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arXiv:cond-mat/0407118 · Journal de Physique IV (Proceedings)

Simple theoretical tools for low dimension Bose gases

Yvan Castin

Published Oct 1, 2004• 117 citations
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Abstract

We first consider an exactly solvable classical field model to understand the coherence properties and the density fluctuations of a one-dimensional (1D) weakly interacting degenerate Bose gas with repulsive interactions at temperatures larger than the chemical potential. In a second part, using a lattice model for the quantum field, we explain how to carefully generalize the usual Bogoliubov approach to study a degenerate and weakly interacting Bose gas in 1D, 2D or 3D in the regime of weak density fluctuations. In the last part, using the mapping to an ideal Fermi gas in second quantized formalism, we calculate and discuss physically the density fluctuations and the coherence properties of a gas of impenetrable bosons in 1D.

DOI
10.1051/jp4:2004116004

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We first consider an exactly solvable classical field model to understand the coherence properties and the density fluctuations of a one-dimensional (1D) weakly interacting degenerate Bose gas with repulsive interactions at temperatures larger than the chemical potential. In a second part, using a lattice model for the quantum field, we explain how to carefully generalize the usual Bogoliubov approach to study a degenerate and weakly interacting Bose gas in 1D, 2D or 3D in the regime of weak density fluctuations. In the last part, using the mapping to an ideal Fermi gas in second quantized formalism, we calculate and discuss physically the density fluctuations and the coherence properties of a gas of impenetrable bosons in 1D.
This is an introductory lecture to the theory of degenerate Fermi gases, in the context of present experiments on atomic Fermi gases. In part one, some properties of the ideal Fermi gas are presented, including a discussion of the fluctuations of the number of fermions in a given spatial zone in 1D, 2D and 3D. In part two, two-body aspects of the interaction potential are discussed and several possible models for the interaction are analyzed, including the two-channel model for the Feshbach resonance. In part three, basic predictions of zero temperature BCS theory are presented, including a derivation of superfluid hydrodynamic equations from time dependent BCS theory.
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.
We present an extension of the well-known Bogoliubov theory to treat low-dimensional degenerate Bose gases in the limit of weak interactions and low density fluctuations. We use a density-phase representation and show that a precise definition of the phase operator requires a space discretization in cells of size l. We perform a systematic expansion of the Hamiltonian in terms of two small parameters, the relative density fluctuations inside a cell and the phase change over a cell. The resulting macroscopic observables can be computed in one, two, and three dimensions with no ultraviolet or infrared divergence. Furthermore, this approach exactly matches Bogoliubov's approach when there is a true condensate. We give the resulting expressions for the equation of state of the gas, the ground state energy, and the first order and second order correlation functions of the field. Explicit calculations are done for homogeneous systems.
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.
This thesis is devoted to the study of the effect of disorder on low-dimensional weakly interacting Bose gases. In particular, the disorder triggers a quantum phase transition in one dimension at zero temperature that is investigated here through the study of the long-range behaviour of the one-body density matrix. An algebraic spatial decay of the coherence marks the quasicondensate, whereas, in the case of strong disorder, an exponential decay is recovered and it characterizes the insulating Bose-glass phase. This analysis is performed using an extended Bogoliubov theory to treat low dimensional Bose gases within a density-phase approach. A systematic numerical study allowed to draw the phase diagram of 1D weakly interacting bosons. The phase boundary obeys two different power laws between interaction and disorder strength depending on the regime of the gas where the transition occurs. These relations can be explained by means of scaling arguments valid in the white noise limit and in the Thomas-Fermi regime of the Bose gas. The phase transition to a quasicondensed phase comes along with the onset of superfluidity: the inspection of the superfluid fraction of the gas is consistent with these predictions for the boundary. The finite temperature case and the scenario in two dimensions are briefly discussed. The quantum phase transition is caused by low-energy phase fluctuations that destroy the quasi-long-range order characterizing the uniform system. Within the approach presented here, the phase fluctuations are identified as the low-lying Bogoliubov modes. Their properties have been investigated in detail to understand which changes trigger the phase transition and we found that the transition to the insulating phase is accompanied by a diverging density of states and a localization length, measured through the inverse participation ratio, that diverges as a power-law with power – 1 for vanishing energy. The fragmentation of the gas is also studied: this notion is very often associated with the onset of the insulating phase. The characterization of the density fragmentation is performed by analyzing the probability distribution of the density. A density profile is defined as fragmented when the probability distribution at vanishing density is finite or divergent and this happens for a gas in the Bose-glass phase. On the contrary, the superfluid phase is characterized by a zero limiting probability of having vanishing densities. This definition is derived analytically, and confirmed by a numerical study. This fragmentation criterion is particularly suited for detecting the phase transition in experiments: when a harmonic trap is included, the transition to the insulating phase can be extracted from the statistics of the local density distribution.
This Thesis discusses approaches to calculating static and dynamic correlation functions in onedimensional gases of interacting bosons. The first part of the Thesis deals with the momentummomentum correlation function in the weakly interacting 1D Bose gas. In the regime of phase fluctuating quasicondensate this correlator is found to differ qualitatively from the form predicted by Bogoliubov theory of the true condensate in that correlations between any two values of momentum become finite. Linear hydrodynamics used to calculate the quasicondensate correlation function prove to be adequate. A classical field approximation is used to smoothly interpolate between the quasicondensate and strongly degenerate gas regimes. In the second part the focus is shifted to the dynamical structure factor and it is shown that hydrodynamics is generally inapplicable to calculating the dynamical correlators. The hydrodynamic treatment is enhanced using impurity theory and an exact model-dependent boundary is obtained for the region of applicability of hydrodynamics on the momentum-energy plane. Numerical estimates show that non-hydrodynamic behavior should be observable for the currently available values of experimental parameters.