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Physical Review Letters

Antiadiabatic theorem for crossing levels

Francis E. Low

Published Nov 20, 1989• 5 citations
Publisher

Abstract

We consider in this note the time dependence of a system for which two levels (which are not in general decoupled) accidentally cross (or almost cross) for some value of the parameters of the Hamiltonian. We show that for a well-defined class of crossings or narrowly avoided crossings, in which the Hamiltonian changes slowly on its own scale, the system will behave oppositely to the usual adiabatic rule.

DOI
10.1103/physrevlett.63.2322

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We consider in this note the time dependence of a system for which two levels (which are not in general decoupled) accidentally cross (or almost cross) for some value of the parameters of the Hamiltonian. We show that for a well-defined class of crossings or narrowly avoided crossings, in which the Hamiltonian changes slowly on its own scale, the system will behave oppositely to the usual adiabatic rule.
The adiabatic theorem states that an initial eigenstate of a slowly varying Hamiltonian remains close to an instantaneous eigenstate of the Hamiltonian at a later time. We show that a perfunctory application of this statement is problematic if the change in eigenstate is significant, regardless of how closely the evolution satisfies the requirements of the adiabatic theorem. We also introduce an example of a two-level system with an exactly solvable evolution to demonstrate the inapplicability of the adiabatic approximation for a particular slowly varying Hamiltonian.
We numerically show the changing of eigenstates of the Hamiltonian as a function of a control parameter governing the strength of a non-integrable perturbation. The eigenstates change abruptly at avoided level crossing. After the avoided level crossing, they tend to restore their original form before the avoided level crossing, and exchange their distribution, especially for those which can he treated as isolated two-body collisions. But in general, the eigenstates have some changings after the avoided crossing, and the symmetries of the system are broken down locally at this kind of avoided level crossings.
The evolution mod Psi (t)) of a system with slowly-varying Hamiltonian H( Omega t) depends not only on the slowness parameter Omega but also on Planck's constant h(cross). For systems with two (or more) classically separated phase-space regions which are quantally connected by tunnelling, the curves of instantaneous energy levels display hyperbolic near-degeneracies rather than crossings. In such cases the limits Omega to 0 (h(cross) fixed and small) and h(cross) to 0 ( Omega fixed and small) lead to opposite behaviour of mod Psi (t)). As an illustration, the uniform semiclassical adiabatic behaviour ( Omega and h(cross) small, Omega /h(cross) arbitrary) is calculated exactly for a double-well potential for which one well gets shallower as the other gets deeper.
We explore the dynamics of mixed classical-quantum Hamiltonians, a class of systems that frequently appears in the study of physical systems characterized by both slow and fast modes. Traditionally such systems are studied within the adiabatic approximation and as a rule they display level crossings. We show that level crossings can have peculiar effects on the dynamics and, in the regime between the adiabatic and diabatic limits in both classical and quantum approaches, the full dynamics has a series of non-intuitive features.
In this paper we find the quantities that are adiabatic invariants of any desired order for a general slowly time-dependent Hamiltonian. In a preceding paper, we chose a quantity that was initially an adiabatic invariant to first order, and sought the conditions to be imposed upon the Hamiltonian so that the quantum mechanical adiabatic theorem would be valid to mth order. [We found that this occurs when the first (m − 1) time derivatives of the Hamiltonian at the initial and final time instants are equal to zero.] Here we look for a quantity that is an adiabatic invariant to mth order for any Hamiltonian that changes slowly in time, and that does not fulfill any special condition (its first time derivatives are not zero initially and finally).