Properties of a coupled two-species atom–heteronuclear-molecule condensate

We study the coherent association of a two-species atomic condensate into a condensate of heteronuclear diatomic molecules, using both a semiclassical treatment and a quantum mechanical approach. The differences and connections between the two approaches are examined. We show that, in this coupled nonlinear atom-molecule system, the population difference between the two atomic species plays a significant role in the ground-state stability properties as well as in coherent population oscillation dynamics.

Figure 1

(Color online) Ground-state molecular population $y_0$ and energy gap $\Delta E$ as functions of $\Delta$. For (a) and (b), $d=0$, the thin black lines represent the semiclassical results, dashed green lines are quantum results for $N=10$, and thick red lines are quantum result for $N=100$. For (c) and (d), $d=0.2$ and only the semiclassical results are shown.

Figure 2

(Color online) Left panel shows an example of dynamical instability with $d=0$ and $\Lambda =-5$. (a) Molecular population in the low-lying semiclassical steady states as functions of $\Delta$; the state represented by the red dashed curve is dynamically unstable. (b) Corresponding dimensionless semiclassical mean-field energies as calculated using Eq. (5). (c) Molecular population as $\Delta$ is linearly swept. (d) The corresponding quantum many-body energy spectrum for $N=20$, only the lowest five eigenenergies are shown. As the classical Hamiltonian (5) represents the energy per pair of atoms, the quantum eigenenergy (in units of $G$) has been rescaled by a factor of ${(N∕2)}^{-1}$. The right panel is the same as the left except for $\Lambda =5$ where there is no dynamical instability.

Figure 3

Stability phase diagram in $\Lambda \text{−}\Delta$ parameter space. The black regions are dynamically unstable. (a) $d=0$ and (b) $d=0.2$.

Figure 4

(a) and (b) Molecular population oscillation amplitude and period, respectively. The three curves correspond to $d=0$, 0.2, and 0.5 in descending order. (c) Molecular population dynamics for $\Lambda =\Delta =d=0$. Solid line: semiclassical result. Dashed and dotted lines: quantum result for $N=100$ and $N=1000$, respectively. (d) On resonance semiclassical oscillation period as a function of $d$.