Metastable Bose-Einstein condensation in a strongly correlated optical lattice

We experimentally and theoretically study the peak fraction of a Bose-Einstein condensate loaded into a cubic optical lattice as the lattice potential depth and entropy per particle are varied. This system is well described by the superfluid regime of the Bose-Hubbard model, which allows for comparison with mean-field theories and exact quantum Monte Carlo (QMC) simulations. By correcting for systematic discrepancies between condensate and peak fraction, we find that the QMC simulations and measured peak fraction agree at low entropies per particle. At high entropy, however, we discover that the experiment consistently shows the presence of a condensate at temperatures higher than the critical temperature predicted by QMC simulations. This metastability suggests that turning on the lattice potential is nonadiabatic. To confirm this behavior, we compute the time scales for relaxation in this system, and find that equilibration times are comparable with the known heating rates. The similarity of these time scales implies that turning on the lattice potential adiabatically may be impossible. Our results point to the urgent need for a better theoretical and experimental understanding of the time scales for relaxation and adiabaticity in strongly interacting quantum gases, and the importance of model-independent probes of thermometry in optical lattices.

Figure 1

High $\text{OD}$ TOF images for a variety of lattice depths (in units of $E_R)$ and initial entropies averaged over seven iterations. The top line shows the gas released from the harmonic trap; the starting conditions were almost identical for all lattice depths. The $\text{OD}$ saturates at approximately 3 because of nonabsorbable light in the laser beam used for imaging.

Figure 2

Illustration of the fitting procedure used to determine peak fraction from a high $\text{OD}$ image [(a) and (b)] and from a low $\text{OD}$ image [(c) and (d)]. Here we use an averaged image for $s=10$ and $S/N{k}_B\approx 1.35$ (middle image in Fig. 1). For all these images, we have first subtracted off a Gaussian fit as described in the main text. In (a), we show the high OD image with the peaks masked and the nonpeak distribution coincident with the peak assumed to be a uniform distribution. Note that we have masked the nine peaks that correspond to zero momentum and the first- and second-order diffraction peaks. We ignore the higher-order peaks, which contain a negligible number of atoms. A slice along the red dashed line is displayed in (b). The black line is the nonpeak distribution, and the red line is the complete image including the peaks. In (c), we show the low $\text{OD}$ image used to determine the peak number. A slice along the red dashed line is displayed in (d). The black line is the TF fit to the peaks and the red line is the data from the image. Note that there is a scale factor between the high and low OD images of approximately 5.75.

Figure 3

Comparison of condensate $n_0$ and peak $f_0$ fraction obtained from QMC simulations for $s=6,8,10,12$ [(a)–(d), respectively]. $n_0$ was obtained directly by diagonalizing the single-particle density matrix. $f_0$ was calculated by fitting QMC momentum distributions for a finite TOF of 20 ms using experimental fitting procedures. The monotonic dependence is clear, but it is to be noted that the procedure cannot be used when $n_0<0.05$. The uncertainty in these values is smaller than $1\%$. The maximum $n_0$ and $f_0$ is smaller at higher $s$ because of quantum depletion.

Figure 4

The energy per particle for varying lattice depths $s=\{6,8,10,12\}$ [(a)–(d), respectively] for a system of $N\sim 200000$ particles in a trap. These points were fitted using a procedure outlined in the text to obtain the entropy per particle that can be compared with experiment. Temperatures only below the critical temperature for superfluidity are sampled.

Figure 5

Comparison between different types of mean-field theories against QMC results. Panels (a)–(d) correspond to $s=6$, 8, 10, and 12, respectively. The particle number was kept constant at $N\sim 200000$. All other parameters were matched to experimental specifications presented in Fig. 7. Green squares (dashed line) correspond to HF, orange triangles (dotted line) to HFBP, red circles (solid line) to Gutzwiller, and black diamonds (with error bars) to QMC. The error bars show the statistical uncertainty in the QMC results.

Figure 6

Comparison between Gutzwiller (red triangles) and QMC (black squares) results for a strongly interacting system $\left(s=13.6\right)$ at low filling. The central density for these results is approximately one particle per site with $\omega =2\pi \times 55.6\text{Hz}$ and $N\sim 60000$. The line is a guide to the eye.

Figure 7

Experimentally determined peak fraction at each lattice depth as a function of the initial entropy per particle in the harmonic trap (circles). Corresponding peak fraction from QMC simulations determined by fitting the momentum distribution using the same procedure used in experiments in the lattice are also shown (triangles). The number of atoms at each experimental point (from lowest to highest entropy) is (a) $s=6,N=\{2.38\pm 0.03,2.08\pm 0.06,1.64\pm 0.03,2.04\pm 0.07,2.06\pm 0.09\}\times {10}^5$, (b) $s=8,N=\{2.36\pm 0.06,2.05\pm 0.11,1.66\pm 0.07,2.02\pm 0.09,1.73\pm 0.05\}\times {10}^5$, (c) $s=10,N=\{2.41\pm 0.08,2.15\pm 0.11,1.68\pm 0.09,2.03\pm 0.16,1.85\pm 0.09\}\times {10}^5$, and for (d) $s=12,N=\{2.55\pm 0.12,1.98\pm 0.10,1.65\pm 0.04,2.06\pm 0.11,1.74\pm 0.11\}\times {10}^5$. QMC curves are constrained to a constant number, which is the mean of the number at each lattice depth $(N\sim 200000)$. The tunneling, interaction energies, and trap frequencies for $s=\{6,8,10,12\}$ are, respectively, $t=\{1.17,0.711,0.443,0.283\}\times {10}^{-31}\mathrm{J},U=\{3.92,5.23,6.48,7.67\}\times {10}^{-31}\mathrm{J}$, and $\omega =2\pi \times \{51.27,55.48,59.4,63.07\}\mathrm{Hz}.$ The error bars in the experimental data show the standard error of the mean for the average across seven measurements. The horizontal error bars for the measurements also include the influence of the uncertainty in the trap frequency and number of atoms and do not impact the QMC results. The uncertainty in the QMC results is too small to be visible.

Figure 8

Peak fraction for different lattice turn-on times $\tau$ at $s=6$. The exponential time constant used for ramping on the lattice light is set to $2\tau$. Data and theoretical predictions are shown for $S/N=0.44$ (triangles, solid), 1.2 (squares, dashed), and 2.2 (circles, dotted) $k_B$. The lines are the equilibrium $f_0$ predicted by QMC simulations taking into account heating from the lattice light. At the highest $S/N$, QMC predicts that a condensate should be absent in the lattice.

Figure 9

Schematic representation of Landau damping in one dimension. The dispersion $E\left(k\right)$ is shown using a solid line. We consider the process whereby a quasiparticle $\left(\gamma \right)$ can decay by coupling to a resonant transition between two quasiparticles $(\nu$ and $\mu )$ mediated by the underlying condensate. By conservation of total particle number, this reduces the overall condensate density.

Figure 10

Estimates of the lower (empty circles) and upper bound (filled circles) of the decay time associated with Landau damping vs entropy per particle for $s=6,8,10,12$ [(a)–(d)].