Abstract An exactly solvable model of a trapped interacting Bose–Einstein condensate (BEC) coupled in the dipole approximation to a quantized light mode in a cavity is presented. The model can be seen as a generalization of the harmonic-interaction model for a trapped BEC coupled to a bosonic bath. After obtaining the ground-state energy and wavefunction in closed form, we focus on computing the correlations in the system. The reduced one-particle density matrices of the bosons and the cavity are constructed and diagonalized analytically, and the von Neumann entanglement entropy of the BEC and the cavity is also expressed explicitly as a function of the number and mass of the bosons, frequencies of the trap and cavity, and the cavity-boson coupling strength. The results allow one to study the impact of the cavity on the bosons and vice versa on an equal footing. As an application we investigate a specific case of basic interest for itself, namely, non-interacting bosons in a cavity. We find that both the bosons and the cavity develop correlations in a complementary manner while increasing the coupling between them. Whereas the cavity wavepacket broadens in Fock space, the BEC density saturates in real space. On the other hand, while the cavity depletion saturates, and hence does the BEC-cavity entanglement entropy, the BEC becomes strongly correlated and eventually increasingly fragmented. The latter phenomenon implies single-trap fragmentation of otherwise ideal bosons, where their induced long-range interaction is mediated by the cavity. Finally, as a complimentary investigation, the mean-field equations for the BEC-cavity system are solved analytically as well, and the breakdown of mean-field theory for the cavity and the bosons with increasing coupling is discussed. Further applications are envisaged.
Abstract An exactly solvable model of a trapped interacting Bose–Einstein condensate (BEC) coupled in the dipole approximation to a quantized light mode in a cavity is presented. The model can be seen as a generalization of the harmonic-interaction model for a trapped BEC coupled to a bosonic bath. After obtaining the ground-state energy and wavefunction in closed form, we focus on computing the correlations in the system. The reduced one-particle density matrices of the bosons and the cavity are constructed and diagonalized analytically, and the von Neumann entanglement entropy of the BEC and the cavity is also expressed explicitly as a function of the number and mass of the bosons, frequencies of the trap and cavity, and the cavity-boson coupling strength. The results allow one to study the impact of the cavity on the bosons and vice versa on an equal footing. As an application we investigate a specific case of basic interest for itself, namely, non-interacting bosons in a cavity. We find that both the bosons and the cavity develop correlations in a complementary manner while increasing the coupling between them. Whereas the cavity wavepacket broadens in Fock space, the BEC density saturates in real space. On the other hand, while the cavity depletion saturates, and hence does the BEC-cavity entanglement entropy, the BEC becomes strongly correlated and eventually increasingly fragmented. The latter phenomenon implies single-trap fragmentation of otherwise ideal bosons, where their induced long-range interaction is mediated by the cavity. Finally, as a complimentary investigation, the mean-field equations for the BEC-cavity system are solved analytically as well, and the breakdown of mean-field theory for the cavity and the bosons with increasing coupling is discussed. Further applications are envisaged.
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 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.
Quasi-particle excitation in a Bose-Einstein condensate leads to quantum entanglement between real bosonic atoms in the system. By using spectral expansion method, the eigenvalues and eigenstates of Bogoliubov-de Gennes equation are numerically calculated in a quasi-one-dimensional infinite square well potential. For the low-energy collective excitations of the quasi-particles, we explore the dependence of quantum entanglement entropy of the Bose-Einstein condensate on scattering length. Our results show that the entanglement entropy increases slowly with the increase of the scattering length, and such an increasing trend can be well described by a power function. These results are analogous to those in a one-dimensional uniform BEC, where the entanglement entropy of the Bogoliubov ground state is approximately proportional to the square root of the scattering length. This work provides a viable way for investigating many-particle entanglement in a quasi-one-dimensional trapped Bose-Einstein condensate where the quantum entanglement is closely related to the interaction strength between particles.
We investigate a trapped mixture of Bose-Einstein condensates consisting of a multiple number of P species. To be able to do so, an exactly solvable many-body model is called into play. This is the P-species harmonic-interaction model. After presenting the Hamiltonian, the ground-state energy and wavefunction are explicitly calculated. All properties of the mixture's ground state can, in principle, be obtained from the many-particle wavefunction. A scheme to integrate the all-particle density matrix is derived and implemented, leading to closed-form expressions for the reduced one-particle density matrices. Of particular interest is the infinite-particle-number limit, which is obtained when the numbers of bosons are taken to infinity while keeping the interaction parameters fixed. We first prove that at the infinite-particle-number limit all the species are 100% condensed. The mean-field solution of the P-species mixture is also obtained analytically and is used to show that the energy per particle and densities per particle computed at the many-body level of theory boil down to their mean-field counterparts. Despite these, correlations in the mixture exist at the infinite-particle-number limit. To this end, we obtain closed-form expressions for the correlation energy, namely, the difference between the mean-field and many-body energies, and the depletion of the species, i.e., the number of particles residing outside the condensed modes, at the infinite-particle-number limit. The depletion and the correlation energy per species are shown to critically depend on the number of species. Of separate interest is the entanglement between one species of bosons and the other P - 1 species. This quantity is governed by the coupling of the center-of-mass coordinates of the species and is obtained by the respective Schmidt decomposition of the P-species wavefunction. Interestingly, there is an optimal number of species, here P = 3, where the entanglement is maximal. Importantly, the manifestation of this interspecies entanglement in an observable is possible. It is the position-momentum uncertainty product of one species in the presence of the other P - 1 species, which is derived and demonstrated to correlate with the interspecies entanglement. All in all, we show and explain how correlations at the infinite-particle-number limit of a trapped multiple-species bosonic mixture depend on the interactions and how they evolve with the number of species. Generalizations and implications are briefly discussed.
Abstract In this paper we consider a system consisting of a number of two-level atoms in a Bose–Einstein condensate (BEC) and a single-mode quantized field, which interact with each other in the presence of two different damping sources, i.e. cavity and atomic reservoirs. The reservoirs which we consider here are thermal and squeezed vacuum ones corresponding to field and atom modes. Strictly speaking, by considering both types of reservoirs for each of the atom and field modes, we investigate the quantum dynamics of the interacting bosons in the system. Then, via solving the quantum Langevin equations for such a dissipative BEC system, we obtain analytical expressions for the time dependence of atomic population inversion, mean atom as well as photon number and quadrature squeezing in the field and atom modes. Our investigations demonstrate that for modeling the real physical systems, considering the dissipation effects is essential. Also, numerical calculations which are presented show that the atomic population inversion, the mean number of atoms in the BEC and the photons in the cavity possess damped oscillatory behavior due to the presence of reservoirs. In addition, non-classical squeezing effects in the field quadrature can be observed especially when squeezed vacuum reservoirs are taken into account. As an outstanding property of this model, we may refer to the fact that one can extract the atom–field coupling constant from the frequency of oscillations in the mentioned quantities such as atomic population inversion.