Statistical properties of one-dimensional Bose gas

The Monte Carlo method within the so-called classical field approximation is applied to one-dimensional, weakly interacting, repulsive Bose gas trapped in a harmonic potential. Equilibrium statistical properties of the condensate are calculated within a canonical ensemble. We also calculate experimentally relevant, low-order correlation functions of the whole gas.

Figure 1

Zero-temperature spatial distribution of the ground state for various values of the parameter $g$. Comparison between the classical field approximation (data from Metropolis algorithm) and Gross-Pitaevski equation (from imaginary-time evolution). The solid red line corresponds to the exact result for noninteracting gas.

Figure 2

Relative average number of atoms in the ground state as a function of temperature $T$. The solid red line corresponds to the exact result for noninteracting gas; points correspond to the results of the classical field approximation.

Figure 3

Relative fluctuation of number of atoms in the ground state. The solid red line corresponds to the exact result for noninteracting gas; points correspond to the results of the classical field approximation.

Figure 4

Probability distribution of ground-state occupation at the temperature $T=0.2T_C$. The solid red line corresponds to the exact results for noninteracting gas; symbols correspond to the results of the classical field approximation.

Figure 5

The first-order correlation function and the density of the ground state. For easier comparison, the quasicondensate density profile has been rescaled by the values at the center of the trap.

Figure 6

The correlation length and the size of the ground-state wave function. Both lengths defined as full width at half maximum. The solid line is the formula for correlation length from [20]. The dashed line is the estimation from [13].

Figure 7

Second-order correlation function at the center of the trap as a function of the temperature. The solid red line corresponds to the exact result for noninteracting gas; points correspond to the results of the classical field approximation. The solid black line corresponds to the estimation from [21] corresponding to $g=0.02$ and low temperatures. Note the imperfect agreement of the classical field approximation for an ideal gas.

Figure 8

Fluctuations of number of atoms found in each bin vs number of atoms in each bin at the temperature $T=0.2T_C$. As the strength of interaction increases, the bosonic gas enters the quasicondensate regime.