Controllability of a quantum particle in a moving potential well

Abstract

We consider a nonrelativistic charged particle in a 1D moving potential well. This quantum system is subject to a control, which is the acceleration of the well. It is represented by a wave function solution of a Schrödinger equation, the position of the well together with its velocity. We prove the following controllability result for this bilinear control system: given 0 close enough to an eigenstate and f close enough to another eigenstate, the wave function can be moved exactly from 0 to f in finite time. Moreover, we can control the position and the velocity of the well. Our proof uses moment theory, a Nash-Moser implicit function theorem, the return method and expansion to the second order.

Implicit Lyapunov control of finite dimensional Schrödinger equations

“…A test case to be looked is the control of a quantum particle in a moving potential well. It is proved in [20] that the first order approximation for this system is not controllable and in [3,4] that the nonlinear system is locally controllable around any eigenstate (we are in the same situation as in the 5-level system of the last section). It seems that, for such a system, the assumptions of Theorem 3 are fulfilled.…”

Limitations on the control of Schrödinger equations

“…to the space of controls; controllability may be recovered by working in a different (higher-regularity) state space. This phenomenon occurs in two recent papers by Beauchard and Coron [8,9]. The bilinear control problem considered there falls in the scope of the noncontrollabilty result by Turinici if the state space is chosen to be H 2 .…”

“…It is also interesting to compare Theorem 6 (in the linear case f ≡ 0) to the results in [8,9], which show that (local) controllability is recovered if the quadratic potential is replaced with an "infinite" confining potential ("particle in a box"). This illustrates the subtle dependence of the controllability properties on the external potential.…”

“…Results on distributed and/or boundary control for linear Schrödinger equations are the subjects of references [15, 25, 27-29, 33, 34, 36]; local exact controllability for the linear Schrödinger equation with bilinear control is shown in [8,9]. A small-data result on distributed exact control for the nonlinear Schrödinger equation was presented in [20] (for completeness, this result is listed in Sect.…”

“…In the sequel we state two conditions each one of which implies the impossibility of exact controllability for (9).…”