Doppler cooling of three-level $\Lambda$ systems by coherent pulse trains

We explore the possibility of decelerating and Doppler cooling an ensemble of three-level $\Lambda$-type atoms by a coherent train of short, nonoverlapping laser pulses. We show that $\Lambda$ atoms can be Doppler cooled without additional repumping of the population from the intermediate ground state. We derive an analytical expression for the scattering force in the quasi-steady-state regime and analyze its dependence on pulse-train parameters. Based on this analysis we propose a method of choosing pulse-train parameters to optimize the cooling process.

Figure 1

Energy levels of $\Lambda$-type system and positions of frequency-comb teeth. The comb is Doppler shifted in the atomic frame moving with velocity $v$.

Figure 2

The dependence of the fractional momentum kick $\Delta p/p_r$ on the phase offset $\overline{\eta }$ at different values of pulse repetition period $T$. Solid purple line, $T=4$ ns; dashed purple line, $T=50$ ns. The parameters of the system are: $\gamma =0.05$ GHz, $\Theta =\pi /4$, $\kappa =0.12$.

Figure 3

The dependencies of the quasi-steady-state values of the postpulse excited-state population ${\left({\rho }_{ee}^s\right)}_r$ and single-pulse momentum kick $\Delta p/p_r$ on effective single-pulse area $\Theta$ at different values of $\mu =\gamma T$: $\mu =10$ (short-dashed pink line), $\mu =1/2$ (long-dashed blue line), $\mu =1/100$ (solid purple line), and optimal parameters $\overline{\eta }=-\pi {\kappa }^{\mathrm{opt}}$, where ${\kappa }^{\mathrm{opt}}$ is obtained from Eq. (20).

Figure 4

Dependence of the friction coefficient $\beta /\hslash k_c^2$ on phase detuning $\overline{\eta }$ at (a) $\gamma T=1/2$ and (b) $\gamma T=1/100$. Each panel has three curves with different values of pulse area $\Theta$: $\Theta =\pi /10$ (solid purple line), $\Theta =\pi /2$ (thick dashed purple line), and $\Theta =\pi$ (thin dashed pink line).

Figure 5

The dependence of the scattering force on the Doppler-shifted phase offset $\overline{\eta }$ at the optimal value of $\kappa ={\kappa }^{\mathrm{opt}}$, chosen according to Eq. (21) at fixed value of $\mu =\gamma T$: (a) $\mu =1/2$, (b) $\mu =1/100$ and different values of effective single pulse area $\Theta$. Different curves correspond to the distinct values of single pulse areas: $\Theta =\frac{\pi }{10}$ (solid purple line), $\Theta =\frac{\pi }{2}$ (long-dashed blue line), $\Theta =\pi$ (short-dashed pink line).

Figure 6

Time evolution of the velocity distribution for a thermal beam subjected to a coherent train of laser pulses. The pulse-to-pulse phase offset of the train is varied linearly in time as prescribed by Eq. (40). $N$ is the number of pulses. (a) Atomic and pulse-train parameters are $\gamma T=0.25$, $\Theta =\pi /2$. The optimal phase detuning is $\overline{\eta }=-1.23$. The center of the initial velocity distribution is $v_{mp}=500$ m/s.