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arXiv:cond-mat/9812260

Phase coherence and ``fragmented'' Bose condensates

Daniel S. Rokhsar

Published Dec 15, 1998• 2 citations
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Abstract

We show that a ``fragmented Bose condensate'' in which two or more distinct single-particle states are macroscopically occupied by the same species of boson is inherently unstable to the formation of a conventional Bose condensate whose macroscopically occupied state is a linear combination of the ``fragments'' with definite relative phases. A related analysis shows that a reproducible relative phase develops when two initially decoupled condensates are placed in contact.

DOI
10.48550/arxiv.cond-mat/9812260

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We show that a fragmented Bose condensate'' in which two or more distinct single-particle states are macroscopically occupied by the same species of boson is inherently unstable to the formation of a conventional Bose condensate whose macroscopically occupied state is a linear combination of the fragments'' with definite relative phases. A related analysis shows that a reproducible relative phase develops when two initially decoupled condensates are placed in contact.
We consider two dilute gas Bose-Einstein condensates with opposite velocities from which a monochromatic light field detuned far from the resonance of the optical transition is coherently scattered. In the thermodynamic limit, when the relative fluctuations of the atom number difference between the two condensates vanish, the relative phase between the Bose-Einstein condensates may be established in a superposition state by detections of spontaneously scattered photons, even though the condensates have initially well-defined atom numbers. For a finite system, stochastic simulations show that the measurements of the scattered photons lead to a randomly drifting relative phase and drive the condensates into entangled superpositions of number states. This is because according to Bose-Einstein statistics the scattering to an already occupied state is enhanced.
Abstract Fragmented Bose–Einstein condensates are large systems of identical bosons displaying multiple macroscopic occupations of one-body states, in a suitable sense. The quest for an effective dynamics of the fragmented condensate at the leading order in the number of particles, in analogy to the much more controlled scenario for complete condensation in one single state, is deceptive both because characterising fragmentation solely in terms of reduced density matrices is unsatisfactory and ambiguous, and because as soon as the time evolution starts the rank of the reduced marginals generically passes from finite to infinite, which is a signature of a transfer of occupations on infinitely many more one-body states. In this work we review these difficulties, we refine previous characterisations of fragmented condensates in terms of marginals, and we provide a quantitative rate of convergence to the leading effective dynamics in the double limit of infinitely many particles and infinite energy gap.
Condensation of bosons in Bose-Einstein condensates or Cooper pairs in superconductors refers to a macroscopic occupation of a few single- or two-particle states. A condensate is called "fragmented" if not a single, but multiple states are macroscopically occupied. While fragmentation is known to occur in particular Bose-Einstein condensates, we propose that fragmentation naturally takes place in striped superconductors. To this end, we investigate the nature of the superconducting ground state realized in the two-dimensional t-t^{'}-J model. In the presence of charge density modulations, the condensate is shown to be fragmented and composed of partial condensates located on the stripes. The fragments of the condensates hybridize to form an extended macroscopic wave function across the system. The results are obtained from evaluating the singlet-pairing two-particle density matrix of the ground state on finite cylinders computed via the density matrix renormalization group method. Our results shed light on the intricate relation between stripe order and superconductivity in systems of strongly correlated electrons.
Recently it has been predicted that if a gas of spin-1/2 bosonic atoms is cooled down while conserving its total spin, it can form a Bose-Einstein condensate in a new type of fragmented state. In that fragmented state two orbital wave functions are macroscopically occupied. We derive a set of Gross-Pitaevskii equations that describe the two occupied wave functions, taking into account the effects of interatomic interactions. We then analyze the solutions of those equations in some special cases. In particular, we find that if the fragmented state is realized in a toroidal trap, under certain conditions the system will spontaneously acquire macroscopic orbital angular momentum. Finally, we discuss the nature of the phase transition from an uncondensed gas to a fragmented-state condensate.
Motivated by a recent experiment [Chikkatur et al., Science, 296, 2193 (2002)] on the merging of atomic condensates, we investigate how two independent condensates with random initial phases can develop a unique relative phase when we move them together. In the adiabatic limit, the uniting of independent condensates can be understood from the eigenstate evolution of the governing Hamiltonian, which maps degenerate states (corresponding to fragmented condensates) to a single state (corresponding to a united condensate). In the nonadiabatic limit corresponding to the practical experimental configurations, we give an explanation on why we can still obtain a large condensate fraction with a unique relative phase. Detailed numerical simulations are then performed for the nonadiabatic merging of the condensates, which confirm our explanation and qualitative estimation. The results may have interesting implications for realizing a continuous atom laser based on the merging of condensates.