Entropy of the Bose-Einstein-condensate ground state: Correlation versus ground-state entropy

Moochan B. Kim, Anatoly Svidzinsky, Girish S. Agarwal, and Marlan O. Scully

Phys. Rev. A 97, 013605 – Published 5 January 2018

Abstract

Calculation of the entropy of an ideal Bose-Einstein condensate (BEC) in a three-dimensional trap reveals unusual, previously unrecognized, features of the canonical ensemble. It is found that, for any temperature, the entropy of the Bose gas is equal to the entropy of the excited particles although the entropy of the particles in the ground state is nonzero. We explain this by considering the correlations between the ground-state particles and particles in the excited states. These correlations lead to a correlation entropy which is exactly equal to the contribution from the ground state. The correlations themselves arise from the fact that we have a fixed number of particles obeying quantum statistics. We present results for correlation functions between the ground and excited states in a Bose gas, so as to clarify the role of fluctuations in the system. We also report the sub-Poissonian nature of the ground-state fluctuations.

Received 6 June 2017
Revised 23 August 2017

DOI:https://doi.org/10.1103/PhysRevA.97.013605

Physics Subject Headings (PhySH)

Quantum thermodynamics

Bose-Einstein condensates

Statistical Physics & Thermodynamics

Authors & Affiliations

Moochan B. Kim¹, Anatoly Svidzinsky¹, Girish S. Agarwal¹, and Marlan O. Scully^1,2

¹Department of Physics, Texas A&M University, College Station, Texas 77843, USA
²Department of Physics, Baylor University, Waco, Texas 76706, USA

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Vol. 97, Iss. 1 — January 2018

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Images

Figure 1
Entropy of ground state in an ideal Bose gas, which is trapped in a 3D harmonic trap. The total number of particles is $N = 200$ . The critical temperature for a 3D harmonic trap is $T_{c} = ℏ Ω / k_{B} {[N / ζ (3)]}^{1 / 3}$ , with harmonic trap oscillation frequency $Ω$ , and Riemann's zeta function $ζ (s)$ . This exact result on entropy is calculated by using the canonical ensemble partition function, which is explained in Appendixes pp1 and pp2 and is drawn as a solid red line. From the approximate density matrix, Eq. (1), the corresponding von Neumann's entropy is plotted as a dashed blue line.
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Figure 2
Entropy for ideal Bose gas which is trapped in a three-dimensional harmonic trap. The detailed parameters are as in Fig. 1. The total entropy is drawn with a dashed blue line, using the procedure in Appendixes pp1 and pp2, and the entropy of the ground state is in the solid red line. In this picture, the entropy for the ground state is multiplied by 100. From the behavior of the occupation number in the ground state, we can see the entropy contribution of the ground state is important below the critical temperature. In a similar way, in terms of correlation entropy the relevant range of the correlation is also below the critical temperature. The inset shows both entropies below $T / T_{c} = 0.2$ .
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Figure 3
The system is an ideal Bose gas trapped in a 3D harmonic trap with 200 particles. The parameters are the same as in Fig. 1. The normalized correlation function $C_{1}$ , Eq. (18), between the ground-state occupation number and that of excited states is plotted as a dashed blue line. $C_{2}$ , Eq. (20), is plotted as a dotted green line. The correlation entropy, Eq. (12), or the entropy of the ground state, is also drawn as a solid red line. In the figure we also show the sub-Poissonian nature of fluctuations by plotting the parameter $Δ n_{0} / \sqrt{〈 n_{0} 〉}$ [dashed brown line ( $\times 5$ )]. The strong sub-Poissonian region corresponds to $Δ n_{0} / \sqrt{〈 n_{0} 〉} ≪ 1$ .
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Figure 4
The correlation function ${\tilde{C}}_{1}$ , Eq. (21), is drawn among the three lowest states. ${\tilde{C}}_{1} (n_{0}, n_{1})$ is drawn as a solid red line, ${\tilde{C}}_{1} (n_{0}, n_{2})$ as a blue dashed line, and ${\tilde{C}}_{1} (n_{1}, n_{2})$ as a green dashed-dotted line. Since the occupation of the ground state is macroscopic in low temperature, the correlation function is noticeable below $T / T_{c} \sim 0.2$ . The correlation between the first- and the second-excited states is negligible, since the occupation number in each state is small compared with the total number of particles. The system is an ideal Bose gas trapped in a 3D harmonic trap, and the parameters are the same as in Fig. 1.
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