Magneto-optical trapping of a diatomic molecule

Abstract

Laser cooling and trapping are central to modern atomic physics. The most used technique in cold-atom physics is the magneto-optical trap (MOT), which combines laser cooling with a restoring force from radiation pressure. For a variety of atomic species, MOTs can capture and cool large numbers of particles to ultracold temperatures (less than ∼1 millikelvin); this has enabled advances in areas that range from optical clocks to the study of ultracold collisions, while also serving as the ubiquitous starting point for further cooling into the regime of quantum degeneracy. Magneto-optical trapping of molecules could provide a similarly powerful starting point for the study and manipulation of ultracold molecular gases. The additional degrees of freedom associated with the vibration and rotation of molecules, particularly their permanent electric dipole moments, allow a broad array of applications not possible with ultracold atoms¹. Spurred by these ideas, a variety of methods has been developed to create ultracold molecules. Temperatures below 1 microkelvin have been demonstrated for diatomic molecules assembled from pre-cooled alkali atoms^2,3, but for the wider range of species amenable to direct cooling and trapping, only recently have temperatures below 100 millikelvin been achieved^4,5. The complex internal structure of molecules complicates magneto-optical trapping. However, ideas and methods necessary for creating a molecular MOT have been developed^{6,7,8,9,10,11} recently. Here we demonstrate three-dimensional magneto-optical trapping of a diatomic molecule, strontium monofluoride (SrF), at a temperature of approximately 2.5 millikelvin, the lowest yet achieved by direct cooling of a molecule. This method is a straightforward extension of atomic techniques and is expected to be viable for a significant number of diatomic species^6,7. With further development, we anticipate that this technique may be employed in any number of existing and proposed molecular experiments, in applications ranging from precision measurement¹² to quantum simulation¹³ and quantum information¹⁴ to ultracold chemistry¹⁵.

Main

In laser cooling, the species of interest is illuminated by counter-propagating pairs of laser beams. The laser frequency is detuned just below resonance with an electronic transition. Owing to the Doppler effect, a particle is more likely to absorb a photon from the beam that opposes the particle velocity, slowing the particle. If the subsequent spontaneous emission returns the particle to the same initial state (or another state also excited by the lasers), then this process of absorption and spontaneous emission can be repeated many times (a ‘cycling transition’). This provides a damping force. However, this force does not act to spatially confine the particles. In a MOT, cooling and confinement are produced simultaneously. Three orthogonal pairs of laser beams are spatially overlapped with a quadrupole magnetic field. For a pair of ground and excited state Zeeman sublevels, a deviation from the trap centre generally induces a Zeeman shift that moves the transition closer to or further from resonance with the lasers. For a small deviation along a given laser axis, the transition shifted closest to resonance can be driven by a particular laser polarization; this polarization is chosen for the laser counter-propagating to the direction of the deviation, while the co-propagating laser has the orthogonal polarization. Hence, on average there is a confining force restoring particles towards the centre of the trap.

The SrF MOT described in this work uses techniques very similar to those used for standard atomic MOTs. Anti-Helmholtz coils create a static quadrupole magnetic field (with an axial field gradient, dB_z/dz, equal to twice the radial field gradient, dB_ρ/dρ), and pairs of circularly polarized laser beams pass through the centre of this field along three orthogonal axes. The trap is loaded with pulses of SrF from a cryogenic buffer gas beam source¹⁶ that have been slowed using radiation pressure¹⁰. However, the level structure of SrF dictates that our MOT is similar to a rarely used and poorly understood configuration of atomic MOT (known as type II), which differs in certain characteristics from the most common atomic MOTs (type I).

The type-I MOT employs an F → F′ = F + 1 cycling transition, where F is the total angular momentum quantum number and the prime indicates the excited state. For a given polarization, particles in all ground-state Zeeman sublevels are optically coupled to the excited state (all states are ‘bright’). The type-II MOT operates on an F → F′ = F or F → F′ = F − 1 transition, where certain ‘dark’ ground state sublevels are not optically coupled to the excited state. The presence of dark states reduces the spontaneous photon scattering rate; this rate can go to zero in the absence of a mechanism to ‘remix’ dark states with the bright states. Moreover, scattering alone does not ensure a confining force: the scattering rate from the laser counter-propagating to a particle’s deviation from the trap centre must exceed the scattering rate from the laser co-propagating. In type-II systems, the level structure makes it possible for particles to be pumped into a state dark to the counter-propagating laser but bright to the co-propagating laser, so a confining force is not guaranteed^17,18. Consequently the damping rate and restoring force may be significantly smaller for type-II MOTs than for those of type I. There appears to be no widely accepted understanding of the mechanisms responsible for generating a restoring force in type-II MOTs. Nevertheless, type-II MOTs have been demonstrated in several atomic systems^18,19,20,21; for these, relatively weak confinement and slightly elevated temperature are typically observed. The rotational structure of diatomic molecules generically requires their cycling transitions to correspond to a type-II MOT system; the SrF MOT described here is hence also of type II.

We use a previously demonstrated scheme for creating a cycling transition in SrF (refs 8,9,10) on the X²Σ⁺ → A²Π_1/2 electronic transition (see Methods). Calculated vibrational branching fractions , for decay of the excited state with vibrational quantum number v′ to the ground state with vibrational quantum number v, suggest that only three vibrationally excited levels in the X state (v = 1, 2, 3) are significantly populated after ∼10⁶ photon scattering events, corresponding to ∼1 s of optical cycling for typical scattering rates (see below). Hence, three vibrational repumping wavelengths are expected to be sufficient to trap SrF for the ∼1 s timescale typical of atomic MOTs. We denote the laser addressing the X(v = i) → A(v′ = j) transition by , so that the three repump lasers are labelled , and ; the primary and secondary trapping lasers are denoted and respectively. (The need for a second trapping laser is explained below.) Radio-frequency sidebands on the , , and lasers address spin-rotation/hyperfine (SR/HF) substructure in the X²Σ state (Fig. 1a). Rotational branching is eliminated by driving an N = 1(J = 3/2, 1/2) → J′ = 1/2 transition⁷, where N is the total angular momentum excluding electronic and nuclear spin and J is the total angular momentum excluding nuclear spin. Driving these transitions optically pumps population into dark ground-state Zeeman sublevels not excited by the laser²². These dark states must be remixed with the bright states for cycling to continue. In this work, remixing occurs both due to Larmor precession in the quadrupole magnetic field and due to optical pumping as molecules move through the complicated optical polarization gradients arising from the orthogonal pairs of circularly polarized laser beams²³.

Figure 1: Experimental set-up.

a, Optical addressing scheme. Relevant energy levels are shown for a positive B-field; m_F is the angular momentum projection. The laser (red dashed lines) primarily addresses the |J = 3/2, F = 2〉, |J = 3/2, F = 1〉 and |J = 1/2, F = 0〉 states, while the laser (orange dashed line) addresses the |J = 1/2, F = 1〉 state. Arrows show σ⁻ transitions (red) driven by the laser light and σ⁺ transitions (orange) driven by the laser light. Transitions from |J = 3/2, F = 1〉, |J = 1/2, F = 0〉 and |J = 1/2, F = 1〉 are marked with transparent arrows for clarity; each line width is proportional to the transition strength. The lasers are drawn at the ground state energy with which they would be resonant. b, Experiment schematic showing the MOT (red) and slowing (green) laser beam paths. The line widths indicate beam diameters, and the grey arrows illustrate the default magnetic field gradient. The waveplates and mirrors used to create the vertical MOT beam (dashed line) are not shown.

PowerPoint slide

The optimal polarization of the trapping light depends both on the sign of the difference in magnetic moment between the ground and excited states of the cycling transition and on the orientation of the quadrupole magnetic field. Within the sublevels of the SrF X²Σ(N = 1) state, two of the four SR/HF manifolds have positive magnetic g-factors, one has g = 0, and the remaining manifold has g < 0 (Fig. 1a); the A²Π_1/2 state has g ≈ 0. The presence of both negative and positive g-factors means that laser frequencies addressing the different SR/HF manifolds must have different polarizations for optimal trapping. This is achieved by combining the light with single-frequency light of the opposite polarization from the secondary trapping laser (see Methods).

The pulse of molecules from the beam source begins with laser ablation of an SrF₂ target at t = 0 ms. The slowing is applied from t = 0 ms to t = 40 ms (see Methods). Molecules in the trapping region are detected via laser-induced fluorescence (LIF) from the X → A cycling transition at λ₀₀ = 663.3 nm and imaged onto a CCD (Fig. 1b; see Methods).

Realization of magneto-optical trapping results in increased LIF from a small area in the trapping region near the B-field zero. This localized LIF persists for an increased duration compared to the spatially broad LIF from the untrapped, slowed molecular beam and suggests that molecules are confined in this region. To observe the MOT, the , , and lasers must be present with the proper detunings (and, for the trapping lasers, polarizations), the B-field gradient must be present (dB_z/dz ≠ 0) and the laser slowing must be applied. The laser is not necessary to observe the MOT but results in increased LIF. Maximum LIF is observed with the laser detunings (where Γ = 2π × 7 MHz is the natural linewidth) and dB_z/dz = 15 G cm⁻¹ (see Methods); these parameters are similar to those for standard atomic MOTs. Unless stated otherwise, measurements are made with these default parameters.

The proper polarization for the trapping light depends on the sign of dB_z/dz. Reversing either the sign of dB_z/dz or the circular polarization of the MOT trapping light should create an anti-restoring force and prevent MOT formation. Reversing both the sign of dB_z/dz and the polarization in tandem, however, should return the system to a restoring configuration, and the MOT should be realized again. We observe the expected behaviour for these four states of the system as shown in Fig. 2, confirming magneto-optical trapping of SrF molecules. From these images we also determine the MOT cloud position and size by fitting the LIF intensity profile to a two-dimensional Gaussian; we find typical r.m.s. widths of ρ_rms = 4.1(1) mm (radial) and z_rms = 2.6(1) mm (axial).

Figure 2: Magneto-optical trapping of SrF.

Shown are images (averaged over 600 pulses) of LIF in the trapping region for different polarizations and signs of dB_z/dz. The grey crosses mark the position of the magnetic field zero. Gravity is in the z-direction. For clarity, data in the images are smoothed with a Gaussian of width σ = 0.7 mm.

PowerPoint slide

To probe the confining and damping forces in the MOT, the molecular cloud’s response to a rapid displacement of the trap centre is measured. During loading, a magnetic bias field offsets the MOT centre radially, along the axis of the molecular beam. The bias field is then switched off, releasing the trapped molecules into the unbiased potential. Using a short camera exposure (Δt_exp = 5 ms), the molecular cloud’s position is measured as a function of time (Fig. 3a; see Methods). The cloud exhibits damped harmonic motion as it moves towards the equilibrium position, with oscillation frequency ω_ρ = 2π × 17.2(6) Hz and damping coefficient α/m_SrF = 140(10) s⁻¹, where m_SrF is the mass. With the measured radial width (ρ_rms), the equipartition theorem is used to find the radial MOT temperature: , where k_B is the Boltzmann constant. If we assume the relation holds, as for a standard atomic MOT in a quadrupole field²⁴, we find the axial oscillation frequency ω_z = 2π × 24.3(9) Hz. The measured MOT axial width z_rms then corresponds to an axial temperature T_z = 2.0(1) mK.

Figure 3: Measurement of MOT properties.

a, MOT cloud response to rapid displacement of the trap centre. Top, LIF images averaged over 1,600 pulses (images are placed at the appropriate position on the x axis); middle, 2D Gaussian fits to LIF images; bottom (main panel), the extracted radial position as a function of time. The fit is to the motion of a damped harmonic oscillator. Zero is set at the position of the MOT with no displacement. b, Free expansion of the MOT following release. For a given position on the x axis (free expansion time), insets show LIF images averaged over 2,000 pulses (top) and 2D Gaussian fits (bottom), and data points show measured MOT radial (black squares) and axial (red circles) widths. In both a and b, images are rescaled to the maximum value at each time. c, Main panel; LIF in the trapping region versus time t for MOT with (red squares) and without (purple circles) the repump laser, for untrapped damping (blue triangles), and for damping/anti-restoring (black triangles) configurations (see Methods). Overlaid are single exponential fits. Inset, MOT lifetime versus MOT laser beam diameter. Error bars, ±1σ confidence interval from a χ² analysis of the fits. LIF images are smoothed as in Fig. 2; all fits and analyses are performed using the unsmoothed data.

PowerPoint slide

To verify the MOT temperature, ballistic expansion measurements are performed. Trapped molecules are released at t_rel = 90 (by extinguishing the laser). After a time of flight Δt_TOF, the light is restored, and the resulting LIF is imaged onto the CCD. A short imaging time (Δt_exp = 5 ms) is used to accurately determine the expanded cloud’s size (see Methods). For an initial Gaussian spatial distribution and a Boltzmann distribution of velocities (with no correlation between position and velocity), the widths z_rms and ρ_rms of the expanding cloud are given by:

The data and associated fits are shown in Fig. 3b. The slopes of the fits give the temperatures, which are then used with the intercepts to determine ω_ρ and ω_z. This treatment plots the measured cloud width (which is an average over the camera exposure duration) at the start time of imaging/illumination. Monte Carlo simulations for the measured trap frequencies suggest that the cloud continues to expand during the short illumination interval, and therefore the extracted width is an upper bound for the actual width at the imaging time, t_im = t_rel + Δt_TOF. Hence this treatment of the data yields upper limits on the MOT temperature. The fits give T_ρ ≤ 2.7(3) mK and T_z ≤ 2.1(1) mK, with trap frequencies ω_ρ = 2π × 19(1) Hz and ω_z = 2π × 29(1) Hz, corresponding to spring constants κ_ρ = 2.5(3) × 10⁻²¹ N m⁻¹ and κ_z = 5.9(4) × 10⁻²¹ N m⁻¹. These values are in good agreement with the values from the MOT oscillation measurement. The temperatures are roughly an order of magnitude greater than the SrF Doppler temperature, T_D = Γ/(2k_B) = 160 µK. Temperatures well above the Doppler temperature are also reported in work with atomic type-II MOTs²¹. These spring constants are two to three orders of magnitude smaller than for typical type-I atomic MOTs²⁵ and approximately one order of magnitude smaller than reported values for atomic type-II MOTs²¹, though measurements of the spring constants in type-II atomic MOTs are so few that ‘typical’ behaviour is difficult to characterize. A third measurement of the MOT temperature T_MOT is performed using the release-and-recapture method, yielding a temperature in good agreement with the other methods (see Methods).

Measurement of the spontaneous photon scattering rate for trapped molecules, R_sc, allows the number of trapped molecules N_MOT to be determined via fluorescence detection. We find (see Methods). Based on the efficiency of the LIF detection system, measured to be ∼0.8%, the MOT population is estimated at N_MOT ≈ 300 SrF molecules, corresponding to a peak trap density of n_MOT ≈ 600 cm⁻³.

The MOT lifetime, τ_MOT, is obtained by measuring LIF in the trapping region as a function of time and fitting a single exponential decay curve to the data after t = 67 ms, as shown in Fig. 3c. This start time avoids significant contributions to the LIF signal from the slowed but ultimately untrapped part of the molecular beam. We find τ_MOT = 56(4) ms, significantly shorter than is typically seen in atomic MOTs. When the repump laser is not present, τ_MOT = 27(2) ms. We have verified that neither collisions with ballistic helium from the buffer-gas beam nor collisions with background gases are the primary loss mechanism from the trap. Optical pumping into the dark X²Σ(v = 4) state would result in τ_MOT ≈ 1 s for the measured value of R_sc, and off-resonant excitation populating dark rotational levels is found to be insignificant (see Methods).

The strikingly low restoring force measured suggests another explanation for the low value of τ_MOT: the trap depth is not large compared to k_BT_MOT, as in typical atomic MOTs, so a significant fraction of molecules can escape the trap simply by being in the high-energy tail of the Boltzmann distribution. The MOT trap depth U_MOT can be estimated using , assuming that κ_ρ is constant to the edges of the MOT beam (d_λ is the beam diameter). This gives U_MOT/k_B = 10(1) mK ≈ 4T_MOT, in contrast to atomic MOTs where U_MOT/k_B ≈ 1 K ≈ 1,000T_MOT. We presume that rapid molecule–light interactions maintain a constant temperature in the MOT, leading to continuous loss rather than evaporative cooling as in a conservative trap (where the trap leaves the total energy of a trapped sample unchanged). A simple model for the rate of particle escape under these conditions lends credence to this explanation for the short MOT lifetime (see Methods). Additional support comes from the observation that τ_MOT depends strongly on the MOT beam diameter (Fig. 3c, inset). Reducing the beam diameter d_λ from 23 mm to 21 mm (a <1% decrease in power) reduces τ_MOT by ∼30%. We are unaware of any other trap loss mechanism that might exhibit this behaviour.

For our cycling transition, the maximum restoring force F_max = κ_zd_λ/2 corresponds to a confining photon scattering rate R_con = F_max/(k) = 5(2) × 10⁴ s⁻¹ (where k = 2π/λ₀₀ is the wavenumber) confining photons from a single MOT beam, only ∼1% of R_sc. The small value of R_con/R_sc may be understood qualitatively by noting that in a simple one-dimensional model, the angular momentum level structure of our system (J = 3/2, 1/2 → J′ = 1/2) ensures that each photon scattered in the ‘correct’ (confining) direction on average must be followed by a photon scattered in the ‘incorrect’ (anti-confining) direction in order to resume optical cycling¹⁸. In three dimensions, with complicated polarization gradients and other means of remixing, this relation no longer holds exactly. Nonetheless, the mechanism behind the slight bias of scattering events towards the trap centre that leads to the weak, yet non-zero confining force is not well understood. This same type of level structure is also the defining characteristic of atomic type-II MOTs, which exhibit qualitatively similar characteristics to our SrF MOT (such as extended spatial extent and elevated temperature) although with reported stronger confinement^19,20,21. Hence the weak trapping and only moderately low temperature observed in our SrF MOT are believed to be due to the angular momentum level structure rather than any other issues related specifically to using a diatomic molecule rather than an atom. Despite these limitations, our method succeeds in trapping and cooling molecules to the lowest temperature reported for any direct-cooling method to date.

Future work is expected to allow substantial increases in the density and the number of molecules trapped. For example, the trappable flux may be increased by implementing a more efficient slowing method²⁶ or by transversely confining the molecular beam as it is slowed²⁷. A variety of methods may enable increased trap depth by increasing the fraction of scattered photons contributing to the confining force (R_con/R_sc), which could in turn increase trap lifetime, capture velocity, density, and number of molecules trapped. This could be accomplished, for example, by rapid synchronous reversing of the MOT magnetic field gradient and the laser circular polarizations, as recently demonstrated in two-dimensional magneto-optic compression of a molecular beam¹¹, or alternatively by using a microwave electric field to pump molecules in anti-trapped Zeeman sublevels into trapped levels by driving transitions through other rotational states^7,8.

Although magneto-optical trapping of molecules is in its infancy, our results demonstrate that this technique could be applied in a straightforward way to a significant number of diatomic species^6,7. The MOT has proved indispensable for cooling and trapping many atomic species; with further development, we expect that it may prove similarly useful for producing ultracold gases of diatomic molecules. Such an advance is expected to enable a wide range of new experiments including tests of the standard model of particle physics^28,29, sensitive searches for variations of fundamental constants³⁰, and studies of novel chemical dynamics in the ultracold temperature regime⁷.

Methods

Cycling scheme and level structure

The X²Σ⁺ → A²Π_1/2 electronic transition employed in this work has a lifetime τ_A = 1/Γ = 24.1 ns. The vibrational branching for this excited state is shown in Extended Data Fig. 1a and dictates that only four lasers (one main cycling laser and three vibrational repump lasers) are required to cycle ≳ photons. In practice we use three repump lasers (, and ) at wavelengths λ₁₀ = 686.0 nm, λ₂₁ = 685.4 nm and λ₃₂ = 684.9 nm, in addition to the primary and secondary trapping lasers ( and ) at .

The SR/HF structure for the X²Σ(v = 0, N = 1) state in the presence of a weak magnetic field is shown in Extended Data Fig. 1b. To address each SR/HF manifold, the primary trap laser () light and the repump laser (, and ) light are phase-modulated with electro-optic modulators (EOMs). For the three repump lasers, the modulation frequency f_mod = 42.5 MHz is chosen so the first- and second-order sidebands address all four SR/HF transitions. The value of f_mod = 40.4 MHz for the light is chosen to minimize the root-mean-squared (r.m.s.) value of the detuning for the upper three SR/HF levels at B = 0 G, while a separate laser () addresses the lowest SR/HF level. The modulation depth M_mod = 2.6 for all these lasers provides equal power in each of the four near-resonant sidebands while minimizing power in other sidebands. For each repump laser, zero detuning (Δ_ij = 0) is defined as the frequency which produces maximal LIF from the molecular beam when the light is applied (and retro-reflected) perpendicular to the molecular beam. For the primary and secondary trapping lasers, first the relative position of is fixed 9 MHz above the highest-frequency sideband; these lasers are then scanned in tandem, and zero detuning () is defined in the same manner as for the repump lasers.

The primary laser light is combined on a polarizing beam splitter with single frequency light of the opposite polarization from the secondary trapping laser. The laser is tuned closer to resonance with the |N = 1, J = 1/2, F = 1〉 state than the closest laser sideband. With sufficient laser intensity, this ensures that molecules in the |N = 1, J = 1/2, F = 1〉 state feel a restoring force from the light that is greater than the anti-restoring force from the nearby frequency component. This additional frequency and polarization component of the light is not used for the vibrational repumping transitions since the radiative forces derived from these lasers are small. (Only 1 − b₀₀ ≈ 1/50 of the photons scattered are from the repump lasers.) Trapping and repump light is delivered to the experiment via a single-mode polarization-maintaining fibre.

At the optimal trap laser detunings, , all trap laser frequencies are detuned to lower frequency than the primary J,F sublevel they address. By default, all repump lasers are tuned to the field-free resonance (Δ₁₀ = Δ₂₁ = Δ₃₂ = 0).

A Breit-Rabi diagram showing the energy dependence of each sublevel versus magnetic field is shown in Extended Data Fig. 1c. The level crossings in the range B = 15–25 G may limit the effective trap radius for a given B-field gradient since, at sufficiently high fields, the trap light frequency addressing the |J = 3/2, F = 1〉 manifold becomes anti-trapping for the |J = 3/2, F = 2〉 manifold. Note that other trapping/anti-trapping level crossings are located at higher magnetic bias fields.

As discussed in refs 29, 31, laser cooling schemes with a large number of resolved ground states can require significantly more power than those employing a two-level system with the same wavelength and electronic excited state lifetime. Briefly, a F = 1 → F′ = 0 type transition will have a saturation intensity 3× higher than the saturation intensity for a two-level system of the same wavelength and lifetime. Resolved ground state energy levels also increase the required intensity by dictating that total laser power be divided up among several frequencies, each driving a weaker transition. Hence, as anticipated, our molecular MOT requires substantially more laser power than standard atomic MOTs. Upon exiting the MOT fibre, the , , , and laser powers are typically 210 mW, 50 mW, 170 mW, 5 mW and 3 mW respectively.

Radiation pressure slowing

The molecular beam is slowed by three lasers, denoted , and (where the ‘s’ superscript indicates slowing), which have powers of 205 mW, 185 mW and 35 mW, respectively. The and lasers are horizontally polarized while the laser is vertically polarized. These lasers are spatially overlapped to produce a single beam with 1/e² intensity diameter d ≈ 3 mm, applied counter-propagating to the molecular beam. A uniform field B^s ≈ 9 G is applied at an angle θ = 45° relative to the linear polarizations of the lasers over the distance ≲ ≲ , where z′ = 0 marks the exit of the cell in the molecular beam source and z′ denotes the downstream distance along the molecular beam. The magnetic field for the slowing is applied only when the slowing lasers are present.

The conditions of the slowing (laser centre frequencies and frequency extents, application time and duration of the slowing, and value of B^s) are optimized by imaging the MOT after the slowed molecular beam pulse has fully subsided (here from t = 80 to t = 110 ms). The optimized frequency detunings of the , and lasers are , and . The spectra are broadened to address a wide range of Doppler shifts associated with the broad velocity spread from the molecular beam source. The spectral widths are 340 MHz, 440 MHz and 570 MHz for the , and lasers respectively (Extended Data Fig. 2). Further details on the slowing can be found in refs 10, 31. Further details on the beam source can be found in ref. 32.

Extended Data Fig. 3 shows a sample slowed velocity profile used to load the MOT, along with the unslowed velocity profile of the source; both profiles are detected upstream of the trapping region at .

Trapping region

The trapping region for the MOT is centred at z′ = 1,382 mm and is separated from the beam propagation region by a differential pumping tube (127 mm long, 12.7 mm diameter) beginning at z′ ≈ 900 mm. In the trapping region, the pressure of all background gas excluding He is P_BG ≈ 4 × 10⁻¹⁰ torr while the helium background pressure is P_He ≈ 2 × 10⁻⁹ torr.

Upon exiting the fibre, the , and lasers are vertically polarized, while the and lasers are horizontally polarized. The beam of all combined frequencies and polarizations, which we refer to as the MOT light, is expanded to a 1/e² intensity diameter of 14 mm. A λ/2 waveplate can rotate all polarizations by 90°.

The MOT light passes six times through the vacuum chamber and is applied at all times. Prior to the first pass, the beam is circularly polarized by a λ/4 waveplate and apertured to a d_λ = 23 mm diameter. Upon exiting the chamber after this first pass, the MOT light polarization is returned to linear by a second λ/4 waveplate. This process is then repeated for the remaining radial, and thereafter, axial dimension. After the axial pass through the chamber, the MOT light is reflected back along its initial path. In this way we provide confinement in three dimensions while using the limited laser power efficiently.

A diagram of the trapping region can be found in the main text (Fig. 1b).

MOT optimization

The optimum magnetic field gradient for the MOT is found to be dB_z/dz = 15 G cm⁻¹; the MOT is visible between 4 and 30 G cm⁻¹. The MOT is sensitive to the values of the laser detunings Δ₀₀ and (and less sensitive to the value of Δ₁₀), as shown in Extended Data Fig. 4, but insensitive to the detunings of the and lasers.

Spontaneous scattering rate for trapped molecules

The spontaneous scattering rate R_sc is measured by blocking the repump light at t_bl = 58.6 ms and observing the LIF decay constant, denoted τ_{v = 2}, as molecules are optically pumped into the now-dark X²Σ(v = 2) state. LIF is recorded as a function of imaging start time t_im, which is scanned from t_im = 54 ms to t_im = 62 ms. The finite duration of the camera exposure, Δt_exp = 1 ms, results in a recorded LIF signal Y(t_im) that is a convolution of the real instantaneous LIF intensity, denoted X(t), and the camera exposure time, that is:

Given the comparatively long unperturbed MOT lifetime, X(t) is modelled as a linear function before the blocking of the repump light (t = 54 ms to t₀ = t_bl − Δt_exp = 57.6 ms), followed by (from t₀) an exponential decay plus an additional linear background term. This background term, included to account for LIF from the tail of the slowed but untrapped molecular beam, is deduced from a fit to the data from t = 59 ms to t = 62 ms. This function has the form

where m_MOT and c_MOT (m_bg and c_bg) are the gradient and intercept respectively of the linear fit to the LIF from the MOT (background), D₀ is the amplitude coefficient of the exponential decay term, and H(t) denotes the Heaviside step function.

From the fitted value and the calculated vibrational branching fraction b₀₂ ≈ 0.0004, we estimate .

The measured scattering rate is close to the maximum scattering rate for this system, (ref. 33). The value of R_sc is similar to those measured in atomic MOTs. This observation suggests the possibility of producing strong confining and damping forces, roughly comparable to those in atomic MOTs, if the fraction of scattered photons contributing to the confining force can be greatly increased.

An independent measure of the scattering rate is obtained from the MOT lifetime measurements. The values of τ_MOT with and without the repump laser allow the loss rate into the X²Σ(v = 3) state to be isolated. This, together with the vibrational branching fraction b₀₃ into v = 3, yields a scattering rate R_sc ≈ 2 × 10⁶ s⁻¹, with uncertainty of ∼100% due to the large uncertainty in b₀₃.

Uncertainties in the calculated vibrational branching fractions stem largely from uncertainties in the molecular constants for the A²Π_1/2 state used to calculate Franck-Condon factors. Although the errors in these constants are small, the resulting fractional uncertainty in calculated values of may be significant for off-diagonal terms (v ≠ v′) where Frank-Condon factors are strongly suppressed and vibrational branching fractions are small³¹.

MOT detection

Molecules in the trapping region are detected via laser-induced fluorescence (LIF) from the X→A cycling transition at λ₀₀ = 663.3 nm and imaged onto a CCD. The camera field-of-view encompasses the majority of the trapping region and is approximately centred on the zero of the quadrupole field. Unless otherwise noted, MOT imaging starts at time t_im = 60 ms after ablation, the camera exposure duration is Δt_exp = 60 ms, and the signal is integrated over the entire camera field-of-view. The imaging start time and duration are chosen so that the vast majority of LIF recorded (∼90%) comes from trapped molecules rather than the temporal tail of the slowed molecular beam pulse.

The LIF collection optics consist of a 150 mm focal-length spherical-singlet lens, followed by a F/0.95 camera lens, then a 650-nm-bandpass interference filter, and finally a CCD camera. The interference filter reflects all repump light at λ₁₀ = 686.0 nm, λ₂₁ = 685.4 nm and λ₃₂ = 684.9 nm for any angle of incidence (AOI) and transmits >99% of the λ₀₀ = 663.3 nm light at normal incidence; however transmission at λ₀₀ is reduced for ≳ .

Using the MOT chamber geometry and assuming the distribution of LIF from the MOT is isotropic, we calculate the geometric collection efficiency of the LIF optics to be η_geo = 1.1%. The amount of light reaching the CCD is further reduced by transmission losses (characterized by η_tra) and by AOI-dependent losses of the bandpass filter (characterized by η_fil).

We measure η_tra = 0.84 by tabulating the transmission efficiency of 663.3-nm light through each element of the collection optics at normal incidence. The value of η_fil is measured as follows. Light emission from the MOT is simulated by back-illuminating a thick piece of white Delrin with 663.3 nm light. The front surface of the Delrin is covered except for a 5-mm hole. This creates approximately uniform emission of light over the range of angles incident on the collection optics. The total number of photons hitting the CCD is measured in the presence of all collection optics and again with only the interference filter and CCD present. In this latter configuration, reflection of 663.3-nm light by the interference filter is negligible since all light is near normal incidence. The ratio of these two numbers is then divided by the ratio of solid angle subtended by the collection optics versus by the CCD sensor alone. Finally, dividing by the transmission losses through the lenses gives the filter transmission efficiency η_fil = 0.82 for this geometry. We measure the CCD gain to be G ≈ 5.5 counts per photoelectron and assume the manufacturer-specified quantum efficiency η_qe = 0.53 for 663.3-nm light.

The magnification of our imaging system, M_mag, is measured using a grid of black squares back-illuminated with 663-nm light and placed at the appropriate distance from the collection optics. We measure M_mag = 0.45, giving a 19.9 mm (horizontal) × 14.9 mm (vertical) field-of-view.

Owing to the high power of 663.3-nm laser light passing through the MOT chamber, scattered light is the primary noise source for the imaging. Several steps are taken to minimize the amount of scattered light reaching the camera. High-quality ultraviolet fused silica windows are used on all laser windows. These windows are antireflection-coated for 663-nm light and mounted on vacuum nipples far (∼260 mm) from the MOT. Scattered light is further reduced by lining the vacuum system with UHV-compatible black copper(ii) oxide³¹. We form and blacken copper sheets in various shapes to line the nipples and the region of the trapping chamber directly in the field-of-view of the camera. Also placed in the nipples are 26-mm-diameter apertures, machined with sharp edges and blackened. At atmospheric pressure, the scattered light is dominated by Rayleigh scattering from air; after pumping down to vacuum, the scattered light signal decreases by ∼ 50×, to a total detected value of ∼1.4 × 10⁵ photons ms⁻¹ across the entire field-of-view.

Trapped molecule number

The number of molecules observed in the MOT is given by

where N_c ≈ 7 × 10⁵ is the (background-subtracted) number of counts registered on the camera over the entire field-of-view for a single pulse of molecules, and N_per is the number of photons scattered per molecule during the camera exposure. For the default exposure duration, this last factor is given by

where the integral accounts for the decay of the trapped population (with τ_MOT = 56 ms) during the Δt_exp = 60 ms camera exposure. Since MOT loading is essentially complete when the slowing phase ends at t = 40 ms, and the camera exposure begins Δt = 20 ms later, the initial trapped population is given by:

Forced MOT oscillation

The confining and damping forces within the MOT are measured by observing the trapped cloud’s response to a rapid displacement of the trap centre. At t = 0 ms, before the loading/slowing phase, a shim coil applies an ∼4 G bias field to offset the trap centre by Δρ ≈ 5 mm radially, downstream along the axis of the molecular beam. When this bias field is switched off at t_off = 58 ms, the centre-of-mass radial position of the trapped molecules exhibits damped harmonic motion described by:

Assuming that the cloud is initially at rest, , the centre-of-mass position versus time is given by

where ρ₀ = Δρ is the initial displacement and is the observed angular oscillation frequency.

Extracting spatial information using LIF detection

The weak confinement of the MOT is crucial in order for our LIF-based detection method to extract certain spatial information from the cloud. For the forced MOT oscillation measurement, the camera exposure duration (Δt_exp) must be short compared to the radial oscillation period (2π/ω_ρ) to precisely determine the position of the cloud. We observe 2π/ω_ρ = 58(2) ms and set Δt_exp = 5 ms; this satisfies the short exposure condition while also allowing the camera to collect enough LIF to accurately measure the spatial distribution.

Similarly, the ballistic expansion measurement uses a camera exposure duration Δt_exp = 5 ms. This duration is short compared to 2π/ω_z = 41(1) ms, which avoids recapture and compression of the cloud by the trap light during illumination. The maximum time of flight used, Δt_TOF = 7 ms, is capped by the imaging field-of-view rather than the LIF signal-to-noise ratio (in contrast to the case for the MOT oscillation measurement).

Release and recapture

In the release-and-recapture method, trapped molecules are released and then expand freely for a variable time of flight Δt_TOF before the trap is turned back on, recapturing a fraction of the initial molecules that depends on their average velocity.

In order to avoid LIF from the untrapped molecular beam, the MOT is released at a fixed release time t = t_rel = 90 ms and Δt_TOF is varied from 0 to 50 ms. After each free expansion the MOT is recaptured at t = t_rel + Δt_TOF, and imaging begins at t = t_rel + Δt_TOF + 3 ms using Δt_exp = 30 ms. In contrast to the free-expansion measurement, this method uses a longer exposure time that gives enhanced sensitivity to the recaptured number of molecules but erases any spatial information about the cloud before recapture.

A cloud temperature is determined by comparing the measured recaptured fraction to that of a Monte Carlo simulation, as a function of Δt_TOF. The model assumes isotropic expansion and a spherical trap volume with radius r_cap; molecules inside this radius are assumed to be recaptured with 100% efficiency and those outside to be lost. The uncertainty in r_cap is a well-known limitation of the release-and-recapture method³⁴; we set r_cap = d_λ/2 to obtain an upper limit on the isotropic temperature T_iso. In the Monte Carlo simulation, initial velocities are drawn from a Boltzmann distribution and the effects of gravity are included. The initial spatial distribution is inferred from LIF images of the MOT, with the assumption that the MOT is radially symmetric. This procedure gives T_iso < 2.7(6) mK, in good agreement with the geometric mean of the axial and radial MOT temperatures derived from the MOT oscillation measurement .

Heating and cooling rates

The measured value of R_sc makes it possible to estimate a lower limit on the MOT heating rate (additional heating may be caused by stimulated emission, which is neglected here). This minimum heating rate is (dE/dt)_h = 2R_scE_r where E is the molecule energy and E_r = (k)²/(2m_SrF) is the photon recoil energy. By equating the heating rate to the rate of cooling from velocity damping, (dE/dt)_c = −2αk_BT_MOT/m_SrF (ref. 24), an independent lower limit on the damping coefficient α is obtained. We find , in agreement with the value from the MOT oscillation measurement.

Diffusion lifetime

We estimate the lifetime that would be measured for an SrF cloud in the presence of only optical molasses to cross-check the measured values of α and T_MOT (given the MOT beam diameter d_λ) and to further verify that a trapping force (rather than simply the cooling effect of optical molasses) is necessary to explain our observations. Here the motion of the molecules is treated as Brownian motion within a viscous fluid^35,36. The position diffusion constant is given by

and, using our measured values of α and T_MOT, we calculate . The molasses lifetime τ_mol is then given by:

The calculated lifetime τ_mol is in agreement with the fits to the data in the presence only of optical molasses. This lifetime is short compared to typical atomic molasses lifetimes (where ≳ ) due both to the small damping coefficient α and the relatively high MOT temperature T_MOT. Furthermore, the measured MOT lifetime τ_MOT ≈ 6 × τ_mol is consistent with our observations that molecules are confined in the MOT.

MOT lifetime

Although the measured MOT lifetime τ_MOT = 56(4) ms is short compared to those of typical atomic MOTs, the lifetime is ∼ 5× longer than the observed lifetimes of the molasses (dB_z/dz = 0) and damping/anti-restoring ( and polarizations reversed) configurations which have apparent lifetimes of 11(1) ms and 10(3) ms, respectively (see Fig. 3c). These apparent lifetimes for untapped molecules should be taken as upper limits due to the temporal and spatial extent of the slowed molecular beam.

Before reaching the conclusion that the lifetime is limited by ‘boil-off’ of molecules with energy greater than the trap depth, several other possible effects that could limit the MOT lifetime were explored. For example, off-resonant excitation into the A²Π_1/2(v′ = 0, J = 3/2) state could lead to decay into the dark X²Σ(v = 0, N = 3) state. To investigate this loss mechanism, a laser was added to repump population from the X²Σ(v = 0, N = 3) state. This did not change the measured MOT lifetime, indicating that losses due to off-resonant excitations are negligible.

MOT loss may be caused by collisions with residual ballistic He from the buffer gas beam source, with background (diffuse) He, or with other gases in the trapping region. We test for attenuation by ballistic He by increasing the flow rate of He into the buffer gas beam source from the default value of standard cubic centimetres per minute (sccm) to sccm. This increases the flux of ballistic He incident on the MOT by a factor of 4. In this configuration we measure τ_MOT to decrease by only ∼20%, suggesting that collisions with ballistic helium are not the primary loss mechanism. With still at 20 sccm, we reduce the rotation speed of the turbomolecular pumps in the trapping region by a factor of 5, resulting in an increase in all background gas pressures by ∼5×. In this configuration we measure τ_MOT to decrease by only ∼25%, indicating that collisions with background gases are not the primary loss mechanism.

Modelling trap loss

The measured MOT lifetime τ_MOT = 56(4) ms corresponds to a total loss rate 1/τ_MOT = 18(1) s⁻¹, log(1/τ_MOT) = 1.25(3). The main loss mechanism is attributed to a shallow trapping potential relative to the MOT temperature, leading to molecules escaping the trap by simply being in the high energy tail of the Boltzmann distribution. Such escape rates depend exponentially on the ratio U_MOT/(k_BT_MOT) (ref. 37). The uncertainty in U_MOT results in predicted loss rates having inherently large associated errors. We have no direct method to measure U_MOT. Instead, U_MOT is estimated by assuming that the spring constant κ_ρ has a constant value all the way to the edges of the MOT light. This estimate yields U_MOT/(k_BT_MOT) ≈ 4. This is likely to be an overestimate of U_MOT, since the MOT light intensity is smaller by a factor of ∼200 at ρ = d_λ/2 (trap edge) versus at ρ = 0 (trap centre).

We crudely model the trap loss using a simple Van’t Hoff–Arrhenius rate in the low damping limit () (ref. 37):

Here we multiply the standard one-dimensional prefactor ω_ρ/(2π) by a factor of 4 to account for the two trap edges visited per oscillation and the two radial dimensions; we neglect loss along the deeper axial dimension. Note that the low damping condition is only marginally satisfied, so the prefactor must also be considered as only approximate. Using U_MOT/(k_BT_MOT) ≤ 4, this yields an estimated loss rate of log(1/τ_MOT) ≥ 0, an order of magnitude smaller than the measured loss rate. If U_MOT is instead assumed to have a smaller but realistic value, for example, consistent with a linear restoring force only out to the measured 1/e² radius (7 mm) of the MOT beams, then U_MOT/(k_BT_MOT) ≈ 1.6, and log(1/τ_MOT) ≈ 1.1 in fair agreement with the measured loss rate. Hence we believe that this mechanism can plausibly account for the loss rate observed in our experiment, although the evidence is not definitive.