Continuous guided strontium beam with high phase-space density

TL;DR

This paper demonstrates a continuous guided strontium atomic beam with significantly higher phase-space density than previous steady-state beams, suitable for advanced quantum applications and potentially leading to an atom laser.

Contribution

The authors present a novel ultracold strontium beam source with high phase-space density and flux, advancing steady-state atomic beam technology.

Findings

Phase-space density exceeds 10^{-4} in the moving frame.

Flux of 3 x 10^7 atoms per second achieved.

Radial temperature below 1 microkelvin.

Abstract

A continuous guided atomic beam of $^{88}\mathrm{Sr}$ with a phase-space density exceeding $10^{-4}$ in the moving frame and a flux of $3 \times 10^{7} \, \mathrm{at \, s^{-1}}$ is demonstrated. This phase-space density is around three orders of magnitude higher than previously reported for steady-state atomic beams. We detail the architecture necessary to produce this ultracold atom source and characterize its output after $\sim 4 \, \mathrm{cm}$ of propagation. With radial temperatures of less than $1 \,\mu \mathrm{K}$ and a velocity of $8.4 \, \mathrm{cm \, s^{-1}}$ this source is ideal for a range of applications. For example, it could be used to replenish the gain medium of an active optical superradiant clock or be employed to overcome the Dick effect that can limit the performance of pulsed-mode atom interferometers, atomic clocks and ultracold atom based sensors in general. Finally, this result represents a significant step towards the development of a steady-state atom laser.

Tables

Table 1. Table 1: Characteristics of the beam, for both Sr 88 superscript Sr 88 {}^{88}\mathrm{Sr} and Sr 84 superscript Sr 84 {}^{84}\mathrm{Sr} , measured 37   mm times 37 millimeter 37\text{\,}\mathrm{mm} away from the MOT. The symbols and their expressions are detailed in the main text. All uncertainties are taken as the standard deviation from the fitted data. The quantities v ¯ z subscript ¯ 𝑣 𝑧 {\bar{v}_{z}} and Δ ​ v z Δ subscript 𝑣 𝑧 \Delta v_{z} could not be measured for Sr 84 superscript Sr 84 {}^{84}\mathrm{Sr} as the flux is not large enough. We assume the same values as for Sr 88 superscript Sr 88 {}^{88}\mathrm{Sr} , scaled by the mass ratio.

Parameter	${}_{}{}^{88}\mathrm{Sr}$ beam	${}_{}{}^{84}\mathrm{Sr}$ beam
Axial temperature $T_z$ [$\mu K$]	$29\left(2\right)$	$29\left(2\right)$
Rad. temperature $T_r$ [$\mu K$]	$0.89\left(4\right)$	$2.0\left(1\right)$
Axial velocity ${\overline{v}}_z$ [$\mathrm{cm}{s}^{-1}$]	$8.4\left(4\right)$	$8.8\left(4\right)$
Vel. spread $\Delta v_z$ [$\mathrm{cm}{s}^{-1}$]	$5.2\left(2\right)$	$5.3\left(2\right)$
Vel. spread $\Delta v_r$ [$\mathrm{cm}{s}^{-1}$]	$0.92\left(2\right)$	$1.41\left(4\right)$
Spatial spread $\Delta r$ [$\mu m$]	$23.3\left(4\right)$	$19.7\left(1.0\right)$
Lin. density ${\rho }_L$ [at $m^{-1}$]	$3.88\left(8\right)\times {10}^8$	$1.04\left(5\right)\times {10}^7$
Peak density $n_0$ [at $m^{-3}$]	$1.14\left(4\right)\times {10}^{17}$	$4.2\left(5\right)\times {10}^{15}$
Flux $\Phi$ [at $s^{-1}$]	$3.25\left(14\right)\times {10}^7$	$9.1\left(6\right)\times {10}^5$
Flux dens. ${\rho }_{\Phi }$ [at $s^{-1}{m}^{-2}$]	$1.02\left(5\right)\times {10}^{20}$	$2.9\left(4\right)\times {10}^{17}$
Trap freq. ${\omega }_r/2\pi$ [$\mathrm{Hz}$]	$185\left(10\right)$	$185\left(10\right)$
Collision rate ${\Gamma }_{\mathrm{el}}$ [$\mathrm{Hz}$]	$3.5\left(2\right)\times {10}^{-4}$	$0.11\left(2\right)$
PSD ${\rho }_{\mathrm{PSD}}$	$1.5\left(2\right)\times {10}^{-4}$	$2.7\left(4\right)\times {10}^{-6}$
Alternate PSD ${\rho }_{\mathrm{PSD},\mathrm{therm}}$	$1.3\left(2\right)\times {10}^{-3}$	$7.1\left(1.2\right)\times {10}^{-6}$
Bright. $ℛ$ [at $s^{-1}{m}^{-2}{\mathrm{sr}}^{-1}$]	$5.0\left(7\right)\times {10}^{17}$	$9\left(2\right)\times {10}^{15}$
Brill. $ℬ$ [at $s^{-1}{m}^{-2}{\mathrm{sr}}^{-1}$]	$2.5\left(5\right)\times {10}^{18}$	$5\left(1\right)\times {10}^{16}$
Vel. bright. ${ℛ}_v$ [at $s^2{m}^{-5}$]	$4.3\left(4\right)\times {10}^{21}$	$7\left(1\right)\times {10}^{19}$

Equations

$$\begin{array}{rl}\Phi & ={\int }_0^{\infty }{n}_0\exp \left(-\frac{r^2}{2\Delta r^2}\right)2\pi rdr{\bar{v}}_z\\ & =2\pi \Delta r^2{n}_0{\bar{v}}_z={\rho }_L{\bar{v}}_z\end{array},$$

$$\begin{array}{rl}{\rho }_{\mathrm{PSD}}& =n_0{\lambda }_{\text{dB},r}^2{\lambda }_{\text{dB},z}\\ & =n_0{\left(\frac{h}{\sqrt{2\pi m{k}_B{T}_r}}\right)}^2\frac{h}{\sqrt{2\pi m{k}_B{T}_z}}\end{array},$$

$$\begin{array}{rl}{\rho }_{\mathrm{PSD},\mathrm{therm}}& =n_{0,\mathrm{therm}}{\lambda }_{\text{dB},r}^2{\lambda }_{\text{dB},z}\\ & ={\rho }_L{\left(\frac{h{\omega }_r}{2\pi k_B{T}_r}\right)}^2\frac{h}{\sqrt{2\pi m{k}_B{T}_z}}\end{array}.$$

$$\mathcal{R}=\frac{\Phi }{\pi \Delta r^2\Omega },$$

$$\mathcal{B}=\mathcal{R}\frac{{\bar{v}}_z}{\Delta v_z}=\frac{\Phi {{\bar{v}}_z}^3}{{\pi }^2\Delta r^2\Delta v_z\Delta v_r^2},$$

$${\mathcal{R}}_v=\frac{\Phi }{\pi \Delta r^2\Delta v_z\Delta v_r^2}.$$

Full text

I Introduction

From atomic clocks Ludlow et al. (2015) to atom interferometers Cronin et al. (2009) cold and ultracold atom devices are defining the state of the art of precision measurement. Cold atom sensors are tackling fundamental questions like detecting dark matter or dark energy Wcisło et al. (2018); Hees et al. (2016); Geraci and Derevianko (2016); Jaffe et al. (2017), gravitational waves Canuel et al. (2018); Graham et al. (2013); Hogan et al. (2011); Kolkowitz et al. (2016) and variations of fundamental constants Martins (2017); Godun et al. (2014), as well as making precision measurements of physical constants Parker et al. (2018); Bertoldi et al. (2006); Fixler et al. (2007); Rosi et al. (2014). In the applied domain, optical atomic clocks continue to set new records in timekeeping Campbell et al. (2017); McGrew et al. (2018); Nemitz et al. (2016), while cold atom gravimeters, gravity gradiometers, gyroscopes and accelerometers are of growing importance for geology and navigation Bidel et al. (2018); Ménoret et al. (2018); Dutta et al. (2016); Cheiney et al. (2018). Yet almost all these cold atom sensors and the atom sources they rely on operate in pulsed mode, which poses a fundamental limitation. The Dick effect Dick (1987); Quessada et al. (2003), where frequency noise aliasing arises from the dead time between sample interrogations, intrinsically limits the performance of a pulsed device.

Atomic clocks now reach sensitivities where the Dick effect limits performance Takamoto et al. (2011); Al-Masoudi et al. (2015). Improvements in the optical clock local oscillator has allowed them to better preserve phase across the dead time Kessler et al. (2012); Häfner et al. (2015). Others synchronize multiple copies of the same apparatus to avoid dead time Biedermann et al. (2013); Schioppo et al. (2016); Dutta et al. (2016), or increase the duty cycle by performing multiple measurements after a single sample preparation phase Kohlhaas et al. (2015); Norcia et al. (2019); Westergaard et al. (2010). Hybridization of a cold atom interferometer with other devices can combine the low offsets of atom interferometers with the higher bandwidth of classical devices in a single apparatus Cheiney et al. (2018). A fundamentally simpler approach would be to create a fully continuous device Keith et al. (1991); Durfee et al. (2006); Jallageas et al. (2018); Xue et al. (2015). Active optical clocks Chen (2009); Meiser et al. (2009) are a promising proposal for producing a new generation of optical clocks that is inherently continuous, circumventing both the Dick effect as well as other challenges now limiting optical lattice clocks, like the thermal noise of the local oscillator. These are based on the principle of superradiant lasing of ultracold atoms inside a “bad” optical cavity. The operating principle has been demonstrated Bohnet et al. (2012), even on strontium’s optical clock transition Norcia et al. (2016), but what is desirable is a high phase-space density (PSD) continuous atom source in order to run the clock steady-state Muniz et al. (2019); iqC . Similarly, other cold atom sensors would benefit from continuous operation, like inertial sensors for navigation that could feature both absolute calibration and continuous measurement Jekeli (2005).

The development of atomic beams with high phase-space density has historically been closely tied with efforts to produce a steady-state atom laser Robins et al. (2013), perhaps the ultimate source for many cold atom sensors. Previous work by Lahaye et al. (2004, 2005) demonstrated steady-state beams with a PSD of $10^{-7}$, by repeatedly outcoupling rubidium atoms from a MOT and evaporatively cooling them as they traversed a $4.5\text{\,}\mathrm{m}$ long magnetic waveguide. A chromium beam of $1\times 10^{7}\,\mathrm{at}\,${\mathrm{s}}^{-1} with a PSD of $3\times 10^{-8}$ was produced in Aghajani-Talesh *et al.* ([2010](#bib.bib49)); Falkenau *et al.* ([2011](#bib.bib50)), and Knuffman *et al.* ([2013](#bib.bib51)) produced a caesium beam of $5\times 10^{10}\,\mathrm{at}\,${\mathrm{s}}^{-1} with a PSD of $4\times 10^{-8}$ for a focused ion beam source.

Here we present a continuous 88Sr source delivering an atomic beam of $3\times 10^{7}\,\mathrm{at}\,${\mathrm{s}}^{-1} with a phase-space density of more than $10^{-4}$ in the moving frame. In order to enable an extremely low forward velocity of $\sim$10\text{\,}\mathrm{cm}\text{\,}{\mathrm{s}}^{-1}, this beam is supported against gravity by an optical dipole guide. This source could feed an atom interferometer in continuous operation mode, in particular using the ${{}^{1}\mathrm{S}_{0}}\mathrm{-}{{}^{3}\mathrm{P}_{0}}$ clock transition in a magic wavelength guide Akatsuka et al. (2017); Hu et al. (2017). Moreover, this high-PSD atomic beam could provide the gain medium for a steady-state superradiant laser Chen (2009); Meiser et al. (2009); Norcia et al. (2016), and produce a clock laser with linewidth substantially narrower than the transition linewidth Debnath et al. (2018). Last, this beam could possibly be the source for a continuous atom laser Hagley et al. (1999); Guerin et al. (2006); Mandonnet et al. (2000); Olson et al. (2014).

This paper is structured as follows. Section II describes the experimental setup and the various steps and methods necessary to produce the beam. In section III, we discuss figures of merit for characterizing cold atomic beams. We then present our measurement protocols and the results we obtain for two strontium isotopes in section IV. Lastly, in section V we discuss possible applications for our continuous high phase-space density atomic beam and conclude.

II Experiment

Our approach to produce a continuous, cold, bright atomic beam is based on flowing gaseous strontium through a series of spatially distributed laser cooling and guiding stages. Our scheme is illustrated in Fig. 1. The first stages are responsible for cooling atoms beginning with an $\sim$800\text{,}\mathrm{K} oven and finishing with a $\sim$10\text{\,}\mathrm{\SIUnitSymbolMicro K} steady-state magneto-optical trap (MOT). We have previously reported on this MOT architecture in Bennetts et al. (2017). We next outcouple atoms from the MOT into an optical dipole guide creating a bright, slow atomic beam. Transverse cooling of this atomic beam together with measures to detune and prevent MOT light from interacting with the beam are critical to achieve high performance. We characterize the resulting atomic beam at a location $\sim$37\text{,}\mathrm{mm}$$ from the MOT. In the following section, we explain the details of this apparatus.

Steady-state MOT atom source — Beginning from an $800\text{\,}\mathrm{K}$ hot oven, atoms are transversely cooled, Zeeman slowed and captured by a 2D continuous magneto-optical trap (MOT) with “push” beams, all operated on the $30\text{\,}\mathrm{MHz}$ linewidth ${{}^{1}\mathrm{S}_{0}}-{{}^{1}\mathrm{P}_{1}}$ transition. This first stage creates a beam with a flux of 2.7\text{\times}{10}^{9}$\,^{88}\mathrm{Sr}\,${\mathrm{s}}^{-1} at $\sim$10\text{,}\mathrm{mK} with a vertical downward velocity of a few meters per second. This beam is transversely cooled by a molasses operating on the $7.4\text{\,}\mathrm{kHz}$ linewidth ${{}^{1}\mathrm{S}_{0}}-{{}^{3}\mathrm{P}_{1}}$ transition reducing the radial temperature to $\sim$10\text{\,}\mathrm{\SIUnitSymbolMicro K} and allowing the beam to efficiently propagate to a second, lower chamber through a baffle. This baffle and dual chamber design was implemented to prevent ultracold atoms in the lower chamber from being heated by the continuously operating cooling light in the upper chamber. This can be critical depending on the time spent by the atoms in the lower chamber.

The falling atomic beam is captured by a steady-state 3D MOT operating on the $7.4\text{\,}\mathrm{kHz}$ ${{}^{1}\mathrm{S}_{0}}-{{}^{3}\mathrm{P}_{1}}$ transition. The MOT geometry consists of five beams in orthogonal configuration. In the vertical axis, we shine a single MOT beam from below and rely on gravity to provide the downward restoring force. The MOT quadrupole magnetic field has gradients of $\{0.55,-0.32,-0.23\}\,$\mathrm{G}\text{,}{\mathrm{cm}}^{-1}$$ in the $\{x,y,z\}$ axis, respectively. Atoms in the MOT typically have a temperature ranging from $5$ to $30\text{\,}\mathrm{\SIUnitSymbolMicro K}$. The MOT laser detunings and intensities are adjusted to maximize the performance of the atomic beam at the region of interest (ROI, see Fig. 1) rather than the MOT itself.

In order to address atoms with Doppler shifts much larger than the $7.4\text{\,}\mathrm{kHz}$ atomic linewidth, we use acousto-optic modulators to frequency broaden the MOT beams to a comb-like structure with a spacing $\delta\sim$20\text{,}\mathrm{kHz} (corresponding to $2\mathrm{-}3\,\times\Gamma_{{{}^{1}\mathrm{S}_{0}}-{{}^{3}\mathrm{P}_{1}}}/2\pi$). The detuning ranges $\left(\Delta_{1};\delta;\Delta_{2}\right)$ are $\left(-2.2;0.015;-0.66\right)$, $\left(-5.2;0.02;-0.95\right)$ and $\left(-2.2;0.016;-0.82\right)\,$\mathrm{MHz} for the $x$, $y$ and $z$ axis, respectively. The power in each $\{x,y,z\}$ axis MOT beam is $\{1.2,10.8,1.14\}\,$\mathrm{mW} and the $1/e^{2}$ beam diameter is ${\{47,68,48\}}\,$\mathrm{mm}. The single beam in the $y$ axis is focused $22\text{\,}\mathrm{cm}$ above the MOT quadrupole magnetic field center and its $1/e^{2}$ diameter is $\sim$35\text{,}\mathrm{mm}$$ at the MOT location.

The MOT beams along the $z$ axis provide confinement that can prevent the emission of an atomic beam along the $z$ direction. To mitigate this problem, we bore an $8\text{\,}\mathrm{mm}$ diameter hole in the center of the two mirrors directing the $z$ MOT beams from each side of the vacuum chamber, see Fig. 1. This allows the insertion of an extra pair of low intensity MOT beams down the $z$ axis of the MOT, strong enough to trap the cold MOT but weak compared to the other MOT beams that are optimized to capture hot incoming atoms. These additional beams fill up the holes entirely with a $1/e^{2}$ diameter of $\sim$8\text{,}\mathrm{mm}, and they have a smaller detuning range of $\left(-1.25;0.017;-0.85\right)\,$\mathrm{MHz} and a much lower power of $5\text{\,}\mathrm{\SIUnitSymbolMicro W}$. In addition to facilitating the outcoupling of a guided atomic beam the resulting MOT cloud is also elongated along the $z$ axis resulting in a better spatial overlap with the guide. Further details describing the steady-state 3D MOT can be found in Bennetts et al. (2017).

Transport guide — We continuously load atoms from the steady-state MOT into a “transport” guide, formed by an optical dipole beam overlapped with the MOT cloud and propagating along the $z$ axis. The guide is produced by focusing $12\text{\,}\mathrm{W}$ from a $1070\text{\,}\mathrm{nm}$ ytterbium fiber laser (IPG YLR-20-LP with $1.1\text{\,}\mathrm{nm}$ linewidth) to a $92\text{\,}\mathrm{\SIUnitSymbolMicro m}$ $1/e^{2}$ radius waist at the location of the MOT. In order to extend the guide length and improve the uniformity of the guide potential depth we retro-reflect the (incoherent) beam, focusing its second pass $\sim$35\text{,}\mathrm{mm} away from the MOT on the $z$ axis in the direction of the atomic beam propagation, with the same waist as in the first pass. The $35\text{\,}\mathrm{mm}$ distance between focii is chosen to be on the same order as the Rayleigh length ($25\text{\,}\mathrm{mm}$) of both of these beams. By adapting the power of the retro-reflected beam with a polarizing beam splitter and a $\lambda/2$ waveplate, the potential landscape along the guide can be tuned and flattened. The effective trap depth at the MOT location is $\sim 35-$40\text{\,}\mathrm{\SIUnitSymbolMicro K}, and $\sim$25\text{,}\mathrm{\SIUnitSymbolMicro K} $37\text{\,}\mathrm{mm}$ away, where the potential is flattened in the propagation direction, and where the radial trapping frequency is $\omega_{r}=2\pi\times$185(10)\text{\,}\mathrm{Hz}. This deep, large volume trap at the MOT location improves the loading efficiency, and the flat potential for the final beam helps with further laser cooling stages by reducing light shift variations. The off-resonant scattering rate in the guide is negligible at $\sim$0.1\text{,}\mathrm{Hz}$$.

Dark SPOT cylinder — In our implementation, two MOT beams are overlapped and co-linear with the transport guide in the $z$ axis. It typically takes $\sim$0.4\text{,}\mathrm{s}$$ for atoms in the atomic beam to propagate the $37\text{\,}\mathrm{mm}$ from the MOT to the characterization location. Over such a long interaction time even a very small amount of resonant MOT light would be sufficient for the MOT restoring force to return atoms from the atomic beam to the MOT, thus devastating the beam’s flux and temperature. Two factors mitigate this effect. Firstly the MOT light intensity is much lower in the $8\text{\,}\mathrm{mm}$ core beam. Secondly the ${{}^{3}\mathrm{P}_{1}}$ $m_{J}=-1$ state experiences a $0.48\text{\,}\mathrm{MHz}\text{\,}{\mathrm{cm}}^{-1}$ Zeeman shift due to the MOT’s quadrupole magnetic field, quickly shifting the atoms propagating in the beam out of resonance with the MOT light. However, these measures alone are not sufficient.

To further reduce the interaction between the MOT beams and the atomic beam we adapt the dark SPOT technique Ketterle et al. (1993). Along both MOT beams in the $z$ axis, we image a $20\text{\,}\mathrm{cm}$-long, $600\text{\,}\mathrm{\SIUnitSymbolMicro m}$ outer diameter cylinder. By overlapping this shadow with the transport guide, we further darken the MOT region extending far beyond the characterization location $37\text{\,}\mathrm{mm}$ from the MOT. The cylinder is assembled by suspending a stainless steel capillary within each inner $z$ axis MOT beam. The capillary has a $400\text{\,}\mathrm{\SIUnitSymbolMicro m}$ inner diameter through which three $50\text{\,}\mathrm{\SIUnitSymbolMicro m}$ diameter twisted wires pass. At each cylinder end the three thin wires are pulled taut triangularly sideways forming a tetrahedron shape and glued to an XY translator (Thorlabs, CXY1, 30 mm Cage XY Translator), see Fig. 2(a) and (b). The XY translators are used to precisely position each cylinder within a MOT beam. In front of each collimated MOT beam, we use a simple two-lens system in $f-2f-f$ configuration ($f=$500\text{,}\mathrm{mm}$$) to image the cylinder’s shadow onto the transport guide with a magnification of one.

We characterize the performance of these dark SPOT cylinders by imaging the shadow actually produced on the atoms, by collecting leakage light transmitted through MOT mirrors and imaging it onto a camera. By varying the image plane on the camera chip, we can select a specific object plane along the transport guide, as shown in the example of Fig. 2(c). Using this method, we checked the alignment of the dark volume across the atom’s whole $37\text{\,}\mathrm{mm}$ travelling distance. We measure an attenuation of the MOT light by a factor $\sim 30-40$ along the center of the transport guide due to the dark SPOT cylinders. The darkness is ultimately limited by imperfections of the cylinder’s surface and imaging, and by Poisson’s spot from diffraction. Another MOT beam geometry, without a beam in the $z$ axis, could be envisioned, but some sort of dark slit would still be required to darken the region where the guide crosses out of the MOT beam.

Launch beam — With both the reduced MOT beam intensities on the $z$ axis and the effect of the dark SPOT cylinders, we observe outcoupling of atoms into the transport guide followed by propagation across $37\text{\,}\mathrm{mm}$. However, the atoms’ speed is extremely low, dictated mainly by the MOT temperature, and outside the experimentalist’s control. The resulting flux varies strongly, and any imperfection in the engineered darkness result in a beam that appears to stop at seemingly random places. Moreover, low propagation speeds render the beam more vulnerable to losses such as background gas collisions and off-resonant scattering from the transport guide. To remedy this situation, we add a “launch” laser beam resonant with the ${{}^{1}\mathrm{S}_{0}}-{{}^{3}\mathrm{P}_{1}}$ $\pi$ transition. This $250\text{\,}\mathrm{\SIUnitSymbolMicro m}$-waist beam shines $30\text{\,}\mathrm{nW}$ of light at the overlap between the MOT and transport guide, forming an angle with the guide of $6\text{\,}\mathrm{\SIUnitSymbolDegree}$, see Fig. 1. With the help of this launch beam, we can outcouple MOT atoms into the guide with a well-controlled mean velocity ranging from 8 to $25\text{\,}\mathrm{cm}\text{\,}{\mathrm{s}}^{-1}$, see Section IV. During the remainder of this work, we typically operate with a launch beam intensity corresponding to a measured velocity of $8-$9\text{,}\mathrm{cm}\text{,}{\mathrm{s}}^{-1}$$.

Transverse cooling — By applying transverse cooling with the ${{}^{1}\mathrm{S}_{0}}-{{}^{3}\mathrm{P}_{1}}$ $\pi$ transition to atoms propagating along the guide, we can both minimize the atomic beam’s transverse temperature and optimize flux by removing evaporative losses. To this end, we place three single frequency laser beams with propagation axes perpendicular to the atomic beam, one $36\text{\,}\mathrm{mm}$ $1/e^{2}$ diameter beam propagating upward along the $y$ axis, and a counter-propagating $28.8\text{\,}\mathrm{mm}$ $1/e^{2}$ diameter beam pair along the $x$ axis, as shown in Fig. 1. These beams are centred around $35\text{\,}\mathrm{mm}$ from the MOT center and have powers of $1.4\text{\,}\mathrm{\SIUnitSymbolMicro W}$ and $6.75\text{\,}\mathrm{\SIUnitSymbolMicro W}$ for the horizontal and vertical axis, respectively. This gives a peak combined intensity of $\sim 0.5$ times the saturation parameter. These transverse cooling beams have a frequency $80\text{\,}\mathrm{kHz}$ blue detuned from the ${{}^{1}\mathrm{S}_{0}}-{{}^{3}\mathrm{P}_{1}}$ $\pi$ transition for free atoms. Due to the differential light shift from the guide this corresponds to a $\sim$200\text{,}\mathrm{kHz}$$ red detuning for atoms passing along the center of the guide. With this transverse cooling, the radial temperature is ultimately lowered to about $1\text{\,}\mathrm{\SIUnitSymbolMicro K}$ and the flux is increased by a factor of $\sim 2$.

III Figures of merit - an overview

Within the literature a variety of measures have been employed to characterize the performance of atomic beams, with each measure optimized for different applications. In this section we shall introduce and summarize these figures of merit as well as put them into context for applications such as interferometry, gain media for superradiant lasers and atom lasers.

Assuming a Gaussian density distribution of the atoms in the radial direction, the beam flux $\Phi$ can be represented by

[TABLE]

with $\Delta r$ the root-mean-squared 1D spatial spread, $n_{0}$ the peak density, $\rho_{L}$ the linear density and ${\bar{v}_{z}}$ the mean longitudinal velocity. All these parameters can be directly measured on our experiment. The flux density is the flux per unit cross-section area, given by $\rho_{\Phi}=\Phi/\pi\Delta r^{2}$.

We also give the beam performance in terms of the gas phase-space density, usually employed for ultracold and quantum degenerate gases. The PSD is expressed in the moving frame as

[TABLE]

where $h$ is the Planck constant, $k_{B}$ the Boltzmann constant, $m$ the atomic mass and $\lambda_{\textrm{dB},r/z}$ the thermal de Broglie wavelength associated with the 1D temperature $T_{r/z}$ in the radial/axial direction. Since we observe that the velocities follow Gaussian distributions, these effective 1D temperatures are directly related to the measured root-mean-squared 1D velocity spreads by the relations $\Delta v_{r,z}=\sqrt{k_{B}T_{r,z}/m}$.

There are two ways in which we estimate the peak atom number density $n_{0}$. Firstly, we can use absorption imaging and fit a Gaussian profile to estimate the peak density. Alternatively, we may assume that the atom density inside the guide follows a Boltzmann distribution with radial temperature $T_{r}$. The density distribution then follows $n(r)=n_{0,\mathrm{therm}}\exp\left(-\frac{U(r)}{k_{B}T_{r}}\right)$, where $U(r)$ is the potential energy due to the transport guide. This is valid in the case of a gas in thermal equilibrium thanks to a high collision rate. For a guide that is deep compared to the radial temperature, its radial potential can be approximated by a harmonic oscillator potential with trapping frequency $\omega_{r}/2\pi$. Using eq. (1), the peak density in the thermalized case is then related to the linear density by $\rho_{L}=n_{0,\mathrm{therm}}\,2\pi k_{B}T_{r}/m\omega_{r}^{2}$, and the expression of the phase-space density of eq. (2) can be written as

[TABLE]

The temperatures $T_{r}$ and $T_{z}$, the trapping frequency $\omega_{r}/2\pi$ and the linear density $\rho_{L}$ are directly accessible from the experimental data.

Another quantity of interest for establishing the usefulness of a beam source is its brightness (or radiance). In the literature it is often expressed by the flux density divided by the solid angle $\Omega$ of the beam divergence, and it is of primary interest to characterize ion beams McClelland et al. (2016). Since our beam is strongly confined by the transport guide and since the axial speed is very low, the beam divergence can be negligible between, for example, two regions of interrogation in an interferometry scheme. The brightness is thus not the most suitable quantity to characterize our beam. We nonetheless provide it for completeness in the case of a cold atomic beam Lison et al. (1999); Glover and Bastin (2015)

[TABLE]

with $\Omega=\pi(\Delta v_{r}/{\bar{v}_{z}})^{2}$. Similarly, the brilliance $\mathcal{B}$ of the beam is given by

[TABLE]

Since our beam is guided, a better suited figure of merit is the velocity brightness $\mathcal{R}_{v}$, expressed as the flux per unit of beam area per three dimensional velocity spread,

[TABLE]

This expression is commonly used to characterize atomic beams for interferometry based precision measurement Treutlein et al. (2001); Miossec et al. (2002); Riis et al. (1990); Chen and Riis (2000) and to characterize atom lasers Robins et al. (2013); Bloch et al. (1999); Robins et al. (2006).

IV Results

We now present the results from characterizing the atomic beam at a location $37\text{\,}\mathrm{mm}$ away from the MOT center, in a region out of the MOT laser beam and out of resonance with scattered light from these beams. We measure the axial mean velocity and velocity spread, and the radial velocity spread. We also measure the density, linear density and transverse spatial spread. From these we infer the beam flux, phase-space density and brightness. All these quantities are summarized in Table 1.

Axial velocity and velocity spread — Due to the extremely low temperatures reached in a strontium MOT operated on the narrow ${{}^{1}\mathrm{S}_{0}}-{{}^{3}\mathrm{P}_{1}}$ line, the resulting atomic beam can be extremely slow. This often welcome feature prevents us from characterizing the beam velocity and velocity spread by the conventional method of Doppler sensitive laser-induced fluorescence (LIF) Phillips and Metcalf (1982). In order to have a LIF signal with enough velocity resolution, the Doppler shift has to be large compared with the fluorescence light transition linewidth. The velocity of our atomic beam is on the order of $10\text{\,}\mathrm{cm}\text{\,}{\mathrm{s}}^{-1}$, corresponding to a Doppler shift of only $217\text{\,}\mathrm{kHz}$ for the $30\text{\,}\mathrm{MHz}$-wide ${{}^{1}\mathrm{S}_{0}}-{{}^{1}\mathrm{P}_{1}}$ transition, clearly an insufficient resolution. Alternatively, the weak $7.4\text{\,}\mathrm{kHz}$-wide ${{}^{1}\mathrm{S}_{0}}-{{}^{3}\mathrm{P}_{1}}$ transition would give $145\text{\,}\mathrm{kHz}$ of shift but insufficient fluorescence signal due to the low scattering rate.

In order to measure the axial atomic beam velocity and velocity spread, we instead apply a $1\text{\,}\mathrm{ms}$ pulse of light resonant with the ${{}^{1}\mathrm{S}_{0}}-{{}^{1}\mathrm{P}_{1}}$ transition, with a horizontal beam perpendicular to the guided atomic beam. This pulse ejects a packet of atoms out of the guide, see Fig. 3(a). We assume that spontaneously emitted photons are equally distributed in all directions during the ejection process, so that the $z$-axis mean velocity is not affected. We infer the mean velocity ${\bar{v}_{z}}$ and velocity spread $\Delta v_{z}$ by examining the evolution of the atom packet propagating alongside the transport guide. We perform this characterization for atoms in the transport guide located $37\text{\,}\mathrm{mm}$ away from the MOT, within the $\sim$4\text{,}\mathrm{mm} long region of interest (ROI) of our imaging setup. Thanks to our tunable transport guide architecture (see Sec. [II](#S2)), the potential landscape along the guide axis is essentially flat across the ROI. We therefore assume the beam velocity to be constant throughout the ROI. We measure the velocity for several launch beam intensities, and confirm that we can adjust the mean velocity within the range of $8$ to $25\text{\,}\mathrm{cm}\text{\,}{\mathrm{s}}^{-1}$, see Fig. [3](#S4.F3)(b). We observe that for all launch beam intensities the atomic beam has a similar velocity spread of $\Delta v_{z}=$5.2(2)\text{\,}\mathrm{cm}\text{\,}{\mathrm{s}}^{-1} (see Fig. 3(c)). This velocity spread corresponds to a 1D temperature of $T_{z}=$29(2)\text{,}\mathrm{\SIUnitSymbolMicro K}$$. By looking at the momentum imparted to the atomic packet by the ejection pulse in the radial direction, we check that the heating due to this pulse is negligible compared to the measured axial temperature.

Radial velocity spread — We measure the radial velocity spread in a more conventional way, by switching off the transport guide and measuring the atomic beam size after ballistic expansion, see Fig. 4(a). We obtain a radial temperature of $1.93(5)\text{\,}\mathrm{\SIUnitSymbolMicro K}$ averaged over the entire ROI. With the addition of transverse cooling light (see Section II), the flux increases by a factor of $\sim 2$ and the radial temperature reduces to $T_{r}=$0.89(4)\text{,}\mathrm{\SIUnitSymbolMicro K}, corresponding to a velocity spread of $\Delta v_{r}=$0.92(2)\text{\,}\mathrm{cm}\text{\,}{\mathrm{s}}^{-1}, see Fig. 4(b).

Within the ROI, we observe a mild variation of both the density and the radial velocity spread along the transport guide axis. More precisely, as the atoms propagate, both the flux and the radial temperature reduce. Without transverse cooling light, this can be explained by radial losses, which could be enhanced by eventual corrugations of the guide potential that transfer momentum from the axial to the radial direction. When the transverse cooling light is applied, the radial temperature slowly decreases, as expected, with the atoms’ travelling time. The losses are thus strongly reduced, leading to a slower decrease of the flux.

Density, flux, brightness — Our imaging system resolution of $4.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ is sufficient to image the beam density profile by absorption imaging (with a negligible $0.1\text{\,}\mathrm{ms}$ time of flight) while staying within the dynamic range of the camera. The linear density $\rho_{L}$ can be estimated by integrating the atom number over the radial direction and along the full length of the ROI. We fit the beam averaged along the propagation axis to the Gaussian profile of eq. (1), in order to estimate the peak density $n_{0}$ and the spatial spread $\Delta r$. From all the measured quantities and the expressions given in Section III, we extract the results summarized in Table 1 for both ${}^{88}\mathrm{Sr}$ and ${}^{84}\mathrm{Sr}$ isotopes. From the measured flux captured by the MOT and the flux observed at the end of the guide, we estimate the transfer efficiency from the MOT to the beam to be $\sim$30\text{,}\mathrm{\char 37\relax}$$.

Thermalization — As shown in Table 1, there is a clear discrepancy between the PSD values of $\rho_{\mathrm{PSD}}$ and the alternate $\rho_{\mathrm{PSD,therm}}$, with the latter being higher by a factor of $\sim 10$ for ${}^{88}\mathrm{Sr}$ and $\sim 3$ for ${}^{84}\mathrm{Sr}$. The expression for $\rho_{\mathrm{PSD}}$ estimates the density based on the measured atomic beam radius rather than assuming a Boltzmann distribution in a harmonic trap. We can therefore understand this discrepancy from the absence of thermalization, which can be seen from the differences between axial and radial temperatures. Without a thermalized sample, $\rho_{\mathrm{PSD,therm}}$ is not reliable so we shall keep the smaller value for PSD given by $\rho_{\mathrm{PSD}}$.

Following the work of Anderlini et al. (2005), we can estimate the elastic collision rate within the beam. From our measurements of the anisotropic density and velocity distributions, we can estimate the elastic collision rates to be $\Gamma_{\mathrm{el,88}}=$3.5(2)\text{\times}{10}^{-4}\text{,}\mathrm{Hz} and $\Gamma_{\mathrm{el,84}}=$0.11(2)\text{\,}\mathrm{Hz}. Given the propagation time from the MOT of less than $0.5\text{\,}\mathrm{s}$, the scattering rates for both isotopes are insufficient to thermalize the atoms.

Let us note that, despite the much higher density, the collision rate for ${}^{88}\mathrm{Sr}$ is $\sim 300$ times smaller than for ${}^{84}\mathrm{Sr}$, due to the scattering length $a_{\mathrm{scat},88}=-1.4\,a_{0}$ being much smaller than $a_{\mathrm{scat},84}=122.7\,a_{0}$, with $a_{0}$ the Bohr radius. These rates can be expressed as $\Gamma_{\mathrm{el}}=n_{0}\sigma_{\mathrm{cross}}v_{\mathrm{coll}}$, where $\sigma_{\mathrm{cross}}=8\pi a_{\mathrm{scat}}^{2}$ is the elastic cross-section and $v_{\mathrm{coll}}=\sqrt{2k_{B}T_{\mathrm{eff}}/\pi m}$ is the mean relative velocity, for an effective isotropic temperature $T_{\mathrm{eff,88}}\sim$8\text{,}\mathrm{\SIUnitSymbolMicro K} and $T_{\mathrm{eff,84}}\sim$10\text{\,}\mathrm{\SIUnitSymbolMicro K}.

V Discussion

The steady-state beam we demonstrate here has a phase-space density around three orders of magnitude higher than any previous steady-state atomic beam. In fact, the velocity brightness of $\mathcal{R}_{v}=$4.3(4)\text{\times}{10}^{21} for ${}^{88}\mathrm{Sr}$ is approaching what has been reached by pulsed quasi-CW atom lasers Robins *et al.* ([2013](#bib.bib46), [2006](#bib.bib69)), for example $\mathcal{R}_{v}=$2\text{\times}{10}^{24} in Ref. Bloch et al. (1999).

An immediate application for such a system might be continuously replenishing the gain medium of a steady-state superradiant active clock. There has been significant interest in the development of active optical clocks in recent years Chen (2009); Meiser et al. (2009). There have been pulsed demonstrations on the strontium clock transition Norcia et al. (2016) and a great deal of theoretical work Meiser and Holland (2010); Kazakov and Schumm (2017); Debnath et al. (2018); Zhang and Mølmer (2019); Schäffer et al. (2019); Hotter et al. (2019), but any active clock requires replenishment of the atoms used for the gain medium. Modelling suggests for the ${{}^{1}\mathrm{S}_{0}}\mathrm{-}{{}^{3}\mathrm{P}_{0}}$ clock transition that an ideal atomic source would consist of a guided continuous beam with a flux of $>$1\text{\times}{10}^{7}$\,^{88}\mathrm{Sr}\,${\mathrm{s}}^{-1} and a velocity of $\sim$10\text{\,}\mathrm{cm}\text{\,}{\mathrm{s}}^{-1}, criteria fulfilled by the beam we have demonstrated.

Another interesting application is steady-state interferometry schemes that aim to operate in a continuous mode, eliminating the Dick effect. This might be particularly important for long interrogation times, for example making use of the ${{}^{1}\mathrm{S}_{0}}\mathrm{-}{{}^{3}\mathrm{P}_{0}}$ transition Akatsuka et al. (2017); Hu et al. (2017) that has been proposed for gravitational wave detectors Hogan et al. (2011), although most likely an additional cooling stage is necessary for interferometry applications.

While the low axial velocity is ideal for some applications, it reduces both the brightness and brilliance figures of merit typically used for applications such as milling with ion beams. This, along with the reduced flux compared to other systems McClelland et al. (2016), would likely limit direct use in applications such as high current ion beam sources or high rate doping. However, it can be a good starting point for ion beam sources where the goal is to achieve the highest resolution. Similarly, this continuous, low spread beam is a good source for high repetition rate deterministic single ion sources Ates et al. (2013); Sahin et al. (2017).

For many applications the ultimate atomic beam source would be a steady-state continuous atom laser and, in the past, efforts towards this goal have produced the highest phase-space density beams reported Lahaye et al. (2004, 2005). The work here represents a major step towards this goal even though it is preliminary in many aspects. A better control of the axial beam velocity is possible in several ways, one of which has been recently demonstrated in our group Chen et al. (2019). By reducing the velocity and increasing the density, it might be possible with the ${}^{84}\mathrm{Sr}$ isotope to increase the elastic collision rate $\Gamma_{\mathrm{el}}$ and reach a regime where collisions dominate Lahaye et al. (2004); Olson et al. (2014). This would enable evaporative cooling along the dipole guide Lahaye et al. (2005); Mandonnet et al. (2000); Olson et al. (2006), which would increase the beam PSD and hopefully produce a continuous atom laser Robins et al. (2013); Bloch et al. (1999).

To summarize, we have demonstrated a continuous guided atomic beam of 88Sr with a phase-space density more than three orders of magnitude higher than in previously reported systems. Our beam has an extremely low mean velocity ${\bar{v}_{z}}=$8.4(4)\text{,}\mathrm{cm}\text{,}{\mathrm{s}}^{-1}, radial spatial spread $\Delta r=$23.3(4)\text{\,}\mathrm{\SIUnitSymbolMicro m} and radial velocity spread $\Delta v_{r}=$0.92(2)\text{,}\mathrm{cm}\text{,}{\mathrm{s}}^{-1}. This corresponds to a radial temperature of just $0.89(4)\text{\,}\mathrm{\SIUnitSymbolMicro K}$. The beam flux is $\Phi=$3.25(14)\text{\times}{10}^{7}$\,\mathrm{at}\,${\mathrm{s}}^{-1} and it reaches a PSD $\rho_{\mathrm{PSD}}=$1.5(2)\text{\times}{10}^{-4}$$. Using the $0.56\text{\,}\mathrm{\char 37\relax}$ abundant 84Sr isotope, we obtain a reduced phase-space density, but the higher scattering length means that our beam is approaching the collisionally dense regime, where evaporative cooling can be used to rapidly improve phase-space density. This represents a significant step towards the demonstration of a steady-state atom laser. Moreover, this beam is likely to find immediate application in efforts to demonstrate a steady-state superradiant active optical clock. A beam with such output performance could fulfill the demands of other applications requiring both ultracold atoms and uninterrupted operation, such as continuous atom interferometers, clocks, ion sources and steady-state atom lasers.

Acknowledgements.

We thank Andrea Bertoldi for careful reading of the manuscript and providing insightful comments. We thank the Netherlands Organisation for Scientific Research (NWO) for funding through Vici grant No. 680-47-619 and the European Research Council (ERC) for funding under Project No. 615117 QuantStro. This project has received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No 820404 (iqClock project). B.P. thanks the NWO for funding through Veni grant No. 680-47-438. C.-C. C. thanks the Ministry of Education of the Republic of China (Taiwan) for a MOE Technologies Incubation Scholarship.