Quantum heating of a parametrically modulated oscillator: Spectral signatures

Dykman, M. I.; Marthaler, Michael; Peano, Vittorio

doi:10.1103/physreva.83.052115

Cited by 53 publications

(89 citation statements)

References 36 publications

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“…It occurs when the natural frequency of a system depends on a parameter oscillating (modulated) at twice the natural system's frequency [1][2][3][4][5]. In the well studied stationary case, the modulation frequency is constant.…”

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confidence: 99%

“…The deviation at large times is due to higher order corrections of the energy levels not included in Eqs. (3) and (4). The fact that the classical results can be reconstructed by solving the quantum equations implies that the correspondence principle is satisfied in the limit of small anharmonicity (β ≪ 1), where many energy levels are coupled simultaneously [17,18].…”

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Quantum Phenomena in a Chirped Parametric Anharmonic Oscillator

Barth

Frièdland

2014

Phys. Rev. Lett.

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The parametric ladder climbing (successive Landau-Zener-type transitions) and the quantum saturation of the threshold for the classical parametric autoresonance due to the zero point fluctuations at low temperatures are discussed. The probability for capture into the chirped parametric resonance is found by solving the Schrodinger equation in the energy basis and the associated resonant phase space dynamics is illustrated via the Wigner distribution. The numerical threshold for the efficient capture into the resonance is compared with the classical and quantum theories in different parameter regimes.PACS numbers: 42.50.Lc, 05.45.Xt, 85.25.Cp Introduction.-The parametric resonance is one of the most interesting and frequently used phenomena in classical and quantum dynamics. It occurs when the natural frequency of a system depends on a parameter oscillating (modulated) at twice the natural system's frequency [1][2][3][4][5]. In the well studied stationary case, the modulation frequency is constant. However, in nonlinear systems the stationary parametric amplification is restricted to small amplitudes, since at larger amplitudes the resonance (phase-locking) is destroyed due to the nonlinear frequency shift [1]. A robust method to overcome this limitation is to slowly vary the modulation frequency so that the resonance condition is preserved despite the increase of the amplitude of oscillations. This nonstationary phenomenon is called parametric autoresonance (PAR). The PAR was studied in such classical oscillatory systems as the anharmonic oscillator [6,7], Faraday waves [8,9], and plasmas [10]. A related, but different control method is the direct autoresonance (AR), where instead of parametric modulations, a chirped external driving force is applied. The direct AR was extensively studied and implemented in various classical and quantum physical systems [10][11][12][13][14][15].When studying the classical to quantum transitions in the direct chirped-driven oscillator, one identifies two limits, where quantum dynamical effects are significant. The first is the saturation of temperature-dependent classical observables at small temperatures due to the zero point quantum fluctuations [16,17]. In the second limit, at sufficiently large anharmonicity, the smooth classical AR dynamics of many simultaneously coupled energy levels transforms into a quantum ladder climbing involving successive two-level Landau-Zener (LZ) transitions [17][18][19][20]. These two quantum limits were also studied recently in the case of the direct chirped subharmonic (twophoton) resonance [21]. This letter, for the first time, discusses the analogous quantum phenomena in application to the PAR.The model.-The simplest system exhibiting nonlinear parametric resonance is the anharmonic oscillator governed by Hamiltonian

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Quantum Phenomena in a Chirped Parametric Anharmonic Oscillator

Barth

Frièdland

2014

Phys. Rev. Lett.

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“…Using the Bose-Einstein distribution for n th then allows us to define an effective temperature T eff for the system due to squeezing of the resonator field. This corresponds to the essential prediction of the quantum heating theory [23][24][25][26]. As discussed in more details in Ref.…”

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confidence: 99%

“…red and blue sideband transitions. Relaxing the second approximation allows us to take into account squeezing of the resonator field and, as predicted by the theory of quantum heating [23][24][25][26], this leads to an effective temperature of the resonator field. Using sideband spectroscopy, a standard tool in ion-trapping experiments [27,28], we then discuss how the qubit can act as an absolute thermometer of this effective temperature.…”

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Superconducting qubit as a probe of squeezing in a nonlinear resonator

et al. 2014

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“…10 Recent theoretical studies of their quantum dynamics show that it differs from the classical motion 11,12 and the nonlinearity can be exploited in the detection of the quantum signatures. The amplitudes needed to observe nonlinear effects are often orders of magnitude larger than the quantum zero-point fluctuations x 0 = √h /mω 0 .…”

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Generating macroscopic superposition states in nanomechanical graphene resonators

2012

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A theoretical study of the quantum dynamics of a symmetric nanomechanical graphene resonator with degenerate flexural modes is presented. Applying voltage pulses to two back gates, flexural vibrations of the membrane can be selectively actuated and manipulated. For graphene, nonlinear response becomes important for amplitudes comparable to the magnitude of zero-point fluctuations. We show, using analytical and numerical methods, that this allows for creation of catlike superpositions of coherent states as well as superpositions of coherent catlike nonproduct states. DOI: 10.1103/PhysRevB.85.205415 PACS number(s): 85.85.+j, 42.50.Dv, 73.50.Fq Coherent superposition of states are characteristic traits of quantum mechanics. These phenomena have already been realized in many-particle contexts such as trapped ultracold atoms, 1 superconductors, 2 and photonic systems. 3 A current challenge is to observe these effects for collective degrees of freedom in a macroscopic context in, e.g., mechanical resonators. 4 Recent advances in cooling mechanical resonators and sensitive displacement detection have allowed reaching the motional ground state and observing zero point fluctuations of center of mass. 5,6 Active manipulation and characterization of the quantum state of these systems, as already achieved with photons, 7 seem to be within reach. For a mechanical system, a desirable state to generate is a "cat" state. This is a coherent superposition of two minimum uncertainty wave packets separated by more than their individual quantum fluctuations.For the harmonic oscillator, a minimum uncertainty wave packet is a coherent state |α = exp[αa † − α * a]|0 generated by displacing the oscillator ground state. 8 As shown by Yurke and Stoler, 9 for a nonlinear oscillator H =hω 0n +h n 2 , an initial coherent state |α will after a time t = π/(2 ) evolve into the cat state (1/ √ 2)[e −iπ/4 |α + e iπ/4 |−α ] with the maximum spatial separation = 2|α|.Nanoelectromechanical resonators are typically intrinsically nonlinear. 10 Recent theoretical studies of their quantum dynamics show that it differs from the classical motion 11,12 and the nonlinearity can be exploited in the detection of the quantum signatures. The amplitudes needed to observe nonlinear effects are often orders of magnitude larger than the quantum zero-point fluctuations x 0 = √h /mω 0 . A cat state obtained due to this nonlinearity would have a separation 1. As the decoherence rate scales as 2 (see Refs. 13 and 14), this has, until now, been unfeasible. Instead, coupling to auxiliary quantum systems has been proposed to engineer the nonlinearity. 15,16 We show how, in the limit k B T hω 0 , the intrinsic nonlinearity of a graphene membrane resonator can be used to prepare cat states by applying voltage pulses to local backgates. The reason for using graphene is the ultralow mass of the graphene sheet, which leads to a large x 0 , and an onset of nonlinear response at small amplitudes. 17,18 This implies that cat states with moderate can be constructed without the...

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Quantum heating of a parametrically modulated oscillator: Spectral signatures

Cited by 53 publications

References 36 publications

Quantum Phenomena in a Chirped Parametric Anharmonic Oscillator

Quantum Phenomena in a Chirped Parametric Anharmonic Oscillator

Superconducting qubit as a probe of squeezing in a nonlinear resonator

Generating macroscopic superposition states in nanomechanical graphene resonators

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