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

arXiv:cond-mat/0011440

The Quantum Josephson Hamiltonian In The Phase Representation

J. R. Anglin, P. D. Drummond, Augusto Smerzi

Published Nov 26, 2000• 0 citations
arXivHTMLPDF

Abstract

The quantum Josephson Hamiltonian of two weakly linked Bose-Einstein condensates is written in an overcomplete phase representation, thus avoiding the problem of defining a Hermitian phase operator. We discuss the limit of validity of the standard, non-rigorous Mathieu equation, due to the onset of a higher order $\cos 2 ϕ$ term in the Josephson potential, and also to the overcompleteness of the representation (the phase $ϕ$ being the relative phase between the two condensates). We thereby unify the Boson Hubbard and Quantum Phase models.

DOI
10.48550/arxiv.cond-mat/0011440

Related papers

Exact quantum phase model for mesoscopic Josephson junctions

J. R. Anglin, P. D. Drummond, Augusto Smerzi

Starting from the two-mode Bose-Hubbard model, we derive an exact version of the standard Mathieu equation governing the wave function of a Josephson junction. For a finite number of particles N, we find an additional $\mathrm{cos}2\ensuremath{\varphi}$ term in the potential. We also find that the inner product in this representation is nonlocal in $\ensuremath{\varphi}.$ Our model exhibits phenomena, such as $\ensuremath{\pi}$ oscillations, which are not found in the standard phase model, but have been predicted from Gross-Pitaevskii mean-field theory.

Physical Review A•Nov 13, 2001•91 citations

On the Bosonic Phase Operator Realization for Josephson Hamiltonian Model

Fan Hong-Yi, Fan Yue

On the assumption that a Cooper pair acts as a Bose particle and based on the newly established representation, which is the common eigenvector of two particles' relative position and total momentum, we introduce a mesoscopic Josephson junction Hamiltonian constituted by two-mode Bose phase operator and number-difference operator. The number-difference-phase uncertainty relation can then be set up, which implies the existence of Josephson current.

Communications in Theoretical Physics•Jan 15, 2001•6 citations

Two mode theory of Bose–Einstein condensates: interferometry and the Josephson model

Bryan J. Dalton, Saeed Ghanbari

This topical review provides an overview of the key theoretical features of Bose–Einstein condensates (BECs) in cold atomic gases at near zero temperature in the situation where all the bosons occupy at most two single particle states or modes. This situation applies to single-component BECs in double well trap potentials and to two-component BEC in single well trap potentials, such as occur when BEC are used in interferometry experiments. The Hamiltonian is introduced in terms of field operators and mode expansions are restricted to a total of two modes. Spin operators and their eigenstates are introduced as the fundamental basis states for describing the two-mode N boson quantum system. The spin states have a macroscopic angular momentum quantum number of N/2 and the magnetic quantum number k specifies the relative number of bosons in the two modes. The treatment presented involves an extensive use of angular momentum theory, including unitary rotation operators. Important states of the two-mode system such as binomial or coherent states, relative phase eigenstates are discussed. Boson position measurements are specified via quantum correlation functions, and the use of these functions in describing coherence properties, interference patterns and fragmentation effects in BECs is presented. The Bloch vector is defined and related to the quantum correlation functions, with quantum fluctuations of the Bloch vector being treated in terms of the covariance matrix. Applications to important two-mode states are made. Spin squeezing is discussed. Based on applying variational principles, the general dynamical behaviour of the two-mode BEC is determined via generalised Gross–Pitaevskii equations for the modes and matrix mechanics equations for the probability amplitudes of the relative number basis states, the mode and amplitude equations being coupled and self-consistent. The single mode equations are also presented. The Hamiltonian is written in terms of the spin operators and the Josephson Hamiltonian obtained as a simplification in which the dynamical behaviour of the mode functions is ignored – for the one-component case the mode functions are also required to be localised and separate. Coefficients in the Josephson Hamiltonian describe tunneling/intercomponent coupling, asymmetry and collisions and these are defined via integrals involving the mode functions. The Josephson model involves using the Josephson Hamiltonian to give simple predictions of the energy states and dynamical behaviour of the two-mode system, dynamical effects on the mode functions being ignored. The three regimes – Rabi, Josephson and Fock are described, and the energy states obtained for the Fock and Rabi regimes. Dynamical behaviour treatments based on the Josephson model are outlined. In the situation where all bosons are in the same single particle state, semi-classical Bloch equations are derived and their solutions given in terms of elliptic functions. The quantum regime is treated using matrix mechanics equations for the probability amplitudes. Two representative applications of the Josephson model dynamics are treated, with graphs showing the results of numerical work being displayed. The first is in describing Heisenberg limited BEC interferometry for a single-component BEC in a double well, the treatment showing collapses and revivals in the probability distribution for the relative phase. The second treats Ramsey interferometry for a two-component BEC in a single well, the study revealing that oscillations of the Bloch vector collapse and revive, with the Bloch vector's departure from the Bloch sphere during the collapse period revealing that the BEC has fragmented. In both cases collisions cause the dephasing effects that result in the collapse, revival phenomena. The review ends with a brief outline of phase space and other approaches that extend the treatment beyond the two-mode theory, enabling decoherence effects associated with bosons in non-condensate modes to be studied. A summary of the review contents is included. Detailed mathematical derivations are included in several appendices, available as online supplementary material.

Journal of Modern Optics•Nov 18, 2011•48 citations

Quantum state of two trapped Bose-Einstein condensates with a Josephson coupling

Michael J. Steel, Matthew J. Collett

We consider the precise quantum state of two trapped, coupled Bose-Einstein condensates in the two-mode approximation. We seek a representation of the state in terms of a Wigner-like distribution on the two-mode Bloch sphere. The problem is solved using a self-consistent rotation of the unknown state to the south pole of the sphere. The two-mode Hamiltonian is projected onto the harmonic-oscillator phase plane, where it can be solved by standard techniques. Our results show how the number of atoms in each trap and the squeezing in the number difference depend on the physical parameters. Considering negative scattering lengths, we show that there is a regime of squeezing in the relative phase of the condensates which occurs for weaker interactions than the superposition states found by Cirac et al. (quant-ph/9706034). The phase squeezing is also apparent in mildly asymmetric trap configurations.

arXiv:cond-mat/9710124•Physical Review A•Apr 1, 1998•174 citations

Macroscopic quantum fluctuations in the Josephson oscillations between two weakly linked Bose-Einstein condensates

Augusto Smerzi, S. Raghavan

Summary form only given. The recent experimental observation of Bose-Einstein condensation (BEC) of trapped, dilute gas of alkali atoms, and their manipulation have opened a new chapter in the physics of systems involving the interplay of macroscopics and quantum coherence. Here, we study the 'secondary' quantum effects, i.e., the quantum corrections to the semi-classical Gross-Pitaevskii equation, for two weakly-linked Bose-Einstein condensates. We hence generalize the Josephson current-phase relations by including the effects of quantum fluctuations of the pertinent observables. Such equations will also describe the quantum fluctuations in the Josephson dynamics between two weakly linked mesoscopic superconducting grains.

Quantum Electronics and Laser Science Conference•Jan 20, 2003•0 citations

Coherent oscillations between two weakly coupled Bose-Einstein Condensates: Josephson effects, $π$-oscillations, and macroscopic quantum self trapping

S. Raghavan, Augusto Smerzi, Stefano Fantoni +1 more

We discuss the coherent atomic oscillations between two weakly coupled Bose-Einstein condensates. The weak link is provided by a laser barrier in a (possibly asymmetric) double-well trap or by Raman coupling between two condensates in different hyperfine levels. The Boson Josephson Junction (BJJ) dynamics is described by the two-mode non-linear Gross-Pitaevskii equation, that is solved analytically in terms of elliptic functions. The BJJ, being a $neutral$, isolated system, allows the investigations of new dynamical regimes for the phase difference across the junction and for the population imbalance, that are not accessible with Superconductor Josephson Junctions (SJJ). These include oscillations with either, or both of the following properties: 1) the time-averaged value of the phase is equal to $π$ ($π-phase$ oscillations); 2) the average population imbalance is nonzero, in states with ``macroscopic quantum self-trapping'' (MQST). The (non-sinusoidal) generalization of the SJJ `ac' and `plasma' oscillations and the Shapiro resonance can also be observed. We predict the collapse of experimental data (corresponding to different trap geometries and total number of condensate atoms) onto a single universal curve, for the inverse period of oscillations. Analogies with Josephson oscillations between two weakly coupled reservoirs of $^3$He-B and the internal Josephson effect in $^3$He-A are also discussed.

arXiv:cond-mat/9706220•Jun 23, 1997•2 citations
The quantum Josephson Hamiltonian of two weakly linked Bose-Einstein condensates is written in an overcomplete phase representation, thus avoiding the problem of defining a Hermitian phase operator. We discuss the limit of validity of the standard, non-rigorous Mathieu equation, due to the onset of a higher order cos⁡2ϕ\cos 2 ϕcos2ϕ term in the Josephson potential, and also to the overcompleteness of the representation (the phase ϕϕϕ being the relative phase between the two condensates). We thereby unify the Boson Hubbard and Quantum Phase models.
Starting from the two-mode Bose-Hubbard model, we derive an exact version of the standard Mathieu equation governing the wave function of a Josephson junction. For a finite number of particles N, we find an additional $\mathrm{cos}2\ensuremath{\varphi}$ term in the potential. We also find that the inner product in this representation is nonlocal in $\ensuremath{\varphi}.$ Our model exhibits phenomena, such as $\ensuremath{\pi}$ oscillations, which are not found in the standard phase model, but have been predicted from Gross-Pitaevskii mean-field theory.
This topical review provides an overview of the key theoretical features of Bose–Einstein condensates (BECs) in cold atomic gases at near zero temperature in the situation where all the bosons occupy at most two single particle states or modes. This situation applies to single-component BECs in double well trap potentials and to two-component BEC in single well trap potentials, such as occur when BEC are used in interferometry experiments. The Hamiltonian is introduced in terms of field operators and mode expansions are restricted to a total of two modes. Spin operators and their eigenstates are introduced as the fundamental basis states for describing the two-mode N boson quantum system. The spin states have a macroscopic angular momentum quantum number of N/2 and the magnetic quantum number k specifies the relative number of bosons in the two modes. The treatment presented involves an extensive use of angular momentum theory, including unitary rotation operators. Important states of the two-mode system such as binomial or coherent states, relative phase eigenstates are discussed. Boson position measurements are specified via quantum correlation functions, and the use of these functions in describing coherence properties, interference patterns and fragmentation effects in BECs is presented. The Bloch vector is defined and related to the quantum correlation functions, with quantum fluctuations of the Bloch vector being treated in terms of the covariance matrix. Applications to important two-mode states are made. Spin squeezing is discussed. Based on applying variational principles, the general dynamical behaviour of the two-mode BEC is determined via generalised Gross–Pitaevskii equations for the modes and matrix mechanics equations for the probability amplitudes of the relative number basis states, the mode and amplitude equations being coupled and self-consistent. The single mode equations are also presented. The Hamiltonian is written in terms of the spin operators and the Josephson Hamiltonian obtained as a simplification in which the dynamical behaviour of the mode functions is ignored – for the one-component case the mode functions are also required to be localised and separate. Coefficients in the Josephson Hamiltonian describe tunneling/intercomponent coupling, asymmetry and collisions and these are defined via integrals involving the mode functions. The Josephson model involves using the Josephson Hamiltonian to give simple predictions of the energy states and dynamical behaviour of the two-mode system, dynamical effects on the mode functions being ignored. The three regimes – Rabi, Josephson and Fock are described, and the energy states obtained for the Fock and Rabi regimes. Dynamical behaviour treatments based on the Josephson model are outlined. In the situation where all bosons are in the same single particle state, semi-classical Bloch equations are derived and their solutions given in terms of elliptic functions. The quantum regime is treated using matrix mechanics equations for the probability amplitudes. Two representative applications of the Josephson model dynamics are treated, with graphs showing the results of numerical work being displayed. The first is in describing Heisenberg limited BEC interferometry for a single-component BEC in a double well, the treatment showing collapses and revivals in the probability distribution for the relative phase. The second treats Ramsey interferometry for a two-component BEC in a single well, the study revealing that oscillations of the Bloch vector collapse and revive, with the Bloch vector's departure from the Bloch sphere during the collapse period revealing that the BEC has fragmented. In both cases collisions cause the dephasing effects that result in the collapse, revival phenomena. The review ends with a brief outline of phase space and other approaches that extend the treatment beyond the two-mode theory, enabling decoherence effects associated with bosons in non-condensate modes to be studied. A summary of the review contents is included. Detailed mathematical derivations are included in several appendices, available as online supplementary material.
We consider the precise quantum state of two trapped, coupled Bose-Einstein condensates in the two-mode approximation. We seek a representation of the state in terms of a Wigner-like distribution on the two-mode Bloch sphere. The problem is solved using a self-consistent rotation of the unknown state to the south pole of the sphere. The two-mode Hamiltonian is projected onto the harmonic-oscillator phase plane, where it can be solved by standard techniques. Our results show how the number of atoms in each trap and the squeezing in the number difference depend on the physical parameters. Considering negative scattering lengths, we show that there is a regime of squeezing in the relative phase of the condensates which occurs for weaker interactions than the superposition states found by Cirac et al. (quant-ph/9706034). The phase squeezing is also apparent in mildly asymmetric trap configurations.
Summary form only given. The recent experimental observation of Bose-Einstein condensation (BEC) of trapped, dilute gas of alkali atoms, and their manipulation have opened a new chapter in the physics of systems involving the interplay of macroscopics and quantum coherence. Here, we study the 'secondary' quantum effects, i.e., the quantum corrections to the semi-classical Gross-Pitaevskii equation, for two weakly-linked Bose-Einstein condensates. We hence generalize the Josephson current-phase relations by including the effects of quantum fluctuations of the pertinent observables. Such equations will also describe the quantum fluctuations in the Josephson dynamics between two weakly linked mesoscopic superconducting grains.
Coherent oscillations between two weakly coupled Bose-Einstein Condensates: Josephson effects, πππ-oscillations, and macroscopic quantum self trapping
We discuss the coherent atomic oscillations between two weakly coupled Bose-Einstein condensates. The weak link is provided by a laser barrier in a (possibly asymmetric) double-well trap or by Raman coupling between two condensates in different hyperfine levels. The Boson Josephson Junction (BJJ) dynamics is described by the two-mode non-linear Gross-Pitaevskii equation, that is solved analytically in terms of elliptic functions. The BJJ, being a neutralneutralneutral, isolated system, allows the investigations of new dynamical regimes for the phase difference across the junction and for the population imbalance, that are not accessible with Superconductor Josephson Junctions (SJJ). These include oscillations with either, or both of the following properties: 1) the time-averaged value of the phase is equal to πππ (π−phaseπ-phaseπ−phase oscillations); 2) the average population imbalance is nonzero, in states with ``macroscopic quantum self-trapping'' (MQST). The (non-sinusoidal) generalization of the SJJ `ac' and `plasma' oscillations and the Shapiro resonance can also be observed. We predict the collapse of experimental data (corresponding to different trap geometries and total number of condensate atoms) onto a single universal curve, for the inverse period of oscillations. Analogies with Josephson oscillations between two weakly coupled reservoirs of 3^33He-B and the internal Josephson effect in 3^33He-A are also discussed.