Collective Dissipative Molecule Formation in a Cavity

We propose a mechanism to realize high-yield molecular formation from ultracold atoms. Atom pairs are continuously excited by a laser, and a collective decay into the molecular ground state is induced by a coupling to a lossy cavity mode. Using a combination of analytical and numerical techniques, we demonstrate that the molecular yield can be improved by simply increasing the number of atoms, and can overcome efficiencies of state-of-the-art association schemes. We discuss realistic experimental setups for diatomic polar and nonpolar molecules, opening up collective light matter interactions as a tool for quantum state engineering, enhanced molecule formation, collective dynamics, and cavity mediated chemistry.

Figure 1

Basic setup for collective dissipative molecule formation. (a) Feshbach molecules are trapped in a lattice inside a cavity and brought into deeply bound states by photoassociation. An angle $\theta$ between the lattice laser beams and the cavity ($z$) axis ensures mode matching. (b) Scheme of energy levels and their coupling for a single molecule. For RbCs the potential energy curves can be identified with the ground state potential $X^1{\mathrm{\Sigma }}^{+}$ (continuous line; dissociates to $5s+6s$), triplet ground state potential $a^3{\mathrm{\Sigma }}^{+}$ (dash-dotted line; dissociates to $5s+6s$) and excited state potential $(A^1{\mathrm{\Sigma }}^{+}-b^3\mathrm{\Pi })0^{+}$ (dashed line; dissociates to $5s+6p$). A molecule prepared in a Feshbach state $|i⟩$ is laser excited (coupling strength $\mathrm{\Omega }$, detuning $\mathrm{\Delta }$) into the excited state $|e⟩$, which can decay back into $|i⟩$, the rovibrational ground state $|g⟩$ or any other state (bound or not), here collectively called $|x⟩$. (c) Energy levels of a single molecule after adiabatic elimination of the cavity and excited state with decay rates ${\zeta }_{\alpha }$ ($\alpha =\kappa$, $i$, $x$, $g$). (d) Evolution of the target ground state molecular fraction $N_g/N$ as a function of rescaled time $t$ (see text), for different $1\le N\le {10}^5$ (log scale). The red dashed line in the inset indicates the results without cavity.

Figure 2

On bare resonance: (a) Contour plot of the final population fraction in the loss state $|x⟩$, $N_x^{\infty }/N$, as a function of the number $N$ of molecules in the cavity and the single molecule cooperativity $C$. (b) Contour plot of the time $T_{\frac{1}{2}}$ needed to transfer half of the population away from the state $|i⟩$ in $\tau =\mathrm{\Gamma }{\mathrm{\Omega }}^{-2}$. All axes are logarithmic.

Figure 3

Chirped pulse: (a) Contour plot of the decay rate of Feshbach molecules ${\dot{N}}_i$ (in units ${\tau }^{-1}$, for symmetric Dicke states), as a function of the laser detuning and the number of ground state molecules. The cavity is kept at resonance with the transition energy ($\delta =\mathrm{\Delta }$). (b) Simulated time evolution of the ground state population for different cavity decay rates $\kappa$. The parameters are chosen for ${10}^3$ ${\mathrm{Rb}}_2$ molecules inside a cavity [13, 20], i.e., $\mathrm{\Gamma }/2\pi =6\text{MHz}$ and $g/2\pi \approx 50\mathrm{MHz}$. Dashed lines: analytical fits of Eqs. (13) and (14).