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Soviet Physics Uspekhi

"Jarring" of a quantum system and the corresponding stimulated transitions

Александр Михайлович Дыхне, Gennady L. Yudin

Published Jul 31, 1978• 78 citations
Publisher

Abstract

A systematic theory of sudden perturbations is derived for quantum systems whose states are described both by wave functions (a pure ensemble) and by a quantum density operator (a mixed ensemble). A perturbation series is written in powers of the parameter ωτ, which is small when the perturbation is sudden; ω is the typical eigenvalue of the unperturbed system; and τ is the characteristic collision time. When the perturbation (t), taken at different times, commutes with itself, the theory yields a compact analytic expression for the probabilities for stimulated transitions for any value of Vτ /. The results of many cross-section calculations for atomic collision processes are discussed from a common standpoint: the processes are treated as jarring processes which stimulate transitions in the quantum system. If a momentum δp is rapidly transferred to the system in a collision, regardless of the physical nature of the jarring, the probabilities for the stimulated transitions are governed by the parameter N ~δpδR/ where δR is a measure of the uncertainty in the coordinates which is due to the relatively slow motions in the unperturbed system.

DOI
10.1070/pu1978v021n07abeh005663

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This paper treats the problem of basic characteristics of quantum transitions, stimulated by a sudden disturbance V(t). The proper ''shake'' theory for quantum systems is constructed by expanding the time-displacement operator S (t, t') in a power series in ..omega..tau (h..omega.. is the typical eigenvalue of the unperturbed hamiltonian, tau is the characteristic collision time). If (V(t), V(t'))equals 0 at any Vtau/h, the ''shake'' theory gives a compact expression for the transition probabilities. Various atomic collision processes are discussed from the ''shake'' point of view. If an impulse deltap is transferred to the quantum system during the short time (..omega..tau << 11), the transition probabilities are determined by the value of universal parameter N approximately deltap. deltaR/h (uncertainty deltaR results from the slow motion of the unperturbed system).

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A systematic theory of sudden perturbations is derived for quantum systems whose states are described both by wave functions (a pure ensemble) and by a quantum density operator (a mixed ensemble). A perturbation series is written in powers of the parameter ωτ, which is small when the perturbation is sudden; ω is the typical eigenvalue of the unperturbed system; and τ is the characteristic collision time. When the perturbation (t), taken at different times, commutes with itself, the theory yields a compact analytic expression for the probabilities for stimulated transitions for any value of Vτ /. The results of many cross-section calculations for atomic collision processes are discussed from a common standpoint: the processes are treated as jarring processes which stimulate transitions in the quantum system. If a momentum δp is rapidly transferred to the system in a collision, regardless of the physical nature of the jarring, the probabilities for the stimulated transitions are governed by the parameter N ~δpδR/ where δR is a measure of the uncertainty in the coordinates which is due to the relatively slow motions in the unperturbed system.
This paper treats the problem of basic characteristics of quantum transitions, stimulated by a sudden disturbance V(t). The proper ''shake'' theory for quantum systems is constructed by expanding the time-displacement operator S (t, t') in a power series in ..omega..tau (h..omega.. is the typical eigenvalue of the unperturbed hamiltonian, tau is the characteristic collision time). If (V(t), V(t'))equals 0 at any Vtau/h, the ''shake'' theory gives a compact expression for the transition probabilities. Various atomic collision processes are discussed from the ''shake'' point of view. If an impulse deltap is transferred to the quantum system during the short time (..omega..tau << 11), the transition probabilities are determined by the value of universal parameter N approximately deltap. deltaR/h (uncertainty deltaR results from the slow motion of the unperturbed system).
The orders of the perturbation approximation to the Schroedinger-picture time evolution operator in powers of the presumed-small perturbation part of the Hamiltonian operator are developed by the iteration of an identity; the possibly confusing switch to the interaction picture isn't needed. Those time-evolution operator approximations are sandwiched between two orthogonal, normalized eigenstates of the unperturbed part of the Hamiltonian operator to produce the transition amplitude approximations, whose calculation is reduced to quadrature when every occurrence of the perturbation part of the Hamiltonian operator in them is expanded in the unperturbed basis. That expansion also reveals the time-dependent parts of those approximations to be multiply nested integrals which in the long-time limit approach simple products of non-singular inverses (principle value plus delta function) of differences of unperturbed energy eigenvalues. Closely related to the long-time limits of transition amplitudes are the long-time averages of transition rates. When the unperturbed basis is that of free-particle states, sums over relevant final states of transition rates from an initial state, divided by the initial state's particle flux, produce cross sections. Here the perturbation approximations to generic quantum transition rates are parlayed to the corresponding approximations to differential cross sections for nonrelativistic-particle potential scattering.
We now consider systems for which the Hamiltonian contains time-dependent interaction terms. In such cases we cannot reduce the time-dependent Schro¨dinger equation to an eigenvalue equation. However, if the time-dependent terms can be regarded as small perturbations, we can develop a form of perturbation theory that can be used to calculate the effects of time-dependent terms. Consider a system for which the Hamiltonian may be written as the sum of a dominant unperturbed part Hˆ0 and a small perturbation Hˆ ′ as Hˆ = Hˆ0 + λHˆ ′(t) , (10.1.1) where the unperturbed Hamiltonian Hˆ0 is independent of time, whereas the perturbation Hˆ ′ may depend on time. The parameter 0 < λ < 1 has been introduced to keep track of various orders of approximations as in the case of time-independent perturbation theory discussed in Chapter 8. The basic idea is that a small perturbation to the Hamiltonian will produce a small change in the wave function. For a time-dependent perturbation, this change means that a system, initially in a particular eigenstate of Hˆ0, will find itself in a time-dependent admixture of the eigenstates. Thus the time-dependent perturbation causes the system to undergo transition to different eigenstates of Hˆ0. So for time-dependent perturbations our interest lies in calculating the probabilities for transition to different states.
Perturbation theory for the interaction of light with a binary collisional system is applied to the calculation of line profiles, spontaneous emission rates and absorption or stimulated emission coefficients. For the usual radiation fields are perturber densities, this approximation is valid at all detunings Delta omega except the very narrow resonance region. Appropriate expansions of the total binary system wavefunctions give the emission and absorption profiles or rates, for atom-electron systems as well as for atom-atom systems, from the impact regime at small Delta omega up to the Born-Oppenheimer regime at very large Delta omega in collision-induced radiative transitions, the transitions being asymptotically allowed or forbidden. Emphasis is given to the proper averages and sums over initial and final states of the binary system which give rise to non-adiabatic effects. One shows that it is of crucial importance to include all dynamical effects during one collisional event before undertaking any calculation at small Delta omega where simultaneous collisions may occur.
If a particle which can decay spontaneously is subjected to collisions with other particles the transition probability may be changed. The calculation of such an effect by the usual time-dependent perturbation methods is complicated by the circumstance that energy can be conserved in some intermediate states, because of the possibility of elastic scattering. The perturbation calculation is here modified in such a way that the resulting expression for the whole transition probability does not contain any divergences or spurious effects. By suitable approximations the expression is separated into two parts, one of which gives the probability of elastic scattering followed by a spontaneous transition, and the other the true stimulated transition probability which causes a change in the lifetime.
A small parameter is used to identify the orders of perturbation. The unperturbed Hamiltonian is assumed time-independent and can be diagonalized. The interaction of an electron with an applied electromagnetic field is governed by the gauge-invariant Hamiltonian. If both the initial and final energy levels are discretized, the delta function indicates that the transition happens only when the energy difference between the two levels matches the frequency of the perturbation. The delta function suggests that the absorption or stimulated emission happens only when the energy difference between the final and initial states matches precisely the frequency of the applied electromagnetic wave. The quantum transition induced by a monochromatic perturbation is determined by the matrix element between the initial and final states.