Spin-1 microcondensate in a magnetic field

We study a spin-1 Bose-Einstein condensate small enough to be treated as a single magnetic domain: a system that we term a microcondensate. Because all particles occupy a single spatial mode, this quantum many-body system has a well-defined classical limit consisting of three degrees of freedom, corresponding to the three macroscopically occupied spin states. We study both the classical limit and its quantization, finding an integrable system in both cases. Depending on the sign of the ratio of the spin interaction energy and the quadratic Zeeman energy, the classical limit displays either a separatrix in phase space or Hamiltonian monodromy corresponding to nontrivial phase space topology. We discuss the quantum signatures of these classical phenomena using semiclassical quantization as well as an exact solution using the Bethe ansatz.

Figure 1

Example of the geometric construction for $\mathrm{q̃}=0.5$ giving the trajectory (dark line) as the intersection of a hyperboloid, the reduced phase space of fixed $s_z=0.2$, and the surface of fixed energy $h=0.7$.

Figure 2

For positive $\mathrm{q̃}$ (here $\mathrm{q̃}=0.5$) the asymptotic planes $k_0=1/2$, $k_0=k_x+\mathrm{q̃}/2$ (black dotted line) of the hyperbolic sheets of constant energy cut the hyperboloid of fixed $s_z=0.2$ (blue dashed line), forming a separatrix between energies greater (red line to the left) or less (green line to the right) than $h=(1/2)s_z^2+\mathrm{q̃}$.

Figure 3

Spectrum of Eq. (5) for $N=30$ particles with $\mathrm{q̃}=0.3$ (units with $e_2=1$). The solid lines correspond to the boundaries given by $[s_z(k_x,k_0),h(k_x,k_0)]$ evaluated on $k_{x,0}(\lambda )$ from Eq. (22) for the ranges in Eq. (23). The dashed line corresponds to the separatrix $h=(1/2)s_z^2+\mathrm{q̃}$.

Figure 4

For negative $\mathrm{q̃}$ (here $\mathrm{q̃}=-0.5$), there are two distinct classes of trajectories that either do (green line to the right, $h<0$) or do not (red line to the left, $h>0$) encircle the apex of the cone corresponding to $s_z=0$.

Figure 5

Spectrum of Eq. (5) for $N=30$ particles with $\mathrm{q̃}=-0.5$ (units with $e_2=1$). The solid (upper boundary) is found as in the $\mathrm{q̃}>0$ case, and the lower boundary (dashed) corresponds to trajectories on the asymptotic plane $k_0=1/2$ of energy $h=(1/2)s_z^2+\mathrm{q̃}$. The feature at the origin is a signature of the phenomenon of monodromy.

Figure 6

Spectrum of Eq. (5) for $N=30$ particles with $\mathrm{q̃}=-2.2$ (units with $e_2=1$). For $\mathrm{q̃}<-2$ the origin passes through the upper boundary, which becomes cusped.

Figure 7

The ground-state phase diagram for $e_2<0$. The vectors denote the spinor $(a_1,a_0,a_{-1})$. The shaded region corresponds to the dark upper boundary in Figs. 5 and 6.

Figure 8

The ground-state phase diagram for $e_2>0$. The shaded region is where the separatrix (dashed line in Fig. 3) is the lowest energy.

Figure 9

Trajectory on the reduced phase space in the half-plane model. The trajectory is first projected to the hemisphere of radius $s_z/2$ from the point $(0,0,-s_z/2)$, and from there it is projected to the plane $k_x=-s_z/2$ from the point $(s_z/2,0,0).$

Figure 10

Level sets of the actions $I_3/N$ (red, cusped below $h=0$) and $I_3^{\prime }=I_3/N+\left|s_z\right|/2$ (blue, cusped above $h=0$) for $\mathrm{q̃}=-0.3$.

Figure 11

Branch cuts (red solid segments) of the integrand and integration contour (dashed) in Eq. (42) for $\mathrm{q̃}<0$, $h<0$, and $s_z\to 0$; $z_{<}$ and the unmarked root are both $O(s_z)$.

Figure 12

Level sets of $h$ on the half-plane for $s_z=0.45$, $\mathrm{q̃}=0.3$. The dark line is the separatrix corresponding to $h=(1/2)s_z^2+\mathrm{q̃}$.

Figure 13

(left) As we circle the origin in $S_z$, $H$ space for $\mathrm{q̃}<0$ the period lattice is deformed continuously, returning to its original form, but after shifting the lattice vector corresponding to $I_3$ by $2\pi$ in the $\varphi$ direction. (right) Schematic illustration of the rotation angle. While executing a single period $T$ of motion on the reduced phase space, the system rotates by an angle $\Phi$.

Figure 14

After one period of motion on the reduced phase space the system returns close to the pinch but rotates by the rotation angle $\Phi$. The angle between $a_{+}$ and $a_{-}$ remains constant. We can circle the origin $h=s_z=0$ by winding $a_{-}(0)$ as shown, keeping $a_{+}(0)$ fixed. By the argument in the text, $a_{+}(T)$ must wind in the opposite direction, showing that the rotation angle changes by $2\pi$.

Figure 15

Evolution of transverse magnetization for $\mathrm{q̃}=-0.3$. (left) Initial conditions $a_1=0.4$, $a_{-1}=0.3.$ (right) Initial conditions $a_1=0.4$, $a_{-1}=-0.3$, rotation angle approaching $+\pi$.

Figure 16

Spectrum of Eq. (5) with $N=20$ particles and $\mathrm{q̃}=-0.3$ (units with $e_2=1$). The dark lines are the integral contours of the action $I_3$. Transporting an elementary cell of the lattice around the origin leads to nontrivial monodromy.