Grand-canonical condensate fluctuations in weakly interacting Bose-Einstein condensates of light

Grand-canonical fluctuations of Bose-Einstein condensates of light are accessible to state-of-the-art experiments [J. Schmitt et al., Phys. Rev. Lett. 112, 030401 (2014).]. We phenomenologically describe these fluctuations by using the grand-canonical ensemble for a weakly interacting Bose gas at thermal equilibrium. For a two-dimensional harmonic trap, we use two models for which the canonical partition functions of the weakly interacting Bose gas are given by exact recurrence relations. We find that the grand-canonical condensate fluctuations for weakly interacting Bose gases vanish at zero temperature, thus behaving qualitatively similarly to an ideal gas in the canonical ensemble (or microcanonical ensemble) rather than the grand-canonical ensemble. For low but finite temperatures, the fluctuations remain considerably higher than for the canonical ensemble, as predicted by the ideal gas in the grand-canonical ensemble, thus clearly showing that we are not in a regime in which the ensembles are equivalent.

Figure 1

The fluctuation $\mathrm{\Delta }N$ in the total number of particles in the subsystem within the LTI model. Thick magenta (dark gray) dash-dotted curve: grand canonical ensemble $N_{\max }=2000, \langle N\rangle =100, \alpha =1/4000000$. For very low temperatures, this agrees with the analytic prediction (46) [green (light gray) solid curve] and for higher temperatures with the noninteracting grand-canonical ensemble [(black) dashed curve]. Thin magenta (dark gray) dashed curve: grand canonical ensemble for the same parameters as the thick dash-dotted curve except for a reduced $\alpha =1/8000000$. For both interaction parameters predictions of all three versions of the LTI model are plotted; the data lie on top of each other.

Figure 2

The fluctuation $\mathrm{\Delta }N$ in the total number of particles in the subsystem within the BDL model, the grand-canonical ensemble is plotted here as a function of temperature. The chemical potential is set such that the average number of particles in the subsystem is $〈n〉=100$. The temperature unit is $T_0=\hslash w_{100}\sqrt{6\times 100}/\pi$, where $w_{100}=\sqrt{{\mathrm{\Omega }}^2+100{\omega }^2}$ and $\mathrm{\Omega }=1,\omega =0.01$. The dashed black curve keeps $w=w_{100}$ constant, and the full red (black) curve takes into account interactions and uses $w_n$.

Figure 3

The statement that the stronger the interaction the smaller are interactions is also true in the BDL model. The top curve shows the condensate fraction, and all other curves show the condensate fluctuations for (from top to bottom): $\omega =0.01,0.02,0.03,0.04$, and 0.05.

Figure 4

In each panel, results within the BDL model are plotted as a function of temperature. The dashed black line shows the number of condensate atoms $n_0$ and the full blue line shows the fluctuation $\mathrm{\Delta }{n}_0$ of this condensate number; both are plotted as a function of temperature. The top panel shows the result in the canonical ensemble with 100 particles. The middle panel shows the results in the grand-canonical ensemble; for the case without interactions, the fluctuations are as large as the condensate number. The bottom panel illustrates the effect of interactions ($\omega =0.01$ in the model) on the grand-canonical fluctuations. The dotted black line in panel (a) shows the result for $n_0$ in the thermodynamic limit. The dash-dotted blue (black) line in the bottom two panels shows the fluctuations for the total number of particles in the grand-canonical ensemble.

Figure 5

The condensate number fluctuation $\mathrm{\Delta }{n}_0$ within the LTI model. Thick light blue (light gray) dash-dotted curve: grand canonical ensemble $N_{\max }=2000, \langle N\rangle =100, \alpha =1/4000000$. For higher temperatures, this again agrees with the non-interacting grand-canonical predictions (black dashed curve). As for the fluctuations of the total number, for lower temperature the system behaves more like a canonical ensemble [black (brown) dotted curves] for which interacting and noninteracting curves lie on top of each other. Thin light blue (light gray) dashed curve: Decreasing the interaction in the grand canonical enesmble by a factor of 2 again shifts the curve toward higher fluctuations (cf. Fig. 1). As in Fig. 1, the predictions of all three versions or the LTI model lie on top of each other.

Figure 6

Kurtosis (49) of the total number fluctuation distribution in the grand-canonical ensemble with the BDL model as a function of temperature and for $〈n〉=100$. The full line is for the case with interactions ($\omega =0.01$), whereas the dashed line is for the ideal gas. The value $\kappa =3$ would correspond to Gaussian fluctuations; it is highligthed via the shaded region.

Figure 7

Thick dash-dotted purple (black) line: Kurtosis (49) of the total number fluctuation distribution in the grand-canonical ensemble with the LTI model. The shaded green (black) area indicates if the kurtosis lies above or below the result for Gaussian fluctuations. Interaction parameters as in Fig. 1; the thin dashed purple (black) line again represents the parameters for which the interaction has been decreased by a factor of 2; for very low temperatures this decreases the kurtosis, and for larger temperatures it increases the kurtosis. Qualitatively, the LTI model again predicts a behavior similar to the BDL model (Fig. 6). As in Figs. 1 and 5, the predictions of all three versions or the LTI model lie on top of each other.