Remarks on the quantum adiabatic theorem

Mirjana Božić, R.J. Lombard, Z. Marić

Pierre Lochak, Claude Meunier

Kurt O. Friedrichs

Jenn‐Tai Hwang, Philip Pechukas

This paper is concerned with the problem of calculating amplitudes for nonadiabatic transitions induced by a time-dependent Hamiltonian, in the semiclassical limit h/→0, with emphasis on questions relevant to semiclassical theories of electronically inelastic scattering. For this problem the semiclassical limit is mathematically equivalent to the adiabatic limit, and the adiabatic theorem says that all these transition amplitudes vanish in the limit; the question is, what is the asymptotic form of the nonadiabatic amplitudes, as they go to zero? We consider Hamiltonia that are analytic matrix functions of time. We prove a generalization of the adiabatic theorem to the complex time plane; paradoxically, the adiabatic theorem in the complex plane gives us directly the nonadiabatic amplitudes along the real time axis. We derive Dykhne’s remarkable formula for the two-state case, which says that the limiting form of the transition amplitude depends only on the energy curves of the two states, not on the nonadiabatic coupling which is responsible for transition between them. We discuss the three-state problem at length and show that the obvious generalization of the Dykhne formula is sometimes true, sometimes false. To indicate the scope of methods based on the adiabatic theorem in the complex plane, we give an elementary proof of the semiclassical formula for above-barrier reflection of a one-dimensional particle.

Tosio Kato

N. Il’in, A. Aristova, Oleg Valentinovich Lychkovskiy

The evolution of a driven quantum system is said to be adiabatic whenever the state of the system stays close to an instantaneous eigenstate of its time-dependent Hamiltonian. The celebrated quantum adiabatic theorem ensures that such pure state adiabaticity can be maintained with arbitrary accuracy, provided one chooses a small enough driving rate. Here, we extend the notion of quantum adiabaticity to closed quantum systems initially prepared at finite temperature. In this case adiabaticity implies that the (mixed) state of the system stays close to a quasi-Gibbs state diagonal in the basis of the instantaneous eigenstates of the Hamiltonian. We prove a sufficient condition for the finite temperature adiabaticity. Remarkably, it turns out that the finite temperature adiabaticity can be more robust than the pure state adiabaticity with respect to increasing the system size. This can be the case for one-body systems with large Hilbert spaces, such as a particle in a large box, as well as for certain many-body systems. In particular, we present an example of a driven many-body system where, in the thermodynamic limit, the finite temperature adiabaticity is maintained, while the pure state adiabaticity breaks down. On the other hand, for generic many-body systems the scaling of the finite temperature adiabatic condition with the system size is exponential, analogously to pure state adiabatic conditions.