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Chapman & Hall/CRC eBooks

Projection Operator Technique

Biman Bagchi

Published Oct 2, 2023• 0 citations
Publisher

Abstract

Overview Projection operator technique is simple in purpose. It allows us to separate the dynamical evolution of a quantity into two parts: known and unknown, the latter to be evaluated. However, this technique serves multiple purposes in TDSM. It is used to derive a reduced equation of motion for the relaxation of a dynamical variable. It is also used to derive expression for an experimentally observed TCF in terms of theoretically known TCFs. This technique was developed by Zwanzig, who was inspired by Kubo’s approach of “reduction in degrees of freedom” in an interacting many-body system. In the projection operator method, we divide the time evolution of a dynamical variable A ( t ) into two parts: (i) one through a relevant subspace that is closely related to A and varies on a similar time scale, and (ii) an orthogonal subspace, which could be fast and therefore can be averaged out to obtain a time-dependent friction kernel. The idea then is that the first contribution is known while the second may be described accurately by projecting it onto the relevant subspace [that is, (i)]. In the derivation of the reduced equation of motion, one starts with the Liouville equation and projects out irrelevant degrees of freedom to obtain, after specific algebraic manipulations, an equation of motion of the dynamical variable. The final relaxation equation has the form of the Langevin equation with microscopic expressions for the terms involved (frequency term, the friction, and the random force term). This final equation of motion is often termed relaxation equation or memory function equation. It finds wide use in the study of time-dependent statistical mechanics. In the present chapter we derive the memory function equation and discuss a few applications.

DOI
10.1201/9781003157601-14

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In a previous paper we showed how time-dependent projection operators may be employed to enable the Nakajima-Zwanzig projection-operator technique for deriving exact master equations to deal efficiently with two or more coupled classical or quantum systems, neither of which is reservoir like. We considered in detail the case where the relevant part of the classical-system probability-density function (PDF) or quantumsystem density operator (DO) is a product of the PDF's or DO's for the separate subsystems, and we applied the techniques developed to problems in quantum optics and in the kinetic theory of dilute nonideal gases. In this paper we make the time-dependent projection-operator approach useful for a greater variety of systems by allowing the relevant part of the PDF or DO to include correlations between two of the interacting subsystems. This extension allows us to describe well the dynamics of strongly interacting systems in a low degree of approximation while avoiding the use of infinite resummations. We derive exact generalized master equations in this manner for the same two cases as in our earlier work, namely a classical gas of $N$ molecules interacting via a two-body potential and a quantum-optical system of $N$ two-level atoms interacting with an electromagnetic field. In the former case, the relevant part of the PDF contains two-body correlations, and we obtain two exact coupled master equations for the singlet and doublet PDF's ${F}_{1}(t)$ and ${F}_{2}(t)$. In the latter case, the relevant part of the DO contains atom-field correlations, and the result is three exact coupled equations for the single-atom DO ${\ensuremath{\rho}}_{1}(t)$, the field DO $R(t)$, and the DO for one atom plus the field $\ensuremath{\chi}(t)$. From the exact equations we derive approximate equations by making simplifying assumptions. In the case of the gas we carry out a straightforward density expansion of the ${\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{F}}_{2}$ master equation and obtain a set of kinetic equations for dense gases, which have been derived previously by Frieman, Goldman, and Dorfman and which describe well effects due to the finite mean free path. In the quantum-optical case we derive kinetic equations for a single-mode laser by treating the $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\chi}}$ equation in the Born-Markoff approximation. The equations describe gain saturation and other effects on the field dynamics which are of infinite order in the atom-field coupling constant. It is shown that these equations reduce to previously derived laser master equations if atom-atom correlations can be neglected. Finally, we mention possible generalizations of the time-dependent projection-operator approach used in this paper and briefly discuss its application to other problems, including the kinetics of liquids, collisional line broadening, superradiance, and amplified spontaneous emission.

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We extend the Nakajima-Zwanzig projection operator technique to the determination of multitime correlation functions of open quantum systems. The correlation functions are expressed in terms of certain multitime homogeneous and inhomogeneous memory kernels for which suitable equations of motion are derived. We show that under the condition of finite memory times, these equations can be used to determine the memory kernels by employing an exact stochastic unraveling of the full system-environment dynamics. The approach thus allows us to combine exact stochastic methods, feasible for short times, with long-time master equation simulations. The applicability of the method is demonstrated by numerical simulations of two-dimensional spectra for a donor-acceptor model, and by comparison of the results with those obtained from the reduced hierarchy equations of motion. We further show that the formalism is also applicable to the time evolution of a periodically driven two-level system initially in equilibrium with its environment.

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Overview Projection operator technique is simple in purpose. It allows us to separate the dynamical evolution of a quantity into two parts: known and unknown, the latter to be evaluated. However, this technique serves multiple purposes in TDSM. It is used to derive a reduced equation of motion for the relaxation of a dynamical variable. It is also used to derive expression for an experimentally observed TCF in terms of theoretically known TCFs. This technique was developed by Zwanzig, who was inspired by Kubo’s approach of “reduction in degrees of freedom” in an interacting many-body system. In the projection operator method, we divide the time evolution of a dynamical variable A ( t ) into two parts: (i) one through a relevant subspace that is closely related to A and varies on a similar time scale, and (ii) an orthogonal subspace, which could be fast and therefore can be averaged out to obtain a time-dependent friction kernel. The idea then is that the first contribution is known while the second may be described accurately by projecting it onto the relevant subspace [that is, (i)]. In the derivation of the reduced equation of motion, one starts with the Liouville equation and projects out irrelevant degrees of freedom to obtain, after specific algebraic manipulations, an equation of motion of the dynamical variable. The final relaxation equation has the form of the Langevin equation with microscopic expressions for the terms involved (frequency term, the friction, and the random force term). This final equation of motion is often termed relaxation equation or memory function equation. It finds wide use in the study of time-dependent statistical mechanics. In the present chapter we derive the memory function equation and discuss a few applications.
Abstract This chapter is devoted to the foundations and to the basic mathematical structure of the non-Markovian quantum dynamics of open systems. It gives a survey of the Nakajima–Zwanzig projection operator methods with the help of which one derives so-called generalized master equations for the reduced system dynamics. In the non-Markovian regime, these master equations involve a retarded memory kernel, i.e. a time-convolution integral taken over the history of the reduced system. The chapter also describes an alternative method of particular relevance in many applications, which is based on a time-local quantum master equation, and which is known as the time-convolutionless projection operator method. This method serves as a starting point for a systematic expansion in terms of ordered cumulants and forms the basis for a non-Markovian stochastic unravelling of the reduced system dynamics.
The Mori-Zwanzig projection operator formalism is a powerful method for the derivation of mesoscopic and macroscopic theories based on known microscopic equations of motion. It has applications in a large number of areas including fluid mechanics, solid-state theory, spin relaxation theory, and particle physics. In its present form, however, the formalism cannot be directly applied to systems with time-dependent Hamiltonians. Such systems are relevant in many scenarios such as driven soft matter or nuclear magnetic resonance. In this article we derive a generalization of the present Mori-Zwanzig formalism that is able to treat also time-dependent Hamiltonians. The extended formalism can be applied to classical and quantum systems, close to and far from thermodynamic equilibrium, and even in the case of explicitly-time-dependent observables. Moreover, we develop a variety of approximation techniques that enhance the practical applicability of our formalism. Generalizations and approximations are developed for both equations of motion and correlation functions. Our formalism is demonstrated for the important case of spin relaxation in a time-dependent external magnetic field. The Bloch equations are derived together with microscopic expressions for the relaxation times.
A projection operator method in Heisenberg picture is developed. Corresponding to the equations for the reduced density matrix, there are two types of Langevin-like equations; time convolution (TC) and time convolutionless (TCL) equations. When the whole system is composed of relevant system and irrelevant system, unnecessary information is eliminated by projection operator. Explicit expansion formulae are obtained for the both types of equations. Relaxation operators and fluctuating forces are expressed in terms of certain cumulants. The method is illustrated by simple examples.
Reduced equations of motion (REM) are derived for the reduced density matrix of some of the degrees of freedom taking part in a scattering process. The derivation is made making use of the Zwanzig–Mori projection operator technique and utilizing a partial time ordering prescription (POP) which results in simple REM which contain no convolution in time. The present formulation is convenient for the direct calculation of reduced information on scattering events since it avoids the necessity of calculating the complete S matrix. The entire dynamical information relevant for the present description is expressed in terms of a hierarchy of correlation functions which in turn depend on the potential surface only in the region of interest. We further develop an exponential approximation for the reduced tetradic scattering S matrix in Liouville space, which forms a convenient framework for systematic approximations.
In a previous paper we showed how time-dependent projection operators may be employed to enable the Nakajima-Zwanzig projection-operator technique for deriving exact master equations to deal efficiently with two or more coupled classical or quantum systems, neither of which is reservoir like. We considered in detail the case where the relevant part of the classical-system probability-density function (PDF) or quantumsystem density operator (DO) is a product of the PDF's or DO's for the separate subsystems, and we applied the techniques developed to problems in quantum optics and in the kinetic theory of dilute nonideal gases. In this paper we make the time-dependent projection-operator approach useful for a greater variety of systems by allowing the relevant part of the PDF or DO to include correlations between two of the interacting subsystems. This extension allows us to describe well the dynamics of strongly interacting systems in a low degree of approximation while avoiding the use of infinite resummations. We derive exact generalized master equations in this manner for the same two cases as in our earlier work, namely a classical gas of NNN molecules interacting via a two-body potential and a quantum-optical system of NNN two-level atoms interacting with an electromagnetic field. In the former case, the relevant part of the PDF contains two-body correlations, and we obtain two exact coupled master equations for the singlet and doublet PDF's F1(t){F}_{1}(t)F1​(t) and F2(t){F}_{2}(t)F2​(t). In the latter case, the relevant part of the DO contains atom-field correlations, and the result is three exact coupled equations for the single-atom DO ${\ensuremath{\rho}}_{1}(t)$, the field DO R(t)R(t)R(t), and the DO for one atom plus the field $\ensuremath{\chi}(t)$. From the exact equations we derive approximate equations by making simplifying assumptions. In the case of the gas we carry out a straightforward density expansion of the ${\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{F}}_{2}$ master equation and obtain a set of kinetic equations for dense gases, which have been derived previously by Frieman, Goldman, and Dorfman and which describe well effects due to the finite mean free path. In the quantum-optical case we derive kinetic equations for a single-mode laser by treating the $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\chi}}$ equation in the Born-Markoff approximation. The equations describe gain saturation and other effects on the field dynamics which are of infinite order in the atom-field coupling constant. It is shown that these equations reduce to previously derived laser master equations if atom-atom correlations can be neglected. Finally, we mention possible generalizations of the time-dependent projection-operator approach used in this paper and briefly discuss its application to other problems, including the kinetics of liquids, collisional line broadening, superradiance, and amplified spontaneous emission.
We extend the Nakajima-Zwanzig projection operator technique to the determination of multitime correlation functions of open quantum systems. The correlation functions are expressed in terms of certain multitime homogeneous and inhomogeneous memory kernels for which suitable equations of motion are derived. We show that under the condition of finite memory times, these equations can be used to determine the memory kernels by employing an exact stochastic unraveling of the full system-environment dynamics. The approach thus allows us to combine exact stochastic methods, feasible for short times, with long-time master equation simulations. The applicability of the method is demonstrated by numerical simulations of two-dimensional spectra for a donor-acceptor model, and by comparison of the results with those obtained from the reduced hierarchy equations of motion. We further show that the formalism is also applicable to the time evolution of a periodically driven two-level system initially in equilibrium with its environment.