Loading and cooling in an optical trap via hyperfine dark states

D. S. Naik,1,* H. Eneriz-Imaz,1 M. Carey ,1,2 T. Freegarde,2 F. Minardi ,3,4 B. Battelier,1 P. Bouyer,1 and A. Bertoldi 1,† 1LP2N, Laboratoire Photonique, Numérique et Nanosciences, Université Bordeaux-IOGS-CNRS:UMR 5298, F-33400 Talence, France 2School of Physics & Astronomy, University of Southampton, Highfield, Southampton SO17 1BJ, United Kingdom 3Istituto Nazionale di Ottica, INO-CNR, Sesto Fiorentino 50019, Italy 4Dipartimento di Fisica e Astronomia, Università di Bologna, Bologna 40127, Italy

We present a novel optical cooling scheme capable of loading and cooling atoms directly inside deep optical dipole traps utilizing hyperfine dark states. In the presence of strong light shifts of the upper excited states, this allows the velocity selective dark-state cooling of atoms into the conservative potential without loss of atoms. We report the lossless optical cooling inside the trap with a seven-fold increase in the number of atoms loaded. Our findings open the door to all-optical cooling of trapped atoms and molecules which lack the closed cycling transitions normally needed to achieve low temperatures and the high initial densities required for evaporative cooling.
Ultra-cold quantum gases have attracted much interest in recent decades as versatile platforms for investigating strongly correlated quantum systems [1] and as the basis for a new class of quantum technologies based on atomic interferometry [2,3]. Cooling of an atomic gas to the required temperatures requires a multi-stage cooling process, beginning with laser cooling in a magneto-optical trap (MOT), followed by sub-Doppler cooling, loading into a conservative magnetic or optical dipole trap, and finally by evaporative cooling. Although quite efficient, this is only possible for a small subset of alkali and alkaliearth-like atoms that can be initially cooled to low temperature by optical means.
Indeed, reaching these temperatures relies on sub-Doppler cooling, which utilizes light red detuned from the F → F = F + 1 cycling transition of the D 2 line nS 1/2 → nP 3/2 with particular polarizations [4]. At its heart is a Sisyphus-like cooling effect [5][6][7][8] in which atoms climb potential hills created by polarization gradients, thereby losing kinetic energy before being optically pumped to a lower energy level. Sub-Doppler cooling schemes based on the same polarization gradient potentials but now coupled to dark states [9] have emerged as a powerful alternative to standard Sisyphus cooling; they are known as gray molasses. Very recently they have been pivotal in obtaining an all-optical BEC in microgravity [10] and a degenerate Fermi gas of polar molecules [11]. The dark states decoupled from the optical field are coherent superpositions of internal states and external momentum states for the atoms; their creation does not require cycling F → F = F + 1 transitions, but can rely on any transitions of the F → F ≤ F form.
To prevent expansion of the cold atom cloud, it is tempting to combine dark-state sub-Doppler cooling with * devang.naik@institutoptique.fr † andrea.bertoldi@institutoptique.fr spatial confinement in a far-off resonant optical dipole trap (FORT). However, standard dark states composed by superposing different momenta [12] are eigenstates of the free Hamiltonian but not of the Hamiltonian in the trapped case. This leads to a finite lifetime of the dark states and their eventual coupling to the light field, i.e. excitation and heating, while the spatial dependence of the trapping potential complicates Sisyphus-like cooling mechanisms. We propose a novel scheme not subject to these limitations enabling sub-Doppler cooling via hyperfine dark states (HDSs) to be implemented directly inside a deep FORT, leading to an order of magnitude improvement in the number of atoms.
In addition, we explore the possibility of using the HDSs to cool atoms directly inside the trap. Our telecom wavelength FORT at 1560 nm is close to resonance with transitions to the rubidium excited state 5P 3/2 → 4D 3/2;5/2 at 1529 nm [13,14]. The resulting light shift breaks the degeneracy of the m F magnetic sublevels of the 5P 3/2 excited state, rendering the Sisyphus-like cooling effect inefficient [15] and preventing the formation of certain classes of dark states. We show that the HDSs are not only resilient to the presence of excited state light shifts, but can exploit them to implement a velocity sensitive optical cooling scheme immune to excited state effects.

I. HYPERFINE DARK STATES
The first experiments using gray molasses were based on the creation of Zeeman dark states (ZDSs) that use counter-propagating circularly polarized light on F → F ≤ F transitions, and blue-detuned lasers so that the bright states, due to their positive light shift, have energy always higher than the unshifted dark states. The commonly used F = 2 → F = 2 atomic transition, for example, can lead to a M type dark state, i.e. a linear superposition of F = 2 Zeeman sublevels with different momenta |m; p of the form [12]: (1) where q indicates the mean momentum and k the wavevector of one of the two counterpropagating laser beams while (F, m F ) denotes the relevant manifold. The dipole matrix elements coupling |ZDS(q) to any excited state |F = 2, m ; q vanish, making that state effectively dark ( p being quantized momentum states centered around the momentum p).
However, |ZDS(q) is not an eigenstate of the kinetic energy operator, and free evolution turns it into a bright, absorbing state. In addition, the presence of a repumper laser, essential to counter off-resonant optical pumping into the F = 1 hyperfine manifold and keep atoms resonant with the F = 2 → F = 2 transition, spoils these ZDSs, limiting the cooling efficacy [16,17].
A striking effect arises, however, when the repump laser is resonant with the Raman transition to the F = 1 manifold: a broad range of dark states is created between the different ground state hyperfine manifolds. These HDSs have been rigorously studied in the context of coherent population trapping and electromagnetically induced transparency [9,18,19], however their role in gray molasses cooling deserves further investigation.
Three types of dark states can in principle exist in these systems: (i) those whose darkness depends only on the laser frequencies matching the Raman condition (occurring for N g = N e + 1 and N g ≥ 2, N e = 1, where N g,e are the number of degenerate Zeeman levels) [9], (ii) those, for the more general case of N g ≤ N e , whose darkness requires the amplitude of the various Rabi frequencies to cancel [9,12], and (iii) those in the latter category whose diagram of all possible transitions form a closed loop: for these specific dark states, the Rabi frequencies constraints imply that the phases of the complex Rabi frequencies around the loop sum to an integer multiple of 2π [9,20,21].
In our configuration, retro-reflecting pairs of lasers tuned to the cooling F = 2 → F = 2 and repumping F = 1 → F = 2 transitions in the σ + ⊥ σ − polarization condition impinge on atoms at zero magnetic field. Dark states of the class (ii) and (iii) can in principle exist, with the phase conditions relevant for class (iii) being satisfied via our use of EOM sidebands to create the repumper. The 1560 nm trapping beam puts further restrictions on the actual dark states present, because the induced vector light shifts break the degeneracy of the excited Zeeman states |F = 2, m . For this reason, we used a numerical approach to ascertain the exact dark states involved (see Supplemental Materials [22]).
Unlike ZDSs, HDSs can exist in forms that are also eigenstates of the kinetic energy operator and, thus, remain dark even during free evolution. Since these truly dark states have null momentum, cooling occurs due to the accumulation in low momenta states typical of coherent population trapping.

II. LIGHT SHIFTS
The light shifts are due to a resonance of the 5P 3/2 to the higher lying excited states 4D 3/2,5/2 at 1529 nm and are shown in Fig. 1(a) [23]. The moderate detuning of the trapping laser with respect to the 5P → 4D transitions, compared with the 5S → 5P , leads to a scalar polarizability of the excited 5P 3/2 level greater than that of the ground level by a factor of α e /α g = 42.6, and in turn to large light shifts of the excited level compared to the natural linewidth. Moreover, the excited-level light shifts are m F -dependent, complicating not only the mechanisms for sub-Doppler cooling, but the formation of dark states themselves. This can be seen by examining the frequency shift of the excited state m F levels, as shown in Fig. 1(b). HDSs relying on transitions that experience the same excited state light shifts (symmetric transitions such as m F = ±1 or m F = ±2) are still formed, but non-symmetric combinations such as m F = 2, 0 or m F = −2, 0 involve differential excited state light shifts that prevent the dark states from forming.
III. EXPERIMENTAL SCHEME 87 Rb atoms are loaded and cooled in the center of an optical cavity via a 2D/3D MOT setup. A twodimensional MOT provides a source of cold atoms, with a flux of 5 × 10 8 s −1 , directly into the cavity crossing region where we create a 3D MOT and load atoms into the resonator-enhanced FORT [24]. During the entire MOT stage, the FORT trap depth is maintained at ∼27 µK. After 2 s of MOT loading, a compressed MOT phase (CMOT) to increase the atomic density is realized by de-tuning the MOT beams to -6 Γ from the F = 2 → F = 3 atomic resonance for 40 ms and decreasing the optical power by a factor 10. Throughout this process, the MOT region overlapping the FORT remains effectively dark to the atoms due to the large excited state light shifts. The result is a larger density in this region, in a similar fashion to what reported for the dark-SPOT MOT [25].

IV. RESULTS
At this point we begin the sub-Doppler cooling phase. The FORT is ramped up to 170 µK to trap atoms and, simultaneously the detuning of the cooling beams is shifted from the CMOT detuning to the desired molasses detuning in 200 µs and the beams remain illuminated for 1 ms.

A. Loading
The number of atoms remaining trapped in the FORT after 0.5 s are plotted in Fig. 2(a) as a function of the molasses stage detuning. Three regions are identified in relation to the dominant coupling transition: (i) F = 2 → F = 3 in the blue-shaded region, relative to standard polarization-gradient cooling; (ii) F = 2 → F = 2 in the red-shaded region; (iii) F = 2 → F = 1 in the greenshaded region.
It is important to note that we are not adiabatically turning on our FORT to optimize the transition between the sub-Doppler phase and the trapping phase of the experiment, but rather that the deep FORT potential is present throughout the loading. When a deep conservative potential is present during the optical cooling phase, a mechanism is required to take away the excess energy of the atoms for loading into the conservative potential. This can be accomplished via collisions where the colliding partner carries away the excess energy. Here, it is the spontaneous emission involved in the sub-Doppler cooling which removes the excess energy, conserving atom number. Other methods are also being explored to similar ends [26].
The blue data points in Fig. 2 represent the molasses phase with cooler and repump beams generated from separate lasers and the repump tuned on-resonance to the F = 1 → F = 2 transition, i.e. no Raman condition. Loading can be seen arising from the standard molasses at -15 Γ. In this condition, ZDSs can form, resulting in an additional loading peak located at -40 Γ that is a factor of 2.2 greater than the peak Sisyphus loading. The position of this peak corresponds to the detuning identified by Rosi el. al. [7] as that which maximises the central atomic density of 87 Rb atoms, similarly cooled on the F = 2 → F = 2 transition with independent repumper in the absences of any excited-state light shifts.
The red data points in Fig. 2 (a) Atoms loaded into the FORT from the sub-Doppler cooling phase. The x-axis lists the detuning of the Raman beams from the F = 2 → F = 3 resonance and the yaxis shows the atoms loaded. Blue data points correspond to an independent repump laser locked on the F = 2 → F = 2 transition, and the red data points to a repump in Raman condition generated by modulating the main beam at ω hf with an EOM. The three relevant transitions and where they dominate are denoted in blue (F = 2 → F = 3), red (F = 2 → F = 2), and green (F = 2 → F = 1). (b) Population fraction of atoms in the F = 2 manifold for the relevant regimes above; N1,2 indicate the atomic populations of the F = 1, 2 manifolds. The violet points result from a numerical evaluation of the dark states forming at each detuning (see Supplemental Material [22]).
laser at the hyperfine splitting frequency of Rb -equal to ω hf ≡ 6.834 GHz -with an EOM to create repumping sidebands. The repumper is no longer fixed on the F = 1 → F = 2 transition but is instead in Raman configuration with the cooler (see Fig. 1(b)). This condition determines a sevenfold increase in atomic loading at a detuning of -43.5 Γ due to the formation of HDSs. Further increasing the detuning shows another loading feature at -69 Γ, which corresponds to HDSs on the F = 2 → F = 1 transition. The lower loading efficiency of the gray molasses in this second region is due to the ramping process of cooler+repump laser frequencies from the MOT frequencies to F = 2 → F = 1 transition frequencies, which involves crossing the F = 2 → F = 2 transition with related heating. The loading near the F = 2 → F = 2 transition shows a sharp, high peak at -43.5 Γ when using the EOMgenerated coherent repumper. As for pure ZDSs, such optimal loading is found to coincide with the peak in atomic density already reported when using a coherent repumper and in absence of FORT-induced light shifts [7]. We ascribe this behavior to optimal HDS cooling of atoms outside and on the edges of the FORT. Such condition maximises the overlap between the FORT volume and the central atomic density of atoms on the exterior of the trap.
However, this peak sits atop a broader feature which, surprisingly, does not abruptly end at the position of the F = 2 → F = 2 resonance. To comprehend this behavior, we repeated the measurement for different FORT depths: in Fig. 3, the broader feature shifts and broadens with increasing FORT power while the position of the sharp peak at -43.5 Γ remains constant. The broader peak is explained in terms of atoms being loaded from within the trapping volume of the FORT; optimal detuning to cool/load these atoms will be larger and position dependent, as they experience the excited state light shifts that follow the spatial profile of the FORT (Fig. 1). As a consequence, atoms from different parts of the atomic distribution overlapping with the FORT can be loaded by changing the detuning. The shape of the broader loading curve is a convolution of the atomic density distribution and the spatial profile of the FORT: the deeper the FORT, the broader the loading curve. The sharpness of the peak at -43.5 Γ occurs due to a competition between the enhanced atomic densities outside the FORT and the effect of these light shifts on atoms approaching the FORT frontier, and its position at -43.5 Γ does not change with the FORT power. The peak ultimately arises from a volumetric effect.
A similar double structure can be resolved in the blue data points, where only ZDSs can form, but the associated peak at -40 Γ is much less intense, because the ZDSs are not truly dark. The HDSs sub-Doppler cooling, on the other hand, results in true dark states centered around p = 0, allowing us to accumulate atoms at p = 0 while going closer to the F = 2 → F = 2 resonance, resulting in higher atomic densities and much more effective loading. The possibility to spatially address atoms within the FORT, as evidenced in Fig. 3, suggests a mechanism for further cooling trapped atoms that we explore below.

B. HDS and Light Shifts
Increasing the power of the dipole trap changes the working of grey molasses, which is reflected in the loading curves shown in Fig. 3. Indeed numerical calculations reveal that the light shifts caused by the trapping light at 1560 nm impact the existence of our dark states (see Supplemental Materials [22]).
The effect of the excited state light shifts on the HDSs can be seen in the graph of Fig. 2(b) where the atomic fraction in the F = 2 manifold is plotted after 200 ms of hold time in the FORT. The data shown gives hints and restrictions to the types of HDSs that we have. At the beginning of the HDS cooling in the shaded red region (F = 2 → F = 2 transition) and all throughout the shaded green region (F = 2 → F = 1 transition), a little under one-half of the atoms can be found in the F=2 manifold (the rest residing in F=1), indicating that all atoms exist in HDSs of equal proportions across the 5S 1/2 level. Of the various possible HDSs [9] some have excited states composed of combinations of symmetric m F levels (m F = ±1 or m F = ±2) and others excited state combinations of non-symmetric m F levels (any combinations of m F = ±2, ±1, 0). Symmetric m F levels undergo the same detunings under the influence of the FORT, leaving the basic excitation structure unchanged. The nonsymmetric m F levels will instead undergo differential excited state light shifts adding extra phase factors in the Rabi frequencies of the various transitions: the related HDSs are as a result increasingly inhibited when entering the FORT. Increasing the detuning in the red shaded region, so as to address atoms further inside the trap, de- stroys non-symmetrically mediated HDSs until only the symmetric mediated HDSs reduces to roughly ∼25%. For the shaded green region (F = 2 → F = 1 transition) the population of atoms in the F=2 manifold remains ∼50% for all detunings; all atoms are in balanced HDSs, mediated either by the symmetric excited state m F = ±1 or via m F =0 . In the Zeeman molasses scenario however (blue data points in Fig. 2), all atoms are found in the F = 1 manifold: the ZDSs are in the F = 2 manifold, however the presence of a non-Raman resonant repumper reduces the darkness of the F = 2 manifold states [16,17], leading to eventual coupling to the light and optical pumping to F = 1.
We observed the deleterious effects of the 1560 nm on the formation of the HDSs can be partly offset by shifting the Raman frequency, ω EOM , from the value of the unperturbed hyperfine splitting ω hf . To demonstrate this behavior, we first measure the cooling efficiency as a function of the Raman detuning. To this purpose, we measure the central density of the atomic cloud in free space and without 1560 nm trapping light after 4 ms of gray molasses, while varying the Raman detuning δ R = ω EOM − ω hg (see Fig. 4, inset). The exact Raman condition, δ R = 0, gives optimal cooling -that is the lowest temperature and highest central density, and the cooling efficiency falls when the Raman detuning is changed, reaching a minimum at δ R = 80 kHz.
Remarkably, the distance between the constructive and destructive regions is much smaller than Γ as in similar work with Cs atoms [27]. These results are in contrast to Ref. [6], where, however, the analysis focused on the interplay between the dark states and the depth of the polarization potentials, which is of the order of Γ.
The presence of light at 1560 nm with the associated light shifts translates horizontally, along the axis of Raman detuning, the dispersive curve seen in Fig. 4 Inset. However, by changing the Raman detuning, δ R , we can shift the curve to remain in the optimal condition. Fig. 4 shows the Raman detuning needed to obtain an optimal cooling at different FORT powers, measured in terms of the peak atomic density in the dipole trap. The red points refer to the hyperfine gray molasses on the F = 2 → F = 2 transition where an increasing compensation shift is required. The blue points refer to the hyperfine gray molasses on the F = 2 → F = 1 transition. On the first transition we measure a small but detectable shift of the optimal Raman detuning when we increase the FORT power, while on the latter transition cooling is optimal at δ R = 0, independently of the FORT power. Since for the ground level the vector light shifts is zero, the reason of the observed shift must arise due to the excited state light shifts effects on the HDSs, however the exact mechanism is still unclear.
It is important to mention that in our particular case, no state is perfectly dark. Consider, for example, any given ZDS on the F = 2 → F = 2 transition: if we take into account also the (off-resonant) F = 3 level, we conclude that the ZDS is not truly dark: its scattering rate being γ ∼ Γ(Ω 3 /∆ 3 ) 2 , with ∆ 3 indicating the typical detuning from the F = 3 level and Ω 3 the typical Rabi frequency. When the laser is tuned close to the F = 2 → F = 2 transition only the large detuning ∆ 3 makes the scattering rate negligible, γ Γ. This is also true for the HDS which are completely dark if we only consider a specific excited level. However the presence of the 1560 nm light shifts and mixes the excited hyperfine levels, which can both be deleterious for the HDSs.

C. Velocity-Tuned HDS Cooling
Optically trapped atoms experience position dependent excited state light shifts, in relation to the local FORT power. Changing the detuning of HDS cooling will hence address atoms at different places in the optical potential creating spatially dependent HDSs. The position of an atom within a conservative trap is also energy dependent, with higher momentum/energy states spending more time near the edges -where the light shift is smaller -and lower momentum/energy states residing near the center -where the light shift is larger. Therefore at any particular detuning only atoms of a particular momentum/energy will be optimally cooled. This opens up the possibility of an altogether different type of cooling: by starting with small detunings, hotter atoms at the edges of the FORT will selectively cool, falling into the FORT where a further shift in the Raman beams to larger detunings will cool the atoms to even lower momentum/energy states (see Fig. 3, inset). A single sweep on the Raman beam detuning can progressively cool all the atoms in the FORT, potentially down to the recoil temperature 2 k 2 /(2mk B ). Excited state effects will be minimized, since the protocol relies on dark states, and, remarkably, no atoms are lost in the process, given that the energy is carried away via spontaneous emission and not by evaporation.
After loading the atoms in the FORT, the trap depth is adiabatically increased to ∼1 mK, to create larger excited state light shifts and allow higher momentum/energy selectivity. The atoms have a temperature of 198 µK, primarily given by the trap depth. Sweeping the detuning of the Raman beams in 6 ms from -55 Γ to -68 Γ, and then measuring the temperature in time-of-flight, we observe cooling to 48 µK (Fig. 5(a)). If the detuning is further increased, all atoms are lost as the laser frequency becomes resonant with the F = 2 → F = 2 transition at the center of the FORT, where the detuning is maximal.
A higher momentum/energy sensitivity can be achieved inside the trap by increasing the FORT depth, as shown in Fig. 3. However, 267 MHz above the relevant F = 2 → F = 2 transition the 5P 3/2 excited state contains the F = 3 hyperfine level (see Fig 1), which also experiences the same light shifts. While the Raman beams are detuned to cool atoms near the edges of the FORT, atoms in the center find themselves closer to the F = 2 → F = 3 transition which can impede the formation of HDSs [12] and eventually cause heating. This effect represents the main limitation to our present scheme but it also points to a straightforward remedy, i.e. perform the scheme on the 5S 1/2 → 5P 1/2 D1 transition which contains no higher F = 3 state. A similar process hinders cooling on our previously discussed F = 2 → F = 1 transition, now with the F = 2 state lying 156 MHz above. Although this would not exclude the formation of dark states, being detuned to the red of the F = 2 → F = 2 transition will lead to heating as the polarization gradient potentials are now inverted. Performing the cooling scheme on the F = 2 → F = 2 D1 transition should permit recoil temperatures in timescales of ms without loss of atoms.

V. CONCLUSION
We have demonstrated the potential of a new type of all-optical cooling performed inside an optical trap using hyperfine dark states. The method could open new avenues in the production of ultra-cold atomic and molecular gases in relation to its key features: high speed, minimal scattering, no atom loss and no need of cycling transitions. Remarkably, cooling and loading happen simultaneously, which solves the mode matching issue for the atomic transfer in the final conservative potential and skips altogether the intermediate step often relying on magnetic trapping. The cooling scheme looks optimal for the rapid production of ultra-cold gases in unusual geometries [28,29] and environments [30,31].

VI. ACKNOWLEDGMENTS
We thank Leonardo Ricci for kindly providing a Rb reservoir in a critical phase of the experiment. This work was partly supported by the "Agence Nationale pour la Recherche" (grant EOSBECMR # ANR-18-CE91-0003-01), Laser and Photonics in Aquitaine (grant OE-TWC), Horizon 2020 QuantERA ERA-NET (grant TAIOL # ANR-18-QUAN-00L5-02), and and the Aquitaine Region (grants IASIG-3D and USOFF To investigate the formation of dark states, we use a 1-dimensional model of the gray molasses [6,32] that includes the presence of two counterpropagating beams with opposite circular polarizations (σ + /σ − configuration), each carrying both cooler and repumper frequency, and the light shifts due to the trapping potential.
We consider the Hamiltonian as a sum of three terms: (i) the atomic Hamiltonian, taking into account only the 5 2 S 1/2 and 5 2 P 3/2 states connected by the D2 transition; (ii) the interaction of the atom with the cooler and repumper light; (iii) the interaction of the atom with the dipole trap light at 1560 nm, causing scalar and vector light shifts, especially large for to the states of the 5 2 P 3/2 level: In the atomic Hamiltonian H at , E(n, F, m) denotes the energies of the ground and excited hyperfine manifolds, hereafter indicated with 5S and 5P , with the definition E(5S, 1, m) = 0.
In the molasses Hamiltonian H mol , Ω R(C) ≡ Γ I R(C) / (2I S ) is the repumper (cooler) Rabi frequency in terms of the saturation intensity I S = 1.67 mW/cm 2 and the excited state linewidth Γ/(2π) = 6.065 MHz, a are the raising operators of atomic levels whose matrix elements are the 6-j Wigner coefficient and, finally, ω R(C) the repumper (cooler) angular frequency. Each molasses beam carries the repumper and cooler frequency ω R , ω C , with Rabi frequencies Ω C = 4.2 Γ, Ω R = 1.2 Γ, corresponding to the total intensities, i.e. summed on the six beams in our experiment. Eq. (4) is further simplified by neglecting the coupling of the cooler with the F = 1 → F transitions, due to very large detuning (∼ 10 3 Γ), and likewise the coupling of the repumper with F = 2 → F transitions: Ω R e −iω R t P e aP 1 + Ω C e −iω C t P e aP 2 × ê + e ikz +ê − e −ikz + h.c. .
Finally, H dip is the second-order perturbation Stark Hamiltonian for the 1560 nm light E(t) = 1 2 E 0ˆ e −iω d t + c.c., with d µ , µ (µ = 0, ±1) being the spherical components of the electric dipole operator and of the polarization vector; E v is the energy of the right ket and k labels the intermediate state.
One can apply a unitary transformation We set the frequency difference ω R − ω C to match the hyperfine separation of a free atom and then diagonalize the Hamiltonian for different values of the position z. At each position, we identify the dark states as those whose projection in the excited level 5P is below an arbitrary threshold value, chosen equal to 0.1; in practice a given eigenstate |ψ of the total Hamiltonian is considered dark if ψ| P e |ψ < 0.1, which is equivalent to require a scattering rate below 0.1 Γ. We find that, in the σ + /σ − configuration of our 1D model, the light shifts are uniform in space, i.e. independent of z. Indeed at each point the polarization of the molasses lasers field is linear, with direction varying periodically in space: the polarization vector winds like a helix. As a consequence, it is expected that the number of dark states does not depend on the position z.
In the set of dark states, we operate a further selection by keeping only those composed of hyperfine states |F, m = ±1 and discarding those formed by |F, 2 and |F, 0 . The former are linear superpositions of states with momenta p ± k 0 , that can be stable under kinetic energy mixing for sufficiently small p. On the other hand, the latter are superpositions of states with momenta p, p±2 k 0 , that cannot be eigenstates of the kinetic energy. For molasses times longer than the inverse kinetic energy associated to the two-photon transitions, t mol > (2 k 2 0 /m) −1 10 µs, the latter states become bright.
In order to calculate the expected relative populations of the F = 1 and F = 2 hyperfine levels at the end of the molasses, we take into account only the dark states selected as described. On each of these states |DS j , we evaluate p 1,j = DS j | P 1 |DS j and p 2,j = DS j | P 2 |DS j and define the relative population N 2 /(N 1 +N 2 ) = j p 2,j /(p 1,j +p 2,j ). The purple points in Fig. 2 of the main text report this quantity in the specific case for zero power of the FORT.