Controlling the Quantum State with a time varying potential

Abstract

The problem of controlling the quantum state of a system is investigated using a time varying potential. As a concrete example we study the problem of a particle in a box with a periodically oscillating infinite square-well potential, from which we obtain results that can be applied to systems with periodically oscillating boundary conditions. We derive an analytic expression for the frequencies of resonance between states, and against standard intuition, we show how to use this behavior to control the quantum state of the system at will. In particular, we offer as an example the transition from the ground state to the first excited state of the square well potential. At first sight, it may be counter intuitive that we can control the state of such a quantum Hamiltonian, as the Schrödinger equation conserves the norm of the wave function. In this manuscript, we show how that can be achieved.

Speeding up maximum population transfer in periodically driven multi-level quantum systems

“…As an example, Doescher and Rice were the first to propose the problem of the infinite square well potential with moving wells, which was then analyzed with different approximations in several publications 1–4 . Recently, researchers have shown, using an analytical exacta approach, that it is possible to control the transition between two predefined states by a particular time-dependent wall motion of this system 6 , which extended the results obtained by Lenz et al . 7 for a different system.…”

“…Hence, the problem of controlling the system has been systematically addressed, now we will address the efficiency of the process problem, namely, how to induce faster transitions in multilevel quantum systems. In such context, a seemingly conflicting result is that such Rabi-like behavior produced by the moving walls involve more states as the oscillation amplitude grow, which compromises the fidelity when we want to drive the system to a specific level 6 . However, the well-known Rabi model suggests that larger amplitudes may induce faster oscillations between states which is certainly of interest for any application.…”

Comparison between time-independent and time-dependent quantum systems in the context of energy, Heisenberg uncertainty, average energy, force, average force and thermodynamic quantities

“…Solution of time-dependent Schrödinger equation is a challenging problem in physics, mathematics and chemistry in presence of time-dependent potential [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]. But some particular problems can be solved analytically with the help of point transformation and separation of variables.…”

Resonant Transitions Due to Changing Boundaries

“…Several other works appeared, dealing with specific box shapes [44,45], aimed at giving proper mathematical treatment of the Schrödinger problem in the presence of moving boundaries [46], and exploring the raise of correlations between different particles confined in the same time-dependent potential [47]. Very recently, some works reporting on the numerical resolution of the dynamics of a particle confined in a one-dimensional box with moving walls has appeared [48,49].…”