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arXiv:cond-mat/0602288 · Physical Review A

Switching via quantum activation: A parametrically modulated oscillator

Michael Marthaler, M. I. Dykman

Published Apr 20, 2006• 130 citations
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Abstract

We study switching between period-two states of an underdamped quantum oscillator modulated at nearly twice its natural frequency. For all temperatures and parameter values switching occurs via quantum activation: it is determined by diffusion over oscillator quasienergy, provided the relaxation rate exceeds the rate of interstate tunneling. The diffusion has quantum origin and accompanies relaxation to the stable state. We find the semiclassical distribution over quasienergy. For $T=0$, where the system has detailed balance, this distribution differs from the distribution for $T\ensuremath{\rightarrow}0$; the $T=0$ distribution is also destroyed by small dephasing of the oscillator. The characteristic quantum activation energy of switching displays a typical dependence on temperature and scaling behavior near the bifurcation point where period doubling occurs.

DOI
10.1103/physreva.73.042108

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We study switching between period-two states of an underdamped quantum oscillator modulated at nearly twice its natural frequency. For all temperatures and parameter values switching occurs via quantum activation: it is determined by diffusion over oscillator quasienergy, provided the relaxation rate exceeds the rate of interstate tunneling. The diffusion has quantum origin and accompanies relaxation to the stable state. We find the semiclassical distribution over quasienergy. For T=0T=0T=0, where the system has detailed balance, this distribution differs from the distribution for T\ensuremath→0T\ensuremath{\rightarrow}0T\ensuremath→0; the T=0T=0T=0 distribution is also destroyed by small dephasing of the oscillator. The characteristic quantum activation energy of switching displays a typical dependence on temperature and scaling behavior near the bifurcation point where period doubling occurs.
Period tripling in driven quantum oscillators reveals unique features absent for linear and parametric drive, but generic for all higher-order resonances. Here we focus at zero temperature on the relaxation dynamics towards a stationary state starting initially from a domain around a classical fixed point in phase space. Beyond a certain threshold for the driving strength, the long-time dynamics is governed by a single time constant that sets the rate for switching between different states with broken time translation symmetry. By analyzing the lowest eigenvalues of the corresponding time evolution generator for the dissipative dynamics, we find that near the threshold the gap between these eigenvalues nearly closes. The closing becomes complete for a vanishing quantum parameter. We demonstrate that this behavior, reminiscent of a quantum phase transition, is associated with a transition from a stationary state which is localized in phase space to a delocalized one. We further show that switching between domains of classical fixed points happens via quantum activation, however, with rates that differ from those obtained by a standard semiclassical treatment. As period tripling has been explored with superconducting circuits mainly in the quasiclassical regime recently, our findings may trigger new activities towards the deep quantum realm.
We study parametrically driven quantum oscillators and show that, even for weak coupling between the oscillators, they can exhibit various many-body states with broken time-translation symmetry. In the quantum-coherent regime, the symmetry breaking occurs via a nonequilibrium quantum phase transition. For dissipative oscillators, the main effect of the weak coupling is to make the switching rate of an oscillator between its period-2 states dependent on the states of other oscillators. This allows mapping the oscillators onto a system of coupled spins. For identical oscillators, the stationary state can be mapped on that of the Ising model with an effective temperature $\ensuremath{\propto}\ensuremath{\hbar}$, for low temperature. If the oscillators are different and are away from the bifurcation parameter values where the period-2 states emerge, the stationary state corresponds to having a microscopic current in the spin system. Close to the bifurcation point the coupling cannot be considered weak and the system maps onto coupled overdamped Brownian particles performing quantum diffusion in a potential landscape.
Motivated by recent experimental progress to readout quantum bits implemented in superconducting circuits via the phenomenon of dynamical bifurcation, transitions between steady orbits in a driven anharmonic oscillator, the Duffing oscillator, are analyzed. In the regime of weak dissipation a consistent diffusion equation in the semiclassical limit is derived to capture the intimate relation between finite tunneling and reflection and bath induced quantum fluctuations. From the corresponding steady-state distribution an analytical expression for the switching probability is obtained. It is shown that a reduction of the transition rate due to finite reflection at the phase-space barrier is overcompensated by an increase due to environmental quantum fluctuations that are specific for diffusion processes over dynamical barriers. The scaling behavior of the rate is discussed and it is revealed that close to the bifurcation threshold the escape dynamics enters an overdamped domain such that the quantum-mechanical energy scale associated with friction even exceeds the thermal energy scale.
Abstract The influence of quantum tunneling effects on the bistability of nonlinear resonantly driven oscillators in the presence of external noise is analyzed by means of the generalized Fokker–Planck equation in quasi-energy space. Two time scales responsible for different stages of bistable system relaxation have been revealed. It was found that quantum tunneling effects could strongly change the switching rates between different stable states and considerably enhance occupation of the stable state with higher amplitude. It was also revealed that tunneling between nearly degenerate quasi-energy states and resonant multi-photon transitions between the genuine eigenstates of the nonlinear oscillator are just similar effects.
The spontaneous switching of a quantum particle between the wells of a double-well potential is a phenomenon of general interest to physics and chemistry. It was broadly believed that the switching rate decreases steadily as a function of the size of the energy barrier. This view was challenged by a recent experiment on a driven superconducting Kerr nonlinear oscillator (often called the Kerr-cat qubit or the Kerr parametric oscillator), whose energy barrier can be continuously increased by ramping up the drive. Remarkably, as the drive amplitude increases, the switching rate exhibits a steplike decrease termed the ``staircase.'' The view challenged by the experimental staircase demands a deep review of our understanding of the role of quantum effects in double wells. In this work, we use a Lindbladian model of dissipation to derive a semianalytical formula for the switching rate, resolving a continuous transition between tunneling-dominated dynamics and dissipation-dominated dynamics. These two dynamics are observed respectively in the flat part and the steep part of each step in the staircase. Our formula exposes two distinct dissipative processes that limit tunneling: the dephasing of interwell superpositions freezes tunneling via the quantum Zeno effect, and the decay from one excited state to another limits the time during which tunneling can be attempted. This physical understanding allows us to pinpoint the critical drive amplitude that separates the flat and steep part of each step using an equation involving the rates of the two dissipative processes and the tunnel splitting. In addition, analyzing the transition matrix elements in the formula shows that in the regime of a few ($\ensuremath{\lesssim}10$) states in the well and under moderate to low temperatures, highly excited states are populated predominantly via cascaded and direct thermal heating rather than quantum heating. At very low temperatures, we find that the perturbation induced by the non-Hermitian Hamiltonian part of the Lindbladian model becomes increasingly important and facilitates a form of quantum heating that has not been identified before. We numerically map the activation mechanism as a function of drive amplitude, damping rate, and temperature. Our theory deepens the understanding of switching dynamics between metastable quantum states, highlights the importance of a general interplay between tunneling and dissipation, and identifies a novel quantum regime in activated transitions.
We study a nonlinear oscillator, which is parametrically driven at a frequency close to twice its eigenfrequency. By judiciously choosing the frequency detuning and linearly increasing the driving amplitude, one can prepare any even quasienergy state starting from the oscillator ground state. Such state preparation is effectively adiabatic. We find the Wigner distribution of the prepared states. For a different choice of the frequency detuning, the adiabaticity breaks down, which allows one to prepare on demand a superposition of quasienergy states using Landau-Zener-type transitions. We find the characteristic spectrum of the transient radiation emitted by the oscillator after it has been prepared in a given quasienergy state.