1D to 3D Crossover of a Spin-Imbalanced Fermi Gas

We have characterized the one-dimensional (1D) to three-dimensional (3D) crossover of a two-component spin-imbalanced Fermi gas of ${}^6\mathrm{Li}$ atoms in a 2D optical lattice by varying the lattice tunneling and the interactions. The gas phase separates, and we detect the phase boundaries using in situ imaging of the inhomogeneous density profiles. The locations of the phases are inverted in 1D as compared to 3D, thus providing a clear signature of the crossover. By scaling the tunneling rate $t$ with respect to the pair binding energy ${\epsilon }_B$, we observe a collapse of the data to a universal crossover point at a scaled tunneling value of ${\tilde{t}}_c=0.025(7)$.

Figure 1

(a) Schematic of an array of 1D coupled tubes formed by a 2D optical lattice. The tunneling rate $t$ between the tubes increases with decreasing optical lattice depth. (b) Schematic of phase separation for a trapped spin-imbalanced Fermi gas in 1D (top) and in 3D (bottom) at zero temperature. In 1D, the central region is an FFLO partially polarized superfluid (${\mathrm{SF}}_{\mathrm{P}}$), with balanced superfluid (${\mathrm{SF}}_0$) wings for small polarization $P$. In 3D, for $P<P_c^{3\mathrm{D}}$, a central ${\mathrm{SF}}_0$ core is surrounded by an ${\mathrm{SF}}_{\mathrm{P}}$ or normal partially polarized (${\mathrm{N}}_{\mathrm{PP}}$) phase depending on interactions, and finally an ${\mathrm{N}}_{\mathrm{FP}}$ outer shell. The arrows indicate phase boundaries.

Figure 2

(a),(b) Column density profiles $n_c(0,0,z)$ of spin-imbalanced gases. The $n_c$ are smoothed in $x$ and $z$ using a Gaussian function with a width of $5.3\mu \mathrm{m}$ before taking a cut along the $z$ axis. Both data sets were taken at $B=890\mathrm{G}$, corresponding to $a_{3\mathrm{D}}=-8610a_0$. The scaled tunneling (defined in text) is $t/{\epsilon }_B=0.004$ for the first column and 0.065 for the second. The $|\uparrow ⟩$, $|\downarrow ⟩$, and the difference distributions are indicated by the black, blue, and red curves, respectively. The radii are extracted on both sides of the cloud by finding the radius at which a phenomenological fit to the $n_c$ rises by one standard deviation above the mean background level. The radii extracted from each side are averaged together. (c),(d) The corresponding local polarization $p(0,0,z)$ profiles are found using a weighted average of the central 18 tubes. $p_0$ is the average of the central $13\mu \mathrm{m}$ region along $z$. $N_{\downarrow }$ is consistent with the background noise in the gray region and thus, the local polarization is poorly defined there. The entire cloud in (a) and (c) is ${\mathrm{SF}}_{\mathrm{P}}$, and $R_{\downarrow }\simeq R_d$ as a consequence, while in (b),(d) there is an extended region of ${\mathrm{SF}}_0$ in the center of the cloud ($p_0=0$), then a partially polarized region, ${\mathrm{SF}}_{\mathrm{P}}$ or ${\mathrm{N}}_{\mathrm{PP}}$. $R_{\downarrow }\simeq R_d$ in this 3D-like example since $P_t$ is small.

Figure 3

(a) 1D- and (b) 3D-like phase diagrams for $B=940\mathrm{G}$. $R_{\downarrow }$ (solid triangles) and $R_d$ (solid circles) are scaled by $N^{1/2}{l}_z$ [6, 8], where $l_z=\sqrt{\hslash /m{\omega }_z}$ is the axial harmonic oscillator length and $N=N_{\uparrow }+N_{\downarrow }$. The colored regions correspond to the indicated phases. In (b), the open circle indicates the measured $P_c^{3\mathrm{D}}$ from (d). The dotted line is an extrapolation from $P_c^{3\mathrm{D}}$. (c),(d) The local central polarization $p_0$ vs $P_t$, used to find $P_c^{3\mathrm{D}}$. The insets show the central region near $P_c^{3\mathrm{D}}$. The solid red line is a fit to the data to find $P_c^{3\mathrm{D}}$, using a function with a bilinear slope [27]. The green vertical arrow indicates $P_c^{3\mathrm{D}}$. Each data point is the average of $\sim 10$ experimental realizations, binned with width $\mathrm{\Delta }{P}_t=0.005$.

Figure 4

(a) $P_c^{3\mathrm{D}}$ and (b) $\overline{R}$ vs $t$. Ordered from lowest to highest field, the corresponding $a_{3\mathrm{D}}$ are $6170a_0$, unitarity, $-8610a_0$, $-5360a_0$, and $-4340a_0$, in units of the Bohr radius $a_0$. The corresponding ranges of ${\epsilon }_B$, depending on lattice strength, are $3.8-5.2E_{\mathrm{r}}$, $2.5-3.7E_{\mathrm{r}}$, $1.9-2.9E_{\mathrm{r}}$, $1.6-2.5E_{\mathrm{r}}$, and $1.4-2.3E_{\mathrm{r}}$, respectively. (c) $P_c^{3\mathrm{D}}$ and (d) $\overline{R}$ vs the scaled tunneling rate $\tilde{t}=t/{\epsilon }_B$, showing data collapse. The dotted line in (c) indicates ${\tilde{t}}_{3\mathrm{D}}=0.021(5)$, the value above which the gas has an ${\mathrm{SF}}_0$ core. The suppression of 1D behavior occurs at ${\tilde{t}}_{1\mathrm{D}}=0.029(5)$, indicated by the dotted line in (d). The gray band indicates the uncertainty range in locating ${\tilde{t}}_{3\mathrm{D}}$ and ${\tilde{t}}_{1\mathrm{D}}$. These uncertainties result from the indicated vertical error bars (a few representative examples are shown) which arise from the fits, as well as systematic uncertainty in $P_t$ which is estimated from the standard error of the mean of 10 images known to be balanced.