Collisional Cooling of Ultracold Molecules

TL;DR

This paper demonstrates collisional cooling of ultracold NaLi molecules via collisions with ultracold Na atoms, achieving temperatures as low as 220 nK and increasing phase-space density, paving the way for quantum degeneracy.

Contribution

First successful demonstration of collisional cooling of ultracold molecules using atom-molecule collisions, with high elastic to inelastic collision ratio and significant phase-space density increase.

Findings

Achieved molecular temperatures down to 220 nK.

Elastic to inelastic collision ratio exceeds 50.

Increased phase-space density by a factor of 20.

Abstract

Since the original work on Bose-Einstein condensation, quantum degenerate gases of atoms have allowed the quantum emulation of important systems from condensed matter and nuclear physics, as well as the study of novel many-body states with no analog in other fields of physics. Ultracold molecules in the micro- and nano-Kelvin regimes promise to bring powerful new capabilities to quantum emulation and quantum computing, thanks to their rich internal degrees of freedom compared to atoms. They also open new possibilities for precision measurement and the study of quantum chemistry. Quantum gases of atoms were made possible by collision-based cooling schemes, such as evaporative cooling. For ultracold molecules, thermalization and collisional cooling have not been realized. With other techniques such as supersonic jets and cryogenic buffer gases, studies have been limited to temperatures above 10 mK. Here we show cooling of NaLi molecules at micro- and nano-Kelvin temperatures through collisions with ultracold Na atoms, both prepared in their stretched hyperfine spin states. We find a lower bound on the elastic to inelastic collision ratio between molecules and atoms greater than 50 -- large enough to support sustained collisional cooling. By employing two stages of evaporation, we increase the phase-space density (PSD) of the molecules by a factor of 20, achieving temperatures as low as 220 nK. The favorable collisional properties of a Na and NaLi mixture show great promise for making deeply quantum degenerate dipolar molecules and suggest the potential for such cooling in other systems.

Equations

$$v_{\mathrm{rel}}$$

$$n_{\mathrm{ov}}$$

$$l_{\text{eff}}=2\frac{{\int }_0^{\infty }z\exp (-z^2/2{\sigma }^2)dz}{{\int }_0^{\infty }\exp (-z^2/2{\sigma }^2)dz}=\sqrt{\frac{8}{\pi }}\sigma ,$$

$${\Gamma }_{\mathrm{th}}\approx \frac{(I/N_{\text{NaLi}}){\sigma }_{\mathrm{el}}{v}_{\mathrm{rel}}}{3/\xi },$$

$$\frac{1}{\Delta T}\frac{d(\Delta T)}{dt}=-{\Gamma }_{\mathrm{th}}=-\frac{\xi }{3}{n}_{\mathrm{ov}}{\sigma }_{\mathrm{el}}{v}_{\mathrm{rel}}.$$

Full text

The full potential of ultracold atoms was not realized until the advent of collision-based cooling methods such as evaporative and sympathetic cooling. Although atomic systems have been recently used to demonstrate laser cooling to quantum degeneracy, these schemes still require collisional thermalizationSchreckLaserCool ; VuleticLaserCooling . Therefore, there has been much work over the past 15 yearsHutsonFirstSymp to achieve collisional cooling of ultracold molecules. Buffer gas coolingBufferGasCaH cannot push below $100\;{\rm mK}$ owing to the rapidly diminishing vapour pressure of buffer gases at such temperatures. Supersonic expansionSupersonic_CO can produce temperatures around $100\;{\rm mK}$. Controlled collisions in crossed molecular beamsAr-NO_Collisions can decrease the laboratory-frame velocity of particles while narrowing the velocity distribution. However, this technique has not been demonstrated below about $500\;{\rm mK}$. Merged supersonic beams can be used to study collisions at energies equivalent to a temperature of $10\;{\rm mK}$ (ref. mergedbeams ).

Cooling below $100\;{\rm mK}$ calls for trapping molecules in magnetic or electrostatic traps and for good collisional properties (that is, a ratio of elastic to inelastic collisions much greater than 1). Such traps require preparing molecules in weak-field seeking states, which are never the absolute ground state, allowing inelastic state-changing collisions to eject the cold molecules from the trap. A variety of systems have been proposed for evaporative or sympathetic cooling of molecules HutsonFirstSymp ; HutsonPolandSymp ; HutsonHSymp ; TarbuttCaFSymp ; TscherbulSrFSymp ; BohnCaOH . So far, elastic collisions have been observed clearly in ${\rm O}_{2}$ at temperatures below $1\;{\rm K}$NareviciusO_2 and possibly in OH radicals around $10\;{\rm mK}$ (ref. JunOHRevised corrects an earlier report JunOHEvap ). In the ${\rm O}_{2}$ case, inelastic collisions prevent thermalization and collisional cooling.

In recent years, the assembly of molecules from ultracold atomsJinYeKRb and the direct laser cooling of moleculesDeMilleMOT ; DoyleMOT ; TarbuttMOT ; JunMOT have both expanded to new molecules and temperature regimes. These techniques have achieved molecular systems at temperatures less than $100\;{\rm mK}$, raising the challenge of collisional cooling in the micro- and nanokelvin regimes. Optical traps enable trapping of the absolute ground state, which removes the concern of state-changing collisions. However, collisional cooling in the absolute ground state of chemically stable systems has not yet been realizedHutsonAlkaliChem . By contrast, chemically stable molecular species have shown anomalously high inelastic loss rates that preclude collisional cooling, possibly owing to collision complex formationStickyBohn or interactions with optical trapping beamsKarmanMolLoss .

In this study we observe sympathetic cooling of rovibrational ground-state triplet ${}^{23}{\rm Na}^{6}{\rm Li}$ molecules by ${}^{23}{\rm Na}$ atoms, both of which are prepared in their upper stretched hyperfine spin states (that is, states with both nuclear and electronic spins aligned along the direction of the magnetic bias field). Although sympathetic cooling of one atomic species by another has been observed in various ultracold atomic mixturesWiemanSympBECs , triplet NaLi was considered unlikely to have sufficiently good collisional properties to support such cooling. NaLi has energetically allowed chemical reactions, even in the electronic ground state (that is, a singlet state), and the triplet state has an electronic excitation energy of $0.8\;{\rm eV}$ or $10,000\;{\rm K}$.

Furthermore, theoretical studies on various systems have explored the possibility of suppressing inelastic collisions and reactions by spin polarization, giving pessimistic predictions for triplet molecules and more favourable ones for doublet moleculesNHChemReacts ; collisionTheoryKrems ; HutsonNHpN ; TscherbulSrFSymp ; TscherbulCaHLi . Nonetheless, here we report clear thermalization and sympathetic cooling of triplet NaLi to 220 nK. We observe 20-fold increases of phase-space density (PSD), opening the possibility of collisional cooling to deep quantum degeneracy.

Our experimental setup is summarized in Fig. 1. Similarly to our previous workNaLiFesh ; NaLiGround , we produce about $3.0\times 10^{4}$ NaLi molecules in the rovibrational ground state of the triplet potential using a mixture of Na and Li atoms (further details in Methods). We also prepare about $1.0\times 10^{5}$ Na atoms in the upper stretched hyperfine state (in the low-field basis, $\ket{F,m_{F}}=\ket{2,2}$, where $F$ and $m_{F}$ are the hyperfine and magnetic quantum numbers, respectively) in a one-dimensional (1D) optical lattice formed by 1,596-nm light. Owing to the differential polarizability at 1,596-nm, molecules feel a deeper trapping potential than atoms. This results in a sudden increase of the potential energy as atoms associate and form molecules (see Fig. 2a). Immediately after production, the effective temperature of the molecules is $2.80(6)\;\mu{\rm K}$ and the temperature of Na atoms is $2.42(3)\;\mu{\rm K}$ (all uncertainties are as defined in the legend of Fig. 2). As the molecules thermalize with the Na atoms and a hot fraction of atoms evaporates out of the trap, the temperatures of both particles settle to $2.23(6)\;\mu{\rm K}$ (see Methods for molecular thermometry).

Although this initial settling of temperatures hints at sympathetic cooling, we are able to see much stronger effects by forced cooling and heating of Na atoms. We evaporate Na atoms with almost no loss of molecules by taking advantage of the particles’ different polarizabilities: $\alpha_{\text{NaLi}}/\alpha_{\text{Na}}=(m\omega^{2})_{\text{NaLi}}/(m\omega^{2})_{\text{Na}}\approx 2.6$, where $\alpha_{i}$ is the polarizability of particle $i$, $\omega$ is the angular frequency for oscillation in the trap and $m$ is the particle mass. The curve in Fig. 2c shows the thermalization between molecules and atoms with a large initial temperature difference after an exponential evaporative cooling ramp followed by a recompression of the trap (see Fig. 2 legend). As we hold the particles in the trap, their temperatures approach each other. Owing to the large particle number ratio, $N_{\text{\rm Na}}/N_{\text{\rm NaLi}}\approx 7$, the molecule temperature decreases by $0.68(9)\;\mu{\rm K}$ after thermalization, whereas the temperature of Na atoms increases only by $70(30)\;{\rm nK}$. However, if the Na is removed immediately before the hold time, the molecule temperature remains fixed during the same period.

As further evidence of thermalization, the sympathetic heating of molecules with hot atoms is shown in Fig. 3. We first prepare the atom–molecule mixture fully thermalized at $1.5\;\mu{\rm K}$, after a 10-ms-long exponential evaporative ramp, followed by a 200-ms-long recompression of the trap to the initial trap power of $1.5\;{\rm W}$ ($50\;\mu{\rm K}$ trap depth). Then, we selectively heat the atoms to $2.07(5)\;\mu{\rm K}$ by sinusoidally modulating the trap amplitude at twice the sodium trap frequency ($2\omega_{Na}=2\pi\times 920\;{\rm Hz}$) with a modulation amplitude of $20\%$ of the trap depth for 100 ms. After 200 ms, particles thermalize and the molecule temperature rises to the temperature of the heated Na atoms. When the trap amplitude is modulated in the same manner without the Na atoms, the temperature of molecules remains at $1.59(5)\;\mu{\rm K}$. The heating process for sodium also induces centre-of-mass motion and breathing oscillations that cause the Na density to depend on time in the early stages of sympathetic heating, which is the reason for the delay in heating and for the non-exponential thermalization curve.

To measure the rate of thermalization, we return to the simpler situation of cooling in Fig. 2. We fit the temperature to a simple exponential model, $T(t)=(T_{0}-T_{\infty})\exp(-\Gamma_{\rm th}t)+T_{\infty}$, where $T_{0}$ and $T_{\infty}$ represent the initial and infinite-time-limit temperature of molecules, respectively, and obtain the thermalization rate, $\Gamma_{\text{th}}=26(5)\;\text{s}^{-1}$. Considering that thermalization requires about $3/\xi$ collisions (for Na–NaLi, $\xi\approx 1$; see Methods), we obtain the average elastic collision rate per particle, $\Gamma_{\text{el}}$, from the measured thermalization rate: $\Gamma_{\text{el}}\approx(3/\xi)\Gamma_{\text{th}}=80(14)\;\text{s}^{-1}$. In the presence of Na atoms, the initial loss rate of molecules is $\Gamma_{\text{inel}}=1.29(6)\;\text{s}^{-1}$, as obtained from a fit to the exponential loss model $N(t)=N_{0}\exp(-\Gamma_{\text{inel}}t)$ (red dashed line in Fig. 2b). Comparing the average elastic collision rate to the total loss rate, we obtain the ratio of elastic to inelastic collisions for NaLi, $\gamma\gtrsim\Gamma_{\text{el}}/\Gamma_{\text{inel}}=62(12)$. Without Na atoms, the molecular loss follows a two-body loss model, $N(t)=N_{0}/(\beta t+1)$ (blue solid line in Fig. 2b), from which we obtain an initial loss rate of $\beta=1.02(6)\;\text{s}^{-1}$. By considering the difference between $\Gamma_{\text{inel}}$ and $\beta$ as the effective inelastic loss rate for collisions between Na and NaLi, we obtain the ratio of “good” to “bad” collisions, $\gamma\approx\Gamma_{\rm el}/(\Gamma_{\rm inel}-\beta)\approx 300$ with an uncertainty of $40\%$. The Na–NaLi loss rate constant is $4.0\pm 1.3\times 10^{-13}\;{\rm cm^{3}\ s^{-1}}$; this is more than two orders of magnitude smaller than the universal loss model rate constant for Na–NaLi $s$-wave collisionsunivlossrate , which is $1.7\times 10^{-10}\;{\rm cm^{3}\ s^{-1}}$ (see Methods). Whereas Na and NaLi in their upper stretched states form a stable mixture, as shown, preparing Na atoms in their lowest hyperfine state ($\ket{F,m_{F}}=\ket{1,1}$) gives a loss rate consistent with the universal loss modelNaLiGround .

We consider the two different evaporation ramps shown in Fig. 4. The molecule numbers, temperatures and PSDs resulting from these ramps are displayed in Fig. 5. For a single-species system with losses, optimal evaporation can be achieved by a single, continuous decrease of the trap depth. We achieve an increase in PSD by a factor of $7$ by using a single stage of evaporation (Fig. 4a, red squares in Fig. 5). This increase is limited by the low initial Na density. In our system, thermalization is dominated by the Na density, whereas loss is dominated by the NaLi density. To overcome the low Na density, we shorten the initial evaporation and recompression cycle ((1) and (2) in Fig. 4b). This cools the Na atoms and quickly increases the density of the cold Na for fast thermalization. After the cold, dense Na efficiently pre-cools the molecules during a short hold of 30 ms in the tight trap (3), we apply the second evaporation ramp (4). In this double evaporation, we achieve a peak PSD of $2.0(2)\times 10^{-2}$ (blue circles in Fig. 5), which is $20$ times higher than the initial PSD. Before the final recompression of the trap, the lowest temperature of the molecules is $220(20)\;{\rm nK}$.

With Na present, more molecules survive the evaporation sequence as we evaporate Na atoms to a lower trap depth. This is mainly due to suppressed two-molecule loss. The loss rate constant scales linearly with the temperature owing to the p-wave character of the collisionsunivlossrate . Evaporation to too low trap depth decreases the Na density, which makes the sympathetic cooling of molecules inefficient, and anti-evaporation (heating due to the preferential loss of the coldest molecules) dominates. This appears as a temperature increase below a trap power of $0.17\;{\rm W}$ ($5.7\;{\rm\mu K}$ trap depth) for a single stage of evaporation, or $0.14\;{\rm W}$ ($4.5\;{\rm\mu K}$ trap depth) for two-stage evaporation. Without Na, the ramp-down in trap depth, (i), is adiabatic to a trap power of around $0.3\;{\rm W}$, as evidenced by the constant molecule numbers and temperatures. Below that trap power, the ramp-down starts becoming non-adiabatic. As a result, a hot fraction of molecules escape, and both the molecule number and the temperature start decreasing. We note that because fermionic molecules do not thermalize by themselves, this temperature should be regarded as the average kinetic energy of molecules that are not in thermal equilibrium.

The favourable collisional properties of the spin-polarized Na–NaLi mixture in fully stretched hyperfine states result from strong suppression of electronic spin flips during collisions, which could otherwise lead to fast reactive or inelastic losses. In fully stretched states, direct spin exchange is forbidden and residual spin flips can occur only by weaker interactions. The main contribution to the weak spin flips is from dipolar relaxation due to direct coupling between the magnetic dipole moments of the spins of Na and NaLi. Studies in other spin-polarized systemsDoyleNHpN ; TscherbulSrFSymp ; TscherbulCaHLi predict spin flips induced by anisotropic electrostatic interaction at short range and intramolecular spin–spin and spin–rotation couplings to be weaker at ultracold temperatures. Dipolar relaxation can be eliminated by using the strong-field-seeking stretched state, but this was not necessary in our work. It is possible that the Na–NaLi system is favourable owing to the low reduced and molecular masses and the small relativistic effects, which result in a large rotational constantWesleySpinRelax ; collisionTheoryKrems , low density of statesStickyBohn and small spin–orbit couplingSpinOrbitSpinRelax , respectively. Further, ref. NHChemReacts has shown that intramolecular spin–spin coupling causes spin flips, leading to chemical reactions for triplet NH molecules, thwarting evaporative cooling. Our results show a clear need for further work to determine what other molecules—and what conditions of internal states and external fields—would be suitable for collisional cooling at ultracold temperatures.

Technical upgrades to our apparatus should allow cooling into the quantum degenerate regime by increasing the initial Na density. The dominant loss mechanism—$p$-wave collisions between molecules—is expected to slow down compared to thermalization at lower temperatures, improving the efficiency of cooling. Further, the atomic and molecular states used in this work are magnetically trappable. Magnetic trapping would allow the use of much larger samples with perfect spin polarization. This would also eliminate the major concern of molecular loss induced by trapping lightKarmanMolLoss , making NaLi an ideal system for studying quantum chemistry. Concerns about sticky collisionsStickyBohn , which can lead to a long-lived collision complex with high density of states, have led to pessimism over the prospects of achieving collisional cooling of ultracold molecules. This work demonstrates that sticky collisions do not limit NaLi, and probably other low-mass systems, suggesting a bright future for cooling light molecules.