We investigate the long-range phase coherence of homogeneous and trapped Bose gases as a function of the geometry of the trap, the temperature, and the mean-field interactions in the weakly interacting limit. We explicitly take into account the (quasi)condensate depletion due to quantum and thermal fluctuations, i.e., we include the effects of both phase and density fluctuations. In particular, we determine the phase diagram of the gas by calculating the off-diagonal one-particle density matrix and discuss the various crossovers that occur in this phase diagram and the feasibility of their experimental observation in trapped Bose gases.We investigate the long-range phase coherence of homogeneous and trapped Bose gases as a function of the geometry of the trap, the temperature, and the mean-field interactions in the weakly interacting limit. We explicitly take into account the (quasi)condensate depletion due to quantum and thermal fluctuations, i.e., we include the effects of both phase and density fluctuations. In particular, we determine the phase diagram of the gas by calculating the off-diagonal one-particle density matrix and discuss the various crossovers that occur in this phase diagram and the feasibility of their experimental observation in trapped Bose gases.

Dec 31, 20030 citations

Universal correlations and coherence in quasi-two-dimensional trapped Bose gases

Markus Holzmann, Maguelonne Chevallier, Werner Krauth

We study the quasi-two-dimensional Bose gas in harmonic traps at temperatures above the Kosterlitz-Thouless transition, where the gas is in the normal phase. We show that mean-field theory takes into account the dominant interaction effects for experimentally relevant trap geometries. Comparing with quantum Monte Carlo calculations, we quantify the onset of the fluctuation regime, where correlations beyond mean-field become important. Although the density profile depends on the microscopic parameters of the system, we show that the correlation density (the difference between the exact and the mean-field density) is accurately described by a universal expression, obtained from classical-field calculations of the homogeneous strictly two-dimensional gas. Deviations from universality, due to the finite value of the interaction or to the trap geometry, are shown to be small for current experiments. We further study coherence and pair correlations on a microscopic scale. Finite-size effects in the off-diagonal density matrix allow us to characterize the crossover from Kosterlitz-Thouless to Bose-Einstein behavior for small particle numbers. Bose-Einstein condensation occurs below a characteristic number of particles which rapidly diverges with vanishing interactions.We study the quasi-two-dimensional Bose gas in harmonic traps at temperatures above the Kosterlitz-Thouless transition, where the gas is in the normal phase. We show that mean-field theory takes into account the dominant interaction effects for experimentally relevant trap geometries. Comparing with quantum Monte Carlo calculations, we quantify the onset of the fluctuation regime, where correlations beyond mean-field become important. Although the density profile depends on the microscopic parameters of the system, we show that the correlation density (the difference between the exact and the mean-field density) is accurately described by a universal expression, obtained from classical-field calculations of the homogeneous strictly two-dimensional gas. Deviations from universality, due to the finite value of the interaction or to the trap geometry, are shown to be small for current experiments. We further study coherence and pair correlations on a microscopic scale. Finite-size effects in the off-diagonal density matrix allow us to characterize the crossover from Kosterlitz-Thouless to Bose-Einstein behavior for small particle numbers. Bose-Einstein condensation occurs below a characteristic number of particles which rapidly diverges with vanishing interactions.

arXiv:0911.1704Physical Review AApr 20, 201032 citations

Phase fluctuations of a Bose-Einstein condensate in low-dimensional geometry

Demascoth Kadio, Mariusz Gajda, Kazimierz Rza̧żewski

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.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.

Physical Review AJul 11, 200530 citations

Condensation and quasicondensation in an elongated three-dimensional Bose gas

Michael C. Garrett, Tod M. Wright, Matthew J. Davis

We study the equilibrium correlations of a Bose gas in an elongated three-dimensional harmonic trap using a grand-canonical classical-field method. We focus in particular on the progressive transformation of the gas from the normal phase, through a phase-fluctuating quasicondensate regime to the so-called true-condensate regime, with decreasing temperature. Choosing realistic experimental parameters, we quantify the density fluctuations and phase coherence of the atomic field as functions of the system temperature. We identify the onset of Bose condensation through analysis of both the generalized Binder cumulant appropriate to the inhomogeneous system, and the suppression of the effective many-body $T$ matrix that characterizes interactions between condensate atoms in the finite-temperature field. We find that the system undergoes a second-order transition to condensation near the critical temperature for an ideal Bose gas in the strongly anisotropic three-dimensional geometry but remains in a strongly phase-fluctuating quasicondensate regime until significantly lower temperatures. We characterize the crossover from a quasicondensate to a true condensate by a qualitative change in the form of the nonlocal first-order coherence function of the field and compare our results to those of previous works employing a density-phase Bogoliubov--de Gennes analysis.We study the equilibrium correlations of a Bose gas in an elongated three-dimensional harmonic trap using a grand-canonical classical-field method. We focus in particular on the progressive transformation of the gas from the normal phase, through a phase-fluctuating quasicondensate regime to the so-called true-condensate regime, with decreasing temperature. Choosing realistic experimental parameters, we quantify the density fluctuations and phase coherence of the atomic field as functions of the system temperature. We identify the onset of Bose condensation through analysis of both the generalized Binder cumulant appropriate to the inhomogeneous system, and the suppression of the effective many-body $T$ matrix that characterizes interactions between condensate atoms in the finite-temperature field. We find that the system undergoes a second-order transition to condensation near the critical temperature for an ideal Bose gas in the strongly anisotropic three-dimensional geometry but remains in a strongly phase-fluctuating quasicondensate regime until significantly lower temperatures. We characterize the crossover from a quasicondensate to a true condensate by a qualitative change in the form of the nonlocal first-order coherence function of the field and compare our results to those of previous works employing a density-phase Bogoliubov--de Gennes analysis.

arXiv:1209.3297Physical Review AJun 12, 201314 citations

Density fluctuations in a quasi-one-dimensional Bose gas as observed in free expansion

Krzysztof Gawryluk, Mariusz Gajda, Mirosław Brewczyk

We study, within the framework of the classical-field approximation, the density correlations of a weakly interacting expanding Bose gas for the whole range of temperatures across the Bose-Einstein condensation threshold. We focus on elongated quasi-one-dimensional systems where there is a huge discrepancy between the existing theory and experimental results [A. Perrin et al., Nat. Phys. 8, 195 (2012)]. We find that the density correlation function is not reduced for temperatures below the critical one as it is predicted for the ideal gas or for a weakly interacting system within the Bogoliubov approximation. This behavior of the density correlations agrees with the above-mentioned experiment with the elongated system. Although the system was much larger than that studied here, we believe that the behavior of the density correlation function found there is quite generic. Our theoretical study indicates also large density fluctuations in the trap in the quasicondensate regime where only phase fluctuations were expected. We argue that the enhanced density fluctuations can originate in the presence of interactions in the system, or more precisely in the presence of spontaneous dark solitons in the elongated gas at thermal equilibrium.We study, within the framework of the classical-field approximation, the density correlations of a weakly interacting expanding Bose gas for the whole range of temperatures across the Bose-Einstein condensation threshold. We focus on elongated quasi-one-dimensional systems where there is a huge discrepancy between the existing theory and experimental results [A. Perrin et al., Nat. Phys. 8, 195 (2012)]. We find that the density correlation function is not reduced for temperatures below the critical one as it is predicted for the ideal gas or for a weakly interacting system within the Bogoliubov approximation. This behavior of the density correlations agrees with the above-mentioned experiment with the elongated system. Although the system was much larger than that studied here, we believe that the behavior of the density correlation function found there is quite generic. Our theoretical study indicates also large density fluctuations in the trap in the quasicondensate regime where only phase fluctuations were expected. We argue that the enhanced density fluctuations can originate in the presence of interactions in the system, or more precisely in the presence of spontaneous dark solitons in the elongated gas at thermal equilibrium.

arXiv:1504.05422Physical Review AOct 7, 20153 citations

Quantum Phase Transition in a Low-Dimensional Weakly Interacting Bose Gas

Luca Fontanesi

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 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.

Jan 1, 20111 citation

T

matrix that characterizes interactions between condensate atoms in the finite-temperature field. We find that the system undergoes a second-order transition to condensation near the critical temperature for an ideal Bose gas in the strongly anisotropic three-dimensional geometry but remains in a strongly phase-fluctuating quasicondensate regime until significantly lower temperatures. We characterize the crossover from a quasicondensate to a true condensate by a qualitative change in the form of the nonlocal first-order coherence function of the field and compare our results to those of previous works employing a density-phase Bogoliubov--de Gennes analysis.