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arXiv:0905.2491 · Physical Review A

Quantum noise thermometry for bosonic Josephson junctions in the mean-field regime

Alex D. Gottlieb, Thorsten Schumm

Published Jun 1, 2009• 7 citations
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Abstract

Bosonic Josephson junctions can be realized by confining ultracold gases of bosons in multiwell traps and studied theoretically with the $M$-site Bose-Hubbard model. We show that canonical equilibrium states of the $M$-site Bose-Hubbard model may be approximated by mixtures of coherent states, provided the number of atoms is large and the total energy is comparable to ${k}_{B}T$. Using this approximation, we study thermal fluctuations in bosonic Josephson junctions in the mean-field regime. Statistical estimates of the fluctuations of relative phase and number, obtained by averaging over many replicates of an experiment, can be used to estimate the temperature and the tunneling parameter or to test whether the experimental procedure is effectively sampling from a canonical thermal equilibrium ensemble.

DOI
10.1103/physreva.79.063601

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Bosonic Josephson junctions can be realized by confining ultracold gases of bosons in multiwell traps and studied theoretically with the MMM-site Bose-Hubbard model. We show that canonical equilibrium states of the MMM-site Bose-Hubbard model may be approximated by mixtures of coherent states, provided the number of atoms is large and the total energy is comparable to kBT{k}_{B}TkB​T. Using this approximation, we study thermal fluctuations in bosonic Josephson junctions in the mean-field regime. Statistical estimates of the fluctuations of relative phase and number, obtained by averaging over many replicates of an experiment, can be used to estimate the temperature and the tunneling parameter or to test whether the experimental procedure is effectively sampling from a canonical thermal equilibrium ensemble.
We use the Bose-Hubbard Hamiltonian to study quantum fluctuations in canonical equilibrium ensembles of bosonic Josephson junctions at relatively high temperatures, comparing the results for finite particle numbers NNN to the classical limit that is attained as NNN approaches infinity. We consider both attractive and repulsive atom-atom interactions, with special focus on the behavior near the T=0T=0T=0 quantum phase transition that occurs, for large enough NNN, when attractive interactions surpass a critical level. Differences between Bose-Hubbard results for small NNN and those of the classical limit are quite small even when $N\ensuremath{\sim}100$, with deviations from the limit diminishing as 1/N1/N1/N.
We investigate the finite-temperature properties of a bosonic Josephson junction composed of N interacting atoms confined by a quasi-one-dimensional asymmetric double-well potential, modeled by the two-site Bose-Hubbard Hamiltonian. We compute numerically the spectral decomposition of the statistical ensemble of states, the thermodynamic and entanglement entropies, the population imbalance, the quantum Fisher information, and the coherence visibility. We analyze their dependence on the system parameters, showing in particular how finite temperature and on-site energy asymmetry affect the entanglement and coherence properties of the system. Moreover, starting from a quantum phase model which accurately describes the system over a wide range of interactions, we develop a reliable description of the strong tunneling regime, where thermal averages may be computed analytically using a modified Boltzmann weight involving an effective temperature. We discuss the possibility of applying this effective description to other models in suitable regimes.
We analyze the effects of temperature on a bosonic Josephson junction realized with ultracold and dilute atoms in a double-well potential. Starting from the eigenstates of the two-site Bose–Hubbard Hamiltonian, we calculate the coherence visibility and the fluctuation of the on-site occupation number and study them as functions of temperature. We show that, contrary to naive expectations, when the boson–boson interaction is suitably chosen thermal effects can increase the coherence visibility and reduce the on-site number fluctuation.
We investigate the thermodynamic properties of a bosonic Josephson junction in the full quantum approach and, in particular, we concentrate in studying the thermal averages of one- and two-body properties below and above the transition from delocalized to self-trapped regimes. This temperature dependence is determined by using the fact that at equilibrium the microcanonical and canonical ensembles should be equivalent. To establish the robustness of the equilibrium state, we first study a one body property and show numerically that any arbitrary state localized in energy, when evolved, reaches a stationary or equilibrium state. Comparison among averages of one- and two-body properties in the microcanonical and canonical ensembles reveals discrepances, thus leading to non-equivalence among these ensembles. Such averages differences can be attributed to the fact that the Hilbert space of the system scales as its size N, and consequently, the entropy does not scale as N. We further find as a natural consequence of studying the finite bosonic Josephson junction in the two-mode Bose Hubbard context, that positive and negative temperatures are obtained. This result can be generalized for any finite optical lattice.
We investigate the superfluid dynamics of a Josephson junction beyond the mean-field description, incorporating the role of thermal fluctuations as well as quantum fluctuations. Using a formalism that accounts for the fluctuations in a homogeneous gas, and under the assumption that the transport of the non-condensed component is negligible, we derive a corrected equation of motion within the two-site approximation. The resulting corrections for the typical dynamical quantities, like the Josephson frequency, the strength of macroscopic quantum self-trapping, and the threshold for spontaneous symmetry breaking, allow us to predict the effects of both types of fluctuations and assess their relative importance in different regimes in a semianalytical fashion. For all the dynamical quantities, the quantum fluctuations are shown to play an opposite role with respect to the thermal fluctuations. Josephson frequency is decreased by thermal fluctuations and both the critical strenghts of macroscopic quantum self trapping and spontaneous symmetry breaking are increased. We assess the experimentally accessible regimes by calculating the relevant parameters of recent experimental realizations of Bosonic Josephson junction and show that the expected regime is dominated by quantum fluctuations.
We realize a one-dimensional Josephson junction using quantum degenerate Bose gases in a tunable double well potential on an atom chip. Matter wave interferometry gives direct access to the relative phase field, which reflects the interplay of thermally driven fluctuations and phase locking due to tunneling. The thermal equilibrium state is characterized by probing the full statistical distribution function of the two-point phase correlation. Comparison to a stochastic model allows us to measure the coupling strength and temperature and hence a full characterization of the system.