A nonlinear quantum adiabatic approximation

Abstract

This paper is devoted to a generalisation of the quantum adiabatic theorem to a nonlinear setting. We consider a Hamiltonian operator which depends on the time variable and on a finite number of parameters and acts on a separable Hilbert space of which we select a fixed basis. We study an evolution equation in which this Hamiltonian acts on the unknown vector, while depending on coordinates of the unknown vector in the selected basis, thus making the equation nonlinear. We prove existence of solutions to this equation and consider their asymptotics in the adiabatic regime, i.e. when the Hamiltonian is slowly varying in time. Under natural spectral hypotheses, we prove the existence of normalised time dependent vectors depending on the Hamiltonian only, called instantaneous nonlinear eigenvectors, and show the existence of solutions which remain close to these vectors, up to a rapidly oscillating phase, in the adiabatic regime. We first investigate the case of bounded operators and then exhibit a set of spectral assumptions under which the result extends to unbounded Hamiltonians.

Parallel transport dynamics for mixed quantum states with applications to time-dependent density functional theory

“…Using the commutator type error bound, in the near adiabatic regime when the a priori estimate is available from the quantum adiabatic theorem, our new result shows the PT dynamics gains one extra order of accuracy in terms of the singularly perturbed parameter than the Schrödinger dynamics, which reproduces the previous result [1]. Recently, the quantum adiabatic theorem has been extended to certain weakly nonlinear systems [8,10]. Our commutator type error analysis can be directly combined with such analysis leading to results comparable to that of [1] in the weakly nonlinear regime.…”

“…The adiabatic theorem can also be generalized to certain linear systems without a gap condition [3,35], and for some weakly nonlinear systems [8,10]. A detailed discussion of the technical conditions for the adiabatic approximation is beyond the scope of this paper.…”