Figure 1

Phase-contrast imaging of the density difference of two spin states. (a) The probe beam is tuned to the red for the $|1⟩\to |e⟩$ transition and to the blue for the $|2⟩\to |e⟩$ transition. The resulting optical signal in the phase-contrast image is proportional to the density difference $n_d\equiv n_1-n_2$, where $n_1$ and $n_2$ are the densities of the states $|1⟩$ and $|2⟩$, respectively. (b) Phase-contrast images of trapped atomic clouds in state $|1⟩$ (left) and state $|2⟩$ (right) and of an equal mixture of the two states (middle).Reuse & Permissions

Figure 2

In situ direct imaging of trapped Fermi gases for various population imbalance $\delta$. The integrated 2D distributions of the density difference ${\tilde{n}}_d(x,z)\equiv \int n_d(\overrightarrow{r})dy$ were measured using phase-contrast imaging at $B=834\mathrm{G}$ for total atom number $N_t\approx 1\times {10}^7$. For $\delta \le 75\%$, a distinctive core was observed showing the shell structure of the cloud. The field of view for each image is $160\mu \mathrm{m}\times 800\mu \mathrm{m}$. The three leftmost images are displayed with different contrast levels for clarity. The image with $\delta =5(3)\%$ was taken for $N_t\approx 1.7\times {10}^7$.Reuse & Permissions

Figure 3

Reconstruction of 3D distributions from their integrated 2D distributions. (a) An integrated 2D distribution ${\tilde{n}}_d$ with $\delta =58\%$ at $B=834\mathrm{G}$. (b) A less noisy distribution was obtained by averaging four quadrants with respect to the central dashed lines in (a). (d) The 1D profile obtained by integrating the averaged distribution along the $z$ direction shows the “flattop” feature. (c) The 2D cut $n_d(x,y=0,z)$ of the 3D distribution $n_d(\overrightarrow{r})$ was reconstructed by applying the inverse Abel transformation to (b). (e) The radial profile of the central section of the reconstructed 3D distribution in the $xy$ plane. The dashed lines in (d) and (e) are fits to the profiles’ wings using a TF distribution.Reuse & Permissions

Figure 4

Phase separation in a strongly interacting Fermi gas. Normalized central density difference $\eta$ (black circles) and condensate fraction (red triangles) as functions of imbalance $\delta$ for various interaction strengths at our lowest temperatures ($T\lesssim 0.06T_F$). $\eta \equiv n_{d0}/n_0$, where $n_{d0}$ was measured as the average over the central region of $7\mu \mathrm{m}\times 40\mu \mathrm{m}$ in a 2D cut of the 3D distribution [Fig. 3c] and $n_0=3.3\times {10}^{12}{\mathrm{cm}}^{-3}$ is the calculated central density of a fully polarized Fermi gas with $N_t=1\times {10}^7$. The condensate fraction was determined from the minority component (see the text). The solid line is a fit for the condensate fraction to a threshold function $\propto (1-|\delta /{\delta }_c{|}^n)$. The critical imbalances ${\delta }_c$ indicated by the vertical dashed lines were 96% (exponent $n=21$), 77% ($n=3.1$), and 51% ($n=3.4$) for $B=780\mathrm{G}$, $B=834\mathrm{G}$, and $B=884\mathrm{G}$, respectively.Reuse & Permissions

Figure 5

Characterization of the shell structure. (a) Reconstructed 3D radial profiles at $B=834\mathrm{G}$. (b) Radius of the majority component $R$ (black circles), peak position of the density difference $R_p$ (red circles), and radius of the empty core $R_c$ (blue triangles) as a function of population imbalance $\delta$. $R$ was determined from the profiles’ wings using a fit to a zero-temperature TF distribution. The solid line indicates the TF radius of the majority component in an ideal noninteracting case. $R$ at $\delta =0$ (open circle) was measured from an image taken with a probe frequency preferentially tuned to one component. $R_c$ is defined as the position of the half-peak value in the empty core region for $\delta <{\delta }_c$. (c) Visibility of the core region is defined as $\alpha \equiv (n_d(R_p)-n_{d0})/(n_d(R_p)+n_{d0})$. For low $\delta$, $n_d(R_p)$ is small, and small fluctuations of $n_{d0}$ around zero lead to large fluctuations in $\alpha$.Reuse & Permissions

Figure 6

Emergence of phase separation in an imbalanced Fermi gas. The temperature of the cloud was controlled by varying the final value of the trap depth $U_f$ in the evaporation process. Phase-contrast images were taken after adiabatically ramping the trap depth up to $k_B\times 3.7\mu \mathrm{K}$ ($f_r=192\mathrm{Hz}$). The whole evaporation and imaging process was performed at $B=834\mathrm{G}$ ($N_t\approx 1.7\times {10}^7$, $\delta \approx 56\%$). The field of view for each image is $160\mu \mathrm{m}\times 940\mu \mathrm{m}$. The vertical and the horizontal scale of the images differ by a factor of 1.5.Reuse & Permissions

Figure 7

Phase transition in an imbalanced Fermi gas. The phase transition shown in Fig. 6 was characterized by measuring (a) population imbalance $\delta$, (b) temperature $T$ (black circles), ${\chi }^2$ for fitting the minority cloud with a finite temperature Fermi-Dirac distribution (red triangles), (c) central density difference $n_{d0}$ (black circles), and condensate fraction (red triangles). The solid line is a guide to the eye for ${\chi }^2$. $\delta$ decreased mainly due to loss in the majority component, which was reduced by 14% between $U_f/k_B=3$ and $1\mu \mathrm{K}$ and more rapidly below $1\mu \mathrm{K}$. The number of minority atoms was almost constant ($N_2\approx 3.7\times {10}^6$). $T$ was determined from the noninteracting outer region of the majority cloud after 10 ms of ballistic expansion [7]. $T$ and $n_{d0}$ are averaged values of three independent measurements. The open (solid) arrow in (c) indicates the position for $T_c$ ($T^{*}$). See the text for the definitions of $T_c$ and $T^{*}$.Reuse & Permissions