Driven Quantum Systems Research Articles

ERRATUM: "QUANTUM MACROSTATISTICAL THEORY OF NONEQUILIBRIUM STEADY STATES"

We provide a general macrostatistical formulation of nonequilibrium steady states of reservoir driven quantum systems. This formulation is centred on the large scale properties of the locally conserved hydrodynamical observables, and our basic assumptions comprise (a) a chaoticity hypothesis for the nonconserved currents carried by these observables, (b) an extension of Onsager's regression hypothesis to the fluctuations about nonequilibrium states, and (c) a certain mesoscopic local equilibrium hypothesis. On this basis we obtain a picture wherein the fluctuations of the hydrodynamical observables about a nonequilibrium steady state execute a Gaussian Markov process of a generalized Onsager-Machlup type, which is completely determined by the position dependent transport coefficients and the equilibrium entropy function of the system. This picture reveals that the transport coefficients satisfy a generalized form of the Onsager reciprocity relations in the nonequilibrium situation and that the spatial correlations of the hydrodynamical observables are generically of long range. This last result constitutes a model-independent generalization of that obtained for special classical stochastic systems and marks a striking difference between the steady nonequilibrium and equilibrium states, since it is only at critical points that the latter carry long range correlations.

Effective Hamiltonian approach to periodically perturbed quantum optical systems

We apply the method of Lie-type transformations to Floquet Hamiltonians for periodically perturbed quantum systems. Some typical examples of driven quantum systems are considered in the framework of this approach and corresponding effective time dependent Hamiltonians are found.

QUANTUM MACROSTATISTICAL THEORY OF NONEQUILIBRIUM STEADY STATES

We provide a general macrostatistical formulation of nonequilibrium steady states of reservoir driven quantum systems. This formulation is centered on the large scale properties of the locally conserved hydrodynamical observables, and our basic physical assumptions comprise (a) a chaoticity hypothesis for the nonconserved currents carried by these observables, (b) an extension of Onsager's regression hypothesis to fluctuations about nonequilibrium states, and (c) a certain mesoscopic local equilibrium hypothesis. On this basis, we obtain a picture wherein the fluctuations of the hydrodynamical variables about a nonequilibrium steady state execute a Gaussian Markov process of a generalized Onsager–Machlup type, which is completely determined by the position dependent transport coefficients and the equilibrium entropy function of the system. This picture reveals that the transport coefficients satisfy a generalized form of the Onsager reciprocity relations in the nonequilibrium situation and that the spatial correlations of the hydrodynamical observables are generically of long range. This last result constitutes a model-independent quantum mechanical generalization of that obtained for special classical stochastic systems and marks a striking difference between the steady nonequilibrium and equilibrium states, since it is only at critical points that the latter carry long range correlations.

Noise-induced transition in a quantum system

We examine the noise-induced transition in a fluctuating bistable potential of a driven quantum system in thermal equilibrium. Making use of a Wigner canonical thermal distribution for description of the statistical properties of the thermal bath, we explore the generic effects of quantization like vacuum field fluctuation and tunneling in the characteristic stationary probability distribution functions undergoing transition from unimodal to bimodal nature and in signal-to-noise ratio characterizing the cooperative effect among the noise processes and the weak periodic signal.

Adiabatic Theorems and Reversible Isothermal Processes

Isothermal processes of a finitely extended, driven quantum system in contact with an infinite heat bath are studied from the point of view of quantum statistical mechanics. Notions like heat flux, work and entropy are defined for trajectories of states close to, but distinct from states of joint thermal equilibrium. A theorem characterizing reversible isothermal processes as quasi-static processes (“isothermal theorem”) is described. Corollaries concerning the changes of entropy and free energy in reversible isothermal processes and on the 0th law of thermodynamics are outlined.

Entanglement within the quantum trajectory description of open quantum systems.

The degree of entanglement in an open quantum system varies according to how information in the environment is read. A measure of this contextual entanglement is introduced based on quantum trajectory unravelings of the open system dynamics. It is used to characterize the entanglement in a driven quantum system of dimension 2 x infinity where the entanglement is induced by the environmental interaction. A detailed mechanism for the environment-induced entanglement is given.

Quantitative model for the effective decoherence of a quantum computer with imperfect unitary operations

The problem of the quantitative degradation of the performance of a quantum computer due to noisy unitary gates (imperfect external control) is studied. It is shown that quite general conclusions on the evolution of the fidelity can be reached by using the conjecture that the set of states visited by a quantum algorithm can be replaced by the uniform (Haar) ensemble. These general results are tested numerically against quantum computer simulations of two particular periodically driven quantum systems.

Electromagnetically induced quantum memory.

We discuss the problem of creating coherence in an optically driven quantum system in conditions where decoherence is caused by the laser field itself, due to coupling of the system to a rapidly decaying state or continuum. It is shown that by applying an additional laser field between this state and a bound state the relaxation channel can be suppressed as a result of a "dark state" formation, giving rise to long living Rabi oscillations in the system. It is found that the same mechanism of preserving coherence exists in systems with level splitting or degeneracy, where the driving field interacts with multiple resonant sublevels simultaneously. We also show that specific coherent propagation phenomena assisted by the interference suppression of decoherence can be observed under these conditions.

Classical versus quantum coherence.

Quantum dots are structures engineered to have desirable quantum-mechanical properties. Much of their potential usefulness stems from the ability to design such structures so that their passive properties can be exploited. However, there are many plans to use dots as the basis of a more active quantum engineering, in which the detailed quantum states of the dot's components are manipulated by externally imposed fields. This leads to a new requirement: our ability to model the time development of individual driven quantum systems. I shall discuss some of the problems which arise when single systems, rather than ensembles, are considered and give examples which illustrate the effects of classical and quantum incoherences. I propose a new classification of incoherent processes, based on their effect on individual wave functions.

Computingquantum eigenvalues made easy

An extremely simple and convenient method is presented forcomputing eigenvalues in quantum mechanics by representingposition and momentum operators in matrix form. The simplicityand success of the method is illustrated by numerical resultsconcerning eigenvalues of bound systems and resonances forHermitian and non-Hermitian Hamiltonians as well as drivenquantum systems. Various MATLAB program codes are listed.

Dynamical localization in quasiperiodic driven systems.

We investigate how the time dependence of the Hamiltonian determines the occurrence of dynamical localization (DL) in driven quantum systems with two incommensurate frequencies. If both frequencies are associated to impulsive terms, DL is permanently destroyed. In this case, we show that the evolution is similar to a decoherent case. On the other hand, if both frequencies are associated to smooth driving functions, DL persists although on a time scale longer than in the periodic case. When the driving function consists of a series of pulses of duration sigma, we show that the localization time increases as sigma(-2) as the impulsive limit, sigma-->0, is approached. In the intermediate case, in which only one of the frequencies is associated to an impulsive term in the Hamiltonian, a transition from a localized to a delocalized dynamics takes place at a certain critical value of the strength parameter. We provide an estimate for this critical value, based on analytical considerations. We show how, in all cases, the frequency spectrum of the dynamical response can be used to understand the global features of the motion. All results are numerically checked.

Wiener–Hermite expansion formalism for the stochastic model of a driven quantum system

The time evolution of a simple two-level system with a periodic modulation of the energy levels superimposed with a stochastic fluctuation is investigated. The Wiener–Hermite expansion method is successfully applied to the numerical solution of the equation of motion. In this method, we transform the stochastic equation into an infinite set of simultaneous deterministic equations, which can be solved by truncating it into a finite one. Two types of mechanisms of phase relaxation are exhibited to modify the population dynamics. The one is the homogeneous phase relaxation in the rapid fluctuation limit, and the other is the inhomogeneous phase relaxation in the slow fluctuation limit.

Autocorrelation Scaling and Fourier Transform of Non-Autonomous Systems

Upper bounds for the (strong) Fourier transform,of a rather general sequence of unitary operators, are related to the uniform α-Holder continuity of its autocorrelation measure. It is a natural generalization of the "Dynamical Bombieri-Taylor Conjecture". Immediate applications include driven quantum systems, classical and quantum harmonic oscillators, and non-autonomous twisted generalized random walks in Hilbert spaces.

Driven systems and the Lax formula

One of the most widely used tools in quantum optics, the Lax formula for two-time correlations, is studied. The formula is then applied to two driven quantum systems: the oscillator and the two-level atom. Through a comparison with exact results for the oscillator, it is shown explicitly that the Lax formula is a weak coupling approximation that does not include quantum corrections to the Onsager regression hypothesis of classical statistical physics. For the two-level atom, the calculation of the well known Burshtein–Mollow spectrum of resonance fluorescence is extended to the case of non-zero temperature.

The Lax–Onsager regression `theorem' revisited

It is stated by Ford and O'Connell in this festschrift issue and elsewhere that “there is no quantum regression theorem” although Lax “obtained a formula for correlation in a driven quantum system that has come to be called the quantum regression theorem”. This produces a puzzle: “How can it be that a non-existent theorem gives correct results?” Clarification will be provided in this paper by a description of the Lax procedure, with a quantitative estimate of the error for a damped harmonic oscillator based on expressions published in the 1960's.

Quasistationary distributions of dissipative nonlinear quantum oscillators in strong periodic driving fields

The dynamics of periodically driven quantum systems coupled to a thermal environment is investigated. The interaction of the system with the external coherent driving field is taken into account exactly by making use of the Floquet picture. Treating the coupling to the environment within the Born-Markov approximation one finds a Pauli-type master equation for the diagonal elements of the reduced density matrix in the Floquet representation. The stationary solution of the latter yields a quasistationary, time-periodic density matrix which describes the long-time behavior of the system. Taking the example of a periodically driven particle in a box, the stationary solution is determined numerically for a wide range of driving amplitudes and temperatures. It is found that the quasistationary distribution differs substantially from a Boltzmann-type distribution at the temperature of the environment. For large driving fields it exhibits a plateau region describing a nearly constant population of a certain number of Floquet states. This number of Floquet states turns out to be nearly independent of the temperature. The plateau region is sharply separated from an exponential tail of the stationary distribution which expresses a canonical Boltzmann-type distribution over the mean energies of the Floquet states. These results are explained in terms of the structure of the matrix of transition rates for the dissipative quantum system. Investigating the corresponding classical, nonlinear Hamiltonian system, one finds that in the semiclassical range essential features of the quasistationary distribution can be understood from the structure of the underlying classical phase space.

Quantum averaging for driven systems with resonances

We discuss the effects of resonances in driven quantum systems within the context of quantum averaging techniques in the Floquet representation. We consider in particular iterative methods of KAM type and the extensions needed to take into account resonances. The approach consists in separating the coupling terms into resonant and nonresonant components at a given scale of time and intensity. The nonresonant part can be treated with perturbative techniques, which we formulate in terms of KAM-type unitary transformations that are close to the identity. These can be interpreted as averaging procedures with respect to the dynamics defined by effective uncoupled Hamiltonians. The resonant parts are treated with a different kind of unitary transfomations that are not close to the identity, and are adapted to the structure of the resonances. They can be interpreted as a renormalization of the uncoupled Hamiltonian, that yields an effective dressed Hamiltonian, around which a perturbation expansion can be developed. The combination of these two ingredients provides a strongly improved approximation technique.

First-order coherent THz optical sideband generation from asymmetric QW intersubband transitions

We have generated terahertz (THz) optical sidebands on a near-infrared probe beam by driving an excitonic intersubband resonance with THz electric fields. We use THz radiation polarized along the non-centro-symmetric axis of a quantum well system to generate a comb of sidebands ω sideband= ω NIR+ nω THz. In exploring the rich polarization of the process's power, and NIR frequency dependences we encounter both an efficient perturbative regime modeled by a X (2) non-linear susceptibility, and a non-perturbative regime which has not been previously explored in driven quantum systems.

Odd terahertz optical sidebands from asymmetric excitonic intersubband excitation

We have generated terahertz (THz) optical sidebands on a near-infrared probe beam by driving an excitonic intersubband resonance with THz electric fields. We use THz radiation polarized along the non-centrosymmetric axis of a quantum well system to generate a comb of sidebands ω sideband= ω NIR+ nω THz. In exploring the process's rich polarization, power, and NIR frequency dependences we encounter both an efficient perturbative regime modeled by a χ (2) non-linear susceptibility, and a non-perturbative regime which has not been previously explored in driven quantum systems.

Control of dynamical localization by an additional quantum degree of freedom

We identify a new parameter that controls the localization length in a driven quantum system. This parameter results from an additional quantum degree of freedom. The center-of-mass motion of a two-level ion stored in a Paul trap and interacting with a standing wave laser field exhibits this phenomenon. We also discuss the influence of spontaneous emission.