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

Persistence of mean-field features in the energy spectrum of small arrays of Bose-Einstein condensates

Pierfrancesco Buonsante, Franzosi, R., Vittorio Penna

Published Jan 1, 2004• 7 citations
OpenAlex

Abstract

The Bose-Hubbard Hamiltonian capturing the essential physics of the arrays of interacting Bose-Einstein condensates is addressed, focusing on arrays consisting of two (dimer) and three (trimer) sites.In the former case, some results concerning the persistence of mean-field features in the energy spectrum of the symmetric dimer are extended to the asymmetric version of the system, where the two sites are characterized by different on-site energies.Based on a previous systematic study of the mean-field limit of the trimer, where the dynamics is exhaustively described in terms of its fixed points for every choice of the significant parameters, an interesting mapping between the dimer and the trimer is emphasized and used as a guide in investigating the persistence of mean-field features in the rather complex energy spectrum of the trimer.These results form the basis for the systematic investigation of the purely quantum trimer extending and completing the existing mean-field analysis.In this respect we recall that, similar to larger arrays, the trimer is characterized by a non-integrable mean-field dynamics featuring chaotic trajectories.Hence, the correspondence between mean-field fixed points and quantum energy levels emphasized in the present work may provide a key to investigate the quantum counterpart of classical instability.

Related papers

Persistence of mean-field features in the energy spectrum of small arrays of Bose–Einstein condensates

P Buonsante, R Franzosi, V Penna

The Bose-Hubbard Hamiltonian capturing the essential physics of the arrays of interacting Bose-Einstein condensates is addressed, focusing on arrays consisting of two (dimer) and three (trimer) sites. In the former case, some results concerning the persistence of mean-field features in the energy spectrum of the symmetric dimer are extended to the asymmetric version of the system, where the two sites are characterized by different on-site energies. Based on a previous systematic study of the mean-field limit of the trimer, where the dynamics is exhaustively described in terms of its fixed points for every choice of the significant parameters, an interesting mapping between the dimer and the trimer is emphasized and used as a guide in investigating the persistence of mean-field features in the rather complex energy spectrum of the trimer. These results form the basis for the systematic investigation of the purely quantum trimer extending and completing the existing mean-field analysis. In this respect we recall that, similar to larger arrays, the trimer is characterized by a non-integrable mean-field dynamics featuring chaotic trajectories. Hence, the correspondence between mean-field fixed points and quantum energy levels emphasized in the present work may provide a key to investigate the quantum counterpart of classical instability.

arXiv:cond-mat/0312596•Journal of Physics B Atomic Molecular and Optical Physics•Mar 24, 2004•12 citations

Spectral properties and Self-trapping effects in coupled Bose-Einstein Condensates

Roberto Franzosi, Vittorio Penna

We study the energy spectrum structure of a system of two interacting bosonic wells (dimer model) occupied by N bosons and compare it with the phase-space topology of the same model within the mean-field approach. To this end, we characterize the structure of energy eigenstates by using the symmetry properties of the Hamiltonian, and show that the energy levels are nondegenerate. The presence of doublets leads to recover in the classical limit the self-trapping effect exhibited by the mean-field dimer model. Finally, we do some comments on trimer dynamics to show how the interaction with a third well can cause both dimer-like regimes and strongly chaotic behaviors.

Laser Physics•Jan 1, 2002•1 citation

Dynamical Instability in a Trimeric Chain of Interacting Bose-Einstein Condensates

Pierfrancesco Buonsante, Roberto Franzosi, Vittorio Penna

We analyze thoroughly the mean-field dynamics of a linear chain of three coupled Bose-Einstein condensates, where both the tunneling and the central-well relative depth are adjustable parameters. Owing to its nonintegrability, entailing a complex dynamics with chaos occurrence, this system is a paradigm for longer arrays whose simplicity still allows a thorough analytical study. We identify the set of dynamics fixed points, along with the associated proper modes, and establish their stability character depending on the significant parameters. As an example of the remarkable operational value of our analysis, we point out some macroscopic effects that seem viable to experiments.

arXiv:cond-mat/0610532•Physical Review Letters•Feb 5, 2003•63 citations

Fragmented condensation in Bose–Hubbard trimers with tunable tunnelling

Albert Gallemí, M. Guilleumas, Joan Martorell +3 more

We consider a Bose–Hubbard (BH) trimer, i.e. an ultracold Bose gas populating three quantum states. The latter can be either different sites of a triple-well potential or three internal states of the atoms. The bosons can tunnel between different states with variable tunnelling strength between two of them. This will allow us to study; (i) different geometrical configurations, i.e. from a closed triangle to three aligned wells and (ii) a triangular configuration with a π -phase, i.e. by setting one of the tunnellings negative. By solving the corresponding three-site BH Hamiltonian we obtain the ground state of the system as a function of the trap topology. We characterize the different ground states by means of the coherence and entanglement properties. For small repulsive interactions, fragmented condensates are found for the π -phase case. These are found to be robust against small variations of the tunnelling in the small interaction regime. A low-energy effective many-body Hamiltonian restricted to the degenerate manifold provides a compelling description of the π -phase degeneration and explains the low-energy spectrum as excitations of discrete semifluxon states.

arXiv:1501.03306•New Journal of Physics•Jul 14, 2015•37 citations

Comparison of classical and quantum properties in an extended Bose-Hubbard model

Albaro Vega Gutierrez de Pineres

In order to explore a quantum version of a discrete nonlinear Schrodinger equation (DNLS), we quantize one nonlinear Schrodinger model, which is used to study different physical systems, e.g. coupled Bose-Einstein condensates. We will focus on small systems, like Dimer and Trimer.In our efforts to solve this quantum problem, we develop a Mathematica routine that implements the Number State Method and solves the corresponding Schrodinger equation. We calculate analytically and numerically the energy spectrum of the Dimer and Trimer systems. Those eigenenergies depend on the parameter set Q=Q1, Q2, Q3, Q4, Q5 and by adjusting this set Q, we can obtain the desired results and examine their effects. After the quantization of the extended DNLS we obtain a quantum DNLS, also known as an extended Bose-Hubbard (BH) model. The aim of this Master's thesis is to study the differences and similarities between the classical DNLS and the extended BH model, and what happens when we approach from the quantum regime to the classical one. Taking into account that the Hamiltonian has an important conserved quantity, the number operator, enables the total Hamiltonian to be block-diagonalized. This can be accomplished by taking advantage of additional symmetries, such as translational symmetry, which will simplify the analysis of the Hamiltonian matrix. In our results we discuss several effects that break the lattice symmetry, as the intersection between symmetric and antisymmetric states. We also compare our results with those obtained in previous works for the classical model, and we find some similarities, e.g. the transition of the highest-energy state from a one-site solution to a two-site solution depending on which Q parameters we vary, but also differences, as the appearance of a three-site solution, in a Trimer system.

•Jan 1, 2011•1 citation

Quantum-classical correspondence for a non-Hermitian Bose-Hubbard dimer

Eva-Maria Graefe, Hans Jürgen Korsch, Astrid Elisa Niederle

We investigate the many-particle and mean-field correspondence for a non-Hermitian $N$-particle Bose-Hubbard dimer where a complex on-site energy describes an effective decay from one of the modes. Recently a generalized mean-field approximation for this non-Hermitian many-particle system yielding an alternative complex nonlinear Schr\"odinger equation was introduced. Here we give details of this mean-field approximation and show that the resulting dynamics can be expressed in a generalized canonical form that includes a metric gradient flow. The interplay of nonlinearity and non-Hermiticity introduces a qualitatively new behavior to the mean-field dynamics: The presence of the non-Hermiticity promotes the self-trapping transition, while damping the self-trapping oscillations, and the nonlinearity introduces a strong sensitivity to the initial conditions in the decay of the normalization. Here we present a complete characterization of the mean-field dynamics and the fixed point structure. We also investigate the full many-particle dynamics, which shows a rich variety of breakdown and revival as well as tunneling phenomena on top of the mean-field structure.

arXiv:1003.3355•Physical Review A•Jul 21, 2010•143 citations
The Bose-Hubbard Hamiltonian capturing the essential physics of the arrays of interacting Bose-Einstein condensates is addressed, focusing on arrays consisting of two (dimer) and three (trimer) sites.In the former case, some results concerning the persistence of mean-field features in the energy spectrum of the symmetric dimer are extended to the asymmetric version of the system, where the two sites are characterized by different on-site energies.Based on a previous systematic study of the mean-field limit of the trimer, where the dynamics is exhaustively described in terms of its fixed points for every choice of the significant parameters, an interesting mapping between the dimer and the trimer is emphasized and used as a guide in investigating the persistence of mean-field features in the rather complex energy spectrum of the trimer.These results form the basis for the systematic investigation of the purely quantum trimer extending and completing the existing mean-field analysis.In this respect we recall that, similar to larger arrays, the trimer is characterized by a non-integrable mean-field dynamics featuring chaotic trajectories.Hence, the correspondence between mean-field fixed points and quantum energy levels emphasized in the present work may provide a key to investigate the quantum counterpart of classical instability.
The Bose-Hubbard Hamiltonian capturing the essential physics of the arrays of interacting Bose-Einstein condensates is addressed, focusing on arrays consisting of two (dimer) and three (trimer) sites. In the former case, some results concerning the persistence of mean-field features in the energy spectrum of the symmetric dimer are extended to the asymmetric version of the system, where the two sites are characterized by different on-site energies. Based on a previous systematic study of the mean-field limit of the trimer, where the dynamics is exhaustively described in terms of its fixed points for every choice of the significant parameters, an interesting mapping between the dimer and the trimer is emphasized and used as a guide in investigating the persistence of mean-field features in the rather complex energy spectrum of the trimer. These results form the basis for the systematic investigation of the purely quantum trimer extending and completing the existing mean-field analysis. In this respect we recall that, similar to larger arrays, the trimer is characterized by a non-integrable mean-field dynamics featuring chaotic trajectories. Hence, the correspondence between mean-field fixed points and quantum energy levels emphasized in the present work may provide a key to investigate the quantum counterpart of classical instability.
We study the energy spectrum structure of a system of two interacting bosonic wells (dimer model) occupied by N bosons and compare it with the phase-space topology of the same model within the mean-field approach. To this end, we characterize the structure of energy eigenstates by using the symmetry properties of the Hamiltonian, and show that the energy levels are nondegenerate. The presence of doublets leads to recover in the classical limit the self-trapping effect exhibited by the mean-field dimer model. Finally, we do some comments on trimer dynamics to show how the interaction with a third well can cause both dimer-like regimes and strongly chaotic behaviors.
We analyze thoroughly the mean-field dynamics of a linear chain of three coupled Bose-Einstein condensates, where both the tunneling and the central-well relative depth are adjustable parameters. Owing to its nonintegrability, entailing a complex dynamics with chaos occurrence, this system is a paradigm for longer arrays whose simplicity still allows a thorough analytical study. We identify the set of dynamics fixed points, along with the associated proper modes, and establish their stability character depending on the significant parameters. As an example of the remarkable operational value of our analysis, we point out some macroscopic effects that seem viable to experiments.
We consider a Bose–Hubbard (BH) trimer, i.e. an ultracold Bose gas populating three quantum states. The latter can be either different sites of a triple-well potential or three internal states of the atoms. The bosons can tunnel between different states with variable tunnelling strength between two of them. This will allow us to study; (i) different geometrical configurations, i.e. from a closed triangle to three aligned wells and (ii) a triangular configuration with a π -phase, i.e. by setting one of the tunnellings negative. By solving the corresponding three-site BH Hamiltonian we obtain the ground state of the system as a function of the trap topology. We characterize the different ground states by means of the coherence and entanglement properties. For small repulsive interactions, fragmented condensates are found for the π -phase case. These are found to be robust against small variations of the tunnelling in the small interaction regime. A low-energy effective many-body Hamiltonian restricted to the degenerate manifold provides a compelling description of the π -phase degeneration and explains the low-energy spectrum as excitations of discrete semifluxon states.
In order to explore a quantum version of a discrete nonlinear Schrodinger equation (DNLS), we quantize one nonlinear Schrodinger model, which is used to study different physical systems, e.g. coupled Bose-Einstein condensates. We will focus on small systems, like Dimer and Trimer.In our efforts to solve this quantum problem, we develop a Mathematica routine that implements the Number State Method and solves the corresponding Schrodinger equation. We calculate analytically and numerically the energy spectrum of the Dimer and Trimer systems. Those eigenenergies depend on the parameter set Q=Q1, Q2, Q3, Q4, Q5 and by adjusting this set Q, we can obtain the desired results and examine their effects. After the quantization of the extended DNLS we obtain a quantum DNLS, also known as an extended Bose-Hubbard (BH) model. The aim of this Master's thesis is to study the differences and similarities between the classical DNLS and the extended BH model, and what happens when we approach from the quantum regime to the classical one. Taking into account that the Hamiltonian has an important conserved quantity, the number operator, enables the total Hamiltonian to be block-diagonalized. This can be accomplished by taking advantage of additional symmetries, such as translational symmetry, which will simplify the analysis of the Hamiltonian matrix. In our results we discuss several effects that break the lattice symmetry, as the intersection between symmetric and antisymmetric states. We also compare our results with those obtained in previous works for the classical model, and we find some similarities, e.g. the transition of the highest-energy state from a one-site solution to a two-site solution depending on which Q parameters we vary, but also differences, as the appearance of a three-site solution, in a Trimer system.
We investigate the many-particle and mean-field correspondence for a non-Hermitian NNN-particle Bose-Hubbard dimer where a complex on-site energy describes an effective decay from one of the modes. Recently a generalized mean-field approximation for this non-Hermitian many-particle system yielding an alternative complex nonlinear Schr\"odinger equation was introduced. Here we give details of this mean-field approximation and show that the resulting dynamics can be expressed in a generalized canonical form that includes a metric gradient flow. The interplay of nonlinearity and non-Hermiticity introduces a qualitatively new behavior to the mean-field dynamics: The presence of the non-Hermiticity promotes the self-trapping transition, while damping the self-trapping oscillations, and the nonlinearity introduces a strong sensitivity to the initial conditions in the decay of the normalization. Here we present a complete characterization of the mean-field dynamics and the fixed point structure. We also investigate the full many-particle dynamics, which shows a rich variety of breakdown and revival as well as tunneling phenomena on top of the mean-field structure.