Direct laser cooling to Bose-Einstein condensation in a dipole trap

TL;DR

This paper demonstrates a laser cooling method to produce Bose-Einstein condensates in a dipole trap, enabling rapid quantum degenerate gas creation without evaporative cooling.

Contribution

The authors introduce a laser cooling technique that achieves Bose-Einstein condensation directly in a dipole trap, bypassing traditional evaporative cooling.

Findings

Condensation achieved with 2.5×10^4 atoms at 0.6 μK after 1.4 s of cooling.

Temperatures below the recoil limit were reached.

Careful tuning of trap parameters is essential at high densities.

Abstract

We present a method for producing three-dimensional Bose-Einstein condensates using only laser cooling. The phase transition to condensation is crossed with $2.5 {\times} 10^{4}$ $^{87}\mathrm{Rb}$ atoms at a temperature of $T_{\mathrm{c}} = 0.6\ \mu\mathrm{K}$ after 1.4 s of cooling. Atoms are trapped in a crossed optical dipole trap and cooled using Raman cooling with far-off-resonant optical pumping light to reduce atom loss and heating. The achieved temperatures are well below the effective recoil temperature. We find that during the final cooling stage at atomic densities above $10^{14}\ \mathrm{cm}^{-3}$, careful tuning of trap depth and optical-pumping rate is necessary to evade heating and loss mechanisms. The method may enable the fast production of quantum degenerate gases in a variety of systems including fermions.

Equations

$$\Delta K_{\mathrm{Raman}}$$

$$=\hslash {\delta }_{\mathbf{v}}-2E_{\mathrm{r}}$$

$$\Delta K_{\mathrm{total}}$$

$$=\hslash {\delta }_{\mathbf{v}}-8E_{\mathrm{r}}$$

$$K_{\Delta \mathbf{k}}$$

$${\sigma }_{\mathrm{reabs}}\approx 4\pi {\text{lambdabar}}^2\frac{\sqrt{\pi }{\Gamma }_{\mathrm{sc}}}{2\sqrt{2}{\omega }_{\mathrm{D}}}$$

$$p_{\mathrm{reabs}}\approx {\sigma }_{\mathrm{reabs}}⟨nl⟩$$

$$⟨nl⟩=\sqrt{\frac{\pi }{8}}n(0){\sigma }_z\frac{\mathrm{artanh}[\sqrt{1-1/{\epsilon }^2}]}{\sqrt{{\epsilon }^2-1}}$$

Full text

Quantum degenerate gases provide a flexible platform with applications ranging from quantum simulations of many-body interacting systems Bloch et al. (2012) to precision measurements Becker et al. (2018). The standard technique for achieving quantum degeneracy is laser cooling followed by evaporative cooling Ketterle and Druten (1996) in magnetic Anderson et al. (1995); Davis et al. (1995); DeMarco and Jin (1999) or optical traps Barrett et al. (2001). Evaporation is a powerful tool, but its performance depends strongly on atomic collisional properties and it requires removal of most of the initially trapped atoms. Laser cooling gases to degeneracy could alleviate those issues, but it has proven difficult to implement.

The elusiveness of laser cooling to Bose-Einstein condensation (BEC) Anderson et al. (1995); Davis et al. (1995) for more than two decades Aspect et al. (1988); Lee et al. (1996); Kerman et al. (2000); Han et al. (2000) can be understood as follows: optical cooling requires spontaneous photon scattering that moves entropy from the atomic system into the light field. Such scattering sets a natural atomic temperature scale $T_{\mathrm{r}}$ associated with the recoil momentum from a single photon of wavelength $2\pi\lambdabar$ and an associated atomic thermal de Broglie wavelength $\lambda_{\mathrm{dB}}=\sqrt{2\pi\hbar^{2}/(mk_{\mathrm{B}}T_{\mathrm{r}})}\sim\lambdabar$. Here $\hbar$ is the reduced Planck constant, $m$ the atomic mass, and $k_{\mathrm{B}}$ the Boltzmann constant. BEC must then be achieved at relatively high critical atomic density $n_{\mathrm{c}}\sim\lambda_{\mathrm{dB}}^{-3}\sim\lambdabar^{-3}$, where inelastic collisions result in heating and trap loss. In particular, light-induced collisional loss becomes severe when $n\gtrsim\lambdabar^{-3}$ Burnett et al. (1996); Ido et al. (2000).

For strontium atoms, where cooling on a spectrally narrow transition is available, a strongly-inhomogeneous trapping potential has been used to cool a lower-density thermal bath while decoupling the emerging condensate from the cooling light Stellmer et al. (2013). Very recently, based on similar principles to the ones presented here, an array of small, nearly one-dimensional condensates has been prepared using degenerate Raman sideband cooling Hamann et al. (1998); Vuletić et al. (1998); Han et al. (2000) of ${}^{87}\mathrm{Rb}$ atoms in a two-dimensional optical lattice Hu et al. (2017).

In this Letter, we demonstrate Raman cooling Kasevich and Chu (1992); Reichel et al. (1995); Lee et al. (1996); Perrin et al. (1999); Boyer et al. (2004) of an ensemble of ${}^{87}\mathrm{Rb}$ atoms into the quantum degenerate regime, without evaporative cooling. Starting with up to $1{\times}10^{5}$ atoms in an optical dipole trap, the transition to BEC is reached with up to $2.5{\times}10^{4}$ atoms within a cooling time of ${\sim}1\ \mathrm{s}$. As discussed in detail below, the essential components of our technique are (i) the use of carefully far-detuned cooling light to reduce atom loss and heating at high atomic densities Burnett et al. (1996); Hijmans et al. (1996); Rührig et al. (2015), (ii) a reduced optical pumping rate during the final stage to avoid heating by photon reabsorption (festina lente regime Cirac et al. (1996); Castin et al. (1998); Santos et al. (2000)), (iii) a final cooling of atoms in the high-energy wings of the thermal velocity distribution to achieve sub-recoil cooling, and (iv) careful control of the final trap depth to reduce heating induced by inelastic three-body collisions.

Raman cooling of the optically trapped atoms is a two-step process where kinetic energy is first removed via a stimulated two-photon Raman transition that simultaneously changes the internal atomic state. Subsequently, entropy is removed in an optical pumping process that restores the original atomic state via the spontaneous scattering of a photon [see Fig. 1(c)-(d)]. The optical pumping into the state $|5\mathrm{S}_{1/2};F{=}2,m_{F}{=}-2\rangle$ along the $z$-axis is performed with $\sigma^{-}$-polarized light. We reduce light-induced loss by using optical pumping light with large negative detuning $\Delta/(2\pi)=-4.33\ \mathrm{GHz}$ from the $|5\mathrm{S}_{1/2};F{=}2\rangle$ to $|5\mathrm{P}_{1/2};F^{\prime}{=}2\rangle$ transition of the $\mathrm{D_{1}}$-line, choosing a detuning far from molecular resonances [see Fig. 1(b-c) and Sup ]. The far-detuned $\sigma^{-}$-polarized beam and a $\pi$-polarized beam of similar detuning which propagates in the $x-y$ plane [see Fig. 1(a)], drive the stimulated Raman transition to the state $|5\mathrm{S}_{1/2};2,-1\rangle$, simultaneously changing the atomic momentum by the two-photon recoil $\hbar(\Delta\mathbf{k})$ Sup . To cool all three directions, we choose $\hbar(\Delta\mathbf{k})$ to have a non-zero projection along any trap eigenaxis.

Cooling is performed in several stages to allow optimization of the cooling as the atomic temperature and density change. Within each stage, the trapping beam powers, optical pumping rate $\Gamma_{\mathrm{sc}}$, Raman coupling Rabi frequency $\Omega_{\mathrm{R}}$, and Raman resonance detuning $\delta_{\mathrm{R}}$ are held constant. The first two cooling stages, S1 and S2, are performed in a single-beam optical dipole trap (sODT), after which the atoms are transferred to a crossed optical dipole trap (xODT), where we perform three more cooling stages [X1 to X3, see Fig. 2(a)]. During each stage, we characterize the cooling performance using time-of-flight absorption imaging, extracting the atom number $N$ and temperature $T$. (For the final cooling stage close to the BEC threshhold, we exclude the central part of the time-of-flight image from the temperature fit.) We quantify the cooling performance by the classical phase space density $\mathrm{PSD_{c}}=N\left(\hbar\bar{\omega}/(k_{\mathrm{B}}T)\right)^{3}$, where $\bar{\omega}=(\omega_{x}\omega_{y}\omega_{z})^{1/3}$ is the geometric mean of the three trapping frequencies. Far from degeneracy, i.e. for a classical gas, $\mathrm{PSD_{c}}$ is equal to the true PSD at the center of the cloud, $\mathrm{PSD}=n(0)\lambda_{\mathrm{dB}}^{3}$. The parameters of each stage are optimized to yield the highest $\mathrm{PSD_{c}}$ at the end of the stage Sup .

We load $1{\times}10^{5}\ \mathrm{atoms}$ from a magneto-optical trap into the sODT propagating along the $y$-direction with a $10\ \mathrm{\mu{}\mathrm{m}}$ waist [Fig. 1(a)]. After cooling in stage S1 (see Fig. 2), the trap power and vibration frequencies are reduced, thereby lowering the density and therefore the loss in stage S2. For all stages, we verify that the trap remains sufficiently deep to keep evaporative cooling negligible.

During stages S1 and S2 we perform fast cooling over $500\ \mathrm{ms}$ from $30\ \mathrm{\mu K}$ down to $1.5\ \mathrm{\mu K}$, and up to $\mathrm{PSD_{c}}$ just below unity. The larger loss rate during S1 relative to the other stages [see Fig. 2(c)] occurs because the Raman cooling cycle, and hence light-induced collision rate, is faster. The initial cooling at high temperatures $T{\approx 30}\ \mathrm{\mu K}$ and densities $n<7{\times}10^{13}\ \mathrm{cm^{-3}}$ is quite efficient, with a logarithmic slope of $\gamma=-d(\ln{\mathrm{PSD_{c}}})/d(\ln{N})=7.2$ [see Fig. 3(a)].

After stage S2, the ensemble is sufficiently cold so that it can be efficiently transferred in the xODT by ramping up the second trapping beam ($18\ \mathrm{\mu{}\mathrm{m}}$ waist, propagating along $x$), and applying a short initial cooling (stage X1). In a similar fashion to the S1-S2 sequence, we reduce the confinement of the xODT and cool further during X2, after which the ensemble is at the threshold to condensation. No condensate appears in X2 despite $\mathrm{PSD_{c}}$ reaching the ideal-gas value of 1.2 Dalfovo et al. (1999), which we attribute to a combination of the finite size effect Ketterle and van Druten (1996), the interaction shift Smith et al. (2011) and small calibration errors. After further reduction of the xODT, we are able to cross the BEC transition during X3, as shown by the appearance of a condensed fraction in the velocity distribution [see Fig. 3(b)]. The onset of BEC is further confirmed by the anisotropic expansion of the central part of the cloud due to trap confinement anisotropy [see Fig. 3(c)].

In order to achieve BEC, the trap depth during X3 must not be too large. For trap depths much larger than ${\sim}k_{B}\times 20\ \mathrm{\mu{}K}$ we observe a strong, density-dependent anomalous heating when the Raman cooling is turned off. (Heating rates of up to $10\ \mathrm{\mu{}K/s}$ are observed at $n\approx 2{\times}10^{14}\ \mathrm{cm^{-3}}$ with a trap depth of $k_{B}\times 250\ \mathrm{\mu{}K}$.) We surmise that at high atomic densities $n\gtrsim 10^{14}\ \mathrm{cm^{-3}}$, recombination products of inelastic three-body collisions undergo grazing collisions with trapped atoms, depositing heat in the cloud Sup . This limit on the maximum trap depth is akin to the necessity to maintain, even in the absence of evaporative cooling, a sufficiently low trap depth by applying a so-called “RF-shield” in magnetic traps which allows highly energetic nonthermal atoms to escape Mewes et al. (1996); Burt et al. (1997).

The BEC transition is crossed with $N{\approx 2.5}{\times}10^{4}\ \mathrm{atoms}$ at a critical temperature of $T_{\mathrm{c}}{=0.61(4)}\ \mathrm{\mu K}$. We are able to reach condensate fractions $N_{0}/N$ up to $7\%$. We also verify that if the Raman cooling is turned off during stage X3, the PSD does not increase, and a condensate does not appear. Furthermore the condensate can be maintained for ${\sim}1\ \mathrm{s}$ after creation if the cooling is left on, but if the cooling is turned off, the condensate decays within ${\sim}100\ \mathrm{ms}$. This confirms that evaporation is insufficient to create or maintain a condensate in this trap configuration, and that the laser cooling is responsible for inducing the phase transition.

For most laser cooling methods, the requisite spontaneous photon scattering sets a recoil temperature limit. Nonetheless, we achieve sub-recoil temperatures by addressing atoms in the high-energy wings of the thermal distribution. The optical pumping $|2,-1\rangle{\rightarrow}|2,-2\rangle$ requires on average the spontaneous scattering of three photons, and therefore imparts $6E_{\mathrm{r}}$ of energy, where $E_{\mathrm{r}}=\hbar^{2}/(2m\lambdabar^{2})=h{\times 3.6}\ \mathrm{kHz}$ is the recoil energy of a $795\ \mathrm{nm}$ photon. As a result, only atoms with $K_{\Delta\mathbf{k}}/h\gtrsim 29\ \mathrm{kHz}$ of kinetic energy along the $\Delta\mathbf{k}$-direction can be cooled at all Sup . This sets an effective recoil temperature $T_{\mathrm{r}}^{\mathrm{eff}}=2.8\ \mathrm{\mu K}$. We achieve cooling below this effective recoil limit down to $0.5\ \mathrm{\mu{}K}$, i.e. a mean kinetic energy $\langle K_{\Delta\mathbf{k}}\rangle/h=5.2\ \mathrm{kHz}$, by detuning the Raman coupling so that atoms with more than the average kinetic energy are addressed by the cooling light Sup . However, this slows down the cooling for temperatures below $T_{\mathrm{r}}^{\mathrm{eff}}$ (stage S2 onwards), while inelastic collisions add an increased heat load at high densities. In X3 when we cross $T_{\mathrm{c}}$, we find that under optimized cooling the Raman transition removes $15\ \mathrm{kHz}$ of kinetic energy, 30% less than the expected $6E_{\mathrm{r}}$ of heating (see SM Sup ). This could indicate that the cooling is aided by bosonic stimulation into the condensate during the photon scattering process, in a similar fashion to Ref. Robins et al. (2008).

The improved performance of our scheme compared to previous Raman cooling results Lee et al. (1996); Perrin et al. (1999) is primarily due to the flexibility to perform optical pumping to a dark state at large detuning from atomic resonances by operating on the $\mathrm{D_{1}}$-line. To identify suitable detunings, we separately characterized light-induced losses over a $16\ \mathrm{GHz}$ frequency range to the red of the bare atomic transition, as shown in Fig. 4(a), and further detailed in the SM Sup . Fig. 4(b) displays the final atom number and condensate fraction of the optimized sequence as a function of detuning around the value of $-4.33\ \mathrm{GHz}$ chosen for the experiment. We find that a suitable detuning has to be maintained within $\pm 50$ MHz to ensure good cooling performance.

Another parameter that needs to be optimized is the photon scattering rate $\Gamma_{\mathrm{sc}}$ for optical pumping into the $|2,-2\rangle$ dark state. Despite the large detuning, the reabsorption of spontaneously scattered optical pumping photons by other atoms is a resonant two-photon process that can lead to excess heating. However, it was shown theoretically Cirac et al. (1996); Castin et al. (1998), and confirmed experimentally Wolf et al. (2000), that the excess heating can be suppressed at sufficiently low scattering rate $\Gamma_{\mathrm{sc}}$, such that the confinement and two-photon Doppler broadening reduce the reabsorption probability. This limit is known as the festina lente regime. The degradation of the performance at larger $\Gamma_{\mathrm{sc}}$ in Fig. 4(c) is consistent with increased rescattering, as the calculated reabsorption probability approaches unity Sup . A too small value of $\Gamma_{\mathrm{sc}}$, on the other hand, leads to higher temperatures as parasitic heating mechanisms cannot be compensated when the cooling is too slow.

While ${}^{87}\mathrm{Rb}$ has relatively favorable collision properties (low two-body inelastic loss rate coefficient, and moderate three-body loss rate coefficient in the upper hyperfine manifold), these properties are not unique, and other atomic species may also be suitable for direct laser cooling to BEC. Since the cooling is not deeply subrecoil, relatively high densities $n\sim\lambdabar^{-3}$ are required for reaching BEC. Thanks to the fast cooling, the effect of inelastic loss is small enough if a cloud is stable at these densities (typ. lifetime $\gtrsim 1\ \mathrm{s}$ at $10^{14}\ \mathrm{cm^{-3}}$). Inelastic processes can be further reduced in an effectively one-dimensional geometry Hu et al. (2017), where fermionization of the bosonic wavefunction reduces collisional processes. The demonstrated technique could also be directly applied to fermionic atoms DeMarco and Jin (1999), as well as to laser cooled molecules Barry et al. (2014); Anderegg et al. (2018).

In conclusion, we have realized the decades-old goal of BEC purely by laser cooling by creating a single, moderately sized Bose-Einstein condensate in a standard crossed optical dipole trap. Notably, the method is consistent with the general theoretical recipe put forward by Santos et al. (2000). Further work is needed to explore the limits of this new technique in terms of speed, condensate size and final temperature. [We observe that the performance is still density-limited, see Fig. 3(a), and verified that when allowing for moderate final evaporation, nearly pure condensates can be created within a cooling time of $1\ \mathrm{s}$.] It may also be interesting to investigate if this technique can be used to experimentally realize an atom laser, where the condensate is created by bosonic stimulation into the atomic final state during the spontaneous photon scattering Spreeuw et al. (1995); Olshanii et al. (1996), rather than through thermalizing elastic collisions.

I Optical Pumping Detuning

To find an optical pumping detuning that avoids losses from molecular resonances as predicted in Refs. Burnett et al. (1996); Hijmans et al. (1996), we start with a cold cloud of atoms and measure the loss spectrum below threshold (i.e. bare atomic resonance) for light-atom detunings $\Delta$ above $-18\ \mathrm{GHz}$. This is closer to resonance than previously explored experimentally on the $\mathrm{D_{1}}$-line of ${}^{87}\mathrm{Rb}$ Jelassi et al. (2006a, b); Jelassi and Pruvost (2014). After executing a sequence which scatters the same number of photons, about 100, regardless of detuning, we obtain a fraction of surviving atoms as given in Fig. 4(a) of the main text, with frequency regions where the remaining atom number is relatively large shown in Fig. S1. The scan is performed with a resolution of $10\ \mathrm{MHz}$. The optimum around $-4.33\ \mathrm{GHz}$ in Fig. S1(b) was chosen due to its good performance. Other optima could be used as well, and better optima may exist outside the range of this scan.

II Experimental details

The cloud is prepared in a similar way as in Ref. Hu et al. (2017). Rb atoms are loaded into a MOT from a thermal vapor, followed by a compressed MOT stage where the optical pumping power is strongly reduced, such that the atoms occupy the $F=1$ ground state manifold. The ODT is turned on at all times and about $1{\times}10^{5}\ \mathrm{atoms}$ are loaded into it when the MOT fields are switched off.

The ODT propagating in the $y$-direction (see Fig. 1 in the main text), in which the atoms are originally loaded, is focused to a waist of about $10\ \mathrm{\mu{}\mathrm{m}}$, while the second ODT propagating in the $x$-direction has a waist of about $18\ \mathrm{\mu{}\mathrm{m}}$ at the position of the atoms. To avoid interference, the beams differ in frequency by $160\ \mathrm{MHz}$. The powers of each trapping beam throughout the sequence are shown in Fig. S2. The (calculated) total trap depths, excluding the influence of gravity but including counter-rotating terms, are $430\ \mathrm{\mu{}K}$ in stage S1 and $14\ \mathrm{\mu{}K}$ in stage X3.

The imaging axis is the same as that of the $\pi$ beam, which is slightly rotated by $\approx 17^{\circ}$ from the $x$-axis in the $x-y$ plane. In Fig. 3(b)-(c), we denote the horizontal axis of the image as $y^{\prime}$, which is rotated from the $y$-axis in the $x-y$ plane by the same angle.

The $\sigma^{-}$ optical pumping beam at $795\ \mathrm{nm}$ has a highly elliptical shape at the position of the atoms, with waists of $30\ \mathrm{\mu{}\mathrm{m}}$ along the $x$-direction and $\sim 1\ \mathrm{mm}$ along the $y$-direction, to optimally address atoms along the sODT. The pumping beam creates a sizable light shift $\delta_{\mathrm{LS}}$ of the $|2,-1\rangle$ state, given by $\delta_{\mathrm{LS}}/\Gamma_{\mathrm{sc}}=\Delta/\Gamma$, where $\Gamma$ is the natural linewidth of the $5\mathrm{P}_{1/2}$ excited state. Since the state $|2,-2\rangle$ is dark for the $\sigma^{-}$ light, there is no appreciable light shift on this state. The light shift is at its largest in stage S1, reaching $\delta_{\mathrm{LS}}/(2\pi)=500\ \mathrm{kHz}$. It is determined experimentally by measuring the shift in the Raman resonance. From the measured values of $\delta_{\mathrm{LS}}$ we deduce the scattering rates $\Gamma_{\mathrm{sc}}$ for $|2,-1\rangle$, shown in Fig. S2. Since the detuning $\Delta$ is comparable to the hyperfine splitting of the ground state, the same light also pumps atoms out of the $|5\mathrm{S}_{1/2};F{=}1\rangle$ manifold.

The $\pi$ Raman beam propagates in the $x-y$ plane (see Fig. 1 in the main text), with a waist of $0.5\ \mathrm{mm}$. Its light is derived from the same laser that generates the $\sigma^{-}$ beam, but it is detuned by $2\ \mathrm{MHz}$ from the $\sigma^{-}$ using acousto-optic modulators to avoid interference. This makes laser frequency noise common mode between the two beams, thereby loosening the requirements on laser linewidth necessary to have a narrow Raman transition. The Raman beam follows the same path as the light used for absorption imaging, and therefore it is circularly polarized. Over the course of the cooling only its $\pi$-polarized component (polarization along the $z$-axis, i.e. half its power), contributes to the Raman coupling. The other half of the power only adds a negligible amount of scattering and light shift to the $|2,-2\rangle$ state. Very little power ($\lesssim 100\ \mathrm{\mu{}W}$) is required to obtain significant Raman coupling (several kHz, see Fig. S2), so the light shift and scattering rate induced on the $|2,-2\rangle$ state are limited to $\lesssim 0.3\ \mathrm{kHz}$ and $\lesssim 2\ \mathrm{s^{-1}}$, respectively, even at the largest powers used in the cooling sequence.

For all the data presented, each data point is evaluated as an average of 3 to 5 time-of-flight absorption images.

III Effective recoil limit

At trap frequencies in the range $0.1{-}5\ \mathrm{kHz}$, smaller than the Raman coupling, the cooling operates mostly in the free space limit with unresolved motional sidebands and outside the Lamb-Dicke regime Kasevich and Chu (1992); Lee et al. (1996); Perrin et al. (1999). Each Raman transition from $|2,-2\rangle$ to $|2,-1\rangle$ transfers $-\hbar\Delta\mathbf{k}$ of momentum to the atoms, where $\Delta\mathbf{k}$ is the difference between the wave vectors of the $\sigma^{-}$ and $\pi$ photons. The kinetic energy removed during one Raman transition for an atom of initial velocity $\mathbf{v}$ is:

[TABLE]

where $\delta_{\mathbf{v}}=\Delta\mathbf{k}\cdot\mathbf{v}$ is the atom’s two-photon Doppler shift. The orthogonality of the $\pi$ and $\sigma^{-}$ beams results in $|\Delta\mathbf{k}|^{2}=2|\lambdabar|^{-2}$, and $E_{\mathrm{r}}=\hbar^{2}/(2m\lambdabar^{2})$ is the recoil energy of a $795\ \mathrm{nm}$ photon. The branching ratio of the optical pumping to the dark state $|2,-2\rangle$ is $1/3$, which results in the scattering of an average of 3 $\sigma^{-}$ photons to optically pump the atoms following a Raman transition. In the limit of low scattering rate ($\Gamma_{\mathrm{sc}}\ll\omega_{x,y,z}$, which is required to limit reabsorption heating, see below), only the recoil energies of the scattered $\sigma^{-}$ photons add, on top of those of the spontaneously emitted photons, resulting in an average net recoil heating of $6E_{\mathrm{r}}$ per optical pumping cycle. The average net energy removed per cooling cycle is then given by:

[TABLE]

Therefore there is net cooling if and only if the two-photon Doppler shift is larger than $8E_{\mathrm{r}}$, which translates to a kinetic energy in the $\Delta\mathbf{k}$ direction of:

[TABLE]

As a result, for temperatures below $T_{\mathrm{r}}^{\mathrm{eff}}=2.8\ \mathrm{\mu{}K}$ ($\langle K_{\Delta\mathbf{k}}\rangle=\frac{1}{2}k_{\mathrm{B}}T_{\mathrm{r}}^{\mathrm{eff}}$ along $\Delta\mathbf{k}$), the cooling speed drops since only a small fraction of the atoms have Doppler shifts above $8E_{\mathrm{r}}$ where they can be cooled.

IV Optimization of the cooling and limiting factors

For each stage the Raman transitions are tuned to a particular velocity class by adjusting the magnetic field, which differentially shifts the $|2,-2\rangle$ and $|2,-1\rangle$ states. In stage S1, where the mean Doppler shift is much larger than the $8E_{\mathrm{r}}$ limit, we find as expected that the optimal Raman detuning is near the rms Doppler shift, see Fig. S3(a). This is a compromise between removing a large amount of energy per cooling cycle ($\Delta K_{\mathrm{total}}\approx\hbar\delta_{\mathrm{R}}$) and having a sufficiently large probability of finding atoms at those velocities (which drops as $\delta_{\mathrm{R}}$ increases). In stage X3 the optimization yields an optimal resonant Doppler shift of $\delta_{\mathrm{R}}/(2\pi)=29\ \mathrm{kHz}$. As shown in Fig. S3(b), the probability of finding an atom at these velocities is very small, and the amount of energy removed $\Delta K_{\mathrm{total}}$ also becomes very small. By averaging $\Delta K_{\mathrm{Raman}}$ over the actual distribution of addressed atoms [green shaded in Fig. S3(b)] in stage X3, we obtain $\langle\Delta K_{\mathrm{Raman}}\rangle/h=15\ \mathrm{kHz}$, well below the recoil heating of $6E_{\mathrm{r}}/h=22\ \mathrm{kHz}$, which should lead to heating. Yet cooling is observed, which suggests bosonic enhancement of the branching ratio into the $|2,-2\rangle$ state due to the emerging condensate. A better branching ratio would reduce the average recoil heating during optical pumping, and so cooling could be achieved even when the Raman transition removes less than $6E_{\mathrm{r}}$ of kinetic energy.

At each stage, the strength of the Raman coupling $\Omega_{\mathrm{R}}$ is optimized by scanning the power of the $\pi$ beam. Too small $\Omega_{\mathrm{R}}$ lead to a narrow excitation profile, and therefore only a small fraction of the atoms undergo a Raman transition. However, if $\Omega_{\mathrm{R}}$ is too large, already “cold” atoms undergo a Raman transition, due to the broadened excitation profile, and are heated during the optical pumping.

Initially in the sODT it is favorable to have a fairly large scattering rate for cooling speed. The smaller number of atoms available to cool in the Boltzmann tail above the recoil energy in the later stages results in a lower optimal scattering rate as it becomes more favorable to decrease reabsorption heating as seen in Fig. 4(c) and discussed below.

For the ODT powers, the main considerations are finding a balance between low density in order to limit inelastic loss and heating, and maintaining large enough trapping frequencies and therefore critical temperatures. Additionally in the final stage, having a low trap depth has proven crucial to avoid an observed density-dependent heating that increases with trap depth. As pointed out in Ref. Schuster et al. (2001), the products of three-body recombination of ${}^{87}\mathrm{Rb}$ atoms in the $|2,-2\rangle$ state to the least-bound molecular state ($h{\times}24\ \mathrm{MHz}$ of binding energy Wolf et al. (2017)) can collide with the cold sample before they leave the trap. This was shown to lead to large loss for a collisionally opaque ensemble. In our case, the sample is not collisionally opaque (collision probability $\sim 0.1$ for $s$-wave scattering only), nor is the trap deep enough to directly hold recombination products which would dump their energy into the cloud. The presence of a $d$-wave shape resonance at the energy of the least-bound state for a ${}^{87}\mathrm{Rb}_{2}$ molecule is expected to enhance the collisional cross-section of recombination products with the trapped atoms, which could lead to strong losses in the collisionally thick regime. However, we mostly observe heating, which we suspect arises from recombination products undergoing grazing collisions with trapped atoms, with the latter remaining trapped and depositing heat into the cloud. We found that lowering the trap power as much as possible during the final stage X3 to minimize heating was necessary to reach condensation.

The optimized values of the relevant parameters throughout the sequence, namely trap power, Raman detuning $\delta_{\mathrm{R}}$, Raman coupling $\Omega_{\mathrm{R}}$ and scattering rate $\Gamma_{\mathrm{sc}}$, are shown in Fig. S2 for reference.

V Estimation of the reabsorption probability

The reabsorption of a scattered photon causes excess recoil heating. This is especially of concern since the reabsorption cross-section, corresponding to a two-photon resonant process, can take on its maximum possible value $\sim 6\pi\lambdabar^{2}$. Several strategies have been laid out for suppressing this Cirac et al. (1996); Castin et al. (1998); Santos et al. (2000), boiling down to the use of a low scattering rate $\Gamma_{\mathrm{sc}}$ for optical pumping, a regime known as festina lente. With our optimal parameters, we have $\Gamma_{\mathrm{sc}}\ll\omega_{\mathrm{D}},\omega_{x,y,z}$, leading to a suppression of the reabsorption process on the order of $\Gamma_{\mathrm{sc}}/\omega_{\mathrm{D}}$, where $\omega_{\mathrm{D}}$ represents the Doppler width. The reabsorption cross-section $\sigma_{\mathrm{reabs}}$ is given by Castin et al. (1998):

[TABLE]

and the reabsorption probability $p_{\mathrm{reabs}}$ is given by:

[TABLE]

where $\langle nl\rangle$ is the mean column density of the cloud. We use the following formula for the mean column density of a thermal cloud, taken from Schuster et al. (2001):

[TABLE]

where $n(0)$ is the peak density, $\sigma_{z}$ is the cloud’s waist along the short axis of an elongated trap ($z$-axis here), and $\varepsilon=\omega_{y}/\omega_{z}$ is the aspect ratio of the trap. Assuming a classical cloud at the condensation point and a scattering rate $\Gamma_{\mathrm{sc}}=0.59\ \mathrm{ms^{-1}}$, we obtain that a probability for reabsorption $p_{\mathrm{reabs}}\sim 0.1$. Hence reabsorption is not expected to be a significant factor, but it could explain why the performance deteriorates at larger $\Gamma_{\mathrm{sc}}$ in Fig. 4(c) of the main text, since $p_{\mathrm{reabs}}$ varies linearly with $\Gamma_{\mathrm{sc}}$.