Dynamics of the quantum Duffing oscillator in the driving induced bistable regime

Abstract

We investigate the nonlinear response of an anharmonic monostable quantum mechanical resonator to strong external periodic driving. The driving thereby induces an effective bistability in which resonant tunneling can be identified. Within the framework of a Floquet analysis, an effective Floquet-Born-Markovian master equation with time-independent coefficients can be established which can be solved straightforwardly. Various effects including resonant tunneling and multi-photon transitions will be described. Our model finds applications in nano-electromechanical devices such as vibrating suspended nano-wires as well as in non-destructive readout procedures for superconducting quantum bits involving the nonlinear response of the read-out SQUID.

Quasienergy description of the driven Jaynes-Cummings model

“…This is the case when a coherent resonant excitation induces a population of a multiphoton quasienergy state, as shown for the Duffing oscillator [9][10][11] . Since the quasienergy states oscillate with different phases with respect to the external modulation, a resonant or antiresonant response of the oscillator to the modulation may occur at a multiphoton transition.…”

From Coulomb-Blockade to Nonlinear Quantum Dynamics in a Superconducting Circuit with a Resonator

“…Accordingly, the classical Josephson energy reads −E J cos(φ + ω J t) and acts as a nonlinear drive on the resonator. Near resonance ω J ≈ ω 0 , the ansatz φ(t) = Z cos(ω J t+ϕ) for the stationary orbit then leads in the rotating frame to (6). We emphasize that in contrast to most driven nonlinear oscillators, recently realized also with superconducting circuits (see e.g.…”

“…This has led to an unprecedented control of quantum properties such as the creation of catlike states [4] and the observation as well as theoretical description of nonlinear dynamics [5][6][7][8][9][10][11]. While in these circuits no net charge flows through the JJ, dc-voltage biased set-ups, implemented very recently in [12][13][14][15], offer new possibilities to study nonlinear quantum properties in a tunable photon-charge system far from equilibrium [16][17][18].…”

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

“…We find a characteristic asymmetry of the antiresonance lineshape. In contrast to [20,21,22,23], our analytic results are obtained without applying a rotating wave approximation (RWA) on the Duffing oscillator. The paper is organized as follows: In section 2 we introduce the Hamiltonian of the non-dissipative Duffing oscillator and the two Floquet based approximation schemes to treat it.…”

“…In particular Rigo et al [19] demonstrated, based on a quantum diffusion model, that in the steady state the quantum DO does not exhibit any bistability or hysteresis. It was also shown that the response of the Duffing oscillator displays antiresonant dips and resonant peaks [20,21,22,23] depending on the frequency of the driving field, originating from special degeneracies of the eigenenergy spectrum of the nonlinear oscillator [20]. While the antiresonances persist in the presence of a weak Ohmic bath, for strong damping the nonlinear response turns to a resonant behaviour, namely the one of a linear oscillator at a shifted frequency [22].…”