Coherent Optical Transfer of Feshbach Molecules to a Lower Vibrational State

Using the technique of stimulated Raman adiabatic passage (STIRAP) we have coherently transferred ultracold ${}^{87}\mathrm{Rb}_2$ Feshbach molecules into a more deeply bound vibrational quantum level. Our measurements indicate a high transfer efficiency of up to 87%. Because the molecules are held in an optical lattice with not more than a single molecule per lattice site, inelastic collisions between the molecules are suppressed and we observe long molecular lifetimes of about 1 s. Using STIRAP we have created quantum superpositions of the two molecular states and tested their coherence interferometrically. These results represent an important step towards Bose-Einstein condensation of molecules in the vibrational ground state.

Figure 1

(a) Level scheme for STIRAP. Lasers 1, 2 couple the ground state molecular levels $|a⟩$, $|g⟩$ to the excited level $|b⟩$ with Rabi frequencies ${\Omega }_1$, ${\Omega }_2$, respectively. $\Delta$ and $\delta$ denote detunings. ${\gamma }_a$, ${\gamma }_b$, ${\gamma }_g$ give effective decay rates of the levels. (b) Zeeman diagram of relevant energy levels. At 1007.4 G a molecular state crosses the threshold of the unbound two atom continuum (dashed line) giving rise to a Feshbach resonance. From there this molecular state adiabatically connects to the last bound vibrational level $|a⟩$, the state of the Feshbach molecules.

Figure 2

Dark resonance. The data show the remaining fraction of Feshbach molecules in state $|a⟩$, after subjecting them to a $200\mu \mathrm{s}$ square pulse of Raman laser light in a narrow range around 0 of the two-photon detuning $\delta$. The inset shows the scan over the whole line of state $|b⟩$. The strong suppression of loss at $\delta =0$ is due to the appearance of a dark state. The laser intensities are $I_1=2.6\mathrm{mW}/{\mathrm{cm}}^2$, $I_2=13\mathrm{mW}/{\mathrm{cm}}^2$ (×), $I_2=51\mathrm{mW}/{\mathrm{cm}}^2$ (●). $\Delta$ is in general tuned close to zero and for the shown measurements happens to be $\Delta /2\pi \approx 2.5\mathrm{MHz}$, which gives rise to the slightly asymmetric line shape of the dark states. The solid and dashed lines are model calculations (see text).

Figure 3

STIRAP. (a) Counterintuitive pulse scheme. Shown are laser intensities as a function of time (laser 1: dashed line, laser 2: solid line). The first STIRAP pulse with length ${\tau }_p=200\mu \mathrm{s}$ transfers the molecule from state $|a⟩$ to state $|g⟩$. After a holding time ${\tau }_h=5\mathrm{ms}$, the second pulse (identical, but reversed) transfers the molecules back to $|a⟩$. $I_{1,2}^{\max }$ indicates the maximal intensity of laser 1 (2) in the pulse, respectively. (b) Corresponding population in state $|a⟩$ (data points, solid line) and state $|g⟩$ (dashed line). The data points are measurements where at a given point in time the STIRAP lasers are abruptly switched off and the molecule population in state $|a⟩$ is determined. For these measurements $\Delta \approx 0\approx \delta$. The lines are model calculations (see text). (c) Efficiency for population transfer from state $|a⟩$ to state $|g⟩$ and back via STIRAP as a function of the two-photon detuning $\delta$. The line is a model calculation, showing a Gaussian line shape with a FWHM width of $\approx 22\mathrm{kHz}$.

Figure 4

Efficiency for population transfer from state $|a⟩$ to state $|g⟩$ and back with two STIRAP pulses. (a) Efficiency vs the laser intensity $I_2^{\max }$ (fixed pulse length of ${\tau }_p=200\mu \mathrm{s}$). (b) Efficiency vs the pulse length ${\tau }_p$ (fixed laser intensity $I_2^{\max }=44\mathrm{mW}/{\mathrm{cm}}^2$). For (a) and (b) the intensity ratio $I_2^{\max }/I_1^{\max }=1/3.2$. The lines are from calculations without free parameters using Eqs. (1). Setting ${\gamma }_a={\gamma }_g=0$, the efficiency would reach unity for a fully adiabatic transfer (dashed lines). Using for ${\gamma }_a$, ${\gamma }_a$ the experimentally determined values, the calculations (solid lines) are in good agreement with the data. The error bars represent a $1\sigma$ statistical error.

Figure 5

Coherence of the ($|a⟩-|g⟩$) superposition state. Shown is the molecule number in state $|a⟩$ as a function of holding time ${\tau }_h$ for different detunings $\delta$ as indicated. The oscillations indicate coherent flopping of the molecular superposition state between the dark and a bright state. The lines are given by $0.5\exp (-{\tau }_h/\tau )\cos (\delta {\tau }_h)+0.5$, with a damping time $\tau =2\mathrm{ms}$.