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arXiv:1111.0203 · Physical Review A

Back-action of a driven nonlinear resonator on a superconducting qubit

Maxime Boissonneault, Andrew C. Doherty, Florian R. Ong, Patrice Bertet, D. Vion, D. Esteve, Alexandre Blais

Published Feb 3, 2012• 41 citations
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Abstract

We study the backaction of a driven nonlinear resonator on a multilevel superconducting qubit. Using unitary transformations on the multilevel Jaynes-Cummings Hamiltonian and quantum optics master equation, we derive an analytical model that goes beyond linear response theory. Within the limits of validity of the model, we obtain quantitative agreement with experimental and numerical data, both in the bifurcation and in the parametric amplification regimes of the nonlinear resonator. We show in particular that the measurement-induced dephasing rate of the qubit can be rather small at high drive power. This is in contrast to measurement with a linear resonator where this rate increases with the drive power. Finally, we show that, for typical parameters of circuit quantum electrodynamics, correctly describing measurement-induced dephasing requires a model going beyond linear response theory, such as the one presented here.

DOI
10.1103/physreva.85.022305

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We study the backaction of a driven nonlinear resonator on a multilevel superconducting qubit. Using unitary transformations on the multilevel Jaynes-Cummings Hamiltonian and quantum optics master equation, we derive an analytical model that goes beyond linear response theory. Within the limits of validity of the model, we obtain quantitative agreement with experimental and numerical data, both in the bifurcation and in the parametric amplification regimes of the nonlinear resonator. We show in particular that the measurement-induced dephasing rate of the qubit can be rather small at high drive power. This is in contrast to measurement with a linear resonator where this rate increases with the drive power. Finally, we show that, for typical parameters of circuit quantum electrodynamics, correctly describing measurement-induced dephasing requires a model going beyond linear response theory, such as the one presented here.
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.
We analyze the use of a driven nonlinear cavity to make a weak continuous measurement of a dispersively coupled qubit. We calculate the backaction dephasing rate and measurement rate beyond leading-order perturbation theory using a phase-space approach which accounts for cavity noise squeezing. Surprisingly, we find that increasing the coupling strength beyond the regime describable by leading-order perturbation theory (i.e., linear response) allows one to come significantly closer to the quantum limit on the measurement efficiency. We interpret this behavior in terms of the non-Gaussian photon number fluctuations of the nonlinear cavity. Our results are relevant to recent experiments using superconducting microwave circuits to study quantum measurement.
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.
The coupled dynamics of a nanomechanical resonator and superconducting quantum circuits are studied in three experiments, in the context of studying the quantum limit for force detection and quantum physics of macroscopic objects. In the first experiment, the dispersive mechanical resonance shift from the interaction with a Cooper-pair box qubit is studied. The measured coupling strength is large enough to satisfy one of the conditions required to perform many of the proposed quantum nanomechanical measurements. The resonance shift also probes the microwave-driven response of the qubit, showing Rabi oscillation and Landau-Zener tunneling, proving the coherent dynamics of the qubit. Second, the parametric excitation of nanomechanical motion is studied via experiments with a driven qubit. Degenerate parametric amplification and oscillation are demonstrated, with a new observation of nonlinear dissipation. The squeezing of the back-action noise from the detection amplifier is also observed, up to 4dB. It is the first demonstration using a qubit as an auxiliary system to modify the nanomechanical dynamics, showing a possible route for generation of nanomechanical quantum states. Finally, back-action cooling of nanomechanical motion has been investigated, which is implemented by capacitively coupling a high-Q coplanar waveguide microwave resonator to a nanomechanical resonator. The thermal state with 7.5 mechanical quanta on average is reached, a result that is ultimately limited due to increased bath heating with microwave power. The heating is consistent with a model based on two-level systems resonantly coupled to the nanomechanical mode. This additional heating suggests future efforts to improve coupling and for reducing two-level system density in materials employed to reach the motional ground state via back-action cooling.
We study bifurcation measurement of a multilevel superconducting qubit using a nonlinear resonator biased in the straddling regime, where the resonator frequency sits between two qubit transition frequencies. We find that high-fidelity bifurcation measurements are possible because of the enhanced qubit-state-dependent pull of the resonator frequency, the behavior of qubit-induced nonlinearities, and the reduced Purcell decay rate of the qubit that can be realized in this regime. Numerical simulations find up to a threefold improvement in qubit readout fidelity when operating in, rather than outside of, the straddling regime. High-fidelity measurements can be obtained at much smaller qubit-resonator couplings than current typical experimental realizations, reducing spectral crowding and potentially simplifying the implementation of multiqubit devices.
Superconducting electrical circuits can be used to study the physics of cavity quantum electrodynamics (QED) in new regimes, therefore realizing circuit QED. For quantum-information processing and quantum optics, an interesting regime of circuit QED is the dispersive regime, where the detuning between the qubit transition frequency and the resonator frequency is much larger than the interaction strength. In this paper, we investigate how nonlinear corrections to the dispersive regime affect the measurement process. We find that in the presence of pure qubit dephasing, photon population of the resonator used for the measurement of the qubit act as an effective heat bath, inducing incoherent relaxation and excitation of the qubit. Measurement thus induces both dephasing and mixing of the qubit, something that can reduce the quantum nondemolition aspect of the readout. Using quantum trajectory theory, we show that this heat bath can induce quantum jumps in the qubit state. Nonlinear effects can also reduce the achievable signal-to-noise ratio of a homodyne measurement of the voltage.