Stimulated slowing of Yb atoms on the narrow ${}^1{S}_0$ $\to$ ${}^3{P}_1$ transition

We analyzed bichromatic and polychromatic stimulated forces for laser cooling and trapping of Yb atoms using only the narrow ${}^1{S}_0\to {}^3{P}_1$ transition. Our model is based on numerical solutions of optical Bloch equations for two-level atoms driven by multiple time-dependent fields combined with Monte Carlo simulations, which account for realistic experimental conditions such as atomic beam divergence, geometry, and Gaussian laser modes. Using 1 W of laser power, we predict a loading rate of $\approx {10}^8$ atoms/s into a 556-nm magneto-optical trap (MOT) with a slowing force of $\approx 60F_{\text{rad}}$. We show that a square-wave modulation can produce similar stimulated forces with almost twice the velocity range and improve the MOT loading rate of Yb atoms by up to 70%.

Figure 1

Energy-level diagram of Yb showing the ${}^1{S}_0\to {}^3{P}_1$ transition (green solid arrow), which is used for both stimulated slowing and magneto-optical trapping. The stimulated slowing method eliminates the need for the traditional first-stage cooling on the ${}^1{P}_1$ transition and helps avoid population loss via decays from ${}^1{P}_1$ to the lower-lying triplet $D$ states, ${}^3{D}_{2,1}$ ($5d6s$), which can lead to further decays to dark ${}^3{P}_{2,0}$ states where atoms are lost from the cooling cycles or MOTs without repumping.

Figure 2

(a) An atom traveling in counterpropagating bichromatic light fields with a velocity $v$. The laser beam frequencies are detuned by $+kv$ and $-kv$ to compensate for Doppler shifts. (b) A schematic diagram of our experimental setup showing a Yb atomic beam leaving an oven at $450{}^{\circ }\mathrm{C}$ and two counterpropagating laser beams at 556 nm. The atoms are slowed over a distance of 33 cm before reaching the MOT region. Our current Yb atomic beam is partially collimated by a tubular array nozzle and diverges with a half-angle $\approx 17$ mrad.

Figure 3

Bichromatic force as a function of atomic velocity for $\delta =50\gamma$ (blue, lower line), $100\gamma$ (black, middle line), $150\gamma$ (red, upper line), Rabi frequency $\mathrm{\Omega }=\sqrt{3/2}\delta$, $\chi =\pi /4$. The magnitude and range of the force scale linearly with $\delta$. At $\delta =150\gamma$ and $\mathrm{\Omega }=184\gamma$ (red, upper line), the maximum stimulated force is near $90F_{\text{rad}}$ and acts over the velocity range of $\pm 30\gamma /k\approx 3$ m/s for the ${}^1{S}_0\to {}^3{P}_1$ transition in Yb.

Figure 4

Velocity distributions before (black dashed line) and after (red solid line) slowing with (a) a fixed detuning at 240 m/s and (b) chirped detuning from 240 to 10 m/s in 2 ms. Gaussian beam diameter of 8 mm, $\delta =200\gamma$, and peak Rabi frequency $\mathrm{\Omega }=244\gamma$ for both (a) and (b). (c) Same chirp parameters and laser power as (b) but with a flattop beam profile of the same diameter. Rabi frequency $\mathrm{\Omega }=173\gamma$ is constant across the beam profile and $\delta$ was adjusted to $141\gamma$. The unrealistic assumption of a flattop beam profile overestimated the MOT loading rate. It is important to take into account the actual Gaussian beam profile.

Figure 5

Percentage of atoms in the atomic beam that can be trapped by the 556-nm MOT after bichromatic slowing plotted against Rabi frequency $\mathrm{\Omega }$ and laser beam diameter for three different chirp settings: 140 $\to$ 10 m/s in 1.5 ms (square, solid line), 180 $\to$ 10 m/s in 2 ms (circle, dashed line), and 240 $\to$ 10 m/s in 2.5 ms (triangle, dot-dashed line). The error bars represent 95% confidence interval. Using 1 W of laser power, up to 0.24% of atoms can be trapped using $\delta =105\gamma$, $\mathrm{\Omega }=129\gamma$ with a Gaussian beam $1/e$ diameter of 7.6 mm and a laser detuning chirp from 180 to 10 m/s in 2 ms.

Figure 6

Stimulated force from an approximate square-wave modulation with only the first three harmonics ($n=3$) of ${\delta }_{\text{sq}}$ plotted in the parameter plane of the atoms velocity $v$ and phase $\chi$. $\mathrm{\Omega }=\pi {\delta }_{\text{sq}}/4=61\gamma$, $\chi =0.36\pi$ produce the strongest stimulated force with a magnitude of over $35F_{\text{rad}}$, which acts over a broad velocity range of over $\pm 25\gamma /k$.

Figure 7

Time evolution of the Bloch vector (top row) and the excited-state population ${\rho }_{11}$ (bottom row) for a two-level system in the optimal bichromatic (left) and square-wave light fields with the first two harmonics (center), and three harmonics (right). The spacing between dots corresponds to the time interval of 0.4 ns.

Figure 8

Force-velocity plots showing BCF (black dashed line) and the stimulated force from a near-ideal square wave (red solid line). $\mathrm{\Omega }=122\gamma$ for both cases. $\delta =100\gamma$ in the BCF case to keep $\mathrm{\Omega }/\delta =\sqrt{3/2}$ and $\chi =\pi /4$. For the square-wave case, ${\delta }_{\mathrm{sq}}=155\gamma$ and $\chi =0.36\pi$ to optimize the force profile.

Figure 9

Percentage of atoms in the atomic beam that can be trapped by the 556-nm MOT after slowing with the square-wave stimulated force plotted against Rabi frequency $\mathrm{\Omega }$ and laser beam diameter for three different chirp settings: 220 $\to$ 10 m/s in 2 ms (square, solid line), 240 $\to$ 10 m/s in 2.5 ms (circle, dashed line), 260 $\to$ 10 m/s in 3 ms (triangle, dot-dashed line). Using 1 W of laser power, up to 0.40% of atoms can be trapped using ${\delta }_{\text{sq}}=160\gamma$, $\mathrm{\Omega }=128\gamma$ with a Gaussian beam $1/e$ diameter of 11 mm, total laser power of 1 W, a slowing distance of 33 cm, and a laser detuning chirp from 240 to 10 m/s in 2.5 ms.

Figure 10

Force-velocity plots for square-wave phase modulation function (red solid line) and sinusoidal phase modulation function (black dashed line). ${\delta }_{\varphi }=153\gamma ,\mathrm{\Omega }=122\gamma$, and $\chi =0.36\pi$ in both cases.

Figure 11

Bichromatic force-velocity plots centered at zero detuning $v_c=0$ (blue solid line) and two velocity detunings at $v_c=\pm 1.5\delta /k=\pm 150\gamma /k$ (black dashed line).