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Chinese Physics B

Floquet spectrum and universal dynamics of a periodically driven two-atom system

Wenzhu 文柱 Xie 谢, Zhengqiang 正强 Zhou 周, Xuan 轩 Li 李, Simiao 思淼 Cui 崔, Mingyuan Sun

Published Oct 24, 2023• 1 citation
Publisher

Abstract

We investigate the Floquet spectrum and excitation properties of a two-ultracold-atom system with periodically driven interaction in a three-dimensional harmonic trap. The interaction between the atoms is changed by varying the s-wave scattering length in two ways, the cosine and the square-wave modulations. It is found that as the driving frequency increases, the Floquet spectrum exhibits two main features for both modulations, the accumulating and the spreading of the quasienergy levels, which further lead to different dynamical behaviors. The accumulation is associated with collective excitations and the persistent growth of the energy, while the spread indicates that the energy is bounded at all times. The initial scattering length, the driving frequency and amplitude can all significantly change the Floquet spectrum as well as the dynamics. However, the corresponding relation between them is valid universally. Finally, we propose a mechanism for selectively exciting the system to one specific state by using the avoided crossing of two quasienergy levels, which could guide preparation of a desired state in experiments.

DOI
10.1088/1674-1056/ad0623

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•Jan 1, 2017•0 citations
We investigate the Floquet spectrum and excitation properties of a two-ultracold-atom system with periodically driven interaction in a three-dimensional harmonic trap. The interaction between the atoms is changed by varying the s-wave scattering length in two ways, the cosine and the square-wave modulations. It is found that as the driving frequency increases, the Floquet spectrum exhibits two main features for both modulations, the accumulating and the spreading of the quasienergy levels, which further lead to different dynamical behaviors. The accumulation is associated with collective excitations and the persistent growth of the energy, while the spread indicates that the energy is bounded at all times. The initial scattering length, the driving frequency and amplitude can all significantly change the Floquet spectrum as well as the dynamics. However, the corresponding relation between them is valid universally. Finally, we propose a mechanism for selectively exciting the system to one specific state by using the avoided crossing of two quasienergy levels, which could guide preparation of a desired state in experiments.
In this work we establish a general scattering theory for two-dimensional quantum systems with periodically modulated interactions. We show that scattering quantities, such as the scattering cross section, effective scattering length, and effective-range parameter, can be defined in a manner analogous to conventional scattering theory, even in the presence of modulation. Unlike systems with static interactions, the incident wave in our setup can be scattered inelastically into different Floquet channels, absorbing or emitting quanta from the modulation field. A family of Floquet resonances is identified, at which universal recurrence relations are derived for the Floquet scattering amplitudes. These relations are equivalent to a discrete Schr\"odinger equation describing a pointlike source trapped by an effective potential. The resonance mechanism is explained as the coincidence of the point source's energy with the bound states of the effective potential. We discuss the scattering properties both at and near the resonances, finding that the resonance width and the ratio of the inelastic to elastic cross sections can be effectively controlled through the modulation of the interaction. Our results provide a pathway to precisely manipulate many-body interactions in periodically driven quantum systems.
We investigate the excitation properties of finite ultracold bosonic systems in a one-dimensional harmonic trap with a time-dependent interaction strength. The driving of the interatomic coupling induces excitations of the relative motion exclusively with specific and controllable contributions of momentarily excited many-body states. Mechanisms for selective excitation to few-body analogs of collective modes and acceleration occur in the vicinity of resonances. Via the few-body spectrum and a Floquet analysis, we study the excitation mechanisms and the corresponding impact of the driving frequency and strength as well as the initial correlation of the bosonic state. The fundamental case of two atoms is analyzed in detail and forms a key ingredient for the bottom-up understanding of cases with higher atom numbers, thereby examining finite-size corrections to macroscopic collective modes of oscillation.
We study the dynamics and timescales of a periodically driven Fermi-Hubbard model in a three-dimensional hexagonal lattice. The evolution of the Floquet many-body state is analyzed by comparing it to an equivalent implementation in undriven systems. The dynamics of double occupancies for the near- and off-resonant driving regime indicate that the effective Hamiltonian picture is valid for several orders of magnitude in modulation time. Furthermore, we show that driving a hexagonal lattice compared to a simple cubic lattice allows us to modulate the system up to 1 s, corresponding to hundreds of tunneling times, with only minor atom loss. Here, driving at a frequency close to the interaction energy does not introduce resonant features to the atom loss.
The extended quantum Rabi models make a significant contribution to understanding of the quantum nature of the atom-light interaction. We transform two kinds of extended quantum Rabi model, the anisotropic Rabi model and the asymmetric Rabi model, into the rotating frame and regard them as periodically driven quantum systems. The analytical solutions of the quasienergy spectrum as well as the Floquet modes for both models are constructed by applying the Floquet theory and the high-frequency expansion, which is applied to the nonstroboscopic dynamics of physical observables such as atomic inversion, transverse magnetization, and atom-field correlation. For the anisotropic Rabi model, the quasienergy fits well with the numerical results even when the rotating-wave coupling is in the deep strong-coupling regime $g\ensuremath{\simeq}\ensuremath{\hbar}\ensuremath{\omega}$ if the counter-rotating term is small enough compared to the driving frequency. Avoided level crossing may occur for quasienergy with the same parity when the positive branch spectrum lines for the total excitation number NNN cross the negative branch lines for N+2N+2N+2, while the high-frequency expansion fails to predict this due to the conservation of the total excitation number. Furthermore, we present analytical and numerical studies of the long-time evolution of population and conclude that the analytical method is credible for the population dynamics. For the asymmetric Rabi model, we find that the external bias field which breaks the parity symmetry of the total excitation number tends to cluster the upper and lower branches into two bundles, and the detuning induced gap in the first temporal Brillouin zone shows a quadratic dependence on the bias. Fourier analysis is applied to extract the frequency composition and two-frequency driving behavior is revealed. Varying the bias strength will change the time-averaged value of the oscillation, which shows how the bias competes with the detuning and atom-field coupling in the driving dynamics. Both models prove that treating the Hamiltonian in the rotating frame by Floquet theory gives an alternative tool in the study of interaction between atom and light.
We solve the two-particle s-wave scattering for an ultracold-atom gas confined in a quasi-one-dimensional trapping potential which is periodically modulated. The interaction between the atoms is included via Fermi's pseudopotential. For a modulated isotropic transverse harmonic confinement, the atomic center of mass and relative degrees of freedom decouple and an exact solution is possible. The modulation opens additional photon-assisted resonant scattering channels. Applying the Bethe-Peierls boundary condition, we obtain the general scattering solution of the time-dependent Floquet-Schrödinger equation which is universal at low energies. The effective one-dimensional scattering length can be controlled by the external driving.
This dissertation presents a self-contained study of periodically-driven quantum systems. Following a brief introduction to Floquet theory, we introduce the inverse-frequency expansion, variants of which include the Floquet-Magnus, van Vleck, and Brillouin-Wigner expansions. We reveal that the convergence properties of these expansions depend strongly on the rotating frame chosen, and relate the former to the existence of Floquet resonances in the quasienergy spectrum. The theoretical design and experimental realisation (`engineering') of novel Floquet Hamiltonians is discussed introducing three universal high-frequency limits for systems comprising single-particle and many-body linear and nonlinear models. The celebrated Schrieffer-Wolff transformation for strongly-correlated quantum systems is generalised to periodically-driven systems, and a systematic approach to calculate higher-order corrections to the Rotating Wave Approximation is presented. Next, we develop Floquet adiabatic perturbation theory from first principles, and discuss extensively the adiabatic state preparation and the corresponding leading-order non-adiabatic corrections. Special emphasis is thereby put on geometrical and topological objects, such as the Floquet Berry curvature and the Floquet Chern number obtained within linear response in the presence of the drive. Last, pre-thermalisation and thermalisation in closed, clean periodically-driven quantum systems are studied in detail, with the focus put on the crucial role of Floquet many-body resonances for energy absorption.