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arXiv:quant-ph/9908018 · Physical Review A

Nonadiabatic transitions in multilevel systems

Michael Wilkinson, Michael A. Morgan

Published May 16, 2000• 17 citations
arXivHTMLPDF

Abstract

In a quantum system with a smoothly and slowly varying Hamiltonian, which approaches a constant operator at times $\stackrel{\ensuremath{\rightarrow}}{t}\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty},$ the transition probabilities between adiabatic states are exponentially small. They are characterized by an exponent that depends on a phase integral along a path around a set of branch points connecting the energy-level surfaces in complex time. Only certain sequences of branch points contribute. We propose that these sequences are determined by a topological rule involving the Stokes lines attached to the branch points. Our hypothesis is supported by theoretical arguments and results of numerical experiments.

DOI
10.1103/physreva.61.062104

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In a quantum system with a smoothly and slowly varying Hamiltonian, which approaches a constant operator at times t\ensuremath→\ifmmode±\else\textpm\fi\ensuremath∞,\stackrel{\ensuremath{\rightarrow}}{t}\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty},t\ensuremath→​\ifmmode±\else\textpm\fi\ensuremath∞, the transition probabilities between adiabatic states are exponentially small. They are characterized by an exponent that depends on a phase integral along a path around a set of branch points connecting the energy-level surfaces in complex time. Only certain sequences of branch points contribute. We propose that these sequences are determined by a topological rule involving the Stokes lines attached to the branch points. Our hypothesis is supported by theoretical arguments and results of numerical experiments.
A two-state quantum system whose Hamiltonian varies adiabatically has a weak probability of making transitions between its instantaneous energy eigenstates. If the system is started out in an eigenstate and its evolution tracked using suitably-defined 'superadiabatic' states, the transition probability exhibits the error-function growth characteristic of the birth of subdominant exponentials across Stokes lines. Here we study model Hamiltonians associated with pairs of Stokes lines in the complex time-plane. The overlapping of Stokes crossing phenomena influences the final transition probability and also leads to interference effects in the transition history. These effects can be adjusted by turning parameters in the models.
The adiabatic quantum evolution of a two-state system without energy-level crossings is an example of the Stokes phenomenon. In the latter, a small (subdominant) exponential is an asymptotic expansion appears when a Stokes line is crossed; truncating the dominant asymptotic series at its least term causes the multiplier of the subdominant term to rise in a smooth, compact and universal manner across the Stokes line. In quantum evolution this corresponds to a smooth transition, universal in form, between 'superadiabatic' basis states (high-order WKB approximate solutions of the time-dependent Schrodinger equation). The authors give a numerical demonstration of this previously predicted universality by constructing, for two Hamiltonians, the superadiabatic quantum bases asymptotic to the actual evolving state. Universality when a Stokes line is crossed is seen in the changing probability that the system makes a transition away from the superadiabatic state, and occurs at that order of superadiabatic approximation corresponding to truncating the asymptotic series at its least term.
We consider the transition probability for two-level quantum-mechanical systems in the adiabatic limit when the Hamiltonian is analytic. We give a general formula for the leading term of the transition probability when it is governed by N complex eigenvalue crossings. This leading term is equal to a decreasing exponential times an oscillating function of the adiabaticity parameter. The oscillating function comes from an interference phenomenon between the contributions from each complex eigenvalue crossing, and when N=1, it reduces to the geometric prefactor recently studied.
When a time-dependent perturbation acts on a quantum system that is initially in the nondegenerate ground state ∣0> of an unperturbed Hamiltonian H(0), the wave function acquires excited-state components ∣k> with coefficients c(k)(t) exp(-iE(k)t/ℏ), where E(k) denotes the energy of the unperturbed state ∣k>. It is well known that each coefficient c(k)(t) separates into an adiabatic term a(k)(t) that reflects the adjustment of the ground state to the perturbation--without actual transitions--and a nonadiabatic term b(k)(t) that yields the probability amplitude for a transition to the excited state. In this work, we prove that the energy at any time t also separates completely into adiabatic and nonadiabatic components, after accounting for the secular and normalization terms that appear in the solution of the time-dependent Schrödinger equation via Dirac's method of variation of constants. This result is derived explicitly through third order in the perturbation. We prove that the cross-terms between the adiabatic and nonadiabatic parts of c(k)(t) vanish, when the energy at time t is determined as an expectation value. The adiabatic term in the energy is identical to the total energy obtained from static perturbation theory, for a system exposed to the instantaneous perturbation λH'(t). The nonadiabatic term is a sum over excited states ∣k> of the transition probability multiplied by the transition energy. By evaluating the probabilities of transition to the excited eigenstates ∣k'(t)> of the instantaneous Hamiltonian H(t), we provide a physically transparent explanation of the result for E(t). To lowest order in the perturbation parameter λ, the probability of finding the system in state ∣k'(t)> is given by λ(2) ∣b(k)(t)∣(2). At third order, the transition probability depends on a second-order transition coefficient, derived in this work. We indicate expected differences between the results for transition probabilities obtained from this work and from Fermi's golden rule.
The passage through a critical point of a many-body quantum system leads to abundant nonadiabatic excitations. Here, we explore a regime, in which the critical point is not crossed although the system is passing slowly very close to it. We show that the leading exponent for the excitation probability can then be obtained by standard arguments of the Dykhne formula, but the exponential prefactor is no longer simple and behaves as a power law on the characteristic transition rate. We derive this prefactor for the nonlinear Landau-Zener model by adjusting Dykhne's approach. Then, we introduce an exactly solvable model of the transition near a critical point in the Stark ladder. We derive the number of excitations for it without approximations and find qualitatively similar results for the excitation scaling.
We develop a theoretical description of non-Hermitian time evolution that accounts for the breakdown of the adiabatic theorem. We obtain closed-form expressions for the time-dependent state amplitudes, involving the complex eigenenergies as well as interband Berry connections calculated using basis sets from appropriately chosen Schur decompositions. Using a two-level system as an example, we show that our theory accurately captures the phenomenon of ``sudden transitions,'' where the system state abruptly jumps from one eigenstate to another.