Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Small-tunneling-amplitude boson-Hubbard dimer. II. Dynamics

G. Kalosakas, A. R. Bishop, V. M. Kenkre

Published Aug 11, 2003• 45 citations
Publisher

Abstract

We present analytical relations for the quantum evolution of the number difference of bosons between the two sites of a double-well potential, by using perturbative results for small tunneling amplitudes in the two-mode approximation. Results are obtained for two different initial conditions: completely localized states and coherent spin states. In the former case both the short and the long time behavior is investigated and the characteristic Bohr frequencies of the energy spectrum are determined. In the latter case we calculate the short time-scale evolution of the number difference. The analytical expressions compare favorably with direct numerical solutions of the quantum problem. Finally, we discuss the corresponding Gross-Pitaevskii (i.e., mean-field) dynamics and we point out the differences between the full quantum evolution and the mean-field evolution.

DOI
10.1103/physreva.68.023602

Related papers

Quantum Bose–Josephson junction with binary mixtures of BECs

Adele Naddeo, R. Citro

We study the quantum behaviour of a binary mixture of Bose–Einstein condensates in a double-well potential starting from a two-mode Bose–Hubbard Hamiltonian. We focus on the small tunnelling amplitude regime and apply perturbation theory up to second order. Analytical expressions for the energy eigenvalues and eigenstates are obtained. Then, the quantum evolution of the number difference of bosons between the two potential wells is fully investigated for two different initial conditions: completely localized states and coherent spin states. In the first case both the short- and the long-time dynamics is studied and a rich behaviour is found, ranging from small amplitude oscillations and collapses and revivals to coherent tunnelling. In the second case, the short-timescale evolution of number difference is determined and a more irregular dynamics is evidenced. Finally, the formation of Schrödinger cat states is considered and shown to affect the momentum distribution.

arXiv:1003.0123•Journal of Physics B Atomic Molecular and Optical Physics•Jun 14, 2010•20 citations

Small-tunneling-amplitude boson-Hubbard dimer: Stationary states

G. Kalosakas, A. R. Bishop

Analytical expressions for the energy eigenstates of the boson-Hubbard (or quantum discrete-nonlinear Schr\"odinger) dimer are obtained by applying perturbation theory in the small-tunneling-amplitude limit. The results are relevant for a Bose-Einstein condensate trapped in a double-well potential. A detailed comparison with the numerical solutions arising from the direct diagonalization of the Hamiltonian (for even or odd numbers of bosons) is presented, and the limits of validity of the perturbative results are determined. For fixed values of the tunneling amplitude, the accuracy of the perturbative calculation varies with the total number of bosons in different ways, depending on the particular energy level. Results for the fluctuations of the boson-number difference in each eigenstate are also presented.

Physical Review A•Apr 10, 2002•39 citations

Quantum dynamics of a binary mixture of BECs in a double-well potential: a Holstein–Primakoff approach

R. Citro, Adele Naddeo, E. Orignac

We study the quantum dynamics of a binary mixture of Bose-Einstein condensates (BEC) in a double-well potential starting from a two-mode Bose-Hubbard Hamiltonian. Focussing on the regime where the number of atoms is very large, a mapping onto a SU(2) spin problem together with a Holstein-Primakoff transformation is performed. The quantum evolution of the number difference of bosons between the two wells is investigated for different initial conditions, which range from the case of a small imbalance between the two wells to a coherent spin state. The results show an instability towards a phase-separation above a critical positive value of the interspecies interaction while the system evolves towards a coherent tunneling regime for negative interspecies interactions. A comparison with a semiclassical approach is discussed together with some implications on the experimental realization of phase separation with cold atoms.

arXiv:1104.4965•Journal of Physics B Atomic Molecular and Optical Physics•May 19, 2011•5 citations

The open Bose-Hubbard dimer

Tadeusz Pudlik

This dissertation discusses a number of theoretical models of coupled bosonic modes, all closely related to the Bose-Hubbard dimer. In studying these models, we will repeatedly return to two unifying themes: the classical structure underlying quantum dynamics and the impact of weakly coupling a system to an environment. Or, more succinctly, semiclassical methods and open quantum systems. \n \nOur primary motivation for studying models such as the Bose-Hubbard is their relevance to ongoing ultracold atom experiments. We review these experiments, derive the Bose-Hubbard model in their context and briefly discuss its limitations in the first half of Chapter 1. In its second half, we review the theory of open quantum systems and the master equation description of the dissipative Bose-Hubbard model. This opening chapter constitutes a survey of existing results, rather than original work. \n \nIn Chapter 2, we turn to the mean-field limit of the Bose-Hubbard model. After reviewing the striking localization phenomena predicted by the mean-field (and confirmed by experiment), we identify the first corrections to this picture for the dimer. The most interesting of these is the dynamical tunneling between the self-trapping points of the mean-field. We derive an accurate analytical expression for the tunneling rate using semiclassical techniques. \n \nWe continue studying the dynamics near the self-trapping fixed points in Chapter 3, focusing on corrections to the mean-field that arise at larger nonlinearities and on shorter time scales than dynamical tunneling. We study the impact of dissipation on coherence and entanglement near the fixed points, and explain it in terms of the structure of the classical phase space. \n \nThe last chapter of the dissertation is also devoted to a dissipative bosonic dimer model, but one arising in a very different physical context. Abandoning optical lattices, we consider the problem of formulating a quantum model of operation of the cylindrical anode magnetron, a vacuum tube crossed-field microwave amplifier. We derive an effective dissipative dimer model and study its relationship to the classical description. Our dimer model is a first step towards the analysis of solid-state analogs of such devices.

•Jan 1, 2016•0 citations

Coherent destruction of tunneling of two interacting bosons in a tight-binding lattice

Stefano Longhi, Giuseppe Della Valle

The dynamics of two correlated bosons hopping on a one-dimensional lattice and driven by a sinusoidal force is analytically investigated by means of a multiple-time-scale asymptotic analysis of the driven Bose-Hubbard Hamiltonian. By assuming the ratio $\ensuremath{\epsilon}=J/\ensuremath{\omega}$ between the tunneling rate $J$ and modulation frequency $\ensuremath{\omega}$ of the driving field as a small parameter, the perturbative analysis shows that, under appropriate driving conditions, coherent destruction of tunneling can be simultaneously realized for both paired and unpaired states up to the long time scale $\ensuremath{\sim}$$1/(\ensuremath{\omega}{\ensuremath{\epsilon}}^{2})$ in the regime corresponding to an energy quantum $\ensuremath{\hbar}\ensuremath{\omega}$ of the driving field of the same order of magnitude as the particle interaction energy $\ensuremath{\hbar}{U}_{0}$. The predictions of the perturbative analysis are confirmed by direct numerical simulations of the two-particle Bose-Hubbard Hamiltonian and by direct quasienergy-band computation.

arXiv:1203.5033•Physical Review A•Oct 4, 2012•23 citations

Finite-size effects in a bosonic Josephson junction

Sandro Wimberger, Gabriele Manganelli, Alberto Brollo +1 more

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.

arXiv:2101.01009•Physical Review A•Feb 24, 2021•15 citations
We present analytical relations for the quantum evolution of the number difference of bosons between the two sites of a double-well potential, by using perturbative results for small tunneling amplitudes in the two-mode approximation. Results are obtained for two different initial conditions: completely localized states and coherent spin states. In the former case both the short and the long time behavior is investigated and the characteristic Bohr frequencies of the energy spectrum are determined. In the latter case we calculate the short time-scale evolution of the number difference. The analytical expressions compare favorably with direct numerical solutions of the quantum problem. Finally, we discuss the corresponding Gross-Pitaevskii (i.e., mean-field) dynamics and we point out the differences between the full quantum evolution and the mean-field evolution.
We study the quantum behaviour of a binary mixture of Bose–Einstein condensates in a double-well potential starting from a two-mode Bose–Hubbard Hamiltonian. We focus on the small tunnelling amplitude regime and apply perturbation theory up to second order. Analytical expressions for the energy eigenvalues and eigenstates are obtained. Then, the quantum evolution of the number difference of bosons between the two potential wells is fully investigated for two different initial conditions: completely localized states and coherent spin states. In the first case both the short- and the long-time dynamics is studied and a rich behaviour is found, ranging from small amplitude oscillations and collapses and revivals to coherent tunnelling. In the second case, the short-timescale evolution of number difference is determined and a more irregular dynamics is evidenced. Finally, the formation of Schrödinger cat states is considered and shown to affect the momentum distribution.
Analytical expressions for the energy eigenstates of the boson-Hubbard (or quantum discrete-nonlinear Schr\"odinger) dimer are obtained by applying perturbation theory in the small-tunneling-amplitude limit. The results are relevant for a Bose-Einstein condensate trapped in a double-well potential. A detailed comparison with the numerical solutions arising from the direct diagonalization of the Hamiltonian (for even or odd numbers of bosons) is presented, and the limits of validity of the perturbative results are determined. For fixed values of the tunneling amplitude, the accuracy of the perturbative calculation varies with the total number of bosons in different ways, depending on the particular energy level. Results for the fluctuations of the boson-number difference in each eigenstate are also presented.
We study the quantum dynamics of a binary mixture of Bose-Einstein condensates (BEC) in a double-well potential starting from a two-mode Bose-Hubbard Hamiltonian. Focussing on the regime where the number of atoms is very large, a mapping onto a SU(2) spin problem together with a Holstein-Primakoff transformation is performed. The quantum evolution of the number difference of bosons between the two wells is investigated for different initial conditions, which range from the case of a small imbalance between the two wells to a coherent spin state. The results show an instability towards a phase-separation above a critical positive value of the interspecies interaction while the system evolves towards a coherent tunneling regime for negative interspecies interactions. A comparison with a semiclassical approach is discussed together with some implications on the experimental realization of phase separation with cold atoms.
This dissertation discusses a number of theoretical models of coupled bosonic modes, all closely related to the Bose-Hubbard dimer. In studying these models, we will repeatedly return to two unifying themes: the classical structure underlying quantum dynamics and the impact of weakly coupling a system to an environment. Or, more succinctly, semiclassical methods and open quantum systems. \n \nOur primary motivation for studying models such as the Bose-Hubbard is their relevance to ongoing ultracold atom experiments. We review these experiments, derive the Bose-Hubbard model in their context and briefly discuss its limitations in the first half of Chapter 1. In its second half, we review the theory of open quantum systems and the master equation description of the dissipative Bose-Hubbard model. This opening chapter constitutes a survey of existing results, rather than original work. \n \nIn Chapter 2, we turn to the mean-field limit of the Bose-Hubbard model. After reviewing the striking localization phenomena predicted by the mean-field (and confirmed by experiment), we identify the first corrections to this picture for the dimer. The most interesting of these is the dynamical tunneling between the self-trapping points of the mean-field. We derive an accurate analytical expression for the tunneling rate using semiclassical techniques. \n \nWe continue studying the dynamics near the self-trapping fixed points in Chapter 3, focusing on corrections to the mean-field that arise at larger nonlinearities and on shorter time scales than dynamical tunneling. We study the impact of dissipation on coherence and entanglement near the fixed points, and explain it in terms of the structure of the classical phase space. \n \nThe last chapter of the dissertation is also devoted to a dissipative bosonic dimer model, but one arising in a very different physical context. Abandoning optical lattices, we consider the problem of formulating a quantum model of operation of the cylindrical anode magnetron, a vacuum tube crossed-field microwave amplifier. We derive an effective dissipative dimer model and study its relationship to the classical description. Our dimer model is a first step towards the analysis of solid-state analogs of such devices.
The dynamics of two correlated bosons hopping on a one-dimensional lattice and driven by a sinusoidal force is analytically investigated by means of a multiple-time-scale asymptotic analysis of the driven Bose-Hubbard Hamiltonian. By assuming the ratio $\ensuremath{\epsilon}=J/\ensuremath{\omega}$ between the tunneling rate JJJ and modulation frequency $\ensuremath{\omega}$ of the driving field as a small parameter, the perturbative analysis shows that, under appropriate driving conditions, coherent destruction of tunneling can be simultaneously realized for both paired and unpaired states up to the long time scale $\ensuremath{\sim}$$1/(\ensuremath{\omega}{\ensuremath{\epsilon}}^{2})$ in the regime corresponding to an energy quantum $\ensuremath{\hbar}\ensuremath{\omega}$ of the driving field of the same order of magnitude as the particle interaction energy $\ensuremath{\hbar}{U}_{0}$. The predictions of the perturbative analysis are confirmed by direct numerical simulations of the two-particle Bose-Hubbard Hamiltonian and by direct quasienergy-band computation.
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.