Atom-Molecule Dark States in a Bose-Einstein Condensate

We have created a dark quantum superposition state of a Rb Bose-Einstein condensate and a degenerate gas of ${\mathrm{Rb}}_2$ ground-state molecules in a specific rovibrational state using two-color photoassociation. As a signature for the decoupling of this coherent atom-molecule gas from the light field, we observe a striking suppression of photoassociation loss. In our experiment the maximal molecule population in the dark state is limited to about 100 ${\mathrm{Rb}}_2$ molecules due to laser induced decay. The experimental findings can be well described by a simple three mode model.

Figure 1

Dark resonances in two-color photoassociation. (a) Atomic loss signal in one-color photoassociation as a function of the laser detuning from the electronically excited molecular line. (b), (c) When we apply a second laser (fixed frequency) which resonantly ($\Delta =0$) couples the excited molecular state to a long-lived molecular ground state, the losses are strongly suppressed at $\delta =0$. Depending on the intensity of laser 2, this dark resonance can get very narrow. The atom lifetime on the dark resonance in (b) is 140 ms whereas in (a) atoms have an initial decay time of about 2 ms. Intensities of laser 1 ($I_1$) and 2 ($I_2$) are as indicated.

Figure 2

Level scheme. $\Delta$ and $\delta$ denote the detunings. ${\Omega }_1$ and ${\Omega }_2$ are the Rabi frequencies. The excited molecular state $|b⟩$ spontaneously decays with a rate ${\gamma }_b$ to levels outside this scheme. The molecular state $|g⟩$ is attributed a decay rate ${\gamma }_g$ which phenomenologically takes into account losses through inelastic collisions and laser induced dissociation, e.g., when laser 1 couples $|g⟩$ to the unstable state $|b⟩$. In all our measurements laser 1 is scanned (varying $\delta$) while laser 2 is held fixed at a particular detuning $\Delta$.

Figure 3

Two-color photoassociation spectra for various detunings $\Delta$ at a large intensity $I_2=20\mathrm{W}/{\mathrm{cm}}^2$. Here $I_1=80\mathrm{W}/{\mathrm{cm}}^2$. The solid lines are fit curves based on our theoretical model.

Figure 4

Dark resonances [blowup of the central part of Fig. 1c and similar curves] for different intensity ratios $I_2/I_1$ (see legend) at a fixed intensity $I_1=7\mathrm{W}/{\mathrm{cm}}^2$. The solid curves are calculations based on our theoretical model. $\Delta =0$.

Figure 5

Dependence of the decay rate ${\gamma }_g$ of the ground-state molecules on the laser intensity $I_1$, measured with an intensity ratio $I_2/I_1=1/40$. The solid curve is given by ${\gamma }_g=2\pi \times 6\mathrm{kHz}/({\mathrm{W}\mathrm{cm}}^{-2})I_1+2\pi \times 1\mathrm{kHz}$.

Figure 6

(a) Maximum molecular fraction as function of the intensity ratio $I_2/I_1$ at a fixed intensity $I_1=10\mathrm{W}/{\mathrm{cm}}^2$ and (b) as a function of intensity $I_1$ at a fixed intensity ratio $I_2/I_1=1/500$. The solid lines show the molecule fraction for the measured decay rate ${\gamma }_g=2\pi \times 6\mathrm{kHz}/({\mathrm{W}\mathrm{cm}}^{-2})I_1+{\gamma }_{\mathrm{bg}}$. The dashed lines show the molecule fraction assuming a lower decay rate ${\gamma }_g=2\pi \times 1\mathrm{kHz}/({\mathrm{W}\mathrm{cm}}^{-2})I_1+{\gamma }_{\mathrm{bg}}$. The dotted line corresponds to ${\gamma }_g=0$. The calculations are based on Eqs. (1) with $\Delta =0$.