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arXiv:2008.00373 · SciPost Physics

Periodically and quasi-periodically driven dynamics of Bose-Einstein condensates

Pengfei Zhang, Yingfei Gu

Published Nov 23, 2020• 15 citations
arXivHTMLPDF

Abstract

We study the quantum dynamics of Bose-Einstein condensates when the scattering length is modulated periodically or quasi-periodically in time within the Bogoliubov framework. For the periodically driven case, we consider two protocols where the modulation is a square-wave or a sine-wave. In both protocols for each fixed momentum, there are heating and non-heating phases, and a phase boundary between them. The two phases are distinguished by whether the number of excited particles grows exponentially or not. For the quasi-periodically driven case, we again consider two protocols: the square-wave quasi-periodicity, where the excitations are generated for almost all parameters as an analog of the Fibonacci-type quasi-crystal; and the sine-wave quasi-periodicity, where there is a finite measure parameter regime for the non-heating phase. We also plot the analogs of the Hofstadter butterfly for both protocols.

DOI
10.21468/scipostphys.9.5.079

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We study the quantum dynamics of Bose-Einstein condensates when the scattering length is modulated periodically or quasi-periodically in time within the Bogoliubov framework. For the periodically driven case, we consider two protocols where the modulation is a square-wave or a sine-wave. In both protocols for each fixed momentum, there are heating and non-heating phases, and a phase boundary between them. The two phases are distinguished by whether the number of excited particles grows exponentially or not. For the quasi-periodically driven case, we again consider two protocols: the square-wave quasi-periodicity, where the excitations are generated for almost all parameters as an analog of the Fibonacci-type quasi-crystal; and the sine-wave quasi-periodicity, where there is a finite measure parameter regime for the non-heating phase. We also plot the analogs of the Hofstadter butterfly for both protocols.
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Phase transitions from heating to non-heating in SU(1,1)SU(1,1)SU(1,1) quantum dynamics: Applications to Bose-Einstein condensates and periodically driven coupled oscillators
We study the entanglement properties in non-equilibrium quantum systems with the SU(1,1) S U ( 1 , 1 ) structure. Through Möbius transformation, we map the dynamics of these systems following a sudden quench or a periodic drive onto three distinct trajectories on the Poincaré disc, corresponding the heating, non-heating, and a phase boundary describing these non-equilibrium quantum states. We consider two experimentally feasible systems where their quantum dynamics exhibit the SU(1,1) S U ( 1 , 1 ) structure: the quench dynamics of the Bose-Einstein condensates and the periodically driven coupled oscillators. In both cases, the heating, non-heating phases, and their boundary manifest through distinct signatures in the phonon population where exponential, oscillatory, and linear growths classify these phases. Similarly, the entanglement entropy and negativity also exhibit distinct behaviors (linearly, oscillatory, and logarithmic growths) characterizing these phases, respectively. Notibly, for the periodically driven coupled oscillators, the non-equilibrium properties are characterized by two sets of SU(1,1) S U ( 1 , 1 ) generators. The corresponding two sets of the trajectories on two Poincaré discs lead to a more complex phase diagram. We identify two distinct phases within the heating region discernible solely by the growth rate of the entanglement entropy, where a discontinuity is observed when varying the parameters across the phase boundary within in heating region. This discontinuity is not observed in the phonon population.
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In this thesis, we study the parametric excitations of a homogeneous Bose-Einstein condensate(BEC) at finite temperature obtained from modulating the interaction strength. In Bogoliubov theory, a result connecting the real part of the diagonal Bogoliubov coefficient and instability, i.e. exponential growth of excitations, is obtained for general periodic modulations. Consequently, an analytic solution for an arbitrary square wave modulation is derived that can be studied to understand similar drivings like the sinusoidal case. This adds to previous works studying only the case for small driving amplitudes. In the recent experimental and theoretical literature well-established nonclassicality criteria from the field of quantum optics have been directly applied to the case of excitations in matter-waves like those studied here. Among these are violations of Cauchy-Schwarz inequalities, Glauber-Sudarshan P-nonclassicality, sub-Poissonian number difference squeezing (also known as the two-mode variance) and the criterion of nonseparability. We review the strong connection of these criteria and their meaning in quantum optics, and point out differences in the interpretation between light and matter waves. By proposing a classical theory of a thermal BEC that gives the same predictions as the quantum theory in the high temperature limit, and using it as a reference, we can isolate real quantum effects like spontaneous emission caused by quantum noise. We conclude that to date in experiments relevant for this scenario nonclassical effects have not conclusively been observed and conjecture that additional, noncommuting, observables have to be measured to this end. Moreover this has important implications for proposed analog gravity models where the observation of nonclassical effects is a major goal. In the second part of the thesis, we present simulational methods with focus on the Truncated Wigner Approximation (TWA). We take a path integral approach inspired by [Polkovnikov, 2010] and derive a set of classical equations from the semi-classical limit that is more general than TWA. We add some proofs to the literature and identify problems in this approach. We also present a highly optimized software package as an extension of previous work [Jain, 2007]. It can carry out the relevant simulations and we use it to confirm the theoretical predictions of the first part.