Narrow-line absorption at 689 nm in an ultracold strontium gas
Fachao Hu, Canzhu Tan, Yuhai Jiang, Matthias Weidemüller, Bing Zhu
NNarrow-line absorption at 689 nm in an ultracold strontium gas
Fachao Hu,
1, 2, ∗ Canzhu Tan,
1, 2, ∗ Yuhai Jiang,
3, 2, † Matthias Weidem¨uller,
1, 2, 4, ‡ and Bing Zhu
4, 1, 2, § Hefei National Laboratory for Physical Sciences at the Microscale and Shanghai Branch,University of Science and Technology of China, Shanghai 201315, China CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics,University of Science and Technology of China, Shanghai 201315, China Shanghai Advanced Research Institute, Chinese Academy of Sciences, Shanghai 201210, China Physikalisches Institut, Universit¨at Heidelberg, Im Neuenheimer Feld 226, 69120 Heidelberg, Germany (Dated: January 28, 2021)We analyse the spectrum on the narrow-line transition 5s S − P at 689 nm in an ultracoldgas of Sr via absorption imaging. In the low saturation regime, the Doppler effect dominates inthe observed spectrum giving rise to a symmetric Voigt profile. The atomic temperature and atomnumber can accurately be deduced from these low-saturation imaging data. At high saturation, theabsorption profile becomes asymmetric due to the photon-recoil shift, which is of the same orderas the natural line width. The line shape can be described by an extension of the optical Blochequations including the photon recoil. A lensing effect of the atomic cloud induced by the dispersionof the atoms is also observed at higher atomic densities in both the low and strong saturation regimes.
I. INTRODUCTION
The existence of metastable states and narrow-linetransitions among the alkaline-earth and alkaline-earth-like atoms brings new opportunities for studying cold andultracold atoms, for example the optical-lattice clocks[1, 2], the time variation of fundamental constants [3–5], atom interferometers[6–8], nonlinear quantum optics[9–11], and strongly correlated Rydberg gases [12, 13].While many of these applications rely on the clock tran-sition S − P with a linewidth on the level of mHz,the other narrow one S − P with a (1 ∼ S - P transitions in alkali-earth or alkali-earth-like systems,fluorescence signals from these transitions can be em-ployed for studying collective atomic scattering and mo-tional effects [21], measuring atomic transition proper-ties [22–24], and detecting single atoms with high fideli-ties [20]. On the other hand, absorption imaging usingbroad dipole-allowed transitions ( ∼
10 MHz) may be byfar the most widely-used method in diagnosing ultracold-atom systems, providing accurate information on the spa-tial distribution of atoms, the atom number, and theatomic temperature [25, 26]. However, absorption withnarrow-line transitions were rarely studied in the ultra-cold regime, where the photon recoil energy is compa-rable to the absorption linewidth including the Doppler ∗ These authors contributed equally to this work. † [email protected] ‡ [email protected] § [email protected] effect. Oates et al. studied the atomic-recoil-inducedasymmetries in a form of saturation spectroscopy witha Ca optical-clock apparatus [27], and the photon-recoileffect on the dispersion was observed in a Yb vapor cellin Ref. [28]. Stellmer et al. have implemented the ab-sorption imaging on the 7.5-kHz transition at 689 nm toresolve the hyperfine structure of the fermionic Sr ata magnetic field of about 0.5 G [29]. They observed aLorentzian lineshape with a full width at half maximum(FWHM) of about 40 kHz, without discussing furtherdetails on the spectrum.In this work, we study in detail the absorption spec-trum on the narrow transition 5s S − P at 689nm with an ultracold Sr atomic cloud. We measurethe spectrum in both the weak and strong saturationregimes. At low saturations, the absorption lineshape isclose to a Gaussian shape essentially determined by theDoppler effect in the temperature range studied here.Thus, this regime can be exploited for thermometry ofthe atomic sample, which is confirmed by a comparison tothe temperature obtained by the standard time-of-flight(TOF) method [25, 26] using the broadband transition5s S − P . The narrow-line absorption imagingat low saturation also provides information on the atomnumbers and atomic densities with a comparable accu-racy to detection methods based on the broad (blue) line.In the strong saturation regime, an asymmetric lineshapeis observed. We have performed a theoretical simulationbased on the optical Bloch equations (OBEs) involvingthe momentum transfers during the imaging process andconfirmed that the photon recoil has important influenceon the line shape. We also observe a density-dependentlensing effect in the absorption images at large detuningsof the imaging light.The article is organized as follows: We show our experi-mental setup in Sec. II. The low- and high-saturation ab-sorption spectra are described in Secs. III A and III B, re-spectively. The theoretical simulation and comparison toexperiments in the high-saturation regime are discussed a r X i v : . [ phy s i c s . a t o m - ph ] J a n g & B HWP PBS -300 -200 -100 0 100 200 3000.000.050.100.15 P ea k OD Detuning (kHz) m j �� → m j' =+1 m j �� → m j' =0 m j �� → m j' =-1 EM-CCDcamera atomic cloudlens 1lens 2(a)(c)(b) z y t ODTImaging MOT τ exp τ TOF
B field τ exp 。 FIG. 1. (a) Schematic of the top view of experimental setup.HWP: half wave-plate; PBS: polarizing beam-splitter. g and B represent the gravity and magnetic field, respectively. Seetext for more details. (b) Time sequence for absorption imag-ing. See text for explanations of τ TOF and τ exp . (c) Ab-sorption spectrum showing all three Zeeman sublevels of P state when the imaging light polarization is tuned to about45 ◦ angled to the residual magnetic field. Black points arethe measured peak OD, and the red curve is the fit to amulti-peak Gaussian function. The obtained Zeeman split-ting is 167.7(1.2) kHz, corresponding to a magnetic field of79.9(6) mG. in Sec. III C. The observation of lensing effect is pre-sented in Sec. IV. Sec. V concludes the paper. II. EXPERIMENTAL SETUP
Fig. 1(a) shows the experimental setup. The Sratoms are first loaded into a two-stage magneto-opticaltrap (MOT) for the laser cooling and trapping [30, 31],operated on the broad 5s S − P and narrow5s S − P transitions, respectively. We couldcreate an atomic cloud of 10 atoms with a density ofabout 10 cm − and a temperature around 1 µ K. Acigar-shaped optical dipole trap (ODT) formed by twohorizontally propagating beams at the wavelength of532 nm, is simultaneously switched on at the second-stage MOT. The two ODT beams both have a waistof about 60 µ m and cross at an angle of 18 ◦ . Holdingatoms in the ODT for 200 ms to reach equilibrium afterswitching off the MOT, we obtain about (0 . · · · × atoms at a temperature of 0 . · · · µ K depending on theODT power. At a power of 0.6 W for each beam thetrap depth of the ODT is about 6 µ K and the trap fre-quencies are 2 π × (217, 34, 217) Hz along the x , y , and z directions [see Fig. 1(a)], respectively, resulting incloud radii of (27, 69, 27) µ m and a peak density of7 × cm − . The temperatures along the y and z direc-tions are mapped out by the standard TOF method. Theabove-mentioned atom numbers, cloud sizes, and temper-atures are measured using absorption imaging with thebroad 5s S − P transition. The lifetime of theatomic clouds in the ODT is about 2 s, limited by thecollisions with background gas.The imaging light at 689 nm is delivered from a com-mercial tapered amplifier seeded by an external-cavitydiode laser (Toptica TApro), used also for the narrow-line MOT cooling, which is frequency-stabilized to a pas-sive ultra-low expansion cavity with a short-term noise of1 kHz level and a long-term drift of 8 kHz/day [31]. Asshown in Fig. 1(a), the imaging beam propagates alongthe z direction with a tunable linear polarization and hasa 1 /e diameter of 4.2 mm. The imaging pulse length andintensity are controlled by an accousto-optic modulator(not shown in the figure). The imaging system consistsof two achromatic lenses with focal lengths of +200 mmand +300 mm, and maps the absorption to an EM-CCDcamera from Andor with a magnification factor of 1.5.We have an imaging resolution of about 12 µ m.The imaging sequence is described in Fig. 1(b). Theabsorption imaging on the narrow-line transition is per-formed after rapidly switching off the ODT to avoid thedifferential AC Stark shifts on the energy levels. A quan-tization magnetic field along the vertical direction is ap-plied (rising time 2 ms) before the imaging pulse to splitthe Zeeman sublevels of P state, as seen in Fig. 1(c).After a given time-of-flight (TOF) time τ TOF , the atomsare shined by the imaging light with an exposure time τ exp = 200 µ s. By tuning the τ TOF we can tune theatomic density during the absorption, which plays animportant role in observing the dispersive lensing effectdiscussed in Sec. III. As done in a standard absorptionimaging sequence, two additional images with and with-out the imaging light are taken after the first pulse. Thethree images are then processed (see, e.g., [32]) to ob-tain the two-dimensional optical density (OD) distribu-tion (see the insets of Fig. 5).By changing the linear imaging polarization angle inthe x − y plane, all three Zeeman sublevels of the P state are addressable. An example is shown in Fig. 1(d).The peak OD is measured as a function of the imagingdetuning showing three peaks at a magnetic field of about80 mG. The relative line strengths are determined by thepolarization and the different coupling strengths of thethree corresponding transitions (see Fig. 1(d)). We haveused this measurement to optimize the compensation ofthe background magnetic field to be better than 5 mGin our setup and to calibrate the quantization fields. Forthe absorption studies, we apply a field of 4 G to split thesublevels and the imaging polarization is tuned parallelto the quantization axis, so that the system is subjectedonly to the closed π transition ( m j = 0 → m j (cid:48) = 0),which can be treated as a perfect two-level system. III. MEASUREMENTS AND ANALYSIS
Thanks to the high sensitivity and large dynamicalrange of our imaging camera (Andor iXon 897) at 689 nm,we can study the absorption spectrum on the narrow-linetransition with a saturation parameter s ranging from0.01 to more than 100. Meanwhile, the cloud temper-ature and the atomic density can be controlled via theODT depth and the TOF time τ TOF independently.
A. Low-saturation absorption
In Fig. 2(a), we show two measured absorption spec-tra at temperatures of 1.3 µ K (black points) and 5.7 µ K(red points) with a saturation parameter of s = 0 .
1. TheTOF time τ TOF (see Fig. 1(b)) is chosen to be 3.1 msto minimize the lensing effect (see Sec. IV) as well asto keep large enough signal-to-noise ratios (SNRs) in theOD images. The plotted signals in Fig. 1(b) are the ODintegrals over the whole atomic cloud region divided bythe peak cross section σ = 3 λ / π , which is the stan-dard way to calculate the atom number in the absorptionimaging (see the following paragraph for a correction).Symmetric lineshapes are observed in both cases andthe linewidth increases with the increasing tempera-ture. The spectra fit well to Voigt profiles with a fixedLorentzian width of v L = 10 .
01 kHz, resulted from thepower broadening Γ √ s/ π and the detection band-width 0 . /τ exp = 4 . / π = 7 . v G obtained from the Voigt profile fit-ting is used to deduce the temperature T Fit along theimaging propagation direction, from the relation v G = λ (cid:112) k b T Fit /m . Here k b is the Boltzmann constant, λ - 1 0 0 - 5 0 0 5 0 1 0 00123450 . 81 . 01 . 21 . 4 0 2 4 60246 m K 5 . 7 m K Integrated OD/ s D e t u n i n g ( k H z )( a )( b ) N red/ N blue T Fit ( m K) T T O F ( m K )
FIG. 2. (a) Low-saturation absorption spectra at tempera-tures of 1.3 µK (black) and 5.7 µK (red). The integratedabsorption signal over the atomic cloud region is plotted as afunction of the imaging detuning. The solid curves are fits tothe Voigt profile. See text for more details. (b) Spectroscopicthermometry. The fitted Doppler widths from (a) are usedto estimate the temperatures T Fit , which are plotted againstthe TOF measurement results T TOF in the lower panel of (b).A linear fit to the data (black dashed line) gives a slope of1.05(3). The red dashed line represents T Fit = T TOF . In theupper panel we also show the ratio (black open circles) be-tween the atom numbers obtained from the narrow- ( N red )and broad-linewidth ( N blue ) imaging, which is a constant of1.06(2) (gray solid line). is the transition wavelength, and m is the atomic mass. T Fit obtained in this way are compared to those measuredby the TOF method in the lower panel of Fig. 2(b). Thelinear fit between T Fit and T TOF (black dashed line) re-sults in a slope of 1.05(3), which agrees excellently withthe ideal case of T Fit = T TOF (red dashed line). We alsonotice the empirical density broadening in the saturationfluorescence spectroscopy reported in [23]. The linearslope is only modified slightly to 1.05(6) even if we takethe empirical density relation following Ref. [23].In addition to the temperature, the atom numberand atomic density can also be extracted from thenarrow-linewidth absorption imaging in the low satura-tion regime. The broad (blue) transition typically used indetermining the atom number and atomic density has anatural linewidth on the order of 10 MHz, much broaderthan the Doppler width. The absorption cross-section inthe broad-transition imaging can hence be regarded astemperature-independent. However, for the narrow tran-sition with a natural linewidth smaller than the Dopplerwidth (Γ / πv G < OD ( x, y ) = n ( x, y ) σ × C (Γ , v G ) , (1)where C (Γ , v G ) = √ παe α Erfc( α ) is the coefficientwith α = √ ln 2Γ / πv G , OD ( x, y ) are the on-resonanceOD spatial distribution, and n ( x, y ) is the atomic col-umn density. Erfc(x) is the complementary error func-tion. The derivation of the coefficient is presented inthe Appendix V. With the on-resonance OD and thetemperature-dependent v G determined from the spec-trum fitting, the atom number and atomic densitycan be obtained with Eq. (1). The upper panel ofFig. 2(b) shows the ratio of the atom number deter-mined by absorption imaging with the narrow 5s S − P ( N red ) and broad 5s S − P ( N blue ) tran-sitions, which lies close to 1 (gray solid line). B. Strong-saturation absorption
In this section we study the narrow-line absorptionspectrum at strong saturations with s (cid:29)
1. The ques-tion arises, to which extent the photon recoils impactsthe absorption profile, as the the recoil shift is compara-ble to the natural linewidth ( ∼ .
78 kHz vs. 7 . s = 0 .
09 and s = 35 .
8, respectively.We observe a decrease of the integrated OD signal at alldetunings due to the saturation effect (note that the dataat higher saturation is magnified by a factor of 6 for abetter view). More importantly, the lineshape is asym-metric at the high saturation, namely the integrated ODapproaches zero more slowly on the negative-detuningside than that on the positive one, and the absorptionpeak is shifted by a few kHz to the positive detuning. Athigh saturation, differences can already be seen in the ODimages at the two detuning sides [see lower rows in theinsets of Figs. 3(b-d)], namely a wider spatial extensionfor the position detuning than that for the negative one.In this series of experiments, the influence of the lensingeffect on the OD measurement(see Sec. IV) is negligibledue to the low atomic densities involved here.The observed asymmetry and peak shift can be in-terpreted qualitatively by considering the absorptionprocess including the influence of the photon recoil.The photon recoil associated with each absorption-spontaneous emission cycle redistributes the momentumof atoms, which depends strongly on the light detun-ing [35]. Consequently, an asymmetric lineshape and theshift of the maximum of absorption emerges when moreand more photons are scattered due to the momentumredistribution in the atomic cloud. In order to resolvesuch effects, the Doppler width has to be comparable tothe power-broadened line width. In the case of the strongsaturation in Fig. 3 (a), the power-broadened Lorentzianwidth Γ √ s ∼
45 kHz is close to the Doppler one of ∼
40 kHz. In the following subsection a quantitativedescription is presented incorporating the photon-recoileffect in an OBE formalism.
C. Spectrum lineshape simulation
We take an OBE formalism including the method ofso-called ’momentum families’ from Ref. [36], originallydeveloped to understand laser cooling on a narrow-linetransition. The model considers a two-level atom systemwith an initial Maxwell-Boltzmann thermal distribution,interacting with a single near-resonant monochromatichomogeneous probe beam. The state of an atom withmomentum p is expressed in the form of {| g, p (cid:105) , | e, p (cid:105)} ,where | g ( e ) (cid:105) corresponds to the atomic ground (excited)state. The system Hamiltonian driven under a laser beampropagating along the z axis is, H = ˆ p m + (cid:126) ω | e (cid:105) (cid:104) e | − ˆ D · ˆ E (2)where ω , ˆ D , ˆ E are the transition frequency, dipole mo-ment operator, and laser electric field, respectively. In (b) (c) (d)
100 75 50 25 0 25 50 75 100detuning (kHz)0.00.20.40.60.81.0 I n t e g r a t e d O D s=0.09s=35.8 (a) ×6
100 m
100 m
100 m -10 -7.5 -5.0 -2.5 0 2.5 5.0 7.5 100.000.010.020.030.040.05-7.5 -5.0 -2.5 0 2.5 5.0 7.5 10-10 0 100.000.010.020.030.040.05 -10.0 -7.5 -5.0 -2.5 0.0 2.5 5.0 7.5 10.00.000.010.020.030.040.05 momentum ( ħ k) t=0 s=0.09, t=200 μ ss=35.8,t=100 μ ss=35.8,t=200 μ s Δ ρ -80 -60 -40 -20 0 20 40 60 80 Doppler shift (kHz) -80 -60 -40 -20 0 20 40 60 80 -80 -60 -40 -20 0 20 40 60 80
FIG. 3. High-saturation absorption. (a) The measured absorption lineshapes at low (blue dots) and high (red dots) saturations.The data in the high-saturation case ( s = 35 .
8) showing asymmetric profile is fitted to the numerical solution of Eq. (5) (redcurve) and magnified by 6 times to have a better visualization. As a comparison, the low-saturation ( s = 0 .
09) data is symmetricand fits well to the Voigt profile (blue curve). (b) - (d), the population difference ∆ ρ ( p ) obtained from the OBE solutions atthere different detunings [0 , ± µ s and 200 µ s,respectively. As a reference, we also show the initial distribution at t = 0, which is the Maxwell-Boltzmann one determinedby the cloud temperature. The black solid vertical lines mark the resonant momentum positions, where the probe detuningis compensated by the Doppler effect. The inset images show measurements of the two-dimensional OD distributions at low(upper) and high (lower) saturations for their respective detunings. our case, only the π -transition branch m j = 0 → m j (cid:48) = 0is considered, and only momentum along the light prop-agation axis p = p z is preserved, with the other two com-ponents p x , p y traced over. The system Hamiltonian un-der the rotating-wave approximation becomes, H S = ˆ p m − (cid:126) δ | e (cid:105) (cid:104) e | + (cid:126) Ω2 ( e ikz | e (cid:105) (cid:104) g | + | g (cid:105) (cid:104) e | e − ikz ) (3)where δ, Ω are the bare detuning and Rabi frequency.The evolution of states | g, p (cid:105) , | e, p + (cid:126) k (cid:105) with any mo-mentum p remains globally closed under H S when thespontaneous emission is not considered, for which reasonthe states | g, p (cid:105) , | e, p + (cid:126) k (cid:105) are grouped as a family F ( p ).The system density matrix ρ expanded in this basis is, ρ gg ( p ) = (cid:104) g, p | ρ | g, p (cid:105) ρ ee ( p ) = (cid:104) e, p + (cid:126) k | ρ | e, p + (cid:126) k (cid:105) ρ ge ( p ) = ρ ∗ eg ( p ) = (cid:104) g, p | ρ | e, p + (cid:126) k (cid:105) . (4) The equations of evolution under H S together with thespontaneous emission processes are,˙ ρ gg ( p ) = Γ¯ π e ( p − (cid:126) k ) − i Ω2 ( ρ eg ( p ) − ρ ge ( p )) , ˙ ρ ee ( p ) = − Γ¯ π e ( p ) + i Ω2 ( ρ eg ( p ) − ρ ge ( p )) , ˙ ρ ge ( p ) = ˙ ρ ∗ eg ( p )= − ( i (¯ δ − kpm ) + Γ2 ) ρ ge ( p ) + i Ω2 ( ρ gg ( p ) − ρ ee ( p )) , (5)where ¯ δ = δ − (cid:126) k / (2 m ) and the term ¯ π e represents theimpact of spontaneous decay on the system evolution, Relative shift (kHz) s m e a s u r e m e n t f o r t e x p = 1 0 0 m s m e a s u r e m e n t f o r t e x p = 2 0 0 m s c a l c u l a t i o n f o r t e x p = 1 0 0 m s c a l c u l a t i o n f o r t e x p = 2 0 0 m s FIG. 4. Absorption peak position shift. The relative positionof the absorption peak at different saturation parameters s are compared for imaging times of 100 µ s (red circles andcurve) and 200 µ s (black circles and curve). The black andred curves are the calculated results without any free parame-ters, while the black and red circles are the fitted results frommeasurements by using the peak position and height as thetwo free fitting parameters. See text for more discussions. defined as¯ π e ( p ) = + ∞ (cid:90) −∞ dp x + ∞ (cid:90) −∞ dp y + (cid:126) k (cid:90) − (cid:126) k dp (cid:48) N ( p (cid:48) ) (cid:104) e, p x , p y , p z = p + p (cid:48) | ρ | e, p x , p y , p z = p + p (cid:48) (cid:105) . (6)Here N ( p (cid:48) ) = (cid:126) k (1 − p (cid:48) / (cid:126) k ) results from the classicaldipole radiation pattern [36] of the π transition. Withall the atoms initially at the ground state | g (cid:105) with aMaxwell-Boltzmann distribution of temperature T , wenumerically integrate the equations (5) to get the sys-tem evolution. The solution of the off-diagonal elements ρ eg ( p ) results in the susceptibility χ ( p ) ∝ nρ eg ( p ) withthe atomic density n . The absorption profile is then cal-culated by tracing the imaginary part of the susceptibilityover all momenta, i.e. (cid:80) p Im χ ( p ), and then integratingover the interaction duration.For the solid curves in Figs. 3(a), we fit the experi-mental data to the calculated profiles with the maximumintegrated OD and the peak position as the only free pa-rameters. Both the lineshape asymmetry and the shiftof the absorption peak at high saturation can be repro-duced very well by Eq. (5) including the momentumtransfer due to the photon-scattering events. While themodel predicts a significant shift of the absorption peak,its position is still used as a free parameter in the fits toaccount for the possible deviation between the measure-ments and the calculations, as discussed in more detailsin Fig. 4. One can gain further insight into the photon-recoileffects by considering the quasi-steady solution of theoff-diagonal element in Eq. (5), Im ρ eg (p) ∝ ∆ ρ ( p ) = ρ gg ( p ) − ρ ee ( p ). We show from Fig. 3(b) to 3(d) the cal-culated distribution of the population difference ∆ ρ ( p )at two saturation parameters of s ≈ .
09 (blue curves)and s = 35 . µ s atom-light in-teraction time (about 10 / Γ, the imaging pulse length inthis measurement), when the probe laser is detuned by − , , +5Γ from left to right. At the low saturation( s ≈ . ρ ( p ) is only slightly modified compared tothe initial Maxwell-Boltzmann distribution (black dot-dashed lines), remaining almost Gaussian even after longinteraction time, such that the convolution between thevelocity-dependent Lorentzian profile. The momentumdistribution results in a lineshape nearly the Voigt one,as the blue curve seen in Fig. 3(a). When highly sat-urated ( s = 35 . ρ ( p ) distribution isstrongly modified and depleted near the resonant mo-mentum (marked by vertical dashed lines) where theDoppler shift compensates the bare imaging detuning.In Fig. 3(b) with a detuning of − ∼ . (cid:126) k after 200 µ s. While at a detuning of +5Γ inFig. 3(d), two peaks appear on the opposite sides of theresonant momentum. Such a strong dependence on thedetuning leads to the observed asymmetric lineshape andthe peak shift.The effects of the photon recoil can also be revealed bystudying the time evolution of the momentum distribu-tion. In Figs. 3(b-d) the ∆ ρ ( p ) at s = 35 . µ sinteraction (red dash-dot curves) are shown as a com-parison to the 200- µ s case. Small but clear differencesof ∆ ρ ( p ) are observed for all three detunings indicatingthat the momentum distribution undergoes some timeevolution, which may result in a time-dependent absorp-tion lineshape. This is actually demonstrated in Fig. 4by comparing the saturation-dependent shift of the ab-sorption peak position for the 100- and 200- µ s imagingdurations. The peak position is shifted towards the pos-itive detuning when increasing the imaging intensity andsuch a shift becomes larger in the case of a longer expo-sure, i.e. more photons are scattered. The solid curvesrepresent the calculated results without any free param-eters, while the solid dots are from fits with the peak po-sition and height as the free fitting parameters [see Fig.3(a)]. Overall, the fitted shifts agree well with the cal-culations without free parameters, while deviations areseen for some points coming from fluctuations of experi-mental conditions like laser power and atom number, aswell as the low SNR for large saturation parameters. IV. THE LENSING EFFECT
As shown in Fig. 5, we have also experimentallyobserved another phenomenon in the absorption spec-trum at high atomic densities, the so-called lensing ef-
100 m ×3 FIG. 5. Absorption spectrum with s = 17 at two differentatomic densities of 8 . × cm − (red circles) and 2 . × cm − (blue diamonds). The red curve is a fit to thenumerical solution of Eq. (5). We obtain negative peak ODsat some large positive detunings. In the right inset, the lowerOD image measured at a large positive detuning with thehigh atomic density has a dark hole instead of a bright peakin the cloud center, caused by the lensing effect. At the largenegative detuning, the dark position appears at the edges ofthe cloud (left inset). As a comparison, we also show anexample of the OD images for the low-density case with anormal Gaussian distribution in the upper panel of the inset. fect which is well known in standard absorption imaging.The absorption spectra at two different atomic densitiesare compared at a saturation of s = 17. In the low-density case ( n ∼ . × cm − , red dots in the fig-ure), we find a similar asymmetry as that in Fig. 3(a)for the high saturation. With a 3-fold higher density( n ∼ . × cm − , blue diamonds in the figure), anegative peak OD is obtained from the two-dimensionGaussian fit at some large positive detunings. Checkingthe OD images there (one example shown as the right in-set in Fig. 5), a dark hole instead of a bright peak is seenat the central region of the atomic cloud for the largepositive detuning, while at the negative one a dark edgeis observed. This phenomonon is related to the micro-scopic lensing effect studied in e.g. Refs. [37–43], wherea spatial-dependent index of refraction leads to a focus-ing or defocusing effect on the imaging beam dependingon the detuning.The observed lensing effect can be understood from thefollowing equation for describing the phase shift of theimaging field in the transverse plane propagating througha cloud of two-level atoms [39], dφ ( x, y ) = − σ n ( x, y, z ) dz δ/ Γ1 + 4( δ/ Γ) + s ( x, y ) . (7)Here dz is the thickness of the atomic cloud along thelight propagation direction, δ is the detuning, and s ( r )has a spatial dependence due to the intensity distribu-tion of a Gaussian probe beam. Spatial inhomogeneityof the index of refraction can be induced by the spatial distribution of the atomic density, or the probe intensity,or both. For negative (positive) detuning, Eq. (7) leadsto a focusing (defocusing) of the imaging beam.The observed lensing effect here mainly stems fromthe density inhomogeneity as indicated by the density-dependence (see Fig. 5) and the fact that the imagingbeam is much larger than the atomic cloud ( ∼
200 times).The lensing induced by such densitiy inhomogeneity wasobserved in both the weak- [40] and strong-saturation[37–39] regimes. The lensing effect shown in Fig. 5 withstrong saturation is also observable in the weak-probecase in our experiment. However, to quantitatively ex-plain our observation, detailed calculations on the lightpropagation are needed like in Refs. [41, 43], even includ-ing the atom dipolar interactions or multiple scatteringevents (e.g. [21, 44, 45]), which is beyond the scope ofthis paper.
V. CONCLUSION
In conclusion, we have studied both experimentallyand theoretically the absorption spectrum of a narrow-line transition at 689 nm in an ultracold Sr gas. Theatomic cloud temperature down to 1 µ K can be inferredfrom the measured absorption lineshape at low probesaturations ( s (cid:28)
1) if the Doppler width dominatesover other line-broadening effects. Information on theatom number can also be reliably extracted from the low-saturation absorption. In the strongly saturated regime,we observed the photon-recoil-induced asymmetry in theabsorption spectrum, which can be described by two-levelOBEs involving the photon recoils. We also showed alensing effect when probing a high-density sample, whichis due to the spatial-dependent dispersive response of theatomic cloud to the imaging field. It is of strong interestin studying further the weak-probe high-density regimebecause of the collective and cooperative effects that arepredicted theoretically [44, 46–48]. The narrow-line ab-sorption can also be employed as sensitive probe for othercold atom systems with similar narrow-line transitions,like, e.g., Yb. The good resolution also makes the narrow-line absorption applicable to detection of interactions inmore complicated systems, e.g. the spatial correlation[49] due to Rydberg blockade.
ACKNOWLEDGEMENTS
We acknowledge C. Qiao, L. Couturier, and I. Nosskefor their contributions on setting up the experiment atthe early stage of project. F.H acknowledges Yaxiong Liufor helpful discussions on numerical algorithms. M.W.’sresearch activities in China are supported by the 1000-Talent-Program. The work was supported by the Na-tional Natural Science Foundation of China (Grant Nos.11574290 and 11604324) and Shanghai Natural ScienceFoundation (Grant No. 18ZR1443800). Y.H.J. also ac-knowledges support under Grant No. 11827806.
APPENDIX
The low-saturation ( s (cid:28)
1) OD spatial distribution isrepresented as, OD ( x, y ) = (cid:90) + ∞−∞ σ n ( x, y ) f ( v ) L ( δ, v, Γ) dv (8)where L ( δ, v, Γ) = Γ / δ − kv ) +Γ / is the Lorentzian profilewith δ the bare laser detuning, Γ the natural linewidth, k = 2 π/λ the laser wavenumber, and v the atom veloc-ity, f ( v ) = u √ π e − v /u is the Gaussian velocity distribu-tion with u = (cid:112) k B T /m the most probable speed. TheDoppler width v G is related to u , v G = ku √ ln 2 /π . Then Eq. (8) reads OD ( x, y ) = σ n ( x, y ) (cid:90) + ∞−∞ u √ π e − ( v/u ) Γ / δ − kv ) + Γ / dv = σ n ( x, y ) α √ π (cid:90) + ∞−∞ e − ( x (cid:48) + δ/ku ) x (cid:48) + α dx (cid:48) (9)The substitution x (cid:48) = kv − δ is used in the secondstep. Here α = √ ln 2Γ2 πv G represents the ratio betweenthe natural linewidth and the Doppler width. At theon-resonance condition ( δ = 0) we have the Eq. (1).The coefficients C (Γ , v G ) = √ παe α Erfc( α ) for correct-ing the on-resonance absorption cross section are plottedin Fig. for the 5s S − P (black dashed line),5s S − P (blue dashed line) transitions of Srand the D2 transition of Rb (red dotted line) as a com-parison. [1] H. K. Jun Ye, H. J. Kimble, Science (2008), 10.1126/sci-ence.1148259.[2] A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, andP. Schmidt, Reviews of Modern Physics , 637 (2015).[3] M. Safronova, D. Budker, D. DeMille, D. F. J. Kimball,A. Derevianko, and C. W. Clark, Reviews of ModernPhysics (2018), 10.1103/RevModPhys.90.025008.[4] M. S. Safronova, S. G. Porsev, C. Sanner, andJ. Ye, Physical Review Letters (2018), 10.1103/phys-revlett.120.173001.[5] C. J. Kennedy, E. Oelker, J. M. Robinson, T. Bothwell,D. Kedar, W. R. Milner, G. E. Marti, A. Derevianko, andJ. Ye, Physical Review Letters (2020), 10.1103/phys-revlett.125.201302.[6] L. Hu, N. Poli, L. Salvi, and G. M. Tino, Physical ReviewLetters (2017), 10.1103/physrevlett.119.263601.[7] L. Hu, E. Wang, L. Salvi, J. N. Tinsley, G. M. Tino,and N. Poli, Classical and Quantum Gravity , 014001(2019).[8] J. Rudolph, T. Wilkason, M. Nantel, H. Swan, C. M.Holland, Y. Jiang, B. E. Garber, S. P. Carman, andJ. M. Hogan, Physical Review Letters (2020),10.1103/physrevlett.124.083604.[9] J. Ye, L.-S. Ma, and J. L. Hall, Journal of the OpticalSociety of America B , 6 (1998).[10] B. T. R. Christensen, M. R. Henriksen, S. A. Sch¨affer,P. G. Westergaard, D. Tieri, J. Ye, M. J. Holland,and J. W. Thomsen, Physical Review A (2015),10.1103/physreva.92.053820.[11] P. G. Westergaard, B. T. Christensen, D. Tieri, R. Matin,J. Cooper, M. Holland, J. Ye, and J. W. Thom-sen, Physical Review Letters (2015), 10.1103/phys-revlett.114.093002.[12] F. B. Dunning, T. C. Killian, S. Yoshida, andJ. Burgd¨orfer, Journal of Physics B: Atomic, Molecularand Optical Physics , 112003 (2016).[13] I. S. Madjarov, J. P. Covey, A. L. Shaw, J. Choi, A. Kale,A. Cooper, H. Pichler, V. Schkolnik, J. R. Williams, and S r 5 s S - 5 s 5 p P S r 5 s S - 5 s 5 p P R b D 2 l i n e C ( G , v G) T e m p e r a t u r e ( m K )
FIG. 6. The coefficient for correcting the on-resonance ab-sorption cross section due to the Doppler effect. The plottedtemperature range is 0 . − µ K. Three atomic transitionsare compared: the broad 5s S − P (black dashed line)and narrow 5s S − P (blue dashed line) transitionsin Sr, and the D2 line of Rb (red dotted line). This co-efficient is 1 for broad transitions (Γ (cid:29) πv G ) and stronglymodified for narrow ones (Γ (cid:46) πv G ) in the ultracold range.M. Endres, Nature Physics (2020), 10.1038/s41567-020-0903-z.[14] E. A. Curtis, C. W. Oates, and L. Hollberg, Phys. Rev.A , 031403 (2001).[15] T. H. Loftus, T. Ido, A. D. Ludlow, M. M. Boyd, andJ. Ye, Physical Review Letters (2004), 10.1103/phys-revlett.93.073003.[16] A. Guttridge, S. A. Hopkins, S. L. Kemp, D. Boddy,R. Freytag, M. P. A. Jones, M. R. Tarbutt, E. A. Hinds, and S. L. Cornish, Journal of Physics B: Atomic, Molec-ular and Optical Physics , 145006 (2016).[17] S. Stellmer, B. Pasquiou, R. Grimm, and F. Schreck,Physical Review Letters (2013), 10.1103/phys-revlett.110.263003.[18] M. A. Norcia, A. W. Young, and A. M. Kaufman, Phys-ical Review X , 041054 (2018).[19] A. Cooper, J. P. Covey, I. S. Madjarov, S. G. Porsev,M. S. Safronova, and M. Endres, Physical Review X (2018), 10.1103/physrevx.8.041055.[20] S. Saskin, J. Wilson, B. Grinkemeyer, and J. Thomp-son, Physical Review Letters (2019), 10.1103/phys-revlett.122.143002.[21] S. L. Bromley, B. Zhu, M. Bishof, X. Zhang, T. Both-well, J. Schachenmayer, T. L. Nicholson, R. Kaiser, S. F.Yelin, M. D. Lukin, A. M. Rey, and J. Ye, Nature Com-munications (2016), 10.1038/ncomms11039.[22] G. Ferrari, P. Cancio, R. Drullinger, G. Giusfredi,N. Poli, M. Prevedelli, C. Toninelli, and G. M.Tino, Physical Review Letters (2003), 10.1103/phys-revlett.91.243002.[23] T. Ido, T. H. Loftus, M. M. Boyd, A. D. Ludlow, K. W.Holman, and J. Ye, Physical Review Letters (2005),10.1103/physrevlett.94.153001.[24] M. Schmitt, E. A. L. Henn, J. Billy, H. Kadau, T. Maier,A. Griesmaier, and T. Pfau, Opt. Lett. , 637 (2013).[25] W. Ketterle, D. S. Durfee, and D. Stamper-Kurn, arXivpreprint cond-mat/9904034 (1999).[26] W. Ketterle and M. W. Zwierlein, arXiv preprintarXiv:0801.2500 (2008).[27] C. Oates, G. Wilpers, and L. Hollberg, Physical ReviewA (2005), 10.1103/physreva.71.023404.[28] R. Grimm and J. Mlynek, Physical Review Letters ,2308 (1988).[29] S. Stellmer, R. Grimm, and F. Schreck, Physical ReviewA (2011), 10.1103/physreva.84.043611.[30] I. Nosske, L. Couturier, F. Hu, C. Tan, C. Qiao, J. Blume,Y. H. Jiang, P. Chen, and M. Weidem¨uller, PhysicalReview A (2017), 10.1103/physreva.96.053415.[31] C. Qiao, C. Z. Tan, F. C. Hu, L. Couturier, I. Nosske,P. Chen, Y. H. Jiang, B. Zhu, and M. Weidem¨uller,Applied Physics B (2019), 10.1007/s00340-019-7328-3. [32] H. J. Lewandowski, D. Harber, D. L. Whitaker, andE. A. Cornell, Journal of low temperature physics ,309 (2003).[33] C. Foot, Atomic Physics (Oxford University Press, 2004).[34] M. Horikoshi, A. Ito, T. Ikemachi, Y. Aratake,M. Kuwata-Gonokami, and M. Koashi, Journal of thePhysical Society of Japan , 104301 (2017).[35] S. Stenholm, Applied Physics , 287 (1978).[36] Y. Castin, H. Wallis, and J. Dalibard, Journal of theOptical Society of America B , 2046 (1989).[37] G. Labeyrie, T. Ackemann, B. Klappauf, M. Pesch,G. Lippi, and R. Kaiser, The European Physical Jour-nal D-Atomic, Molecular, Optical and Plasma Physics , 473 (2003).[38] Y. Wang and M. Saffman, Phys. Rev. A , 013801(2004).[39] G. Labeyrie, G. Gattobigio, T. Chaneli`ere, G. Lippi,T. Ackemann, and R. Kaiser, The European PhysicalJournal D , 337 (2007).[40] S. Roof, K. Kemp, M. Havey, I. M. Sokolov, and D. V.Kupriyanov, Opt. Lett. , 1137 (2015).[41] J. Han, T. Vogt, M. Manjappa, R. Guo, M. Kiffner,and W. Li, Physical Review A (2015), 10.1103/phys-reva.92.063824.[42] M. Noaman, M. Langbecker, and P. Windpassinger, Opt.Lett. , 3925 (2018).[43] J. R. Gilbert, C. P. Roberts, and J. L. Roberts, J. Opt.Soc. Am. B , 718 (2018).[44] B. Zhu, J. Cooper, J. Ye, and A. M. Rey, Physical Re-view A (2016), 10.1103/physreva.94.023612.[45] J. Chab´e, M.-T. Rouabah, L. Bellando, T. Bienaim´e,N. Piovella, R. Bachelard, and R. Kaiser, Physical Re-view A (2014), 10.1103/physreva.89.043833.[46] T. Bienaim´e, R. Bachelard, N. Piovella, and R. Kaiser,Fortschritte der Physik , 377 (2013).[47] D. Kupriyanov, I. Sokolov, and M. Havey, Physics Re-ports , 1 (2017).[48] R. J. Bettles, M. D. Lee, S. A. Gardiner, and J. Ru-ostekoski, Communications Physics , 1 (2020).[49] G. G¨unter, M. R. de Saint-Vincent, H. Schempp,C. S. Hofmann, S. Whitlock, and M. Weidem¨uller,Physical Review Letters108