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arXiv:1810.08898 · Journal of Physics B Atomic Molecular and Optical Physics

Nutation dynamics and multifrequency resonance in a many-body seesaw

Hong-Ze Xu, Shun-Yao Zhang, Yukai Lu, Guang‐Can Guo, Ming Gong

Published Feb 17, 2021• 0 citations
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Abstract

Abstract The multifrequency resonance has been widely explored in the context of single-particle models, of which the modulating Rabi model has been the most widely investigated. It has been found that with diagonal periodic modulation, steady dynamics can be realized in some well-defined discrete frequencies. These frequencies are independent of off-diagonal couplings. In this work, we generalize this physics to the many-body seesaw realized using the tilted Bose–Hubbard model. We find that the wave function will recover to its initial condition when the modulation frequency is commensurate with the initial energy level spacing between the ground and the first excited levels. The period is determined by the driving frequency and commensurate ratio. In this case, the wave function will be almost exclusively restricted to the lowest two instantaneous energy levels. By projecting the wave function to these two relevant states, the dynamics is exactly the same as that for the spin precession dynamics and nutation dynamics around an oscillating axis. We map out the corresponding phase diagram, and show that, in the low-frequency regime, the state is thermalized, and in the strong modulation limit, the dynamics is determined by the effective Floquet Hamiltonian. The measurement of these dynamics from the mean position and mean momentum in phase space are also discussed. Our results provide new insights into multifrequency resonance in the many-body system.

DOI
10.1088/1361-6455/abd87b

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Abstract The multifrequency resonance has been widely explored in the context of single-particle models, of which the modulating Rabi model has been the most widely investigated. It has been found that with diagonal periodic modulation, steady dynamics can be realized in some well-defined discrete frequencies. These frequencies are independent of off-diagonal couplings. In this work, we generalize this physics to the many-body seesaw realized using the tilted Bose–Hubbard model. We find that the wave function will recover to its initial condition when the modulation frequency is commensurate with the initial energy level spacing between the ground and the first excited levels. The period is determined by the driving frequency and commensurate ratio. In this case, the wave function will be almost exclusively restricted to the lowest two instantaneous energy levels. By projecting the wave function to these two relevant states, the dynamics is exactly the same as that for the spin precession dynamics and nutation dynamics around an oscillating axis. We map out the corresponding phase diagram, and show that, in the low-frequency regime, the state is thermalized, and in the strong modulation limit, the dynamics is determined by the effective Floquet Hamiltonian. The measurement of these dynamics from the mean position and mean momentum in phase space are also discussed. Our results provide new insights into multifrequency resonance in the many-body system.
We present a new mechanism for realizing of time crystal phase with arbitrary period, termed as commensurate time crystal (CTC), in a quantum seesaw constructed by an interacting Bose-Hubbard model. By periodically modulating the Hamiltonian, we find that the wave function will recover to its initial condition to when the modulating frequency is commensurate with the initial energy level spacing between the ground and first excited levels. The period is determined by the driving frequency and commensurate ratio. In this case, the wave function will almost be restricted to the lowest two instantaneous energy levels. By projecting the wave function to these two relevant states, the dynamics is exactly the same as that for a spin precessing dynamics and nutating dynamics about an rotating axis. The commensurate motion between these two dynamics give rise to CTC. We map out the corresponding phase diagram and show that in the low frequency regime the state is thermalized and in the strong modulating limit, the dynamics is determined by the effective Floquet Hamiltonian. This CTC is realizable between these two limits. The measurement of this dynamics from mean position and mean momentum in phase space is also discussed.
Abstract We study the tunneling dynamics of N = 10 one-dimensional interacting bosons confined in a temporally driven double well potential that imitates a quantum seesaw and how we can manipulate these dynamics by changing the drive of the seesaw potential. We emulate the seesaw with a driven double well potential and consider two driving protocols: an harmonic constant-frequency drive and a chirped drive with linearly increasing frequency. We consider the time-dependent many-body Schrödinger equation of a repulsively interacting quasi-one-dimensional few-boson system. We solve it by using the multiconfigurational time-dependent Hartree method for bosons (MCTDHB) as implemented in the MCTDH-X software. For an harmonic drive and at small values of the driving amplitude, the dynamics of the particles become very slow rendering a stationary-like state. In a phase-space picture the population imbalance between the wells follows a trajectory which is restricted to a comparatively small region of space. For an harmonic drive at intermediate amplitudes, the dynamics become periodic in nature, implying that the bosons populate each of wells periodically. At comparatively large amplitudes of the harmonic drive, the dynamics show features of chaos in phase-space representation. For the chirped drive with a driving frequency increasing linearly in time, the imbalance of the atoms in the seesaw, however, has a temporal evolution that is faster for certain frequency ranges. The tunneling dynamics in such cases, for small amplitudes, show the appearance of quasi-periodicity with simultaneously present slow and fast oscillations. Increasing the amplitude of the chirped drive, we observe that the dynamics, although being periodic, become severely damped in their amplitude. Our study establishes that by tuning the temporal evolution of the quantum seesaw, a precise control of tunneling dynamics of the correlated bosons can be achieved. Since harmonic driving and chirp frequency modulation of the seesaw are experimentally achievable, our simulations can be experimentally realized in laboratories dealing with cold atomic gases.
We theoretically study the dynamics and resonance shift of the Rabi model with frequency modulation, i.e., the Rabi model driven additionally by a slow longitudinal field, by using the counterrotating-hybridized rotating-wave (CHRW) method, aiming to illustrate the effects of the counterrotating (CR) terms of the transverse field. The CHRW method is based on a unitary transformation and reduces the aperiodic Hamiltonian to an effective periodic Hamiltonian that can be efficiently treated by Floquet theory. The validity of the effective Hamiltonian and widely used rotating-wave approximation (RWA) Hamiltonian is carefully examined compared to the numerically exact results over a wide parameter range. It is found that the effective Hamiltonian gives a correct description, while the RWA breaks down in the strong driving regime. Interestingly, we show that under certain conditions the longitudinal field can be used to modify resonance widths such that resonance widths can be comparable to the magnitude of the Bloch-Siegert (BS) shift, which in turn makes the CR-induced BS shift significant and leads to the complete breakdown of the RWA even in a moderately strong driving regime (in which the RWA holds for the Rabi model without frequency modulation). In addition, by using the effective Hamiltonian, we can efficiently access resonance positions for the bichromatically driven qubit and study how the resonance shifts due to the combined effects of the CR terms and frequency modulation. For a weak longitudinal field, we show that resonance positions can be analytically calculated from the effective Rabi frequency for the effective Hamiltonian, which are in excellent agreement with the numerically exact results.
We study the tunneling dynamics in a time-periodically modulated two-mode Bose-Hubbard model using Floquet theory. We consider situations where the system is in the self-trapping regime and either the tunneling amplitude, the interaction strength, or the energy difference between the modes is modulated. In the former two cases, the tunneling is enhanced in a wide range of modulation frequencies, while in the latter case the resonance is narrow. We explain this difference with the help of Floquet analysis. If the modulation amplitude is weak, the locations of the resonances can be found using the spectrum of the nonmodulated Hamiltonian. Furthermore, we use Floquet analysis to explain the coherent destruction of tunneling (CDT) occurring in a large-amplitude modulated system. Finally, we present two ways to create a NOON state (a superposition of NNN particles in mode 1 with zero particles in mode 2 and vice versa). One is based on a coherent oscillation caused by detuning from a partial CDT. The other makes use of an adiabatic variation of the modulation frequency. This results in a Landau-Zener type of transition between the ground state and a NOON-like state.
Experiments on periodically driven quantum systems have effectively realized quasi-Hamiltonians, in the sense of Floquet theory, that are otherwise inaccessible in static condensed matter systems. Although the Floquet quasi-Hamiltonians are time-independent, however, these continuously driven systems can still suffer from heating due to a secular growth in the expectation value of the time-dependent physical Hamiltonian. Here we use an exact space-time mapping to construct a class of many-body systems with rapid periodic driving which we nonetheless prove to be completely free of heating, by mapping them exactly onto time-independent systems. The absence of heating despite the periodic driving occurs in these cases of harmonically trapped dilute Bose gas because the driving is a certain periodic but anharmonic modulation of the gas's two-body contact interaction, at a particular frequency. Although we prove that the absence of heating is exact within full quantum many-body theory, we then use mean-field theory to simulate 'Floquet heating spectroscopy' and compute the heating rate when the driving frequency is varied away from the critical value for zero heating. In both weakly and strongly non-linear regimes, the heating rate as a function of driving frequency appears to show a number of Fano resonances, suggesting that the exactly proven absence of heating at the critical frequency may be explained in terms of destructive interferences between excitation modes.
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.