Matter-wave bistability in coupled atom-molecule quantum gases

We study the matter-wave bistability in coupled atom-molecule quantum gases, in which heteronuclear molecules are created via an interspecies Feshbach resonance involving either two-species Bose or two-species Fermi atoms at zero temperature. We show that the resonant two-channel Bose model is equivalent to the nondegenerate parametric down-conversion in quantum optics, while the corresponding Fermi model can be mapped to a quantum optics model that describes a single-mode laser field interacting with an ensemble of inhomogeneously broadened two-level atoms. Using these analogies and the fact that both models are subject to the Kerr nonlinearity due to the two-body $s$-wave collisions, we show that under proper conditions, the population in the molecular state in both models can be made to change with the Feshbach detuning in a bistable fashion.

Figure 1

(a) For given $\Lambda$ and $\nu$, the thick solid line represents $y_0({\nu }^{\text{'}})$ in Eq. (8) and the thin dashed straight line represents Eq. (7). The intersections are the stationary solutions. Here we take $\Lambda =-5$ and $\nu =-0.4\Lambda$. For this particular value of $\nu$, there are three stationary solutions. (b) Steady-state molecular population $y_0$ as a function of detuning $\nu$ for $\Lambda =-5$. The state represented by the red dashed line is dynamically unstable.

Figure 2

Mapping of the two-channel resonant Fermi superfluid model to the Dicke model. The bosonic molecules and the fermionic atoms in the former are mapped to the cavity laser field and an ensemble of two-level atoms in the latter, respectively. See text for details.

Figure 3

Free energy density $f$ as a function of ${\Delta }_e$ and $\left|c\right|{}^2$ at $\nu =0.2$ (a) and $\nu =0.02$ (b). Extremum points are indicated with “x” (minimum) and “$+$” (saddle point). $f$, ${\Delta }_e$, and $\nu$ are all in units of $E_F=(3{\pi }^{2}n){}^{2/3}/(2m)$, the Fermi energy of the noninteraction system. In all the examples shown in this article, the physical parameters corresponding to $g_0$ and $U_0$ are 1.2 $E_F/k_F^{3/2}$ and $-60.7E_F/k_F^3$, respectively.

Figure 4

Molecular population $\left|c\right|{}^2$ as a function of detuning. The vertical line in (b) indicates the critical point of a first-order phase transition. In (b), the collisional term is included, while it is neglected in (a).

Figure 5

Dynamics of atom-molecule conversion as illustrated by the molecular population when the detuning $\nu$ is slowly swept. Curve (a) is obtained by sweeping $\nu$ from positive to negative values, while curve (b) is obtained by sweeping $\nu$ in the opposite direction. The dotted line is the steady-state molecular population, the same as in Fig. 4b.