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Contemporary Physics

Overdamping of coherently driven quantum systems

B. W. Shore, Nikolay V. Vitanov

Published Nov 1, 2006• 16 citations
Publisher

Abstract

The phenomenon of overdamping, when frictional forces overwhelm the restoring force of an harmonic oscillator, is well known in classical mechanics. Analogous phenomena occur in time-dependent quantum mechanics when probability loss rates (e.g. photoionization rates) become larger than the characteristic inverse times for coherent excitation (the Rabi frequencies). These results often seem surprising, even for simple two-state quantum systems. We discuss examples of this overdamping phenomenon—when an increase of the loss rate actually produces a slower rate of probability loss and even freezes population—for a number of quantum systems: steady and pulsed two-state excitation, time evolution with adiabatic passage, simple multistate chains, and branched chains of excitation linkage. In each situation there occurs for very large damping, often unexpectedly, phenomena akin to the overdamping of the classical harmonic oscillator.

DOI
10.1080/00107510601181159

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The phenomenon of overdamping, when frictional forces overwhelm the restoring force of an harmonic oscillator, is well known in classical mechanics. Analogous phenomena occur in time-dependent quantum mechanics when probability loss rates (e.g. photoionization rates) become larger than the characteristic inverse times for coherent excitation (the Rabi frequencies). These results often seem surprising, even for simple two-state quantum systems. We discuss examples of this overdamping phenomenon—when an increase of the loss rate actually produces a slower rate of probability loss and even freezes population—for a number of quantum systems: steady and pulsed two-state excitation, time evolution with adiabatic passage, simple multistate chains, and branched chains of excitation linkage. In each situation there occurs for very large damping, often unexpectedly, phenomena akin to the overdamping of the classical harmonic oscillator.
Overdamping is a regime in which friction is sufficiently large that the motion either decays to its equilibrium position or it crosses the equilibrium position exactly once before returning monotonically towards the equilibrium position. The phenomena of overdamping has been studied classically and quantum mechanically only for the case of the linear damped harmonic oscillator. Here we study the classical and quantum dynamics of a family of over-damped non linear systems. The main objective of this paper is to find a Lagrangian and Hamiltonian framework to study over-damped non linear systems and to show that a quantum mechanical description can be developed in the momentum representation. Our results reduce to the well known solution of the linear damped harmonic oscillator when the non linear part is set to zero.
A quantum system weakly interacting with a fast environment usually undergoes a relaxation with complex frequencies whose imaginary parts are damping rates quadratic in the coupling to the environment in accord with Fermi's ``golden rule.'' We show for various models (spin damped by harmonic-oscillator or random-matrix baths, quantum diffusion, and quantum Brownian motion) that upon increasing the coupling up to a critical value still small enough to allow for weak-coupling Markovian master equations, a different relaxation regime can occur. In that regime, complex frequencies lose their real parts such that the process becomes overdamped. Our results call into question the standard belief that overdamping is exclusively a strong coupling feature.
The open driven quantum harmonic oscillator is studied for three different possible mechanisms of driving; a generalization of the forced oscillator, driving due to a time-dependent coupling to a two-level system, and driving due to an adiabatic modulation of the frequency. The first case encompasses all possible configurations going from the standard forced harmonic oscillator to the classical pumping type of driving typically used in quantum optics. In the second case, an additional two-level system, coupled to the oscillator at a time-varying rate, is assumed and the reduced dynamics of the oscillator and the qubit is examined. In the driving due to adiabatic modulation, the dynamics of a harmonic oscillator whose frequency slowly varies in time is studied. The study done in this paper is focused on a harmonic type of driving, although the solutions are derived for an arbitrary type of driving. In all the cases, the oscillator is assumed to be in contact with a zero- or finite-temperature environment, which provides dissipation to the system. The treatment done in this paper is fully analytic and the derivation of the solutions is based on the employment of the double Fourier transform of the Wigner function.
We consider the quantum Rabi model as an open system that is subject to dissipation, dephasing, and sinusoidal qubit driving. One can change to an interaction picture where the qubit-driving term disappears at the expense of changing the free energy of the qubit which becomes time dependent. If the driving frequency is large with respect to the rest of the parameters with the exception of the driving strength, then one can obtain an effective Hamiltonian that accurately describes the dynamics of the system. The driving has two effects: the qubit-transition frequency is changed and the qubit has reduced dephasing. The driving strength can be chosen so that the qubit-transition frequency is reduced, made equal to zero, or even made negative so that the excited and ground states of the qubit are interchanged. Therefore, sinusoidal qubit driving offers another method to control the qubit-transition frequency and to reduce qubit dephasing. Adjusting the driving strength allows one to consider a qubit with degenerate energy levels. Not taking dissipation into account, the evolution operator of the qubit-harmonic oscillator system is given by a linear combination of the orthogonal projectors onto the eigenstates of ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}_{x}$ followed by the evolution operator of a forced harmonic oscillator, the harmonic oscillator can be prepared in such a way that it is always found in a Schr\"odinger cat state, and the transition probability of the qubit can exhibit a collapse-revival behavior. In addition, the Born-Markov-secular master equation is deduced and the effects of dissipation are presented. In particular, smaller ultrastrong-coupling values are preferable over larger ultrastrong-coupling values and deep strong-coupling values in order to have long-lived, easily distinguishable Schr\"odinger cat states because the decoherence rate is inversely proportional to the square of the coupling. Finally, the qubit-harmonic oscillator system can be prepared in highly entangled states that are stable under dissipation.
We compare the time evolutions of two quantum systems: a quantum harmonic oscillator driven by a classical force and two coupled quantum harmonic oscillators. We find the conditions under which the first system can be considered as a `mean-field approximation' to the second one. This happens in the weak coupling limit provided that the `master' oscillator was initially in a highly excited quantum state with small fluctuations of the canonical variables (such as the coherent state or displaced thermal state), while the `subordinate' oscillator must not be `too excited'. The essential difference between the resonance and nonresonance cases is discussed. In particular, the effect of the resonance-quantum-state exchange is demonstrated.