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:2311.12192 · Journal of Physics Conference Series

Correlation Effects in a Trapped Bose-Fermi Mixture: Exact Results

Ofir E. Alon, L. S. Cederbaum

Published Nov 1, 2024• 2 citations
arXivHTMLPDF

Abstract

Abstract Many-body properties of a fermionic impurity embedded in a Bose-Einstein condensate are analyzed analytically using a solvable model, the harmonic-interaction model for Bose-Fermi mixtures. The one-particle and two-particle densities, reduced density matrices, and correlation functions of the fermions and bosons, both in position and momentum spaces, are prescribed in closed form. The various coherence lengths are analyzed. We show that the first-order coherence lengths in position and momentum spaces are equal whereas the second-order quantities can differ substantially. Illustrative examples where the sole interaction is between the impurity and the condensate are presented. Implications are briefly discussed.

DOI
10.1088/1742-6596/2894/1/012012

Related papers

Reduced density matrices and coherence of trapped interacting bosons

Kaspar Sakmann, Alexej I. Streltsov, Ofir E. Alon +1 more

The first- and second-order correlation functions of trapped interacting Bose-Einstein condensates are investigated numerically on a many-body level from first principles. Correlations in real space and momentum space are treated. The coherence properties are analyzed. The results are obtained by solving the many-body Schr\"odinger equation. It is shown in an example how many-body effects can be induced by the trap geometry. A generic fragmentation scenario of a condensate is considered. The correlation functions are discussed along a pathway from a single condensate to a fragmented condensate. It is shown that strong correlations can arise from the geometry of the trap, even at weak interaction strengths. The natural orbitals and natural geminals of the system are obtained and discussed. It is shown how the fragmentation of the condensate can be understood in terms of its natural geminals. The many-body results are compared to those of mean-field theory. The best solution within mean-field theory is obtained. The limits in which mean-field theories are valid are determined. In these limits the behavior of the correlation functions is explained within an analytical model.

arXiv:0802.3417•Physical Review A•Aug 8, 2008•150 citations

Analytical and numerical studies of Bose–Fermi mixtures in a one-dimensional harmonic trap

Amin S. Dehkharghani, F. F. Bellotti, Nikolaj Thomas Zinner

Abstract In this paper we study a mixed system of bosons and fermions with up to six particles in total. All particles are assumed to have the same mass. The two-body interactions are repulsive and are assumed to have equal strength in both the Bose–Bose and the Fermi–Boson channels. The particles are confined externally by a harmonic oscillator one-body potential. For the case of four particles, two identical fermions and two identical bosons, we focus on the strongly interacting regime and analyze the system using both an analytical approach and density matrix renormalization group calculations using a discrete version of the underlying continuum Hamiltonian. This provides us with insight into both the ground state and the manifold of excited states that are almost degenerate for large interaction strength. Our results show great variation in the density profiles for bosons and fermions in different states for strongly interacting mixtures. By moving to slightly larger systems, we find that the ground state of balanced mixtures of four to six particles tends to separate bosons and fermions for strong (repulsive) interactions. On the other hand, in imbalanced Bose–Fermi mixtures we find pronounced odd–even effects in systems of five particles. These few-body results suggest that question of phase separation in one-dimensional confined mixtures are very sensitive to system composition, both for the ground state and the excited states.

arXiv:1703.01836•Journal of Physics B Atomic Molecular and Optical Physics•Jun 19, 2017•22 citations

Solvable model of a generic trapped mixture of interacting bosons: reduced density matrices and proof of Bose–Einstein condensation

Ofir E. Alon

Abstract A mixture of two kinds of identical bosons, species 1 with N 1 bosons of mass m 1 and species 2 with N 2 bosons of mass m 2 , held in a harmonic potential of frequency ω and interacting by harmonic intra-species and inter-species particle-particle interactions of strengths λ 1 , λ 2 , and λ 12 is discussed. This is an exactly-solvable model of a generic mixture of trapped interacting bosons which allows one to investigate and determine analytically properties of interest. To start, closed form expressions for the frequencies, ground-state energy, and wave-function of the mixture are obtained and briefly analyzed as a function of the masses, numbers of particles, and strengths and signs of interactions. To prove Bose–Einstein condensation of the mixture three steps are needed. First, we integrate the all-particle density matrix, employing a four-parameter matrix-recurrence relations, down to the lowest-order intra-species and inter-species reduced density matrices of the mixture. Second, the coupled Gross–Pitaevskii (mean-field) equations of the mixture are solved analytically. Third, we analyze the mixture’s reduced density matrices in the limit of an infinite number of particles of both species 1 and 2 (when the interaction parameters, i.e. the products of the number of particles times the intra-species and inter-species interaction strengths, are held fixed) and prove that: (i) both species 1 and 2 are 100% condensed; (ii) the inter-species reduced density matrix per particle is separable and given by the product of the intra-species reduced density matrices per particle; and (iii) the mixture’s energy per particle, and reduced density matrices and densities per particle all coincide with the Gross–Pitaevskii quantities. Finally, when the infinite-particle limit is taken with respect to, say, species 1 only (with interaction parameters held fixed) we prove that: (iv) only species 1 is 100% condensed and its reduced density matrix and density per particle, as well as the mixture’s energy per particle, coincide with the Gross–Pitaevskii quantities of species 1 alone; and (v) the inter-species reduced density matrix per particle is nonetheless separable and given by the product of the intra-species reduced density matrices per particle. The results are compared and discussed with respect to the recent work by Bouvrie et al (2014 Eur. Phys. J . D 68 346 ) who found for attractive mixtures vanishing bipartite entanglement between the two species and between a single particle (of either kind) and the remaining particles in the mixture. Implications are briefly discussed.

arXiv:1702.08219•Journal of Physics A Mathematical and Theoretical•Jun 12, 2017•20 citations

Fermionization of a strongly interacting Bose-Fermi mixture in a one-dimensional harmonic trap

Bess Fang, Patrizia Vignolo, Christian Miniatura +1 more

We consider a strongly interacting one-dimensional (1D) Bose-Fermi mixture confined in a harmonic trap. It consists of a Tonks-Girardeau (TG) gas (1D Bose gas with repulsive hard-core interactions) and of a noninteracting Fermi gas (1D spin-aligned Fermi gas), both species interacting through hard-core repulsive interactions. Using a generalized Bose-Fermi mapping, we determine the one-body density matrices, exact particle density profiles, momentum distributions, and behavior of the mixture under 1D expansion when opening the trap. In real space, bosons and fermions do not display any phase separation: the respective density profiles extend over the same region and they both present a number of peaks equal to the total number of particles in the trap. In momentum space the bosonic component has the typical narrow TG profile, while the fermionic component shows a broad distribution with fermionic oscillations at small momenta. Due to the large boson-fermion repulsive interactions, both the bosonic and the fermionic momentum distributions decay as $C{p}^{\ensuremath{-}4}$ at large momenta, like in the case of a pure bosonic TG gas. The coefficient $C$ is related to the two-body density matrix and to the bosonic concentration in the mixture. When opening the trap, both momentum distributions ``fermionize'' under expansion and turn into that of a Fermi gas with a particle number equal to the total number of particles in the mixture.

arXiv:0809.4419•Physical Review A•Feb 26, 2009•20 citations

Correlation properties of a three-body bosonic mixture in a harmonic trap

R. E. Barfknecht, A S Dehkharghani, Angela Foerster +1 more

Abstract We make use of a simple pair correlated wave function approach to obtain results for the ground-state densities and momentum distribution of a one-dimensional three-body bosonic system with different interactions in a harmonic trap. For equal interactions this approach is able to reproduce the known analytical cases of zero and infinite repulsion. We show that our results for the correlations agree with the exact diagonalization in all interaction regimes and with analytical results for the strongly repulsive impurity. This method also enables us to access the more complicated cases of mixed interactions, and the probability densities of these systems are analyzed.

arXiv:1601.06918•Journal of Physics B Atomic Molecular and Optical Physics•Jun 1, 2016•16 citations

Density-functional theory of two-component Bose gases in one-dimensional harmonic traps

Yajiang Hao, Shu Chen

We investigate the ground-state properties of two-component Bose gases confined in one-dimensional harmonic traps in the scheme of density-functional theory. The density-functional calculations employ a Bethe-ansatz-based local-density approximation for the correlation energy, which accounts for the correlation effect properly from the weakly interacting regime to the strongly interacting regime. For the binary Bose mixture with spin-independent interaction, the homogeneous reference system is exactly solvable by the Bethe-ansatz method. Within the local-density approximation, we determine the density distribution of each component and study its evolution from Bose distributions to Fermi-like distribution with the increase in interaction. For the binary mixture of Tonks-Girardeau gases with a tunable interspecies repulsion, with a generalized Bose-Fermi transformation we show that the Bose mixture can be mapped into a two-component Fermi gas, which corresponds to exact soluble Yang-Gaudin model for the homogeneous system. Based on the ground-state energy function of the Yang-Gaudin model, the ground-state density distributions are calculated for various interspecies interactions. It is shown that with the increase in interspecies interaction, the system exhibits composite-fermionization crossover.

arXiv:0905.3207•Physical Review A•Oct 14, 2009•34 citations
Abstract Many-body properties of a fermionic impurity embedded in a Bose-Einstein condensate are analyzed analytically using a solvable model, the harmonic-interaction model for Bose-Fermi mixtures. The one-particle and two-particle densities, reduced density matrices, and correlation functions of the fermions and bosons, both in position and momentum spaces, are prescribed in closed form. The various coherence lengths are analyzed. We show that the first-order coherence lengths in position and momentum spaces are equal whereas the second-order quantities can differ substantially. Illustrative examples where the sole interaction is between the impurity and the condensate are presented. Implications are briefly discussed.
The first- and second-order correlation functions of trapped interacting Bose-Einstein condensates are investigated numerically on a many-body level from first principles. Correlations in real space and momentum space are treated. The coherence properties are analyzed. The results are obtained by solving the many-body Schr\"odinger equation. It is shown in an example how many-body effects can be induced by the trap geometry. A generic fragmentation scenario of a condensate is considered. The correlation functions are discussed along a pathway from a single condensate to a fragmented condensate. It is shown that strong correlations can arise from the geometry of the trap, even at weak interaction strengths. The natural orbitals and natural geminals of the system are obtained and discussed. It is shown how the fragmentation of the condensate can be understood in terms of its natural geminals. The many-body results are compared to those of mean-field theory. The best solution within mean-field theory is obtained. The limits in which mean-field theories are valid are determined. In these limits the behavior of the correlation functions is explained within an analytical model.
Abstract In this paper we study a mixed system of bosons and fermions with up to six particles in total. All particles are assumed to have the same mass. The two-body interactions are repulsive and are assumed to have equal strength in both the Bose–Bose and the Fermi–Boson channels. The particles are confined externally by a harmonic oscillator one-body potential. For the case of four particles, two identical fermions and two identical bosons, we focus on the strongly interacting regime and analyze the system using both an analytical approach and density matrix renormalization group calculations using a discrete version of the underlying continuum Hamiltonian. This provides us with insight into both the ground state and the manifold of excited states that are almost degenerate for large interaction strength. Our results show great variation in the density profiles for bosons and fermions in different states for strongly interacting mixtures. By moving to slightly larger systems, we find that the ground state of balanced mixtures of four to six particles tends to separate bosons and fermions for strong (repulsive) interactions. On the other hand, in imbalanced Bose–Fermi mixtures we find pronounced odd–even effects in systems of five particles. These few-body results suggest that question of phase separation in one-dimensional confined mixtures are very sensitive to system composition, both for the ground state and the excited states.
Abstract A mixture of two kinds of identical bosons, species 1 with N 1 bosons of mass m 1 and species 2 with N 2 bosons of mass m 2 , held in a harmonic potential of frequency ω and interacting by harmonic intra-species and inter-species particle-particle interactions of strengths λ 1 , λ 2 , and λ 12 is discussed. This is an exactly-solvable model of a generic mixture of trapped interacting bosons which allows one to investigate and determine analytically properties of interest. To start, closed form expressions for the frequencies, ground-state energy, and wave-function of the mixture are obtained and briefly analyzed as a function of the masses, numbers of particles, and strengths and signs of interactions. To prove Bose–Einstein condensation of the mixture three steps are needed. First, we integrate the all-particle density matrix, employing a four-parameter matrix-recurrence relations, down to the lowest-order intra-species and inter-species reduced density matrices of the mixture. Second, the coupled Gross–Pitaevskii (mean-field) equations of the mixture are solved analytically. Third, we analyze the mixture’s reduced density matrices in the limit of an infinite number of particles of both species 1 and 2 (when the interaction parameters, i.e. the products of the number of particles times the intra-species and inter-species interaction strengths, are held fixed) and prove that: (i) both species 1 and 2 are 100% condensed; (ii) the inter-species reduced density matrix per particle is separable and given by the product of the intra-species reduced density matrices per particle; and (iii) the mixture’s energy per particle, and reduced density matrices and densities per particle all coincide with the Gross–Pitaevskii quantities. Finally, when the infinite-particle limit is taken with respect to, say, species 1 only (with interaction parameters held fixed) we prove that: (iv) only species 1 is 100% condensed and its reduced density matrix and density per particle, as well as the mixture’s energy per particle, coincide with the Gross–Pitaevskii quantities of species 1 alone; and (v) the inter-species reduced density matrix per particle is nonetheless separable and given by the product of the intra-species reduced density matrices per particle. The results are compared and discussed with respect to the recent work by Bouvrie et al (2014 Eur. Phys. J . D 68 346 ) who found for attractive mixtures vanishing bipartite entanglement between the two species and between a single particle (of either kind) and the remaining particles in the mixture. Implications are briefly discussed.
We consider a strongly interacting one-dimensional (1D) Bose-Fermi mixture confined in a harmonic trap. It consists of a Tonks-Girardeau (TG) gas (1D Bose gas with repulsive hard-core interactions) and of a noninteracting Fermi gas (1D spin-aligned Fermi gas), both species interacting through hard-core repulsive interactions. Using a generalized Bose-Fermi mapping, we determine the one-body density matrices, exact particle density profiles, momentum distributions, and behavior of the mixture under 1D expansion when opening the trap. In real space, bosons and fermions do not display any phase separation: the respective density profiles extend over the same region and they both present a number of peaks equal to the total number of particles in the trap. In momentum space the bosonic component has the typical narrow TG profile, while the fermionic component shows a broad distribution with fermionic oscillations at small momenta. Due to the large boson-fermion repulsive interactions, both the bosonic and the fermionic momentum distributions decay as $C{p}^{\ensuremath{-}4}$ at large momenta, like in the case of a pure bosonic TG gas. The coefficient CCC is related to the two-body density matrix and to the bosonic concentration in the mixture. When opening the trap, both momentum distributions ``fermionize'' under expansion and turn into that of a Fermi gas with a particle number equal to the total number of particles in the mixture.
Abstract We make use of a simple pair correlated wave function approach to obtain results for the ground-state densities and momentum distribution of a one-dimensional three-body bosonic system with different interactions in a harmonic trap. For equal interactions this approach is able to reproduce the known analytical cases of zero and infinite repulsion. We show that our results for the correlations agree with the exact diagonalization in all interaction regimes and with analytical results for the strongly repulsive impurity. This method also enables us to access the more complicated cases of mixed interactions, and the probability densities of these systems are analyzed.
We investigate the ground-state properties of two-component Bose gases confined in one-dimensional harmonic traps in the scheme of density-functional theory. The density-functional calculations employ a Bethe-ansatz-based local-density approximation for the correlation energy, which accounts for the correlation effect properly from the weakly interacting regime to the strongly interacting regime. For the binary Bose mixture with spin-independent interaction, the homogeneous reference system is exactly solvable by the Bethe-ansatz method. Within the local-density approximation, we determine the density distribution of each component and study its evolution from Bose distributions to Fermi-like distribution with the increase in interaction. For the binary mixture of Tonks-Girardeau gases with a tunable interspecies repulsion, with a generalized Bose-Fermi transformation we show that the Bose mixture can be mapped into a two-component Fermi gas, which corresponds to exact soluble Yang-Gaudin model for the homogeneous system. Based on the ground-state energy function of the Yang-Gaudin model, the ground-state density distributions are calculated for various interspecies interactions. It is shown that with the increase in interspecies interaction, the system exhibits composite-fermionization crossover.