Painlevé transcendent evaluations of finite system density matrices for 1d impenetrable Bosons

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 $\sigma$-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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References in corpus (8)

• Realization of Bose-Einstein condensates in lower dimensions
• Exploring phase coherence in a 2D lattice of Bose-Einstein condensates
• Observation of Phase Fluctuations in Bose-Einstein Condensates
• Application of the $\tau$-Function Theory of Painlevé Equations to Random Matrices: PIV, PII and the GUE
• Finite one dimensional impenetrable Bose systems: Occupation numbers
• Distribution of the first particle in discrete orthogonal polynomial ensembles
• Application of the $\tau$-function theory of Painlevé equations to random matrices: \PVI, the JUE, CyUE, cJUE and scaled limits
• New Transformations for Painlevé's Third Transcendent

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