Atom-Molecule Dark State: The Exact Quantum Solution

Developments in ultracold atomic physics have enabled the generation of an atom-molecule dark state in dilute quantum gases. Previous studies of this dark state were all carried out in the mean-field regime. Here we present an exact quantum many-body wave function for the atom-molecule dark state for an ideal system (in the absence of losses and two-body collisions). Using this exact quantum wave function, we are able to validate the mean-field solution to be a good approximation when the particle number $N$ is large. For small $N$, unique quantum features will become important.

Figure 1

Schematic level diagram of the atom-molecular system. ${\Omega }_p$ and ${\Omega }_d$ are the two Rabi frequencies. $\Delta$ is the one-photon detuning of the pump field with respect to the $|a⟩-|e⟩$ transition, while $\delta$ is the two-photon detuning of the Raman transition between $|a⟩$ and $|g⟩$.

Figure 2

Population in atomic state $|a⟩$, normalized by the total atomic number $N$, as a function of Rabi frequency ratio $\chi$.

Figure 3

The correlation function $g^{(2)}$ and $C_{ag}$ as functions of $\chi$.

Figure 4

(a) The energy spectrum as a function of single-photon detuning $\Delta$. Solid lines are quantum results, while the dots represent mean-field energy spectrum. Parameters used are $N=6$ and ${\Omega }_p={\Omega }_d=1$. (b) The degree of degeneracy at $\Delta =0$ as a function of $N$.

Figure 5

(a) The overlap function $|⟨\psi (t)|D(t){⟩}_{\mathrm{Q}}|$ as a function of time, for $\Delta =0$ (dashed lines, $N=6,10,20$ in descending order) and $\Delta =0.5$ (solid lines, indistinguishable for different $N$’s). Other parameters: ${\Omega }_0=5$, $T=100$, $t_d=250$, and $t_p=450$. (b) Coupling function between the dark state with other degenerate states. $N=6,10,20$ in ascending order.