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Coupled Quantum Kicked Rotors Under Lindbladian Dynamics

Adam Bäck, Philip Santi Straihamer

Published Jan 1, 2026• 0 citations
OpenAlex

Abstract

Lindblad dynamics is one of the most popular approaches to study open quantum systems. This paper studies a kicked rotor; in particular, a two rotor system where one of the rotors is treated as the environment and the other is studied with the Lindblad master equation. The energy levels are solved explicitly in terms of Bessel functions and the results show that the energy levels exhibit dissipative behavior with variance in momentum increasing as $\langle\hat{p}^2\rangle=2Dt$ similar to the classical kicked rotor. This result shows us how interaction between systems can break down quantum phenomena and result in more classical results, bridging the gap between the quantum and the classical world.

Related papers

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A quantum mechanical system of two coupled rotors (particles constrained to move on a circle) is studied from an open quantum systems point of view. One of the rotors is integrated out and the reduced density operator of the other rotor is studied. Its eigenvalues are worked out explicitly using the properties of Mathieu functions, and the von Neumann entropy, which is a standard measure of entanglement, is computed in terms of the Fourier coefficients defining the Mathieu functions. Furthermore, upon introducing a time-periodic delta kick and making one of the rotors much heavier than the other, the two-rotor system can be interpreted as a system-bath model, allowing us to introduce a series of approximations to derive a master equation of the Lindblad type describing the time-evolution of the reduced density operator.

arXiv:2505.13686•Modern Physics Letters A•Jan 31, 2026•1 citation

Chaos and localized phases in a two-body linear kicked rotor system

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Despite the periodic kicks, a linear kicked rotor (LKR) is an integrable and exactly solvable model in which the kinetic energy term is linear in momentum. It was recently shown that spatially interacting LKRs are also integrable, and results in dynamical localization in the corresponding quantum regime. Similar localized phases exist in other nonintegrable models such as the coupled relativistic kicked rotors. This work, using a two-body LKR, demonstrates two main results; first, it is shown that chaos can be induced in the integrable linear kicked rotor through interactions between the momenta of rotors. An analytical estimate of its Lyapunov exponent is obtained. Second, the quantum dynamics of this chaotic model, upon variation of kicking and interaction strengths, is shown to exhibit a variety of phases: classically induced localization, dynamical localization, subdiffusive, and diffusive phases. We point out the signatures of these phases from the perspective of entanglement production in this system. By defining an effective Hilbert space dimension, the entanglement growth rate can be understood using appropriate random matrix averages.

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Observation of many-body dynamical localization

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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.

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Coupled quantum kicked rotors: a study about dynamical localization, slow heating and thermalization

Simone Notarnicola

Periodically driven systems are nowadays a very powerful tool for the study of condensed quantum matter: indeed, they allow the observation of phenomena in huge contrast with the expected behavior of their classical counterparts. An outstanding example is dynamical localization: this phenomenon consists in the prevention of heating despite an external periodic perturbation. It has been defined for the first time in a single particle model, the kicked rotor. This chaotic system undergoes an unbounded heating in time in its classical limit; on the opposite, in the quantum regime, the kinetic energy grows until a saturation value is reached and the system stops heating. In my thesis I consider a chain of coupled quantum kicked rotors in order to investigate the fate of dynamical localization in presence of interactions. I remarkably show that a dynamically localized phase persists in the quantum system also in presence of interactions. This is an unexpected behavior since periodically driven, interacting, non-integrable quantum systems heat up to an infinite temperature state. Moreover, I find a genuine quantum dynamics also in the delocalized phase: the heating is not diffusive, as it happens in the classical system, but it follows a sub-diffusive power law. A focus on the properties of the Floquet eigenstates and operators matrix properties gives interesting hints, still under investigation, for a possible justification of the above mentioned slow heating.

•Oct 25, 2018•0 citations

Quantum to Classical Transition by a Classically Small Interaction

Wen-Lei Zhao, Jie Quan-Lin, Bo Zhou

We investigate the quantum-classical transition of a kicked rotor (KR) under perturbation by a second one. The influence of such a chaotic KR makes decoherence of the first one, resulting in the emergence of classical diffusion from its quantum dynamics. Such quantum-classical transition persists by decreasing the effective Planck's constant ℏ, and at the same time, decreasing the mass of the second KR and the interaction strength proportionally. In the limit of ħ → 0, due to vanishing small mass and interaction, the second KR has almost no effect on the classical dynamics of the first one. We demonstrate this via two different coupling potentials.

Communications in Theoretical Physics•Aug 1, 2010•10 citations

Quantum to Classical Transition in a System of Two Coupled Kicked Rotors

Wen-Lei Zhao, Jie Quan-Lin

We investigate the quantum-classical transition in a system of two coupled kicked rotors. We find that when the mass of one kicked rotor is much smaller than the other's, the influence of the light kicked rotor is still able to make decoherence of the heavy one. This leads to the quantum-classical transition of the heavy kicked rotor. We demonstrate this by two different coupling potentials.

Communications in Theoretical Physics•Mar 1, 2009•12 citations
Lindblad dynamics is one of the most popular approaches to study open quantum systems. This paper studies a kicked rotor; in particular, a two rotor system where one of the rotors is treated as the environment and the other is studied with the Lindblad master equation. The energy levels are solved explicitly in terms of Bessel functions and the results show that the energy levels exhibit dissipative behavior with variance in momentum increasing as ⟨p^2⟩=2Dt\langle\hat{p}^2\rangle=2Dt⟨p^​2⟩=2Dt similar to the classical kicked rotor. This result shows us how interaction between systems can break down quantum phenomena and result in more classical results, bridging the gap between the quantum and the classical world.
A quantum mechanical system of two coupled rotors (particles constrained to move on a circle) is studied from an open quantum systems point of view. One of the rotors is integrated out and the reduced density operator of the other rotor is studied. Its eigenvalues are worked out explicitly using the properties of Mathieu functions, and the von Neumann entropy, which is a standard measure of entanglement, is computed in terms of the Fourier coefficients defining the Mathieu functions. Furthermore, upon introducing a time-periodic delta kick and making one of the rotors much heavier than the other, the two-rotor system can be interpreted as a system-bath model, allowing us to introduce a series of approximations to derive a master equation of the Lindblad type describing the time-evolution of the reduced density operator.
Despite the periodic kicks, a linear kicked rotor (LKR) is an integrable and exactly solvable model in which the kinetic energy term is linear in momentum. It was recently shown that spatially interacting LKRs are also integrable, and results in dynamical localization in the corresponding quantum regime. Similar localized phases exist in other nonintegrable models such as the coupled relativistic kicked rotors. This work, using a two-body LKR, demonstrates two main results; first, it is shown that chaos can be induced in the integrable linear kicked rotor through interactions between the momenta of rotors. An analytical estimate of its Lyapunov exponent is obtained. Second, the quantum dynamics of this chaotic model, upon variation of kicking and interaction strengths, is shown to exhibit a variety of phases: classically induced localization, dynamical localization, subdiffusive, and diffusive phases. We point out the signatures of these phases from the perspective of entanglement production in this system. By defining an effective Hilbert space dimension, the entanglement growth rate can be understood using appropriate random matrix averages.
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.
Periodically driven systems are nowadays a very powerful tool for the study of condensed quantum matter: indeed, they allow the observation of phenomena in huge contrast with the expected behavior of their classical counterparts. An outstanding example is dynamical localization: this phenomenon consists in the prevention of heating despite an external periodic perturbation. It has been defined for the first time in a single particle model, the kicked rotor. This chaotic system undergoes an unbounded heating in time in its classical limit; on the opposite, in the quantum regime, the kinetic energy grows until a saturation value is reached and the system stops heating. In my thesis I consider a chain of coupled quantum kicked rotors in order to investigate the fate of dynamical localization in presence of interactions. I remarkably show that a dynamically localized phase persists in the quantum system also in presence of interactions. This is an unexpected behavior since periodically driven, interacting, non-integrable quantum systems heat up to an infinite temperature state. Moreover, I find a genuine quantum dynamics also in the delocalized phase: the heating is not diffusive, as it happens in the classical system, but it follows a sub-diffusive power law. A focus on the properties of the Floquet eigenstates and operators matrix properties gives interesting hints, still under investigation, for a possible justification of the above mentioned slow heating.
We investigate the quantum-classical transition of a kicked rotor (KR) under perturbation by a second one. The influence of such a chaotic KR makes decoherence of the first one, resulting in the emergence of classical diffusion from its quantum dynamics. Such quantum-classical transition persists by decreasing the effective Planck's constant ℏ, and at the same time, decreasing the mass of the second KR and the interaction strength proportionally. In the limit of ħ → 0, due to vanishing small mass and interaction, the second KR has almost no effect on the classical dynamics of the first one. We demonstrate this via two different coupling potentials.