Simultaneous sub-Doppler laser cooling of fermionic ${}^6\mathrm{Li}$ and ${}^{40}\mathrm{K}$ on the $D_1$ line: Theory and experiment

We report on simultaneous sub-Doppler laser cooling of fermionic ${}^6\mathrm{Li}$ and ${}^{40}\mathrm{K}$ using the $D_1$ optical transitions. We compare experimental results to a numerical simulation of the cooling process applying a semiclassical Monte Carlo wave-function method. The simulation takes into account the three-dimensional optical molasses setup and the dipole interaction between atoms and the bichromatic light field driving the $D_1$ transitions. We discuss the physical mechanisms at play, identify the important role of coherences between the ground-state hyperfine levels, and compare $D_1$ and $D_2$ sub-Doppler cooling. In 5 ms, the $D_1$ molasses phase greatly reduces the temperature for both ${}^6\mathrm{Li}$ and ${}^{40}\mathrm{K}$ at the same time, with final temperatures of 44 and $11\mu \mathrm{K}$, respectively. For both species this leads to a phase-space density close to ${10}^{-4}$. These conditions are well suited to direct loading of an optical or magnetic trap for efficient evaporative cooling to quantum degeneracy.

Figure 1

Cooling scheme on the ${}^6\mathrm{Li}$ and ${}^{40}\mathrm{K}$ $D_1$ lines. The cooling beam (blue) is blue detuned by ${\delta }_{\mathrm{cool}}$ from the $|F=3/2\rangle \to |F_h^{\prime }=3/2\rangle$ $\left(|F=9/2\rangle \to |F_h^{\prime }=7/2\rangle \right)$ transition, where $F_h^{\prime }$ $\left(F_l^{\prime }\right)$ is the upper (lower) excited-state level. The repumping beam (red) is blue detuned by ${\delta }_{\mathrm{rep}}$ from the $|F=1/2\rangle \to |F_h^{\prime }=3/2\rangle$ $\left(|F=7/2\rangle \to |F_h^{\prime }=7/2\rangle \right)$ transition. Detuning from the Raman condition is denoted $\Delta ={\delta }_{\mathrm{rep}}-{\delta }_{\mathrm{cool}}$.

Figure 2

Fluorescence (squares) and temperature (triangles; logarithmic scale) of the ${}^6\mathrm{Li}$ atomic cloud after a 100-$\mu \text{s}$ pulse of $D_1$ light with variable Raman detuning $\Delta$. Experiment: (a) low intensity; (b) high intensity. Simulation: (c) low intensity; (d) high intensity. Experimental and simulation results (a, c) for $I_{\mathrm{cool}}=2.7I_{\mathrm{sat}},I_{\mathrm{rep}}=0.13I_{\mathrm{sat}}$ and (b, d) for $I_{\mathrm{cool}}=9I_{\mathrm{sat}},I_{\mathrm{rep}}=0.46I_{\mathrm{sat}}$ per beam.

Figure 3

Temperature of the ${}^6\mathrm{Li}$ $D_1$ molasses after a 100-$\mu \text{s}$ pulse with variable Raman detuning $\Delta$ for different cooling and repumping intensities. (a) Experiment; (b) simulation. Standard intensities [(red) circles]: $I_{\mathrm{cool}}=9I_{\mathrm{sat}},I_{\mathrm{rep}}=0.46I_{\mathrm{sat}}$. Equal cooling/repumping ratio (black squares): $I_{\mathrm{cool}}=I_{\mathrm{rep}}=9I_{\mathrm{sat}}$. Inverted cooling/repumping ratio [(blue) triangles]: $I_{\mathrm{cool}}=0.6I_{\mathrm{sat}},I_{\mathrm{rep}}=4.6I_{\mathrm{sat}}$.

Figure 4

Experiment: Atom number and equilibrium temperature of the ${}^{40}\mathrm{K}$ $D_1$ molasses as functions of the Raman detuning $\Delta$. ${\delta }_{\mathrm{cool}}=3\Gamma ,I_{\mathrm{cool}}=6I_{\mathrm{sat}},I_{\mathrm{rep}}/I_{\mathrm{cool}}=7.6\%,t_{\mathrm{m}}=5\mathrm{ms}$. In the constant-temperature regions below $-0.1\Gamma$ and above $2\Gamma$ gray molasses cooling involves coherences between Zeeman states in a given hyperfine state but not between hyperfine states. At the exact Raman condition $\Delta =0$, long-lived coherences between hyperfine states are established, as shown in the simulation in Fig. 5. In a narrow detuning range, the temperature [(red) triangles] drops to $20\mu \mathrm{K}$. Inset: Expanded scale.

Figure 5

Simulation: Hyperfine coherence and $\Lambda$-enhanced cooling for the ${}^{40}\mathrm{K}$ $D_1$ molasses. Simulation time, 2 ms; ${\delta }_{\mathrm{cool}}=3\Gamma$; $I_{\mathrm{cool}}=6I_{\mathrm{sat}}$; $I_{\mathrm{rep}}/I_{\mathrm{cool}}=7.6\%$. (a) Fluorescence (squares) and temperature (triangles) as functions of the Raman detuning $\Delta$. (b) Coherence $4·\langle {\rho }_{F=7/2,F=9/2}^2\rangle$ between the two hyperfine ground states, $F=7/2$ and $F=9/2$ (see Sec. 2e). The coherence is peaked under the Raman-resonance condition, with a width matching the temperature dip.

Figure 6

Cooling mechanism around the Raman condition in a simplified model. Optical Bloch equation simulation for ${}^6\mathrm{Li}$ subjected to a 1D bichromatic lattice with linear orthogonal polarizations near $D_1$ resonance. The cooling lattice and repumping lattice are displaced by $\pi$ $\left(\lambda /4\right)$. $I_{\mathrm{cool}}=15I_{\mathrm{sat}},I_{\mathrm{rep}}=0.75I_{\mathrm{sat}},{\delta }_{\mathrm{cool}}=4\Gamma$. (a–c) Dressed states as functions of the position, in units of the $D_1$ optical wavelength. The two dressed $F=1/2$ levels (light blue and orange) are nearly flat in all the graphs due to the small $I_{\mathrm{rep}}$. (d–f) Decay rate of dressed states as a function of their energy shifts. Here the two dressed $F=1/2$ levels span a very small energy range and have a low decay rate. (g–i) Velocity-dependent optical force for an atom dragged at velocity $v$. (a, d, g) $\Delta =0$; (b, e, h) $\Delta =\Gamma /2$; (c, f, i) $\Delta =-\Gamma /4$. Note the negative sign of the force in (g) and (i), implying cooling, and the anticooling force for velocities near 0.5 m/s in (h).

Figure 7

Optical scheme of the $D_1$ molasses. The 3D MOT light and the $D_1$ cooling light are superposed using a D-shaped mirror $\left(M_{\mathrm{D}}\right)$. Afterwards a dichroic mirror $\left(M_{\mathrm{dichroic}}\right)$ combines the lithium and potassium light, which is subsequently expanded and distributed to the three perpendicular axes of the 3D MOT.

Figure 8

Number of atoms captured in the ${}^6\mathrm{Li}$ $D_1$ molasses (circles) and their temperature (triangles) as functions of the molasses duration, with a $1/e$ cooling time constant ${\tau }_{\mathrm{cool}}=0.6$ ms. The number of atoms in the compressed MOT is $8\times {10}^8$.

Figure 9

Number of atoms captured in the ${}^6\mathrm{Li}$ $D_1$ molasses (circles) and their temperature (triangles) after a 3-ms capture phase at a high intensity, $I_{\mathrm{cool}}=14.6I_{\mathrm{sat}}$, as functions of the global detuning $\delta ={\delta }_{\mathrm{cool}}={\delta }_{\mathrm{rep}}$. The number of atoms in the compressed MOT is $8\times {10}^8$.

Figure 10

Number of atoms captured in the ${}^6\mathrm{Li}$ $D_1$ molasses (circles) and their temperature (triangles) as functions of the $D_1$ cooling beam intensity for ${\delta }_{\mathrm{cool}}=4\Gamma$ and $I_{\mathrm{rep}}=I_{\mathrm{cool}}/20$. The number of atoms in the compressed MOT is $8\times {10}^8$. The atom number (temperature) increases linearly with a slope of $4\times {10}^7$ atoms/$I_{\mathrm{sat}}$ $(6.5\mu \text{K}/I_{\mathrm{sat}})$.

Figure 11

Number of atoms captured in the ${}^6\mathrm{Li}$ $D_1$ molasses (circles) and their temperature (triangles) after a 3-ms capture phase at a high intensity, $I_{\mathrm{cool}}=14.6I_{\mathrm{sat}}$, followed by a 2-ms linear intensity ramp to an adjustable value. The temperature increases linearly for a higher intensity, with a slope of $\sim 6\mu \mathrm{K}/I_{\mathrm{sat}}$. The detuning is fixed to ${\delta }_{\mathrm{cool}}=4\Gamma$. The number of atoms in the compressed MOT is $8\times {10}^8$.