Effective Schrödinger equation for fast driven quantum systems

Abstract

Effective Schrödinger equations, governing the mean movement of a system influenced by fast oscillating external perturbation, are derived. The method represents a direct and straightforward quantum extension of the Kapitza’s approach provided in the Schrödinger picture. First-order terms (over inverse frequency of external driving) in effective Hamiltonians that cannot be eliminated by a unitary transformation, have been derived. Additional terms, in comparison with a standard equation, are similar to those found in the classical case and they may be responsible for a variety of new interesting effects.

Fast driven quantum systems: interaction picture and boundary conditions

“…As noted in [1], in quantum dynamics, even a small number of periodically-driven spins lead to complicated dynamics. The study of the dynamics of such systems (in particular, with high-frequency external influence, the so-called fast driven systems) has become an integral part of quantum physics [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. As stated in [18], the analysis of a physical problem may be simplified by taking into account the presence of substantially different characteristic scales, not only temporal, but also spatial or others.…”

“…The idea and the algorithm of calculations are as follows: (a) Schrödinger equation for a particle in a potential box with moving walls is reduced to an equation with stationary boundary conditions by unitary transformation proposed by Di Martino and Facchi in [20]; (b) from the resulting equation, in the high-frequency limit, the effective Schrödinger equation is derived, according to the method proposed in a number of papers [3][4][5][6][7]. Namely, in case that the Hamiltonian contains rapidly oscillating parts:…”

“…the solution of the Schrödinger equation, in our approach [5], is searched in the form ψ = Φ + μj, where μ = E 0 /ÿΩ = 1, where E 0 = ÿω 0 is the characteristic energy of the unperturbed (stationary) system, and the wave function j(x, t) 'rapidly' changes with time (a linear approximation in μ is considered). If the operators h n and b n in (1) are not commuting, then the effective Schrodinger equation for the 'slow' wave function takes the form:…”

“…Interestingly, the works [3][4][5][6][7] use different approaches and notation (most authors use the so-called Floquet approach, while we do not use it), but the resulting effective equations (in particular, (2)) are the same. This, of course, should be considered as a strong verification of these results.…”

Rapidly oscillating scatteringless non-Hermitian potentials and the absence of Kapitza stabilization

“…pendulum, i.e., stabilization of an inverted pendulum when the point of suspension is being displaced periodically along the vertical direction [13,14]. The original idea of the Kapitza pendulum has been successfully extended into quantum [23][24][25][26][27][28][29][30] and nonlinear [31][32][33] physics, with important applications in Paul traps for charged particles [34], in driven bosonic Josephson junctions [35], in nonlinear optical dispersion management [31,33], and in light guiding and diffraction control in optics [36][37][38][39][40]. We note that Kapitza stabilization can be found for ac oscillations that are periodic, quasi-periodic or even stochastic.…”