Quantum and thermal fluctuations in bosonic Josephson junctions

We use the Bose-Hubbard Hamiltonian to study quantum fluctuations in canonical equilibrium ensembles of bosonic Josephson junctions at relatively high temperatures, comparing the results for finite particle numbers $N$ to the classical limit that is attained as $N$ approaches infinity. We consider both attractive and repulsive atom-atom interactions, with special focus on the behavior near the $T=0$ quantum phase transition that occurs, for large enough $N$, when attractive interactions surpass a critical level. Differences between Bose-Hubbard results for small $N$ and those of the classical limit are quite small even when $N\sim 100$, with deviations from the limit diminishing as $1/N$.

Figure 1

Energy spectra for $N=100$ atoms and several values of $\gamma =-6,-1.5,0,1.5$. Only the excitation energies are plotted. The inset focuses on the lower eigenstates.

Figure 2

Husimi distributions of some eigenvectors of the BH for $\gamma =1.5$. (a)–(f) The distributions corresponding to the states with, respectively, index $k=0,2,10,90,98$, and 101, where $k=0$ is the ground state. $N=100$ particles.

Figure 3

Husimi distributions for $\gamma =1.5$ (repulsive interactions). (a)–(f) The respective temperatures are $k_{B}T/\left(NJ\right)=0,0.25,0.5,0.75,1,1.25$ and 1.5. $N=100$.

Figure 4

Husimi distributions for $\gamma =-1.5$ (attractive interactions). (a)–(f) The respective temperatures are $k_{B}T/\left(NJ\right)=0,0.25,0.5,0.75,1,1.25$ and 1.5. $N=100$.

Figure 5

Thermal averages for $\gamma =-20,0$, and 20. Black lines show the prediction with the classical $N\to \infty$ approximation of Ref. [26], while red ones are the BH results with $N=100$.

Figure 6

Thermal averages for $\gamma =-1.5,0$, and $1.5$. Red lines are the prediction with the classical $N\to \infty$ approximation of Ref. [26], while black lines are the BH results with $N=100$. The blue dot-dashed lines are the asymptotic GS predictions for large temperature given in (16) and (17).

Figure 7

Contour plots of $\Delta {\widehat{y}}^2$, $\Delta {\widehat{z}}^2$, $\langle {\widehat{x}}^2\rangle$, and $\langle \widehat{x}\rangle$ computed for Bose-Hubbard model with $N=100$, showing dependence on $T$ and $\gamma$.

Figure 8

Same description as Fig. 7, but computed with the GS approximation.

Figure 9

Coherence factor $\alpha =\langle \widehat{x}\rangle$ as a function of $1/N$ for two values of $k_{B}T/\left(NJ\right)=0.5$ and 1, and for two different values of $\gamma =\pm 1.5$. The thin lines are linear regression fits to the first three data points.

Figure 10

(a) Convergence of the BH Husimi distributions (18) to the classical limit (19) in terms of the averaged deviation (20). The solid lines are linear fits to the numerical results. (b) Large-temperature behavior of $N{L}_1$ for three different $N$ and $\gamma =0,\pm 1.5$. Note that we are plotting $N{L}_1$ against $\delta =NJ/k_{B}T$, not $J/k_{B}T$. The case $\gamma =0$ is computed with $N=100$.

Figure 11

Coherence factor of the system as a function of the temperature for several repulsive interactions, $\gamma =1,10$ and 100, for $N=100$ atoms, focusing on the low-temperature region where $k_{B}T\sim J$. The symbols represent the BH results, and the solid curves are obtained from (a) Eq. (15) and (b) Eq. (22).