arXiv:1210.1707 · Physical Review A

Analytical results for Josephson dynamics of ultracold bosons

Lena Simon, Walter T. Strunz

Published Nov 26, 2012• 33 citations

Abstract

We study the dynamics of ultracold bosons in a double-well potential within the two-mode Bose-Hubbard model by means of semiclassical methods. By applying a Wentzel-Kramers-Brillouin (WKB) quantization we find analytical results for the energy spectrum, which are in excellent agreement with numerically exact results. They are valid in the energy range of plasma oscillations, both in the Rabi and the Josephson regime. Adopting the reflection principle and the Poisson summation formula we derive an analytical expression for the dynamics of the population imbalance depending only on the few relevant parameters of the system. This allows us to discuss its characteristic dynamics, especially the oscillation frequency and the collapse and revival time, as a function of the model parameters, leading to a deeper understanding of Josephson physics. We find that our formulas match previous experimental observations.

DOI: 10.1103/physreva.86.053625

Related papers

Time-dependent semiclassics for ultracold bosons

Lena Simon, Walter T. Strunz

We study the out-of-equilibrium dynamics of ultracold bosons in a double- and triple-well potential within the Bose-Hubbard model by means of the semiclassical Herman-Kluk propagator and compare the results to the frequently applied ``classical dynamics'' calculation in terms of the truncated Wigner approximation (TWA). For the double-well system we find the semiclassical results in excellent agreement with the numerically exact ones, while the TWA is not able to reproduce any revivals of the wave function. The triple-well system turns out to be more difficult to handle due to the irregularity of the corresponding classical phase space. Here, deviations of the TWA from the exact dynamics appear even for short times, while better agreement is obtained using the semiclassical approach presented in this article.

arXiv:1403.1096Physical Review AMay 13, 201419 citations

Wentzel-Kramers-Brillouin approach and quantum corrections to classical dynamics in the Josephson problem

Felix Nissen, Jonathan Keeling

We apply a many-body Wentzel-Kramers-Brillouin (WKB) approach to determine the leading quantum corrections to the semiclassical dynamics of the Josephson model, describing interacting bosons able to tunnel between two localized states. The semiclassical dynamics is known to divide between regular oscillations and self-trapped oscillations where the sign of the imbalance remains fixed. In both cases, the WKB wave functions are matched to Airy functions, yielding a modified Bohr-Sommerfeld quantization condition. At the critical energy dividing normal and self-trapped oscillations, the WKB wave functions should instead be matched to parabolic cylinder functions, leading to a quantization formula that is not just the Bohr-Sommerfeld formula, and recovering the known logarithmic quantum break times at this energy. This work thus provides another illustration of the usefulness of the WKB approach in certain many-body problems.

arXiv:0912.0189Physical Review AJun 23, 201022 citations

0 Nonlinear quantum model for atomic Josephson junctions with one and two bosonic species

G. Mazzarella, Marco Moratti, Luca Salasnich +1 more

We study atomic Josephson junctions (AJJs) with one and two bosonic species confined by a double-well potential. Proceeding from the second quantized Hamiltonian, we show that it is possible to describe the zero-temperature AJJ microscopic dynamics by means of extended Bose–Hubbard (EBH) models, which include usually neglected nonlinear terms. Within the mean-field approximation, the Heisenberg equations derived from such two-mode models provide a description of AJJ macroscopic dynamics in terms of ordinary differential equations (ODEs). We discuss the possibility of distinguishing the Rabi, Josephson and Fock regimes in terms of the macroscopic parameters which appear in the EBH Hamiltonians, and then in the ODEs. We compare the predictions for the relative populations of the Bose gas atoms in the two wells obtained from the numerical solutions of the two-mode ODEs, with those deriving from the direct numerical integration of the Gross–Pitaevskii equations (GPEs). Our investigations show that the nonlinear terms of the ODEs are crucial to achieve a good agreement between the ODE and GPE approaches, and in particular to give quantitative predictions of the self-trapping regime.

Aug 18, 201614 citations

Nonlinear quantum model for atomic Josephson junctions with one and two bosonic species

Giovanni Mazzarella, Marco Moratti, Luca Salasnich +1 more

arXiv:1002.2881Journal of Physics B Atomic Molecular and Optical PhysicsMar 9, 201029 citations

Nonlinear mixing of Bogoliubov modes in a bosonic Josephson junction

S. Martínez-Garaot, Giulio Pettini, Michele Modugno

We revisit the dynamics of a Bose-Einstein condensate in a double-well potential, from the regime of Josephson plasma oscillations to the self-trapping regime, by means of the Bogoliubov quasiparticle projection method. For a very small imbalance between the left and right wells only the lowest Bogoliubov mode is significantly occupied. In this regime the system performs plasma oscillations at the corresponding frequency, and the evolution of the condensate is characterized by a periodic transfer of population between the ground and the first excited state. As the initial imbalance is increased, more excited modes---though initially not macroscopically occupied---get coupled during the evolution of the system. Since their population also varies with time, the frequency spectrum of the imbalance turns out to be still peaked around a single frequency, which is continuously shifted towards lower values. The nonlinear mixing between Bogoliubov modes eventually drives the system into the the self-trapping regime, when the population of the ground state can be transferred completely to the excited states at some time during the evolution. For simplicity, here we consider a one-dimensional setup, but the results are expected to hold also in higher dimensions.

arXiv:1807.06682Physical Review AOct 24, 201819 citations

Josephson Oscillations of Bose-Einstein Condensate in Double-Well Potential: Two-Mode Model and Beyond

Haoyu Wang, Guangyu Liu, Lizhen Sun +1 more

Abstract The Josephson oscillation of a Bose-Einstein condensate (BEC) in a double-well potential is typically characterized by the population imbalance between the two wells. Within the two-mode approximation, the dynamics of the center of mass of the condensate is predicted to coincide with that of the population imbalance. However, numerical simulations reveal a discrepancy between the dynamics of the center of mass and that of the population imbalance, indicating that the two-mode model fails to fully capture the dynamics of BEC Josephson oscillations. To address this limitation, a four-mode approximation is employed to describe the dynamics of the system. The four-mode model not only describes the dynamics of the population imbalance but also accounts for the oscillation of the center of mass, thereby explaining the origin of the discrepancy between the two-mode model and the simulation results. Moreover, we find that although higher-order excitation modes are always present in Josephson oscillations, the two-mode model can still describe the dynamics of the population imbalance. This phenomenon is explained by the interference of population oscillations between local states. PACS: 03.75.Lm, 03.75.Kk, 67.85.De

Journal of Low Temperature PhysicsSep 18, 20260 citations