Creating an ultracold 84Sr beam

We use the experimental scheme developed in our previous work7,39 to create an ultracold 84Sr beam propagating within a dipole trap guide. The scheme begins with strontium atoms emitted by an 850-K oven. They then travel through a succession of laser-cooling stages arranged along several connected vacuum chambers using first the 1S01P1 and then the 1S03P1 transitions. Using the 30-MHz-wide 1S01P1 transition is necessary to efficiently slow and cool the fast atoms from the oven. However, this strong transition cannot be used in the last chamber in which the BEC is located, owing to the probable heating of the BEC from scattered near-resonant photons. Cooling using the narrow 1S03P1 transition is, however, made possible in this last chamber thanks to the addition of a transparency beam (see below).

To form a guided beam, atoms arriving in the final vacuum chamber are first captured and cooled in a narrow-line MOT. They are then outcoupled into a long, horizontal dipole guide with a 92-μm waist. The 84Sr atoms propagate along the guide with a velocity vG = 8.8(8) cm s−1, a Gaussian velocity spread ΔvG = 5.3(2) cm s−1 and a flux ΦG = 8.6(1.0) × 106 atoms s−1.

Making the reservoir and dimple traps

The 11.5-μK-deep reservoir is produced by a right circularly polarized 1,070-nm laser beam propagating in the z direction. It uses 540 mW of power focused to an elliptical spot with waists of wy = 14.5 μm vertically and wx = 110 μm horizontally. The guided atomic beam and the reservoir intersect with a horizontal angle of 6° about 1 mm from the reservoir centre and 37 mm from the MOT quadrupole centre. The reservoir beam crosses approximately 45(10) μm below the guide beam and descends with a vertical tilt of around 1.2(1)° as it separates from the guide beam. A secondary 250-mW beam of waist 175(25) μm runs parallel to the guide and points at the reservoir region. The fine adjustment of these beams is used to optimize the flow of atoms from the guide to the reservoir.

The dimple region has a 7 μK deeper potential located at the centre of the reservoir. This is mainly produced by a vertically propagating 1,070-nm ‘dimple beam’, although 1 μK is due to the vertically propagating transparency beam. The dimple beam uses 130 mW of power linearly polarized along the z axis with a 27-μm waist in the plane of the reservoir. The dimple trap frequencies are (ωDx, ωDy, ωDz) = 2π × (330, 740, 315) Hz, whereas the reservoir beam alone produces a trap with frequencies (ωRx, ωRy, ωRz) = 2π × (95, 740, 15) Hz.

Zeeman slower on the 1S03P1 transition

To load the guided atomic beam into the reservoir, it must first be slowed and pushed into the reservoir. To perform this task, we implement a Zeeman slower using the 1S03P1 transition starting around 3 mm before the guide–reservoir intersection. The slower makes use of the quadrupole magnetic field of the narrow-line MOT to provide a magnetic gradient along the axis of the guide. The quadrupole field of the MOT has gradients of −0.55, 0.32 and 0.23 G cm−1 in the x, y and z directions respectively. The slower is displaced by 37 mm along the z axis with respect to the quadrupole centre, resulting in a magnetic field offset of 0.85 G. The slower uses a counterpropagating 200-μm-waist laser beam that crosses the guide at a shallow horizontal angle of 4°. We modulate the laser frequency to broaden its effective linewidth to 50 kHz. This makes the slowing robust to potential fluctuations in the effective detuning (see Extended Data Table 1). The light intensity corresponds to 2.2 Isat when not frequency-broadened, in which Isat ≈ 3 μW cm−2 is the saturation intensity of the transition. We choose the laser detuning to match the Zeeman shift of the ({}^{3}{rm{P}}_{1}|{rm{J}}{prime} =1,{m}_{{rm{J}}}^{{prime} }=-1rangle ) state at the intersection between the guide and the reservoir. This way, atoms reach zero axial velocity at the intersection before being pushed back and into the reservoir.

Loading the reservoir

Because the reservoir is a conservative trap, efficiently loading atoms from the guide requires a dissipative mechanism. This is provided in two ways by laser cooling on the 1S03P1 transition. The first is a ‘counter Zeeman slower’ beam propagating approximately along the z axis opposing the Zeeman slower beam. This beam addresses the ({}^{3}{rm{P}}_{1}|{rm{J}}{prime} =1,{m}_{{rm{J}}}^{{prime} }=-1rangle ) state with a peak intensity of about 8 Isat and has a waist of 150 μm. Making use of this magnetic transition, we choose the light detuning such as to address the atoms near the guide–reservoir intersection and thus compensate the backwards acceleration of the Zeeman slower beam. This allows atoms to gradually diffuse towards the reservoir centre, in which collisions and the second laser-cooling mechanism will further lower their temperature.

The second cooling mechanism consists of a molasses on the radial axes (x, y) addressing the magnetically insensitive π transition. Using a magnetically insensitive transition avoids affecting cooling by the spatial inhomogeneities in the effective detuning owing to magnetic field variation across the extent of the laser-cooled cloud. Another cause of spatial inhomogeneities, which does affect the molasses cooling efficiency, is the differential light shift induced by the reservoir trap. This shift is around +55 kHz, many times larger than the linewidth of the transition. The optimal molasses cooling frequency is found to be 42 kHz higher than the unperturbed transition. This partially accommodates for the differential light shifts and preferentially cools atoms located near the bottom of the reservoir. To reach the lowest temperature and enable condensation in the dimple, we also apply a very low total light intensity of 0.4 Isat. With this choice of detuning and intensity, some of the incoming atoms reach the reservoir centre, in which they are radially cooled to TRr = 0.85(7) μK. Other atoms might be heated out of the 9-μK-evaporation-threshold trap by the blue-detuned light in the outer trap region.

Minimizing heating and loss in the reservoir

The atoms in the reservoir have a lifetime of 7 s, limited by collisions with the background gas of the vacuum chamber. However, these losses can be overwhelmed by optical effects such as photoassociation or heating by photon scattering. It is therefore critical to minimize the exposure of the reservoir to unnecessary light, and we address this point by implementing four techniques.

First, the 37-mm offset between the MOT and reservoir centres allows us to avoid any direct illumination from the x, y MOT beams on the reservoir; see Extended Data Fig. 1. On the z axis, the influence of the MOT beams is greatly reduced by using a ‘dark cylinder’, as described in ref. 39.

Second, we optimize the cooling spectrum and intensity of each laser-cooling beam entering the last vacuum chamber. By separately measuring their influence on the reservoir atom number, we optimize on a compromise between the lifetime of atoms and the loading flux. The results are illustrated in Extended Data Fig. 1 and Extended Data Table 1.

Third, we maximize the π polarization component of the molasses beams that illuminate both the guided beam and the reservoir, thus minimizing the effects of unwanted transitions. Unavoidably, beams along the y axis possess admixtures of σ and σ+ owing to the orientation of the local magnetic field.

Finally, we purify the spectrum of the light used to address the 1S03P1 cooling transition. Our cooling light is produced by several injection-locked diode lasers beginning from a single external-cavity diode laser (ECDL). We reduce the linewidth of this ECDL to 2 kHz by locking it onto a cavity with a finesse of approximately 15,000, whose spectrum has a full width at half maximum of about 100 kHz. By using the light transmitted through this cavity to injection lock a second diode laser, we can filter out the amplified spontaneous emission of the ECDL and servo bumps. This filtering is critical to increase the lifetime of the atoms inside the dimple by reducing resonant-photon scattering.

Without the dimple and transparency beams, individual laser-cooling beams reduce the lifetime of atoms in the reservoir to no shorter than about 1.5 s. With the dimple, transparency and all laser-cooling beams on, atoms in the reservoir have a 1/e lifetime of 420(100) ms, as determined from the fits shown in Extended Data Fig. 2.

Transparency beam

To minimize the destructive effects of resonant light on the BEC and atoms within the dimple, we render this region locally transparent to light on the 1S03P1 cooling transition. By coupling light to the 3P13S1 transition, we induce a light shift on the 3P1 state, as illustrated in Extended Data Fig. 3a, b. Owing to the extreme sensitivity of the BEC to photon scattering, all sub-levels of the 3P1 state must be shifted markedly. This requires using at least two of the three transition types (σ±, π) in this J = 1–J′ = 1 structure. However, when polarizations at the same frequency are combined, quantum interference between sub-levels always produces a dark state in the dressed 3P1 manifold. In this case, the energy of this dark state can only be shifted between ±ΔZeeman, in which ΔZeeman is the Zeeman shift of the ({}^{3}{rm{P}}_{1}{m}_{{rm{J}}}^{{prime} }=1) state. This corresponds to ΔZeeman = 1.78 MHz at the dimple location, giving a light shift that is insufficient to protect the BEC. Thus it is necessary to use different frequencies for the different polarization components of the transparency beam, as illustrated in Extended Data Fig. 3c.

The transparency beam is implemented by a single beam propagating vertically and focused on the dimple location with a 23-μm waist. This geometry aims to minimize the overlap of the transparency beam with the reservoir volume. In this way, we protect atoms at the dimple location without affecting the laser cooling taking place in the surrounding reservoir. This is necessary to maintain the high phase-space flux of the reservoir. The transparency laser light is blue detuned by 33 GHz from the 3.8-MHz-wide 3P13S1 transition at 688 nm. This detuning is chosen to be as large as possible while still enabling sufficient light shift with the available laser power. The light contains two frequency components: 7 mW of right-hand circularly polarized light and 3 mW of left-hand circularly polarized light, separated by 1.4 GHz. The relative detuning is chosen to be large enough to avoid dark states while remaining experimentally easy to implement. It is small compared with the absolute detuning to obtain similarly good protection by each component. The relative intensity is chosen to shift all 3P1 states by a similar magnitude. The magnetic field at the dimple location lies in the (y, z) plane and has an angle of 60° with respect to the vertical y axis along which the transparency beam propagates. This leads to a distribution of the light intensity onto the transitions {σ+, σ, π} of {1, 9, 6} for the left-hand and {9, 1, 6} for the right-hand circular polarization.

The light is produced from a single ECDL, frequency shifted by acousto-optic modulators and amplified by several injection-locked laser diodes and a tapered amplifier. Because the 1S03P1 and 3P13S1 lines are less than 1.5 nm apart, it is crucial to filter the light to prevent amplified spontaneous emission from introducing resonant scattering on the 1S03P1 transition. This filtering is performed by a succession of three dispersive prisms (Thorlabs PS853 N-SF11 equilateral prisms), followed by a 2.5-m (right-hand circular) or 3.9-m (left-hand circular) propagation distance before aperturing and injection into the final optical fibre.

Characterizing the transparency beam protection

The transparency-beam-induced light shifts on the 1S03P1 transition were measured spectroscopically by probing the absorption of 88Sr samples loaded into the dimple. 88Sr is used instead of 84Sr because the higher natural abundance improves signal without affecting the induced light shifts. Spectra are recorded for various transparency beam laser intensities at the magnetic field used for the CW BEC experiments. The results are shown in Extended Data Fig. 3c for one and then two polarization components.

The observed light shifts are consistent with the calculated dressed states for the six coupled sub-levels of the 3P1 and 3S1 states. This is evaluated by solving the Schrödinger equation in the rotating frame of the light field for a transparency beam consisting of a single-frequency, right-hand circular laser beam in the presence of the measured external magnetic field. The theoretical results are given in Extended Data Fig. 3c (solid lines, left side), with no adjustable parameters. We find a reasonable agreement with the observed shifts and reproduce the expected saturation of the light shift owing to the presence of a dark state. An optimized fit can be obtained with a slightly higher intensity corresponding to a waist of 21 μm instead of 23 μm, and a slightly modified polarization distribution. In this fitted polarization distribution, the contribution of the weakest component, σ, is enhanced by a factor of roughly 2.5. Both differences can be explained by effects from the vacuum chamber viewports and dielectric mirrors.

When the left-hand circular polarization component of the transparency beam is added, we observe in Extended Data Fig. 3c (right side) that the ‘dark’ state shifts linearly away. In this manner, all sub-levels of 3P1 can be shifted by more than 4 MHz, more than 500 times the linewidth of the laser-cooling transition. For comparison, the light shift on the 1S0 ground state from the transparency beam is 20 kHz, and at most 380 kHz by all trapping beams, about one order of magnitude smaller than the shift on 3P1 states from the transparency beam.

We demonstrate the protection achieved by the transparency beam in two ways. First, we measure the lifetime of a pure BEC inside the dimple in the presence of all light and magnetic fields used in the CW BEC experiments. This pure BEC is produced beforehand using time-sequential cooling stages. Once the pure BEC is produced, we apply the same conditions as used for the CW BEC, except that the light addressing the 1S01P1 transition is off, to prevent new atoms from arriving. Without the transparency beam, the 1/e lifetime of a pure BEC in the dimple cannot even reach 40 ms, whereas with the transparency beam, it exceeds 1.5 s.

Second, we show the influence of the transparency beam on the existence of a CW BEC. Beginning with the same configuration as the CW BEC but without the transparency beam, steady state is established after a few seconds, with no BEC formed. We then suddenly switch the transparency beam on and observe the evolution of the sample as shown in Extended Data Fig. 4. Although the reservoir sample seems unaffected, the dimple atom number increases by a factor of 6.4(1.8), indicating fewer losses. At the same time, the sample (partially) thermalizes and a BEC appears after about 1 s. No BEC is formed if only one transparency beam frequency component is present or only one-third of the nominal transparency beam power is applied. This demonstrates the critical importance of the transparency beam.

Characterizing the BEC and thermal cloud

To characterize the CW BEC and surrounding thermal cloud, we switch all traps and beams off and perform absorption imaging. Fitting the distributions of the expanding clouds allows us to estimate atom numbers and temperatures throughout the system, as well as the number of condensed atoms, all from a single image.

We begin with absorption images typically recorded after an 18-ms time-of-flight expansion. The observed 2D density distribution can be fitted by an ensemble of four thermal components plus an extra Thomas–Fermi distribution when a BEC is present. Three independent 2D Gaussian functions represent atoms originating from the dimple, the reservoir and the crossing between the guide and the reservoir. Atoms originating from the guide are represented along the guide’s axis by a sigmoid that tapers off owing to the effect of the Zeeman slower and in the radial direction by a Gaussian profile. Examples are shown in Extended Data Fig. 5.

We found this fit function with 18 free parameters to be the simplest and most meaningful one capable of representing the data. By combining knowledge of their distinct locations and/or momentum spreads, we can determine individually the populations and their characteristics. We find that the uncertainty in the fitted parameters is mostly unimportant compared with shot-to-shot variations in the data. An exception is distinguishing the population in the reservoir from that in the guide–reservoir crossing region, in which there is some ambiguity, resulting in higher uncertainties. In both the main text and Methods, the error bars indicate the standard deviation σ calculated from several images. Although it is possible to estimate the temperatures in the y axis from a single fitted image, the initial cloud sizes in the z direction are large compared with the ballistic expansion. Thus we use a set of measurements with varying times of flight to estimate z-axis temperatures.

When a BEC is present, it is necessary to add a Thomas–Fermi profile to the previously discussed fit function. The only other free parameter used in the fit is the number of atoms in the BEC. We assume that the BEC position is the same as that of the non-condensed atoms in the dimple and we calculate the radii of the BEC from the BEC atom number, the s-wave scattering length, the trap frequencies in the dimple and the expansion time59. These frequencies are calculated from the knowledge of the waists of each relevant beam and of the powers used. The waists are either directly measured or extracted from observations of dipole oscillation frequencies of a pure BEC in the trap for several beam powers.

Adding an extra fitting parameter can lead to overfitting. To rigorously determine whether including this Thomas–Fermi distribution provides a significantly better fit of the data, we use a statistical F-test. This allows us to determine a BEC atom number threshold above which the fit is statistically better than that without the Thomas–Fermi distribution. For this F-test, we isolate a region of interest (ROI) in the image containing both thermal and BEC atoms. We then calculate the value (F=frac{({{rm{RRS}}}_{1}-{{rm{RSS}}}_{2})}{{p}_{2}-{p}_{1}}/frac{{{rm{RSS}}}_{2}}{n-{p}_{2}},) in which RRSi is the residual sum of squares over the ROI for model i with pi parameters and n is the number of pixels of the ROI. The fit including the Thomas–Fermi distribution is significantly better than that without only if F is higher than the critical value of an F-distribution with (p2 − p1, n − p2) degrees of freedom, with a desired confidence probability. By applying this test to the data of Fig. 3, we find that the BEC model fits better, with a confidence greater than 99.5%, when the BEC atom number exceeds 2,000. This sets our detection limit, above which we are confident a BEC exists. Notably, this limit is lower than the BEC atom number, corresponding to a −2σN shot-to-shot fluctuation. This shows that, at all times after steady state is reached, a BEC exists.

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