Semiclassical quantization for a bosonic atom-molecule conversion system

We consider a simple quantum model of atom-molecule conversion where bosonic atoms can combine into diatomic molecules and vice versa. The many-particle system can be expressed in terms of the generators of a deformed $\text{su}\left(2\right)$ algebra, and the mean-field dynamics takes place on a deformed version of the Bloch sphere, a teardrop shaped surface with a cusp singularity. We analyze the mean-field and many-particle correspondence, which shows typical features of quantum-classical correspondence. We demonstrate that semiclassical methods can be employed to recover full many-particle features from the mean-field description in cold atom systems with atom-molecule conversion, and we derive an analytic expression for the many-particle density of states in the limit of large particle numbers.

Figure 1

Eigenvalues of ${\widehat{K}}_x$ (and ${\widehat{K}}_y$) for $N=50$.

Figure 2

Many-particle spectrum in dependence on $\epsilon$ for $v=1$ and $N=10$ (top) and $N=50$ (bottom) particles.

Figure 3

Mean-field dynamics on the deformed Bloch sphere for $v=1$ and $\epsilon =0,1,2$ (from top to bottom). The right panel shows the dynamics of $s_z$ for selected initial conditions.

Figure 4

Many-particle [blue (dark gray)] and mean-field [magenta (light gray)] energies in dependence on $\epsilon$ for $v=1$ and $N=30$ particles.

Figure 5

Mean-field (black) and many-particle dynamics for $v=1$ and $\epsilon =1$ (top) and $\epsilon =0$ (bottom), and different particle numbers. The blue (dark gray), green (gray), and yellow (light gray) curves correspond to $N=20,100,$ and 500, respectively. The initial states are chosen as the ground states of $\pm {\widehat{K}}_x$ in the top figure and $-{\widehat{K}}_z$ in the bottom figure, and the initial conditions are marked by a star. The right figure on the top shows the dynamics of the $x$ component for $\epsilon =1$, and the figure on the bottom right shows the dynamics of the $z$ component for $\epsilon =0$.

Figure 6

Expectation values of ${\widehat{K}}_x$ and ${\widehat{K}}_z$ for the ground states of the Hamiltonian (32) where $c=0,b$ varies from $-0.5$ to $0.5$ and $a=\pm \frac{1}{2}\sqrt{(1-2b){(1+2b)}^2}$, for different particle numbers $N=2$ (blue dash-dotted line), $N=4$ (red dashed line), $N=10$ (magenta dotted line), and $N=100$ (black solid line).

Figure 7

Mean-field phase-space portraits (top) and potential curves (bottom) for $v=1$ and $\epsilon =0$ (left) and $\epsilon =2$ (right). The false colors in the phase-space plots represent the energy values. An orbit belonging to one selected energy each (marked by a black line in the bottom pictures) is highlighted in each of the phase-space plots by a thick black line.

Figure 8

Many-particle (solid blue lines) and semiclassical (dashed magenta lines) eigenvalues in dependence on $\epsilon$ for $v=1$ and two different particle numbers. The upper panel corresponds to $N=4$ particles; the lower panel corresponds to $N=20$.

Figure 9

Density of states $\frac{dn}{dE}$ for the many-particle system in comparison with the mean-field periods for $v=1$ and different values of $\epsilon$. The many-particle density of states is approximated by the normalized histogram of the energies, for $N=10000$ particles. The mean-field result is depicted by the solid magenta line. The values of $\epsilon$ are $\epsilon =0,1,2,5$ from top left to bottom right.

Figure 10

Exact $n\mathrm{th}$ many-particle eigenvectors (green circles) with the WKB wave function (red) for $N=40,\epsilon =0.5$, and $v=1$ for $n=1$ (top), $n=3$ (middle), and $n=10$ (bottom). The red crosses indicate the semiclassical wave function (52). The solid red line, indicating the continuous semiclassical wave function according to Eq. (51), is added to guide the eye.