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arXiv:0807.4627 · Physical Review A

Fragility of fragmentation in Bose-Einstein condensates

Andrew David Jackson, G. M. Kavoulakis, Manolis Magiropoulos

Published Dec 29, 2008• 12 citations
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Abstract

A Bose-Einstein condensate produced by a Hamiltonian which is rotationally or translationally symmetric may be fragmented as a direct result of these symmetries. A corresponding mean-field unfragmented state, with an identical energy to leading order in the number of particles, can generally be constructed. As a consequence, vanishingly weak symmetry-breaking perturbations destabilize the fragmented state, which would thus be extremely difficult to realize experimentally and lead to an unfragmented condensate.

DOI
10.1103/physreva.78.063623

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A Bose-Einstein condensate produced by a Hamiltonian which is rotationally or translationally symmetric may be fragmented as a direct result of these symmetries. A corresponding mean-field unfragmented state, with an identical energy to leading order in the number of particles, can generally be constructed. As a consequence, vanishingly weak symmetry-breaking perturbations destabilize the fragmented state, which would thus be extremely difficult to realize experimentally and lead to an unfragmented condensate.
Weakly interacting Bose gases usually form Bose-Einstein condensates in which most particles occupy the same single-particle state. However, when this state cannot realize a continuous symmetry of the many-body Hamiltonian, a fragmented condensate exhibiting the expected symmetry may emerge. Here, we produced a three-fragment condensate for a mesoscopic spin-1 gas of about 100 atoms, with anti-ferromagnetic interactions and vanishing collective spin. Using a spin-resolved detection approaching single-atom resolution, we show that the reconstructed state is close to the expected many-body ground state, whereas one-body observables are the same as for a completely mixed state. Our results highlight how the interplay between symmetry and interactions generates entanglement in a mesoscopic quantum system.
The theoretical investigation of rotating Bose-Einstein condensates has mainly focused on the emergence of quantum vortex states and the condensed properties of such systems. In the present work, we concentrate on other facets by examining the impact of rotation on the ground state of weakly interacting bosons confined in anharmonic potentials computed both at the mean-field level and particularly at the many-body level of theory. For the many-body computations, we employ the well-established many-body method known as the multiconfigurational time-dependent Hartree method for bosons. We present how various degrees of fragmentation can be generated following the breakup of the ground state densities in anharmonic traps without ramping up a potential barrier for strong rotations. The breakup of the densities is found to be associated with the acquisition of angular momentum in the condensate due to the rotation. In addition to fragmentation, the presence of many-body correlations is examined by computing the variances of the many-particle position and momentum operators. For strong rotations, the many-body variances become smaller than their mean-field counterparts, and one even finds a scenario with opposite anisotropies of the mean-field and many-body variances. Further, it is observed that for higher discrete symmetric systems of order k, namely three-fold and four-fold symmetry, breakup to k sub-clouds and emergence of k-fold fragmentation take place. All in all, we provide a thorough many-body investigation of how and which correlations build up when a trapped Bose-Einstein condensate breaks up under rotation.
We study spin fragmentation of an antiferromagnetic spin 1 condensate in the presence of a quadratic Zeeman (QZ) effect breaking spin rotational symmetry. We describe how the QZ effect turns a fragmented spin state, with large fluctuations of the Zeeman populations, into a regular polar condensate, where the atoms all condense in the m = 0 state along the field direction. We calculate the average value and variance of the population of the Zeeman state m = 0 to illustrate clearly the crossover from a fragmented to an unfragmented state. The typical width of this crossover is q ∼ k B T / N , where q is the QZ energy, T the spin temperature and N the atom number. This shows that the spin fluctuations are a mesoscopic effect that will not survive in the thermodynamic limit N → ∞ , but are observable for a sufficiently small atom number.
We investigate dynamical behavior of fragmentation of a spinor Bose-Einstein condensate in both the semiclassical and quantum regimes. We show that an unfragmented condensate can become fragmented at later times due to the inherent interactions of elastic atomic collisions, when a certain phase relationship exists among the state amplitudes in different components initially. A close relationship between the fragmentation and population dynamics of atomic distributions within different spin components is found for certain initial states with zero magnetization. The type of fragmentation regime is characterized via atom-number fluctuations. We also address complications of our study when the condensate spatial mode function is different for each spin component.
We consider two models of interacting Bose gases: a gas of spin one particles in the ground state of a cubic box and a one-dimension Bose gas with contact interactions. We show how to calculate exact eigenstates of the corresponding N-body Hamiltonians. Both models share the property of not leading to the formation of a Bose-Einstein condensate, even at zero temperature, in the strict sense of the existence of a single one-particle state with a macroscopic population. We show that a lot of physical insight can be gained on these two model systems by using the usual Hartree-Fock mean field approach: in this approximation, that we test against the exact result, everything happens as if a single realization of the system was a Bose-Einstein condensate in a state phi breaking the rotational or translational symmetry, and varying in a random way for any new experimental realization.
In this thesis, we present some experiments realized with ultracold gases of sodium atoms, trapped at the intersect between two laser beams. At very low temperature, the discretization of energy and the indistinguishability of particles, lead to a new state of matter, a Bose-Einstein condensate. This remarkable phenomenon was initially introduced to describe an ideal gas, that is to say with no interactions between its constituents. Here, we are interested in the effects of the interactions between the atoms. More precisely, our atoms carry a spin 1, and we focus on the collective spin state, in a regime where the spatial degrees of freedom are frozen. Two important results that we present were obtained by embedding the condensate in a nearly vanishing magnetic field. In that regime, interactions dominate and favor the emergence of strongly correlated states. In a first series of experiment, the magnetic field is suddenly decreased to bring the system out-of-equilibrium. The ensuing relaxation dynamics leads to a stationnary state that can be well described by a Gibbs ensemble. In a second experiment, the field is slowly reduced, in order to follow the ground state of the system. We thereby produce a fragmented condensate, which possesses the remarkable feature of being invariant upon spin rotations. The restoration of this symmetry, always broken by single (i.e. nonfragmented) condensates, is driven by the pairing of atoms in singlet states.