Enhancement of Ultracold Molecule Formation Using Shaped Nanosecond Frequency Chirps

We demonstrate that judicious shaping of a nanosecond-time-scale frequency chirp can dramatically enhance the formation rate of ultracold ${{}^{87}\mathrm{Rb}}_2$ molecules. Starting with ultracold ${}^{87}\mathrm{Rb}$ atoms, we apply pulses of frequency-chirped light to first photoassociate the atoms into excited molecules and then, later in the chirp, deexcite these molecules into a high vibrational level of the lowest triplet state $a$ ${{}^3\mathrm{\Sigma }}_u^{+}$. The enhancing chirp shape passes through the absorption and stimulated emission transitions relatively slowly, thus increasing their adiabaticity, but jumps quickly between them to minimize the effects of spontaneous emission. Comparisons with quantum simulations for various chirp shapes support this enhancement mechanism.

Figure 1

Long-range portion of molecular potentials and the vibrational levels relevant to the frequency-chirped molecule formation. Transitions between the lower $a$ ${{}^3\mathrm{\Sigma }}_u^{+}$ state and $0_g^{-}$ excited state are also shown: photoassociation (PA) and stimulated emission (STE) induced by the chirped pulse, and spontaneous emission (SPE) of the excited state.

Figure 2

Measured frequencies versus time, smoothed with a 2 ns FWHM Gaussian, for the various chirps used for molecule formation: unchirped (UC), negative linear (NL), positive linear (PL), positive linear plus Gaussian (PLG), and positive piecewise linear (PPL). The solid and dotted horizontal lines represent the PA transition to $0_g^{-}$ ($v^{\prime }=78$) and the STE transition to $a$ ${{}^3\mathrm{\Sigma }}_u^{+}$ ($v^{\prime \prime }=39$), respectively, as shown in Fig. 1. The timing of the intensity pulse is indicated by the double-ended arrow ($\text{length}=15\mathrm{ns}$ FWHM).

Figure 3

Measured (points) and simulated (curves) molecule formation rates versus peak intensity for the various chirps in Fig. 2. The theory curves have been scaled by a factor of 1.997.

Figure 4

(a) Temporal evolution of dressed-state molecular energies for the cases of the positive linear (PL) and positive piecewise linear (PPL) chirps. Horizontal lines indicate the energies of the $v^{\prime }=77–79$ vibrational levels of the $0_g^{-}$ excited state. Upper and lower sloped lines indicate the energies of the continuum and the $v^{\prime \prime }=39$ bound level of the $a$ ${{}^3\mathrm{\Sigma }}_u^{+}$ state, respectively, with the energy of the chirped photon added. The dashed line denotes the adiabatic path from the free-atom continuum to the bound $v^{\prime \prime }=39$ molecule for the PPL chirp. The double-ended arrow ($\text{length}=15\mathrm{ns}$ FWHM) indicates the timing of the intensity pulse. Time-dependent populations for a peak pulse intensity of $150\mathrm{W}/{\mathrm{cm}}^2$ for the (b) $v^{\prime }=78$ excited state, (c) $v^{\prime \prime }=39$ populations resulting from stimulated emission (STE), and (d) $v^{\prime \prime }=39$ populations resulting from spontaneous emission (SPE). At long times, the excited states have completely decayed and the SPE populations in (c) reach $6\times 10^{-10}$ and $3\times 10^{-10}$ for the PL and PPL chirps, respectively. For the PPL chirp, note the enhanced population from STE and diminished population from SPE relative to the PL chirp.