Lattice thermodynamics for ultracold atoms

We measure the temperature of ultracold ${}^{87}\text{R}\text{b}$ gases transferred into an optical lattice and compare to noninteracting thermodynamics for a combined lattice-parabolic potential. Absolute temperature is determined at low temperature by fitting quasimomentum distributions obtained using band mapping, i.e., turning off the lattice potential slowly compared with the band gap. We show that distributions obtained at high temperature employing this technique are not quasimomentum distributions through numerical simulations. To overcome this limitation, we extract temperature using the in-trap size of the gas.

Figure 1

Spatial (left column) and quasimomentum (right column) probability distributions for a one-dimensional (1D) combined lattice-parabolic potential (light gray curve in top left image) for different quantum numbers $n$ and $\alpha =352.175$. For low $n$, the states are similar to discretized harmonic-oscillator wave functions, but for higher energies the states become localized away from the center of the parabolic potential.

Figure 2

Examples of predicted images from Eq. (26) for $k_{B}T/J=0.5$, $k_{B}T/J=2$, and $k_{B}T/J=15$ (in the thermal limit). Cross sections through the distribution are also shown. Along cross section (a), the distribution has a sharp edge at the Brillouin-zone momentum at high $k_{B}T/J$. For low values of $k_{B}T/J$, this edge vanishes and the distribution is similar to a momentum distribution for atoms confined in a harmonic trap. The direction of gravitational acceleration $g$ relative to the imaging plane for our experiment is labeled by an arrow.

Figure 3

Measured temperatures $T$ in the lattice obtained by fitting the quasimomentum distributions for $s=2$ (top panel) and $s=6$ (bottom panel) as the temperature $T_{ho}$ in the harmonic trap is varied. The vertical error bars are determined by the standard deviation in several measurements that are averaged for each point and a 10% uncertainty in the Brillouin-zone width after time of flight. The error bars in $T_{ho}$ represent statistical uncertainty. There is also a 5% systematic uncertainty in the TOF, which is not included in the error bars. The curve is the thermodynamic prediction for isentropic transfer into the lattice and noninteracting particles. The width of the theory curve reflects errors in $N$ and $\omega$.

Figure 4

Numerical simulation of band mapping for the $n=0$ state. The single-particle eigenstate is shown at the top left. The wave function after 20 ms of TOF, revealing the corresponding momentum distribution, is displayed at the bottom left. The wave function after band mapping is displayed at the top right, with the wave function after 20 ms TOF shown at the bottom right.

Figure 5

(Color online) Failure of band mapping for high $k_{B}T/J$. In the top two panels, the black line is a finite-temperature band-mapped distribution, the red (dark gray) line is a finite-temperature quasimomentum distribution, and the cyan (light gray) line is a semiclassical fit to the band-mapped distribution. The bottom panel shows the temperature $T^{\prime }$ obtained using a semiclassical fit to a 1D simulation of imaging at temperature $T$ after band mapping and TOF. The red (dark gray) line has a slope of 1 and is present to guide the eye.

Figure 6

Temperature measured using the in-trap size for $s=6$. The curve is the theoretical prediction assuming adiabatic transfer into the lattice. The width of this curve is determined by uncertainty in $N$ and $\omega$. The error bars in $T$ reflect the uncertainty in the extrapolation to the in-trap size. The error bars in $T_{ho}$ represent the uncertainty in using standard TOF expansion to measure the temperature of the gas before loading into the lattice. Inset (a) shows the exact calculation of the width in one dimension (black line) from Eq. (29) versus the semiclassical value of $\sqrt{kT/m{\omega }^2}$ (dashed gray line). The agreement is exact except at very low temperatures. Inset (b) shows a sample set of expansion data used to determine the in-trap size. The line is a fit to the data of the form $\sigma =\sqrt{{\sigma }_0^2+A^2{t}^2}$, where $A$ and ${\sigma }_0$ are fit parameters.