Dynamical Tunneling in Macroscopic Systems

Serban, I.; Wilhelm, Frank K.

doi:10.1103/physrevlett.99.137001

Cited by 42 publications

(68 citation statements)

References 19 publications

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“…For the anharmonic case they follow within the RWA from the extended Pauli-master equation. It is thus convenient to introduce the concept of an effective temperature, which is defined according to (12) by exp(−β xy ω) =D xy (−ω)/D yx (ω), with x, y = Q, P . Now, a quantum of energy ω ≥ 0 emitted from the system reaches the bath in the lab frame either with energy ω + = (ω F + ω) or ω − = (ω F − ω), thus determining two distinct regimes.…”

Section: Effective Temperaturementioning

confidence: 99%

“…Theoretically, a general approach is provided by the Floquet representation [7], but in the above situation a more powerful procedure for analytical investigations is to describe the driven dynamics in a frame rotating with a frequency equal to the response frequency of the system as already analyzed in the classical regime by Dykman and co-workers [8,9]. The extension to the quantum regime has been given in [10][11][12]. In essence, one arrives at a time-independent description with a non-standard Hamiltonian though.…”

mentioning

confidence: 99%

“…For driven systems the concept of an effective temperature has been introduced in [10], but it was shown only in certain limits [11,15] that this temperature significantly differs from that in the laboratory. In these and related works [12,16] the focus has been on the switching between driving induced states, while an analysis of the structure of the dissipative dynamics in the rotating frame has seen less attention. Here we fill this gap in consistently deriving a corresponding master equation and, importantly for ongoing experimental activities, in providing an analysis to what extent a unique effective temperature can actually be introduced.…”

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confidence: 99%

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Dissipation in a rotating frame: Master equation, effective temperature, and Lamb shift

Verso

Ankerhold

2010

Phys. Rev. A

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Motivated by recent realizations of microwave-driven nonlinear resonators in superconducting circuits, the impact of environmental degrees of freedom is analyzed as seen from a rotating frame. A system plus reservoir model is applied to consistently derive in the weak coupling limit the master equation for the reduced density in the moving frame and near the first bifurcation threshold. The concept of an effective temperature is introduced to analyze to what extent a detailed balance relation exists. Explicit expressions are also found for the Lamb-shift. Results for ohmic baths are in agreement with experimental findings, while for structured environments population inversion is predicted that may qualitatively explain recent observations. The prospect to tailor quantum devices on ever growing scales has stimulated major experimental research in the last years. In solid state physics various types of nonlinear resonators have been fabricated such as mechanical beams [1] and superconducting tunnel junctions embedded in cavities [2], partly to study fundamental physical phenomena, partly to develop highly sensitive amplifiers for detection schemes towards the quantum limit. These systems can be tuned over broad ranges of parameter space and particularly between the domains of classical and quantum behavior since they interact inevitably with surrounding degrees of freedom. Theory is challenged as now dynamical processes like bifurcation, parametric amplification, and period doubling appear on a level where tends to play a decisive role. A particular example of this type are microwave-driven Josephson junctions (JJ), recently realized in form of the Josephson Bifurcation Amplifier [3,4] and the Cavity Bifurcation Amplifier [2]. They are based on the fact that near the first bifurcation threshold the switching between the two driving-induced states is extremely sensitive to parameters of the JJ. The latter ones may be influenced by the coupling to a quantum system of interest, e.g. a Cooper pair island acting as artificial atom [5], meaning that the state of the quantum system can be retrieved from the measurement of the switching rate of the JJ. It has been shown that this strategy allows for a single shot read-out close to the quantum non-demolition (QND) limit [6]. Theoretically, a general approach is provided by the Floquet representation [7], but in the above situation a more powerful procedure for analytical investigations is to describe the driven dynamics in a frame rotating with a frequency equal to the response frequency of the system as already analyzed in the classical regime by Dykman and co-workers [8,9]. The extension to the quantum regime has been given in [10][11][12]. In essence, one arrives at a time-independent description with a non-standard Hamiltonian though. A complete understanding, however, must include also the bath degrees of freedom residing in the lab frame.The common formulation for dissipative systems in the weak coupling regime is provided by so-called master equations [13,14]. It i...

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Section: Effective Temperaturementioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Dissipation in a rotating frame: Master equation, effective temperature, and Lamb shift

Verso

Ankerhold

2010

Phys. Rev. A

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“…Most of the features of Josephson bifurcation amplifiers, in fact, only rely on (and can consequently be described by) any type of nonlinearity [17][18][19][20][21][22] , e.g., Duffing-type models, so that only recently the full nonlinear potential of the JJ has become of interest in this field 23 .…”

Section: Introductionmentioning

confidence: 99%

Resonators coupled to voltage-biased Josephson junctions: From linear response to strongly driven nonlinear oscillations

et al. 2015

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Motivated by recent experiments, where a voltage biased Josephson junction is placed in series with a resonator, the classical dynamics of the circuit is studied in various domains of parameter space. This problem can be mapped onto the dissipative motion of a single degree of freedom in a nonlinear time-dependent potential, where in contrast to conventional settings the nonlinearity appears in the driving while the static potential is purely harmonic. For long times the system approaches steady states which are analyzed in the underdamped regime over the full range of driving parameters including the fundamental resonance as well as higher and sub-harmonics. Observables such as the dc-Josephson current and the radiated microwave power give direct information about the underlying dynamics covering phenomena as bifurcations, irregular motion, up-and down conversion. Due to their tunability, present and future set-ups provide versatile platforms to explore the changeover from linear response to strongly nonlinear behavior in driven dissipative systems under well defined conditions. I. INTRODUCTIONThe nonlinear properties of Josephson junctions (JJs) have made such devices a key circuit element for classical and quantum electronics. Accordingly, there has been a long tradition of studying non-linear phenomena in driven superconducting circuits, starting as early as the 1960s with the discovery of Shapiro steps 1 . While Shapiro-steps have remained a tool in exploring new directions in Josephson physics, for instance in atomic point contacts 2-6 , other nonlinear phenomena, like synchronization, have been investigated in arrays of JJs: as a test-bed for generic theory models to capture synchronization phenomena 7,8 , but as importantly with the prospect of applications as sources of more intense coherent radiation 9 , cf. also new developments using intrinsic arrays 10-12 .More recently, the nonlinearity of the JJ was exploited as the crucial factor in enabling the high sensitivity of Josephson bifurcation amplifiers, achieving substantial improvements towards reaching quantum-limited measurement processes [13][14][15][16] . Most of the features of Josephson bifurcation amplifiers, in fact, only rely on (and can consequently be described by) any type of nonlinearity 17-22 , e.g., Duffing-type models, so that only recently the full nonlinear potential of the JJ has become of interest in this field 23 .A recent addition to the field of driven nonlinear JJs 24-26 are experiments on a dc-biased JJ connected to a resonator 27-29 . In this sort of setup, charge transfer through the JJ leads to excitations in the resonator, and therefore allows to convert energy carried by charge quanta into quantum microwave photons. In these devices measurement of both the Josephson current and the emitted microwave radiation is possible, a distinct advantage in comparison to other recently proposed transport setups 30,31 , which show similar nonlinear features like bifurcations, period multiplication and up-and downconversion 32-34 . S...

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“…Composed qubit-Josephson bifurcation amplifier systems have been considered in [13,14]. Dynamical tunneling in a Duffing oscillator was accounted for in [15] within a semiclassical WKB scheme, while in [16,17,18] a Wigner function analysis near the bifurcation point is put forward. The behaviour of the Duffing oscillator (DO) in the deep quantum regime has attracted recently lot of interest.…”

Section: Introductionmentioning

confidence: 99%

The dissipative quantum Duffing oscillator: A comparison of Floquet-based approaches

Vierheilig¹,

Grifoni²

2010

Chemical Physics

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We study the dissipative quantum Duffing oscillator in the deep quantum regime with two different approaches: The first is based on the exact Floquet states of the linear oscillator and the nonlinearity is treated perturbatively. It well describes the nonlinear oscillator dynamics away from resonance. The second, in contrast, is applicable at and in the vicinity of a N-photon resonance and it exploits quasi-degenerate perturbation theory for the nonlinear oscillator in Floquet space. It is perturbative both in driving and nonlinearity. A combination of both approaches yields the possibility to cover the whole range of driving frequencies. As an example we discuss the dissipative dynamics of the Duffing oscillator near and at the one-photon resonance.

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Dynamical Tunneling in Macroscopic Systems

Cited by 42 publications

References 19 publications

Dissipation in a rotating frame: Master equation, effective temperature, and Lamb shift

Dissipation in a rotating frame: Master equation, effective temperature, and Lamb shift

Resonators coupled to voltage-biased Josephson junctions: From linear response to strongly driven nonlinear oscillations

The dissipative quantum Duffing oscillator: A comparison of Floquet-based approaches

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