Spin dynamics and structure formation in a spin-1 condensate in a magnetic field

We study the dynamics of a trapped spin-1 condensate in a magnetic field. First, we analyze the homogeneous system, for which the dynamics can be understood in terms of orbits in phase space. We analytically solve for the dynamical evolution of the populations of the various Zeeman components of the homogeneous system. This result is then applied via a local-density approximation to trapped quasi-one-dimensional condensates. Our analysis of the trapped system in a magnetic field shows that both the mean-field and Zeeman regimes are simultaneously realized, and we argue that the border between these two regions is where spin domains and phase defects are generated. We propose a method to experimentally tune the position of this border.

Figure 1

(Color online) Energy (in units of $\left|c_2\right|n$) of the homogeneous system for the cases (a) $\mathcal{M}=0$, $B=0$ and (b) $\mathcal{M}=0.3$, $B=1\text{mG}$, as given by Eq. (4) for a spin-1 condensate of ${}^{87}\text{R}\text{b}$.

Figure 2

(Color online) Contour plot of the energy surface corresponding to $\mathcal{M}=0.3$ and $B=100\text{mG}$. The white line shows the orbit corresponding to the initial conditions ${\lambda }_0^{\text{in}}=1/2$, ${\theta }_{\text{in}}=\pi /2$ (indicated by the white dot). The minimum of $\mathcal{E}$ is at ${\lambda }_0\approx 0.455$, $\theta =0$. Note the presence of open orbits for energies above that of the indicated white line.

Figure 3

(Color online) Evolution of the population of the $m=0$ Zeeman component, ${\lambda }_0\left(t\right)$, for the cases (a) $\mathcal{M}=0$, $B=0$ and (b) $\mathcal{M}=0.3$, $B=100\text{mG}$, starting in both instances from ${\lambda }_0^{\text{in}}=0.5$, ${\theta }_{\text{in}}=\pi /2$. In both panels, the solid line corresponds to the analytic result, (a) Eq. (16) or (b) (12), while the circles are a numerical integration of the differential equation for $\dot{{\lambda }_0}$. The dashed line gives the expected average value of ${\lambda }_0$, ${\lambda }_0^{\text{av}}\approx 0.433$. The arrows indicate the amplitude and period as predicted by the analytical results. In the bottom plot, also the value of the equilibrium population ${\lambda }_0^{\text{eq}}\approx 0.455$ is indicated by a dotted line, while the dashed-dotted line stands for a fit to Eq. (21) (displaced vertically by 0.1 for clarity).

Figure 4

(Color online) Phase-space plot $\left({\lambda }_0,\theta \right)$ corresponding to (a) the evolution shown in Fig. 3a and (b) Fig. 3b (compare with Fig. 2). The solid line is the analytic result in (a) Eq. (16) and (b) Eq. (12), while the circles are the solution of the differential equations for $\dot{{\lambda }_0}$ and $\dot{\theta }$. The vertical dashed lines stand for the average value ${\lambda }_0^{\text{av}}$ in each case.

Figure 5

(Color online) Time evolution of the integrated $|m=0⟩$ population (normalized to the total population, $N=20000$) for ${\lambda }_0^{\text{in}}=0.5$, ${\theta }_{\text{in}}=\pi /2$, $\mathcal{M}=0.3$, and $B=100\text{mG}$. The solid line shows the LDA result, Eq. (20), with ${\lambda }_0\left(x,t\right)={\lambda }_0^{n\left(x\right)}\left(t\right)$. The circles stand for the analytic estimate of Eq. (22), and the dashed line is the result of integrating the set of coupled Gross-Pitaevskii Eqs. (24).