Introduction

Methods of cooling atoms and molecules play a crucial role in building quantum simulators for studying many-body physics, and help to explore the fundamental principles of chemical reactions in the lowest temperature regime. Historically, the advances of the principles and techniques of cooling have benefited the developments of cold atomic physics. For example, magneto-optical trapping relies on the Doppler cooling mechanism1,2, Bose-Einstein condensations are produced by evaporative cooling3,4, and degenerate Fermi gases are generated by unitary-limited elastic collisions near the Feshbach resonance5,6,7,8. Most of these cooling methods fall into two categories: one utilizes laser-atom interactions to reduce the kinetic energy of the atomic system. The other one removes the atoms with higher kinetic energy from the trapping potential, lowering the temperature through the thermal equilibrium process. For the latter, the most popular technique is evaporative cooling by lowering the trapping potential as well as other variants9,10,11,12. One of the common feature for these evaporative processes is that they all take advantage of the different spatial positioning of hotter and colder atoms in a trapping potential.

Is there a method to direct remove the atoms in the higher momentum states regardless of the spatial difference between hotter and colder atoms? Recently, collisional cooling has been proposed for this purpose, which uses inelastic scattering processes near a narrow magnetic Feshbach resonance to enhance the kinetic energy-dependent loss13,14. Near the Feshbach resonance, the atoms with higher kinetic energy are closer to the threshold energy, and therefore have a higher probability of colliding. The collisions could change the internal states of the atoms, and the energy stored in the electronic state is converted into the kinetic energy of the colliding atoms, resulting the evaporation when the kinetic energy gained is much larger than the trapping potential. Since the lost atoms usually have higher kinetic energy than the thermal average value, the temperature of the atoms will be reduced. Experimentally, this method does not need to vary the trapping potential, but requires exquisite tuning of the magnetic field for precisely controlling the atomic interaction15.

In this paper, we present the observation of collisional cooling with ultracold 6Li fermionic atoms near a narrow Feshbach resonance. Our scheme is based on the framework proposed in the previous theoretical work13,14, where the significant difference between our scheme and these proposals is that we utilize three-body recombination16 instead of inelastic two-body collision to selectively remove hotter atoms. Compared with the two-body collision scheme, three-body recombination requires only one open channel for generating collisional cooling without the need for a second one, which makes it more feasible in experiments. In a one-dimensional Bose gas, the cooling has recently been observed with three-body recombination17. However, the cooling is not based on removing hot atoms through energy-dependent loss, but is because of the loss of phonon modes, which is quite different with the physics presented in this paper.

Results

Schematic of collisional cooling

The scheme of the collisional cooling near the narrow Feshbach resonance of 6Li is shown in Fig. 1. From the Bardeen-Cooper-Schrieffer side of the Feshbach resonance, when the molecular bound state in a closed scattering channel approaches the scattering state in an open channel, the threshold energy of the molecule Et is positive and a quasi-bound molecule is temporarily formed, where the relative kinetic energy of the two colliding atoms matches the threshold energy18,19. Once the molecule collides with the third incoming atom, three-body recombination produces a deeper-bound molecule20,21. The gap energy between the quasi-bounded and the deeply-bounded molecular states is converted into the kinetic energy of the atoms. The gained kinetic energy will be higher than the trapping potential, leading to the loss of all three atoms. The three-body recombination involves the total loss of kinetic energy of all three atoms. Particularly, when the loss of kinetic energy per particle is greater than the average kinetic energy of the cloud 3/2kBT, the process leads to cooling, where kB is the Boltzmann constant and T is the gas temperature. The threshold energy of Et for the formation of quasi-molecules thus acts like a knife to selectively remove the hotter atoms.

Fig. 1: Schematic of three-body recombination near an s-wave narrow Feshbach resonance of a 6Li Fermi gas.
figure 1

The black curves are the potentials of the open and closed scattering channels, respectively. A quasi-bound molecular state with a threshold energy of Et is shown in the closed channel. A deeper-bound molecule is generated from the upper state with a rate of \({K}_{ad}^{m}\) through the three-body process. The quasi-bound molecule decays back to the open channel with a rate of Γ(E).

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It should be emphasized that quasi-bound molecules also decay back into the original open channel, which can be considered as an elastic collision for cloud thermalization13. In the Feshbach resonance, the thermalization is usually guaranteed because the three-body recombination rate \({\Gamma }_{0}\propto {K}_{ad}^{m}\) is much smaller than the formation rate of the quasi-bound molecule Γ(E), where \({K}_{ad}^{m}\) is the rate coefficient of the atom-dimer interaction leading to the formation of a deeper molecule20.

Although the principle of the three-body cooling is well understood, the observation of the signatures of cooling is rather difficult in experiments, since Et should be controlled strictly precisely near a narrow Feshbach resonance. For 6Li atoms, it needs at least part-per-million (ppm) stability of the bias magnetic field around 543.3 G. Such a cooling effect has been noticed previously22,23, but the convincing evidence is still lacking. In this paper, we verify that the inelastic collisional cooling is actually from three-body recombination, and systematically study the magnetic field and temperature dependence of this collisional cooling. Our investigation shows the cooling efficiency becomes higher with the decrease of the atom temperature, which could support a runaway evaporative cooling down to extremely low temperature.

Heuristic explanation of this cooling

We use a noninteracting Fermi gas in a harmonic trap to explain the collisional cooling. According to the virial theorem for a harmonically trapped gas (The cooling regime of the three-body recombination is away from the resonance point, so the s-wave scattering length is small and the gas is a weakly interacting Fermi gas. The virial theory works reasonably under this case)24, the average kinetic energy per particle 〈Ek〉 = 3/2 kBT is equal to the average potential energy 〈Ep〉 = U(σx, σy, σz), where σx,y,z is the Gaussian width of n(x, y, z). The lost potential energy per lost atom is \(\dot{{E}_{3p}}/\dot{{N}_{3}}=U({\sigma }_{x},{\sigma }_{y},{\sigma }_{z})\). Then, \(\dot{{E}_{3p}}/\dot{{N}_{3}}=\langle {E}_{p}\rangle =3{k}_{B}T/2\). Therefore, the Boltzmann equation can be applied to a harmonically trapped gas as those done in the 3D homogeneous gases13,25. The collisional process in terms of density and temperature is given by (Details in Supplementary Note 3)

$$\begin{array}{c}\frac{\partial n}{\partial t}=-{L}_{3}{n}^{3},\\ \frac{\partial T}{\partial t}=-{L}_{3}{n}^{2}\left(\frac{{E}_{t}}{3{k}_{B}}-\frac{T}{2}\right),\end{array}$$
(1)

with three-body loss rate15,20,22

$${L}_{3}({E}_{t},T)=3{K}_{ad}^{m}{\left(\sqrt{2}{\lambda }_{T}\right)}^{3}{e}^{-{E}_{t}/{k}_{B}T}.$$
(2)

The thermal wavelength \({\lambda }_{T}={(2\pi {\hslash }^{2}/m{k}_{B}T)}^{1/2}\) and the threshold energy Et = 2μB(B − B0) for the binding energy of the Feshbach molecule with μB as the Bohr magneton. Eq. (1) can be solved analytically,

$$\begin{array}{c}\frac{1}{{n}^{2}(t)}=2{L}_{3}(B)t+\frac{1}{{n}^{2}(0)},\\ T(t)=2\left\{\frac{{E}_{t}}{3{k}_{B}}-\left[\frac{{E}_{t}}{3{k}_{B}}-\frac{T(0)}{2}\right]{\left[\frac{n(0)}{n(t)}\right]}^{1/2}\right\}.\end{array}$$
(3)

For convenience, we define η0 = Et/kBT(0). When Et = 3kBT(0)/2, the temperature will remain constant, where the leaving atom carries away the average energy of 3kBT from the gas, the sum of average kinetic energy and potential energy. This is the turning point from heating to cooling. In the cooling regime, the lower temperature could result in better cooling efficiency. As shown in S.12, when T is lower, L3 becomes larger and n(t)/n(0) becomes lower, then the phase-space density ratio Fρ = ρ(t)/ρ(0) gets higher since it is monotonically decreased when η0 > 3.5, where the phase-space density is \(\rho (t)={(2\pi \hslash )}^{3}n(t)/{(2\pi m{k}_{B}T(t))}^{3/2}\).

Collisional cooling of a thermal gas

We present the magnetic field dependence of the atom number N/N(0) and its density n, temperature T, phase-space density ρ, and the three-body loss coefficient L3 in Fig. 2a–e. The simulation explains the experimental result very well. When η0 approaches zero, the lower energy atoms are expelled from the trap due to three-body recombination, which increases the average energy of the rest gas and results in heating. Once Et exceeds 3kBT/2, the leaving atoms carry more energy (>3kBT) away from the gas, resulting in cooling. As shown in Fig. 2c, the best cooling rate is achieved around η0 ~ 3, and the temperature decreases from 5.1 μK to 4 μK in 450 ms. However, ρ keeps roughly constant around η0 = 3. Above this point, ρ slightly increases with time showing a weak cooling signature. ρ reaches a maximum when η0 ~ 9/2. This finding agrees with our numerical simulation. In our simulation, the energy difference ΔE between the two molecular states is much larger than the trap depth, so we assume that three scattering atoms leave the trap quickly without energy exchanging collisions with other atoms.

Fig. 2: Collisional cooling of a thermal Fermi gas of T(0) = 5.10 ± 0.04μK and T/TF(0) = 1.50 ± 0.01.
figure 2

a–e show the dependence of N/N(0), n, T, ρ, and L3 on magnetic field B, respectively. The black dashed lines are guiding for the η0 = 0, 3/2, 3, 9/2. Solid curves in (a–d) are the simulation by Eq. (3). The red solid curve in (e) is the fitting result of Eq. (2) with \({K}_{ad}^{m}=1.69\pm 0.01\times 1{0}^{-16}{m}^{3}/s\). The raw data of the L3 and the fitting procedure are included in Supplementary Note 2. The vertical error bars are the standard derivation for 2-3 measurements, and the horizontal error bars stand for the uncertainty of the magnetic field.

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Collisional cooling of a degenerate gas

We measure the collisional cooling of the gas when the temperature is close to the degenerate regime, as shown in Fig. 3a–e. Since the three-body loss rate \({L}_{3}\propto 1/{T}^{3/2}\exp (-{E}_{t}/{k}_{B}T)\), it is expected that a higher cooling rate could be obtained in the degenerate regime compared to the thermal one. This feature is rather attractive for its role in a potential runaway cooling. In Fig. 3a–e, the initial temperature T(0) = 0.60 ± 0.01 μK, and T/TF(t = 0) is 1.00 ± 0.02, where \({T}_{F}={(6N)}^{1/3}\hslash \bar{\omega }/{k}_{B}\) is the Fermi temperature. During a cooling process at η0 = 4.5, ρ increases from 0.25–0.5 and the lowest T/TF reaches 0.79. The maximum increase of Fρ is about 2 around η0 = 9/2, which is larger than the maximum value of the thermal one. At this point, the corresponding cooling efficiency \(\chi \equiv -\ln [\rho (t)/\rho (0)]/\ln [N(t)/N(0)]\) is about 1.3. When η0 gets larger, χ can be significantly increased, which should enhance the cooling. But, L3 also decreases with larger η0, resulting in the degraded of the overall cooling effect. It is noted that our experiments have not optimized the cooling process which requires a dynamic magnetic sweep that we discuss later.

Fig. 3: Collisional cooling of a near-degenerate Fermi gas of T(0) = 0.60 ± 0.01μK and T/TF(0) = 1.00 ± 0.02.
figure 3

a–e, the black dashed lines, and solid curves are defined the same as those in Fig. 2. The red solid curve in (e) is the fitting results of Eq. (2) with \({K}_{ad}^{m}=2.02\pm 0.02\times 1{0}^{-15}{m}^{3}/s\). The vertical error bars and the horizontal error bars are the same as those in Fig. 2.

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Our simulation shows that Fρ increases when η0 > 3.5 (in Supplementary Note 3), which agrees with the behavior of ρ in Fig. 3d. For the maximum increase of Fρ, the simulation indicates that η must be time-dependent. As η0 approaches the resonance, the measured L3 deviates from the perdition of Eq. (2), which is related to the unitary-limited behavior26,27. Such derivation is not predicted by our heuristic model, because many-body calculation is required to predict the three-body cooling behavior in the unitary regime.

Discussion

We first discuss a scheme with a time-dependent magnetic sweep to enhance the cooling efficiency. Following the optimization approach in ref. 14 and using Eq. (1), we obtain the time-dependent of η(τ) as

$$\frac{d\eta }{d\tau }=\frac{1}{2}\left(\eta -\frac{13}{2}\right)\frac{{n}^{{\prime} 2}}{{T}^{{\prime} 3/2}}{e}^{-\eta +\frac{13}{2}},$$
(4)

with dimensionless parameters \(\tau =6\sqrt{2}{e}^{-13/2}{K}_{ad}^{m}n(0)\rho (0)t\), \({n}^{{\prime} }=n/n(0)\), and \({T}^{{\prime} }=T/T(0)\). From this equation, the optimal η0 for the different cooling time τ are shown in Fig. 4a, b, which are different from the two-body collisional cooling14, where the optimal η0 is constant with a value of 9/2. For a specific example η0 = 6.0, we present the time-evolution of the atom density n0 and the phase space density ρ0 in the center of the trap, the atom number N, and the temperature T in the inset figure of Fig. 4b. It shows that three-body cooling has the potential to directly cool a thermal gas to a deeply degenerate regime. Moreover, the collision rate \(\gamma \propto {n}_{0}^{2}\exp (-\eta )/{T}^{3/2}\) (black line) is increased during the cooling, indicating a typical runaway scheme based on our model.

Fig. 4: Optimization the path of the three-body collisional cooling.
figure 4

a The phase-space density ρ/ρ(0) versus η0 for different cooling duration τ. b The optimized η0 for the maximum ρ/ρ(0), which is a time-dependent value. The inset figure is the time evolution of ρ0 and n0 in the center of the trap, γ, η, N, and T with η0 = 6. All the curves in the inset figure are normalized by their initial condition.

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Note that the densities in Figs. 2b and 3b are defined as n(t) = N(t)/V(0), where V(0) is approximately the initial cloud size (see Supplementary Note 2 for details). By using V(0) instead of V(t) in these figures, we obtain a relatively simpler theoretical model to derive the optimal cooling curve for collisional cooling (details in the Supplementary Note 3). On the other hand, the peak density n0(t) can be obtained by n0(t) = N(t)/V(t), where V(t) refers to the time-dependent cloud size σx,y,z(t). As V(t) decreases during the cooling, n0(t) will be significantly larger than n(t), which indicates a better cooling capacity of three-body collisional cooling.

Second, we discuss the experimental limitations for three-body cooling. In our experiment, the range of cooling regime is only about kBT/2μB ~ 50 mG. To implement the cooling in this regime, it requires the stability of the magnetic field better than 10 mG. As shown in Supplementary Note 1, the fluctuation of our magnetic field is about σB = 1.6 mG at 543.3 G, giving the fluctuation of the threshold energy. This fluctuation limits the lowest temperature that we could achieve 2μBσB/(3kB/2) = 0.14 μK. Third, we could improve the theoretical model for three-body cooling. The current model is based on the classical kinetic theory, which agrees with our observation in the weakly interacting regime qualitatively, but deviates from the data near the resonance. We suggest to consider the entropy exchange and use a quantum kinetic model to improve the discrepancies of the whole cooling process25,27,28.

In conclusion, we have verified that a 6Li Fermi gas can be cooled to a higher phase-space density through three-body recombination near a narrow Feshbach resonance. Different from the standard evaporative cooling, our collisional cooling could use Feshbach resonance to selectively expel the atoms with certain kinetic energy. The cooling efficiency of collisional cooling increases as temperature decreases, which enables it as a useful tool for cooling novel systems, such as dipole molecules29 or mixed atomic species30,31,32,33.

Methods

Gas preparation

We prepare a two-component (two lowest hyperfine states of F = 1/2, mF = ± 1/2) 6Li Fermi gas in a crossed-beam optical dipole trap made by a 100 W 1064 nm fiber laser. The trapping potential of the dipole trap we can generate at the highest power is about 5.6 mK. A bias magnetic field of 300 G is added to generate a weakly interacting Fermi gas after the atoms are loaded from the magneto-optical trap. Before starting the evaporative cooling, a radio-frequency pulse is implemented to balance the spin mixture to 50:50. We force evaporatively cool the gas by lowering the trap to a different trap potential to generate different temperatures20. Then, the magnetic field is fast swept over the narrow Feshbach resonance to 570 G to calibrate the initial temperature T(0) and the initial atom number N(0). Notice that the two-component gas is tested to be stable at 570 G and its s − wave scattering length is finite. In our experiments, a 5.10 μK gas is prepared with a trap depth of 33.6 μK, and a 0.60 μK gas is generated with a trap depth of 2.8 μK.

Timing sequence of the magnetic field

The two lowest-energy hyperfine ground-state mixtures of ultracold 6Li Fermi gases have an s-wave narrow Feshbach resonance around 543.3 G with a resonance width of 0.1 G. We adopted the same timing sequence as Fig. 1 of our previous experiment15 to measure the three-body loss rate. The magnetic field is swept from 570 G to a target field Bf near the narrow Feshbach resonance in about 50 ms due to the eddy effect of our setup15. Then, the atoms are held at Bf for a time duration of t. After the three-body recombination, the left atoms are swept back to the initial magnetic field 570 G and wait several hundred milliseconds for a gas re-thermalization. Then, we acquire the time-of-flight absorption images to avoid the high column density induced by the error of the atom number. The fluctuation of our magnetic field is controlled within a ppm-level. More details about the stability of our magnetic field are presented in Supplementary Note 1.