Controlling quantum chaos: Time-dependent kicked rotor

Abstract

One major objective of controlling classical chaotic dynamical systems is exploiting the system's extreme sensitivity to initial conditions in order to arrive at a predetermined target state. In a recent Letter [Phys. Rev. Lett. 130, 020201 (2023)], a generalization of this targeting method to quantum systems was demonstrated using successive unitary transformations that counter the natural spreading of a quantum state. In this paper further details are given and an important quite general extension is established. In particular, an alternate approach to constructing the coherent control dynamics is given, which introduces a time-dependent, locally stable control Hamiltonian that continues to use the chaotic heteroclinic orbits previously introduced, but without the need of countering quantum state spreading. Implementing that extension for the quantum kicked rotor generates a much simpler approximate control technique than discussed in the Letter, which is a little less accurate, but far more easily realizable in experiments. The simpler method's error can still be made to vanish as h → 0.

Controlling many-body quantum chaos: Bose–Hubbard systems

“…Similar to [11,12], the protocol for Bose-Hubbard systems is designed to control states that are initially localized in the quadrature phase space. Glauber coherent states provide a natural and optimal choice with which to embark as they are minimum uncertainty states and thus the most classical possible.…”

“…However, this generally introduces terms in the time evolution that are not particle number conserving and thus challenging, if not impossible, with respect to experimental realization. There exists a much simpler approach developed in [12] based on introducing a new time-dependent simulation control Hamiltonian, Ĥc , that is uniquely designed for some particular targeting problem with fixed initial and final quadrature conditions; note that a minimal time related to heteroclinic motion could be the desired goal or any specific time longer than that. Consider Hamilton's equations for each component using equation (7),…”

“…Recently, the translation of classical targeting into the quantum realm has been introduced and applied to a longstanding paradigm of simple chaotic systems [11,12], the kicked rotor [13,14]. For quantum systems with a well defined classical analog, it turns out that the same heteroclinic motion can be exploited both to arrive at a predetermined, possibly exotic, target or final state, and greatly accelerate the way there.…”

“…The first technique follows the local stability analysis and counteracts from time to time the inevitable spreading of the density [11]. The second technique is best conceived of as a form of quantum simulation [17] in which a quite different Hamiltonian simulates the heteroclinic evolution in a stable way [12]. In this work, the second technique is being applied to an important, experimentally realizable many-body system, i.e.…”

Dynamical localization in a non-Hermitian Floquet synthetic system

“…Quantum diffusion of chaotic systems with unique quantum coherence effects is a fundamental problem, [1][2][3][4] which has potential applications in quantum control [5][6][7][8] and quantum communication. [9][10][11][12][13] Understanding the fingerprint of the quantum coherence in chaotic diffusion is critical for the resolution of the elusive issue of the quantum-classical transition.…”