Quenched binary Bose-Einstein condensates: Spin-domain formation and coarsening

We explore the time evolution of two-component Bose-Einstein condensates (BECs), quasi one dimensional with respect to their spinor dynamics, following a quench from one component BECs with a $\mathrm{U}\left(1\right)$ order parameter into two-component condensates with a $\mathrm{U}\left(1\right)\times Z_2$ order parameter. In our case, these two spin components have a propensity to phase separate, i.e., they are immiscible. Remarkably, these spin degrees of freedom can equivalently be described as a single-component attractive BEC. A spatially uniform mixture of these spins is dynamically unstable, rapidly amplifying any quantum or pre-existing classical spin fluctuations. This coherent growth process drives the formation of numerous spin-polarized domains, which are far from the system's ground state. At much longer times these domains grow in size, coarsening, as the system approaches equilibrium. The experimentally observed time evolution is consistent with our fully three-dimensional stochastic-projected Gross–Pitaevskii calculation.

Figure 1

Magnetization $M_z\left(\mathbf{r}\right)$. (a) Schematic illustrating a spinor BEC with domains in an anisotropic crossed-dipole trap. (b) Images showing the progression from a uniformly magnetized condensate (short times) in which domains appear (intermediate times) and then grow spatially (long times); during this process the condensate slowly decays away. (c) Color scale indicating the degree of magnetization (colors from blue to red), and the density (intensity from black to colored).

Figure 2

Time evolution of magnetization $M_z\left(z\right)$. (a) Experimental data and (b) finite-temperature simulation using the SP-GPE method. In both simulation and experiment, the spatial structure of $M_z\left(z\right)$ coarsens after an initial growth period as domains coalesce.

Figure 3

Number of domains as a function of $t_{\mathrm{hold}}$. The red symbols depict the experimentally observed number of domains (typical uncertainty plotted on the leftmost point) and the blue curve plots the results of our SP-GPE simulation (uncertainties denoted by the blue band). In both cases, the uncertainties reflect the standard deviation over many realizations. In addition, the red curve fits the data to a model assuming exponential growth along with a nonzero observation threshold, in the LDA. The gray symbols correspond to the ratio $R_z/\left(2{\xi }_{1\mathrm{D}}\right)$: an estimate of domain number, assuming the system with length $2R_z$ is partitioned into domains of local size $\pi {\xi }_{1\mathrm{D}}\left(z\right)$ (the size at which domains initially form); the weighted average of this over our system is about $4{\xi }_{1\mathrm{D}}$.

Figure 4

Power spectral density (PSD). (a) ${\mathrm{PSD}}_z\left(k\right)$ as a function of $t_{\mathrm{hold}}$ showing the formation of a peak at finite wave vector $k$, followed by the gradual movement of that peak to smaller $k$ as the spin domains expand. Each vertical slice represents a single experimental realization, i.e., no averaging. The color scale depicts increasing spectral power with darker color. (b) Wave-vector of peak of ${\mathrm{PSD}}_z\left(k\right)$. (c) Width of ${\mathrm{PSD}}_x\left(k\right)$, which always peaked around zero. In panels (b) and (c), the red symbols depict the experimentally observed peak location (typical uncertainty plotted on the leftmost point) and the blue curve plots the results of our SP-GPE simulation (uncertainties denoted by the blue band). In these three cases, the uncertainties reflect the standard deviation over eleven realizations, i.e., panels (b) and (c) show averaged data. The gray symbols mark $1/{\xi }_{1\mathrm{D}}$, the homogenous-system wave vector of maximum gain (the uncertainties are comparable to the symbol size).

Figure 5

Ratio of peak in ${\mathrm{PSD}}_z\left(k\right)$ over $1/{\xi }_{1\mathrm{D}}$ plotted on a linear time scale. We use only the seven data runs that include $t_{\mathrm{hold}}>6sec$. Unlike the rest of the paper, the uncertainties given here represent the standard deviation of the mean. The red symbols depict the experimentally observed peak location and the blue curve shows the results of our SP-GPE simulation (uncertainties denoted by the blue band).