Degenerate Raman sideband cooling of ⁴⁰K atoms

Ultracold atomic gases are used in a wide range of applications, from studying many-body physics [1, 2, 3], through quantum simulation and computation [4, 5, 6, 7], and all the way to precision metrology [8, 9, 10]. Most such experiments require several cooling stages before the atoms can be used for the intended purpose. Laser cooling and evaporative cooling are the working horses of this preparation procedure. The former has the advantage of being efficient, namely, being relatively fast and incurring only minor loss of atoms. A natural limitation of laser cooling is the recoil energy an atom acquires following an emission of a single photon. To mitigate this effect, the cooling rate has to be reduced as the cooling process progresses. One way to achieve this is to design the process such that atoms will accumulate in a low energy final state which is almost decoupled from all the cooling lights. The buildup of population in this final dark state achieves another desirable goal – spin polarization. The alternative route to preparing a sample in a well-defined internal state is by optical pumping, but it leads to heating due to spontaneous photon scattering. A cooling scheme that ends with all atoms being in a single dark state at a very low temperature solves this problem.

One of the most successful and widely used methods to approach and go below the recoil limit is to employ Raman transitions. Originally, the technique was developed in continuous space [11], with a series of velocity-selective Raman pulses. Later, it was adopted to a periodic potential, where it was called Raman sideband cooling (RSC) [12, 13]. In this scenario, the two Raman photons drive side-band transitions $n\rightarrow n-1$, where $n$ is the vibrational state in any given lattice site. Since the transition induced by a continuous Raman process is reversible, it cannot by itself lead to cooling. To break time-reversal symmetry, one can either make the Hamiltonian time-dependent (e.g., pulsing the Raman beams [11]), or introduce a dissipative irreversible process, such as spontaneous emission. In the case of Raman cooling in an optical lattice, the latter is achieved by optical pumping. In the Lamb-Dicke regime, the spontaneous emission is not likely to change the vibrational state. Thus, the cooling cycle is closed with some of the vibrational energy removed.

A particularly clever implementation of Raman sideband cooling uses the lattice light itself to drive the Raman transitions [14]. That means that the transition $n\rightarrow n-1$ needs to be degenerate, since the two Raman photons have the same frequency. Degenerate Raman sideband cooling (dRSC) was first realized with Cs atoms in a 1D geometry [14], and later extended to 3D [15, 16]. It achieved atomic densities that approach the lattice site density and a temperature of $1.5$T${}_{\text{rec}}$, where T${}_{\text{rec}}=\hbar^{2}k^{2}/mk_{B}$ is the recoil temperature. dRSC was successfully implemented with the bosonic isotopes ⁸⁷Rb [17, 18], ⁸⁵Rb [19], and ³⁹K [20]. To the best of our knowledge, dRSC was not implemented with fermionic atoms. Here, we report on a realization of dRSC with the fermionic isotope of potassium (⁴⁰K). Non-degenerate RSC was previously used to image single ⁴⁰K atoms in an optical lattice [21]. In our implementation, we start with a non-degenerate gas and optimize the scheme to achieve the highest phase space density and prepare the gas for further cooling.

Our complete cooling sequence begins with loading a 3D magneto-optical trap (MOT) with approximately $10^{7}$ atoms, followed by gray molasses cooling (GMC) [22, 23] on the D₁ line. The GMC stage cools the atoms to $\sim$$5\mu$K, while leaving them occupying all spin states of the $F=9/2$ ground state manifold. Then, we load the cloud into the Raman optical lattice, and within a few milliseconds of dRSC we achieve a spin polarized sample with a temperature as low as $1\mu$K, which is around 1.2T${}_{\text{rec}}$. The highest phase space density (PSD) we measure is around $10^{-3}$, two orders of magnitude higher than that of the cloud immediately after the GMC stage. The dRSC leaves over $80\%$ of the atoms in the $F=9/2,m_{F}=-9/2$ ground state, and the rest are at m${}_{F}=-7/2$ (see Fig. 2). These conditions significantly improve the starting point and efficiency of subsequent forced evaporation, which we perform in a crossed far-off-resonance dipole trap. Since identical ultracold fermions cannot interact through symmetric s-wave scattering, the remaining $20\%$ of the atoms in the $m_{F}=-7/2$ state are essential for fast thermalization and efficient evaporation [24, 25]. Moreover, a mixture of these two spin states is a natural choice for quantum simulation experiments since they are stable against spin-exchange collisions and offer a convenient Feshbach resonance around $202.14$G [26], which allows tuning the strength of their interaction [27]. It is also possible to change the spin composition by driving spin rotations with a radio-frequency radiation that matches the energy difference between the two spin states [28].

A sketch of the experimental setup is presented in Fig. 1. The 3D lattice is created by four beams: one beam, propagating along the $x$-axis in the sketch, is retro-reflected, while the two others, propagating along the $y$ and $z$ axes, are free running. All beams are linearly polarized; The retro-reflected beam is polarized at an angle of $45^{\circ}$ relative to the $z$-axis, as depicted in Fig. 1. This makes the dipole potential induced by the lattice isotropic in two out of three axes ($y$ and $z$). All polarizations of the lattice beams, as well as the magnetic field, lie in the same $y$-$z$ plane.

The source of the lattice beams is a distributed Bragg reflector (DBR) laser, whose frequency is red detuned $-11$ GHz with respect to the D₂ line. By deriving all lattice beams from the same source we ensure that relative phase shifts do not modify the lattice geometry, but only translate the entire lattice, which does not harm the cooling process [29, 20]. The lattice laser is amplified by a home-built tapered amplifier (TA) and then passed through an acousto-optic modulator (AOM), for fast intensity control. It is also passed through a temperature stabilized solid etalon, to filter out the broadband amplified spontaneous emission (ASE). The etalon has a free spectral range (FSR) of $60$ GHz and a bandwidth of $1.6$ GHz.

After the etalon, we split the light once using a $\lambda/2$ plate and a polarizing beam splitter (PBS), and then a second time using a non-polarizing beam splitter (NPBS). The $\lambda/2$ plate allows us control over the ratio between the power of the retro-reflected beam ($\hat{x})$ and the $\hat{y}$ and $\hat{z}$ beams, which is typically set such that the former has twice the power of the latter. This ratio was optimized experimentally at a fixed lattice detuning. All beams are then sent to the experiment via three $5$ m single-mode polarization maintaining optical fibers. We sample and monitor the power of one of the beams with a photodiode at the fiber output, and then use a servo loop and the AOM before the fibers to stabilize it. The total power of all three beams is kept at $100$ mW with a relative stability of $\sim 10$ ppm, which yields a peak intensity of $1590\,\text{mW}/\text{cm}^{2}$ ($795\,\text{mW}/\text{cm}^{2}$) and thus a calculated depth of 586$\mu$K (238$\mu$K) in the $\hat{x}$ ($\hat{y}$ and $\hat{z}$) direction. With these parameters, the calculated harmonic trapping frequencies in each of the lattice sites are $344$ kHz in the $\hat{x}$ direction and $138$ kHz in the $\hat{y},\;\hat{z}$ directions. The magnetic field is applied using two pairs of external Helmholtz coils, such that the total field points in the desired direction.

There are two optical pumping beams in our setup: m_F-pump, and F-pump. The m_F-pump is set as close as possible to perfect circular polarization. Due to a small angle between it and the magnetic field, it carries a strong (weak) $\sigma^{-}$ ($\pi$) component. The strong $\sigma^{-}$ component pumps the atoms to the $|F=\frac{9}{2},m_{F}=-\frac{7}{2}\rangle$ state. Because the entire process is performed in the Lamb-Dicke regime, that is, the Lamb-Dicke parameter $\eta=\sqrt{\frac{E_{R}}{\hbar\omega_{\textrm{lattice}}}}=0.31\;(0.49)$ for the $\hat{x}$ ($\hat{y}$ and $\hat{z}$) direction ($E_{R}$ is the photon recoil energy), this process is unlikely to change the vibrational state of the atom. This means that atoms undergo several cycles which consist of a Raman transition that lowers the vibrational state and an m_F-pump which resets the atoms to the m${}_{F}=-7/2$, until they reach the ground state. The weak $\pi$ component is used to pump the atoms to m${}_{F}=-9/2$, at which point they are decoupled from both the Raman and optical pumping processes. The second optical pumping beam (F-pump) is used to pump atoms which decay from the m_F-pumping process to the $F=7/2$ ground state manifold, which is not part of the cooling process. The F-pumping is performed through the $F=9/2$ state of the D₁ level. We send both optical pumping lights through the same optical fiber. The m_F-pump is locked to the D₁ line using a saturated absorption spectroscopy (SAS) setup [30], and the F-pump frequency is stabilized relative to the m_F-pump by monitoring their beat signal [31]. The required frequency shifts which bring each of the optical pumping lights to the desired detunings are achieved using several AOMs.

The full loading sequence is thus as follows. ⁴⁰K atoms are dispensed from home-built dispensers, then trapped and cooled in a 2D MOT [32, 33] and pushed into the main vacuum chamber. There, a 3D MOT captures and cools them further during $6$ seconds of loading time. Following the 3D MOT, the atoms are cooled for $8$ ms with a gray molasses D₁ cooling scheme [22, 23], during which the magnetic field is set to zero using three pairs of Helmholtz coils. We then turn off the D₁ cooling light and simultaneously ramp up in $0.8$ ms the Raman lattice power and the magnetic field. The magnetic field is ramped to $500$ mG using two of the three Helmholtz coils, such that it points in the desired direction, as depicted in Fig. 1. With the lattice and magnetic fields ready, we turn on the m_F-pump and F-pump beams. During optimization, we found that optimal cooling is achieved when the m_F-pump is linearly ramped up from a low initial intensity of $0.1$ mW/cm² to $2.1$ mW/cm² during the $10$ ms of dRSC. We discuss this observation in the results section.

After $10$ ms of cooling, we turn off the optical pumping lights and ramp down the lattice for another $0.8$ ms. The next step in our experiment is loading the cloud to a crossed dipole trap. We find no significant difference in cooling efficiency if the crossed trap is kept on during the GMC and dRSC processes, and this is indeed what we do. Once the lattice is ramped down, we capture more than $10^{6}$ atoms (12% of the atoms in the dRSC) in the crossed dipole trap. Since this process compresses the atoms in real space, and the phase space density is preserved, the temperature immediately after their loading is $7\mu$K. The results presented below, however, were taken without the crossed dipole trap, therefore they give a direct characterization of only the dRSC process.

To characterize the gas conditions, we measure the integrated density distribution of the cloud using absorption imaging, both in situ and after a time-of-flight of typically $30$ ms. We fit the distributions measured at the two times with a Gaussian, and from the change in their widths we extract the cloud temperature. In addition, integration over the density gives us the total number of atoms. We can then calculate the PSD given by $n\lambda_{\text{db}}^{3}$, where $n$ is the density and $\lambda_{\text{db}}$ is the thermal de-Broglie wavelength. We use the number of atoms, and the volume as approximated by a sphere of a radius $\frac{\sigma_{x}+\sigma_{y}}{2}$, where $\sigma_{i}$ is the in situ Gaussian width in direction $i$, to calculate $n$.

We now turn to characterize the dependence of the cooling sequence on various parameters. First, we calibrate the optimal magnetic field magnitude required for cooling. To this end, we fix all other parameters, scan the magnetic field, and extract the PSD. As can be seen in Fig. 3, The optimum fits the theoretical lattice frequency in the two isotropic directions. There is no distinct peak corresponding to the trapping frequency in the $\hat{x}$ direction because the Raman coupling in that direction is vanishingly small due to the standing wave [14].

The second parameter we scan to optimize the dRSC is the m_F-pump intensity, with the detuning fixed at $-8$ MHz. The result is presented in Fig. 4. Working with constant intensity, we observe an initial improvement of the PSD with increasing optical pumping intensity. The PSD saturates around $0.5\times 10^{-3}$ for intensity in the range of $0.5-1$ mW/cm². At even higher intensity, the PSD decreases. We attribute this behavior to the interplay between the Raman and optical pumping transition rates. When the latter is much higher than the former, transitions that increase the vibrational quantum number become too frequent, and atoms eventually become untrapped and leave the lattice. This is particularly detrimental when there is a substantial occupation of atoms at $m_{\text{F}}>-5/2$ states, as is the case immediately at the beginning of the dRSC process, following the GMC stage.

Instead, we find that it is beneficial to initiate the optical pumping at low intensity ($0.1$ mW/cm²) and then ramp it linearly to a higher value. Results following this procedure are shown in Fig. 4 versus the final intensity. With ramping we obtain a two-fold improvement in the final PSD. There are two plausible explanations for this effect. First, a higher efficiency of the optical pumping with the $\pi$ component at the final stages of the cooling process. Second, the linear ramp induces a smooth change of the light shift ($\Delta E_{\text{LS}}$ in Fig. 2), which is better suited to the non-uniform and anharmonic lattice potential. In what follows, we fix the intensity profile of the m_F-pump to linear ramping with a final intensity of $2.1$ mW/cm².

Next, we study the dependence of the cooling process on the m_F-pump detuning (Fig. 5). Previous implementations of dRSC with other atomic species found that it is advantageous to work with a blue detuned m_F-pump [15, 20]. In contrast, we observe that the dependence is almost symmetric with respect to the sign of the detuning, with slightly better results obtained when working with a red detuned light. As for the F-pump, we find that the detuning and power do not affect the final temperature, or increase the loss, as long as the intensity is above $4$ mW/cm² and the detuning is at least $-4$ MHz. This behavior is inline with previous results obtained with ³⁹K [20].

One of the advantages of dRSC is its ability to spin polarize the ensemble. To determine the final spin composition, we perform microwave (MW) spectroscopy after the cooling process is done. First, we load the atoms from the lattice to a crossed optical dipole trap. Then, we apply a MW field whose frequency we ramp over $50$kHz to compensate for magnetic field fluctuations. The MW field drives a ${|F=\frac{9}{2},m_{F}\rangle\rightarrow|F=\frac{7}{2},m_{F}^{\prime}\rangle}$ transition in the ground state manifold. The MW central frequency determines which $m_{F},m_{F}^{\prime}$ levels we address, based on the Zeeman splitting. Then, we turn on a quadruple magnetic field which traps only the atoms transferred by the MW pulse while the other atoms are ejected out [26]. Finally, we detect the atoms captured in the quadruple trap using MOT fluorescence. The inset of Fig. 6 shows an example of this spectroscopy, from which we extract the relative population in each state. In the main graph of Fig. 6 we present the spin composition for different durations of the dRSC. One can see that after a few milliseconds, over 95% of the atoms occupy the two m${}_{F}=-9/2,-7/2$ states. In about $10$ms, their fraction saturate towards $80\%$ and $20\%$, respectively. In an ideal dRSC, the m${}_{F}=-9/2$ population should ultimately approach unity. In practice, off-resonant excitations to the $F^{\prime}=9/2$ state, imperfect polarization of the $m_{F}$-pump, and single photon scattering from the Raman lattice beams contribute to the observed asymptotic spin composition.

Finally, we measure the lifetime of the atoms in the lattice. This measurement is done in the following way: the first $10$ ms are performed in the same manner as described before, including ramping the m_F-pump. For longer durations, we keep all lights and the magnetic field at a fixed value after the initial $10$ ms. The resulting number of atoms and temperature versus the total duration are shown in Fig. 7. The inset shows the PSD in the first $12$ ms. Based on this measurement, we choose the cooling time to be $10$ ms. We fit the number of atoms data with an exponentially decaying function and find a timescale of $\tau=22\pm 2$ ms, much shorter than the vacuum limited lifetime of $6.5\pm 0.5$ sec. Since the cooling achieves optimal conditions in $10$ ms, this lifetime does not harm the usefulness of the dRSC technique. We attribute the loss to the spatial Gaussian profile of the lattice beams, which gives rise to a spatially varying lattice depth and harmonic trapping frequency. Specifically, the lattice beams’ waist radius is around $1$ mm while the atomic cloud radius is around $0.5$ mm. This means that in the outer regions of the lattice the Raman degeneracy condition is not met and atoms are heated and lost. We calculate the heating rate due to single photon scattering by the lattice beams to be $\sim$$2\mu$K/ms, which is of the same order as the cooling rate. The loss rate could be reduced by increasing the lattice detuning and working with larger lattice beams. Both of these solutions, however, require considerably more laser power.

We have presented a first experimental implementation of dRSC with the fermionic isotope of potassium ⁴⁰K. The cooling scheme achieves a six fold reduction in temperature, compared to gray molasses cooling, and concurrently spin polarizes the sample. By ramping the m_F-pump light intensity up, we achieve a two-fold improvement in phase space densities achieved by this scheme when compared to using a constant intensity. The dRSC scheme greatly improves the loading conditions to a crossed dipole trap for subsequent evaporative cooling. Our implementation employs an optical lattice close to the D₂ line and optical pumping beams red detuned relative to the D₁ line. In the future, this separation of wavelengths can allow sensitive detection of the atoms position using the dRSC scheme and filtering out efficiently unwanted D₂ light scattering.