Diagnostics for the ground-state phase of a spin-2 Bose-Einstein condensate

We propose a method of using spin exchange dynamics to determine the coefficient of the spin-singlet pair term of a spin-2 Bose-Einstein condensate (BEC) by preparing the initial populations in the magnetic sublevels 0 and $\pm 2$ with appropriate relative phases. This method is suitable for a BEC with a short lifetime, since only the initial change in the population of each magnetic sublevel is needed. Our method therefore enables the determination of the ground-state phase of a spin-2 ${}^{87}\mathrm{Rb}$ BEC at zero magnetic field, which is considered to lie in the immediate vicinity of the boundary between the antiferromagnetic and cyclic phases. We propose a scheme to prepare the initial state in which the relative phases between the magnetic sublevels are controlled. We also propose a simple scheme to distinguish between the antiferromagnetic and cyclic phases under a situation of very low bias magnetic field.

Figure 1

The ground-state phase diagrams for $p=0$ and $q<0$ (a) with respect to ${\tilde{c}}_1$ and ${\tilde{c}}_2$ and (b) with respect to $\mid q\mid$ and ${\tilde{c}}_1$. The symbols F, AF, C, and M indicate the states ${\mathbf{\zeta }}_{\mathrm{F}}$, ${\mathbf{\zeta }}_{\mathrm{AF}}$, ${\mathbf{\zeta }}_{\mathrm{C}}$, and ${\mathbf{\zeta }}_{\mathrm{M}}$, respectively. In (b), ${\tilde{c}}_1>0$ is assumed.

Figure 2

Time evolution of the relative population of each spin component for the density of atoms of $2.1\times {10}^{14}{\mathrm{cm}}^{-3}$ [11] and a magnetic field of $250\mathrm{mG}$. The scattering lengths are assumed to be $(c_1,c_2)M∕\left(4\pi {\hslash }^2\right)=(a_{\mathrm{B}},a_{\mathrm{B}})$ in (a) and $(a_{\mathrm{B}},-a_{\mathrm{B}})$ in (b), where $a_{\mathrm{B}}$ is the Bohr radius. The initial state is chosen to be ${\zeta }_0=\sqrt{0.3}$, ${\zeta }_2=-{\zeta }_{-2}=\sqrt{0.349}$, and ${\zeta }_{\pm 1}=\sqrt{0.001}$. The dashed curves are drawn according to Eq. (37).

Figure 3

Time evolution of ${\mid {\zeta }_0\mid }^2$ with the same parameters as in Figs. 2a, 2b. The initial state is given by ${\zeta }_0=\sqrt{0.3}$, ${\zeta }_{\pm 1}=0$, ${\zeta }_2=\sqrt{0.35}$, and ${\zeta }_{-2}=\exp (i\pi j∕4)\sqrt{0.35}$, where $j$ is an integer ranging from 0 to 4.

Figure 4

(a) Time evolution of the spin populations with a magnetic field of $100\mathrm{mG}$. The other parameters are the same as in Fig. 2a. The initial state is given by ${\zeta }_0=\sqrt{0.01}$, ${\zeta }_{\pm 1}=\sqrt{0.001}$, ${\zeta }_2=-{\zeta }_{-2}=\sqrt{0.494}$. The dashed curve is drawn according to Eq. (43). (b) Time evolution of ${\mid {\zeta }_0\mid }^2$. The initial phase of the $m=-2$ component is given as ${\zeta }_{-2}=\exp (i\pi j∕4)\sqrt{0.494}$, where $j$ is an integer ranging from 0 to 4 (bottom to top at $t\lesssim 100\mathrm{ms}$). (c) Time evolution of the spin populations for the scattering lengths $(c_1,c_2)M∕\left(4\pi {\hslash }^2\right)=(a_{\mathrm{B}},-a_{\mathrm{B}})$. The other conditions are the same as in (a).

Figure 5

Time evolution of the relative population $\int d\mathit{r}{\mid {\psi }_m\mid }^2∕N$ of each spin component $m$ in the trap with ${\omega }_{\perp }=237\mathrm{Hz}$ and ${\omega }_z=21\mathrm{Hz}$ [11]. The scattering lengths are assumed to be $(c_0,c_1,c_2)M∕\left(4\pi {\hslash }^2\right)=(100a_{\mathrm{B}},a_{\mathrm{B}},a_{\mathrm{B}})$ in (a) and $(100a_{\mathrm{B}},a_{\mathrm{B}},-a_{\mathrm{B}})$ in (b), where $a_{\mathrm{B}}$ is the Bohr radius. The strength of the magnetic field is $250\mathrm{mG}$, and the two-body loss coefficient is $K_2=1.4\times {10}^{-13}{\mathrm{cm}}^3{\mathrm{s}}^{-1}$ [11]. The $m=-2$ ground state ${\psi }_g$ is prepared, and is initially transferred to each spin component as ${\psi }_0=\sqrt{0.298}{\psi }_g$, ${\psi }_{\pm 1}=\sqrt{0.001}{\psi }_g$, and ${\psi }_2=-{\psi }_{-2}=\sqrt{0.35}{\psi }_g$. The insets show the column density distributions of the $m=-2,\dots ,2$ components from top to bottom at $t=100\mathrm{ms}$, each having a $70\times 8\text{−}\mu \mathrm{m}$ field of view.

Figure 6

(a) Schematic illustration of the energy levels of the $f=2$ and $f^{\prime }=2$ $D_1$ states. The circularly polarized light ${\sigma }_{\pm }$ changes $m$ by $\pm 1$ and hence starting from the $f=2$, $m=0$ state, the system undergoes transitions only to $f^{\prime }=2$, $m=\pm 1$ and $f=2$, $m=\pm 2$ states with the Clebsch-Gordan coefficients given by $\pm 1∕2$ and $\mp 1∕\sqrt{6}$, respectively. (b) The $f=2$ and $f^{\prime }=1$ $D_1$ transitions used for the polarization-dependent phase-contrast imaging. The ${\sigma }_{-}$ light makes transitions between $f=2$, $m=2$ and $f^{\prime }=1$, $m=1$, between $f=2$, $m=1$ and $f^{\prime }=1$, $m=0$, and between $f=2$, $m=0$ and $f^{\prime }=1$, $m=-1$ with the Clebsch-Gordan coefficients given by $1∕\sqrt{6}$, $1∕2$, and $1∕\sqrt{2}$, respectively.