Phase Coherence of Strongly Interacting Bosons in One-Dimensional Optical Lattices

Abstract

Ultracold Bose gases in one-dimensional optical lattices constitute an important benchmark problem in the study of strongly interacting many-body quantum phases. Here we present a combined experimental and theoretical study of their phase-coherence properties over a wide range of lattice depths. Experimentally, we extract the single-particle correlation function directly from the measured momentum distribution. Theoretically, we perform tensor-network simulations of the Bose-Hubbard model that incorporate all relevant experimental parameters. For deep lattices well within the Mott insulator regime, the experimental results are in good agreement with the expected zero-temperature behavior, with only small temperature-dependent corrections. As the lattice depth is reduced, finite-temperature effects become increasingly important. We find that the experimental data are quantitatively described by an effective temperature extracted from the tensor-network simulations, and that this effective temperature decreases markedly with increasing lattice depth. Rather than indicating actual cooling, we interpret this behavior as evidence of inhibition of thermalization caused by the formation of Mott domains that suppress heat transport. Counterintuitively, the inhibition of thermalization favors the preparation of an effectively low-entropy quantum gas in the trap center for large lattice depths.

Figures (12)

• Figure 1: a: Sketch of the experimental system. We realize a two-dimensional array of independent 1d gases using very anisotropic optical lattices. For typical parameters, we populate about $\sim 12$ planes along the vertical $z$ direction, with $\sim 100$ 1d systems in the central plane and with $\sim 150$ atoms in the most populated 1d system in the trap center. b: Calculated density profiles $\bar{f}(x)$ of the 1d system at the center of the array (chemical potential $\mu = 2.3\,U$), shown for lattice depths $V_x = 7.3,\, 9.0,\, 10.8,\,13.3,\, 15.9\,E_{\mathrm{R}}$ from right to left. All profiles are plotted for $\beta U =164$, with $\beta =1/(k_{\mathrm{B}} T)$ the inverse temperature. c: Measured momentum distribution for $V_x \simeq 9.9\,E_{\mathrm{R}}$. The solid line shows a fit using Eq. \ref{['eq:nkexp']}. The dashed line shows the Wannier envelope $\mathcal{W}_0(k)$. Both $n(k)$ and $\mathcal{W}_0(k)$ are normalized to unity. The orange line shows the residuals of the fit. d: Remainder $R_0 =n(k) - \mathcal{W}_0(k)$ after removing the first (zero order) Fourier component. e: Remainder $R_1 =n(k) -\mathcal{W}_0(k) [1+ 2 C_1(1) \cos( k)]$ after removing the first two Fourier components.
• Figure 2: Single-particle correlation function $\overline{C_1}(s)$ for $V_x=7.2$ (a), $V_x=12.4$ (b), and $V_x=15.0\,E_{\mathrm{R}}$ (c). Solid circles show the experimental data, the dashed lines exponential fits to the data, and the solid lines numerical TN calculations for varying inverse temperatures in the range $\beta U = 1.3-164$ identified by the color code. Three indicative values (for clarity) in the sequence $1.28,2.56,5.12,10.24,20.48,40.96,81.92,163.84$ are shown next to the curves. Exponential fits include a constant background shown by the shaded areas.
• Figure 3: Exponential coherence length deduced from experimental data (solid circles). The dashed and solid lines show the $T=0$, strong-coupling coherence length in the Mott insulator regime [Eq. (\ref{['eq:LcMott']})] for $f_0=2,3$, respectively. The shaded area indicates the region of validity $V_x \geq V_x^\ast$ of the single-band Bose-Hubbard model.
• Figure 4: a-b: Low-temperature behavior for deep lattices $V_x > 15\,E_{\mathrm{R}}$: single-particle correlators (filled circles) for $V_x = 15.9, 17.3\,E_{\mathrm{R}}$, respectively, restricted to short distances $s \leq 4$. The solid lines show the TN simulations $\overline{C_1}_{\mathrm{TN}}$ for the experimental geometry and low temperatures $\beta U = 40.96,81.92,163.84,327.68$. The dotted lines show the $T=0$ strong-coupling prediction $C_1^{\mathrm{MI}}$ [Eq. (\ref{['eq:CsMott']})] for a uniform system with $f_0=3$ atoms per site. c: Finite temperature behavior for $V_x \leq 15\,E_{\mathrm{R}}$: effective temperature $T_{\mathrm{eff}}$ in units of the single-particle tunneling energy $t_{x}$.
• Figure 5: Charateristic sizes $r_{x/y}=\sqrt{U/\kappa_{x/y}}$ (and their geometric average) in the strongly interacting regime $U \gg J$.