Sideband cooling of molecules in optical traps

Sideband cooling is a popular method for cooling atoms to the ground state of an optical trap. Applying the same method to molecules requires a number of challenges to be overcome. Strong tensor Stark shifts in molecules cause the optical trapping potential, and corresponding trap frequency, to depend strongly on rotational, hyperfine, and Zeeman states. Consequently, transition frequencies depend on the motional quantum number and there are additional heating mechanisms, either of which can be fatal for an effective sideband cooling scheme. We develop the theory of sideband cooling in state-dependent potentials, and derive an expression for the heating due to photon scattering. We calculate the ac Stark shifts of molecular states in the presence of a magnetic field, and for any polarization. We show that the complexity of sideband cooling can be greatly reduced by applying a large magnetic field to eliminate electron- and nuclear-spin degrees of freedom from the problem. We consider how large the magnetic field needs to be, show that heating can be managed sufficiently well, and present a simple recipe for cooling to the ground state of motion.

Figure 1

Schematic of the two steps of Raman sideband cooling. (a) Step 1: two-photon Raman transition with detuning set to change internal state from $g_A$ to $g_B$ while removing one quantum of motional energy. This step is forbidden when the molecule reaches the motional ground state. (b) Step 2: optical pumping via an electronic excited state returns the molecule to its original internal state. For cooling to work efficiently, the optical pumping should preserve the motional quantum number.

Figure 2

Schematic of the coordinate system used in this paper. The tweezer light propagates along $x$ and is (except where otherwise specified) linearly polarized along $z$. The $y$ axis is into the page. The quantization axis of the molecule $Z$, taken along the $\mathcal{B}$ field in Sec. 5 onward, makes an angle $\beta$ to the $z$ axis. Inset: For the optical pumping step we consider the angles ${\theta }_{\mathrm{abs}}$ and ${\theta }_{\mathrm{sp}}$ that the incoming and outgoing photons make with one of the three trap axes, chosen to be $z$ in our illustration.

Figure 3

Energies (in units of $w$) of the $N=1$ states of the model molecule as a function of ${\alpha }_2^{\prime }I/w$, in the case where $\beta =\pi /24$. The horizontal axis is proportional to the intensity of the light and to the tensor polarizability of the molecule. The scalar Stark shift, which shifts all states equally, has been subtracted.

Figure 4

Ratio of tensor Stark shift to scalar Stark shift for the $N=1$ states of the model molecule, as a function of polarization angle $\beta$ from the $Z$ axis. In all cases, ${\alpha }_2^{\prime }=-0.6{\alpha }_0^{\prime }$. Dashed lines show the case where $|w/({\alpha }_2^{\prime }I\left)\right|\gg 1$. Solid, colored lines are for (a) $w=4{\alpha }_2^{\prime }I$, (b) $w={\alpha }_2^{\prime }I$.

Figure 5

Ratio of tensor Stark shift to scalar Stark shift for the $N=1$ states of the model molecule, as a function of ellipticity parameter $\xi$, defined in the text and shown schematically in inset. $\xi =0$ corresponds to linear polarization, and $\xi =\pi /4$ to circular polarization. The light propagates along $Z$ and we have chosen ${\alpha }_2^{\prime }=-0.6{\alpha }_0^{\prime }$. Dashed lines show the case where $|w/({\alpha }_2^{\prime }I\left)\right|\gg 1$. Solid, colored lines are for $w={\alpha }_2^{\prime }I$.

Figure 6

Mean change in $n$ during optical pumping $\overline{\mathrm{\Delta }{n}^{\mathrm{op}}}$ as a function of $n$ and ${\omega }_{\mathrm{t}0}/{\omega }_{\mathrm{t}1}$ for ${\eta }_1=0.2$. The circularly polarized optical pumping beam is assumed parallel to the trap axis and the linearly polarized beam perpendicular.

Figure 7

Branching ratio as a function of magnetic field for decay from each spin manifold of the ${\mathrm{B}}^2{\mathrm{\Sigma }}^{+}(N=0)$ states of CaF to a different spin manifold within ${\mathrm{X}}^2{\mathrm{\Sigma }}^{+}(N=1)$. In order of increasing branching ratio at $\mathcal{B}=800\mathrm{G}$ the states have $(m_S,m_I)=(-\frac{1}{2},-\frac{1}{2}), (-\frac{1}{2},\frac{1}{2}), (\frac{1}{2},\frac{1}{2}), (\frac{1}{2},-\frac{1}{2})$. We have set the light intensity to zero.

Figure 8

Energy levels of CaF as a function of light intensity, in a magnetic field of 300 G. All levels have $N=1,m_S=\frac{1}{2}$. For clarity, the rotational energy and the scalar Stark shift, which are common to all the levels, have been subtracted. The light is linearly polarized at an angle of $\pi /24$ to the magnetic field axis. In ascending order of energy, the states have $(m_N,m_I)=(-1,-\frac{1}{2}), (0,-\frac{1}{2}), (1,-\frac{1}{2}), (-1,\frac{1}{2}), (0,\frac{1}{2}), (1,\frac{1}{2})$. The color of the lines indicates $(m_S,m_I)$ and follows the same scheme used in Fig. 7. Dashed lines show the intensities where the three-level model predicts avoided crossings between $m_N=1$ and 0 levels. The pattern of levels with $m_S=-\frac{1}{2}$ is almost identical, but with states in the opposite order.

Figure 9

Ratio of tensor Stark shift to scalar Stark shift for the $N=1$ states of CaF. (a), (b) As a function of polarization angle $\beta$ with magnetic field of (a) 30 G, (b) 300 G. (c) As a function of magnetic field with polarization parallel to magnetic field. The light intensity is $I=25$ GW/${\mathrm{m}}^2$ throughout. The color of the lines indicates the value of $(m_S,m_I)$ and follows the same scheme used in Fig. 7.

Figure 10

Calculated elements of the $\mathcal{S}$ tensor, defined in the text. (a) ${\mathcal{S}}_0^0$, (b) $\frac{1}{\sqrt{2}}\mathrm{Im}({\mathcal{S}}_{-1}^1+{\mathcal{S}}_1^1)$, (c) ${\mathcal{S}}_0^2$, (d) $\frac{1}{\sqrt{2}}\mathrm{Re}({\mathcal{S}}_{-2}^2+{\mathcal{S}}_2^2)$. The elements shown in (b) and (d) have been scaled by factors of 4 and 40, respectively, in order to be shown on the same scale.