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Physical Review A

Fluctuations and correlations in quantum-optical systems: An alternative computational approach

Aaron S. Manka, E. J. D’Angelo, L. M. Narducci, Marlan O. Scully

Published May 1, 1993• 10 citations
Publisher

Abstract

We discuss a procedure for the calculation of operator correlation functions that arise frequently in quantum-optics problems. The starting point of our method is the tradiational quantum regression theorem, and its implementation requires only the solution of the master equation for the density operator of the system of interest. An attractive feature of this scheme of calculation is that it offers some distinctive advantages over the conventional Langevin-equations approach or other procedures that rely on the mapping of the density operator into appropriate phase-space distribution functions. Although the practical implementation of this method is likely to require numerical computations for virtually every model of physical interest, the procedure is sufficiently general to allow consideration of both field and atomic correlation functions, as well as multitime expectation values of mixed products of atomic and field operators. In this paper we provide a fairly detailed derivation of the formulas of interest, and illustrate our results with specific examples and tests of the procedure.

DOI
10.1103/physreva.47.4236

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We discuss a procedure for the calculation of operator correlation functions that arise frequently in quantum-optics problems. The starting point of our method is the tradiational quantum regression theorem, and its implementation requires only the solution of the master equation for the density operator of the system of interest. An attractive feature of this scheme of calculation is that it offers some distinctive advantages over the conventional Langevin-equations approach or other procedures that rely on the mapping of the density operator into appropriate phase-space distribution functions. Although the practical implementation of this method is likely to require numerical computations for virtually every model of physical interest, the procedure is sufficiently general to allow consideration of both field and atomic correlation functions, as well as multitime expectation values of mixed products of atomic and field operators. In this paper we provide a fairly detailed derivation of the formulas of interest, and illustrate our results with specific examples and tests of the procedure.
Beyond the conventional quantum regression theorem, a general formula for non-Markovian correlation functions of arbitrary system operators both in the time- and frequency-domain is given. Weapproach the problem by transforming the conventional time-non-local master equation into dispersed time-local equations-of-motion. The validity of our approximations is discussed and we find that the non-Markovian terms have to be included for short times. While calculations of the density matrix at short times suffer from the initial value problem, a correlation function has a well defined initial state. The resulting formula for the non-Markovian correlation function has a simple structure and is as convenient in its application as the conventional quantum regression theorem for the Markovian case. For illustrations, we apply our method to investigate the spectrum of the current fluctuations of interacting quantum dots contacted with two electrodes. The corresponding non- Markovian characteristics are demonstrated.
Beyond the conventional quantum regression theorem, a general formula for non-Markovian correlation functions of arbitrary system operators both in the time- and frequency-domain is given. We approach the problem by transforming the conventional time-non-local master equation into dispersed time-local equations-of-motion. The validity of our approximations is discussed and we find that the non-Markovian terms have to be included for short times. While calculations of the density matrix at short times suffer from the initial value problem, a correlation function has a well defined initial state. The resulting formula for the non-Markovian correlation function has a simple structure and is as convenient in its application as the conventional quantum regression theorem for the Markovian case. For illustrations, we apply our method to investigate the spectrum of the current fluctuations of interacting quantum dots contacted with two electrodes. The corresponding non-Markovian characteristics are demonstrated.
Using the recently developed phase-space techniques for the treatment of quantum-mechanical problems, we set up a procedure for calculating multitime-correlation functions in terms of the joint distribution functions. The correlation functions are then expressed simply as integrals over the associated phase space. Explicit expressions are given for these joint distribution functions, in terms of Green's functions of the ccc-number equations of motion for the phase-space equivalent of the density operator. Using these joint distribution functions, an exact regression theorem is rederived, and the connection with the multitime correspondence between classical and quantum stochastic processes is discussed.
Very recently there have been a number of papers discussing the applicability of stochastic methods as an alternative to the usual treatment of master equations in quantum mechanics.1-5 In this paper we present a wave function approach including a stochastic element to study the evolution of a small system when it is coupled to a reservoir. We show how fluctuations and dissipation originate from quantum jumps occuring randomly during the time evolution of the system, and we present a physical interpretation of the procedure. We demonstrate how this treatment may replace a wide class of master equations, and its use in problems with a large number of states N is discussed. Since the density matrix has N ×N elements, where a wave function is determined by only N variables, there is a computational advantage in using the wave function procedure. This is illustrated by calculations on cooling atoms in a laser field.
The author considers a dynamic system interacting with a special dissipative system composed of many harmonic oscillators. Using the Feynman disentangling theorem and a bath approximation he derives differential equations for correlation functions of dynamic operators (CFD) concerning the maximum time argument t which in general occurs in more than one operator. They must be solved with appropriate initial conditions at time t', the second in height time argument in the CFD. This procedure allows the successive calculation of CFD with an arbitrary time arrangement (e.g. CFD of three operators, where the central one carries the smallest time argument). The exact equations of motion for dynamic operators may also be written as Langevin equations. The author defines the corresponding fluctuation operators and derives compact expressions for their correlation functions (CFF). In general the fluctuation operators show no Gaussian behaviour. CFF of second order are delta -correlated and CFF of third order are only delta -correlated with respect to the two highest time arguments.
Using the recently discussed quantum dynamics in phase space, we derive a master equation, starting from the phase-space equivalent to the Schr\"odinger equation of motion for the density operator. Use is made of Zwanzig's projection-operator techniques and some explicit realizations of the projection operators are given. The master equation is then applied to show that the time-correlation functions, as defined in the text, satisfy an integral equation of the Volterra type. Next, a master equation for a system interacting with a large system is derived. As an illustration, we determine the lowest-order Born approximation and carry out a short-memory-approximation calculation for an oscillator coupled to a reservoir and for a two-level system interacting with an oscillator heat bath; we obtain equations of the Fokker-Planck type. Some physical implications of these equations are also discussed.