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arXiv:1901.09362

Many-body dynamical localisation of coupled quantum kicked rotors

L. A. Toikka, Alexei Andreanov

Published Jan 27, 2019• 1 citation
arXivHTMLPDF

Abstract

The quantum motion of $N$ coupled kicked rotors is mapped to an interacting $N$-particle Anderson-Aubry-Andr$é$ tight-binding problem supporting many-body localised (MBL) phases. Interactions in configuration space are known to be insufficient for destroying Anderson localisation in a system in the MBL phase. The mapping we establish here predicts that a similar effect takes place in momentum space and determines the quantum dynamics of the coupled kicked rotors. Due to the boundedness of the Floquet quasi-energy spectrum there exists limitations on the interacting lattice models that can be mapped to quantum kicked rotors; in particular, no extensive observable can be mapped in the thermodynamic limit.

DOI
10.48550/arxiv.1901.09362

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The quantum motion of NNN coupled kicked rotors is mapped to an interacting NNN-particle Anderson-Aubry-Andreˊéeˊ tight-binding problem supporting many-body localised (MBL) phases. Interactions in configuration space are known to be insufficient for destroying Anderson localisation in a system in the MBL phase. The mapping we establish here predicts that a similar effect takes place in momentum space and determines the quantum dynamics of the coupled kicked rotors. Due to the boundedness of the Floquet quasi-energy spectrum there exists limitations on the interacting lattice models that can be mapped to quantum kicked rotors; in particular, no extensive observable can be mapped in the thermodynamic limit.
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We study the effect of many-body quantum interference on the dynamics of coupled periodically kicked systems whose classical dynamics is chaotic and shows an unbounded energy increase. We specifically focus on an N-coupled kicked rotors model: We find that the interplay of quantumness and interactions dramatically modifies the system dynamics, inducing a transition between energy saturation and unbounded energy increase. We discuss this phenomenon both numerically and analytically through a mapping onto an N-dimensional Anderson model. The thermodynamic limit N→∞, in particular, always shows unbounded energy growth. This dynamical delocalization is genuinely quantum and very different from the classical one: Using a mean-field approximation, we see that the system self-organizes so that the energy per site increases in time as a power law with exponent smaller than 1. This wealth of phenomena is a genuine effect of quantum interference: The classical system for N≥2 always behaves ergodically with an energy per site linearly increasing in time. Our results show that quantum mechanics can deeply alter the regularity or ergodicity properties of a many-body-driven system.
The quantum kicked rotor is a paradigmatic model system in quantum physics. As a driven quantum system, it features dynamical localization, specifically Anderson localization in momentum space. However, the interacting many-body kicked rotor is believed to break localization. Here, we present evidence for many-body dynamical localization for the Lieb-Liniger version of the many-body quantum kicked rotor. After some initial evolution, the momentum distribution of interacting quantum-degenerate bosonic atoms in one-dimensional geometry, kicked hundreds of times by means of a pulsed sinusoidal potential, stops spreading. Our results shed light on the boundary between the classical, chaotic world and the realm of quantum physics.
The kicked rotor system is a textbook example of how classical and quantum dynamics can drastically differ. The energy of a classical particle confined to a ring and kicked periodically will increase linearly in time whereas in the quantum version the energy saturates after a finite number of kicks. The quantum system undergoes Anderson localization in angular-momentum space. Conventional wisdom says that in a many-particle system with short-range interactions the localization will be destroyed due to the coupling of widely separated momentum states. Here we provide evidence that for an interacting one-dimensional Bose gas, the Lieb-Liniger model, the dynamical localization can persist at least for an unexpectedly long time.
We map the infinite-range coupled quantum kicked rotors over an infinite-range coupled interacting bosonic model. In this way we can apply exact diagonalization up to quite large system sizes and confirm that the system tends to ergodicity in the large-size limit. In the thermodynamic limit the system is described by a set of coupled Gross-Pitaevskii equations equivalent to an effective nonlinear single-rotor Hamiltonian. These equations give rise to a power-law increase in time of the energy with exponent $\ensuremath{\gamma}\ensuremath{\sim}2/3$ in a wide range of parameters. We explain this finding by means of a master-equation approach based on the noisy behavior of the effective nonlinear single-rotor Hamiltonian and on the Anderson localization of the single-rotor Floquet states. Furthermore, we study chaos by means of the largest Lyapunov exponent and find that it decreases towards zero for portions of the phase space with increasing momentum. Finally, we show that some stroboscopic Floquet integrals of motion of the noninteracting dynamics deviate from their initial values over a timescale related to the interaction strength according to the Nekhoroshev theorem.
We study the fate of dynamical localization of two quantum kicked rotors with contact interaction, which relates to experimental realizations of the rotors with ultra-cold atomic gases. A single kicked rotor is known to exhibit dynamical localization, which takes place in momentum space. The contact interaction affects the evolution of the relative momentum k of a pair of interacting rotors in a non-analytic way. Consequently the evolution operator U is exciting large relative momenta with amplitudes which decay only as a power law 1/k4. This is in contrast to the center-of-mass momentum K for which the amplitudes excited by U decay superexponentially fast with K. Therefore dynamical localization is preserved for the center-of-mass momentum, but destroyed for the relative momentum for any nonzero strength of interaction.