Spin Echo in Spinor Dipolar Bose-Einstein Condensates

We theoretically propose and numerically realize spin echo in a spinor Bose-Einstein condensate (BEC). We investigate the influence on the spin echo of phase separation of the condensate. The equation of motion of the spin density exhibits two relaxation times. We use two methods to separate the relaxation times and hence demonstrate a technique to reveal magnetic dipole-dipole interactions in spinor BECs.

Figure 1

Time development of the spin density vectors $\mathbf{S}$ projected on the $x\mathrm{\text{−}}y$ plane in the spin echo with ${\omega }_{\perp }\tau =5$. The magnitude of $\mathbf{S}$ depends on length of the arrow.

Figure 2

Time development of $⟨S_y⟩/\hslash$ (black line) and $⟨S_z⟩/\hslash$ (white circles). The symbols (a)–(h) indicate the times shown in Fig. 1. The gray zones represent the intervals of the pulses. The behavior of $⟨{\widehat{S}}_z⟩/\hslash$ shows that the $T_1$ relaxation is irrelevant to the dynamics $⟨{\widehat{S}}_y⟩/\hslash$.

Figure 3

Time ${\omega }_{\perp }{t}_{\mathrm{peak}}$ at which echoes appear versus the interval ${\omega }_{\perp }\tau$ between $\pi /2$ and $\pi$ pulses. The circles and triangles are the results for $|c_{dd}/c_2|=0$ and 0.02, respectively.

Figure 4

Density profiles $|{\tilde{\psi }}_{\alpha }(y=0){|}^2=a_h^2|{\psi }_{\alpha }(y=0){|}^2/N$ with $a_h=\sqrt{\hslash /M{\omega }_{\perp }}$ and $\tilde{x}=x/a_h$ without the MDDI just before the $\pi$ pulse [(a), (b), (c)] and just after the $\pi$ pulse [(d), (e), (f)] for ${\omega }_{\perp }\tau =5$, 7.5, and 12.5.

Figure 5

(a) Results of method $A$, showing multiple exposures of $⟨{\widehat{S}}_y⟩/\hslash$ with (circles) and without (triangles) the MDDI at $t_{\mathrm{peak}}$ for various $\tau$, where ${\omega }_{\perp }\tau =3.5,4.0,\dots ,7.5,10,12.5,15,17.5$. The black line shows the Carr-Purcell form $\exp [-bt-c^3{t}^3]$ with $b/c=0.072$. (b) Results of method $B$, showing a single exposure of the signal of peaks with ${\omega }_{\perp }\tau =3.5$ and $n=17$. The line $\exp (-bt)$ is fit to the early peaks, which are little affected by $\exp (-c^3{t}^3/{17}^2)$. We obtain ${\omega }_{\perp }{T}_{dd}\sim 500$.