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arXiv:physics/0102074 · Physical Review A

Effective theory of systems coupled strongly to rapidly varying external sources

R. Huerta, José Wudka

Published May 9, 2001• 0 citations
arXivHTMLPDF

Abstract

We consider quantum systems that interact strongly with a rapidly varying environment and derive a Schr\"odinger-like equation that describes the time evolution of the average wave function. We show that the corresponding Hamiltonian can be taken to be Hermitian provided all states are rotated using an appropriate-unitary transformation. The formalism is applied to a variety of systems and is compared and contrasted with related results describing stochastic resonances.

DOI
10.1103/physreva.63.062104

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We consider quantum systems that interact strongly with a rapidly varying environment and derive a Schr"odinger-like equation that describes the time evolution of the average wave function. We show that the corresponding Hamiltonian can be taken to be Hermitian provided all states are rotated using an appropriate-unitary transformation. The formalism is applied to a variety of systems and is compared and contrasted with related results describing stochastic resonances.
It is shown that evolution of an open quantum system can be exactly described in terms of wave function which obeys Schrodinger equation with randomly varying parameters whose statistics is universally determined by separate dynamics of the system's environment. Corresponding stochastic evolution of the wave function is unitary on average, and this property implies optical theorem for inelastic scattering as demonstrated by the example of one-dimensional conducting channel with thermally fluctuating potential perturbation.
Abstract The work continues systematic applications of the method developed by Di Martino and Facchi (2015) for obtaining, for systems with variable boundary conditions, an equivalent Schrödinger equation with constant boundary conditions. From the resulting equation with rapidly varying terms, an effective equation is then derived for the averaged (slowly varying) wave functions. In this work, we obtain this effective equation for rapidly driven systems in the interaction picture (in first order in reverse frequency of external driving). This is the most suitable picture for systems with external driving. As an example, an extended two-state fast driven system is considered. As a second example, we consider the one-dimensional motion of a particle in a potential box with high-frequency oscillations of its width. Oscillations of width lead to additional terms in the effective equation proportional not only to the second derivative of the potential, but to the first derivative as well. In the case of width oscillations, compared to the oscillations of the box as a whole, the changes are even more significant. In our examples, phenomena such as slow averaged oscillations, resonance, the appearance of volcano potentials are observed, as is often the case in driven systems.
Explicitly time-dependent Hamiltonians are ubiquitous in applications of quantum theory. It is therefore necessary to solve the time-dependent Schrödinger equation directly. The system’s dynamics is associated with a unitary time-evolution operator (a propagator), formally given as an infinite Dyson series. Time-dependent observables are invariant under unitary time-dependent transformations, where it is sometimes useful to transform the time-evolution from the states into the corresponding operators. This is carried out in part (in full) by transforming to the interaction (Heisenberg) picture. The corresponding equations of motion for the time-dependent operators are introduced. For quadratic potential energy functions, the time evolution of quantum expectation values coincides with the corresponding classical dynamics. This is demonstrated and analyzed in detail for Gaussian wave packets and a coherent state. Finally, we derive exact and approximate expressions for time-dependent transition probabilities and transition rates between quantum states. The validity of time-dependent perturbation theory is analyzed by comparison to exact dynamics.
The Schrodinger equation with Gaussian dynamic fluctuating potentials is considered. This model gives a convenient example of an open quantum system coupled to a given external environment. The evolution equations for averaged correlators are derived and it is shown that a system coupled to a thermoequilibrium environment tends asymptotically with time to a thermodynamic density matrix. At high temperatures the quantum evolution equation for the density matrix can be approximated by the quasiclassical Fokker-Planck equation. The generalisation for multiparticle systems is also given. The results are illustrated by the particular examples of coupling with phonons and of two-level systems. Finally, it is shown how the problem can be reformulated in terms of continuous stochastic evolution in the space of psi functions and a corresponding functional formalism based on the continuous Fokker-Planck equation in the space of psi functions.
Quantum mechanics is essentially described in terms of complex quantities like wave functions. The interesting point is that phase and amplitude of the complex wave function are not independent of each other, but coupled by some kind of conservation law. This coupling exists in time-independent quantum mechanics and has a counterpart in its time-dependent form. It can be traced back to a reformulation of quantum mechanics in terms of nonlinear real Ermakov equations or equivalent complex nonlinear Riccati equations, where the quadratic term in the latter equation explains the origin of the phase-amplitude coupling. Since realistic physical systems are always in contact with some kind of environment this aspect is also taken into account. In this context, different approaches for describing open quantum systems, particularly effective ones, are discussed and compared. Certain kinds of nonlinear modifications of the Schrödinger equation are discussed as well as their interrelations and their relations to linear approaches via non-unitary transformations. The modifications of the aforementioned Ermakov and Riccati equations when environmental effects are included can be determined in the time-dependent case. From formal similarities conclusions can be drawn how the equations of time-independent quantum mechanics can be modified to also incluce the enviromental aspects.
Stochastic methods are ubiquitous to a variety of fields, ranging from physics to economics and mathematics. In many cases, in the investigation of natural processes, stochasticity arises every time one considers the dynamics of a system in contact with a somewhat bigger system, an environment with which it is considered in thermal equilibrium. Any small fluctuation of the environment has some random effect on the system. In physics, stochastic methods have been applied to the investigation of phase transitions, thermal and electrical noise, thermal relaxation, quantum information, Brownian motion and so on. In this review, we will focus on the so-called stochastic Schrödinger equation. This is useful as a starting point to investigate the dynamics of open quantum systems capable of exchanging energy and momentum with an external environment. We discuss in some detail the general derivation of a stochastic Schrödinger equation and some of its recent applications to spin thermal transport, thermal relaxation, and Bose-Einstein condensation. We thoroughly discuss the advantages of this formalism with respect to the more common approach in terms of the reduced density matrix. The applications discussed here constitute only a few examples of a much wider range of applicability.