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arXiv:chao-dyn/9805018 · The European Physical Journal D

WKB for a damped spin

P.A. Braun, Daniel Braun, F. Haake

Published Jan 1, 1998• 9 citations
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Abstract

The master equation for a damped spin well known from the theory of superradiance, is written as a finite-difference equation and solved by a WKB-like method. The propagator thus obtained looks like the van Vleck propagator of a certain classical Hamiltonian system with one degree of freedom. A new interpretation is provided of the temporal broadening of initially sharp probability distributions as the analogue of the spreading of the quantum mechanical wave packet.

DOI
10.1007/s100530050142

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The master equation for a damped spin well known from the theory of superradiance, is written as a finite-difference equation and solved by a WKB-like method. The propagator thus obtained looks like the van Vleck propagator of a certain classical Hamiltonian system with one degree of freedom. A new interpretation is provided of the temporal broadening of initially sharp probability distributions as the analogue of the spreading of the quantum mechanical wave packet.
This paper presents a general expression for the WKB approximation to the propagator corresponding to an arbitrary Hamiltonian operator H. For example, if the correspondence rule used to pass from the classical Hamiltonian Hc to H is such that it associates aPiQ j +(1−a)Q jPi to piq j, then the formula gives KWKB=KVVexp {( 1/2 −a)ℱT(∂2Hc/∂qi∂pi )(qc(t),pc(t),t) dt}, where KVV≡(2πih/)−n/2 (det M)1/2exp(iSc/h/) is Van Vleck’s well-known formula, Sc being the action functional evaluated at the classical path (qc,pc) and Mij ≡−∂2Sc/∂qa i∂qbj. More generally, the formula presented here applies to any system with n degrees of freedom described by a function f(x,t) whose time evolution is given by (H(x,k∂/∂x,t)+k∂/∂t) f(x,t)=0, regardless of the form of H. The Schrödinger equation of quantum mechanics and the Fokker–Planck equation of diffusion are obvious examples. Many examples are discussed. This generalizes results obtained in a previous publication [J. Math. Phys. 18, 786–90 (1977)].
We test the ability of semiclassical theory to describe quantitatively the revival of quantum wave packets-a long time phenomena-in the one dimensional quartic oscillator (a Kerr type Hamiltonian). Two semiclassical theories are considered: time-dependent WKB and Van Vleck propagation. We show that both approaches describe with impressive accuracy the autocorrelation function and wave function up to times longer than the revival time. Moreover, in the Van Vleck approach, we can show analytically that the range of agreement extends to arbitrary long times.
\n \nA well known description of superradiance from pointlike collections of many\natoms involves the dissipative motion of a large spin. The pertinent\n"superradiance master equation" allows for a formally exact solution which\nwe subject to a semiclassical evaluation. The clue is a saddle-point\napproximation for an inverse Laplace transform. All previous approximate\ntreatments, disparate as they may appear, are encompassed in our systematic\nformulation. A byproduct is a hitherto unknown rigorous relation between\ncoherences and probabilities. Our results allow for generalizations to spin\ndynamics with chaos in the classical limit. \n\n\n\n
A well known description of superradiance from pointlike collections of many atoms involves the dissipative motion of a large spin. The pertinent ``superradiance master equation'' allows for a formally exact solution which we subject to a semiclassical evaluation. The clue is a saddle-point approximation for an inverse Laplace transform. All previous approximate treatments, disparate as they may appear, are encompassed in our systematic formulation. A byproduct is a hitherto unknown rigorous relation between coherences and probabilities. Our results allow for generalizations to spin dynamics with chaos in the classical limit.
A system of spins coupled to a bath is a traditional setup in open quantum systems. Through Heisenberg's equation, the spin dynamics can be modeled by a set of first-order differential equations. Interpreting the terms as colored noise and non-Markovian damping, one can write them as quantummechanical Heisenberg-Langevin (HL) equations. These are notoriously difficult to solve because of the high dimensionality of the Hilbert space. Classical generalized Langevin equations, involving non-Markovian damping and colored noise, are well understood and can be treated numerically with relative ease. Thus, a classical ansatz can be made by substituting quantum expectation values with classical functions. This allows the application of standard methods developed for classical stochastic dynamical systems to tackle spin dynamics. However, this approach is uncontrolled and should be benchmarked against known quantum dynamics. In this investigation, a Hamiltonian for spin dynamics is modified to obtain a setup analogous to the Weisskopf-Wigner (WW) theory of spontaneous emission, enabling a comparison of the results. This will be compared for T = 0 and with a slight adaptation in the high-temperature limit.
In this work, a suitable Hamiltonian that describes the damped harmonic oscillator is constructed. Besides, the Hamilton-Jacobi equation for this dissipative system is written and the action function is obtained. Then, the system is quantized using two methods: the WKB approximation and the canonical quantization. In addition, evolution of the probability density is plotted at different times, and it is observed that the width of these densities decreases with time. The average energy of the system is also calculated and found to decrease with time.