Implementation of a single femtosecond optical frequency comb for rovibrational cooling

We show that a single femtosecond optical frequency comb may be used to induce two-photon transitions between molecular vibrational levels to form ultracold molecules (e.g., KRb). The phase across an individual pulse in the pulse train is sinusoidally modulated with a carefully chosen modulation amplitude and frequency. Piecewise adiabatic population transfer is fulfilled to the final state by each pulse in the applied pulse train, providing a controlled population accumulation in the final state. Detuning the pulse train carrier and modulation frequency from one-photon resonances changes the time scale of molecular dynamics but leads to the same complete population transfer to the ultracold state. A standard optical frequency comb with no modulation is shown to induce similar dynamics, leading to rovibrational cooling.

Figure 1

The three-level $\lambda$ system modeling the molecular energy levels involved in cooling dynamics. State $|1\rangle$ is the Feshbach state, state $|2\rangle$ is the transitional, electronic excited state, and state $|3\rangle$ is the cold molecular state.

Figure 2

Population transfer in the three-level $\lambda$ system, achieved via the resonant Raman transitions and using an optical frequency comb as in Eq. (1) having the carrier frequency and the modulation frequency in resonance with one-photon transitions in the $\lambda$ system. The values of the parameters are the carrier frequency ${\omega }_L=5.9$ (410.7 THz), the modulation frequency $\Omega =4.9$, (340.7 THz), the modulation amplitude ${\Phi }_0=4$, and the peak Rabi frequency ${\Omega }_R=1$ (70 THz); the system one-photon transition frequencies are ${\omega }_{21}=\Omega$, ${\omega }_{32}={\omega }_L$. Stepwise, adiabatic accumulation of the population is observed in state $|3\rangle$ [green (light gray)], which is the ultracold KRb state. The population of the Feshbach state $|1\rangle$ (black) reduces gradually to zero, while the excited state manifold $|2\rangle$ [red (gray)] is only slightly populated during the transitional time. Full population transfer is accomplished in 112 pulses.

Figure 3

Population transfer in the three-level $\lambda$ system, achieved using an optical frequency comb as in Eq. (1) with the field parameters detuned off resonance with the frequencies of the $\lambda$ system. The detuning $\delta$ is equal to ${\omega }_{31}/2$. The carrier frequency is ${\omega }_L=5.4$, the modulation frequency is $\Omega =4.4$, the Rabi frequency is ${\Omega }_R=1$, and the modulation amplitude is ${\Phi }_0=4$. The $\lambda$ system transition frequencies are ${\omega }_{21}=4.9$, ${\omega }_{32}=5.9$ (with all frequency parameters in the units of frequency ${\omega }_{31}=70$ THz). Full population transfer to the final, cold state [green (light gray)] occurs within 42 pulses. During this time, the population of the initial Feshbach state (black) reduces to zero and the excited state [red (gray)] gets substantially populated during the transitional time. The population is reversed by the next 42 pulses.

Figure 4

Population dynamics in the three-level $\lambda$ system, reached using an optical frequency comb as in Eq. (1) with different values of the modulation amplitude equal to (a) ${\Phi }_0=3$, (b) ${\Phi }_0=5$, and (c) ${\Phi }_0=8$. Other parameters are the effective Rabi frequency ${\Omega }_R=1$ (70 THz), the carrier frequency ${\omega }_L=5.9$ (410.7 THz), and the modulation frequency $\Omega =4.9$ (340.7 THz). For all three values of ${\Phi }_0$, there is only partial population transfer to the final state $|3\rangle$ [green (light gray)], unlike the solution for ${\Phi }_0=4$ in Fig. 2. Black line shows the population of state $|1\rangle$, and red (gray) lines show population of state $|2\rangle .$

Figure 5

The envelope of the power spectrum of the optical frequency comb as in Eq. (1) (in which no dense radio frequency comb lines are included). The pulse train parameters are the effective Rabi frequency ${\Omega }_R=1$ (70 THz), the carrier frequency ${\omega }_L=5.9$ (410.7 THz), the modulation frequency $\Omega =4.9$ (340.7 THz), and the modulation amplitudes ${\Phi }_0=3$ [red (gray)], ${\Phi }_0=4$ [green (light gray)], ${\Phi }_0=5$ [blue (closest to black)], and ${\Phi }_0=8$ (black).

Figure 6

Population transfer in the three-level $\lambda$ system using a standard optical frequency comb characterized by a radio-frequency equal to 5 GHz and zero offset frequency. The black curve shows the population dynamics of the Feshbach state, the red (gray) curve shows that of the excited electronic state, and the green (light gray) curve shows that for the final, ultracold state. Parameters of the pulse train are the carrier frequency ${\omega }_L={\omega }_{32}=434.8$ THz, and the pulse duration ${\tau }_0=3$ fs.

Figure 7

Population transfer in the three-level $\lambda$ system, achieved using an optical frequency comb as in Eq. (3) with the carrier frequency of the electric field detuned off resonance with the ${\omega }_{32}$ in the $\lambda$ system. The detuning $\delta$ is $3/2{\omega }_{31}$. Bold black, bold red (bold gray), and bold green (bold light gray) lines show populations of the Feshbach state, the excited electronic state, and the final, ultracold state for ${\Omega }_R=1.26$ THz ($~{10}^{10}$ W/cm${}^2$); thin lines show respective populations for ${\Omega }_R=12.55$ THz ($~{10}^{11}$ W/cm${}^2$). The pulse train period is $T=6400\tau =0.02$ ns.