two different physical systems, namely superconducting circuits and optomechanical cavities. \nIn the first part of the thesis, we study superconducting qubits and resonators and their potential to implement quantum information processing tasks. We propose a circuit quantum electrodynamics realization of a protocol to generate a Greenberger-Horne-Zeilinger (GHZ) state for transmon qubits homogeneously coupled to a microwave cavity in the dispersive limit. We derive an effective Hamiltonian with pairwise qubit exchange interactions of the XY type that can be globally controlled. Starting from a separable initial state, these interactions allow to generate a multi-qubit GHZ state within a time that does not depend on the number of qubits. We discuss how to probe the non-local nature and the genuine multipartite entanglement of the generated state. Finally, we investigate the stability of the proposed scheme to inhomogeneities in the physical parameters and the weak anharmonicity of transmon qubits. \nIn the second part of the thesis, we study optomechanical systems in which the position of a mechanical resonator modulates the resonance frequency of an optical cavity. The resulting radiation-pressure interaction is intrinsically nonlinear and can be used to implement strong Kerr nonlinearities and an effective interaction between photons. We investigate the optical bistability of such a system. The steady-state mean-field equation of the optical mode is identical to the one for a Kerr medium, and thus we expect it to have the same characteristic behavior with a lower, a middle, and an upper branch. However, the presence of position fluctuations of the mechanical resonator leads to a new feature: the upper branch will become unstable at sufficiently strong driving in certain parameter regimes. We identify the appropriate parameter regime for the upper branch to be stable, and we confirm, by numerical investigation of the quantum steady state, that the mechanical mode indeed acts as a Kerr nonlinearity for the optical mode in the low-temperature limit. This equivalence of the optomechanical system and the Kerr medium will be important for future applications of cavity optomechanics in quantum nonlinear optics and quantum information science. \n
We have performed spectroscopic measurements of a superconducting qubit dispersively coupled to a nonlinear resonator driven by a pump microwave field. Measurements of the qubit frequency shift provide a sensitive probe of the intracavity field, yielding a precise characterization of the resonator nonlinearity. The qubit linewidth has a complex dependence on the pump frequency and amplitude, which is correlated with the gain of the nonlinear resonator operated as a small-signal amplifier. The corresponding dephasing rate is found to be close to the quantum limit in the low-gain limit of the amplifier.
This chapter investigates the coupling of a two-level system to a nonlinear resonator, one of the simplest extensions to the well-known Jaynes–Cummings model. In the experiment, the two-level system is a superconducting transmon qubit, and the resonator is a coplanar resonator with a Kerr nonlinearity induced by a Josephson junction. Effects arising from the resonator nonlinearity (bistability, parametric amplification, squeezing) qualitatively modify the dispersive qubit-resonator coupling. This makes it possible to design a qubit readout which has both a high fidelity and the potential of being QND. Motivated by fundamental and practical issues concerning the quantum limit of such a measurement, the chapter also studies spectroscopically the quantum back action exerted by the intraresonator field onto the qubit. Qualitative agreement is obtained with analytical results and quantitative agreement with numerical simulations.
Abstract We explore the dissipative dynamics of nonlinearly driven oscillator systems tuned to resonances where multiple excitations are generated. Such systems are readily realised in circuit QED systems combining Josephson junctions with a microwave cavity and a drive achieved either through flux or voltage bias. For resonances involving three or more photons the system undergoes a sequence of two closely spaced dynamical transitions (the first one discontinuous and the second continuous) as the driving is increased leading to steady states that form complex periodic structures in phase space. In the vicinity of the transitions the system displays interesting bistable behaviour: we find that coherent effects can lead to surprising oscillations in the weight of the different dynamical states in the steady state of the system with increasing drive. We show that the dynamics is well-described by a simple effective rate model with transitions between states localised at different points in the phase space crystal. The oscillations in the weights of the dynamical states is reflected in corresponding oscillations in a time-scale that describes transitions between the states.