Demagnetization cooling utilizes dipolar relaxations that couple the internal degree of freedom (spin) to the external (angular momentum) in order to cool an atomic cloud efficiently. Optical pumping into a dark state constantly recycles the atoms that were thermally excited to higher spin states. The net energy taken away by a single photon is very favorable since the lost energy per atom is the Zeeman energy rather than the recoil energy. As the density of the atomic sample rises the presence of the photons leads to limiting processes. In our previous publication [Volchkov et al. (2014)] we have shown that light-assisted collisions are such an important limiting process. In this paper we suppress light-assisted collisions by detuning the optical pumping light such that the Condon point coincides with the first node of the ground state wave function of two colliding atoms. This leads to an increased cooling efficiency χ ≥ 17 as well as to increased maximum densities of n ≈ 1 · 1020 m−3. However, due to the high number of involved molecular states the net suppression is not strong enough to reach quantum degeneracy.
© 2015 Optical Society of America
Recently, Bose–Einstein condensation of non-alkali elements has experienced growing interest due to their magnetic and electronic properties [1–6]. One reason for this is that the highly magnetic species chromium, erbium and dysprosium extend the area of study to dipolar interacting and strongly correlated quantum matter [7–9]. Renewed interest in optical cooling methods originates from progress in laser cooling of molecules [10, 11] and the possibility of using narrow linewidth transitions for standard laser cooling techniques [12–15]. On the one hand, optical cooling may pave the way to degeneracy of ground-state molecules - a goal pursued by many groups. On the other hand, new lossless laser cooling methods may be considered as an alternative to evaporative cooling allowing to boost the number of atoms of a degenerate gas. Finally, Bose–Einstein condensation by optical means only [16, 17] is a long-standing goal in the field. To the present day density-dependent processes like reabsorption and light-assisted collisions (LAC) have prevented condensation with photons being involved. To prevent such collisions in experiments using laser cooling as the only cooling mechanism , the BEC region has to be well shielded from photons. The key difference of demagnetization cooling compared to standard laser cooling techniques which rely on the photon recoil, is the large net energy that is carried away by a single photon. This energy is on the order of the Zeeman energy Ez. From a different point of view the number of scattered photons for a certain temperature reduction is decreased by a factor ∼ Ez/Erecoil which is ∼ 50 for our experimental parameters. In this paper we show that the performance of demagnetization cooling does indeed depend on the detuning of the optical pumping light, which is an insight that early experiments did not observe . Because of the rapidly rising densities in the cloud LAC become an efficiency limiting process in demagnetization cooling . However, as LAC and the involved transition probabilities to excited molecular potentials depend on Franck–Condon factors it is possible to find detunings with suppressed losses. This suppression originates from a vanishing Franck-Condon factor at the first node in the ground state scattering wave function. We experimentally determine the node position by scanning the detuning of the optical pumping light and show that LAC are indeed suppressed at the node position. We confirm the prediction that red detunings lead to superior cooling performance compared to blue detunings . The cooling efficiency for optimized parameters is significantly better than in previous demagnetization experiments and, to our knowledge, the highest efficiency of any cooling mechanism for ultracold atoms.
This paper is organized as follows. After a brief introduction into the theoretical description of LAC and the method to suppress them in section 2 we introduce the experimental setup and sequence in section 3. We continue with the experimental results beginning with the coarse determination of the node position by a heating measurement on the blue detuned side in section 4. In sections 5 and 6 we examine the theoretical predictions obtained in section 2. Finally we compare the optimized conditions to earlier experiments in section 7.
In this paper we will only shortly recapitulate the most important equations necessary to describe demagnetization cooling. We refer the interested reader to [19, 20, 22, 23] for a detailed treatment. Without limiting the generality most of the expressions below are given explicitly for our system 52Cr. Demagnetization cooling relies on spin-orbit coupling that enables a thermalization of motional and spin degrees of freedom. For atoms in a dilute gas this is achieved by inelastic collisions of two atoms called dipolar relaxations. Figure 1(b) depicts the Zee-man levels for the ground and excited state when a finite magnetic field is applied. Hot spin polarized atoms in the lowest Zeeman state mJ = −3 (red circle) can be thermally excited to mJ = −2 (blue circle) states by dipolar relaxations. In such an inelastic collision one or both atoms may change their spin state and each spin flip costs the atom the Zeeman energy Ez. Atoms in mJ > −3 couple to the σ− polarized optical pumping light and are thus pumped back to mJ = −3 via the excited state 7P3. Thermalization of the atoms that have been pumped back with the rest of the cloud then leads to the net cooling effect on the whole sample. The single (double) spin flip cross section σ1,(2) for the collision of two magnetic atoms in a stretched state depends on the third power (square) of their spin S and the square of their mass m . These cross sections have to be averaged thermally to obtain the dipolar relaxation rate constant20]. In the saturated regime, when the rate equations can be simplified and the excited state density is independent of the optical pumping rate ΓSC. Here NdrN refers to the total number of atoms, γnat the natural linewidth, V the temperature dependent effective volume and vrel the relative collision velocity.
An important figure of merit for any cooling technique is the efficiency χ = −ln(ρf/ρi)/ln(Nf/Ni). It relates the gain in phase-space density ρ to the loss of atoms N. In the case of evaporative cooling χ ≈ 4 is, to our knowledge, the highest experimentally observed efficiency and can only be reached by substantial effort .
Although demagnetization cooling is in principle lossless experimental observations clearly show that additional loss channels are present . The collisional physics of dipolar relaxations is well understood and does not give rise to such additional loss channels. The inherent involvement of photons is eventually accompanied by the problems known from the first days of laser cooling. Light-assisted collisions constitute such a loss mechanism  and have been identified to limit the performance of demagnetization cooling in previous experiments . This density-dependent process originates from the temporary creation of quasi molecules in unbound / bound states. Because the molecular state is formed between one atom in the ground state 7S3 and one atom in the excited state 7P3 the resulting interaction for homonuclear samples is resonant dipole-dipole interaction (DDI). The resulting R−3 potential depends only on a single parameter the C3 coefficient. Figure 1(a) sketches out such molecular potentials. In the case of positive (blue) detuning Δ = ω − ωA > 0 only unbound states are within reach. Colliding atoms approach each other until they reach the Condon point RC where the laser frequency ω matches the energy difference between ground Vg and excited state potentials Ve, i.e. h̄ω = Ve(RC) − Vg(RC) or21] to overcome losses in a Bose–Einstein Condensate (BEC). The idea behind the method is not restricted to BECs and has been observed in photoassociation and superradiance experiments [26,27]. In the following we will briefly summarize the theoretical foundations found by Burnett et al. . The method makes use of the fact that for weak excitation the probability to excite the molecular state 28]. The result of this reflection approximation is that fC only depends on the square of Ψg: 29]. The ground state wave function can be approximated by Eq. (2) to RC = (h̄Δ/C3)1/3 we see that gC is a function of the detuning Δ. For a constant scattering rate Γsc = const Eq. (9) shows that gC is the only part that modifies the loss probability as a function of Δ. In the experiments we will vary the detuning Δ to probe the nodal structure of the ground state scattering wave function. The scattering rate Γsc will be held constant for all detunings by varying the light intensity accordingly. For red detunings Γbinary will have a superimposed discrete vibrational level structure. For the exact form we refer the interested reader to . The important point is that between two subsequent vibrational levels νn and νn+1 there is additional suppression because the detuning does not match the excitation energy to any of the states. So far we have treated our system by just one ground state potential corresponding to 7S3 +7 S3 and just one (attractive or repulsive) excited state potential corresponding to 7S3 +7 P3 and neglected that each state can have different mJ. However, having one ground state potential is only true for two atoms that are both in mJ = −3 but these atoms will not scatter any photons because of the laser polarization. The most probable incoming channel is |7S3, mJ = −3〉 + |7S3, mJ = −2〉 because the mJ = −2 fraction in the cloud will always be close to zero. Assuming Hund’s coupling a)  and applying the Wigner-Wittmer rules  to this incoming channel, there are two possible molecular states, i.e. , . These states do not have the same scattering length and can thus differ in their respective nodal position. Up to now only the scattering lengths for , and states are known . The Hund’s coupling for the excited state is case c) making Ω, the projection of the total electronic angular momentum on the internuclear axis, the only good quantum number. The exact analytic relation of the C3 and the natural linewidth is, however, only valid for Hund’s case a) coupling, where λ̄ = λ/2π is the reduced wavelength. The mixing of states with the same Ω due to spin-orbit coupling leads to different effective  and thus to different Condon points for the same detuning. Both effects tend to smear out and reduce the expected suppression of light assisted collisions. In the photoassociation spectrum of Cr we clearly observe two distinct series with unequal C3 coefficients .
We started our experiments by loading typically 1.5 · 106 bosonic 52Cr atoms with a temperature of 90μK in a single beam optical dipole trap (ODT) (trapping frequencies: ωx = ωy = 2π · 5.5kHz and ωz = 2π · 40Hz) . The atoms were initially spin polarized in the lowest Zeeman substate mJ = −3 due to the nature of the loading mechanism [36, 37]. Lowering the homogeneous offset magnetic field with a sudden jump from Bx ≈ 1.5G to Bx ≈ 300mG energetically allows dipolar relaxations to occur and thus starts demagnetization. Simultaneously we applied the 427nm optical pumping light with a variable detuning Δ and a constant optical pumping rate of ΓSC. The choice of Bx ≈ 300mG is a good tradeoff between having sufficient initial Γdr and a favorable Ez/Erecoil ratio. Naively one may expect that due to the decreasing temperature a constant magnetic field rapidly stalls demagnetization cooling. This would certainly be true for a gas that does not change its density while cooling. However, due to the density dependance of Γdr the cooling rate |dT/dt| and efficiency χ show a non-trivial dependance on the applied magnetic field B. The experimental results suggest that we probe the density limits that are addressed in this paper before demagnetization is stalled due to a constant magnetic field. This can also be seen in  where the temperatures of the two presented datasets - one with decreasing B and one with constant B - differ only by little. Throughout this paper we will use constant magnetic offset fields during demagnetization cooling. The transversal magnetic fields By & Bz were of negligible size and were scanned separately to maximize the σ− polarization purity. In contrast to previous experiments the large detuning difference for loading and demagnetization cooling required a second independent laser. We used a grating stabilized diode laser second harmonic generation system (SHG) with a maximum blue output power of P427 ≈ 150mW. The master-laser (855nm) was frequency-stabilized to a transfer cavity with a free spectral range FSR = 75MHz and had an experimentally determined linewidth well below 30kHz. The 427nm output entered an accousto-optic modulator (AOM) double pass with a subsequent optical single mode fiber to ensure stable alignment for all detunings and to minimize day to day drifts. The beam was then intensity-stabilized via a tilted partly reflective glass plate and the AOM amplitude modulation before it passed the polarization optics to get to the experimental chamber. At the position of the atomic cloud we measured the optical pumping beam waist to be ω̃y,z ≈ 90μm. The AOM double pass together with the small FSR of the transfer cavity enabled us to scan the laser detuning without any gap. The optical pumping rate ΓSC was chosen such that it exceeded even the largest dipolar relaxation rates to operate in the saturated regime . For every detuning the light intensity was adjusted to maintain the constant ΓSC.
4. Coarse determination of the node position on the blue side
In a first set of experiments we conducted a simple heating measurement similar to  in order to find the coarse position of the node. These experiments were done on the blue detuned side because there are no bound levels. After loading and precooling the sample with the laser used in  we switched to the SHG and tilted the magnetic field by applying additional By & Bz fields to enhance the scattering rate and thus the losses. The cloud initially had a density of n ∼ 3.5 · 1019 m−3 and a temperature of of 21μK. During the exposure/holding time of 2 s we either applied the 427nm SHG light with ΓSC = 2π · 1kHz or we held the cloud without any light. The observed numbers of atoms Nl and Nd were normalized to the number of atoms N0 without any additional holding time. We averaged every Ni (i = l, d, 0) over 5 cycles and obtained the normalized loss by (Nd − Nl)/N0. Figure 2 shows the detuning scan from 750MHz to 12GHz and the observed characteristic change in the loss of atoms. The red curve is a fit of M · gC +C to the data, where M is a scaling amplitude and C is an offset both without deeper physical meaning. The fit should rather be seen as a guide to the eye than a precise determination of the node position. The reason for this is the not spin polarized cloud due to the tilted magnetic fields. Many ground & excited states can participate leading to a washed out minimum at Δ/(2π) ≈ 6.7GHz. The fit was obtained by fixing C3 to a value of 1.53a.u. determined by calculations for photoassociation experiments on the red side . The assumption is that only states with a definite g/u symmetry can be excited. From the good agreement of the observed and calculated photoassociation lines we deduce the validity of the calculated C3 also on the blue side and pick the C3 according to symmetry considerations. However, it should be noted again that the good agreement with this simple fit formula is rather surprising as it only accounts for a single ground and a single excited state channel. We also fixed the C6 coefficient to 733a.u.  and used M, C and As as free parameters. The resulting As = 113a0 agrees with the known scattering length of the Cr ground state potential [33, 39].
5. Red vs. blue detuning
One prediction of  was that Γbinary is smaller on the red side when the detuning is not resonant to a bound level, i.e. Δ ≠ Δν. To check this prediction we cooled the cloud for a fixed time of 4 s and a fixed ΓSC = 2π · 400Hz on the red and on the blue detuned side. The achieved phase-space densities ρ for three detuning ranges are depicted in Fig. 3. In the red detuned case (red curves) several resonance dips corresponding to bound levels occur. In between these bound levels the obtained ρ clearly exceeds the one in the case of blue detunings (blue curve). The effect is visible even though adjacent bound levels do not belong to the same series, which is a consequence of the multiple excited states involved .
6. Temporal correlation of ne and excess losses
In our previous work  we have shown that the increased losses during the demagnetization cooling process have a strong temporal correlation to the calculated excited state density ne. The excess loss rates per atom ζ(t) = (Ṅ(t) − γbgN(t))/N(t) were obtained from the experimental data by subtracting the background losses γbgN(t) from the numerical differential Ṅ(t) = dN(t)/dt and normalizing this to the number of atoms N(t) for every experimental measurement point. The excited state densities ne were then calculated for every experimental point using Eq. (1) and measured temperatures and densities. To show the suppression of light-assisted collisions at the node we cooled our cloud for variable times ranging from 0 – 8s for detunings smaller, equal and bigger than the optimum detuning. The experiments have been carried out with ΓSC = 2π · 400Hz on the blue detuning side. The resulting ne(t) and ζ(t) for different detunings Δ/2π = 2.25, 6.75, 15GHz are depicted in Fig. 4(a)–(c). Figure 4(a) shows that for Δ/2π = 2.25GHz we obtain almost the same values of ne(t) (blue diamonds) and ζ(t) (red squares) as observed in  where Δ/2π was −360MHz. As we increase the detuning, as shown in Fig. 4(b) and (c), we observe a clear speeding up of the cooling process that does not have an optimum at the nodal position. We attribute this to the reduction of a light induced heating rate most likely related to reabsorption of optical pumping photons. To understand the increasing maximum values of ne one has to keep in mind that any reduction of limiting processes - e.g. heating and LAC - will result in an increased density n. In return the excited state density ne is proportional to n2 . However, the figure of merit is achieving high densities n as well as low losses ζ. The excess losses ζ in Fig 4(b) are slightly reduced with respect to the close detuned case and do not have a strong temporal correlation to their respective ne. It is not surprising that there is a remaining weak temporal correlation between ne and ζ for an imperfect suppression of light-assisted collisions. In Figure 4(c), for Δ/2π = 15GHz, ζ(t) exhibits overall higher values and a stronger temporal correlation to ne. The inset in Fig. 4 summarizes the findings explained above. The inset depicts the numerical integral of ne (blue) and ζ (red) over the cooling time. The excess losses ζ show a minimum at the node position while the excited state density ne increases with the detuning.
Finally we compare the performance of demagnetization cooling with suppressed light-assisted collisions to data taken close to the atomic resonance (Δ/2π = −360MHz) . In order to demonstrate the best performance we can achieve, a red detuning Δ/2π = −9GHz is chosen in accordance with section 5. The magnetic field was held constant for the full cooling curve and had a value of B = 280mG. Note that Δ/2π = −9GHz is not the value that is expected from the fit in section 4 but it is well in the region of reduced losses. An optimized value of ΓSC = 2π · 220Hz has been determined in a separate set of experiments where it minimized the temperature T for a constant cooling time of 4s. Figure 5(a) shows the evolution of the cloud temperature for small (grey circles) and optimized (blue squares) Δ. The optimized cooling yields an initial cooling rate |dT/dt| > 23μK/s (red line) leading to significantly faster experimental sequences. The evolution of the cloud density n is depicted in Fig. 5(b). The suppression leads to an increased maximum density of a factor of 2 but qualitatively it has the same shape as in the close detuned case. To illustrate the superior performance of operating at the optimized detuning Fig. 5(c) shows the typical double logarithmic plot of ρ versus the number of atoms N. The black and red lines are a guide to the eye and display efficiencies of χ = 6.5 and χ = 17. For Δ/2π = −9GHz the efficiency follows χ = 17 for roughly one order of magnitude in ρ and then smoothly bends down towards χ = 0. We are not able to maintain a high efficiency for higher ρ by reducing the magnetic field or by changing ΓSC. Possible limiting mechanisms are the imperfect suppression of light-assisted collisions and reabsorption of optical pumping photons. It is very hard to distinguish between the two mechanisms because both are density-dependent and reabsorption can enhance light-assisted collisions.
It should be noted that in the proof of principle experiment  the stated efficiency of χ = 11 has been obtained over a ρ range of roughly half an order of magnitude. Due to weaker confinement the experiment had significantly smaller relaxation rates and, thus, smaller ne. Their final density did not exceed n = 2 · 1019m−3 which is why photo-induced losses were not observed. The detuning of Δ/2π = −200MHz together with the small Γdr resulted in a cooling rate of ∼ 1μK/s.
In conclusion we have shown that the efficiency of demagnetization cooling can be enhanced significantly by detuning the optical pumping light to the optimum position in order to suppress light-assisted collisions. This position originates from the first node of the ground state scattering wave function and is defined by only few parameters As, C6 and C3. Because the cloud is not spin polarized more than one ground / excited state scattering wave function participates, the node position is washed out and the suppression is reduced. The resulting suppression is strong enough to enhance the efficiency χ by a factor > 2.5. More important, however, is the increase of the maximum peak density nmax by a factor of 2. To reach quantum degeneracy the density n has to fulfill nmax > n ≥ nc, where nc is the critical density for condensation. A simple estimate suggests that for nmax on the order of 5 · 1020 m−3 condensation would be possible through demagnetization cooling.
We thank Paul Julienne and Eite Tiesinga for fruitful discussions and for calculating the excited state C3 coefficients. This work was supported by the DFG under Contract No. PF381/11-1.
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