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Physical Review A

Effective action for a nonadiabatic quantum process

Aurel Bulgac

Published Jun 1, 1988• 23 citations
Publisher

Abstract

A sum rule for the so-called quantum phases is established, without assuming adiabatic evolution. By considering two interacting quantum systems, conventionally designated as a boson and a fermionic one, we evaluate the effective bosonic action after integrating out the fermionic variables. This effective action contains a nontrivial topological part, which has the same origin as the quantum adiabatic phase. However, the derivation does not rely on any adiabatic assumption.

DOI
10.1103/physreva.37.4084

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A sum rule for the so-called quantum phases is established, without assuming adiabatic evolution. By considering two interacting quantum systems, conventionally designated as a boson and a fermionic one, we evaluate the effective bosonic action after integrating out the fermionic variables. This effective action contains a nontrivial topological part, which has the same origin as the quantum adiabatic phase. However, the derivation does not rely on any adiabatic assumption.
A dynamical meaning is given for the phase $\ensuremath{\Gamma}$ recently found in a quantum adiabatic process. A noncanonical system, specifically a spin system coupled with a proper intrinsic system, is considered by use of a path integral in a SU (2) coherent-state representation. It is shown that $\ensuremath{\Gamma}$ appears as an addition of topological nature to the conventional action function, which leads to a novel form of the semiclassical quantization rule including the phase $\ensuremath{\Gamma}$.
The effective theory of an open boson or fermion system is studied, which evolves out of equilibrium with time-dependent Hamiltonian H^(t)\hat{H}(t)H^(t). A measure of nonequilibrium temperature for the open system evolving from an equilibrium is proposed as the time-averaged energy expectation value T(t)=Ti(<H^(t)>Ψˉ/<I^(t)>Ψˉ)T(t) = T_i(\bar{< \hat{H} (t) >_Ψ}/ \bar{< \hat{I} (t) >_Ψ})T(t)=Ti​(<H^(t)>Ψ​ˉ​/<I^(t)>Ψ​ˉ​), where I^(t)\hat{I} (t)I^(t), the action operator, satisfies the quantum Liouville-von Neumann equation and determines the true density operator. It recovers the result for the adiabatic (quasi-equilibrium) and nonadiabatic (out of equilibrium) evolution from one static Hamiltonian to another and takes into account the particle production due to the intermediate processes.
A major difficulty in comparing quantum and classical behavior resides in the struc- tural differences between the corresponding mathematical languages. The Heisenberg equations of motion are operator equations only formally identical to the classical equations of motion. By taking the expectation of these equations, the well-known Ehrenfest theorem provides identities which, however, are not a closed system of equations which allows to evaluate the time evolution of the system. The formalism of the effective action seems to offer a possibility of comparing quantum and classical evolutions in a system- atic and logically consistent way by naturally providing approximation schemes for the expectations of the coordinates which at the zeroth order coincide with the classical evolution [1]. The effective action formalism leads to equations of motion which differ from the classical equations by the addition of terms nonlocal in the time variable. This means that for these equations an initial value problem is not meaningful and they have to be interpreted in an appropriate way. Here we analyze situations in which the nonlocal terms can be reasonably approximated by local ones, so that the quantum corrections do not modify the locality of classical equations. In the simplest approximation, the effective Lagrangian differs from the corresponding classical one by a renormalization of both the potential- and the kinetic-energy terms. We shall not discuss the causal formalism used, for example, in refs. [2-4], as in the approximation considered this would lead to the same local equations. The present contribution describes the beginning of a systematic study of semiclassical evolutions using the effective action formalism. In the first part, after introducing the formalism of the effective action and its expansion in powers of hbar (loop expansion) in the context of quantum mechanics, we concentrate on the structure of the first-order corrections in li. These corrections are evaluated to the second order in the derivative expansion [5], by two different methods. The first is based on a Euclidean approach [6], the second one on an adiabatic approximation in evaluating functional determinants. In the second part of the article we put the formalism at work, choosing as our case study a two-dimensional (2D) anharmonic oscillator of the kind considered in molecular physics. The results of the simulations show that by increasing lithe effective dynamics tends to regularize the classical motion and becomes qualitatively very similar to the quantum evolution provided the energy is sufficiently small. The evaluation of the effective dynamics in more general cases will be presented in a forthcoming paper.
We investigate a bosonic Josephson junction by using the path-integral formalism with relative phase and population imbalance as dynamical variables. We derive an effective only-phase action performing functional integration over the population imbalance. We then analyze the quantum effective only-phase action, which formally contains all the quantum corrections. To the second order in the derivative expansion and to the lowest order in $\ensuremath{\hbar}$, we obtain the quantum correction to the Josephson frequency of oscillation. Finally, the same quantum correction is found by adopting an alternative approach. Our predictions are a useful theoretical tool for experiments with atomic or superconducting Josephson junctions.