We report the first experimental observation of the time-driven phase transition in a canonical quantum chaotic system, the quantum kicked rotor. The transition bears a firm analogy to a thermodynamic phase transition, with the time mimicking the temperature and the quantum expectation of the rotor's kinetic energy mimicking the free energy. The transition signals a sudden change in the system's memory behavior: before the critical time, the system undergoes chaotic motion in phase space and its memory of initial states is erased in the course of time; after the critical time, quantum interference enhances the probability for a chaotic trajectory to return to the initial state, and thus the system's memory is recovered.
We report the first experimental observation of the time-driven phase transition in a canonical quantum chaotic system, the quantum kicked rotor. The transition bears a firm analogy to a thermodynamic phase transition, with the time mimicking the temperature and the quantum expectation of the rotor's kinetic energy mimicking the free energy. The transition signals a sudden change in the system's memory behavior: before the critical time, the system undergoes chaotic motion in phase space and its memory of initial states is erased in the course of time; after the critical time, quantum interference enhances the probability for a chaotic trajectory to return to the initial state, and thus the system's memory is recovered.
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.
Decoherence in quantum systems which are classically chaotic is studied. The Arnold cat map and the quantum kicked rotor are chosen as examples of linear and nonlinear chaotic systems. The Feynman-Vernon influence functional formalism is used to study the effect of the environment on the system. It is well known that quantum coherence can obliterate chaotic behavior in the corresponding classical system. But interaction with an environment can under general circumstances quickly diminish quantum coherence and reenact classical chaotic behavior. How effectively decoherence works to sustain chaos, and how the resultant behavior qualitatively differs from the quantum picture, depending on the coupling of the system with the environment and the spectral density and temperature of the environment. We show how recurrence in the quantum cat map is lost and classical ergodicity is recovered due to the effect of the environment. Quantum coherence and diffusion suppression are instrumental to dynamical localization for the kicked rotor. We show how environment-induced effects can destroy this localization. Such effects can also be understood as resulting from external noises driving the system. Peculiar to decohering chaotic systems is the apparent transition from reversible to irreversible dynamics. We show such transitions in the quantum cat map and the kicked rotor and distinguish it from apparent irreversibility originating from dynamical instability and imprecise measurements. By performing a time reversal on and following the quantum kicked rotor dynamics numerically, we show how the otherwise reversible quantum dynamics acquires an arrow of time upon the introduction of noise or interaction with an environment.
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.
Most classical dynamical systems are chaotic. The trajectories of two identical systems prepared in infinitesimally different initial conditions diverge exponentially with time. Quantum systems, instead, exhibit quasiperiodicity due to their discrete spectrum. Nonetheless, the dynamics of quantum systems whose classical counterparts are chaotic are expected to show some features that resemble chaotic motion. Among the many controversial aspects of the quantum-classical boundary, the emergence of chaos remains among the least experimentally verified. Time-resolved observations of quantum chaotic dynamics are particularly rare, and as yet unachieved in a single particle, where the subtle interplay between chaos and quantum measurement could be explored at its deepest levels. We present here a realistic proposal to construct a chaotic driven top from the nuclear spin of a single donor atom in silicon, in the presence of a nuclear quadrupole interaction. This system is exquisitely measurable and controllable, and possesses extremely long intrinsic quantum coherence times, allowing for the observation of subtle dynamical behavior over extended periods. We show that signatures of chaos are expected to arise for experimentally realizable parameters of the system, allowing the study of the relation between quantum decoherence and classical chaos, and the observation of dynamical tunneling.
We have performed an experiment to demonstrate the approximate time reversal of a "chaotic" time evolution of atomic de Broglie waves. We use ultracold atoms from a Bose-Einstein condensate in a quantum δ-kicked rotor experiment, and show that an initial state can be approximately re-created even after a period of "chaotic" evolution (a number of kicks). As this mechanism only works for a very narrow range of momenta, the net effect is a narrowing of the momentum distribution after the kick sequence.