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Journal of Physics A Mathematical and General

Superadiabatic tracking of quantum evolution

R Lim, Michael Berry

Published Jul 21, 1991• 107 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/24/14/014

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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.
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.
Abstract The way in which the transition amplitude to an initially unoccupied state increases to its exponentially small final value is studied in detail in the adiabatic approximation, for a 2-state quantum system. By transforming to a series of superadiabatic bases, clinging ever closer to the exact evolving state, it is shown that transition histories renormalize onto a universal one, in which the amplitude grows to its final value as an error function (rather than via large oscillations as in the ordinary adiabatic basis). The time for the universal transition is of order √ (ћ/δ) where δ is the small adiabatic (slowness) parameter. In perturbation theory the pre-exponential factor of the final amplitude renormalizes superadiabatically from the incorrect value ⅓π (for the ordinary adiabatic basis) to the correct value unity. The various histories could be observed in spin experiments.
For time-dependent two-state quantum systems, the transition probability is exponentially small in the adiabatic parameter in , with the exponent determined by a transition point t c in the complex time plane. The authors study the in -independent prefactors associated with different sorts of transition point (which need not correspond to complex degeneracies of the adiabatic energy). Unlike previous approaches the method they use does not make use of special functions. It consists of applying first-order perturbation theory to the Schrodinger equation obtained by transforming to a series of 'superadiabatic' bases clinging ever more closely to the evolving state. If the original matrix elements share a leading singularity (t-t c ) r , and their fractional deviation from this is (t-t c ) s , the prefactor is 4 sin 2 ( pi s / 2(2 r +s+2) ). This is universal in the sense of being invariant under time reparametrization and quantum changes of frame.
A generalized adiabatic approximation is formulated for a two-state dissipative Hamiltonian which is valid beyond weak dissipation regimes. The history of the adiabatic passage is described by superadiabatic bases as in the nondissipative regime. The topology of the eigenvalue surfaces shows that the population transfer requires, in general, a strong coupling with respect to the dissipation rate. We present, furthermore, an extension of the Davis-Dykhne-Pechukas formula to the dissipative regime using the formalism of Stokes lines. Processes of population transfer by an external frequency-chirped pulse-shaped field are given as examples.
The adiabatic theorem in quantum mechanics implies that if a system is in a discrete eigenstate of a Hamiltonian and the Hamiltonian evolves in time arbitrarily slowly, the system will remain in the corresponding eigenstate of the evolved Hamiltonian. Understanding corrections to the adiabatic result that arise when the evolution of the Hamiltonian is slow, but not arbitrarily slow, has become increasingly important, especially since adiabatic evolution has been proposed as a method of state preparation in quantum computing. This paper identifies two regimes, an adiabatic regime in which corrections are generically small and can depend on details of the evolution throughout the path and a hyperadiabatic regime in which the error is given by a form similar to an asymptotic expansion in the inverse of the evolution time with the coefficients depending principally on the behavior at the endpoints. However, the error in this hyperadiabatic regime is neither given by a true asymptotic series nor solely dependent on the endpoints: The coefficients combine the contributions from both endpoints, with relative phase factors that depend on the average spectral gaps along the trajectory, multiplied by the evolution time. The central result of this paper is to identify a quantity, referred to as the typical error, which is obtained by appropriately averaging the error over evolution times that are small compared to the evolution time itself. This typical error is characterized by an asymptotic series and depends solely on the endpoints of the evolution, remaining independent of the details of the intermediate evolution.
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.