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arXiv:0804.1836 · Physical Review A

Phase coherence and fragmentation in weakly interacting bosonic gases

Gheorghe Sorin Paraoanu

Published Apr 10, 2008• 18 citations
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Abstract

We present a theory of measurement-induced interference for weakly interacting Bose-Einstein-condensed gases. The many-body state resulting from the evolution of an initial fragmented (Fock) state can be approximated as a continuous superposition of Gross-Pitaevskii (GP) states; the measurement breaks the initial phase symmetry, producing a distribution pattern corresponding to only one of the GP solutions. We discuss also analytically solvable models, such as two-mode on-chip adiabatic recombination and soliton generation in quasi-one-dimensional condensates.

DOI
10.1103/physreva.77.041605

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We present a theory of measurement-induced interference for weakly interacting Bose-Einstein-condensed gases. The many-body state resulting from the evolution of an initial fragmented (Fock) state can be approximated as a continuous superposition of Gross-Pitaevskii (GP) states; the measurement breaks the initial phase symmetry, producing a distribution pattern corresponding to only one of the GP solutions. We discuss also analytically solvable models, such as two-mode on-chip adiabatic recombination and soliton generation in quasi-one-dimensional condensates.
The many-body Schr\"odinger dynamics of a one-dimensional bosonic Josephson junction is investigated for up to 10 000 bosons and long times. The initial states are fully condensed and the interaction strength is weak. We report on a universal fragmentation dynamics on the many-body level: systems consisting of different numbers of particles fragment to the same value at constant mean-field interaction strength. The phenomenon manifests itself in observables such as the correlation functions of the system. We explain this universal fragmentation dynamics analytically based on the Bose-Hubbard model. We thereby show that the extent to which many-body effects become important at later times depends crucially on the initial state. Even for arbitrarily large particle numbers and arbitrarily weak interaction strength the dynamics is many-body in nature and the fragmentation universal. There is no weakly interacting limit where the Gross-Pitaevskii mean field is valid for long times.
We have discussed the interference of two independently prepared Bose-Einstein condensates. When the number of atoms in each gas is fixed and the phases of the gases are uncertain, no interference is expected in the single-particle distribution. However, the interference is observed in each single snapshot. We have presented tools for studying such snapshot interference, the measurement-induced interference. The idea is to discard the randomness in the snapshot profiles (the spatial offsets of the interference patterns) and to focus on the typical feature in the interference patterns (the fringe spacing). If the sinusoidal patterns with a definite fringe spacing is actually typical, the fringe spectrum with the random offset discarded does not fluctuate, and we are allowed to speak of the snapshot interference through its average. We have described the two gases as canonical ensembles with the numbers of atoms fixed at N individually. We have shown that the covariance of the fringe spectrum over all possible snapshot profiles is vanishingly small below the critical temperature of the Bose-Einstein condensation: the interference pattern whose fringe contrast is characterized by the average fringe spectrum is certainly observed in every snapshot in the presence of condensates. The fringe contrast becomes stronger as the temperature is lowered and the condensation fraction is increased. This clarifies the importance of the Bose-Einstein condensation to the interference of independent Bose-Einstein condensates. The knowledge I acquired from Prof. Tasaki's lecture on how to characterize the canonical ensemble of bosonic atoms has enabled us to carry out this analysis. Like this example, the coherence phenomena of quantum many-body systems in general would be explained on the basis of the idea of the measurement-indued coherence, without resort to the idea of the spontaneous symmetry breaking. This is actually true also for fermionic systems: the relative phase between two independent superconductors is built up by measurement, even though their U(1) symmetries are not broken and their phases are uncertain beforehand. It would be interesting to explore the possibility of explaining various phenomena, in which the symmetry breaking plays an essential role, on the basis of the measurement-induced coherence.
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.
We investigate the level population statistics and degree of coherence encoded in the single-particle density matrix of harmonically trapped low-dimensional [quasi-one-dimensional (quasi-1D) or quasi-two-dimensional (quasi-2D)] Bose gases with repulsive contact interactions. Using a variational analysis, we derive fragmentation of the condensate in the weakly confining directions into two (quasi-1D) and three (quasi-2D) mutually incoherent macroscopic pieces, upon increasing a dimensionless interaction measure beyond a critical value. Fragmented condensate many-body states in low-dimensional systems therefore occur well before the thermodynamic limit of infinite extension is reached, in which phase fluctuations of the matter wave field create an infinite number of nonmacroscopic fragments.
Quantum phenomena appear in a macroscopic scale in Bose-Einstein condensates. The Gross-Pitaevskii (GP) equation describes the dynamics of the weakly interacting Bose-Einstein condensates. The GP equation has a form of the Schroedinger equation with self-interaction. The coupled Gross-Pitaevskii equations are used to describe some mixtures of Bose-Einstein condensates. In this paper, we will show some numerical results of coupled Gross-Pitaevskii equations without self-interaction, which has a form of nonlinearly-coupled Schroedinger equations. We demonstrate that the macroscopic oscillation and the interference of two quantum wave packets decay in time owing to mutual interaction, which is analogous to the decoherence in quantum mechanics of many particles.