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arXiv:1902.07809 · Physical Review Letters

Integrable Floquet Hamiltonian for a Periodically Tilted 1D Gas

A. Colcelli, Giuseppe Mussardo, Germán Sierra, Andrea Trombettoni

Published Sep 26, 2019• 11 citations
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Abstract

An integrable model subjected to a periodic driving gives rise generally to a nonintegrable Floquet Hamiltonian. Here we show that the Floquet Hamiltonian of the integrable Lieb-Liniger model in the presence of a linear potential with a periodic time-dependent strength is instead integrable and its quasienergies can be determined using the Bethe ansatz approach. We discuss various aspects of the dynamics of the system at stroboscopic times and we also propose a possible experimental realization of the periodically driven tilting in terms of a shaken rotated ring potential.

DOI
10.1103/physrevlett.123.130401

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We present theoretical methods for studying quantum mechanical systems subjected to fast periodic driving and apply them to model systems with long-range interaction. We provide a comparison between the methods and insight facilitated by these methods. The methods recently occurred in scientific papers, which supports the need for a scrutinized exposition of the theory. One of the main objects of the theory is a so-called Floquet Hamiltonian-an artificial stationary Hamiltonian describing important features of a quantum system. The methods construct Floquet Hamiltonians in the form of series in the powers of the time period. We present the spectra of Floquet Hamiltonians-the so-called quasienergy spectra-computed by the methods and computed numerically (with higher precision). The quasienergy spectra were computed using various approximations of Floquet Hamiltonians and compared. We discuss an interesting topic of the classical limit of an artificial stationary system. We also mention the kicked rotor system and its connection with the kicked top system-one of our model systems. In summary, the method characterized by simultaneous construction of a Floquet Hamiltonian and a so-called kick operator (operator capturing fast changes of the system) was found universal and accurate. The thesis presents an...

•Jan 1, 2022•0 citations
An integrable model subjected to a periodic driving gives rise generally to a nonintegrable Floquet Hamiltonian. Here we show that the Floquet Hamiltonian of the integrable Lieb-Liniger model in the presence of a linear potential with a periodic time-dependent strength is instead integrable and its quasienergies can be determined using the Bethe ansatz approach. We discuss various aspects of the dynamics of the system at stroboscopic times and we also propose a possible experimental realization of the periodically driven tilting in terms of a shaken rotated ring potential.
We study a system of one-dimensional interacting quantum particles subjected to a time-periodic potential linear in space. After discussing the cases of driven one- and two-particle systems, we derive the analogous results for the many-particle case in the presence of a general interaction two-body potential and the corresponding Floquet Hamiltonian. When the undriven model is integrable, the Floquet Hamiltonian is shown to be integrable too. We determine the micromotion operator and the expression for a generic time evolved state of the system. We discuss various aspects of the dynamics of the system both at stroboscopic and intermediate times, in particular the motion of the center of mass of a generic wave packet and its spreading over time. We also discuss the case of accelerated motion of the center of mass, obtained when the integral of the coefficient strength of the linear potential on a time period is nonvanishing, and we show that the Floquet Hamiltonian gets in this case an additional static linear potential. We also discuss the application of the obtained results to the Lieb-Liniger model.
When a closed quantum system is driven periodically with period T, it approaches a periodic state synchronized with the drive in which any local observable measured stroboscopically approaches a steady value. For integrable systems, the resulting behavior is captured by a periodic version of a generalized Gibbs ensemble. By contrast, here we show that for generic nonintegrable interacting systems, local observables become independent of the initial state entirely. Essentially, this happens because Floquet eigenstates of the driven system at quasienergy ω(α) consist of a mixture of the exponentially many eigenstates of the undriven Hamiltonian, which are thus drawn from the entire extensive undriven spectrum. This is a form of equilibration which depends only on the Hilbert space of the undriven system and not on any details of its Hamiltonian.
We consider a quantum system periodically driven with a strength which varies slowly on the scale of the driving period. The analysis is based on a general formulation of the Floquet theory relying on the extended Hilbert space. It is shown that the dynamics of the system can be described in terms of a slowly varying effective Floquet Hamiltonian that captures the long-term evolution, as well as rapidly oscillating micromotion operators. We obtain a systematic high-frequency expansion of all these operators. Generalizing the previous studies, the expanded effective Hamiltonian is now time dependent and contains extra terms appearing due to changes in the periodic driving. The same applies to the micromotion operators which exhibit a slow temporal dependence in addition to the rapid oscillations. As an illustration, we consider a quantum-mechanical spin in an oscillating magnetic field with a slowly changing direction. The effective evolution of the spin is then associated with non-Abelian geometric phases reflecting the geometry of the extended Floquet space. The developed formalism is general and also applies to other periodically driven systems, such as shaken optical lattices with a time-dependent shaking strength, a situation relevant to the cold-atom experiments.
We study the fate of interacting quantum systems which are periodically driven by switching back and forth between two integrable Hamiltonians. This provides an unconventional and tunable way of breaking integrability, in the sense that the stroboscopic time evolution will generally be described by a Floquet Hamiltonian which progressively becomes less integrable as the driving frequency is reduced. Here, we exemplify this idea in spin chains subjected to periodic switching between two integrable anisotropic Heisenberg Hamiltonians. We distinguish the integrability-breaking effects of resonant interactions and perturbative (local) interactions, and illustrate these by contrasting different measures of energy in Floquet states and through a study of level spacing statistics. This scenario is argued to be representative for general driven interacting integrable systems.
We give a general overview of the high-frequency regime in periodically driven systems and identify three distinct classes of driving protocols in which the infinite-frequency Floquet Hamiltonian is not equal to the time-averaged Hamiltonian. These classes cover systems, such as the Kapitza pendulum, the Harper–Hofstadter model of neutral atoms in a magnetic field, the Haldane Floquet Chern insulator and others. In all setups considered, we discuss both the infinite-frequency limit and the leading finite-frequency corrections to the Floquet Hamiltonian. We provide a short overview of Floquet theory focusing on the gauge structure associated with the choice of stroboscopic frame and the differences between stroboscopic and non-stroboscopic dynamics. In the latter case, one has to work with dressed operators representing observables and a dressed density matrix. We also comment on the application of Floquet Theory to systems described by static Hamiltonians with well-separated energy scales and, in particular, discuss parallels between the inverse-frequency expansion and the Schrieffer–Wolff transformation extending the latter to driven systems.
We present theoretical methods for studying quantum mechanical systems subjected to fast periodic driving and apply them to model systems with long-range interaction. We provide a comparison between the methods and insight facilitated by these methods. The methods recently occurred in scientific papers, which supports the need for a scrutinized exposition of the theory. One of the main objects of the theory is a so-called Floquet Hamiltonian-an artificial stationary Hamiltonian describing important features of a quantum system. The methods construct Floquet Hamiltonians in the form of series in the powers of the time period. We present the spectra of Floquet Hamiltonians-the so-called quasienergy spectra-computed by the methods and computed numerically (with higher precision). The quasienergy spectra were computed using various approximations of Floquet Hamiltonians and compared. We discuss an interesting topic of the classical limit of an artificial stationary system. We also mention the kicked rotor system and its connection with the kicked top system-one of our model systems. In summary, the method characterized by simultaneous construction of a Floquet Hamiltonian and a so-called kick operator (operator capturing fast changes of the system) was found universal and accurate. The thesis presents an...