Scaling of fluctuations in a trapped binary condensate

We demonstrate that measurements of number fluctuations within finite cells provide a direct means to study susceptibility scaling in a trapped two-component Bose-Einstein condensate. This system supports a second-order phase transition between miscible (cospatial) and immiscible (symmetry-broken) states that is driven by a diverging susceptibility to magnetic fluctuations. As the transition is approached from the miscible side the magnetic susceptibility is found to depend strongly on the geometry and orientation of the observation cell. However, a scaling exponent consistent with that for the homogenous gas ($\gamma =1$) can be recovered, for all cells considered, as long as the fit excludes the region in the immediate vicinity of the critical point. As the transition is approached from the immiscible side, the magnetic fluctuations exhibit a nontrivial scaling exponent $\gamma \simeq 1.30$. Interestingly, on both sides of the transition, we find it best to extract the exponents using an observation cell that encompasses half of the trapped system. This implies that relatively low-resolution in situ imaging will be sufficient for the investigation of these exponents. We also investigate the gap energy and find exponents $\nu z=0.505$ on the miscible side and, unexpectedly, $\nu z=0.60$(3) for the immiscible phase.

Figure 1

Transition properties. Condensate densities of components 1 (dashed line) and 2 (solid line) in (a) miscible and (b) immiscible phases. (c) Component separation in the $x$ direction, where $\langle x_k\rangle =\int d\mathit{}\mathbf{\rho }x{n}_k^0\left(\mathit{}\mathbf{\rho }\right)$. Short red (gray) vertical lines mark the five smallest $g_{12}/g_c$ plotted in Fig. 3. (d) Bogoliubov energy spectrum. The red (gray) solid curves are scaling fits with $\nu z=0.505$ for the miscible region and $\nu z=0.60\left(3\right)$ for the immiscible region (see the text for details). The vertical dashed lines show the interaction parameters for (a) and (b). Here $a_x=\sqrt{\hslash /m{\omega }_x}$.

Figure 2

Magnetic (solid black line) and LDA magnetic [solid red (gray) line] number fluctuations with fits (dashed line), where the fitting regions are $g_{12}/g_c=0.1$–1 (miscible) and 1–1.2 (immiscible). Normal (dot-dashed line) number fluctuations are defined as $\delta N^2=\mathit{}\langle {[{\widehat{N}}_{\sigma }-\langle {\widehat{N}}_{\sigma }\rangle ]}^2\mathit{}\rangle$. Cells are squares of width $L=3a_x$ (cell A), $5a_x$ (cell B), and $9.3a_x$ (cell C). The insets show the cell size and position relative to the condensates for $g_{12}/g_c=0.897$ (left) and 1.094 (right). We consider a temperature $T=7.8\hslash {\omega }_x/k_B$.

Figure 3

Magnetic fluctuations vs cell position for a square cell with $L=5a_x$ on the miscible side. From black to cyan (gray), $g_{12}/g_c=0.01,0.63,0.88,0.97,0.9905,0.9972,0.9991,0.9997$. The five smallest $g_{12}/g_c$ values are marked in Fig. 1. The LDA results for $g_{12}/g_c=0.01$ and $0.9997$ are shown by the short red (gray) curves. The inset shows the cell size and path relative to the condensate for $g_{12}/g_c=0.897$.