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arXiv:2101.01009 · Physical Review A

Finite-size effects in a bosonic Josephson junction

Sandro Wimberger, Gabriele Manganelli, Alberto Brollo, Luca Salasnich

Published Feb 24, 2021• 15 citations
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Abstract

We investigate finite-size quantum effects in the dynamics of $N$ bosonic particles which are tunneling between two sites adopting the two-site Bose-Hubbard model. By using time-dependent atomic coherent states (ACSs) we extend the standard mean-field equations of this bosonic Josephson junction, which are based on time-dependent Glauber coherent states. In this way we find $1/N$ corrections to familiar mean-field (MF) results: the frequency of macroscopic oscillation between the two sites, the critical parameter for the dynamical macroscopic quantum self-trapping (MQST), and the attractive critical interaction strength for the spontaneous symmetry breaking (SSB) of the ground state. To validate our analytical results we perform numerical simulations of the quantum dynamics. In the case of Josephson oscillations around a balanced configuration we also find that, for a few atoms, the numerical results are in good agreement with the predictions of time-dependent ACS variational approach, provided that the time evolution is not too long. Also, the numerical results of SSB are better reproduced by the ACS approach than with the MF approach. Instead, the onset of MQST is correctly reproduced by ACS theory only in the large $N$ regime and, for this phenomenon, the $1/N$ correction to the MF formula is not reliable.

DOI
10.1103/physreva.103.023326

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We investigate finite-size quantum effects in the dynamics of NNN bosonic particles which are tunneling between two sites adopting the two-site Bose-Hubbard model. By using time-dependent atomic coherent states (ACSs) we extend the standard mean-field equations of this bosonic Josephson junction, which are based on time-dependent Glauber coherent states. In this way we find 1/N1/N1/N corrections to familiar mean-field (MF) results: the frequency of macroscopic oscillation between the two sites, the critical parameter for the dynamical macroscopic quantum self-trapping (MQST), and the attractive critical interaction strength for the spontaneous symmetry breaking (SSB) of the ground state. To validate our analytical results we perform numerical simulations of the quantum dynamics. In the case of Josephson oscillations around a balanced configuration we also find that, for a few atoms, the numerical results are in good agreement with the predictions of time-dependent ACS variational approach, provided that the time evolution is not too long. Also, the numerical results of SSB are better reproduced by the ACS approach than with the MF approach. Instead, the onset of MQST is correctly reproduced by ACS theory only in the large NNN regime and, for this phenomenon, the 1/N1/N1/N correction to the MF formula is not reliable.
The mean-field approach to two-site Bose-Hubbard systems is well established and leads to nonlinear classical equations of motion for the population imbalance and the phase difference. It can, e.g., be based on the representation of the solution of the time-dependent Schrodinger equation either by a single Glauber state or by a single atomic (SU(2)) coherent state [S. Wimberger et al., Phys. Rev. A 103, 023326 (2021)]. We demonstrate that quantum effects beyond the mean-field approximation are easily uncovered if, instead, a multi-configuration ansatz with a few time-dependent SU(2) basis functions is used in the variational principle. For the case of plasma oscillations, the use of just two basis states, whose time-dependent parameters are determined variationally, already gives good qualitative agreement of the phase space dynamics with numerically exact quantum solutions. In order to correctly account for more non-trivial effects, like macroscopic quantum self trapping, moderately more basis states are needed. If one is interested in the onset of spontaneous symmetry breaking, however, a multiplicity of two gives a big improvement towards the exact result already. In any case, the number of variational trajectories needed for good agreement with full quantum results is orders of magnitude smaller than in the semiclassical case, which is based on multiple mean-field trajectories.
The mean-field approach to two-site Bose–Hubbard systems is well-established and leads to non-linear classical equations of motion for population imbalance and phase difference. It can, for example, be based on the representation of the solution of the time-dependent Schrödinger equation either by a single Glauber state or by a single atomic (SU(2)) coherent state [S. Wimberger et al., Phys. Rev. A 103, 023326 (2021)]. We demonstrate that quantum effects beyond the mean-field approximation are easily uncovered if, instead, a multiconfiguration ansatz with a few time-dependent SU(2) basis functions is used in the variational principle. For the case of plasma oscillations, the use of just two basis states, whose time-dependent parameters are determined variationally, already gives a good qualitative agreement of the phase space dynamics with numerically exact quantum solutions. In order to correctly account for more non-trivial effects, like macroscopic quantum self-trapping, moderately more basis states are needed. For the onset of spontaneous symmetry breaking, however, a multiplicity of 2 gives a significant improvement already. In any case, the number of variational trajectories needed for good agreement with the full quantum results is orders of magnitude smaller than that in the semi-classical case, which is based on multiple mean-field trajectories.
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 investigate the fully quantum evolution of the population imbalance in a perfectly symmetric Bose-Josephson junction modeled by a two-mode Bose-Hubbard Hamiltonian, focusing on the validity of macroscopic quantum self-trapping beyond the mean-field theory. We show that for any finite number of particles, the exact quantum dynamics leads to the breakdown of macroscopic quantum self-trapping after a finite time, regardless of the initial state. Using the symmetries of the Bose-Hubbard Hamiltonian, we provide a mathematical demonstration of this result and analyze the spectral properties governing the dynamics. We identify a branching behavior in the eigenvalue differences and a nontrivial structure of the population-imbalance amplitudes. These features allow us to distinguish two clearly different dynamical regimes and to elucidate the mechanism leading to the emergence of a quasi–macroscopic quantum self-trapping regime for large particle numbers. These findings bridge the gap between mean-field predictions and exact quantum dynamics and provide insight into the emergence of classical nonlinear behavior from finite quantum many-body systems.
We study an atomic Josephson junction (AJJ) in the presence of two interacting Bose–Einstein condensates (BECs) confined in a double-well trap. We assume that bosons of different species interact with each other. The macroscopic wavefunctions of the two components obey a system of two 3D coupled Gross–Pitaevskii equations (GPEs). We write the Lagrangian of the system, and from this we derive a system of coupled ordinary differential equations (ODEs), for which the coupled pendula represent the mechanical analogues. These differential equations control the dynamical behaviour of the fractional imbalance and of the relative phase of each bosonic component. We perform the stability analysis around the points which preserve the symmetry and get an analytical formula for the oscillation frequency around the stable points. Such a formula could be used as an indirect measure of the inter-species s-wave scattering length. We also study the oscillations of each fractional imbalance around zero and nonzero—the macroscopic quantum self-trapping (MQST)—time averaged values. For different values of the inter-species interaction amplitude, we carry out this study both by directly solving the two GPEs and by solving the corresponding coupled pendula equations. We show that, under certain conditions, the predictions of these two approaches are in good agreement. Moreover, we calculate the crossover value of the inter-species interaction amplitude which signals the onset of MQST.
Abstract We investigate the role of quantum fluctuations in the dynamics of a bosonic Josephson junction in D spatial dimensions, by using beyond-mean-field Gaussian corrections. We derive some key dynamical properties in a systematic way for D = 3 , 2 , 1 . In particular, we compute the Josephson frequency in the regime of low population imbalance. We also obtain the critical strength of the macroscopic quantum self-trapping. Our results show that quantum corrections increase the Josephson frequency in spatial dimensions D = 2 and D = 3, but they decrease it in the D = 1 case. The critical strength of macroscopic quantum self-trapping is instead reduced by quantum fluctuations in D = 2 and D = 3 cases, while it is enhanced in the D = 1 configuration. We show that the difference between the cases of D = 2 and D = 3 on one side, and D = 1 on the other, can be related to the qualitatively different dependence of the interaction strength on the scattering length in the different dimensions.