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Bopp operators and phase-space spin dynamics: Application to rotational quantum brownian motion

David Zueco, Iván Calvo

Published Nov 20, 2006• 17 citations
OpenAlex

Abstract

As already known for nonrelativistic spinless particles, Bopp operators give an elegant and simple way to compute the dynamics of quasiprobability distributions in the phase space formulation of Quantum Mechanics. In this work, we present a generalization of Bopp operators for spins and apply our results to the case of open spin systems. This approach allows to take the classical limit in a transparent way, recovering the corresponding Fokker-Planck equation.

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Bopp operators and phase-space spin dynamics: application to rotational quantum Brownian motion

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As already known for nonrelativistic spinless particles, Bopp operators give an elegant and simple way to compute the dynamics of quasiprobability distributions in the phase space formulation of Quantum Mechanics. In this work, we present a generalization of Bopp operators for spins and apply our results to the case of open spin systems. This approach allows to take the classical limit in a transparent way, recovering the corresponding Fokker-Planck equation.

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CLASSICAL DYNAMICS FOR LINEAR SYSTEMS: THE CASE OF QUANTUM BROWNIAN MOTION

J. R. Anglin, SALMAN HABIB

The dynamics of linear quantum systems is classical in the Wigner representation, yet linear problems are often analyzed using such general techniques as influence functionals and Bogoliubov transformations. In fact, the classical equations of motion provide a simpler and more intuitive formalism for these systems. As an important example, we show that quantum Brownian dynamics in the independent oscillator model is described directly and completely by a c-number Langevin equation. The corresponding Fokker-Planck equation is always local in time, regardless of the environmental spectrum.

arXiv:quant-ph/9507011•Modern Physics Letters A•Oct 30, 1996•14 citations
As already known for nonrelativistic spinless particles, Bopp operators give an elegant and simple way to compute the dynamics of quasiprobability distributions in the phase space formulation of Quantum Mechanics. In this work, we present a generalization of Bopp operators for spins and apply our results to the case of open spin systems. This approach allows to take the classical limit in a transparent way, recovering the corresponding Fokker-Planck equation.
We propose a Wigner transformation of the density matrix and operators of a system with bounded coordinates and discrete momentum spectra (typically, a particle in a box) and prove that all requirements for such a transformation are satisfied. The phase-space mapping adopted here is employed to compute the quasiprobability distributions associated with the pure states of a quantum rotator and the image of the angular momentum operator. The mapping of the dynamics of these objects, both in the reversible and irreversible frame, is constructed as well, and the conditions for the existence of a classical limit in the form of a Fokker-Planck equation of motion are investigated.
We first point out that nonrelativistic quantum mechanics formulated at imaginary time s = it can formally be recognized as the Fokker-Planck description of a classical Brownian motion, which occurs (in general) in a medium where particles are absorbed and emitted. We next offer a new formulation of quantum dynamics, which is basically the Langevin treatment of that Brownian motion. More explicitly, we derive a noise average representation of the transition probability W ( X", s"|X', s' ) in terms of the solutions to a Langevin equation with a Gaussian white noise. Upon analytic continuation back to real time W ( X", s"|X', s' ) becomes the propagator of the original Schrödinger equation, namely we have ⟨ X", t"|X', t' ⟩ = W ( X", it"|X', it' ). The main advantage of this approach is to permit a systematic application to quantum-dynamical problems of the mathematical techniques developed in the theory of classical stochastic processes, even when vector potentials and/or time-dependent potentials are present.
The theory of Brownian motion of a quantum oscillator is developed. The Brownian motion is described by a model Hamiltonian which is taken to be the one describing the interaction between this oscillator and a reservoir. Use is made of the master equation recently derived by the author, to obtain the equation of motion for the various reduced phase-space distribution functions that are obtained by mapping the density operator onto ccc-number functions. The equations of motion for the reduced phase-space distribution functions are found to be of the Fokker-Planck type. On transforming the Fokker-Planck equation to real variables, it is found to have the same form as the Fokker-Planck equation obtained by Wang and Uhlenbeck to describe the Brownian motion of a classical oscillator. The Fokker-Planck equation is solved for the conditional probability (Green's function) which is found to be in the form of a two-dimensional Gaussian distribution. This solution is then used to obtain various time-dependent quantum statistical properties of the oscillator. Next, the entropy for a quantum oscillator undergoing Brownian motion is calculated and we show that this system approaches equilibrium as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. Finally we show that in the weak-coupling limit the Fokker-Planck equation reduces to the one obtained by making the usual rotating-wave approximation.
It has long been recognized that the dynamics of linear quantum systems is classical in the Wigner representation. Yet many conceptually important linear problems are typically analyzed using such generally applicable techniques as influence functionals and Bogoliubov transformations. In this Letter we point out that the classical equations of motion provide a simpler and more intuitive formalism for linear quantum systems. We examine the important problem of Brownian motion in the independent oscillator model, and show that the quantum dynamics is described directly and completely by a c-number Langevin equation. We are also able to apply recent insights into quantum Brownian motion to show that the classical Fokker-Planck equation is always local in time, regardless of the spectral density of the environment.
The dynamics of linear quantum systems is classical in the Wigner representation, yet linear problems are often analyzed using such general techniques as influence functionals and Bogoliubov transformations. In fact, the classical equations of motion provide a simpler and more intuitive formalism for these systems. As an important example, we show that quantum Brownian dynamics in the independent oscillator model is described directly and completely by a c-number Langevin equation. The corresponding Fokker-Planck equation is always local in time, regardless of the environmental spectrum.