Single-Photon Molecular Cooling

We propose a general method to cool the translational motion of molecules. Our method is an extension of single photon atomic cooling which was successfully implemented in our laboratory. Requiring a single event of absorption followed by a spontaneous emission, this method circumvents the need for a cycling transition and can be applied to any paramagnetic or polar molecule. In our approach, trapped molecules would be captured near their classical turning points in an optical dipole or RF-trap following an irreversible transition process.

scheme that overcomes some of the deficiencies of the optical trap based single photon cooling method.
Optical tweezer based single photon molecular cooling begins with a far detuned optical dipole trap placed at the location of the classical turning point of the "hottest" molecules in the trap. As the "hottest" molecules arrive at the classical turning point, they exchange their kinetic energy for potential energy and come to near rest. At that point, we induce a resonant transition to a single rovibrational level of the electronically excited molecule. It decays back to the rovibrational ground state, with a certain probability given by the branching ratios (defined by the Hönl-London and Franck-Condon factors). The selection rules for the total angular momentum projection change in a spontaneous decay are Δm J = 0, ±1. In the case where Δm J = -1 the slope due to the magnetic quadrupole becomes smaller in the final state (assuming that the Lande factor does not change) and the potential well due to the attractive standing wave is deep enough to support a bound state. Since the lifetimes of electric-dipole allowed transitions range from tens to hundreds of nanoseconds, both the position and the kinetic energy of the molecule do not change during the decay (up to a photon recoil). If the spontaneous emission occurs at the location of the optical tweezer, the stationary molecule is trapped. This condition can be fulfilled by making the excitation volume small compared to the optical trap dimensions. The spontaneous decay into the ground state is irreversible since the final state is detuned from the initial level by the rotational energy level separation. In order to accumulate more molecules, we translate the center of the magnetic trap relative to the optical trap and the resonance excitation beam, thus picking up more molecules along the sweep.
We now analyze a single photon cooling processes of the NH radical, that has a 3 Σelectronic configuration in the ground state. The level diagram is displayed in Fig. 1a. Since the orbital momentum of an NH radical is zero in the 3 Σstate it has a particularly simple Zeeman level structure. The ground state NH molecules can be created in an arc discharge that would dissociate a precursor molecule such as NH 3 near the orifice of a supersonic nozzle. Magnetically trapped ground level molecules can be optically pumped into the rotationally excited initial level |ν=0, J=1, N=2> (ν, J and N are the vibration, total rotation and nuclear rotation quantum numbers respectively). We estimate that the optical pumping efficiency via the |ν=0, J=1> level of the 3 ∏ 0 manifold can be as high as 64%, using the absorption and fluorescence branching ratios derived from the Hönl-London factors for 3 ∏← 3 Σtransition given in reference [23]. The projection of the total angular momentum on the quantization axis in the initial state is m Ji =1. In order to confine molecules with a temperature of 50 mK to a 5 mm diameter cloud we need to apply a magnetic quadrupole field with a gradient of 0.14 T/cm.
Since the polarizability of an NH radical is only 1.3*10 -24 cm 3 we need to apply a high intensity optical field to create a deep enough attractive potential. A far detuned standing wave with 1kW of optical power focused to 100 µm beam waist will form a 250 µK deep well. The high optical intensity needed can be achieved by using a buildup cavity with a moderate finesse. We illustrate the combined magnetic and optical potential for the initial level in Fig 1b. The magnetic potential is so steep that the contribution from the optical potential is hardly noticeable. The irreversible transition from the |ν=0, J=1, N=2> level is initiated by electronic excitation along the 3 ∏ ← 3 Σtransition at 336 nm. We selectively populate the |ν=0, J=1> level of the 3 ∏ 1 manifold that decays into the ground |ν=0, J=1 N=0, m J =0> sublevel with a probability of ~35%. Since the projection of the total angular momentum is zero in the final ground level, the magnetic contribution to the overall potential becomes negligible (see Fig 1c) and the NH radical becomes trapped in the standing optical wave. By continuing the cooling process, ground state NH radicals will accumulate in a shallow optical trap. Singlephoton cooled and optically trapped molecules can be further cooled by evaporation. Evaporative cooling can be initiated by reducing the power of the standing wave. Since the molecules are in the overall ground state there will be no losses associated with inelastic collisions. Note, that technique was proposed to allow for multiple loadings of NH radicals into a magnetic trap [24]. During the single photon cooling process cold molecules accumulate in the tweezer after the irreversible transition that lowers the projection of the total angular momentum. The tweezer loading rate is limited by the frequency that molecules encounter the tightly focused electronic excitation beam. It is possible to enhance the loading rate using a different scheme based on accumulation of cold molecules in a trap formed by coupling different magnetic sublevels with an rf-field A shell-like potential trap for neutral atoms formed by an avoided crossing in the RF-dressed magnetic sublevels was proposed by Zobay and Garraway [25]. Colombe et. al [26] have demonstrated atom loading into the rf-trap from the magnetic quadrupole trap and Schumm et. al. [27] demonstrated the formation of a double well RF-trap potential on an atom chip. We will show that single-photon cooling can be used to cool molecules by transfering the cold molecules into a local RF-trap. We use an example of a molecule in the 2 Σ state, particularly a CN radical. have to be slow enough to follow the adiabatic potential curve, otherwise they might be lost due to a non-adiabatic transition. Larger splitting at the avoided crossing point allows higher kinetic energy molecules to stay in the RF-trap.
In order to estimate the amplitude of the magnetic component of the RF field needed we have to compare the coupling strength to the frequency associated with the passage time through the avoided crossing region. We estimate that for a CN radical moving at 1 m/s (corresponding to a temperature of 3 mK) and assuming that the magnetic field gradient in the quadrupole trap is 1000 G/cm, and the amplitude of the magnetic component of the RF field is 1 G, the coupling strength is larger by more than two orders of magnitude compared to the level splitting. The RF field needed can be created by an oscillating electric field with an amplitude of 100 V across two electrodes spaced by 6 mm. In order to accumulate molecules with different energies we have to sweep both the microwave and RF frequencies and move the location of the RF-induced trap to lower energies. The RF-trap "implodes" catching atoms or molecules near their classical turning points.
The loading rate into the optical tweezer can be estimated as ρ V ex ω trap , where ρ is the trapped molecule density, V ex electronic excitation region volume and ω trap is the magnetic (or electrostatic) trap frequency.
Clearly it is limited by the small excitation volume; assuming the excitation region dimensions given by 10 µm x 10 µm x 100 µm (excitation beam waist of 10 µm) and density of the trapped molecules to be 10 8 cm -3 the loading rate approaches a magnetic trap frequency that can be as high as few kHz. The RF-trap loading method overcomes this problem by exciting the molecules from the iso-B surface. This method should allow us to capture nearly all the molecules from the magnetic trap, limited only by the branching ratio of the irreversible step. Even if that branching ratio is not large, single-photon cooling can work well because a transition only needs to happen once. Finally, we note that the RF cooling method presented above can be applied to atoms as well.