Forbidden atomic transitions driven by an intensity-modulated laser trap
FForbidden atomic transitions driven by an intensity-modulated laser trap
Kaitlin R. Moore, Sarah E. Anderson & Georg Raithel
Department of Physics and Program in Applied Physics, University of Michigan, Ann Arbor, Michigan 48109, USA
Abstract
Spectroscopy is an essential tool in understanding and manipulating quantum systems,such as atoms and molecules. The model describing spectroscopy includes a multipole-fieldinteraction, which leads to established spectroscopic selection rules, and an interaction that isquadratic in the field, which is often neglected. However, spectroscopy using the quadratic(ponderomotive) interaction promises two significant advantages over spectroscopy using themultipole-field interaction: flexible transition rules and vastly improved spatial addressabil-ity of the quantum system. For the first time, we demonstrate ponderomotive spectroscopyby using optical-lattice-trapped Rydberg atoms, pulsating the lattice light at a microwave fre-quency, and driving a microwave atomic transition that would otherwise be forbidden by es-tablished spectroscopic selection rules. This new ability to measure frequencies of previouslyinaccessible transitions makes possible improved determinations of atomic characteristics andconstants underlying physics. In the spatial domain, the resolution of ponderomotive spec-troscopy is orders of magnitude better than the transition frequency (and the correspondingdiffraction limit) would suggest, promising single-site addressability in a dense particle arrayfor quantum control and computing applications. Future advances in technology may allowponderomotive spectroscopy to be extended to ground-state atoms and trapped molecules.
Introduction.
Spectroscopy is a well-established, powerful tool in science for characterizing mi-croscopic systems. Fields that use this tool range from precision metrology (development offrequency standards and high-precision sensing, such as gravitometry ) to trace analysis andchemical sensing . Characterizing and manipulating particles using light is also the foundation ofnew fields such as quantum optics , computing , and information processing , . The interactionbetween a particle and a light field is described by the particle-field interaction Hamiltonian ,which includes a mulipole-field interaction term and a ponderomotive (quadratic) interaction term.In the case of direct application of a radiation field to atoms, the dipole-field term typically dom-inates the total atom-field interaction and leads to the electric-dipole selection rules for atomictransitions. In contrast, the ponderomotive term dominates the atom-field interaction when thereis substantial spatial variation of the field intensity within the volume of the atom and when theintensity is modulated in time at the transition frequency of interest. In this case, any applicableselection rules are much less restrictive. 1 a r X i v : . [ phy s i c s . a t o m - ph ] S e p o experimentally demonstrate an atomic transition via the ponderomotive interaction for thefirst time, we have chosen to use Rydberg atoms trapped in an intensity-modulated standing-waveoptical lattice. Rydberg-atom optical lattices are an ideal tool for this demonstration because theatoms’ electronic probability distributions can extend over several wells of the optical lattice , andRydberg-Rydberg transitions are in the microwave regime , a regime in which light-modulationtechnology exists. Using this system, we have demonstrated for the first time a specific case ofponderomotive spectroscopy. The approach used here may be extended to other particles, includingground-state atoms and trapped molecules, with appropriate advances in technology.Advantages of ponderomotive spectroscopy over typical dipole-field spectroscopy includeflexible transition rules and vastly improved spatial addressability. First, ponderomotive spec-troscopy affords single-step access to energy transitions that are forbidden by established dipoleselection rules. When a particle size and a standing-wave period are comparable, we may inprinciple use the standing wave to drive electric-dipole-forbidden transitions in the particle in asingle-step process rather than a multi-step (multi-photon) process . We may achieve this bymodulating the standing-wave intensity at the resonance frequency of the desired transition. In ourexperimental system, we have demonstrated this phenomenon by modulating the intensity of theoptical lattice at the resonant microwave frequency of a Rydberg-Rydberg transition and driving anatomic transition that is typically forbidden. The advantage of the ability to drive forbidden trans-itions in a single step rather than through a multi-photon process is that we may avoid high fieldstrengths that induce large unwanted AC Stark shifts of the observed transition frequencies. Asa consequence, ponderomotive spectroscopy enables the study of otherwise-forbidden electronictransitions, which is very convenient for precision spectroscopy.Furthermore, another powerful innovation afforded by ponderomotive spectroscopy is a spa-tial resolution orders of magnitude better than the frequency of the transition would suggest. Elec-tronic transitions in a particle are typically driven by applying radiation resonant with the trans-ition frequency. The best possible spatial resolution will be at the diffraction limit of the appliedradiation, which is on the order of the wavelength corresponding to the transition frequency and,in most cases, orders of magnitude larger than the particle size. In contrast, in ponderomotive2pectroscopy, the frequency of the applied radiation is very different from the frequency of thetransition being driven. The applied radiation is a standing wave with a wavelength on the orderof the particle size. The frequency resonant with the desired transition is introduced by intensity-modulating the standing wave. In the present report, we demonstrate ponderomotive spectro-scopy by driving microwave-frequency atomic transitions (typical resolution: centimetre-scale) byintensity-modulating a standing-wave optical lattice (typical resolution: micrometre-scale). As aresult, ponderomotive spectroscopy may enable advances in quantum computing
7, 8 , where single-site addressability plays a central role. We report a successful demonstration of this new spectro-scopy. We trap Rb atoms in a one-dimensional standing-wave optical lattice, formed by counter-propagating 1064-nm laser beams. We drive the S / → S / transition by sinusoidally mod-ulating the lattice intensity at the resonant transition frequency, found to be 38.76861(1) GHz. The S / → S / transition has been chosen because it is forbidden under electric-dipole selectionrules and because we expect a particularly high population transfer rate in our intensity-modulatedlattice . Pulsating the optical lattice at a microwave frequency.
To motivate the use of an intensity-modulated optical lattice for a demonstration of ponderomotive spectroscopy, we briefly examinethe physics underlying the interaction between the Rydberg atom and the lattice light. The inter-action between the atom and a light field is described by the following interaction Hamiltonian : V int = 12 m e (cid:0) | e | A · p + e A · A (cid:1) , (1)where p is the Rydberg electron’s momentum operator and A the vector potential of the light(the laser electric field E = − ∂∂t A ) . The A · p term describes most types of atom-field interac-tions and is commonly engaged in spectroscopy . Transitions driven by the A · p term followwell-established spectroscopic selection rules . In the case of direct application of microwaveradiation to Rydberg atoms, the electric-dipole selection rules apply (in first-order perturbationtheory). In contrast, the quadratic A · A (ponderomotive) term allows us to drive transitions farbeyond the electric dipole selection rules in first order by providing a substantial spatial variationof the field intensity within the volume of the atom and by modulating the intensity in time at the3 igure 1: Experimental set-up. A Mach-Zehnder interferometer (top) combines two 1064-nm c.w. beams.One beam is intensity-modulated at frequency Ω by a fibre-based electro-optic modulator (‘Fibre modu-lator’), the operating point of which is set using photo-detector ‘PD1’. Using photo-detector ‘PD2’ and apiezo-electric transducer (‘Piezo’), the interferometer is locked such that the beams add up in phase. Ry-dberg atoms are laser-excited and optically trapped by lattice inversion. The optical lattice (bottom) isformed by retro-reflecting and focusing the 1064-nm laser beam into the magneto-optical trap (‘MOT’).The lattice potential is intensity-modulated with period T . Beam blocks (‘B1’,‘B2’) are used for signalinterpretation. Details in Methods. ,we proposed that, utilizing the A · A term, one can drive a wide variety of transitions beyond theusual spectroscopic selection rules.In Figure 1 we show a schematic of the experimental set-up. A continuous-wave (c.w.) 1064-nm laser beam is split in a Mach-Zehnder interferometer (top) into a low-power and high-powerbeam. The low-power beam is sinusoidally modulated via an electro-optic fibre modulator drivenby a tunable microwave-frequency voltage signal. This intensity-modulated low-power beam iscoherently re-combined with the unmodulated high-power beam at the exit of the interferometer.In the atom-field interaction region (bottom), the intensity ratio between the modulated and un-modulated portions of the incident lattice beam is about 1:100. We form the standing-wave opticallattice by retro-reflecting the lattice beam. Cold Rb Rydberg atoms are trapped with centre-of-mass positions near intensity minima of the lattice (Methods). At these locations, the intensitymodulation of the lattice results, via the A · A term, in a time-periodic atom-field interaction witha leading quadratic dependence on position (needed to drive an S → S transition). The propercombination of temporal and spatial intensity modulation is essential for utilizing the A · A termto realize the type of spectroscopy introduced in this report.Due to the coherent mixing of the low-power, modulated lattice beam with a high-power, un-modulated beam, the time-averaged lattice depth is large enough that most atoms remain trapped inthe lattice while the S / → S / transition is probed. The coherent mixing of the two beamsis also beneficial because it enhances the modulation in the atom-field interaction region, resultingin a much larger S / → S / coupling than would be possible with the weak modulated beamalone. The Rabi frequency for a transition | n, l, m (cid:105) → | n (cid:48) , l (cid:48) , m (cid:48) (cid:105) between two Rydberg states thatare resonantly coupled by the intensity-modulated lattice is χ ≈ √ ε e ¯ hm e c(cid:15) ω I J (cid:18) πV IM V π (cid:19) (cid:34) (cid:115) I I (cid:35) D n (cid:48) ,l (cid:48) ,m (cid:48) n,l,m , (2)where V IM is the amplitude of the microwave voltage signal that drives the fibre modulator, V π isthe voltage difference between minimum and maximum intensity transmission through the modu-5ator (a fixed modulator property), ε is the intensity ratio at the atom location between the returnand incident beams forming the lattice, e is electron charge, m e is electron mass, ω is the angularfrequency of the optical-lattice light, and I and I are the incident intensities of the mod-ulated and unmodulated lattice beams at the atom location, respectively. The transition matrix ele-ment D n (cid:48) l (cid:48) m (cid:48) nlm (unitless) has been derived in previous work and is 0.215 for the S / → S / transition for an atom located at a minimum in our lattice. This value is large compared with thoseof other possible transitions due to a favorable ratio between atom size and lattice period (whichis on the order of one). A detailed derivation of equation (3) is given in Supplementary Inform-ation. The in-phase addition of the fields corresponding to intensities I and I , usingthe Mach-Zehnder beam combination set-up, leads to the enhancement term in square brackets inequation (3). In our experiment, the Rabi frequency is enhanced by a factor of ≈ , which aidssignificantly in observing the transition. While in our experiment the enhancement afforded by theinterferometric set-up is critical for a successful demonstration, different laser wavelengths, mod-ulation schemes, sub-Doppler and evaporative cooling techniques or the use of separate trappingand modulation beams could make the interferometric set-up unnecessary. ResultsSpectroscopy results.
In Figure 2a we show the atomic transition S / → S / driven viaintensity-modulation of the optical lattice. The S / → S / spectrum is obtained by scanningthe frequency of the microwave source driving the fibre modulator across the expected transitionfrequency. The spectral line shown in Figure 2a is proof that the S / → S / transitionhas been driven in a single step (in first order), at the fundamental transition frequency. The in-set in Figure 2a shows a simulated spectrum (details given in Supplementary Information). Thesimulation indicates that the sub-structure in the spectral line originates from different types ofcentre-of-mass trajectories of the atoms in the optical lattice. The dominant central peak resultsmainly from trapped atoms, whereas the small side structures result from atoms that traverse sev-eral lattice wells or that remain nearly stationary close to a lattice maximum during atom-fieldinteraction . The sub-structures, which are observed in the simulation, are not as clearly resolved6 .00.10.2-1000 -500 0 500 1000 b. Transition frequency (kHz)Detuned from 38.768545 GHz F r a c t i ona l popu l a t i on i n S / a. -500 0 500 Figure 2: The / → / transition driven via intensity modulation of an optical lattice. Thefraction of atoms in S / is shown. Spectral-line centres and background offset are determined by Gaus-sian fits (red curves). Gray areas represent s.e.m. of fit results for line centres. a , Spectral line obtainedby scanning the modulation frequency of the intensity-modulated optical lattice. The data are averagesof 28 scans with 200 measurements each, taken under similar experimental conditions. Vertical error bar,s.e.m. of the 28 scans. Line centre is at 38.76861(1) GHz (black solid line). Inset, simulation results. b ,Reference scan performed by conventional two-photon microwave spectroscopy without the optical lattice,plotted versus twice the microwave frequency. The data are averages of 11 scans with 200 measurementseach. Line centre is at 38.768545(5) GHz (black dashed line).
7n the experiment. This is most likely due to inhomogeneous broadening. The simulation motiv-ates us to fit the spectrum in Figure 2a as a triple-Gaussian, which yields a central-peak location of38.76861(1) GHz.For reference, we have also driven the S / → S / transition using direct applicationof microwave radiation at half the transition frequency; in this case the transition results froma two-step (second-order) electric-dipole coupling through the P / off-resonant intermediatestate. In Figure 2b we show the lattice-free S / → S / spectral line, driven as a two-photonelectric-dipole transition at half the transition frequency, using microwaves from a horn directed atthe atom-field interaction region. This reference measurement yields a S / → S / transitionfrequency of 38.768545(5) GHz. The transition frequencies measured in Figures 2a and 2b are ingood agreement with each other. The slight blue-detuning of the central peak in Figure 2a relativeto the peak in Figure 2b is due to a light shift from the optical lattice. Calculations based onpublished quantum defect values predict a transition frequency at 38.7686(1) GHz, which is inagreement with the experimental measurements. We note that for the two-photon transition shownin Figure 2b the microwave-induced AC Stark shift of the transition frequency is unusually small,due to near-cancellation of the upper- and lower-level AC shifts. We have estimated this shift ofthe transition frequency to be less than 500 Hz, which is insignificant in the above comparisons. Testing the nature of the atom-field interaction mechanism.
In Figure 3 we show two tests thatprove the spectral line shown in Figure 2a is indeed caused by a perturbation due to the A · A atom-field interaction term in equation (4), which drives transitions via an intensity-modulated opticallattice. First, in Figure 3a we verify that the transition does not originate from a combination ofmicrowave leakage and stray DC fields. Although we carefully zero DC electric fields, a stray DCelectric field in the atom-field interaction volume could, in principle, weakly perturb the S / and S / levels by adding some P -admixture to these levels. Any microwave-radiation leak-age into the atom-field interaction region under the simultaneous presence of a DC electric fieldwould drive the transition as an electric-dipole transition between the weakly-perturbed S / and S / levels. In order to verify that our spectral line is not due to such a coincidence, we look8 b. F r a c t i ona l popu l a t i on i n S / a. Transition frequency (kHz)Detuned from 38.768545 GHz
Figure 3: Verification of the nature of the atom-field interaction mechanism.
The fraction of atoms in S / is recorded as the modulation frequency is scanned. The absence of signals is important because itverifies the model presented in the text. Background offset is determined by a constant fit (red line). a , Testto exclude dipole transitions caused by microwave leakage into the atom-field interaction region, performedusing beam block ‘B1’ (Figure 1) during frequency scans, while leaving the microwave source at full power.The data are averages of 16 scans with 200 measurements each. Error bar, s.e.m. of the 16 scans. b , Test toexclude a stimulated electric-dipole Raman transition, performed using beam block ‘B2’ (Figure 1) duringfrequency scans. The data are averages of 10 scans with 200 measurements each. Error bar, s.e.m. of the 10scans. A · p and A · A transition mechanismsin equation (4). As detailed in the next paragraph, either mechanism may, in principle, drivethe observed S / → S / transition via lattice modulation. However, these fundamentallydistinct mechanisms differ in ways that can be tested experimentally.The modulated light field contains the frequency ω and frequency sidebands ω ± Ω , where ω is the light frequency, and Ω is the microwave modulation frequency of the light intensity(which is resonant with the S / → S / transition). The presence of multiple frequenciesallows, in principle, for the A · p term to couple S / to S / in a two-step (second-order)process via a stimulated Raman transition through one or more states that have an energy separ-ation ≈ ¯ hω from the S / and S / levels. Following dipole selection rules, this mechan-ism would involve optical S → P and P → S electric-dipole transitions through distant inter-mediate P -states. The Raman coupling mechanism would be effective in both running-wave orstanding-wave laser fields. On the other hand, the coupling due to the A · A term is proportional to (cid:10) S / (cid:12)(cid:12) I ( z ) (cid:12)(cid:12) S / (cid:11) sin (Ω t ) , where I ( z ) is the light intensity (Figure 1). For this coupling tobe effective in first order, two conditions must be simultaneously fulfilled: the light intensity mustsubstantially vary as a function of position z within the volume of the atom, and the modulationfrequency Ω must correspond to the energy-level difference. Without spatial variation of I ( z ) , thecoupling vanishes due to the orthogonality of the atomic states. Hence, the A · A coupling can beturned on or off by providing an intensity-modulated standing wave or a running wave, respectively(where the running-wave beam may or may not be intensity-modulated in time).10 .00.2 a. b. F r a c t i ona l popu l a t i on i n S / -1000 -500 0 500 10000.00.2 c. µ s µ s3 µ sTransition frequency (kHz)Detuned from 38.768545 GHz Figure 4: Dependence of intensity-modulation-driven / → / spectral line on interactiontime t int with the modulated lattice. a-c , For each t int , the fraction of atoms in S / is recorded whilescanning the microwave modulation frequency. The data are averages of 8 scans with 200 measurementseach. Error bar, s.e.m. of the 8 scans. The data for t int = µ s are fit with double- (triple-)peak Gaussians (red solid curves). The FWHM (gray areas) of the dominant Gaussians (red dashed curves)decrease with increasing t int , and the heights increase. To test this, in Figure 3b we exchange the intensity-modulated standing-wave optical latticefor an intensity-modulated running-wave beam by blocking the retro-reflected lattice beam withbeam block ‘B2’ in Figure 1. We then scan the microwave frequency in a manner identical withthe procedure used for Figure 2a. No spectral line is evident in Figure 3b. Therefore, the transitionmechanism responsible for the spectral line observed in Figure 2a is indeed a single-step atom-field interaction arising from the modulated optical standing-wave intensity (via a first-order A · A interaction), not a two-step electric-dipole Raman coupling process arising from the frequencysidebands in the light field (via a second-order A · p interaction).11 ependence on experimental parameters. Here we characterize the dependence of ponderomot-ive spectroscopy on several experimental parameters. In Figure 4 we show the scaling behaviourof the spectral line width and amplitude on the atom-field interaction time. Experimentally, the in-teraction time t int is defined as the time between when the atoms are excited to the S / state andwhen the atoms are ionized for detection (Methods), typically t int = 6 µ s. During the atom-fieldinteraction time, the transitions are driven by the intensity-modulated lattice (which is always on),and the atoms undergo a square-pulse coupling to state S / of duration t int . In the limit of weaksaturation and assuming a Fourier-limited spectral profile of the driving field, the full-width-at-halfmaximum (FWHM) of the spectral line is expected to decrease with increased interaction time as ≈ . /t int . This agrees with the trend observed in Figure 4. A double-Gaussian fit of the 3 µ sspectral line and triple-Gaussian fits of the 6-and-9- µ s spectral lines (red solid curves in Figure 4)indicate that the FWHM of the dominant Gaussian components in each line (red dashed curves)are within 20% of the Fourier limit.Examining the spectral lines in Figure 4 further, we also find that the maximum signal heightapproximately doubles between 3 and 6 µ s; the additional increase between 6 and 9 µ s is relativelyminor. This observation is consistent with a Rabi frequency within the range of 50-100 kHz. Thisresult is in qualitative agreement with our calculated Rabi frequency presented in SupplementaryInformation.In Figure 5 we summarize the dependence of the spectral-line height on additional experi-mental parameters. In Figure 5a we investigate the dependence of spectral-line height on modu-lation strength, which is controlled by varying the amplitude of the microwave voltage signal V IM that drives the fibre modulator. As can be seen in equation (3), the Rabi frequency χ has a first-order Bessel function ( J ) dependence on V IM . Because the height of the spectral line indicates thepeak fraction of population in S / , we expect the height to scale as ( χt int ) ∝ J ( πV IM /V π ) ,for fixed t int and in the limit of weak saturation. In Figure 5a, we plot the spectral line height asa function of the Bessel function argument, πV IM /V π , as we vary V IM . A J ( πV IM /V π ) fit to thedata yields good agreement.The spectral-line height also depends on the distance of the retro-reflector from the atoms.12 .0 0.4 0.8 1.2 1.6 2.00.000.050.100.15 b.a. Retro-reflector displacement (mm) P ea k f r a c t i ona l popu l a t i on i n S / π V IM / V π Figure 5: Dependence of intensity-modulation-driven / → / spectral-line height on ex-perimental parameters. The peak fraction of S / atoms is recorded while varying experimental para-meters. Each data point represents the observed peak height for an average of 5 ( a ) or 3 ( b ) scans with 200measurements each. Vertical error bars, uncertainty of peak height. a , Dependence on πV IM /V π , where wevary the microwave voltage amplitude V IM applied to the fibre modulator ( V π is fixed). Horizontal errorbars are due to the uncertainty in V π . Red curve, fit proportional to J ( πV IM /V π ) . b , Dependence on latticeretro-reflector distance from the atoms. Horizontal error bars, distance uncertainty. A sinusoidal curve (red)with a period of 4 mm is plotted for reference. Ω . The spectral-line height is maximal ifthe pulse trains of the incident and retro-reflected lattice beams arrive synchronously at the atoms’location. Considering the time delay between the retro-reflected and incident pulses, it is seen thatthe spectral-line height should sinusoidally vary with the position of the retro-reflector mirror witha period of 4 mm. In Figure 5b we plot the spectral-line height as a function of retro-reflectordisplacement over a range of a few millimetres. A sinusoidal curve with a period of 4 mm has alsobeen plotted for reference. We observe good qualitative agreement, with deviations attributed toalignment drift of the excitation beams during acquisition of multiple data scans. This test providesanother verification of our interpretation of the transition mechanism.In conclusion, we have successfully demonstrated ponderomotive spectroscopy for the firsttime. We have demonstrated major advantages over standard spectroscopy, including improvedspatial addressability and flexible transition rules, which are relevant in a broad range of applica-tions. Using a temporally- and spatially-modulated ponderomotive interaction, we have driven anatomic microwave transition forbidden by established electric-dipole selection rules, with a spatialresolution in the micrometre range. In our specific case, we have demonstrated ponderomotivespectroscopy using cold Rydberg atoms and an intensity-modulated standing-wave laser beam.One immediate application of this demonstration is in quantum computing , , where single-siteaddressability plays a central role. Another application is in precision measurement of atomiccharacteristics and physical constants (e.g. the Rydberg constant , leading to the proton size );there, flexible spectroscopic transition rules will be very convenient. In the future, one may also ex-plore the possibility of extending ponderomotive spectroscopy to smaller-sized atoms or moleculestrapped in shorter-wavelength optical lattices. 14 ethodsPreparation of cold Rb Rydberg atoms in an optical lattice.
Initially, Rb atoms are cooledto a temperature of about 150 µ K in a magneto-optical trap (MOT). A one-dimensional, 1064-nm optical lattice is applied to the MOT region. The optical lattice is formed by focusing anincident laser beam into the MOT region, retro-reflecting and re-focusing it. The incident beaminterferes with the return beam and forms a standing wave. Pointing stability is greatly improvedby using a retro-reflector rather than a plane mirror. See Figure 1 in the main text for an illustration.The lattice has 880 mW of incident power at the MOT region with an 11-micrometre beam waistradius, and 490 mW of return power with an approximately 37-micrometre beam waist radius.See Supplementary Information for a discussion of these numbers. Both MOT magnetic field andlattice light remain on throughout a data scan, while the MOT light is turned off during Rydbergatom excitation and probing.The Rydberg state is excited via a two-stage excitation S / → P / → S / using780-nm and 480-nm lasers, detuned from the intermediate P / level by ≈ . During a data scan, the Rydbergatom excitation laser wavelength is tuned so as to produce Rydberg atoms in the optical lattice atan average of 0.5 Rydberg atoms per experimental cycle. This ensures a negligible probability ofatom-atom interactions. Zeroing the fields.
The electric field is zeroed prior to a series of data runs by performing Starkspectroscopy of the D / and D / fine-structure levels, while scanning the potentials ap-plied to a set of electric-field compensation electrodes. The residual field is less than 60 mV/cm.The MOT magnetic field, which is on during the experiment, does not cause Zeeman shifts of the S / → S / transition. 15 attice modulation. While S / Rydberg atoms are trapped in optical lattice wells, the intens-ity of the lattice is modulated at the expected S / → S / transition frequency. The electro-optic modulator used to perform the modulation is a fibre-coupled, polarization-maintaining, z-cut lithium niobate modulator, tunable from DC to 40 GHz. The modulator has an input opticalpower limit of 200 mW, which is, in the present experiment, insufficient for Rydberg-atom trap-ping. In a Mach-Zehnder-type interferometric set-up, we split the 1064-nm laser into a high-power(3.9 W), unmodulated beam and a low-power (190 mW), intensity-modulated beam, which ispassed through the fibre modulator. The two beams are coherently re-combined at the exit beam-splitter of the Mach-Zehnder interferometer. The re-combined beam incident at the MOT regionhas approximately 1-W of average power, which is sufficient for Rydberg atom trapping. SeeSupplementary Information. Operating point of the fibre modulator.
The fibre modulator has two voltage inputs, one for themicrowave voltage signal and another for a DC bias voltage. As described by equations (1) and (2)in Supplementary Information, the intensity transmitted through the fibre modulator depends onthe values of these voltage inputs. For most of the work, the amplitude of the microwave signal isset to V IM = V π / , which yields maximal intensity modulation when the DC bias voltage is set fora time-averaged transmission through the modulator that equals 50% of the maximum possible.Due to thermal drifts in the fibre modulator, the DC bias voltage must be actively regulated tomaintain this operating point. The lock circuit utilizes a photo-detector (‘PD1’ in Figure 1) and aPID regulator. Locking the interferometer.
Due to drifts in the optical path length difference between the armsof the Mach-Zehnder interferometer, the path length of one of the interferometer arms must be act-ively regulated to maintain a fixed phase difference at the recombination beam-splitter. This phasedifference is locked so that the intensity sent to the experiment is at a maximum (by maintaining anintensity minimum at the unused output of the recombination beam-splitter). The lock circuit util-izes a photo-detector (‘PD2’ in Figure 1), a mirror mounted on a piezo-electric transducer (‘Piezo’in Figure 1), and a PID regulator. 16 etection.
The spectral line is detected through state-selective field ionization , in which Ry-dberg atoms are ionized by a ramped electric field. Freed electrons are detected by a micro-channelplate, and detections on the micro-channel plate are registered by a pulse counter. Counting gatesare synchronized with the field ionization ramp to enable state-selective detection of the S / and S / Rydberg levels.
Read-out protocol.
During a data scan, the microwave frequency of the intensity-modulation isstepped across the expected S / → S / resonance frequency. At each microwave frequencystep, the pulse counter registers counts for 200 experimental cycles. Average counts per cycleare recorded before advancing to the next frequency step. The interferometer lock status is quer-ied before and after each set of 200 experimental cycles. If either query indicates an unlockedinterferometer, the data for that frequency step is ignored and re-taken.17 eferences
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S.E.A. acknowledges support from DOE SCGF. This work was supported by NSFGrant No. PHY-1205559 and NIST Grant No. 60NANB12D268.
Author Contributions
All authors contributed extensively to the work presented in this paper.
Author Information upplementary Information1 Derivation of the Rabi frequencyIntensity of the modulated running wave. The optical power ( λ = 1064 nm) transmitted throughthe fibre electro-optic modulator is P T = P low sin (cid:18) π V total V π (cid:19) , (3)where P low is the maximum possible output of the fibre modulator, and V π is the voltage differencebetween adjacent power transmission maxima and minima (a fixed property of the fibre modulator). V total , the total voltage on the fibre modulator, is V total = V DC ( t ) + V IM sin (Ω t ) + V drift ( t ) , (4)where V DC ( t ) is the applied DC bias voltage and V IM is the amplitude of the intensity-modulatingmicrowave voltage signal with frequency Ω . Bias drifts due to thermal and environmental effectsare accounted for via a slowly-varying effective drift voltage V drift ( t ) .Experimentally, V DC ( t ) is actively regulated such that V DC ( t ) + V drift ( t ) is locked to theinflection point of the transmission curve given by supplementary equation (3). With the fibremodulator locked, the power transmission curve is P T = P low (cid:18)
12 + 12 sin (cid:18) πV IM V π sin (Ω t ) (cid:19)(cid:19) . (5)Most experiments are performed at full modulation, V IM = V π / .At the location of the atoms, the intensity is determined by the incident (1/e ) beam waistradius w inc = 11 µ m and the incident beam power, which is measured (in the absence of modu-lation) to be P = P low in supplementary equation (3)due to inefficiencies in the optical system.) Therefore, the maximum intensity of the modulatedrunning-wave beam incident at the location of the atoms is I = 2 P πw . (6)1he intensity of the modulated (low-power) running-wave beam incident at the location of theatoms can be determined by supplementary equations (5) and (6), along with a Jacobi-Anger sub-stitution, and is I modrw = I (cid:32)
12 + ∞ (cid:88) q =1 J − (cid:18) πV IM V π (cid:19) sin ((2 q −
1) Ω t ) (cid:33) . (7) Intensity expression for modulated standing wave.
The power transmission efficiency of theoptics in the lattice return beam path has been determined to be 56%. Optical aberrations and adisplacement of the retro-reflector position from the optical axis have been estimated to yield aneffective return beam waist radius of w ret = 37 µ m (also see section 2 of supplementary equations).Therefore, the ratio of the return and incident intensity on the optical axis is ε = 9%. After addingthe fields of the incident and return beams coherently, the intensity of the modulated (low-power)standing-wave beam is I modsw = I (cid:32)
12 + ∞ (cid:88) q =1 J − (cid:18) πV IM V π (cid:19) sin ((2 q −
1) Ω t ) (cid:33) × (cid:16) ε + 2 √ ε cos (2 kz ) (cid:17) , (8)where k is the optical wavenumber k = 2 π/λ . Intensity expression for unmodulated standing wave.
Similar to supplementary equation (6),the intensity of the unmodulated running-wave beam at the location of the atoms is I = 2 P πw , (9)where beam power P is measured to be 900 mW. Therefore, at the location of the atoms, theunmodulated (high-power) beam will form a standing wave, described by an expression similar tosupplementary equation (8), albeit without the time-dependent envelope function. This expressionis I unmodsw = I (cid:0) ε + 2 √ ε cos (2 kz ) (cid:1) . (10)2 otal standing-wave intensity. The fields of the modulated and unmodulated standing waves addup coherently to yield an intensity, I mod+unmodsw = (cid:34) I (cid:32)
12 + ∞ (cid:88) q =1 J − (cid:18) πV IM V π (cid:19) sin ((2 q −
1) Ω t ) (cid:33) + 2 (cid:118)(cid:117)(cid:117)(cid:116) I I (cid:32)
12 + ∞ (cid:88) q =1 J − (cid:18) πV IM V π (cid:19) sin ((2 q −
1) Ω t ) (cid:33) + I (cid:35) × (cid:16) ε + 2 √ ε cos (2 kz ) (cid:17) . (11)This expression for the total standing-wave intensity at the location of the atoms includes anyharmonics of the microwave frequency Ω . Rabi frequency.
We consider a two-level atomic system that has a transition resonant with themodulation frequency Ω and that is insensitive to higher harmonics (a valid consideration for thisexperiment). The light intensity must also be spatially modulated within the volume of the atomfor the A · A term in equation (1) of the main text to drive transitions. After a spectral analysis ofsupplementary equation (11), the relevant spatially- and temporally-modulated component of thestanding-wave intensity is I Ω mod+unmodsw ≈ √ ε I J (cid:18) πV IM V π (cid:19) (cid:32) (cid:115) I I (cid:33) sin (Ω t ) cos (2 kz )=: F Ω sin (Ω t ) cos (2 kz ) . (12)Averaged over one optical cycle, this component of the intensity leads to the term in A · A that isimportant in lattice modulation spectroscopy, A ( z, t ) = I Ω mod+unmodsw c(cid:15) ω = F Ω c(cid:15) ω sin (Ω t ) cos (2 kz ) . (13)It is important to note that the time-averaging needed to arrive at supplementary equation (13) isover one optical period ( ≈ ≈
25 pico-seconds). In the atom-field interaction Hamiltonian (equation (1) in the main text), the correspond-ing term is 3 Ω = e F Ω m e c(cid:15) ω sin (Ω t ) cos (2 kz ) =: B sin (Ω t ) cos (2 kz ) . (14)Following a previous proposal[1 ∗ ], W Ω is the driving term in the spatially- and temporally-modulatedinteraction potential. The variable B in supplementary equation (14) is identical with the spa-tial/temporal modulation amplitude used in Ref. [1 ∗ ]. As shown in Ref. [1 ∗ ], the Rabi frequencyfor an atom is given by χ ≈ √ ε e ¯ hm e c(cid:15) ω I J (cid:18) πV IM V π (cid:19) (cid:32) (cid:115) I I (cid:33) D n (cid:48) ,l (cid:48) ,m (cid:48) n,l,m . (15)Here, D n (cid:48) ,l (cid:48) ,m (cid:48) n,l,m is the (unitless) transition matrix element between states | n, l, m (cid:105) → | n (cid:48) , l (cid:48) , m (cid:48) (cid:105) thatare resonantly coupled by frequency Ω . This matrix element includes a cos (2 kz ) dependence,where z is the centre-of-mass position of the Rydberg atom relative to the nearest optical-latticeintensity minimum. For atoms located at lattice intensity minima (or maxima), the light intensityhas a leading term proportional to z and the matrix element for the transition used in the ex-periment, (cid:10) S / (cid:12)(cid:12) z (cid:12)(cid:12) S / (cid:11) , is large. Numerical calculations[1 ∗ ] show D S S = 0.215 at latticeminima (and -0.215 at maxima). Using the experimental parameters given in this supplementarymaterial, we find a Rabi-frequency estimate of π × kHz and a population inversion time of5 µ s. If the | S (cid:105) population is small, it has a quadratic dependence on the Rabi frequency. Asdescribed in section 2 of supplementary equations below, our atoms have velocity and positiondistributions. After performing a weighted average of the population in the | S (cid:105) target stateover these different velocity and position classes, an approximate dependence on a quadratic first-order Bessel function of the field-modulating voltage amplitude, J ( πV IM /V π ) , remains. This isour motivation for exploring the population dependence on the argument of this quadratic Besselfunction in Figure 5a of the main text. 4 Spectrum simulation
In the simulation, initial positions and velocities of S / atom ensembles are determined froma Maxwell-Boltzmann distribution in the ground-state trapping potential. The ground-state atomtemperature, T , is on the order of the rubidium Doppler cooling temperature ( µ K). The opticalexcitation S / → S / is assumed to be resonant at the lattice intensity maxima, where the S / atoms collect. The atom-lattice interaction times are chosen consistent with the timing usedin the experiment. In the simulation the lattice is inverted upon Rydberg atom excitation (as in theexperiment).The classical centre-of-mass Rydberg-atom trajectories follow from the trapping potentialcalculated for the Rydberg levels[2 ∗ ]. The quantum evolution in the internal state space {| S (cid:105) , | S (cid:105)} driven by the lattice intensity modulation is computed along these trajectories by integrating thetime-dependent Schr¨odinger equation. Both the Rabi frequency and the detuning between modu-lation frequency and atomic transition frequency depend on position (which for a moving atom istime-dependent). In the inset in Figure 2a of the main text, we show a simulated spectrum obtainedfor our experimental conditions. Most parameters in the simulation have values known from theexperiment. The Rabi frequency depends on experimental parameters as specified in supplement-ary equation (15). The measured powers of the incident and return lattice beams and the measured1/e radius of the incident lattice beam (11 µ m) are also entered as fixed values. The only free fitparameters of the simulation are the initial atom temperature T and the effective 1/e radius of thereturn lattice beam, w ret , which could not be measured. Good agreement between experimentaland simulated microwave spectra is found for w ret = 37 µ m and T ≈ µ K, which is in line withreasonable expectations.
Supplementary References [1 ∗ ] Knuffman, B. & Raithel, G. Multipole transitions of Rydberg atoms in modulated pon-deromotive potentials. Phys. Rev. A. , 053401 (2007)[2 ∗ ] Younge, K.C., Anderson, S.E. & Raithel, G. Adiabatic potentials for Rydberg atoms in aponderomotive optical lattice. New J. Phys.12