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Physical Review A

Quantum-statistical mechanics of an atom-dimer mixture: Lee-Yang cluster expansion approach

Takahiro Ohkuma, Masahito Ueda

Published Jun 7, 2006• 5 citations
Publisher

Abstract

We use the Lee-Yang cluster expansion method to study quantum-statistical properties of a mixture of interconvertible atoms and dimers, where the dimers form in a two-body bound state of the atoms. We point out an infinite series of cluster diagrams whose summation leads to the Bose-Einstein condensation of the dimers below a critical temperature. Our theory captures some important features of a cold atom-dimer mixture such as interconversion of atoms and dimers and properties of the mixture at the unitarity limit.

DOI
10.1103/physreva.73.063608

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We use the Lee-Yang cluster expansion method to study quantum-statistical properties of a mixture of interconvertible atoms and dimers, where the dimers form in a two-body bound state of the atoms. We point out an infinite series of cluster diagrams whose summation leads to the Bose-Einstein condensation of the dimers below a critical temperature. Our theory captures some important features of a cold atom-dimer mixture such as interconversion of atoms and dimers and properties of the mixture at the unitarity limit.
The quantum-statistical cluster expansion method of Lee and Yang is extended to investigate off-diagonal long-range order (ODLRO) in one-component and multicomponent mixtures of bosons or fermions. Our formulation is applicable to both a uniform system and a trapped system without local-density approximation and allows systematic expansions of one-particle and multiparticle reduced density matrices in terms of cluster functions, which are defined for the same system with Boltzmann statistics. Each term in this expansion can be associated with a Lee-Yang graph. We elucidate a physical meaning of each Lee-Yang graph; in particular, for a mixture of ultracold atoms and bound dimers, an infinite sum of the ladder-type Lee-Yang 0-graphs is shown to lead to Bose-Einstein condensation of dimers below the critical temperature. In the case of Bose statistics, an infinite series of Lee-Yang 1-graphs is shown to converge and gives the criteria of ODLRO at the one-particle level. Applications to a dilute Bose system of hard spheres are also made. In the case of Fermi statistics, an infinite series of Lee-Yang 2-graphs is shown to converge and gives the criteria of ODLRO at the two-particle level. Applications to a two-component Fermi gas in the tightly bound limit are also made.
We present an exact many-body theory of ultracold fermionic gases for the Bose-Einstein condensation (BEC) regime of the BEC-BCS crossover. This is a purely fermionic approach which treats explicitly and systematically the dimers formed in the BEC regime as made of two fermions. We consider specifically the zero temperature case and calculate the first terms of the expansion of the chemical potential in powers of the density nnn. We derive first the mean-field contribution, which has the expected standard expression when it is written in terms of the dimer-dimer scattering length aM{a}_{M}aM​. We go next in the expansion to the Lee-Huang-Yang order, proportional to n3∕2{n}^{3∕2}n3∕2. We find the far less obvious result that it retains also the same expression in terms of aM{a}_{M}aM​ as for elementary bosons. The composite nature of the dimers appears only in the next term proportional to n2{n}^{2}n2.
In the quantum statistical method of Lee and Yang, the cluster functions of quantum statistics are expressed in terms of the cluster functions of Boltzmann statistics, which in turn are computed in terms of certain two-body functions. In the present paper, following a detailed study of the Boltzmann cluster functions, it is shown that the symmetric representation can be used for the two-body functions and that large classes of diagrams can be summed. This leads to the introduction of linked-pair graphs to describe the functions of quantum statistics. The two-body functions are expressed in terms of two-body wave functions, and are therefore well-defined for hard-core repulsions. For weak potentials the method is shown to be equivalent to the theory of Bloch and DeDominicis.
The binary collision method of Lee and Yang is extended in order to investigate the statistical mechanics of a multicomponent mixture of interacting gases obeying Bose-Einstein or Fermi-Dirac statistics. First, the case of a mixture of two kinds of hard-sphere spinless bosons is studied and then the generalization is carried over to the case of a mixture of several components consisting of particles with arbitrary spin and statistics. An explicit derivation of the fugacity expansion, correct to the second order in the interaction parameters, is given. Expressions are derived for the second and third virial coefficients of the system and a discussion is given of the salient features of the results obtained. Finally, the case of a binary mixture, with one component consisting of bosons and the other of fermions, is considered and the phenomenon of Bose-Einstein condensation in this system is studied.
We study liquid quantum droplets in a mixture of two-component Bose-Einstein condensates under variable confinement introduced along one or two spatial dimensions. Although the atom-atom scattering has a three-dimensional character, discreteness of the available modes in the reduced dimension(s) strongly influences the zero-point energy, which for a weakly interacting three-dimensional system is given by the Lee-Huang-Yang term. Here, we generalize the result of Lee, Huang, and Yang and calculate the higher order corrections to the mean-field energy for a single component Bose-Einstein condensate under condition of a tight confinement in one or two spatial directions. We apply our results to the two-component mixture of bosonic species under conditions favorable for the creation of quantum liquid droplets, which constitute a perfect playground for studies of the beyond mean-field effects. Our findings allow one to analyze the properties of quantum droplets at the dimensional crossover and provide demanding conditions for accessing quasilow dimensions. They also pave a way for experimental exploration of this state of matter in reduced dimensions.
It is shown that a cluster expansion technique, which is usually applied in the high-temperature regime to calcutate virial coefficients, can be applied to evaluate the superfluid transition temperature of the Bardeen-Cooper-Schrieffer (BCS)--Bose-Einstein condensation (BEC) crossover \`a la Lee and Yang. The transition temperature is identified with the emergence of the singularity in the sum of a certain infinite series of cluster functions. In the weak-coupling limit, we reproduce the Thouless criterion and the number equation of Nozi\`eres and Schmitt-Rink, and hence the transition temperature of the BCS theory. In the strong-coupling limit, we reproduce the transition temperature of BEC of noninteracting tightly bound dimers.