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arXiv:1601.06918 · Journal of Physics B Atomic Molecular and Optical Physics

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

R. E. Barfknecht, A S Dehkharghani, Angela Foerster, Nikolaj Thomas Zinner

Published Jun 1, 2016• 16 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/0953-4075/49/13/135301

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We develop an analytical many-body wave function to accurately describe the crossover of a one-dimensional bosonic system from weak to strong interactions in a harmonic trap. The explicit wave function, which is based on the exact two-body states, consists of symmetric multiple products of the corresponding parabolic cylinder functions and respects the analytically known limits of zero and infinite repulsion for arbitrary number of particles. For intermediate interaction strengths we demonstrate that the energies, as well as the reduced densities of first and second order, are in excellent agreement with large scale numerical calculations.
We consider multi-component quantum mixtures (bosonic, fermionic, or mixed) with strongly repulsive contact interactions in a one-dimensional harmonic trap. In the limit of infinitely strong repulsion and zero temperature, using the class-sum method, we study the symmetries of the spatial wave function of the mixture. We find that the ground state of the system has the most symmetric spatial wave function allowed by the type of mixture. This provides an example of the generalized Lieb–Mattis theorem. Furthermore, we show that the symmetry properties of the mixture are embedded in the large-momentum tails of the momentum distribution, which we evaluate both at infinite repulsion by an exact solution and at finite interactions using a numerical DMRG approach. This implies that an experimental measurement of the Tan’s contact would allow to unambiguously determine the symmetry of any kind of multi-component mixture.
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 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.
Abstract For a harmonically trapped system consisting of two bosons in one spatial dimension with infinite contact repulsion (hard core bosons), we derive an expression for the one-body density matrix ρB\rho _\mathrm{B}ρB​ ρ B in terms of center of mass and relative coordinates of the particles. The deviation from ρF\rho _\mathrm{F}ρF​ ρ F , the density matrix for the two fermions case, can be clearly identified. Moreover, the obtained ρB\rho _\mathrm{B}ρB​ ρ B allows us to derive a closed form expression of the corresponding momentum distribution nB(p)n_\mathrm{B}(p)nB​(p) n B ( p ) . We show how the result deviates from the noninteracting fermionic case, the deviation being associated with the short-range character of the interaction. Mathematically, our analytical momentum distribution is expressed in terms of one and two variables confluent hypergeometric functions. Our formula satisfies the correct normalization and possesses the expected behavior at zero momentum. It also exhibits the high momentum 1/p41/p^4 1/p4 1 / p 4 tail with the appropriate Tan’s coefficient. Numerical results support our findings.
Abstract We systematically investigate and illustrate the complete ground-state phase diagram for a one-dimensional, three-species mixture of a few repulsively interacting bosons trapped harmonically. To numerically obtain the solutions to the many-body Schrödinger equation, we employ the improved Exact Diagonalization method (Anh-Tai et al 2023 SciPost Phys. 15 048), which is capable of treating strongly-correlated few-body systems from first principles in an efficiently truncated Hilbert space. We present our comprehensive results for all possible combinations of intra- and interspecies interactions in the extreme limits that are either the ideal limit (g = 0) or close to the hard-core limit ( g → ∞ ). These results show the emergence of unique ground-state properties related to correlations, coherence and spatial localization stemming from strongly repulsive interactions.