Finite one-dimensional impenetrable Bose systems: Occupation numbers

Forrester, Peter J.; Frankel, N. E.; Garoni, Timothy M.; Witte, N. S.

doi:10.1103/physreva.67.043607

Cited by 129 publications

(254 citation statements)

References 38 publications

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“…Accurate values of the λ j have been determined in [69]. In particular, the largest eigenvalue λ 1 was shown to be of order √ N for large N , as in the spatially uniform case.…”

Section: A Impenetrable Bosons (Tg Limit)mentioning

confidence: 98%

Effective interactions, Fermi–Bose duality, and ground states of ultracold atomic vapors in tight de Broglie waveguides

Girardeau

Nguyen

Olshanii

2004

Optics Communications

141

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Derivation of effective zero-range one-dimensional (1D) interactions between atoms in tight waveguides is reviewed, as is the Fermi-Bose mapping method for determination of exact and stronglycorrelated many-body ground states of ultracold bosonic and fermionic atomic vapors in such waveguides, including spin degrees of freedom. Odd-wave 1D interactions derived from 3D p-wave scattering are included as well as the usual even-wave interactions derived from 3D s-wave scattering, with emphasis on the role of 3D Feshbach resonances for selectively enhancing s-wave or p-wave scattering so as to reach 1D confinement-induced resonances of the even and odd-wave interactions. A duality between 1D fermions and bosons with zero-range interactions suggested by Cheon and Shigehara is shown to hold for the effective 1D dynamics of a spinor Fermi gas with both even and odd-wave interactions and that of a spinor Bose gas with even and odd-wave interactions, with even(odd)-wave Bose coupling constants inversely related to odd(even)-wave Fermi coupling constants. Some recent applications of Fermi-Bose mapping to determination of many-body ground states of Bose gases and of both magnetically trapped, spin-aligned and optically trapped, spin-free Fermi gases are described, and a new generalized Fermi-Bose mapping is used to determine the phase diagram of ground-state total spin of the spinor Fermi gas as a function of its even and odd-wave coupling constants.

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“…Accurate values of the λ j have been determined in [69]. In particular, the largest eigenvalue λ 1 was shown to be of order √ N for large N , as in the spatially uniform case.…”

Section: A Impenetrable Bosons (Tg Limit)mentioning

confidence: 98%

Effective interactions, Fermi–Bose duality, and ground states of ultracold atomic vapors in tight de Broglie waveguides

Girardeau

Nguyen

Olshanii

2004

Optics Communications

141

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“…This relates to the behaviour of λ k for k large. A comprehensive study of such issues will be discussed in a forthcoming publication [9].…”

Section: Discussionmentioning

confidence: 99%

“…Note added: Subsequent to the completion of this work an asymptotic form analogous to (1.4) has been derived in [9] for the harmonic well case, and from this it is deduced that, as with periodic boundary conditions, the λ k for fixed k are proportional to √ N .…”

Section: Discussionmentioning

confidence: 99%

“…Thus for large N the fraction of particles in the zero momentum state is proportional to √ N . The result (1.4) can also be used to compute the large N behaviour of λ k for any fixed k ≥ 0 [9]. For the impenetrable Bose gas confined by a harmonic one-body potential, or indeed in other geometries such as Dirichlet or Neumann boundary conditions, no results of this type are known.…”

Section: Introductionmentioning

confidence: 99%

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Painlevé Transcendent Evaluations of Finite System Density Matrices for 1d Impenetrable Bosons

Forrester

Frankel²,

Garoni³

et al. 2003

Commun. Math. Phys.

112

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Abstract:The recent experimental realisation of a one-dimensional Bose gas of ultra cold alkali atoms has renewed attention on the theoretical properties of the impenetrable Bose gas. Of primary concern is the ground state occupation of effective single particle states in the finite system, and thus the tendency for Bose-Einstein condensation. This requires the computation of the density matrix. For the impenetrable Bose gas on a circle we evaluate the density matrix in terms of a particular Painlevé VI transcendent in σ-form, and furthermore show that the density matrix satisfies a recurrence relation in the number of particles. For the impenetrable Bose gas in a harmonic trap, and with Dirichlet or Neumann boundary conditions, we give a determinant form for the density matrix, a form as an average over the eigenvalues of an ensemble of random matrices, and in special cases an evaluation in terms of a transcendent related to Painlevé V and VI. We discuss how our results can be used to compute the ground state occupations.

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“…This theory has been used extensively in analyses of the experiments. Forrester et al [58] have been able to show that, for the 1D Bose gas with infinite δ-function interaction, whether homogeneous or harmonically trapped, all the eigenvalues of the one-body density matrix at T = 0 are of order √ N . Such a result would certainly correspond to a GBEC if we could state that the system was a quantum fluid.…”

Section: The 3d Transition To 1dmentioning

confidence: 99%

Generalized Bose–Einstein Condensation

Mullin

Sakhel

2011

J Low Temp Phys

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Generalized Bose-Einstein condensation (GBEC) involves condensates appearing simultaneously in multiple states. We review examples of the three types in an ideal Bose gas with different geometries. In Type I there is a discrete number of quantum states each having macroscopic occupation; Type II has condensation into a continuous band of states, with each state having macroscopic occupation; in Type III each state is microscopically occupied while the entire condensate band is macroscopically occupied. We begin by discussing Type I or "normal" BEC into a single state for an isotropic harmonic oscillator potential. Other geometries and external potentials are then considered: the "channel" potential (harmonic in one dimension and hard-wall in the other), which displays Type II, the "cigar trap" (anisotropic harmonic potential), and the "Casimir prism" (an elongated box), the latter two having Type III condensations. General box geometries are considered in an appendix. We particularly focus on the cigar trap, which Van Druten and Ketterle first showed had a two-step condensation: a GBEC into a band of states at a temperature Tc and another "one-dimensional" transition at a lower temperature T1 into the ground state. In a thermodynamic limit in which the ratio of the dimensions of the anisotropic harmonic trap is kept fixed, T1 merges with the upper transition, which then becomes a normal BEC. However, in the thermodynamic limit of Beau and Zagrebnov, in which the ratio of the boundary lengths increases exponentially, T1 becomes fixed at the temperature of a true Type I phase transition. The effects of interactions on GBEC are discussed and we show that there is evidence that Type III condensation may have been observed in the cigar trap.

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Finite one-dimensional impenetrable Bose systems: Occupation numbers

Cited by 129 publications

References 38 publications

Effective interactions, Fermi–Bose duality, and ground states of ultracold atomic vapors in tight de Broglie waveguides

Effective interactions, Fermi–Bose duality, and ground states of ultracold atomic vapors in tight de Broglie waveguides

Painlevé Transcendent Evaluations of Finite System Density Matrices for 1d Impenetrable Bosons

Generalized Bose–Einstein Condensation

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