Molecules cooled below the Doppler limit

Journal name:
Nature Physics
Year published:
DOI:
doi:10.1038/nphys4241
Received
Accepted
Published online

Magneto-optical trapping and sub-Doppler cooling have been essential to most experiments with quantum degenerate gases, optical lattices, atomic fountains and many other applications. A broad set of new applications await ultracold molecules1, and the extension of laser cooling to molecules has begun2, 3, 4, 5, 6. A magneto-optical trap (MOT) has been demonstrated for a single molecular species, SrF7, 8, 9, but the sub-Doppler temperatures required for many applications have not yet been reached. Here we demonstrate a MOT of a second species, CaF, and we show how to cool these molecules to 50μK, well below the Doppler limit, using a three-dimensional optical molasses. These ultracold molecules could be loaded into optical tweezers to trap arbitrary arrays10 for quantum simulation11, launched into a molecular fountain12, 13 for testing fundamental physics14, 15, 16, 17, 18, and used to study collisions and chemistry19 between atoms and molecules at ultracold temperatures.

At a glance

Figures

  1. Schematic of the experiment.
    Figure 1: Schematic of the experiment.

    Pulses of CaF molecules, with a mean forward speed of ~150ms−1, are emitted from a cryogenic buffer gas source. The molecules are slowed to ~10ms−1 using frequency-chirped counter-propagating laser light, captured in a MOT, then cooled to sub-Doppler temperatures in an optical molasses.

  2. Characterization of the MOT.
    Figure 2: Characterization of the MOT.

    a, Fluorescence image of the MOT, averaged over 100 shots with 50ms exposure starting at t = 40ms. Also shown are the 1D axial and radial density profiles obtained by integrating the image along each axis. Gaussian fits to these yield the axial and radial centres and r.m.s. widths, and associated 1σ standard deviations. b, MOT lifetime versus scattering rate. The lifetime, and its 1σ standard deviation, are found by fitting the decay of the MOT fluorescence to an exponential decay. The scattering rate is controlled via the intensity of the main MOT laser and determined as described in Methods. c, Radial displacement of the cloud versus time after pushing it. Each point is obtained by summing 50 images each of 1ms exposure. Solid line: fit to the motion of a damped harmonic oscillator. d, Temperature measurement using the free-expansion method. The mean squared width of the cloud is plotted against the square of the free-expansion time, for (i) the axial and (ii) the radial directions. Each point is obtained by summing 50 images each of 1ms exposure. The top row shows typical images. Solid lines: quadratic fit to the data (see Methods). Where error bars are not visible in bd, they are smaller than the point size.

  3. Cooling the MOT by ramping down the intensity.
    Figure 3: Cooling the MOT by ramping down the intensity.

    Temperature (blue filled circles) and geometric mean r.m.s. width of the MOT (red open circles) versus total intensity of the MOT laser at the end of the ramp. Dashed line: Doppler temperature (Methods, equation (6)). Error bars are 1σ standard deviations of fits similar to those in Fig. 2d, and are smaller than the points.

  4. Sub-Doppler cooling.
    Figure 4: Sub-Doppler cooling.

    a, Temperature versus time in molasses, when intensity is 460mWcm−2. Solid line: fit to an exponential decay towards a base temperature, giving a 1/e time constant of 361 ± 2μs. Dashed line: minimum Doppler temperature. b, Temperature versus magnetic field produced at the cloud by one of three shim coils during the molasses phase. Intensity is 460mWcm−2 and molasses hold time is 5ms. Solid line: quadratic fit giving curvature of 5,740 ± 30μKmT−2. c, Temperature versus molasses intensity, with molasses hold time of 5ms. d, Temperature measurement after a period of 5ms in a molasses. Intensity is 93mWcm−2. Inset: images at each time point, from which cloud sizes in the axial (red) and radial (blue) directions are obtained from Gaussian fits. Error bars are 1σ standard deviations. Solid lines: straight line fits. Temperature from six repeated measurements is 55 ± 2μK.

Main

We first focus on the MOT, which is likely to become a workhorse for cooling molecules just as it is for atoms. Previously, only SrF had been trapped this way. For SrF, two types of MOT have been developed, a d.c. MOT where the lifetime was short and the temperature high7, 8, and a radiofrequency (rf) MOT where longer lifetimes and lower temperatures were achieved9, 20. In the rf MOT, optical pumping into dark states is avoided by rapidly reversing the magnetic field and the handedness of the MOT laser. It has been suggested that the detrimental effects of dark states can also be avoided in the d.c. MOT by driving the cooling transition with two oppositely polarized laser components, one red- and one blue-detuned21. We use this dual-frequency technique to make a d.c. MOT of CaF and find that it works just as well as the rf MOT. Thus, we demonstrate a MOT of a second molecular species, which is important for applications of ultracold molecules, and also verify the effectiveness of this new scheme.

Figure 1 illustrates the experiment, which is described in more detail in Methods. A pulse of CaF molecules produced at time t = 0 is emitted from a cryogenic buffer gas source, then decelerated by frequency-chirped counter-propagating laser light, and finally captured in the MOT between t = 16 and 40ms. Figure 2a shows the molecules in the MOT, imaged on a charge-coupled device (CCD) camera by collecting their fluorescence. We estimate that there are (1.3 ± 0.3) × 104 molecules in this MOT (see Methods), with a peak density of n = (1.6 ± 0.4) × 105cm−3. These are similar to the best values achieved for SrF20. To determine the MOT lifetime, we fit the decay of its fluorescence to a single exponential. Figure 2b shows this lifetime as a function of the scattering rate. The lifetime is typically 100ms and decreases with higher scattering rate, suggesting loss by optical pumping to a state not addressed by the lasers. We do not see the precipitous drop in lifetime observed at low scattering rate in the d.c. MOT of SrF9. To watch the molecules oscillate in the trap, we push them radially by pulsing on the slowing laser for 500μs at t = 50ms, and then image them for 1ms after a fixed delay, τ. Figure 2c shows ρ(τ), the mean radial displacement of the molecules as a function of this delay. Describing ρ(τ) by the damped harmonic oscillator equation, ρ′′ + βρ′ + ω2ρ = 0, we determine an oscillation frequency of ω = 2π × (94.4 ± 0.2)Hz and a damping constant of β = 390 ± 4s−1. Similar values were found in SrF MOTs (refs 8,9). The oscillation frequency is within 20% of the value predicted from rate-equation simulations21, where it was established that the dual-frequency mechanism is mainly responsible for the trapping force. By contrast, the damping constant is 14 times smaller than predicted by these simulations. The reduced damping may be caused by polarization-gradient forces which oppose the Doppler cooling force in the molecule MOT and dominate at low velocities22. This is in contrast to normal atomic MOTs, where both forces damp the motion.

Figure 1: Schematic of the experiment.
Schematic of the experiment.

Pulses of CaF molecules, with a mean forward speed of ~150ms−1, are emitted from a cryogenic buffer gas source. The molecules are slowed to ~10ms−1 using frequency-chirped counter-propagating laser light, captured in a MOT, then cooled to sub-Doppler temperatures in an optical molasses.

Figure 2: Characterization of the MOT.
Characterization of the MOT.

a, Fluorescence image of the MOT, averaged over 100 shots with 50ms exposure starting at t = 40ms. Also shown are the 1D axial and radial density profiles obtained by integrating the image along each axis. Gaussian fits to these yield the axial and radial centres and r.m.s. widths, and associated 1σ standard deviations. b, MOT lifetime versus scattering rate. The lifetime, and its 1σ standard deviation, are found by fitting the decay of the MOT fluorescence to an exponential decay. The scattering rate is controlled via the intensity of the main MOT laser and determined as described in Methods. c, Radial displacement of the cloud versus time after pushing it. Each point is obtained by summing 50 images each of 1ms exposure. Solid line: fit to the motion of a damped harmonic oscillator. d, Temperature measurement using the free-expansion method. The mean squared width of the cloud is plotted against the square of the free-expansion time, for (i) the axial and (ii) the radial directions. Each point is obtained by summing 50 images each of 1ms exposure. The top row shows typical images. Solid lines: quadratic fit to the data (see Methods). Where error bars are not visible in bd, they are smaller than the point size.

We turn now to the temperature, which we measure by turning off the MOT and then imaging the cloud after various free-expansion times τ. Figure 2d shows the mean squared widths in the axial and radial directions versus τ2, together with fits to the model described in Methods, which give temperatures of Tz = 11.0 ± 0.3mK and Tρ = 12.3 ± 0.5mK. A 10–20% variation between the two directions is typical of all our data. Hereafter, we use the geometric mean temperature, T = Tρ2/3Tz1/3, giving T = 11.4 ± 0.3mK in this case. This temperature is similar to that observed in SrF MOTs (refs 7,8,9). We may compare this to the Doppler temperature (see Methods), which for our intensity and detuning is 830μK, 14 times lower than measured. The elevated temperature is consistent with the reduced damping constant already noted above.

This MOT has a phase-space density of ρ = h3n/(2πmkBT)3/2 = (1.5 ± 0.4) × 10−15. We now show how we increase this by lowering the temperature. In the rf MOT of SrF, lowering the laser intensity reduced the temperature from 10mK to 400μK without loss of molecules9, and even to 250μK (ref. 20), although at substantial cost to the number and density. This method was not useful for cooling the d.c. MOT of SrF because of its short lifetime at low intensity. By contrast, the lifetime of our d.c. CaF MOT increases at lower intensity, so this method is open to us. We decrease the power of the main MOT laser between t = 50 and 70ms, hold it for 5ms, then measure the MOT temperature. Figure 3 shows both the temperature and the size of the MOT versus final intensity, together with the expected Doppler temperature, TD (see equation (6) in Methods). At 9.2mWcm−2 we find a minimum of 960μK, which is about 4TD. Extending the ramp to lower intensities increases the temperature again. Optimization of the detuning and the bias magnetic field lowers the minimum temperature to about 500μK, but we do not pursue that further here. Figure 3 shows that the MOT size first decreases as the intensity decreases, but then grows once the intensity is below 50mWcm−2. At the minimum MOT size the density is 2.5 × 105cm−3. The maximum phase-space density in the MOT is obtained near 25mWcm−2, where it is ρ = (3.4 ± 0.9) × 10−14, almost the same as the compressed rf MOT of SrF20. This ability to reach similar temperatures and phase-space densities in a d.c. MOT as in an rf MOT is an important advance.

Figure 3: Cooling the MOT by ramping down the intensity.
Cooling the MOT by ramping down the intensity.

Temperature (blue filled circles) and geometric mean r.m.s. width of the MOT (red open circles) versus total intensity of the MOT laser at the end of the ramp. Dashed line: Doppler temperature (Methods, equation (6)). Error bars are 1σ standard deviations of fits similar to those in Fig. 2d, and are smaller than the points.

To lower T and increase ρ further, we transfer into a three-dimensional blue-detuned optical molasses where we find remarkably effective sub-Doppler cooling. Sub-Doppler cooling usually occurs in a light field with non-uniform polarization, where optical pumping between sub-levels sets up a friction force at low velocity much stronger than the Doppler force23, 24. Details of the polarization-gradient cooling mechanism depend on the ground- and excited-state angular momenta, F and F′. In our case FF′ (Supplementary Fig. 1), so there are dark ground states that cannot couple to the local laser polarization, and bright states that can. A bright-state molecule loses kinetic energy on moving into blue-detuned light, and pumps to a dark state when the light has sufficient intensity. The dark molecule moves through changing polarization and switches non-adiabatically back to a bright state, preferentially at low intensity where the energies of dark and bright states are similar. This cooling method25, often called ‘grey molasses’, has been used to cool atoms. Magnetically induced laser cooling24 is similar, but uses a magnetic field instead of polarization gradients so that Larmor precession transfers molecules between dark and bright sub-levels. This mechanism has been used to cool molecules in one dimension2, 6, but sub-Doppler temperatures were not reached.

The procedure we use to transfer from MOT to molasses is detailed in Methods. Figure 4a shows the molecules cooling in the molasses towards a base value of ~100μK, which is half the Doppler limit. The 1/e time constant is 361 ± 2μs, implying a damping constant of 1,400s−1. We find that the base temperature is sensitive to all three components of the background magnetic field. Figure 4b shows the variation of temperature versus one field component after optimizing the other two. We find a quadratic dependence on magnetic field, consistent with the model outlined in Methods. Figure 4c shows the temperature versus the laser intensity during the molasses phase. The temperature has a minimum of 46μK near 100mWcm−2, increases rapidly at lower intensities and more gradually at higher intensities. The temperature dependencies shown in Fig. 4a–c are all similar to those observed in atomic grey molasses25, 26, so we conclude that the cooling process is indeed the grey molasses mechanism. Figure 4d shows the thermal expansion of a cloud after cooling for 5ms in a 93mWcm−2 molasses. The average of six such temperature measurements gives T = 55 ± 2μK. To within our 5% uncertainty, no molecules are lost between the initial MOT and this ultracold cloud, excepting loss due to the MOT lifetime. The cloud now has n = (0.8 ± 0.2) × 105cm−3 and ρ = (2.4 ± 0.6) × 10−12, 1,500 times higher than in the initial MOT and 40 times higher than previously achieved for laser-cooled molecules. It also exceeds that obtained by other molecule cooling methods that have reached sub-millikelvin temperatures13, 27.

Figure 4: Sub-Doppler cooling.
Sub-Doppler cooling.

a, Temperature versus time in molasses, when intensity is 460mWcm−2. Solid line: fit to an exponential decay towards a base temperature, giving a 1/e time constant of 361 ± 2μs. Dashed line: minimum Doppler temperature. b, Temperature versus magnetic field produced at the cloud by one of three shim coils during the molasses phase. Intensity is 460mWcm−2 and molasses hold time is 5ms. Solid line: quadratic fit giving curvature of 5,740 ± 30μKmT−2. c, Temperature versus molasses intensity, with molasses hold time of 5ms. d, Temperature measurement after a period of 5ms in a molasses. Intensity is 93mWcm−2. Inset: images at each time point, from which cloud sizes in the axial (red) and radial (blue) directions are obtained from Gaussian fits. Error bars are 1σ standard deviations. Solid lines: straight line fits. Temperature from six repeated measurements is 55 ± 2μK.

Single molecules from this ultracold gas could be loaded into low-lying motional states of microscopic optical tweezer traps and formed into regular arrays10 for quantum simulation11. They could be loaded into chip-based electric traps and coupled to transmission line resonators, forming the elements of a quantum processor28. By mixing the molecules with atoms, it will be possible to explore collisions, chemistry19 and sympathetic cooling29 in the ultracold regime. Our cooled molecules could be used to search for a time variation of the electron-to-proton mass ratio16, while application of the methods to other amenable molecules will advance measurements of electric dipole moments12, 17 and nuclear anapole moments18. All these applications are feasible with the phase-space densities achieved in the present work. For other applications, it is desirable to increase the density towards 1012cm−3, as often achieved for ultracold atomic gases. Major increases in density are likely to come by compressing the MOT20 and from more efficient slowing methods30 along with transverse cooling2 prior to slowing. The resulting dense, ultracold sample would be an ideal starting point for sympathetic or evaporative cooling to quantum degeneracy.

Methods

Laser cooling scheme.

Supplementary Fig. 1 shows the energy levels in CaF relevant to the experiment, and the branching ratios between them. The two excited states, A2Π1/2 and B2Σ+, have decay rates of Γ = 2π × 8.3MHz (ref. 31) and 2π × 6.3 MHz (ref. 32), respectively. The main slowing laser ( ) drives the B2Σ+(v′ = 0) X2Σ+(v′′ = 0) transition at 531.0nm. Population that leaks to v′′ = 1 during the slowing is returned to the cooling cycle by a repump slowing laser ( ) that drives the A2Π1/2(v′ = 0) X2Σ+(v′′ = 1) transition at 628.6nm. The hyperfine interval of about 20MHz in the lowest level of the B2Σ+ excited state is inconvenient for making a MOT, and our modelling21 suggests that a higher capture velocity is obtained by using the A2Π1/2 state where the hyperfine structure is unresolved. The MOT uses four lasers, denoted , to drive the A2Π1/2(v′ = j) X2Σ+(v′′ = i) transitions. These are at 606.3nm (called the main MOT laser in the main paper), at 628.6nm, at 628.1nm and at 627.7nm. All lasers drive the P(1) component so that rotational branching is forbidden by the electric dipole selection rules33. Each of the levels of the X state shown in Supplementary Fig. 1 is split into four components due to the spin–rotation and hyperfine interactions. The splittings for the v′′ = 0 state are shown in the figure, while those for the other states are similar. Radiofrequency (rf) sidebands are added to each laser (see discussion of Supplementary Fig. 2 below) to ensure that all these components are addressed. The frequency component of that addresses the upper F = 1 state has the opposite circular polarization to the other three. Thus, when the overall detuning of is negative, as in Supplementary Fig. 1, the F = 2 component is driven simultaneously by two frequencies with opposite circular polarization, one red- and the other blue-detuned. This configuration produces the dual-frequency MOT described in ref. 21. The simulations presented there suggest that most of the confining force in the MOT is due to this dual-frequency effect.

Set-up and procedures.

Figure 1 illustrates the experiment. A short pulse of CaF molecules is produced at t = 0 by laser ablation of a Ca target in the presence of SF6. These molecules are entrained in a continuous 0.5sccm flow of helium gas cooled to 4K, producing a pulsed beam with a typical mean forward velocity of 150ms−1. The beam exits the source through a 3.5-mm-diameter aperture at x′ = 0, passes into the slowing chamber through an 8-mm-diameter aperture at x′ = 15cm, and then through a 20-mm-diameter, 200-mm-long differential pumping tube whose entrance is at x′ = 90cm, reaching the MOT at x′ = 130cm. The pressure in the slowing chamber is 6 × 10−8mbar, and in the MOT chamber is 2 × 10−9mbar. The experiment runs at a repetition rate of 2Hz.

The beam is slowed using the methods described in ref. 34. The slowing light is combined into a single beam, containing 100mW of and 100mW of . This beam has a 1/e2 radius of 9mm at the MOT, converging to 1.5mm at the source. The changes in frequency and intensity of are illustrated in the timing diagram in Supplementary Fig. 3. The initial frequency of is set to a detuning of −270MHz so that molecules moving at 145ms−1 are Doppler-shifted into resonance. The light is switched on at t = 2.5ms and frequency chirped at a rate of 23MHzms−1 between 3.4ms and 15ms. The frequency of is not chirped, which differs from the procedure used previously34. Instead, it is frequency broadened as described below, and its centre frequency detuned by −200MHz. Both and are turned off at t = 15ms. A 0.5mT magnetic field, directed along y′, is applied throughout the slowing region and is constantly on.

The MOT light is combined into a single beam containing 80mW of , 100mW of , 10mW of and 0.5mW of . This beam is expanded to a 1/e2 radius of 8.1mm, and then passed through the centre of the MOT chamber six times, first along y, then x, then z, then −z, then −x, then −y. In this paper, intensity refers always to the full intensity at the MOT due to , which is six times the intensity of the beam entering along y. The light is circularly polarized each time it enters the chamber, and returned to linear polarization each time it exits, following ref. 7. For any given frequency component of the light, the handedness is the same for each pass in the horizontal plane, but opposite in the vertical direction. All MOT lasers have zero detuning, apart from , which has variable detuning Δ. The MOT field gradient, which is 2.9mTcm−1 in the axial direction, is produced by a pair of anti-Helmholtz coils inside the vacuum chamber. The MOT vanishes when this field gradient is reversed or turned off. Three bias coils, with axes along x′, y′ and z, are used to tune the magnetic field in the MOT region to trap the most molecules. The MOT fluorescence at 606nm is collected by a lens inside the vacuum chamber and imaged onto a CCD camera with a magnification of 0.5. An interference filter blocks background light at other wavelengths.

Supplementary Fig. 2 shows the frequency spectrum of each laser. For , a 73.5MHz electro-optic modulator (EOM) generates the sidebands that drive the F = 2, F = 0 and lower F = 1 states, while a 48MHz acousto-optic modulator (AOM) generates the light of opposite polarization to address the upper F = 1 state4. The rf sidebands for , , and are generated using 24MHz EOMs. We spectrally broaden to approximately 500 MHz using three consecutive EOMs, one driven at 72MHz, one at 24MHz and one at 8MHz. For each laser, we find the frequency that maximizes the laser-induced fluorescence (LIF) from the molecular beam when that laser is used as an orthogonal probe. These frequencies define zero detuning for each laser, with the exception of . For , we find that there is a critical frequency where an observable MOT is only formed in half of all shots. These large fluctuations in molecule number seem to come from a strong sensitivity to the approximately 1MHz frequency fluctuations of at this frequency. We define this critical frequency to be zero detuning, Δ = 0. At Δ = −0.25Γ the MOT is stable, and at Δ = 0.25Γ there is never a MOT (recall that Γ = 2π × 8.3MHz). We find that this method is more sensitive and reproducible than maximizing the LIF. When is used as an orthogonal probe, the LIF is maximized at Δ = 0.25Γ ± 0.5Γ. We observe MOTs when 0 > Δ > −1.8Γ, and we load the most molecules when Δ = −0.75Γ, the value used for all data in this paper.

The complete procedure for cooling to the lowest temperatures is illustrated in Supplementary Fig. 3. The intensity of is ramped down by a factor 100 to 4.6mWcm−2 between t = 50 and 70ms to lower the MOT temperature (see Fig. 3), while keeping the detuning at −0.75Γ. At t = 72ms, the shim coil currents are switched from those that load the most molecules in the MOT to those that give the lowest temperature in the molasses, thereby optimizing both molecule number and temperature. The MOT coils are turned off at t = 75ms. At t = 76ms we jump to a detuning of +2.5Γ, and to a (variable) higher intensity, to form the molasses (see Fig. 4c). After allowing the molasses to act for a variable time (see Fig. 4a) is turned off so that the cloud can expand for a variable time (see Fig. 4d) before it is imaged for 1ms at full intensity with Δ = −0.75Γ.

Scattering rate and saturation intensity.

Despite the complexity of the multi-level molecule, it is useful to use a simple rate model12 to predict some of the properties of the MOT, as done previously9. In this model, ng ground states are coupled to ne excited states, and the steady-state scattering rate is found to be

Here, Ij is the intensity of the light driving transition j, Δj is its detuning, and Is, j = πhcΓ/(3λj3) is the two-level saturation intensity for a transition of wavelength λj. In applying this model we need to include the ng = 24 Zeeman sub-levels of the v = 0 and v = 1 ground states, all of which are coupled to the same ne = 4 levels of the excited state. The v = 2 and v = 3 ground states can be neglected since they are repumped through other excited states with sufficient intensity that their populations are always small. Because is detuned whereas is not, and because the intensity is always higher than the intensity, the transitions driven by make only a small contribution to the sum in equation (1) and we neglect them. This is a reasonable approximation at full power, and a very good approximation once the power of is ramped down. The 12 transitions driven by have common values for Δ and Is, and the total intensity, I00, is divided roughly equally between them so that we can write Ij = I00/(ng/2). With these simplifications, we can rewrite equation (1) in the form

where

and

Writing seff = I00/Is, eff, we find an effective saturation intensity of

We have measured the scattering rate at various intensities, using the method described in the next paragraph. These measurements show that the scattering rate does indeed follow the form of equation (2), but suggest that Γeff is roughly a factor of 2 smaller than equation (3) while Is, eff is roughly a factor of 2 smaller than equation (5). However, the determination of Is, eff is sensitive to the value of the detuning, which is imperfectly defined for the multi-level molecule. Fortunately, none of our conclusions depend strongly on knowing the values of either Γeff or Is, eff.

Number of molecules.

To estimate the number of molecules in the MOT, we need to know the photon scattering rate per molecule. We measure this by switching off and recording the decay of the fluorescence as molecules are optically pumped into v′′ = 2. The decay is exponential with a time constant of 570 ± 10μs at full intensity. Combining this with our measured branching ratio of (8.4 ± 0.5) × 10−4 to v′′ = 2 (ref. 35) gives a scattering rate of 2.1 × 106s−1. This is about 2.5 times below the value predicted by equation (2). The detection efficiency is 1.6 ± 0.2% and is determined by numerical ray tracing together with the measured transmission of the optics and the specified quantum efficiency of the camera. For the MOT shown in Fig. 2a, the detected photon count rate at the camera is 4.3 × 108s−1. From these values, we estimate that there are (1.3 ± 0.3) × 104 molecules in this MOT. From one day to the next, using nominally identical parameters, and after optimization of the source, the molecule number varies by about 25%. We assign this uncertainty to all molecule number estimates in this paper.

Temperature.

In the standard theory of Doppler cooling, the equilibrium temperature is reached when the Doppler cooling rate equals the heating rate due to the randomness of photon scattering. Because both rates are proportional to the scattering rate, the multi-level system is expected to have the same Doppler-limited temperature as a simple two-level system. This is

We measure the temperature using the standard ballistic expansion method. We turn off the magnetic field and , then turn back on after a delay time τ to image the cloud using a 1ms exposure. For a thermal velocity distribution and an initial Gaussian density distribution of r.m.s. width σ0, the density distribution after a free-expansion time τ is a Gaussian with a mean squared width given by σ2 = σ02 + kBTτ2/m, where m is the mass of the molecule. Thus, a plot of σ2 against τ2 should be a straight line whose gradient gives the temperature.

There are several potential sources of systematic error in this measurement which we address here. First we consider whether the finite exposure time of 1ms introduces any systematic error. While the image is being taken using the MOT light, the magnetic quadrupole field is off. Thus, there is no trapping force, but there is a velocity-dependent force which, according to ref. 22, may either accelerate or decelerate the molecules depending on whether their velocity is above or below some critical value. To quantify the effect, we have made temperature measurements using various exposure times. For a 12mK cloud we estimate that the 1ms exposure time results in an overestimate of the temperature by about 0.3 ± 0.5mK. For a 50μK cloud, the overestimate is about 0 ± 0.3μK. These corrections are insignificant.

With a MOT that is centred on the light beams, the intensity of the imaging light is higher in the middle of the cloud than it is in the wings, making the cloud look artificially small. This skews the temperature towards lower values because the effect is stronger for clouds that have expanded. The error is mitigated by using laser beams that are considerably larger than the cloud and that strongly saturate the rate of fluorescence. Using a three-dimensional model of the MOT beams and equations (2) and (5) for the dependence of the scattering rate on intensity, we have simulated the imaging to determine the functional form of σ2(τ2) expected in our experiment. The model suggests that a simple τ4 correction—σ2 = σ02 + kBTτ2/M + a2τ4—will fit well to all our ballistic expansion data, will recover the correct temperature, and will give a significantly non-zero a2 for our T ~ 10mK data, but a negligible one for all our data where T ≤ 1mK. We have investigated this in detail using σ2 versus τ2 data with a higher density of data points (12 points, instead of our usual 6). For a hot MOT (T ~ 12mK), we see a nonlinear expansion and find that a fit to the above ‘quadratic model’ gives a temperature that is typically about 10% higher than a linear fit. For an ultracold molasses (T ~ 100μK) linear fits to the earliest six points and the latest six points of the ballistic expansion give the same temperature to within the statistical uncertainty of 4%. For the data in Figs 2 and 3 we use the quadratic model, while for the data in Fig. 4 we use the linear model. We determine the minimum temperature of 55 ± 2μK from the weighted mean of six measurements at a molasses intensity of 93mWcm−2. None of these six datasets show a statistically significant nonlinearity in the plots of σ2 versus τ2, so we use the linear model. Using the quadratic model instead of the linear one changes the temperature to 66 ± 7μK.

The magnification of the imaging system may not be perfectly uniform across the field of view. This can alter the apparent size of the cloud as it drops under gravity and expands. We have measured the magnification across the whole field of view that is relevant to our data and find that the uniformity is better than 3%. At this level, the effect on the temperatures is negligible.

Finally, a non-uniform magnetic field can result in an expansion that does not accurately reflect the temperature. A magnetic field gradient accelerates the molecules, and since the magnetic moment depends on the hyperfine component and Zeeman sub-level, this acceleration is different for different molecules. We calculate that this effect contributes a velocity spread less than 0.5cms−1 after 10ms of free expansion. This is negligible, even for a 50μK cloud. The second derivative of the magnetic field causes a differential acceleration across the cloud, but this effect is even smaller.

The temperature reached in the molasses depends on the background magnetic field, as shown in Fig. 4a. The sub-Doppler cooling mechanism relies on optical pumping into a dark state near the intensity maxima of the light field, and motion-induced non-adiabatic transitions back to a bright state near the intensity minima25, 36. This process becomes ineffective if the Larmor precession time between bright and dark states is comparable to the time taken by a molecule to move from a minimum to a maximum. Therefore, in the presence of a magnetic field B, the cooling will not reduce the mean velocity much below v μλB/(4h), where μ is the magnetic moment and λ is the wavelength. Converting from mean speed to temperature we find the relation T (πm/8kB)(μλB/4h)2. Taking μ μB we obtain T = aB2 with a 12,500μKmT−2. Our data fits well to this quadratic dependence on B and gives a = 5,740 ± 30μKmT−2, consistent with our model, given its approximate nature.

Data availability.

Data underlying this article can be accessed from Zenodo (https://doi.org/10.5281/zenodo.264440) and may be used under the Creative Commons CCZero licence.

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Acknowledgements

We thank J. Devlin for his assistance and insight. We are grateful to J. Dyne, G. Marinaro and V. Gerulis for technical assistance. The research has received funding from EPSRC under grants EP/I012044, EP/M027716, and EP/P01058X/1, and from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement 320789.

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  1. Centre for Cold Matter, Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2AZ, UK

    • S. Truppe,
    • H. J. Williams,
    • M. Hambach,
    • L. Caldwell,
    • N. J. Fitch,
    • E. A. Hinds,
    • B. E. Sauer &
    • M. R. Tarbutt

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All authors contributed to all aspects of this work.

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The authors declare no competing financial interests.

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