Static structure factors for a spin-1 Bose-Einstein condensate

We consider the total density and spin density fluctuations of a uniform spin-1 Bose-Einstein condensate within the Bogoliubov formalism. We present results for the total density and spin density static structure factors for all four magnetic phases. A key result of our work is a set of analytic predictions for the structure factors in the large- and small-momentum limits. These results will be useful in current experiments aiming to develop a better understanding of the excitations and fluctuations of spinor condensates.

Figure 1

Zero temperature phase diagram of a spin-1 Bose gas for cases with (a) $c_1>0$, and (b) $c_1<0$. Vertical and horizontal axes are the linear and quadratic Zeeman energies (see text) in units of $|c_1|n$, where $n$ is the condensate number density. The phases shown are defined in Table I (see Refs. [3, 12]). The rotational symmetry about the direction of the applied field is spontaneously broken in the AF and BA phases.

Figure 2

Bogoliubov dispersion relations in the F phase. (a) and (b) focus on different ranges of $k$ values. We show the phonon (solid black line), magnon [dashed (blue) line], and transverse magnon [dash-dotted (red) line] branches of the excitation spectra and attribute these the indices $\nu =0$, 1, and 2, respectively. Parameters: $p=1.5c_{1}n$, $q=-c_{1}n$, $c_0=50c_1$, $c_1>0$.

Figure 3

Fluctuation amplitudes for the F phase. (a) $\delta {\tilde{n}}_{\mathbf{k}\nu }$, $\delta {\tilde{f}}_{z,\mathbf{k}\nu }$ and (b) $\delta {\tilde{f}}_{x,\mathbf{k}\nu }$, $\delta {\tilde{f}}_{y,\mathbf{k}\nu }$, as defined in Eq. (20). Modes (index $\nu$) have the same line types as in Fig. 2. Other parameters as in Fig. 2.

Figure 4

Static structure factors for the F phase at various temperatures. Structure factors (a) $S_n$ and $S_z$ (which are identical) and (b) $S_x$ and $S_y$ (which are also identical), as defined in Eq. (21). For temperatures of (from bottom to top curves) $T=\{0,c_1,2c_1,c_0+c_1\}\times n/k_B$, as labeled in the inset in (b). Insets: Additional details for the lower temperature results at small $k{\xi }_s$. Other parameters as in Fig. 2.

Figure 5

Bogoliubov dispersion relations in the P phase. (a) and (b) focus on different ranges of $k$ values. We show the branches of the excitation spectra for the phonon mode (solid black line) and the two magnon modes [dashed (blue) line and dash-dotted (red) line]. Parameters: $q=2.1c_{1}n$, $c_0=50c_1$ and $p=1.5c_{1}n$, $c_1>0$.

Figure 6

Fluctuation amplitudes for the P phase. (a) $\delta {\tilde{n}}_{\mathbf{k}\nu }$ and (b) $\delta {\tilde{f}}_{x,\mathbf{k}\nu }$, $\delta {\tilde{f}}_{y,\mathbf{k}\nu }$, as defined in Eq. (20). Note that $\delta {\tilde{f}}_{z,\mathbf{k}\nu }=0$. Modes (index $\nu$) have the same line types as in Fig. 5. Other parameters as in Fig. 5.

Figure 7

Static structure factors for the P phase at various temperatures. Structure factors (a) $S_n$ and (b) $S_x$, $S_y$, as defined in Eq. (21). Note $S_z=0$. For temperatures of (from bottom to top curves) $T=\{0,c_1,2c_1,c_0\}\times n/k_B$, as labeled in the inset in (b). Insets: Additional details for small $k{\xi }_s$. Other parameters as in Fig. 5.

Figure 8

Bogoliubov dispersion relations in the AF phase. (a) and (b) focus on different ranges of $k$ values. We show the phonon (solid black line), axial magnon [dashed (blue) line], and transverse magnon [dash-dotted (red) line] branches of the excitation spectra and attribute these the indices $\nu =0$, 1, and 2, respectively. Parameters: $q=-c_{1}n$, $c_0=50c_1$, and $p=0.2c_{1}n$, giving $f_z=0.2n$, ${\alpha }_z\approx 0.98$.

Figure 9

Fluctuation amplitudes for the AF phase. (a) $\delta {\tilde{n}}_{\mathbf{k}\nu }$, (b) $\delta {\tilde{f}}_{z,\mathbf{k}\nu }$, (c) $\delta {\tilde{f}}_{x,\mathbf{k}\nu }$, and (d) $\delta {\tilde{f}}_{y,\mathbf{k}\nu }$, as defined in Eq. (20). Modes (index $\nu$) have the same line types as in Fig. 8. Other parameters as in Fig. 8.

Figure 10

Static structure factors for the AF phase at various temperatures. Structure factors (a) $S_n$, (b) $S_z$, (c) $S_x$, and (d) $S_y$, as defined in Eq. (21). For temperatures of (from bottom to top curves) $T=\{0,c_1,2c_1,c_0\}\times n/k_B$, as labeled in the inset in (b). Insets: Lower temperature results at small $k{\xi }_s$. Other parameters as in Fig. 8.

Figure 11

Bogoliubov dispersion relations in the BA phase. (a) and (b) focus on different ranges of $k$ values. We show the branches of the excitation spectra for the phonon mode (solid black line) and the axial [dashed (blue) line; note that the magnetization axis is $x$] and transverse [dash-dotted (red) line] magnon modes. Inset in (a): An avoided crossing between the phonon and the transverse magnon modes. Parameters: $q=0.5|c_1|n$, $c_0=-250c_1$, $p=0$, and $c_1<0$.

Figure 12

Fluctuation amplitudes for the BA phase. (a) $\delta {\tilde{n}}_{\mathbf{k}\nu }$, (b) $\delta {\tilde{f}}_{z,\mathbf{k}\nu }$, (c) $\delta {\tilde{f}}_{x,\mathbf{k}\nu }$, and (d) $\delta {\tilde{f}}_{y,\mathbf{k}\nu }$, as defined in Eq. (20). Modes (index $\nu$) have the same line types as in Fig. 11. Insets in (a) and (c): Additional details for $k{\xi }_s$ close to the avoided crossing. Other parameters as in Fig. 11.

Figure 13

Static structure factors for the BA phase at various temperatures. Structure factors (a) $S_n$, (b) $S_z$, (c) $S_x$, and (d) $S_y$, as defined in Eq. (21). For temperatures of (from bottom to top curves) $T=\{0,|c_1|,2|c_1|,c_0+c_1\}\times n/k_B$, as labeled in the inset in (b). Insets: Additional details for small $k{\xi }_s$. Other parameters as in Fig. 11.