Sub-Doppler laser cooling of $^{39}$K via the 4S$\to$5P transition

We would like to thank the referee for his/her comments. Please find our responses below. 1. As suggested, we have included these points in the abstract.

1. We have changed the 6th sentence of the first paragraph of section 2.1 as follows to explicitly indicate that the Raman couplings are due to the lattice: “The frequency and polarizations of the lattice light beams are chosen such that there are significant Raman couplings due to the lattice light between the states that are brought into degeneracy by the magnetic field.”

2. In section 2.1, the idea is to describe degenerate Raman sideband cooling in a more general context using the example of 39K. We have made changes so that this is more apparent. In this section, we mention that the lattice light frequency is chosen to provide good Raman couplings and provide information that are specific to our experimental sequence, such as the lattice configuration and detuning, in Section 3.1.
   We use the same beam configuration as in Ref. [33], as mentioned in Section 3.1.

3. We have changed our sentence as follows to avoid confusion: “A $second$ $ \sigma^- $-polarized beam (so called ``repumper), which is tuned to the 4S$ _{1/2}$$ F=1 \rightarrow$4P$ _{1/2} $ $ {F=2} $ transition , depletes population in the $F=2 $ manifold"

4. As suggested, we now include the starting state |F=2, mF = -2, v> in our discussion in Section 2.1.

5. We agree with the referee. The order has been switched.

6. The broadening is not always detrimental. At the intensities used in our experiment, it makes the cooling scheme more robust as it allows appropriate Raman transitions to take place in a broader range of magnetic field values than what would be possible without broadening.

7. We have included this information in the caption.

8. We report all experimental parameters in section 3.1 for completeness. We have now included the lattice depth as well in this section. Lamb-Dicke parameters are also provided in section 3.2. The repumper intensity should be large enough to deplete the population in the F=1 manifold, which would otherwise result in heating. We include the powers and intensities as these are experimentally relevant parameters

9. Cooling takes place during the entire cycle, but largely during the ramping of the offset magnetic field to sweep through different vibrational levels.

10. The other spin components are imaged after Stern-Gerlach separation, as was also done in Ref. [33] to determine the fraction of atoms in the desired state. In our case, our total atom number was not enough to have a significant fraction in the other spin components. On the other hand, we do not think that the other spin components will limit the final temperatures since our samples are still quite dilute to allow for any significant interaction to take place.

11. Fig. 4 (b) is used to extract the lattice frequency required to obtain the mean vibrational quantum number plotted in Fig. 4 (a). As opposed to typical measurements [32], we observe a strong broadening of the modulation spectrum due to strong Raman couplings in our system. We give Fig 4 (b) as a separate figure to highlight these points.

12. In our experiments, off-resonant scattering of light due to the closely detuned lattice light limits the lifetime (the off-resonant scattering rate of the lattice being close to 4.6 kHz). We now mention this explicitly in the manuscript.

Responses to requested changes: 1. Section 1. Paragraph 2, “Doppler temperatures in optical lattices”

 -We have made this correction.

1. Section 2.2, Paragraph 4, “levels are 286 nm, 454 nm, ….”

 -Since we refer to $the$ ionization threshold, which is singular, we now write the sentence as: “The ionization threshold from the levels 4S$ _{1/2} $, 4P$ _{1/2} $, 4P$ _{3/2} $, 5S$_{1/2} $, 3D$ _{3/2} $, and 5P$ _{1/2} $ $is$ 286 nm, 454 nm, 455 nm, 715 nm, 742 nm and 972 nm away, respectively."

2. Section 3.2, Paragraph 1, “detunings of -9 MHz and 6.9 MHz” (unit was left off)

 -We have now included the units.

3. Section 3.2, Paragraph 2 “We estimate a systematic..”

 -We rewrite the sentence as “We estimate systematic errors less than 25% in our temperature and atom number measurements.”

We thank the referee for his/her valuable comments. Please find our responses below. 1. The Raman transitions do not lead to cooling by themselves. The Raman transitions are between degenerate energy levels. Cooling happens whenever the atoms travel down the non-degenerate vibrational levels in the F=2, mF=-1,-2 manifolds, which occurs only in combination with the polarizer. In principle, this can happen via the direct decay from the 5P1/2 level to the ground state or after a radiative cascade to the 4P levels from where a decay occurs to the ground state. When directly using the D1 line for the polarizer, only the F=1, mF=-1 state in the 4P1/2 is populated since we use a σ- polarized light field for the polarizer. This would mean an eventual dominant decay into the F=2, mF=-2 dark state of the ground state. However, the radiative cascade populates multiple Zeeman states of the F=1 hyperfine excited state, leading to a less dominant decay into the dark state. Hence, it is not equivalent to the cooling mechanism which occurs when using a polarizer on the D1 line. It is because of these reasons that we have used the terminology “cooling via the 4S-5P” transition.

1. We use bias fields values very close to those in Ref. [33]. We have mentioned this now in the first paragraph in Section 3.2.

2. The densities reached after our cooling procedure is of the order of a few $10^9 cm^{-3}$, which is still very dilute (for comparison, our BEC densities are of the order of $10^{13} cm^{-3}$). Hence, we do not expect multiple atoms to occupy the same lattice site. The question of thermal equilibrium between atoms in the same lattice site does not arise as a consequence. However, we do expect atoms to occupy vibrational levels in separate lattice sites commensurate with their kinetic energies. This assumption has previously been used to calculate $<\nu>$ for laser cooled ions, where the average <> can be over time or over an ensemble [21,48]. In our case, each experimental run involves the realization of an ensemble of atoms trapped in multiple lattice sites.

3. (a) First, we clarify that we have done two independent measurements: 1) Cooling: The polarizer intensity and detuning of 2.9 mW/cm$^2$ and 6.9 MHz, respectively, was the combination we obtained for the best cooling performance – in terms of temperature and atom number. The duration of our sequence was a 6.35 ms. 2) Fluorescence collection: As an independent measurement, we show that it is possible to collect blue fluorescence by carrying out the same experimental sequence but with the polarizer beam at 0.9 MHz detuning and an exposure time of 20.5 ms. The detuning and exposure time were chosen to maximize the fluorescence signal. At the optimum parameters for cooling, most of the atoms end up in the dark state quickly and do not produce a strong fluorescence signal. Since the measurements are independent, the optimum detuning and intensity used for cooling are not relevant for estimating the exposure times mentioned in the last paragraph of Section 3.2.
   At the optimal detuning for cooling (6.9 MHz), we found that the cooling performance increased with increasing polarizer intensity. We work at the maximum intensity we can achieve (2.9 mW/cm$^2$). It may be possible to achieve an improved cooling performance with an even higher polarizer intensity. However, limiting factors could be the radiation pressure due to the optical pumping, which can push the atoms out of the lattice [23]. A higher repumping rate may be required, which can affect the final temperatures achieved. Moreover, for a 1064-nm lattice, the 4P excited states are strongly anti-trapping for potassium. A large scattering rate of the optical pumping beams can hence lead to increased heating and diffusion in such lattices [16].

We clarified that the fluorescence measurement is independent and included our justifications in the last paragraph of Section 3.2.

1. (b) The difference between the two scattering rates is not as large as one would expect from a theoretical calculation since the value of 440 photons/atom/s is obtained from our measured scattering rate of 2.2 kHz, which is only a lower bound as mentioned in the manuscript (we believe that we underestimate our overall losses). The value of 192 photons/atom/s, on the other hand, is the detection rate calculated from the theoretical scattering rate and a NA of 0.8 at 4.9 MHz detuning. In the manuscript, we have now emphasized that the value of 440 photons/atoms/second is a lower bound.

2. (c) We have now provided both the intensity and power of the polarizer beam in the caption of Fig. 5.

We thank the referee for his/her valuable comments. Please see our responses to the requested changes below.
1-For the sake of clarity, as pointed out by the referee, we have changed Fig. 4 (a) to plot the temperature vs the cooling time in place of cloud size vs cooling time. This figure shows a direct measurement of the temperature of the cloud when all other parameters are tuned to their optimum values indicated in Fig. 3. Furthermore, we now mention in the manuscript that a temperature of 4 uK corresponds to cloud radius of about 1.4 mm after a fixed TOF expansion to aid the interpretation of the other figures.
2-We have added a vertical bar in the figure to indicate the value of 6.9 MHz for more clarity. At small detunings, cooling is strongly compromised due to the strong broadening and large light shift of the energy levels by near resonant light, for both red and blue detuned light (as reflected in the reduced atom number). For detunings more than +-5 MHz where there is reasonable atom number, we see a clear asymmetery between the red and blue detunings. We have now included this point in the manuscript.
3-As pointed out by the referee, long exposure time is not possible in near detuned lattices as used in our case. We now mention this explicitly in the manuscript, as this was also pointed out in Report 4, point 13.
On the other hand, changing to a far-detuned lattice will have different challenges. For example, in a 1064-nm lattice, the 4P excited states of K experience strong anti-trapping potentials. A large scattering rate from the optical pumping beams can hence lead to increased heating and diffusion in such lattices [16]. This issue has been tackled by keeping the optical pumping beams far away from resonance as in Ref. [16]. Another factor that needs to be considered would be the strong light shifts, which would arise when using the high intensities required to get good trapping conditions in far-detuned lattices. The differential light shift to the levels involved would need to be taken into account. The use of a ‘magic wavelength’ lattice, which provides the same trapping potential to the ground and excited state, may prove useful in this case. Furthermore, many existing non-degenerate Raman sideband cooling schemes require additional Raman beams to induce Raman transitions, requiring a suitable change of frequencies of both the polarizer and repumper light fields. We now discuss these points in the last paragraph of Section 3.2.
The main advantage of this scheme as compared to existing microscopes would be a significant release on the constraints on the numerical aperture provided by the shorter wavelength.
4-Figure 6 shows the 404.8 nm fluorescence collected from the atoms. We use this figure to experimentally determine the scattering rate. We think that from an experimental perspective, it is an important step towards high-resolution imaging with shorter wavelength transitions.
5-Since the level structure and cascade routes for potassium is similar to those in Na, Rb, Cs and Fr [39], sub-Doppler cooling schemes on the nS->(n+1)P transitions are likely to be possible for all these elements. We have added this comment to our conclusion.