Thermalization in a Bose-Hubbard dimer with modulated tunneling

Abstract

The periodically modulated Bose-Hubbard dimer model offers an experimentally realizable and highly tunable platform for observing the scrambling of quantum information and the apparent thermalization of isolated, interacting quantum many-body systems. In this work we apply fidelity out-of-time correlators (FOTOCs) to establish connections between the thermalization in the Floquet system, the exponential growth of FOTOCs as quantified by a nonzero quantum Lyapunov exponent, and the underlying classical transition from regular to chaotic dynamics in the dimer. Moreover, we demonstrate that a nonzero quantum Lyapunov exponent can also be inferred from measures quantifying the delocalization of the Floquet modes of the system such as the Shannon entropy, which approaches unity if the system thermalizes to the periodic Gibbs ensemble prediction.

Saddle-point scrambling without thermalization

“…First introduced to describe the dynamics of quantum information in black holes, scrambling has since been used to probe the connections among the dynamics of entanglement, chaos, and thermalization [8][9][10][11][12][13]. In particular, studies of chaotic models in periodically driven and undriven systems have used a variety of OTOCs, including the fidelity out-of-time-order correlator (FOTOC), to show that in the chaotic phase the OTOC grows, often exponentially, up to an Ehrenfest time after which it saturates to a steady-state value [8][9][10][11][12][14][15][16][17][18][19]. If the state of an isolated quantum system is sufficiently delocalized in the basis of energy eigenstates, the system is expected to relax towards the "diagonal ensemble" (DE) due to dephasing between energy eigenstates [20].…”

Saddle-point scrambling without thermalisation

“…Furthermore, we work with sufficiently small δ 1, such that the FO-TOC simplifies to the variance of the generator Ŝα , as C(t) ≈ δ 2 var [ Ŝα (t)]+O(δ 3 ) [10,28]. The predicted value of the FOTOC for the infinite temperature 'uniform diagonal ensemble' (UDE) is C UDE = δ 2 N (N + 2)/12 [16].…”

“…1(d). The extent of the chaos can be finely controlled using the modulation constants µ, ω [16,26,27].…”

“…First introduced to describe the dynamics of quantum information in black holes, scrambling has since been used to probe the connections between the dynamics of entanglement, chaos, and thermalisation [7][8][9][10][11][12]. In particular, studies of chaotic models in periodically driven and undriven systems have used a variety of OTOC, including the fidelity out-of-time-order correlator (FO-TOC), to show that, in the chaotic phase, the OTOC grows exponentially up to an Ehrenfest time, after which it saturates to a value predicted by the eigenstate thermalisation hypothesis (ETH) [7][8][9][10][11][13][14][15][16]. According to the ETH, if the state of an isolated quantum system is sufficiently delocalised in the basis of energy eigenstates, the system will relax towards the 'diagonal ensemble' (DE) due to dephasing between energy eigenstates [17,18].…”

“…According to the ETH, if the state of an isolated quantum system is sufficiently delocalised in the basis of energy eigenstates, the system will relax towards the 'diagonal ensemble' (DE) due to dephasing between energy eigenstates [17,18]. Consequently, in the absence of an external drive an initial product state will evolve to a finitetemperature thermal state with volume-law entanglement entropy, whereas a periodically driven system will evolve to an infinite-temperature state [10,13,14,16,17].…”