Quantum Quench of an Atomic Mott Insulator

We study quenches across the Bose-Hubbard Mott-insulator-to-superfluid quantum phase transition by using an ultracold atomic gas trapped in an optical lattice. Quenching from the Mott insulator to the superfluid phase is accomplished by continuously tuning the ratio of Hubbard tunneling to interaction energy. Excitations of the condensate formed after the quench are measured by using time-of-flight imaging. We observe that the degree of excitation is proportional to the fraction of atoms that cross the phase boundary and that the quantity of excitations and energy produced during the quench have a power-law dependence on the quench rate. These phenomena suggest an excitation process analogous to the Kibble-Zurek mechanism for defect generation in nonequilibrium classical phase transitions.

Figure 1

Quench across the MI-SF boundary (a) and excitation measure (b). (a) As shown by the vertical lines, the trapped gas samples a range of densities and effective chemical potentials $\tilde{\mu }$ (in the local-density approximation) [18]; MI regions in the gas are colored red and SF blue. Given the overall confining potential in our experiment and atom number [$(161\pm 13)\times {10}^3$, averaged across all measurements], the maximum $\tilde{\mu }$ is roughly fixed at $2U$ for the measurements described here, which corresponds to a central filling of 4 atoms per site at low $s$ and a central MI with 3 atoms per site at high $s$. A quench is accomplished by rapidly reducing $s$ and thereby increasing $t/U$, as shown by the arrow. (b) A slice (black line) through a typical image (inset) taken after a quench for $s_0=25E_R$ is displayed. The image is fit to a smooth profile (red line), which is used to determine the deviation ${\tilde{\chi }}_{ij}^2$ (blue line) at each pixel in a masked region (gray).

Figure 2

Amount of excitation produced by quenching as the fraction crossing the phase SF-MI boundary is varied. The top left inset shows the experimental timeline; a magnetic field gradient is applied to support the atoms against gravity during TOF. Characteristic TOF images are shown as insets for $s_0=9$, 15, and $25E_R$, and the error bars in this and the next figure are the standard deviation for the average taken over 5 images. There is a 7% systematic uncertainty to $s_0$, which is calibrated by using Kapitza-Dirac diffraction. The overall uncertainty in the MI fraction (red line) ranges from 30% at $s_0=16E_R$ to 10% at $s_0=20E_R$; below $s_0=12E_R$ and above $s_0=22E_R$, the uncertainty in the MI fraction is zero.

Figure 3

Excitation dependence on quench rate. The amount of excitation and kinetic energy determined from TOF images is shown as the quench rate $1/{\tau }_Q$ is varied. The measured offset ${\tilde{\chi }}_0^2$ is subtracted from ${\tilde{\chi }}^2$. Analogously, the measured expansion energy without the quench ${\mathrm{KE}}_0$ (determined by averaging across images with $s_0<13E_R$) is subtracted from KE. For comparison, the critical temperature for condensation in the trap before turning on the lattice and after band mapping is approximately 100 nK. The inset shows the experimental timeline.