Different growth rates for spin and superfluid order in a quenched spinor condensate

In this paper we study the coarsening dynamics of a spinor condensate quenched into an easy-axis ferromagnetic phase by a sudden change in the quadratic Zeeman energy. We show that applying a spin rotation prior to changing the Zeeman energy accelerates the development of local order and reduces heating. We examine the longitudinal spin ordering and the superfluid ordering of the system and show that the respective order parameter correlation functions exhibit dynamic scaling in the late-time dynamics. Our results also demonstrate that these two types of order grow at different rates, i.e., with different dynamic critical exponents. The spin domain area distribution is calculated and is shown to have power-law scaling behavior expected from percolation theory.

Figure 1

Magnetic phase diagram for a ferromagnetic spin-1 condensate with zero $z$ magnetization. The spheres show the direction of magnetization in the three states. (a) For $q<0$ the magnetization lies along the $F_z$ axis and the state is termed the easy-axis phase. (b) For $0<q<q_0$ the magnetization lies in the transverse $F_x-F_y$ plane and the state is termed the easy-plane phase. Quenches to this phase are not analyzed here but can be found in Refs. [18, 19, 38]. (c) For $q>q_0$ the $m=\pm 1$ levels are unoccupied and the system is unmagnetized. This state is termed the polar phase. The directed line on the phase diagram indicates the quench we consider in this paper, where the Zeeman energy is suddenly reduced from an initial value $q_i>q_0$, where the ground state is polar, to a value $q<0$, where the system favors easy-axis spin ordering.

Figure 2

Bogoliubov spectra for the postquench initial states emphasizing the dynamically unstable modes. Spectra of the ${\mathbf{\xi }}_{\mathrm{P}}$ initial state of the Q1 quench to (a) $q=-0.3q_0$ and (b) $q=-q_0$. Spectra of the ${\mathbf{\xi }}_{\mathrm{AF}}$ initial state of the Q2 quench to (c) $q=-0.3q_0$ and (d) $q=-q_0$. The solid lines indicate the real parts of the energies, while the dashed lines indicate the imaginary parts. The phonon (${\epsilon }_0$) and magnon (${\epsilon }_1,{\epsilon }_2$) excitation branches are labeled in the plots, noting that the magnon branches are degenerate in (a) and (b). These results are for the case $g_s=-g_n/3$.

Figure 3

Growth of local magnetization following the Q1 and Q2 quenches to (a) $q=-0.3q_0$ and (b) $q=-q_0$. In all cases the (local) magnetization is calculated as a spatial average over the system at each time, i.e., $\langle F_{\nu }^2\rangle =l^{-2}\int d^2\mathbf{x}{F}_{\nu }^2\left(\mathbf{x}\right)$ for $\nu =z$ (plain lines) and $\nu =\perp$ (lines with symbols). Other parameters are the same as in Fig. 2.

Figure 4

Longitudinal spin density at a time of $t=100t_s$ after the (a) Q1 and (b) Q2 quenches to $q=-0.3q_0$. The white line in each image indicates the spin correlation length $L_z$ defined in the text, with a value of (a) $5.5{\xi }_s$ and (b) $8.0{\xi }_s$.The spin density is only shown over a subregion of the simulation, with other parameters the same as in Fig. 2.

Figure 5

Correlation functions for the Q1 quench to $q=-0.3q_0$. The longitudinal spin correlation function is shown (a) at various times and (b) scaled by the length scale $L_z\left(t\right)$ to reveal correlation function collapse. The superfluid correlation function is shown (c) at various times and (d) scaled by $L_0\left(t\right)$. (e) Evolution of the length scales. Results with symbols out to $t=4\times {10}^3{t}_s$ are for simulations with $l=510{\xi }_s$ and $N=1024$ points, while the dashed lines out to $t={10}^4{t}_s$ are for simulations with $l=800{\xi }_s$ with $N=1024$ points.

Figure 6

Correlation functions for the Q2 quench to $q=-0.3q_0$. The longitudinal spin correlation function is shown (a) at various times and (b) scaled by the length scale $L_z\left(t\right)$ to reveal correlation function collapse. The superfluid correlation function is shown (c) at various times and (d) scaled by $L_0\left(t\right)$. (e) Evolution of the length scales. Results with symbols out to $t=4\times {10}^3{t}_s$ are for simulations with $l=510{\xi }_s$ and $N=1024$ points, while the dashed lines out to $t={10}^4{t}_s$ are for simulations with $l=800{\xi }_s$ with $N=1024$ points.

Figure 7

Domain area distributions for the (a) and (b) Q1 and (c) and (d) Q2 quenches to $q=-0.3q_0$. The domains area distribution is shown at early times as a function of (a) and (c) area and (b) and (d) scaled area $S/L_z^2$, where $L_z$ is the longitudinal spin correlation length evaluated as discussed in Sec. 3b. The insets in (b) and (d) use the same axis range as the main plots and indicate results at later times for comparison.