Effects of thermal and quantum fluctuations on the phase diagram of a spin-1 ${}^{87}$Rb Bose-Einstein condensate

We investigate the effects of thermal and quantum fluctuations on the phase diagram of a spin-1 ${}^{\text{87}}\text{Rb}$ Bose-Einstein condensate (BEC) under the quadratic Zeeman effect. Due to the large ratio of spin-independent to spin-dependent interactions of ${}^{\text{87}}\text{Rb}$ atoms, the effect of noncondensed atoms on the condensate is much more significant than that in scalar BECs. We find that the condensate and spontaneous magnetization emerge at different temperatures when the ground state is in the broken-axisymmetry phase. In this phase, a magnetized condensate induces spin coherence of noncondensed atoms in different magnetic sublevels, resulting in temperature-dependent magnetization of the noncondensate. We also examine the effect of quantum fluctuations on the order parameter at absolute zero and find that the ground-state phase diagram is significantly altered by quantum depletion.

Figure 1

$q$ dependence of longitudinal magnetization per particle $F_z^{\mathrm{c}}/n^{\mathrm{c}}$ and that of transverse magnetization $F_{\perp }^{\mathrm{c}}/n^{\mathrm{c}}$ in the mean-field ground state of a spin-1 ferromagnetic BEC ($c_1<0$). The dashed line and solid curve show the longitudinal and transverse magnetizations, respectively, where $F_{\perp }^{\mathrm{c}}\equiv \left|F_{+}^{\mathrm{c}}\right|={\left[{\left(F_x^{\mathrm{c}}\right)}^2+{\left(F_y^{\mathrm{c}}\right)}^2\right]}^{1/2}$. The labels F, BA, and P stand for ferromagnetic, broken-axisymmetry, and polar phases, which are shaded in light, medium, and dark gray, respectively. The longitudinal magnetization per particle is $F_z^{\mathrm{c}}/n^{\mathrm{c}}=\Theta (-q)$, and the transverse one is $|F_{+}^{\mathrm{c}}|/n^{\mathrm{c}}=\sqrt{1-(q/2|c_1{\left|n\right)}^2}\Theta \left(q\right)\Theta \left(2\right|c_1|n-q)$, where $\Theta \left(q\right)$ is the unit-step function.

Figure 2

Finite-temperature phase diagram of a spin-1 ferromagnetic Bose gas in the first-order self-consistent approximation. We used the interaction parameters for ${}^{\text{87}}\text{Rb}$ atoms, (i.e., $a_0=101.8a_{\mathrm{B}}$ and $a_2=100.4a_{\mathrm{B}}$ [33]), with the total number density (a) $n=1.0\times {10}^{12}{\mathrm{cm}}^{-3}$, and (b) $n=1.0\times {10}^{13}{\mathrm{cm}}^{-3}$. The quadratic Zeeman energy $q$ and temperature $T$ are measured in units of the spin-dependent interaction $|c_1|n$ and the transition temperature of a uniform ideal scalar BEC, $T_0=3.31{\hslash }^2{n}^{2/3}/\left(k_{\mathrm{B}}M\right)$, respectively. Crosses show temperature $T_{\mathrm{c}1}$ below which the condensate density $n^{\mathrm{c}}$ becomes nonzero. Open triangles show temperature $T_{\mathrm{c}2}$ below which the condensate acquires a nonzero transverse magnetization $F_{\perp }^{\mathrm{c}}$. The solid curves are to guide the eye.

Figure 3

Temperature dependence of the condensate fraction $n^{\mathrm{c}}/n$ (squares), transverse magnetizations per particle of the condensate $F_{\perp }^{\mathrm{c}}/n^{\mathrm{c}}$ (circles), and noncondensate $F_{\perp }^{\mathrm{nc}}/n^{\mathrm{nc}}$ (triangles) for $q=|c_1|n$ and $n=1.0\times {10}^{12}{\mathrm{cm}}^{-3}$. The BA, polar (P), and normal (N) phases are shaded in medium, dark, and light gray, respectively. The inset shows enlarged behaviors of these physical quantities near absolute zero. The longitudinal magnetizations of the condensate and noncondensate vanish for $q>0$; thus, for the BA phase their magnetizations are both perpendicular to the external magnetic field. The negative values of the transverse magnetization of the noncondensate at $0.01T_0\lesssim T\lesssim 0.3T_0$ imply that the magnetization vectors of the condensate and noncondensate are antiparallel to each other.

Figure 4

Energy spectra of excitation modes at (a) $T=0$ and (b) $T=0.2T_0$ and the contributions of these modes to the transverse magnetization $F_{\perp }^{\mathrm{nc}}$ of the noncondensate due to (c) thermal fluctuations $F_{\perp ,\nu ,\mathbf{k}}^{\mathrm{tf}}$ ($T=0.2T_0$) and (d) quantum depletion $F_{\perp ,\nu ,\mathbf{k}}^{\mathrm{qd}}$ ($T=0$) [see Eqs. (40) and (41)] for $q=|c_1|n$ and $n=1\times {10}^{12}{\mathrm{cm}}^{-3}$. The magnitude of wave vector $k=\left|\mathbf{k}\right|$ is measured in units of $k_0\equiv \sqrt{2M{k}_{\mathrm{B}}{T}_0}/\hslash$. There are, in total, three excitation modes labeled by $\nu =1,2$, and 3, which are shown by the dotted, solid, and dashed curves, respectively. The energy spectra of $\nu =1$ and $\nu =3$ modes almost coincide in the high-momentum region, as shown in (b). The negative values of $F_{\perp ,\nu =3,\mathbf{k}}^{\mathrm{tf}}$ in (c) imply that the transverse magnetization of this mode is antiparallel to that of the condensate. Note that $F_{\perp ,\nu =1,\mathbf{k}}^{\mathrm{tf}}$ in (c) and $F_{\perp ,\nu =1,3,\mathbf{k}}^{\mathrm{qd}}$ in (d) vanish.

Figure 5

Density dependence of the relative shift of the order parameter from the mean-field value: $\Delta \varphi /\varphi \equiv {\max }_{i,q}|{\varphi }_i^{\left(2\right)}-{\varphi }_i^{\left(0\right)}|/{\max }_{i,q}\left|{\varphi }_i^{\left(0\right)}\right|$ (squares). The dotted line $\Delta \varphi /\varphi =0.1$ gives an estimate of the threshold below which the perturbative expansion gives quantitatively reliable results.

Figure 6

$q$ dependence of longitudinal magnetization per condensate particle $F_z^{\mathrm{c}}/n^{\mathrm{c}}$ and that of transverse magnetization $F_{\perp }^{\mathrm{c}}/n^{\mathrm{c}}$ for a uniform ferromagnetic BEC at $T=0$. The parameters are those of ${}^{\text{87}}\text{Rb}$ with atomic density $n=1\times {10}^{10}{\mathrm{cm}}^{-3}$. The squares show the transverse magnetization numerically calculated by using the ${\chi }^{1/2}$ perturbative expansion, while the solid curve shows the mean-field result given by $F_{\perp }^{\mathrm{c}}/n^{\mathrm{c}}=\sqrt{1-(q/2|c_1{\left|n\right)}^2}\Theta \left(2\right|c_1|n-q)\Theta \left(q\right)$, where $\Theta \left(q\right)$ is the unit-step function. The longitudinal magnetization, which is given by $F_z^{\mathrm{c}}/n^{\mathrm{c}}=\Theta (-q)$ and shown by the dashed lines, is the same for the two approximations. The ferromagnetic (F), BA, and polar (P) phases are shaded in light, medium, and dark gray, respectively. The phase boundary between the BA and polar phases lies at $q=q_{\mathrm{b}}=2.05\left|c_1\right|n$. The point indicated by the single arrow shows the value of $F_{\perp }^{\mathrm{c}}/n^{\mathrm{c}}$ at $q/\left(\right|c_1\left|n\right)=2$, which is to be compared with the mean-field result: $F_{\perp }^{\mathrm{c}}/n^{\mathrm{c}}=0$ at $q/\left(\right|c_1\left|n\right)=2$. The deviation of this point from zero indicates how much the BA phase expands from the mean-field result. The point indicated by the double arrow shows the value of $F_{\perp }^{\mathrm{c}}/n^{\mathrm{c}}$ that lies in the critical parameter region $2|c_1|n<q<q_{\mathrm{b}}$.

Figure 7

Position $q_{\mathrm{b}}$ of the phase boundary between the BA and polar phases versus atomic density $n$. The dotted line shows the mean-field value $q_{\mathrm{b}}=2\left|c_1\right|n$. The deviation from the mean-field value is the effect of quantum depletion. The values of $q_{\mathrm{b}}$ for $n={10}^{11}$ and ${10}^{12}{\mathrm{cm}}^{-3}$, for which the perturbative expansion no longer gives quantitatively reliable results, are plotted only to show their rough estimates.