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:1201.0714 · Physical Review A

Excitation spectra of fragmented condensates by linear response: General theory and application to a condensate in a double-well potential

Julian Grond, Alexej I. Streltsov, L. S. Cederbaum, Ofir E. Alon

Published Dec 6, 2012• 16 citations
arXivHTMLPDF

Abstract

Linear response of simple (i.e., condensed) Bose-Einstein condensates is known to lead to the Bogoliubov--de Gennes equations. Here, we derive linear response for fragmented Bose-Einstein condensates, i.e., for the case where the many-body wave function is not a product of one, but of several single-particle states (orbitals). This gives one access to excitation spectra and response amplitudes of systems beyond the Gross-Pitaevskii description. Our approach is based on the number-conserving variational time-dependent mean-field theory [Alon, Streltsov, and Cederbaum, Phys. Lett. A 362, 453 (2007)], which describes the time evolution of best-mean-field states. Correspondingly, we call our linear-response theory for fragmented states LR-BMF. In the derivation it follows naturally that excitations are orthogonal to the ground-state orbitals. As applications excitation spectra of Bose-Einstein condensates in double-well potentials are calculated. Both symmetric and asymmetric double wells are studied for several interaction strengths and barrier heights. The cases of condensed and twofold fragmented ground states are compared. Interestingly, even in such situations where the response frequencies of the two cases are computed to be close to each other, which is the situation for the excitations well below the barrier, striking differences in the density response in momentum space are found. For excitations with an energy of the order of the barrier height, both the energies and the density response of condensed and fragmented systems are very different. In fragmented systems there is a class of ``swapped'' excitations where an atom is transferred to the neighboring well. The mechanism of its origin is discussed. In asymmetric wells, the response of a fragmented system is purely local (i.e., finite in either one or the other well) with different frequencies for the left and right fragments. This finding is in stark contrast to that for condensed systems.

DOI
10.1103/physreva.86.063607

Related papers

Excitation spectra of many-body systems by linear response: General theory and applications to trapped condensates

Julian Grond, Alexej I. Streltsov, Axel U. J. Lode +3 more

We derive a general linear-response many-body theory capable of computing excitation spectra of trapped interacting bosonic systems, e.g., depleted and fragmented Bose-Einstein condensates (BECs). To obtain the linear-response equations we linearize the multiconfigurational time-dependent Hartree for bosons (MCTDHB) method, which provides a self-consistent description of many-boson systems in terms of orbitals and a state vector (configurations), and is in principle numerically exact. The derived linear-response many-body theory, which we term LR-MCTDHB, is applicable to systems with interaction potentials of general form. For the special case of a $\ensuremath{\delta}$ interaction potential we show explicitly that the response matrix has a very appealing bilinear form, composed of separate blocks of submatrices originating from contributions of the orbitals, the state vector (configurations), and off-diagonal mixing terms. We further give expressions for the response weights and density response. We introduce the notion of the type of excitations, useful in the study of the physical properties of the equations. From the numerical implementation of the LR-MCTDHB equations and solution of the underlying eigenvalue problem, we obtain excitations beyond available theories of excitation spectra, such as the Bogoliubov--de Gennes (BdG) equations. The derived theory is first applied to study BECs in a one-dimensional harmonic potential. The LR-MCTDHB method contains the BdG excitations and, also, predicts a plethora of additional many-body excitations which are out of the realm of standard linear response. In particular, our theory describes the exact energy of the higher harmonic of the first (dipole) excitation not contained in the BdG theory. We next study a BEC in a very shallow one-dimensional double-well potential. We find with LR-MCTDHB low-lying excitations which are not accounted for by BdG, even though the BEC has only little fragmentation and, hence, the BdG theory is expected to be valid. The convergence of the LR-MCTDHB theory is assessed by systematically comparing the excitation spectra computed at several different levels of theory.

arXiv:1307.1667•Physical Review A•Aug 12, 2013•33 citations

Properties of fragmented repulsive condensates

Alexej I. Streltsov, L. S. Cederbaum

Repulsive Bose-Einstein condensates immersed into a double-well trap potential are studied within the framework of the recently introduced mean-field approach which allows for bosons to reside in several different orthonormal orbitals. In the case of a one-orbital mean-field theory (Gross-Pitaevskii) the ground state of the system reveals a bifurcation scenario at some critical values of the interparticle interaction and/or the number of particles. At about the same values of the parameters the two-orbital mean field predicts that the system becomes twofold fragmented. By applying the three-orbital mean field we verify numerically that for the double-well external potential studied here the overall best mean field is achieved with two orbitals. The variational principle minimizes the energy at a vanishing population of the third orbital. To discuss the energies needed to remove a boson from and the energies gained by adding a boson to the condensate, we introduce boson ionization potentials and boson affinities and relate them to the chemical potentials. The impact of the finite number of bosons in the condensate on these quantities is analyzed. We recall that within the framework of the multiorbital mean-field theory each fragment is characterized by its own chemical potential. Finally, the stability of fragmented states is discussed in terms of the boson transfer energy which is the energy needed to transfer a boson from one fragment to another.

arXiv:cond-mat/0412222•Physical Review A•Jun 22, 2005•18 citations

Many-body effects in the excitations and dynamics of trapped Bose-Einstein condensates

Ofir E. Alon, Raphael Beinke, L. S. Cederbaum

This review explores the dynamics and the low-energy excitation spectra of Bose-Einstein condensates (BECs) of interacting bosons in external potential traps putting particular emphasis on the emerging many-body effects beyond mean-field descriptions. To do so, methods have to be used that, in principle, can provide numerically exact results for both the dynamics and the excitation spectra in a systematic manner. Numerically exact results for the dynamics are presented employing the well-established multicongurational time-dependent Hartree for bosons (MCTDHB) method. The respective excitation spectra are calculated utilizing the more recently introduced linear-response theory atop it (LR-MCTDHB). The latter theory gives rise to an, in general, non-hermitian eigenvalue problem. The theory and its newly developed implementation are described in detail and benchmarked towards the exactly-solvable harmonic-interaction model. Several applications to BECs in one- and two-dimensional potential traps are discussed. With respect to dynamics, it is shown that both the out-of-equilibrium tunneling dynamics and the dynamics of trapped vortices are of many-body nature. Furthermore, many-body effects in the excitation spectra are presented for BECs in different trap geometries. It is demonstrated that even for essentially-condensed systems, the spectrum of the lowest-in-energy excitations computed at the many-body level can differ substantially from the standard mean-field description. In general, it is shown that bosons carrying angular momentum are more sensitive to many-body effects than bosons without. These effects are present in both the dynamics and the excitation spectrum.

arXiv:2101.11615•Jan 27, 2021•1 citation

Multiconfigurational Hartree-Fock theory for identical bosons in a double well

David J. Masiello, Sarah B. McKagan, William P. Reinhardt

Multiconfigurational Hartree-Fock theory is presented and implemented in an investigation of the fragmentation of a Bose-Einstein condensate made of identical bosonic atoms in a double-well potential at zero temperature. The approach builds in the effects of the condensate mean field and of atomic correlations by describing generalized many-body states that are composed of multiple configurations which incorporate atomic interactions. Nonlinear and linear optimization is utilized in conjunction with the variational and Hylleraas-Undheim theorems to find the optimal ground and excited states of the interacting system. The resulting energy spectrum and associated eigenstates are presented as a function of double-well barrier height. Delocalized and localized single configurational states are found in the extreme limits of the simple and fragmented condensate ground states, while multiconfigurational states and macroscopic quantum superposition states are revealed throughout the full extent of barrier heights. Comparison is made to existing theories that either neglect mean field or correlation effects and it is found that contributions from both interactions are essential in order to obtain a robust microscopic understanding of the condensate's atomic structure throughout the fragmentation process.

arXiv:cond-mat/0509530•Physical Review A•Dec 29, 2005•40 citations

Excitations of Bose-Einstein condensates at finite temperatures

Martin Rusch, Rusch, Martin

Recent experimental observations of collective excitations of Bose condensed atomic vapours have stimulated interest in the microscopic description of the dynamics of a Bose-Einstein condensate confined in an external potential. We present a finite temperature field theory for collective excitations of trapped Bose-Einstein condensates and use a finite-temperature linear response formalism, which goes beyond the simple mean-field approximation of the Gross-Pitaevskii equation. The effect of the non-condensed thermal atoms we include using perturbation theory in a quasiparticle basis. This presents a simple scheme to understand the interaction between condensate and non-condensed atoms and enables us to include the effect the condensate has on collision dynamics. At first we limit our treatment to the case of a spatially homogeneous Bose gas. We include the effect of pair and triplet anomalous averages and thus obtain a gapless theory for the excitations of a weakly interacting system, which we can link to well known results for Landau and Beliaev damping rates. A gapless theory for trapped systems with a static thermal component follows straightforwardly. We then investigate finite temperature excitations of a condensate in a spherically symmetric harmonic trap. We avoid approximations to the density of states and thus emphasise finite size aspects of the problem. We show that excitations couple strongly to a restricted number of modes, giving rise to resonance structure in their frequency spectra. Where possible we derive energy shifts and lifetimes of excitations. For one particular mode, the breathing mode, the effects of the discreteness of the system are sufficiently pronounced that the simple picture of an energy shift and width fails. Experiments in spherical traps have recently become feasible and should be able to test our detailed quantitative predictions.

•Jan 1, 2000•0 citations

Many-body excitation spectra of trapped bosons with general interaction by linear response

Ofir E. Alon

The linear-response theory of the multiconfigurational time-dependent Hartree for bosons method (LR-MCTDHB) for computing many-body excitations of trapped Bose-Einstein condensates [Phys. Rev. A 88 , 023606 (2013); J. Chem. Phys. 140 , 034108 (2014)] is implemented, for the first time, for systems with general interparticle interaction. This allows us to investigate the many-body excitation spectrum of interacting bosons with, for instance, a long-range interaction. Illustrative numerical examples for repulsive and attractive bosons are provided. The LR-MCTDHB theory is capable of identifying all excitations, including the excitations which are not unraveled within Bogoliubov-de Gennes equations. The theory is herewith benchmarked against the exactly-solvable one-dimensional harmonic-interaction model. As a complementary result, using a complex transformation, we represent LR-MCTDHB in a compact block-diagonal form, opening up thereby an avenue for treating larger many-body systems. We expect the LR-MCTDHB theory and its implementation for general interparticle interaction to provide a proved probe into the many-body excitations involved in the out-ofequilibrium dynamics of trapped interacting bosons.

arXiv:1407.7678•Journal of Physics Conference Series•Mar 18, 2015•11 citations
Linear response of simple (i.e., condensed) Bose-Einstein condensates is known to lead to the Bogoliubov--de Gennes equations. Here, we derive linear response for fragmented Bose-Einstein condensates, i.e., for the case where the many-body wave function is not a product of one, but of several single-particle states (orbitals). This gives one access to excitation spectra and response amplitudes of systems beyond the Gross-Pitaevskii description. Our approach is based on the number-conserving variational time-dependent mean-field theory [Alon, Streltsov, and Cederbaum, Phys. Lett. A 362, 453 (2007)], which describes the time evolution of best-mean-field states. Correspondingly, we call our linear-response theory for fragmented states LR-BMF. In the derivation it follows naturally that excitations are orthogonal to the ground-state orbitals. As applications excitation spectra of Bose-Einstein condensates in double-well potentials are calculated. Both symmetric and asymmetric double wells are studied for several interaction strengths and barrier heights. The cases of condensed and twofold fragmented ground states are compared. Interestingly, even in such situations where the response frequencies of the two cases are computed to be close to each other, which is the situation for the excitations well below the barrier, striking differences in the density response in momentum space are found. For excitations with an energy of the order of the barrier height, both the energies and the density response of condensed and fragmented systems are very different. In fragmented systems there is a class of ``swapped'' excitations where an atom is transferred to the neighboring well. The mechanism of its origin is discussed. In asymmetric wells, the response of a fragmented system is purely local (i.e., finite in either one or the other well) with different frequencies for the left and right fragments. This finding is in stark contrast to that for condensed systems.
We derive a general linear-response many-body theory capable of computing excitation spectra of trapped interacting bosonic systems, e.g., depleted and fragmented Bose-Einstein condensates (BECs). To obtain the linear-response equations we linearize the multiconfigurational time-dependent Hartree for bosons (MCTDHB) method, which provides a self-consistent description of many-boson systems in terms of orbitals and a state vector (configurations), and is in principle numerically exact. The derived linear-response many-body theory, which we term LR-MCTDHB, is applicable to systems with interaction potentials of general form. For the special case of a $\ensuremath{\delta}$ interaction potential we show explicitly that the response matrix has a very appealing bilinear form, composed of separate blocks of submatrices originating from contributions of the orbitals, the state vector (configurations), and off-diagonal mixing terms. We further give expressions for the response weights and density response. We introduce the notion of the type of excitations, useful in the study of the physical properties of the equations. From the numerical implementation of the LR-MCTDHB equations and solution of the underlying eigenvalue problem, we obtain excitations beyond available theories of excitation spectra, such as the Bogoliubov--de Gennes (BdG) equations. The derived theory is first applied to study BECs in a one-dimensional harmonic potential. The LR-MCTDHB method contains the BdG excitations and, also, predicts a plethora of additional many-body excitations which are out of the realm of standard linear response. In particular, our theory describes the exact energy of the higher harmonic of the first (dipole) excitation not contained in the BdG theory. We next study a BEC in a very shallow one-dimensional double-well potential. We find with LR-MCTDHB low-lying excitations which are not accounted for by BdG, even though the BEC has only little fragmentation and, hence, the BdG theory is expected to be valid. The convergence of the LR-MCTDHB theory is assessed by systematically comparing the excitation spectra computed at several different levels of theory.
Repulsive Bose-Einstein condensates immersed into a double-well trap potential are studied within the framework of the recently introduced mean-field approach which allows for bosons to reside in several different orthonormal orbitals. In the case of a one-orbital mean-field theory (Gross-Pitaevskii) the ground state of the system reveals a bifurcation scenario at some critical values of the interparticle interaction and/or the number of particles. At about the same values of the parameters the two-orbital mean field predicts that the system becomes twofold fragmented. By applying the three-orbital mean field we verify numerically that for the double-well external potential studied here the overall best mean field is achieved with two orbitals. The variational principle minimizes the energy at a vanishing population of the third orbital. To discuss the energies needed to remove a boson from and the energies gained by adding a boson to the condensate, we introduce boson ionization potentials and boson affinities and relate them to the chemical potentials. The impact of the finite number of bosons in the condensate on these quantities is analyzed. We recall that within the framework of the multiorbital mean-field theory each fragment is characterized by its own chemical potential. Finally, the stability of fragmented states is discussed in terms of the boson transfer energy which is the energy needed to transfer a boson from one fragment to another.
This review explores the dynamics and the low-energy excitation spectra of Bose-Einstein condensates (BECs) of interacting bosons in external potential traps putting particular emphasis on the emerging many-body effects beyond mean-field descriptions. To do so, methods have to be used that, in principle, can provide numerically exact results for both the dynamics and the excitation spectra in a systematic manner. Numerically exact results for the dynamics are presented employing the well-established multicongurational time-dependent Hartree for bosons (MCTDHB) method. The respective excitation spectra are calculated utilizing the more recently introduced linear-response theory atop it (LR-MCTDHB). The latter theory gives rise to an, in general, non-hermitian eigenvalue problem. The theory and its newly developed implementation are described in detail and benchmarked towards the exactly-solvable harmonic-interaction model. Several applications to BECs in one- and two-dimensional potential traps are discussed. With respect to dynamics, it is shown that both the out-of-equilibrium tunneling dynamics and the dynamics of trapped vortices are of many-body nature. Furthermore, many-body effects in the excitation spectra are presented for BECs in different trap geometries. It is demonstrated that even for essentially-condensed systems, the spectrum of the lowest-in-energy excitations computed at the many-body level can differ substantially from the standard mean-field description. In general, it is shown that bosons carrying angular momentum are more sensitive to many-body effects than bosons without. These effects are present in both the dynamics and the excitation spectrum.
Multiconfigurational Hartree-Fock theory is presented and implemented in an investigation of the fragmentation of a Bose-Einstein condensate made of identical bosonic atoms in a double-well potential at zero temperature. The approach builds in the effects of the condensate mean field and of atomic correlations by describing generalized many-body states that are composed of multiple configurations which incorporate atomic interactions. Nonlinear and linear optimization is utilized in conjunction with the variational and Hylleraas-Undheim theorems to find the optimal ground and excited states of the interacting system. The resulting energy spectrum and associated eigenstates are presented as a function of double-well barrier height. Delocalized and localized single configurational states are found in the extreme limits of the simple and fragmented condensate ground states, while multiconfigurational states and macroscopic quantum superposition states are revealed throughout the full extent of barrier heights. Comparison is made to existing theories that either neglect mean field or correlation effects and it is found that contributions from both interactions are essential in order to obtain a robust microscopic understanding of the condensate's atomic structure throughout the fragmentation process.
Recent experimental observations of collective excitations of Bose condensed atomic vapours have stimulated interest in the microscopic description of the dynamics of a Bose-Einstein condensate confined in an external potential. We present a finite temperature field theory for collective excitations of trapped Bose-Einstein condensates and use a finite-temperature linear response formalism, which goes beyond the simple mean-field approximation of the Gross-Pitaevskii equation. The effect of the non-condensed thermal atoms we include using perturbation theory in a quasiparticle basis. This presents a simple scheme to understand the interaction between condensate and non-condensed atoms and enables us to include the effect the condensate has on collision dynamics. At first we limit our treatment to the case of a spatially homogeneous Bose gas. We include the effect of pair and triplet anomalous averages and thus obtain a gapless theory for the excitations of a weakly interacting system, which we can link to well known results for Landau and Beliaev damping rates. A gapless theory for trapped systems with a static thermal component follows straightforwardly. We then investigate finite temperature excitations of a condensate in a spherically symmetric harmonic trap. We avoid approximations to the density of states and thus emphasise finite size aspects of the problem. We show that excitations couple strongly to a restricted number of modes, giving rise to resonance structure in their frequency spectra. Where possible we derive energy shifts and lifetimes of excitations. For one particular mode, the breathing mode, the effects of the discreteness of the system are sufficiently pronounced that the simple picture of an energy shift and width fails. Experiments in spherical traps have recently become feasible and should be able to test our detailed quantitative predictions.
The linear-response theory of the multiconfigurational time-dependent Hartree for bosons method (LR-MCTDHB) for computing many-body excitations of trapped Bose-Einstein condensates [Phys. Rev. A 88 , 023606 (2013); J. Chem. Phys. 140 , 034108 (2014)] is implemented, for the first time, for systems with general interparticle interaction. This allows us to investigate the many-body excitation spectrum of interacting bosons with, for instance, a long-range interaction. Illustrative numerical examples for repulsive and attractive bosons are provided. The LR-MCTDHB theory is capable of identifying all excitations, including the excitations which are not unraveled within Bogoliubov-de Gennes equations. The theory is herewith benchmarked against the exactly-solvable one-dimensional harmonic-interaction model. As a complementary result, using a complex transformation, we represent LR-MCTDHB in a compact block-diagonal form, opening up thereby an avenue for treating larger many-body systems. We expect the LR-MCTDHB theory and its implementation for general interparticle interaction to provide a proved probe into the many-body excitations involved in the out-ofequilibrium dynamics of trapped interacting bosons.