Periodically driven oscillators are commonly described in a frame corotating with the drive and using the rotating-wave approximation (RWA). This description, however, is known to induce errors for off-resonant driving. Here, we show that the standard quantum description, using the creation and annihilation of particles with the oscillators' natural frequency, necessarily leads to incorrect results when combined with the RWA. We demonstrate this on the simple (quantum) harmonic oscillator and present an alternative operator basis that reconciles the RWA with off-resonant driving. The approach is also applicable to more complex models, where it accounts for known discrepancies. As an example, we demonstrate the advantage of our scheme on a driven quantum Duffing oscillator.
Parametrically driven oscillators can display period tripling in response to a drive at thrice the resonance frequency. In contrast to the parametric instability for period doubling, the symmetric fixed point corresponding to the state of rest remains stable at arbitrary strong driving for the tripling transition. Previously, it has been shown that fluctuations can circumvent this and induce a period-tripling instability. In this article, we explore an alternative way of inducing a period-tripling transition by investigating properties of period tripling due to parametric down-conversion beyond the rotating-wave approximation. We show that despite the absence of an instability threshold, off-resonant frequency contributions can induce a period-tripling transition by activating the parametric down-conversion. Moreover, we study the subsequent period-tripled states of the Josephson potential and discuss the asymmetry between the clockwise and counterclockwise rotating fixed points that only arises beyond the rotating-wave approximation.
We study a quantum many-body variant of the parametric oscillator by investigating the driven sine-Gordon model with a modulated tunnel coupling via a semiclassical truncated Wigner approximation (TWA). We first analyze the parametric resonant regime for driving protocols that retain our model gapped, and compare the TWA to a time-dependent Gaussian variational ansatz (TGVA). We then turn to a drive which closes the gap, resulting in an enhanced energy absorption. While the TGVA approach breaks down in this regime, we can apply TWA to explore the dynamics of the mode-resolved energy density and the higher-order correlations between modes in the prethermal heating regime. For weak driving amplitude, we find an exponentially fast energy absorption in the main resonant mode, while the heating of all remaining modes is almost perfectly suppressed on short timescales. At later times, the highly excited main resonance provides effective resonant driving terms for its higher harmonics through the nonlinearities in the Hamiltonian, and gives rise to an exponentially fast heating in these particular modes. We capture the strong correlations induced by these resonant processes by evaluating higher-order connected correlation functions. Our results can be experimentally probed in ultracold-atomic settings, with parallel one-dimensional quasicondensates in the presence of a modulated tunnel coupling.
We investigate the steady-state energy flow of the periodically driven nonequilibrium spin-boson model by means of combining counter-rotating-hybridized rotating-wave (CHRW) method with full counting statistics. In the case of weak driving $(A<0.1\mathrm{\ensuremath{\Delta}})$, where
A is the driving amplitude and $\mathrm{\ensuremath{\Delta}}$ is the energy gap of the two-level system, our results are consistent with those obtained by traditional rotating-wave approximation (RWA) approach; in the case of high-frequency driving, our results are in agreement with the ones of the secular Floquet-Redfield method. As the driving amplitude increases, the steady-state flow exhibits nonlinear behavior: it reaches a maximum at medium driving, and then decreases to zero, which means that the coherent destruction of tunneling (CDT) appears, and then continues to grow slowly. We also find that, when the steady-state energy flow varies with driving frequency ${\ensuremath{\omega}}_{d}$, comparing with RWA results, the counter-rotating (CR) terms increase the energy flow at low-frequency driving $({\ensuremath{\omega}}_{d}<\mathrm{\ensuremath{\Delta}})$, where ${\ensuremath{\omega}}_{d}$ is the driving frequency, and decreases it at high-frequency driving $({\ensuremath{\omega}}_{d}>\mathrm{\ensuremath{\Delta}})$. In the case of strong driving, when the zero-order Bessel function of the first kind satisfies ${J}_{0}(\frac{A}{{\ensuremath{\omega}}_{d}}\ensuremath{\zeta})=0$, instead of ${J}_{0}(\frac{A}{{\ensuremath{\omega}}_{d}})=0$, the steady-state energy flow is 0, which corresponds to the CDT of the energy flow. The modified parameter $\ensuremath{\zeta}$ mainly defines the contribution of the CR terms. At high-frequency driving, the modified parameter $\ensuremath{\zeta}$ tends to
1, our result is consistent with previous results ${J}_{0}(\frac{A}{{\ensuremath{\omega}}_{d}})=0$. We find that the CDT of flow only occurs when both CR terms of the driving and the dissipation are considered simultaneously. Our results show the influence of the CR terms on the energy flow and the modification of CDT condition and give theoretical guidance for energy transport of small quantum devices.