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Cambridge University Press eBooks

Dynamics for a Correlated Initial State and Various Kinds of Contours in the Complex Time Plane

Giancarlo Calvanese Strinati

Published Feb 6, 2026• 0 citations
Publisher

Abstract

This chapter extends the treatment of the previous chapters to the case when the system is initially prepared in an ensemble average. This requires adding a “vertical track” to the oriented contour. Alternative formalisms are considered in this context, depending on the way the vertical track is dealt with. The relationship between transient phenomena and the adiabatic assumption is also considered.

DOI
10.1017/9781009411509.009

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This chapter extends the treatment of the previous chapters to the case when the system is initially prepared in an ensemble average. This requires adding a “vertical track” to the oriented contour. Alternative formalisms are considered in this context, depending on the way the vertical track is dealt with. The relationship between transient phenomena and the adiabatic assumption is also considered.
This chapter introduces the contour Schwinger–Keldysh method for time-dependent averages, in light of its relevance to nonequilibrium processes. A key feature of this approach is that it leaves open the possibility that no state of a system in the future can be identified with any of its states in the past. This method is here illustrated in detail with reference to time-dependent quantum averages, whereby for definiteness the system is initially prepared at the reference time t ₀ in a definite quantum state.
Similar to Chapter 20 of Part I, this chapter considers the treatment of a few topics, which are relevant to the general purposes of the book but whose inclusion in previous chapters would have diverted the discussion of the main topics of interest therein. Specifically, it considers the Dyson equations for the contour single-particle Green’s function in the Nambu representation, the relative strength of different frequency terms in the derivation of the time-dependent Gross–Pitaevskii equation, the detailed calculation of an integral occurring in the derivation of the time-dependent Ginzburg–Landau equation, the irrelevance of the reference time t ₀ for the convolutions entering the Kadanoff–Baym equations at equilibrium, and the average energy of the system expressed in terms of the lesser Green’s function even in nonequilibrium situations.
This chapter considers the initial preparation of the many-particle system, whose control is achieved before the reference time t ₀ when the time-dependent perturbation begins to act on the system. After t ₀, the system is let to evolve in time according to the full time-dependent Hamiltonian. The initial control can be either full or partial. Full control signifies that at t ₀ the system is prepared in a definite “pure” quantum state (like the ground state), while partial control signifies that initially the system is only known to be in a “mixture” of states with given probabilities, such that the information on the phases of the superposition is lost. These two cases are here treated separately.
Ensembles and expectation values Equations of motion provide predictions of time-evolving variables of a dynamical system, and thereby enable us to predict the changing behavior. These equations are characteristic of a particular type of system, say a classical harmonic oscillator or a two-state quantum system. To complete the description of behavior one must specify initial conditions that select, from the many possible solutions to the equations of motion, a particular realization. In the example of a classical point particle the equation is second order in time, and so we need two initial conditions, of position and velocity, to specify a particular solution. These two numbers, taken with the classical equation of motion, provide a complete description of the behavior of a specific harmonic oscillator. Similarly the two-state quantum system requires two initial values to complete its definition. The initial conditions are never defined with infinite precision, even for classical mechanics. More generally we deal with an ensemble, that is, a collection of similar dynamical systems (e.g. oscillators) that differ in their preparation. The individual cases share some common attributes, but not necessarily all. Thus we have only incomplete information about any single system. We might, for example, consider planetary motion in which we know only the mean energy and the orbital plane. The loci of positions of the planets then form ellipses that make up the ensemble. Although it is possible to prepare a single two-state quantum system in which the initial state is known (apart from an overall phase), often one deals with ensembles of such systems, each of which may differ in some uncontrollable property.
Making use of the inversion method, which is invented to discuss the characteristics of phase transition in a perturbative manner, a fundamental equation for nonequilibrium superconductivity is derived. In the course of the calculation, the generalized contour-ordered Green's-function formalism is used to treat the initial time correlation function and real-time Green's function in a unified manner. The result is a set of self-consistent integral equations and the Gorkov's equation is given a posteriori as one of these components describing the initial time correlation. By virtue of the contour-ordered Green's-function formalism, it is not necessary to employ any assumption to consider the effect of the initial time correlation on the system at subsequent time. Owing to this nature, our result helps in discussing transient behavior of the system induced by a supercritical current that appears in some kinds of superconducting device.