Competition between Factors Determining Bright versus Dark Atomic States within a Laser Mode
CCompetition between Factors Determining Bright versus Dark Atomic States within aLaser Mode
Bryan Hemingway, T. G. Akin, and Steven Peil
Clock Development Division, Precise Time Department,United States Naval Observatory, Washington, DC 20392
J. V. Porto
Joint Quantum Institute, National Institute of Standards and Technologyand the University of Maryland, College Park, Maryland 20742 USA (Dated: May 12, 2020)We observe bimodal fluorescence patterns from atoms in a fast atomic beam when the laserexcitation occurs in the presence of a magnetic field and the atoms sample only a portion of thelaser profile. The behavior is well explained by competition between the local intensity of the laser,which tends to generate a coherent-population-trapping (CPT) dark state in the J = 1 to J (cid:48) = 0system, and the strength of an applied magnetic field that can frustrate the CPT process. Thiswork is relevant for understanding and optimizing the detection process for clocks or other coherentsystems utilizing these transitions and could be applicable to in situ calibration of the laser-atominteraction, such as the strength of the magnetic field or laser intensity at a specific location.Keywords: fluorescence; dark state; CPT PACS numbers:
I. INTRODUCTION
Electric-dipole coupled atomic transitions are widelyused for quantum-state measurement [1] and momentumtransfer [2] in laser interactions because of the high rateof photon scattering when driven with a resonant opti-cal field. Sustained driving of these strong transitionscan be straightforward when the angular momentum ofthe higher-energy state is greater than that of the lowerstate, J (cid:48) > J , supporting a cycling transition betweentwo sublevels. When this condition is not met, an atomcan be optically pumped to a state that no longer couplesto the polarization of the excitation laser, or, if multiplepolarizations are present to couple all of the states in thelower manifold, destructive interference among the vari-ous transition pathways can inhibit excitation via coher-ent population trapping (CPT) [3]. In either case a darkstate arises and photon scattering ceases.Each of these types of dark state has been used in avariety of applications, from eliminating heating due tospontaneous emission in sub-recoil cooling [4] to creatingmicrowave resonances for atomic clocks [5]. But the inhi-bition of desirable scattering is a negative consequence ofdark states for state detection [6] and laser manipulationof external degrees of freedom. In the case of CPT, thedark state can be frustrated by application of a magneticfield, resulting in excited-state population and photonscattering [7]. The field introduces time evolution of themagnetic sublevels, disrupting the steady-state superpo-sition that suppresses laser excitation.The success or failure in suppressing a CPT darkstate with a magnetic field is determined by the relativestrength of the Zeeman frequency shift of the sublevelsand the Rabi frequency, Ω, characterizing the strength ofthe coherent interaction between the lower- and higher- energy states [8]. For a transition used for detection inan atomic clock or other coherent system, the signal-to-noise ratio (SNR) is maximized by collecting the mostphotons possible, necessitating a large enough magneticfield to optimize the excited state population and photonscattering rate. This needs to be balanced with the sizeof the magnetic-field induced frequency shifts impactingthe accuracy, stability or more generally the coherenceof the system. This is a known problem in trapped-ion optical clocks [6, 9]. More recently optical clocksbased on beams of neutral alkaline-earth atoms are uti-lizing (atom-)background-free detection on a transitionwith J (cid:48) < J and therefore vulnerable to CPT [10, 11].In the case of atoms driven in a uniform magnetic fieldby a Gaussian laser mode, the relative strengths of theZeeman shift and Ω can vary over the changing local in-tensity of the spatial mode profile, and both photon scat-tering and dark-state behavior may be present. Coherentprocesses are often obscured, however, in an ensemble ofatoms driven by an optical field due to the varying Rabifrequency across both dimensions of the spatial mode.Mitigating this by simply enlarging the beam comes atthe expense of intensity, so past efforts at observing co-herent processes have used tailored flat-top laser modesto produce a uniform intensity [12, 13]. Efforts to removethe effects of averaging over different intensities couldalso benefit from imaging the result of the interactionto recover some spatial information rather than collect-ing scattered light from atoms sampling different Ωs ona single-pixel photodetector.Here we present observation of the competition be-tween factors leading to strong photon scattering andCPT along the profile of a laser mode. The atomic tran-sition subject to this behavior is only accessible after ini-tially transferring population from the (global) ground a r X i v : . [ phy s i c s . a t o m - ph ] M a y FIG. 1: (Color online.) (a) Calcium energy level diagram,showing the narrow 657 nm transition and strong 423 nmand 431 nm transitions. (b) Illustration of experimental ar-rangement. Atoms interact with linearly polarized 657 nmlaser light in the presence of a magnetic field. The size of the657 nm mode is smaller than the spatial extent of the atomicbeam, and smaller than the mode of the 431 nm laser in thesubsequent interaction region. The 431 nm light is linearlypolarized at 45 degrees with respect to vertical ( y ), providingcoupling among all Zeeman sublevels. A CCD camera belowcaptures fluorescence images in the xz plane. Inset: Exam-ple fluorescence images obtained for laser power of ∼
10 mW(top) and ∼
500 mW (bottom). state to a metastable state with a separate laser excita-tion; the relative size of the spatial modes used for thetwo transitions allows us to sample just a fraction of thelaser mode of interest and remove much of the averagingthat would occur if sampling the entire profile.
II. EXPERIMENT
Our system consists of a thermal (no laser cooling)beam of neutral calcium atoms interacting with lasersof two different wavelengths at two different positions, asillustrated in Fig. 1. Like other alkaline-earth atoms, cal-cium has an inter-combination transition resulting froma change in spin state of the two valence electrons. Thisweak S − s p P transition corresponds to a wave-length of 657 nm and has a natural width of about400 Hz. It is useful as an optical frequency reference andis particularly well suited for spectroscopy with a fastatomic beam. Additionally, there is a strong transitionwith a wavelength of 431 nm that couples the 4 s p P metastable state, populated by atoms excited by 657 nmlight, to a 4 p P state [10, 11].The atomic beam is generated by heating calcium gran-ules in a molecular-beam epitaxy (MBE) cell to 650 ◦ C,creating an effusive source with most probable velocity v ∼
500 m/s. The beam traverses a zone where the S − s p P inter-combination transition is driven with ∼
10 mW of 657 nm laser light. This light is linearly po-larized vertically (along y in Fig. 1(b)), which selectivelyexcites the m J = 0 to m J (cid:48) = 0 transition in the presenceof a ∼ µ T (5 G) vertical magnetic field. The narrow400 Hz resonance is not resolved for fast atoms, where thetime spent in the path of the laser limits the achievablelinewidth to about 500 kHz for a mm-wide beam. The lifetime of the 4 s p P state is on order of 400 µ s,and over the course of a 1 m long beamline only a fractionof the atoms excited will decay back to the S groundstate, yielding a detection signal of fewer than one pho-ton per excited atom. Instead, those atoms excited tothe metastable 4 s p P state can be driven to the short-lived, 5 ns lifetime, 4 p P state using 431 nm light. Thisstrong transition enables many blue photons to be scat-tered by a single atom originally excited by 657 nm light.This transition is driven in the second interaction regionshown in Fig. 1(b). The 431 nm light is a collimatedbeam with an approximately elliptical spatial mode withhorizontal ( z ) and vertical ( y ) sizes (1/e diameter) of2.5 mm and 4 mm. As in the 657 nm excitation region, a ∼ µ T (5 G) vertical magnetic field provides a quanti-zation axis. Fluorescence in this region is maximized fora linear polarization at ∼
45 degrees from vertical. Ver-tical linear polarization is required to drive atoms fromthe 4 s p P m J = 0 state to the 4 s P level. Circu-larly polarized light can then drive σ + and σ − transitionsconnecting 4 s p P m J = − m J = 1 to 4 s P .Laser-induced fluorescence from the 431 nm transitioncan be collected on a detector, such as a photomultipliertube (PMT) or photodiode. In order to acquire spatialinformation, the fluorescence can also be imaged, as seenin the inset to Fig. 1(b), where we show two differentframes taken with a conventional CCD color video cam-era. As illustrated, we will refer to a coordinate systemin which the k -vectors of the laser beams point along x ,the atomic velocity is along z , and we image the fluores-cence from the atoms along y . The fluorescence imagesthen capture the spatial pattern in the xz plane, with thelength of the pattern along z determined by the spatialmode (and intensity) of the 431 nm laser and the thick-ness along x determined by the velocities in the atomicbeam with transverse Doppler shifts of ±
250 kHz or less,which are resonant with the 657 nm light [14].Illumination of an atomic sample with a Gaussianbeam typically results in a similar pattern of fluorescence,as long as the intensity is below saturation. This is thecase for the top image in the inset of Fig. 1(b). Weobserve that when the power in the 431 nm beam is in-creased beyond a certain value, the imaged fluorescencepattern starts to become bimodal, with two spatial lobeslying along the atomic beam trajectory ( z ). The separa-tion of the scattering peaks increases with laser power,with a notable dip in fluorescence in the center (as seenin the image on the bottom of the inset in 1(c)). A se-quence of true-color images obtained at 6 different laserpowers is shown in Fig. 2. III. ANALYSISA. Model for Excited-State Population
For fixed values of experimental parameters and forthe specific angular momenta J = 1 and J (cid:48) = 0 for our FIG. 2: (Color online.) (a) True-color fluorescence imagesacquired at different 431 nm laser powers. The length of theimage ( z direction) is determined by the dimension of the431 nm laser profile along the atomic beam direction (and bythe laser power). Because these pictures are captured frombelow ( y direction), the height of the image ( x direction) isrelated to the frequency resolution of the 657 nm excitation,which is limited to ∼
500 kHz. (b) Waterfall plot of 1D cross-sections of the fluorescence images. One curve for a power of440 mW is included here but not shown as an image in (a).Some saturation exists in the blue channel of the RGB imagesacquired at the higher powers. system, the steady-state J (cid:48) = 0 population, ρ ee , can becalculated exactly, as shown in Ref. [7]. Although the lo-cal value of the laser intensity varies over the spatial ex-tent of the mode, the steady-state population is reachedfast enough that a fixed intensity can be used to calcu-late the excited-state population as a function of position;the 4 s P state lifetime of 5 ns and atomic velocity of500 m/s correspond to an atom traveling ∼ µ m, a smallfraction of the 2.5 mm laser mode, before the populationreaches steady-state.The solution for ρ ee in Ref. [7] is obtained by solv-ing the equation of motion for the atomic density oper-ator with time evolution of the Zeeman sublevels | J =1 , m j = ± (cid:105) due to the energy shift from the appliedmagnetic field. The formalism in that work was devel-oped to quantify the impact of an applied magnetic field(or laser-polarization modulation) on the dark-state thatwould otherwise result. The magnetic field causes thelower-level sublevels to evolve in time at different rates,disrupting the steady-state solution in which no excited-state amplitude is generated. Inserting our experimentalparameters into their Eq. (4.1) gives the following steady-state population in J (cid:48) = 0 as a function of Rabi frequencyΩ, magnetic-field frequency shift δ B = µ B B/ ¯ h ( µ B is theBohr magneton, and B the magnitude of the magneticfield), and natural width of the excited state γ/ π : ρ ee = 34 (cid:18) Ω γ (cid:19) − (cid:16) Ω γ (cid:17) (cid:18) − (cid:16) Ω4 δ B (cid:17) − (cid:0) δ B Ω (cid:1) (cid:19) . (1)This expression is for zero detuning and a linear polar-ization at an angle of π/ FIG. 3: (Color online.) (a) Calculated excited-state popula-tion (solid grey curve, left axis) for Gaussian intensity dis-tribution (dashed black curve, right axis) and magnetic fieldvalue of 0 . γ . For low local intensities, the Rabi frequencyassociated with the optical interaction is not fast enough forthe population to adiabatically follow the time evolution in-troduced by the magnetic field. The vertical (red) dashedline shows the point at which the Rabi frequency and Zee-man detuning are on par. As the intensity increases further,population accumulates in the dark state at the expense of theexcited state. (b) Series of curves showing calculated excited-state population as a function of position ( z ) at the samelaser powers as used for the data in Fig. 2. These are 1Dcross-sections of 2D solutions to Eq. (1). The horizontal axisis scaled to ± magnetic field of δ B = 0 . γ yields ρ ee ≈ (cid:18) Ω γ (cid:19) . − (cid:16) Ω γ (cid:17) + 0 . (cid:16) Ω γ (cid:17) . (2)The competition between laser and magnetic fields indetermining the excited-state population is illustrated inFig 3. In (a) we calculate the solution to Eq. (2) for aGaussian laser mode with waists of 1.3 mm along z and2 mm along y . The total power chosen for this illustrationis 40 mW. The behavior of ρ ee corresponds to transitionsbetween predominantly bright and predominantly darkatomic states. The maximum excited-state population isabout 0.25, half of the steady-state value for a two-statesystem driven to saturation. This maximum occurs whenthe laser intensity is given by Ω = 1 . γ ; at higher intensi-ties the time-evolution due to the optical field dominatesthe dephasing due to the magnetic field and a dark statebegins to arise. The value of the maximum excited-statepopulation and the optical intensity at which the max-imum population occurs depend on both the magneticfield strength and the total laser power. In Fig 3(b) a‘waterfall’ plot is shown for ρ ee for the same laser powersas used for the data in Fig. 2 and with the same horizon-tal scale as in that figure. It can be seen that the peakfluorescence can occur outside the beam waist of the lasermode for large powers. B. Additional Details
Calculating the excited-state population along a 1Dslice of a Gaussian laser mode is a simplification com-pared to the actual experimental arrangement. We de-velop the model further and apply it in Fig. 4 to themaximum power used in our system of 500 mW. On theleft in (a)(i) we show a modeled 2D laser intensity pro-file with waists along z of 1.3 mm and y of 2 mm, thevalues measured for our laser mode using a profile meter.The (left) image in (a)(ii) is a plot of the solution for theexcited-state population ρ ee for all points in this inten-sity distribution, and in (a)(iii) the fluorescence patternfrom atoms traversing only a fraction of the laser modeis shown. This is obtained by multiplying the image in(a)(ii) with a Gaussian distribution (waist of 1.5 mm)along y representing the extent of the atoms in the beamexcited by the 657 nm laser. The size chosen for thiswaist is estimated from the size of the 657 nm modeat the location of the atoms plus the divergence of theatomic beam between the two interaction regions. Thecamera in our experiment observes fluorescence from be-low, collecting light from all atoms emitting. This aspectof the data acquisition is captured in (b), where the flu-orescence integrated along y is shown in the xz plane.The extent of the image along x is modeled by using an-other Gaussian distribution to represent the extent of theatoms in the beam, this time along x ; the spatial extentof the atomic beam in that direction is due to the spreadof transverse velocities corresponding to Doppler shiftswithin ±
250 kHz of the 657 nm resonance.A 1D cross-section of the left image in Fig. 4(b) is plot-ted as the grey-dashed curve in (c). The blue curve inthe plot is the 1D cross section of a 500 mW fluorescenceimage, shown in the xz plot in (d). The location andwidth of the peaks of the calculated fluorescence distri-bution do not agree with observation as well as could beexpected.Better agreement is achieved between the data (bluecurve) in the graph in (c) and the black curve. The blackcurve corresponds to the 1D cross-section of the excited-state population distribution on the right in (b), whichis calculated using the actual laser intensity distributionmeasured with a profile meter, shown on the right in(a)(i). It can be seen that accounting for the deviationof the physical laser mode from a pure Gaussian resultsin far better agreement with observation. The laser lightused in the experiment is the output of a frequency dou-bled Ti:sapphire laser. In order to investigate fluores-cence patterns at the highest powers possible, we avoidedspatial filtering or fiber coupling of the laser light, leavingus with a non-ideal spatial mode.The black curve in Fig. 4(c) is calculated from Eq. (2)using measured experimental inputs and no free param-eters, other than an overall scaling of signal size. Theremaining discrepancy between the model and data, par-ticularly regarding the width of the fluorescence peaks,is likely due to two sources of uncertainty. The first is FIG. 4: (Color online.) Calculations of excited-state popu-lation (proportional to scattering rate) including additionalexperimental details. (a) Series of images in zy plane show-ing laser intensity distribution (i), corresponding excited-statepopulation from Eq. (1) with 500 mW of laser power (ii), frac-tion of distribution that materializes due to limited extent ofbeam of atoms excited by 657 nm laser (iii), using calculatedGaussian distribution (left) and physical laser mode (right).(b) Modeled fluorescence pattern observed with camera in zx plane after integrating the image in (a)(iii) along y and multi-plying by Gaussian distribution of atoms along x . (c) Plot of1D cross-section of observed fluorescence pattern (blue curve)and 1D cross-section of calculated distributions in (b) (dottedgrey curve - Gaussian laser mode; solid black curve - phys-ical laser mode). The horizontal axis extends over 14 mmto account for observed features. The calculations based onthe physical laser mode extend only about 6 mm, the sizeof the detector on the profile meter. (d) Fluorescence imageacquired with 500 mW of laser power. the value of the magnification of the imaging system; cal-ibration of the magnification could not be carried out atthe position of the atoms inside the vacuum chamber,but rather only by using a scale at the nearest view-port. An error in this scaling can change the width ofthe fluorescence peaks compared to the model. The sec-ond source of uncertainty is the spatial variation of thebias magnetic field. The coils used to create the fieldin the 431 nm interaction region are far from the idealHelmholtz arrangement, resulting in a non-uniform field.We have demonstrated that using a non-uniform mag-netic field in the calculation can broaden the modeledfluorescence peaks. C. 423 nm Transition
The 431 nm transition provides a detection signal thatis free from the background of ground-state atoms, i.e. those that do not get excited to the metastable 4 s p P state with the 657 nm laser. Alkaline-earth frequencystandards have more commonly used the S − P tran-sition for detection [15–17]. This cycling transition con-nects the ground state to a short-lived excited state, en-abling photons to be scattered at a high rate. Excitationof the S − s p P is measured as a reduction of the423 nm fluorescence from the ground state atoms. Be-cause all ground state atoms (in the absence of 657 nmlight) contribute to this cycling process, this transition isalso useful for measuring atomic flux and general atomic-beam characterization.In calcium the S − P transition has similar prop-erties to the 4 s p P − s P transition at 431 nm.The transition occurs at a wavelength of 423 nm, has anatural width of γ/ π = 34 MHz and a saturation in-tensity of 30 mW/cm . We can look at the fluorescencepatterns from this similar transition in the same detec-tion region by replacing the 431 nm source with one at423 nm. The mode size and power for the lasers at thetwo wavelengths are similar. Unlike the 431 nm transi-tion, the fluorescence pattern from the 423 nm interac-tion shows a single fluorescence peak for all powers; noreduction in the center of the pattern is observed. Anexample fluorescence image from this transition is shownin Fig. 5.There are two significant differences between excita-tion of the 423 nm and the 431 nm transitions that playa role in the z -dependence of the fluorescence patterns.The first is that the J = 1 to J (cid:48) = 0 transition at 431 nmenables formation of a dark state via CPT, whereas the J = 0 to J = 1 (cid:48) at 423 nm transition does not. The otherdifference is that 431 nm excitation is part of a doubleresonance; only atoms first excited by 657 nm light canbe subsequently driven by the 431 nm field. The aspectof double resonance that is important is the spatial selec-tion of the subsequent 431 nm interaction imposed by thegeometry of the first excitation. Because the mode of the657 nm laser is smaller (2 mm round) than that of the blue laser, the atoms excited by the 657 nm beam that areprepared to interact with the 431 nm light sample only afraction of the spatial mode (see Fig. 1(b)). This removessome of the averaging over Rabi frequencies that wouldoccur if fluorescence from the entire spatial mode werecollected. (The fact the 423 nm fluorescence is collectedfrom atoms sampling all of the spatial mode of the laseris likely the reason that the 1D cross-section in Fig. 5(b)does not exhibit the expected saturation for that highintensity; the wings of the spatial mode contribute a flu-orescence profile that does not exhibit saturation.)For completeness, we point out that the difference inextent of fluorescence for 431 nm and 423 nm along x stems from the different transverse velocity classes withDoppler shifts that are resonant with the excitation laser.The broad 423 nm transition is resonant with the exci-tation laser for all transverse velocities in the divergingatomic beam. Since this laser couples to ground-stateatoms, fluorescence is observed over the entire spatialextent of the atomic beam. The 657 nm transition isresonant with the excitation laser for a smaller range oftransverse velocities, resulting in a small spatial extendalong x for the subsequent 431 nm fluorescence. This isillustrated in Fig. 5(c). IV. CONCLUSIONS
With enough experimental detail included, Eq. (1)reproduces the observed bi-modal fluorescence patternswith reasonable agreement. This model then should en-able optimization of the detection process in systems suchas ours. The amount of laser power available will deter-mine the size of the magnetic field required to optimizethe photon scattering rate, and the impact on coherenceof the size and stability of the required magnetic field canbe empirically investigated. Additionally, elongation ofthe probe laser beam (along z ) and corresponding designof the collection optics can be used to reduce peak inten-sity and avoid a dip in fluorescence while increasing thesize of the region over which photons are scattered. Thebest solution will be specific to the experimental designand constraints of a given system. Additionally, furtherrefinement in applying the model could enable fluores-cence images to be used to calibrate the strength of theapplied magnetic field or laser intensity distribution insitu . This could be useful, for instance, for measuringfairly large magnetic fields without the need to tune thelaser frequency over tens of MHz in order to map out theZeeman resonances.In summary, we have observed competition betweenthe laser intensity driving a CPT dark state and themagnetic field disrupting the CPT process along the in-tensity distribution of a laser mode. This competitioncan be present in state detection for atomic beam clocks,where it can impact the clock frequency if it leads to vari-ations in the fluorescence signal—driven by variations inone of the competing fields—over timescales shorter than FIG. 5: (Color online.) (a) Fluorescence image from the S − P transition, for 700 mW of 423 nm laser power,shown on the left. For comparison, a fluorescence image fromthe 4 s p P − s P transition, for 500 mW of 431 nm laserpower, is shown on the right. The scales of the two imagesare the same. (b) Plot of 1D cross-section of 423 nm fluores-cence pattern in (a) (solid blue curve), showing none of thesuppression of scattering at the center as in the 431 nm tran-sition; for the J = 0 to J (cid:48) = 1, 423 nm transition, no darkstate is expected to form. For comparison, the cross-sectionfor the 431 nm J = 1 to J (cid:48) = 0 transition is also shown(dashed grey line). (c) The different extents of the fluores-cence images along x can be understood from the range oftransverse-velocity Doppler shifts that are resonant with the500-kHz (resolution-time limited) 657 nm interaction versusthose that are resonant with the 34-MHz 423 nm interaction,as illustrated. Ground-state atoms with any transverse ve-locity in the divergent atomic beam are resonant with the423 nm laser, producing fluorescence for the entire width ofthe atomic beam (all x ) (top). Only atoms originally ex-cited by 657 nm light, with Doppler shifts from transversevelocities of ±
250 kHz from resonance, subsequently scatter431 nm; this smaller group of angles translates to a smallerspatial extent in x of fluorescence (bottom). a measurement cycle [18]. These systematic frequencyshifts can be predicted and mitigated by using analysissimilar to that presented here. Future investigation ofthe similar transition in strontium [17] or other atomsused in optical beam clocks would be of interest. V. ACKNOWLEDGEMENTS
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