Statistical properties of cold bosons in a ring trap

Maciej Kruk, Maciej Łebek, and Kazimierz Rzążewski

Phys. Rev. A 101, 023622 – Published 28 February 2020

Abstract

A study of an interacting system of bosons in a ring trap at a finite temperature is presented. We consider a gas with contact and long-range dipolar interactions within a framework of the classical fields approximation. For a repulsive gas we have obtained coherence length, population of the ground state, and its fluctuations as a function of temperature. In the case of an attractive gas we study local density fluctuations. Additionally, we exactly calculate the partition function for the ideal gas in the canonical ensemble and derive several other macroscopic state functions.

Received 23 October 2019
Accepted 4 February 2020

DOI:https://doi.org/10.1103/PhysRevA.101.023622

Physics Subject Headings (PhySH)

Bose gases Cold and ultracold molecules Dipolar gases

Bose-Einstein condensates

Atomic, Molecular & OpticalStatistical Physics & Thermodynamics

Authors & Affiliations

Maciej Kruk ^*, Maciej Łebek ^†, and Kazimierz Rzążewski^‡

Center for Theoretical Physics, Polish Academy of Sciences, Aleja Lotników 32/46, PL-02-668 Warsaw, Poland

^*mkruk@cft.edu.pl
^†mlebek@cft.edu.pl
^‡kazik@cft.edu.pl

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Issue

Vol. 101, Iss. 2 — February 2020

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Images

Figure 1
Comparison of exact distribution $P (N_{e x})$ and discretized classical fields distribution $P_{c l} (N_{e x})$ obtained for different values of cutoff parameter $n_{\max}$ . The inverse temperature equals $β = 0.03625$ . We observe that cutoff obtained from relation (10) provides the best agreement between curves.
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Figure 2
Average occupation of the condensate (left) and its fluctuations (right) as a function of temperature. We have obtained results for ideal and interacting gas with $g = 1.0$ . For the ideal gas we are able to reproduce with MC simulations exact analytic results calculated in the canonical ensemble. We compare contact interactions and dipolar with $l_{⊥} = 0.025$ . Note that fluctuations obtained in time evolution are smaller than those from MC simulations. Contrary to the 1D harmonic trap [8], depletion with the temperature of interacting gas is slower than in the case of the ideal gas.
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Figure 3
Coherence length as a function of temperature for $g = 0.2$ . We compare results for contact and dipolar interactions with $l_{⊥} = 0.06$ . With new values of cutoff parameter $n_{\max}$ obtained by matching HWHM of exact and classical fields correlation function of the ideal gas (a) we are able to reproduce with MC simulations coherence length calculated from analytic formulas. The lower inset (b) presents comparison of cutoff (10) and new cutoff suited for coherence length. From relation (10) we see that $β n_{\max}^{2} = 0.58 = const$ contrary to values of $n_{\max}$ obtained to match HWHMs.
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Figure 4
Results of reproducing the optimal state from Gaussian ansatz (12) with MC simulations. The left panels present data for contact interactions, whereas the right panels correspond to the dipolar interaction with $l_{⊥} = 0.025$ . The first row compares average energy at very low temperature $β = 0.58$ obtained using cutoff criterion (14) with energy of optimal state with the same value of $g$ . The values of interaction strength for which the MC simulations were performed correspond to the values of $n (g) = 1, ..., 12$ . The second row presents comparison between the width of the optimal state and average width from MC simulations. Note the differences for wide states with large $λ$ .
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Figure 5
Average distance traveled by the wave packet maxima and a histogram of the shift in its position after a single MC step with corresponding function fits— $\sqrt{n / a} + b$ and a normal distribution, respectively. Note the spike in the histogram at $Δ x = 0$ ; the fit was done ignoring this spike. Interactions used for simulations were contact with interaction strength $g = - 0.0152$ .
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Figure 6
Local density fluctuations. The left panels present data for $β = 0.58$ , whereas for the right panels we have $β = 0.0644$ . We compare results for contact interaction (upper row) and dipolar with $l_{⊥} = 0.05$ (lower row) with equal interaction strength $g = - 0.0439$ . The trap was divided into 101 bins.
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