Quantum Phenomena in a Chirped Parametric Anharmonic Oscillator

Abstract

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

Capture into resonance and phase-space dynamics in an optical centrifuge

“…By using methods in the theory of AR and analyzing the associated phase space dynamics we will for the first time calculate the efficiency of the OC process. The quantum counterpart of the AR is the quantum energy ladder climbing [29][30][31], but we will show that the classical AR analysis is relevant to many current experimental setups.…”

Bifurcations of autoresonant modes in oscillating systems with combined excitation

“…15). If ( , ) ∈ Ω + , system (1) has four different solutions * ( ), * ( ) with asymptotic expansion in the form (3). If ( , ) ∈ Ω − , system (1) has two different solutions * ( ), * ( ) with asymptotic expansion in the form (3).…”

“…as → ∞ and for all ( , ) ∈  * , where ℎ 0 2 ( , ) = 1∕2 2 + 2 2 2 2 +  ′′ ( ; , ) 3 6 , 0 2 ( , ) = 3 + 2 2 4 2 9 1∕2 +  ′′ ( ; , ) 4 3 27 , ℎ 2 ( , ) = (Δ 2 ), 2 ( , ) = (Δ) as Δ = √ 2 + 2 → 0, and 2 2 = −  ′′ ( ; , ) > 0. We see that the function  2 ( , , ) is suitable for the basis of a Lyapunov function candidate.…”

“…1,2 The autoresonance was first suggested in the problems associated with the acceleration of relativistic particles in the middle of the twentieth century, and nowadays it is considered as a universal phenomenon that plays the important role in a wide range of physical systems. [3][4][5][6][7] The study of relevant mathematical models leads to new and interesting problems in the field of nonlinear dynamics. [8][9][10][11] Mathematical models associated with the autoresonance have been actively studied recently.…”

Capture into parametric autoresonance in the presence of noise

“…The influence of additive noise on the capture into the autoresonance (non-parametric) was analysed in [6], where the effect of perturbations was considered only at the initial time. The problem of capture into the parametric autoresonance for a quantum anharmonic oscillator with initial disturbances was discussed in [7]. The effect of persistent perturbations with random jumps on the stability of autoresonance models was investigated in [8].…”