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arXiv:cond-mat/0107387 · Physical Review A

Quantum fluctuations in one-dimensional arrays of condensates

Alessandro Cuccoli, Andrea Fubini, V. Tognetti, Ruggero Vaia

Published Nov 8, 2001• 4 citations
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Abstract

The effects of quantum and thermal fluctuations upon the fringe structure predicted to be observable in the momentum distribution of coupled Bose-Einstein condensates are studied by the effective-potential method. For a double-well trap, the coherence factor recently introduced by Pitaevskii and Stringari (e-print cond-mat/0104458) is calculated using the effective potential approach and is found in good agreement with their result. The calculations are extended to the case of a one-dimensional array of condensates, showing that quantum effects are essentially described through a simple renormalization of the energy scale in the classical analytical expression for the fringe structure. The consequences for the experimental observability are discussed.

DOI
10.1103/physreva.64.061601

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The effects of quantum and thermal fluctuations upon the fringe structure predicted to be observable in the momentum distribution of coupled Bose-Einstein condensates are studied by the effective-potential method. For a double-well trap, the coherence factor recently introduced by Pitaevskii and Stringari (e-print cond-mat/0104458) is calculated using the effective potential approach and is found in good agreement with their result. The calculations are extended to the case of a one-dimensional array of condensates, showing that quantum effects are essentially described through a simple renormalization of the energy scale in the classical analytical expression for the fringe structure. The consequences for the experimental observability are discussed.
The quantum and thermal fluctuations of the phase are investigated in a cold Bose gas confined by a double well trap. The coherence of the system is discussed in terms of the visibility of interference fringes in both momentum and coordinate space. The visibility is calculated at zero as well as at finite temperature. The thermal fluctuations are shown to affect significantly the transition from the coherent to the incoherent regime even at very low temperatures. The coherence of an array of multiple condensates is also discussed.
The quantum description of a Bose-Einstein condensates in double-well traps connected by a Josephson junction is discussed. The quantization in terms of the number and phase canonically-conjugated operators gives correct energy levels, but results in finite phase fluctuations in the absence of interaction. This seems to contradict the perfect coherence of a system of non-interacting particles. The problem can be solved by calculating the observables that characterize the coherence properties of a system, as, for example, the momentum distribution. It occurs that the visibility of interference fringes depends both on the fluctuations of the phase and the number of atoms. The visibility is calculated in the limit of small interaction and the results exhibit the correct behavior. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Motivated by the MIT experiment [Gorlitz {\em et al.}, Phys. Rev. Lett. {\bf 87}, 130402 (2001)], we analytically study the effect of density and phase fluctuations on various observables in a quasi one-dimensional degenerate Bose gases. Quantizing the Gross-Pitaevskii Hamiltonian and diagonalize it in terms of the normal modes associated with the density and phase fluctuations of a quasi-one dimensional Bose gas. We calculate dynamic structure factor S(q,ω)S(q,ω)S(q,ω) from low-energy condensate density fluctuations and find that there are multiple peaks in S(q,ω)S(q,ω)S(q,ω) for a given momentum qqq due to the discrete energy spectrum. These multiple peaks can be resolved by a two-photon Bragg pulse with a long duration which transfer the momentum to the system. We calculate the momentum transferred $ P_z(t)$ by using the phase-density representation of the Bose order parameter. We also calculate the single-particle density matrix, phase coherence length, and momentum distribution by taking care of the phase fluctuations upto fourth-order term as well as the density fluctuations. Our studies on coherence properties shows that 1D Bose gases of MIT experiment do not form a true condensate, but it can be obtained by a moderate changes of the current experimental parameters.
Motivated by the MIT experiment [Gorlitz {\em et al.}, Phys. Rev. Lett. {\bf 87}, 130402 (2001)], we analytically study the effect of density and phase fluctuations on various observables in a quasi one-dimensional degenerate Bose gases. Quantizing the Gross-Pitaevskii Hamiltonian and diagonalize it in terms of the normal modes associated with the density and phase fluctuations of a quasi-one dimensional Bose gas. We calculate dynamic structure factor S(q,ω)S(q,\omega)S(q,ω) from low-energy condensate density fluctuations and find that there are multiple peaks in S(q,ω)S(q,\omega)S(q,ω) for a given momentum qqq due to the discrete energy spectrum. These multiple peaks can be resolved by a two-photon Bragg pulse with a long duration which transfer the momentum to the system. We calculate the momentum transferred $ P_z(t)$ by using the phase-density representation of the Bose order parameter. We also calculate the single-particle density matrix, phase coherence length, and momentum distribution by taking care of the phase fluctuations upto fourth-order term as well as the density fluctuations. Our studies on coherence properties shows that 1D Bose gases of MIT experiment do not form a true condensate, but it can be obtained by a moderate changes of the current experimental parameters.
So far much theoretical consideration of experiments on the Bose-Einstein condensation (BEC) of alkali-metal atoms in harmonic traps is based on the Gross-Pitaevskii (GP) equation. In this paper, we attempt to formulate the BEC in the language of quantum field theory and to estimate the quantum and thermal fluctuation effects, which are neglected in the approximation using the GP equation. First, the formulation at zero temperature is developed, and then it is extended to the finite-temperature case by means of thermofield dynamics. We treat the zero-energy mode with care, so that the canonical commutation relations hold. As a result, an infrared divergence appears, but it can be renormalized into the observed condensate number. Numerical calculations are performed. For illustration, the corrections at one-loop level to the original GP equation are given. We also calculate numerically the effects of quantum and thermal fluctuations on the distribution of condensed atoms.
We investigate the phase fluctuations of a Bose-Einstein condensate in one- (1D) and three-dimensional (3D) elongated harmonic traps with the help of the numerical simulation using classical field approximation. By calculating the eigenvalues of the one-dimensional single-particle density matrix we show that the phase fluctuations of a Bose-Einstein Condensate in a 1D system are due to the thermal low-energy excitations. We find that the phase fluctuations depend on the temperature as has been predicted [Phys. Rev. Lett. 85, 3745 (2000), Phys. Rev. Lett. 87, 050404 (2001)]. Finally we show that the phase coherence length of a condensate in a harmonic trap depends on the aspect ratio of anisotropy (geometry) of the trap. We determine the border between the 3D and quasi-one-dimensional systems by calculating the phase coherence length of the condensate.