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arXiv:2112.13231 · Physical review. B./Physical review. B

Quantum effective action for the bosonic Josephson junction

Koichiro Furutani, Jacques Tempere, Luca Salasnich

Published Apr 13, 2022• 9 citations
arXivHTMLPDF

Abstract

We investigate a bosonic Josephson junction by using the path-integral formalism with relative phase and population imbalance as dynamical variables. We derive an effective only-phase action performing functional integration over the population imbalance. We then analyze the quantum effective only-phase action, which formally contains all the quantum corrections. To the second order in the derivative expansion and to the lowest order in $\ensuremath{\hbar}$, we obtain the quantum correction to the Josephson frequency of oscillation. Finally, the same quantum correction is found by adopting an alternative approach. Our predictions are a useful theoretical tool for experiments with atomic or superconducting Josephson junctions.

DOI
10.1103/physrevb.105.134510

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We investigate a bosonic Josephson junction by using the path-integral formalism with relative phase and population imbalance as dynamical variables. We derive an effective only-phase action performing functional integration over the population imbalance. We then analyze the quantum effective only-phase action, which formally contains all the quantum corrections. To the second order in the derivative expansion and to the lowest order in \ensuremathℏ\ensuremath{\hbar}\ensuremathℏ, we obtain the quantum correction to the Josephson frequency of oscillation. Finally, the same quantum correction is found by adopting an alternative approach. Our predictions are a useful theoretical tool for experiments with atomic or superconducting Josephson junctions.
ABSTRACT We investigate quantum corrections to the Josephson dynamics of two weakly coupled Bose–Einstein condensates using the population imbalance as the sole dynamical variable. Starting from the two‐variable action, we derive the imbalance‐only Lagrangian with a position‐dependent mass and quantize it via symmetric operator ordering. The leading quantum corrections to the classical potential and mass are computed via the one‐loop quantum effective action, using a covariant background‐field method that fully accounts for the coordinate dependence of the mass. This yields explicit expressions for the effective potential and the effective mass, from which we derive the quantum‐corrected Josephson frequency. Numerical comparison with exact diagonalization of the two‐site Bose–Hubbard model shows that the imbalance‐only formulation performs better than the complementary phase‐only approach in the regime of weak interactions.
The dissipative dynamics of a Josephson junction in Bose gases is considered within the framework of the model of a tunneling Hamiltonian. The effective action that describes the dynamics of the phase difference across the junction is derived using the functional integration method. The dynamic equation obtained for the phase difference across the junction is analyzed for the finite temperatures in the low-frequency limit involving the radiation terms. The asymmetric case of the Bose gases with the different order parameters is calculated as well.
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.
The bosonic Josephson junction, one of the maximally simple models for periodic-driven many-body systems, has been intensively studied in the past two decades. Here, we revisit this problem with five different methods, all of which have solid theoretical reasoning. We find that to the order of ω^{-2} (ω is the modulating frequency), these approaches will yield slightly different effective Hamiltonians. In particular, the parameters in the effective Hamiltonians may be unchanged, increased, or decreased, depending on the approximations used. Especially, some of the methods generate new interactions, which still preserve the total number of particles, and the others do not. The validity of these five effective models is verified using dynamics of population imbalance and self-trapping phase transition, and we find that the method based on the rotating frame by a unitary transformation has the highest accuracy. The difference between these methods will become significant when the driving amplitude or interaction strengths are comparable with the driving frequency. These differences can also be manifested from their long-time dynamics. We demonstrate this physics using a Bose-Josephson junction, and it is to be hoped that the validity of these methods and their tiny differences put forward in this paper can be verified in realistic experiments in the future using quantum simulating platforms, including but not limited to ultracold atoms.
In this paper we consider a bosonic Josephson junction described by a two-mode Bose-Hubbard model, and we thoroughly analyze a quantum phase transition occurring in the system in the limit of infinite bosonic population. We discuss the relation between this quantum phase transition and the dynamical bifurcation occurring in the spectrum of the discrete self-trapping equations describing the system at the semiclassical level. In particular, we identify five regimes depending on the strength of the effective interaction among bosons, and study the finite-size effects arising from the finiteness of the bosonic population. We devote special attention to the critical regime which reduces to the dynamical bifurcation point in the thermodynamic limit of infinite bosonic population. Specifically, we highlight an anomalous scaling in the population imbalance between the two wells of the trapping potential, as well as in two quantities borrowed from quantum information theory, i.e., the entropy of entanglement and the ground-state fidelity. Our analysis is not limited to the zero-temperature case, but considers thermal effects as well.
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.