Raman cooling

I now want to show how Raman transitions can be used to cool the atom. To introduce the basic idea, let us first consider Raman cooling using the three-level model we discussed 1.5.3. In the harmonic approximation, the Hamiltonian for the model system is

H =H0+Hi

where

H0=ωb†b+δA 2 σz

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 -3 -2 -1 0 1 2 3 probability Raman detuning [MHz]

Figure 1.31: Population transfer due to laser noise. The red curve is the observed Raman spectrum for Ω = (2π)(200 kHz); the green curve is the predicted spectrum based on the power spectrum of the beat note.

is the Hamiltonian for the external and internal degrees of freedom for the atom, and

Hi= Ω (1−η2(b+b†)2)σxcosδRt

describes the Raman coupling. The eigenstates of H0 are {|a, ni,|b, ni}, and the eigenvalues are

nω−δA/2 for|a, niandnω+δA/2 for|b, ni. The spectrum ofH0 is shown in Figure 1.32.

Suppose we start the atom in ground state a and vibrational state n. We can lower the vibra- tional quantum number by driving the atom with a Raman pulse that is tuned to the red sideband (δR =δA−2ω). The Raman pulse will transfer some of the population from state |a, nito state |b, n−2i. We can then optically pump the atom back into ground state aby applying a classical field on the b−e transition. Note that because n-changing transitions are suppressed by at least ∼ η√n, for small enough n it is unlikely that the atom will change its vibrational state during the optical pumping process. The net effect of the Raman and optical pumping pulses is to move some of the population from state|a, nito state|a, n−2i. By iterating the pulse sequence, we can cool the atom. Although the scheme described here involves alternating Raman pulses with optical pumping pulses, it is also possible to cool the atom by applying the Raman and optical pumping light simultaneously, and it is this method that we actually use in the lab.

|a,0i |a,1i |a,2i |a,3i |b,0i |b,1i |b,2i |b,3i 6 ? ? 6 H H H H H H H H H H H H Y HH HH HH HH HH HHj ? δA ω Ω2→0 Γ

Figure 1.32: Raman cooling. The cooling operates by interleaving Raman pulses tuned to the red sideband with optical pumping pulses on theb−etransition.

Note that the cooling scheme combines a coherent process (the Raman pulse) with a spontaneous process (the spontaneous decay caused by the optical pumping pulse). The spontaneous process is necessary because when we cool the atom we are trying to map all possible initial states to the same final state, which is impossible with unitary evolution alone.

The cooling rate is determined by the Rabi frequency of the Raman pulses. Recall from section 1.5.3 that if the effective Rabi frequency is Ω, then the Rabi frequency for the |a, ni ↔ |b, n−2i transition is

Ωn→n−2=η2√n √

n−1 Ω

Thus, we can speed up the cooling by increasing the effective Rabi frequency Ω. However, if we try to go too fast we will off-resonantly excite the transition|a, ni ↔ |b, ni, which has Rabi frequency

Ωn→n= Ω−η2(2n+ 1) Ω

Thus, to ensure that we drive the red sideband exclusively, Ω must be smaller than the detuning 2ω

of the carrier. This requirement sets the maximum possible cooling rate. Also, note that Ωn→n−2 is roughly proportional to n. This means that the cooling rate becomes smaller and smaller as the atoms get colder and colder.

So far we have been working in the harmonic approximation, which is only valid near the bottom of the well. When we consider the full sin2kx potential, the situation becomes more complicated, because the vibrational frequency of the atom now depends on the atom’s energy. The potential is shallower than harmonic, so hot atoms see a smaller vibrational frequency than cold atoms (see section 1.5.5). To deal with this problem, one can slowly sweep the Raman detuning over the course

of the cooling procedure.

Now let us consider Raman cooling for a real Cesium atom. We will assume that the magnetic field is nulled, so all the Zeeman transitions pile up around zero detuning. If we choose the quantiza- tion axis to lie along the cavity axis, then state|3, miis coupled to state|4, miby the Raman pulse, and the system can be thought of as a collection of two-level atoms, all with resonant frequency ∆HF. Thus, an arbitrary state|3, miin theF = 3 manifold plays the role of state|aiin the model we just considered, and the corresponding state|4, miin theF = 4 manifold plays the role of state|bi.

To implement Raman cooling in the lab, we apply a series of 2,000 cooling cycles to the atom. Each cooling cycle involves three 100µs pulses of light, so the entire series takes 600 ms to complete. The first pulse consists of blue detuned 4−4′_{light, which is applied from the side of the cavity by pairs of} circularly polarized beams, together with Raman light, which has detuning −2ωa =−(2π)(1 MHz) and Rabi frequency Ω0 = (2π)(200 KHz). The 4−4′ _{light provides polarization gradient cooling} in the radial direction, and also acts in tandem with the Raman light to provide Raman cooling in the axial direction (it is the analog of the b−e beam in the three level model). The second pulse consists of Raman light alone, and the third pulse consists of 4−5′ _{probe light, which is} used to detect if the atom is still present in the cavity. The probe light heats the atom, so without the axial cooling provided by the Raman beam the atom would gradually be heated out of the FORT.

Unfortunately, because of the laser noise issues discussed in the previous section, the Raman beam does not drive coherent Rabi flops at the large Rabi frequencies needed for Raman cooling. However, even with the laser noise, the cooling should still be partially effective. For these initial cooling tests we do not worry about sweeping the Raman detuning to account for the anharmonicity of the well; rather, we keep it set at a fixed detuning for the entire cooling process.

To check that the Raman cooling was indeed cooling the atom, we measured the mean lifetime and the probability that the atom survived all 2,000 cooling cycles for several different values of the Raman detuning. Table 1.3 shows the results we obtained when we tuned the Raman beam to the red sideband, blue sideband, and carrier (−1 MHz, +1 MHz, and 0 MHz). The mean lifetime and survival probability are much larger when the Raman beam is tuned to the red sideband than when it is tuned to the carrier or to the blue sideband, which indicates that the Raman cooling is working as intended.

The idea of using the FORT itself as one leg of a Raman pair means that the FORT and Raman beams are perfectly registered, so the effective Rabi frequencies are the same for all the FORT wells.

Raman detuning [MHz] mean lifetime (# cycles) survival probability

+1.0 331 0%

0.0 443 2%

-1.0 1,317 36%

Table 1.3: Results of Raman cooling experiment.

This is very convenient for state preparation and for diagnostics such as Raman spectroscopy, but it is a disadvantage when it comes to Raman cooling because, as we discussed earlier, it causes a slowdown in the cooling rate. One could get around this by driving Raman transitions using a pair of Raman beams on a different cavity mode from the FORT. The resulting misregistration of the FORT and Raman beams would allow at least some of the FORT wells to have efficient cooling rates. The ideal cavity mode for the Raman beams would be the mode at exactly half the FORT wavelength (that is, mode number 180 at 468 nm), because at this wavelength there would be a node in the Raman beam at every antinode of the FORT beam. This would result in very efficient cooling for every FORT well.

As discussed in the previous section, we are planning to get rid of the FORT/Raman phase lock and to generate the Raman beam by using an EOM to add 9.2 GHz sidebands to the FORT beam. For this new setup, one could take the laser that currently generates the Raman beam, tune it to a different cavity mode, and put its output light through the same EOM. This way, the FORT and FORT sideband could be used to drive well-insensitive Raman transitions, and the second beam and its sideband could be used for efficient Raman cooling.