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arXiv:1205.3511 · Physica Scripta

Second Josephson excitations beyond mean field as a toy model for thermal pressure: exact quantum dynamics and the quantum phase model

M P Strzys, J. R. Anglin

Published Nov 1, 2012• 3 citations
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Abstract

A simple four-mode Bose–Hubbard model with intrinsic time-scale separation can be considered as a paradigm for mesoscopic quantum systems in thermal contact. In our previous work, we showed that in addition to coherent particle exchange, a novel slow collective excitation can be identified by a series of Holstein–Primakoff transformations. This resonant energy exchange mode is not predicted by the linear Bogoliubov theory, and its frequency is sensitive to interactions among Bogoliubov quasi-particles; it may be referred to as a second Josephson oscillation, in analogy to the second sound mode of liquid helium II. In this paper, we will explore this system beyond the Gross–Pitaevskii mean-field regime. We directly compare the classical mean-field dynamics to the exact full quantum many-particle dynamics and show good agreement over a wide range of system parameters. The second Josephson frequency becomes imaginary for stronger interactions however, indicating dynamical instability of the symmetric state. By means of a generalized quantum phase model for the full four-mode system, we then show that, in this regime, high-energy Bogoliubov quasiparticles tend to accumulate in one pair of sites, while the actual particles preferentially occupy the opposite pair. We interpret this as a simple model of thermal pressure.

DOI
10.1088/0031-8949/2012/t151/014046

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A simple four-mode Bose–Hubbard model with intrinsic time-scale separation can be considered as a paradigm for mesoscopic quantum systems in thermal contact. In our previous work, we showed that in addition to coherent particle exchange, a novel slow collective excitation can be identified by a series of Holstein–Primakoff transformations. This resonant energy exchange mode is not predicted by the linear Bogoliubov theory, and its frequency is sensitive to interactions among Bogoliubov quasi-particles; it may be referred to as a second Josephson oscillation, in analogy to the second sound mode of liquid helium II. In this paper, we will explore this system beyond the Gross–Pitaevskii mean-field regime. We directly compare the classical mean-field dynamics to the exact full quantum many-particle dynamics and show good agreement over a wide range of system parameters. The second Josephson frequency becomes imaginary for stronger interactions however, indicating dynamical instability of the symmetric state. By means of a generalized quantum phase model for the full four-mode system, we then show that, in this regime, high-energy Bogoliubov quasiparticles tend to accumulate in one pair of sites, while the actual particles preferentially occupy the opposite pair. We interpret this as a simple model of thermal pressure.
Adiabatic approximations are a powerful tool for simplifying nonlinear quantum dynamics, and are applicable whenever a system exhibits a hierarchy of time scales. Current interest in small nonlinear quantum systems, such as few-mode Bose-Hubbard models, warrants further development of adiabatic methods in the particular context of these models. Here we extend our recent work on a simple four-mode Bose-Hubbard model with two distinct dynamical time scales, in which we showed that among the perturbations around excited stationary states of the system is a slow collective excitation that is not present in the Bogoliubov spectrum. We characterized this mode as a resonant energy exchange with its frequency shifted by nonlinear effects, and referred to it as a second Josephson oscillation, in analogy with the second sound mode of liquid helium II. We now generalize our previous theory beyond the mean-field regime, and construct a general Bogoliubov free quasiparticle theory that explicitly respects the system's adiabatic invariant as well the exact conservation of particles. We compare this theory to the numerically exact quantum energy spectrum with up to 40 particles, and find good agreement over a significant range of parameter space.
Bose–Einstein condensation represents a quintessential manifestation of quantum mechanics at macroscopic scales, where a dilute gas of bosonic atoms undergoes a phase transition to form a coherent matter wave. This study presents a comprehensive theoretical investigation of collective excitations and critical phenomena in interacting Bose–Einstein condensates (BECs). Beginning with the Gross–Pitaevskii equation , we develop the Bogoliubov theory of excitations yielding the spectrum , where is the free-particle energy and characterizes contact interactions. The spectrum reveals phonon behavior at low momenta with sound velocity , transitioning to free-particle dispersion at momenta exceeding the inverse healing length . Collective modes in harmonically trapped condensates are analyzed in the Thomas–Fermi regime, yielding breathing mode frequency and quadrupole frequency . Damping mechanisms including Landau damping ( ) and Beliaev damping ( ) are characterized. The Lee–Huang–Yang correction extends the equation of state beyond mean field. Critical phenomena near are analyzed within the 3D XY universality class, with critical exponents , , and the interaction-induced shift with . All predictions show excellent agreement with experimental measurements on ultracold atomic gases.
Starting from the two-mode Bose-Hubbard model, we derive an exact version of the standard Mathieu equation governing the wave function of a Josephson junction. For a finite number of particles N, we find an additional $\mathrm{cos}2\ensuremath{\varphi}$ term in the potential. We also find that the inner product in this representation is nonlocal in $\ensuremath{\varphi}.$ Our model exhibits phenomena, such as $\ensuremath{\pi}$ oscillations, which are not found in the standard phase model, but have been predicted from Gross-Pitaevskii mean-field theory.
Taking inspiration from the state-of-the art knowledge of the Bose-Hubbard (BH) model and recent methodological developments in its fermionic counterpart, this work deals with the study of the collective dynamics of a lattice Bose gas beyond the mean-field picture through a quantum description of its elementary excitations. The Hamiltonian quantization, performed via a Bogoliubov quadratization of the BH action within the Gutzwiller approach, allows to expand the effective action of the theory up to second order in the fluctuations around the mean-field solution, as well as to prefigure the possibility of identifying the main decay vertices of the collective modes and other effects that are not evident at the second-order level. This quantum description extends the standard Bogoliubov approach to the study of superfluid Bose systems for comprising higher excitation branches, including the Higgs mode in the superfluid phase, and identifies their physical meaning together with appropriate observables which could be taken into consideration for their experimental characterization. The ultimate aim of the quantization procedure is the determination of fundamental quantities as the depletion of the condensate and the effective superfluid fraction, which are not accessible by a mean-field description or not completely characterized in the regime of strong interactions.
In the context of out of equilibrium dynamics of isolated interacting quantum many-body systems, we study the mean-field discretized and nonlinear Gross-Pitaevskii equation (GPE) of motion for a fully connected (or all-to-all coupled) Bose-Hubbard model describing a large population of bosons on a lattice of generic dimension V, with potential experimental applications ranging from ultra-cold atomic gases in optical traps to systems of Josephson junctions. Considering a quench on the system Hamiltonian, we describe the time evolution of a system initialized to the Mott insulating phase and taken in the strong superfluid regime, finding a sharp change in the dynamical behaviour when varying the quench intensity across a critical value, that is a dynamical phase transition. Our results generalize the study presented in some previous papers focusing on the particular cases of V = 2, V = 3 and 1D lattice, and place particular emphasis on the relaxation of some dynamical order parameters in the large V limit. On the side of the dynamical phase transition corresponding to the strong superfluid regime, we find such relaxation to consist of a π-synchronization of the mean field bosonic phases, that we find to compete against the eventual site-dependent disorder of the Hamiltonian in what, in analogy with the notorious Kuramoto model for nonlinear coupled oscillators, can be referred to as a synchronization transition. Additionally, in support of the validity of the mean field approximation far from the transition, we report a study beyond mean field based on the Bogolioubov-de Gennes method.