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arXiv:1203.2491 · Physical Review A

Emission spectrum of the driven nonlinear oscillator

Stephan André, Lingzhen Guo, Vittorio Peano, Michael Marthaler, Gerd Schön

Published May 22, 2012• 26 citations
arXivHTMLPDF

Abstract

Motivated by recent ``circuit QED'' experiments we investigate the noise properties of coherently driven nonlinear resonators. By using Josephson junctions in superconducting circuits, strong nonlinearities can be engineered, which lead to the appearance of pronounced effects already for a low number of photons in the resonator. Based on a master equation approach we determine the emission spectrum and observe for typical circuit QED parameters, in addition to the primary Raman-type peaks, second-order peaks. These peaks describe higher harmonics in the slow noise-induced fluctuations of the oscillation amplitude of the resonator and provide a clear signature of the nonlinear nature of the system.

DOI
10.1103/physreva.85.053825

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Motivated by recent ``circuit QED'' experiments we investigate the noise properties of coherently driven nonlinear resonators. By using Josephson junctions in superconducting circuits, strong nonlinearities can be engineered, which lead to the appearance of pronounced effects already for a low number of photons in the resonator. Based on a master equation approach we determine the emission spectrum and observe for typical circuit QED parameters, in addition to the primary Raman-type peaks, second-order peaks. These peaks describe higher harmonics in the slow noise-induced fluctuations of the oscillation amplitude of the resonator and provide a clear signature of the nonlinear nature of the system.
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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.
In circuit quantum electrodynamical systems, the quasiparticle-related losses in Josephson junctions are suppressed due to the gap in the superconducting density of states which is much higher than the typical energy of a microwave photon. In this work, we show that a strong drive even at a frequency lower than twice the superconductor gap parameter can activate dissipation in the junctions due to photon-assisted breaking of the Cooper pairs. Both the decay rate and noise strength associated with the losses are sensitive to the dc phase bias of the junction and can be tuned in a broad range by the amplitude and the frequency of the external driving field, making the suggested mechanism potentially attractive for designing tunable dissipative elements. We also predict pronounced memory effects in the driven Josephson junctions, which are appealing for both theoretical and experimental studies of non-Markovian physics in superconducting quantum circuits. We illustrate our theoretical findings by studying the spectral properties and the steady-state population of a low-impedance resonator coupled to the driven Josephson junction: we show the emergence of non-Lorentzian spectral lines and broad tunability of effective temperature of the steady state.
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.
A Josephson junction embedded in a dissipative circuit can be externally driven to induce nonlinear dynamics of its phase. Classically, under sufficiently strong driving and weak damping, dynamic multistability emerges, associated with dynamical bifurcations so that the often-used modeling as a Duffing oscillator, which can exhibit bistability at most, is insufficient. The present work analyzes in this regime corresponding quantum properties by mapping the problem onto a highly nonlinear quasienergy operator in a rotating frame. This allows us to identify in detail parameter regions where simplifications such as the Duffing approximation are valid, to explore classical-quantum correspondences, and to study how quantum fluctuations impact the effective junction parameters as well as the dynamics around higher-amplitude classical fixed points.