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Europhysics Letters (EPL)

Invariant Theory and Superadiabatic Bases

Xiao-Chun Gao, Jing‐Bo Xu, Donghui Xu

Published May 15, 1992• 2 citations
Publisher

Abstract

In this letter, the exact evolving state for a 2-state quantum system is found by making use of the Lewis-Riesenfeld invariant theory. Then, the exact evolving state for the system is expanded in terms of the eigenstates of the instantaneous Hamiltonian and the superadiabatic basis first introduced by Berry for the 2-state system is shown to be an approximation of the exact evolving state.

DOI
10.1209/0295-5075/19/2/002

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In this letter, the exact evolving state for a 2-state quantum system is found by making use of the Lewis-Riesenfeld invariant theory. Then, the exact evolving state for the system is expanded in terms of the eigenstates of the instantaneous Hamiltonian and the superadiabatic basis first introduced by Berry for the 2-state system is shown to be an approximation of the exact evolving state.
In this paper, the exact evolving state for a two-level quantum system is found by making use of the Lewis-Riesenfeld invariant theory. In principle, the correction to an arbitrary order in the adiabatic approximation parameter can be obtained from this exact evolving state. To the -th order in , it is shown that the exact evolving state reduces to the superadiabatic basis introduced by Berry recently. The Aharonov-Anandan phase and its adiabatic limit (Berry phase for the system) are also calculated.
Different methods have been recently put forward and implemented experimentally to inverse engineer the time-dependent Hamiltonian of a quantum system and accelerate slow adiabatic processes via nonadiabatic shortcuts. In the ``transitionless quantum driving'' proposed by Berry, shortcut Hamiltonians are designed so that the system follows exactly, in an arbitrarily short time, the approximate adiabatic path defined by a reference Hamiltonian. A different approach is based on first designing a Lewis-Riesenfeld invariant to carry the eigenstates of a Hamiltonian from specified initial to final configurations, again in an arbitrary time, and then constructing from the invariant the transient Hamiltonian that connects these boundary configurations. We show that the two approaches, apparently quite different in form and so far in results, are, in fact, strongly related and potentially equivalent, so that the inverse-engineering operations in one of them can be reinterpreted and understood in terms of the concepts and operations of the other one. We study, as explicit examples, expansions of time-dependent harmonic traps and the state preparation of two-level systems.
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.
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.
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.