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 new 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 $\hbar \rightarrow 0$.
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 new 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 ℏ→0.
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)0031-900710.1103/PhysRevLett.130.020201], 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 ℏ→0.
One of the principal goals of controlling classical chaotic dynamical systems is known as targeting, which is the very weakly perturbative process of using the system's extreme sensitivity to initial conditions in order to arrive at a predetermined target state. It is shown that a generalization to chaotic quantum systems is possible in the semiclassical regime, but requires tailored perturbations whose effects must undo the dynamical spreading of the evolving quantum state. The procedure described here is applied to initially minimum uncertainty wave packets in the quantum kicked rotor, a preeminent quantum chaotic paradigm, to illustrate the method, and investigate its accuracy. The method's error can be made to vanish as ℏ→0.
The kicked rotor provides a simple yet powerful model for introducing many of the central concepts of classical and quantum chaos. Despite its apparent simplicity, it exhibits rich dynamical behavior and has found applications across a wide range of fields, including atomic and optical physics, condensed matter physics, and emerging quantum technologies. This chapter begins by exploring foundational ideas using the kicked rotor as a unifying framework. We first discuss the transition from regular to chaotic motion in the classical system, and then introduce key quantum phenomena such as dynamical localization and quantum resonances. Special attention is devoted to the emergence of characteristic time scales and their role in the quantum-classical correspondence. To make these ideas more concrete, we also provide a brief overview of experimental realizations of the kicked rotor and its variants, illustrating how theoretical concepts are implemented in practice. In the second part of the chapter, we guide the reader toward more recent and advanced developments. Topics include near-resonant dynamics, topological features of kicked systems, the emergence of quantum dynamical phases inferred from classical transport properties, and extensions to non-Hermitian physics. We conclude with a discussion of open problems and future perspectives, outlining directions in which the kicked rotor continues to offer valuable insights.
Using the key properties of chaos, i.e. ergodicity and exponential instability, as a resource to control classical dynamics has a long and considerable history. However, in the context of controlling "chaotic" quantum unitary dynamics, the situation is far more tenuous. The classical concepts of exponential sensitivity to trajectory initial conditions and ergodicity do not directly translate into quantum unitary evolution. Nevertheless properties inherent to quantum chaos can take on those roles: i) the dynamical sensitivity to weak perturbations, measured by the fidelity decay, serves a similar purpose as the classical sensitivity to initial conditions; and ii) paired with the fact that quantum chaotic systems are conjectured to be statistically described by random matrix theory, implies a method to translate the ergodic feature into the control of quantum dynamics. With those two properties, it can be argued that quantum chaotic dynamical systems, in principle, allow for full controllability beyond a characteristic time that scales only logarithmically with system size and ℏ−1. In the spirit of classical targeting, it implies that it is possible to fine tune the immense quantum interference with weak perturbations and steer the system from any initial state into any desired target state, subject to constraints imposed by conserved quantities. In contrast, integrable dynamics possess neither ergodicity nor exponential instability, and thus the weak perturbations apparently must break the integrability for control purposes. The main ideas are illustrated with the quantum kicked rotor. The production of revivals, cat-like entangled states, and the transition from any random state to any other random state is possible as demonstrated.
In this work, we study quantum chaos by focusing on the evolution of initially close states in the dynamics of the Quantum Kicked Rotor (QKR). We propose a novel measure, the Quantum Lyapunov Exponent (QLE), to quantify the degree of chaos in this quantum system, analogous to its classical counterpart. We begin by modeling the momentum space and then the QLE is computed through analyzing the fidelity between evolving states, offering insights into the quantum chaotic behavior. Furthermore, we extend our investigations to various initial states: localized, uniform, spreading, contracting and oscillating in momentum space. Our results unveil a diverse range of dynamical behaviors, highlighting the complex nature of quantum chaos. Finally, we propose an innovative optimization framework to represent a complex state as a superposition of the aforementioned states, which has potential implications for visualizing and understanding the dynamics of multifaceted quantum systems.
Abstract This work develops a quantum control application of many-body quantum chaos for ultracold bosonic gases trapped in optical lattices. It is long known how to harness exponential sensitivity to changes in initial conditions for control purposes in classically chaotic systems. In the technique known as targeting, instead of a hindrance to control, the instability becomes a resource. Recently, this classical targeting has been generalized to quantum systems either by periodically countering the inevitable quantum state spreading or by introducing a control Hamiltonian, where both enable localized states to be guided along special chaotic trajectories toward any of a broad variety of desired target states. Only strictly unitary dynamics are involved; i.e. it gives a coherent quantum targeting. In this paper, the introduction of a control Hamiltonian is applied to Bose–Hubbard systems in chaotic dynamical regimes. Properly selected unstable mean field solutions can be followed particularly rapidly to states possessing precise phase relationships and occupancies. In essence, the method generates a quantum simulation technique that can access rather special states. The protocol reduces to a time-dependent control of the chemical potentials, opening up the possibility for application in optical lattice experiments. Explicit applications to custom state preparation and stabilization of quantum many-body scars are presented in one- and two-dimensional lattices (three-dimensional applications are similarly possible).