Quantum versus classical phase-locking transition in a frequency-chirped nonlinear oscillator

Abstract

Classical and quantum-mechanical phase locking transition in a nonlinear oscillator driven by a chirped frequency perturbation is discussed. Different limits are analyzed in terms of the dimensionless parameters P1 = ε/ √ 2m ω0α and P2 = (3 β)/(4m √ α) (ε, α, β and ω0 being the driving amplitude, the frequency chirp rate, the nonlinearity parameter and the linear frequency of the oscillator). It is shown that for P2 ≪ P1 + 1, the passage through the linear resonance for P1 above a threshold yields classical autoresonance (AR) in the system, even when starting in a quantum ground state. In contrast, for P2 ≫ P1 + 1, the transition involves quantum-mechanical energy ladder climbing (LC). The threshold for the phase-locking transition and its width in P1 in both AR and LC limits are calculated. The theoretical results are tested by solving the Schrodinger equation in the energy basis and illustrated via the Wigner function in phase space.

Quantum Phenomena in a Chirped Parametric Anharmonic Oscillator

“…One can see that the response of the quantum anharmonic oscillator to the chirped parametric modulation involves successive transitions between neighboring even energy levels, i.e., the PLC. We define the anharmonicity parameter P 2 = 2γ/ √ α [17,18] (P 2 = 10 in this example) and observe that the successive n → n + 2 transitions occur at times τ n = 4nP 2 , in agreement with the theory below (dashed black line). To our knowledge, such ladder climbing dynamics in the parametrically modulated anharmonic oscillator is observed for the first time.…”

Quantum and Classical Chirps in an Anharmonic Oscillator

“…4b) and are the result of multi-level Landau-Zener tunneling effects [13]. In the simulation, the amplitude of these oscillations decreases at larger β/ √ α values, converging to the theoretical ladder climbing threshold scaling [13].…”

“…This condition is met when the broadening of the first transition (caused by the drive amplitude) is comparable to the frequency difference between neighboring transitions. This marks the transition between the classical and quantum regimes, where the energy levels are mixed or resolved [12,13]. For comparison, the theoretical threshold lines of autoresonance and ladder climbing are shown on the same axes in red and black respectively.…”

“…The phase-locked wavepacket is centered around n ≈ 18 and is remarkably long-lived, despite the short decay time at these highly excited levels. We define the locking probability P locked as the probability to be in the phase-locked state, taken for this measurement as the integrated probability for levels n > 10 [13,18]. The locking probability decays non-exponentially with a time constant T locked = 1.4 µs, where T locked is defined as the time it takes for the lock-ing probability to decay to half of its initial value.…”

“…In a quantum anharmonic oscillator, the expected time evolution under a similar drive is sequential excitation of single energy levels of the system, or "quantum ladder climbing" [11]. In practice, for a given anharmonicity the drive itself introduces some mixing between the energy levels due to power broadening and finite bandwidth, which may wash out ladder climbing and lead to a classical behavior in a quantum system [12,13]. In this letter, we measure the dynamics in these two distinct regimes in the same system by varying the drive parameters and the system's anharmonicity.…”