Fluctuations of spinor Bose-Einstein condensates

We develop theory for fluctuations in atom number and spin within finite-sized cells of a spinor Bose-Einstein condensate. This theory provides a model of measurements that can be performed in current experiments using finite resolution in situ imaging. We develop analytic results for quantum and thermodynamic limits of the fluctuations and apply our theory to the four equilibrium phases of a spin-1 condensate. We then validate these limits and examine the behavior over a wide parameter regime using numerical calculations specialized to the case of a spinor condensate confined to be quasi-two-dimensional.

Figure 1

Schematic illustration of fluctuation measurement for a quasi-2D spinor condensate. Within a small cell of radius $R$ [indicated by a Gaussian weight function $\sigma$ (green)] an operator of interest is measured. For example, the total number of atoms or the components of spin.

Figure 2

Examples of $T=0$ structure factors showing the comparison to their low-$k$ expansion (20), along with characteristic length scale ${\xi }_c$. Subplots are (a) $j=1$ case of the density structure factor relevant to Sec. 5c. (b) $j=2$ case of the $x$-spin density structure factor relevant to Sec. 5c. (c) $j=1$ case with $S_w\left(0\right)\ne 0$ for the $x$-spin density structure factor relevant to Sec. 5d. For the divergent treatment presented in Eq. (22), (d) a $j=-1$ case for the $y$-spin density structure factor relevant to Sec. 5d. Parameters for the results are those used in Secs. 5c and 5d.

Figure 3

Phases of a uniform spin-1 condensate. The normalized spinor, $\xi ={[{\xi }_1,{\xi }_0,{\xi }_{-1}]}^T$, magnetization characteristics, and spherical-harmonic representation of the four distinct magnetic phases. The spherical-harmonic representation figures are surface plots of $|\Psi \left(\widehat{s}\right){|}^2$, with the surface color indicating $\arg \left[\Psi \left(\widehat{s}\right)\right]$, and the arrow indicates the direction of magnetization (if any). For the BA phase we have restricted our attention to the $p=0$ case where the magnetization is in plane.

Figure 4

Cell fluctuations for the F phase. In (a) and (c) we plot temperature-dependent results for cell sizes of $R\in \{0.2,2,20\}\mu \mathrm{m}$, as labeled in (a). In (b) and (d) we plot fluctuations as a function of cell size $R$ for temperatures $T\in \{0,5,50\}$ nK, as labeled in (b). Also plotted are the QSN limit (dashed blue line), QLC limit (dashed black line), and the TL (dot-dashed red line). Parameters are for ${}^{23}\mathrm{Na}$ as in Table 2, and we also set $f_z=n,p=c_{1}n$, and $q=-c_{1}n$. For ${\widehat{N}}_{\sigma }$ and ${\widehat{F}}_{z,\sigma },{\xi }_c=1.2$ $\mu \mathrm{m}$ [vertical green line in (b)]. For ${\widehat{F}}_{x,\sigma }$ and ${\widehat{F}}_{y,\sigma },l_{\mathrm{F}}$ is shown [vertical green line in (d)] and ${\xi }_c=0$.

Figure 5

Cell fluctuations for the P phase. In (a) and (c) we plot temperature-dependent results for cell sizes of $R\in \{0.2,2,20\}\mu \mathrm{m}$, as labeled in (a). In (b) and (d) we plot fluctuations as a function of cell size $R$ for temperatures $T\in \{0,5,50\}$ nK, as labeled in (b). Also plotted are the QSN limit (dashed blue line), QLC limit (dashed black line), and the TL (dot-dashed red line). Parameters are for ${}^{23}\mathrm{Na}$ as in Table 2, with $f_z=0,p=1.5c_{1}n$, and $q=2.1c_{1}n$. For ${\widehat{N}}_{\sigma },{\xi }_c=1.2$ $\mu \mathrm{m}$ and for ${\widehat{F}}_{x,\sigma }$ and ${\widehat{F}}_{y,\sigma },{\xi }_c=6.6$ $\mu \mathrm{m}$ and $l_{\mathrm{P}}^{\max }$ are shown [vertical green lines in (b) and (d)].

Figure 6

Cell fluctuations for the AF phase as a function of temperature, for cell sizes $R\in \{0.2,2,20\}\mu \mathrm{m}$ as labeled in (c). Also plotted are the QSN limit (dashed blue line), QLC limit (dashed black line), and TL (dot-dashed red line). Parameters are for ${}^{23}\mathrm{Na}$ as in Table 2, with $f_z=0.2n$ and $q=-c_{1}n$.

Figure 7

Cell fluctuations for the AF phase as a function of cell size $R$, for temperatures $T\in \{0,5,50\}$ nK as labeled in (a). Parameters, limits, and line styles are as in Fig. 6. For ${\widehat{N}}_{\sigma },{\xi }_c=1.2$ $\mu \mathrm{m}$; for ${\widehat{F}}_{z,\sigma },{\xi }_c=8.6$ $\mu \mathrm{m}$; for ${\widehat{F}}_{x,\sigma },{\xi }_c=8.3$ $\mu \mathrm{m}$; for ${\widehat{F}}_{y,\sigma },{\xi }_c=11$ $\mu \mathrm{m}$; are shown along with $l_{\mathrm{AF}}$ for ${\widehat{F}}_{x,\sigma }$ and ${\widehat{F}}_{y,\sigma }$ (vertical green lines).

Figure 8

Cell fluctuations for the BA phase as a function of temperature for cell sizes $R\in \{0.2,2,20\}\mu \mathrm{m}$ as labeled in (c). Also plotted are the QSN limit (dashed blue line), QLC limit (dashed black line), and TL (dot-dashed red line). Parameters are for ${}^{87}\mathrm{Rb}$ as in Table 2, with $f_z=0,p=0$, and $q=|c_1|n$. To calculate ${\widehat{F}}_{y,\sigma }$ fluctuations in (d) we use $L_{\mathrm{sys}}=70\mu \mathrm{m}$ (see text).

Figure 9

Cell fluctuations for the BA phase as a function of cell size $R$, for temperatures $T\in \{0,5,50\}$ nK as labeled in (a). Parameters, limits, and line styles are as in Fig. 8. For ${\widehat{N}}_{\sigma },{\xi }_c=0.45$ $\mu \mathrm{m}$; for ${\widehat{F}}_{z,\sigma },{\xi }_c=10$ $\mu \mathrm{m}$; for ${\widehat{F}}_{x,\sigma },{\xi }_c=0.84$ $\mu \mathrm{m}$; and for ${\widehat{F}}_{y,\sigma },{\xi }_c=10$ $\mu \mathrm{m}$ (vertical green lines).