Abstract
Spectroscopy is an essential tool in understanding and manipulating quantum systems, such as atoms and molecules. The model describing spectroscopy includes the multipolefield interaction, which leads to established spectroscopic selection rules, and an interaction that is quadratic in the field, which is not often employed. However, spectroscopy using the quadratic (ponderomotive) interaction promises two significant advantages over spectroscopy using the multipolefield interaction: flexible transition rules and vastly improved spatial addressability of the quantum system. Here we demonstrate ponderomotive spectroscopy by using opticallatticetrapped Rydberg atoms, pulsating the lattice light and driving a microwave atomic transition that would otherwise be forbidden by established spectroscopic selection rules. This ability to measure frequencies of previously inaccessible transitions makes possible improved determinations of atomic characteristics and constants underlying physics. The spatial resolution of ponderomotive spectroscopy is orders of magnitude better than the transition frequency would suggest, promising singlesite addressability in dense particle arrays for quantum computing applications.
Introduction
Spectroscopy is a wellestablished, powerful tool in science for characterizing microscopic systems. Fields that use this tool range from precision metrology^{1} (development of frequency standards^{2} and highprecision sensing, such as gravitometry^{3}) to trace analysis^{4} and chemical sensing^{5}. Characterizing and manipulating particles using light is also the foundation of modern fields such as quantum optics^{6}, computing^{7} and information processing^{8,9}. The interaction between a particle and a light field is described by the particlefield interaction Hamiltonian^{10}, which includes a multipolefield interaction term and a ponderomotive (quadratic) interaction term. In the case of direct application of a radiation field to atoms, the dipolefield term typically dominates the total atomfield interaction and leads to the electricdipole selection rules for atomic transitions. In contrast, the ponderomotive term may dominate the atomfield interaction when there is substantial spatial variation of the field intensity within the volume of the atom and when the intensity is modulated in time at a transition frequency of interest. In this case, any applicable selection rules are much less restrictive.
To experimentally demonstrate an atomic transition via the ponderomotive interaction, we have chosen to use Rydberg atoms trapped in an intensitymodulated standingwave optical lattice. Rydbergatom optical lattices are an ideal tool for this demonstration because the atoms’ electronic probability distributions can extend over several wells of the optical lattice^{11}, and RydbergRydberg transitions are in the microwave regime^{12}, a regime in which lightmodulation technology exists. To further illuminate the use of such an intensitymodulated optical lattice for a demonstration of ponderomotive spectroscopy, we briefly examine the physics underlying the interaction between the Rydberg atom and the lattice light. The interaction between the atom and a light field is described by the following interaction Hamiltonian^{10}:
where p is the Rydberg electron’s momentum operator and A the vector potential of the light (the laser electric field E=−∂A/∂t). The A·p term describes most types of atomfield interactions that are commonly engaged in spectroscopy^{10}. These are classified as E1, M1, E2, etc. transitions, which follow wellestablished spectroscopic selection rules^{13}. In the case of direct application of microwave radiation to Rydberg atoms, the electricdipole (E1) selection rules apply (in firstorder perturbation theory). In contrast, the quadratic A·A (ponderomotive) term allows us to drive transitions far beyond these selection rules by providing a substantial spatial variation of the field intensity within the volume of the atom and by modulating the intensity in time at the transition frequency between the coupled states. This method to drive atomic transitions is particularly effective in an amplitudemodulated optical lattice, as the lattice is spatially modulated with a period that is on the same scale as the Rydbergatom diameter. In earlier work^{14}, we proposed that, utilizing the A·A term, one can drive a wide variety of transitions beyond the usual spectroscopic selection rules.
In this work, we report a successful demonstration of this spectroscopy. We trap ^{85}Rb atoms in a onedimensional standingwave optical lattice, formed by counterpropagating 1,064nm laser beams. We drive the 58S_{1/2}→59S_{1/2} transition by sinusoidally modulating the lattice intensity at the resonant transition frequency, found to be 38.76861(1) GHz. The 58S_{1/2}→59S_{1/2} transition has been chosen because it is forbidden as a firstorder electricdipole (E1) transition, because its E2 coupling (due to the A·p term) is negligible and because for this transition we expect a high population transfer (due to the A·A term) in our intensitymodulated lattice^{14}.
Results
Experiment
In Fig. 1 we show a schematic of the experimental setup. A continuouswave (c.w.) 1,064nm laser beam is split in a MachZehnder interferometer (top) into a lowpower and highpower beam. The lowpower beam is sinusoidally modulated via an electrooptic fibre modulator driven by a tunable microwavefrequency voltage signal. This intensitymodulated lowpower beam is coherently recombined with the unmodulated highpower beam at the exit of the interferometer. In the atomfield interaction region (bottom), we form the standingwave optical lattice by retroreflecting the lattice beam. Cold ^{85}Rb Rydberg atoms are trapped with centreofmass positions near intensity minima of the lattice^{11} (Methods). At these locations, the intensity modulation of the lattice results, via the A·A term, in a timeperiodic atomfield interaction with a leading quadratic dependence on position (needed to couple states with an angular momentum change Δl=0, 2 in first order). The proper combination of temporal and spatial intensity modulation is essential for utilizing the A·A term to realize the type of spectroscopy introduced in this report.
Spectroscopy signal
In Fig. 2a we show the atomic transition 58S_{1/2}→59S_{1/2} driven via intensity modulation of the optical lattice. The 58S_{1/2}→59S_{1/2} spectrum is obtained by scanning the frequency of the microwave source driving the fibre modulator across the expected transition frequency. The spectral line shown in Fig. 2a is proof that the 58S_{1/2}→59S_{1/2} transition has been driven in a single step (in first order), at the fundamental transition frequency. The inset in Fig. 2a shows a simulated spectrum (details given in Supplementary Note 1). The simulation indicates that the substructure in the spectral line originates from different types of centreofmass trajectories of the atoms in the optical lattice. The dominant central peak results mainly from trapped atoms, whereas the small side structures result from atoms that traverse several lattice wells or that remain nearly stationary close to a lattice maximum during atomfield interaction^{15}. The substructures, which are observed in the simulation, are not as clearly resolved in the experiment. This is most likely due to inhomogeneous broadening caused by opticallattice imperfections. The simulation motivates us to fit the spectrum in Fig. 2a as a tripleGaussian, which yields a centralpeak location of 38.76861(1) GHz.
For reference, we have also driven the 58S_{1/2}→59S_{1/2} transition using direct application of microwave radiation at half the transition frequency; in this case, the transition results from a twostep (secondorder) electricdipole (E1) coupling through the 59P_{3/2} offresonant intermediate state. In Fig. 2b we show the latticefree 58S_{1/2}→59S_{1/2} spectral line, driven as a twophoton electricdipole transition at half the transition frequency, using microwaves from a horn directed at the atomfield interaction region. This reference measurement yields a 58S_{1/2}→59S_{1/2} transition frequency of 38.768545(5) GHz. The transition frequencies measured in Fig. 2a,b are in good agreement with each other. The slight bluedetuning of the central peak in Fig. 2a relative to the peak in Fig. 2b is due to a light shift from the optical lattice. A calculation based on published quantum defect values^{16} predicts a transition frequency at 38.7686(1) GHz, which is in agreement with the experimental measurements. We note that for the twophoton transition shown in Fig. 2b the microwaveinduced AC Stark shift of the transition frequency is unusually small, due to nearcancellation of the upper and lowerlevel AC shifts. We have estimated this shift of the transition frequency to be less than 500 Hz, which is insignificant in the above comparisons.
Testing the nature of the atomfield interaction mechanism
In Fig. 3 we show two tests that prove the spectral line shown in Fig. 2a is indeed caused by a perturbation due to the A·A atomfield interaction term in equation (1), which drives transitions via an intensitymodulated optical lattice. First, in Fig. 3a we verify that the transition does not originate from a combination of microwave leakage and stray DC fields. Although we carefully zero DC electric fields, a stray DC electric field in the atomfield interaction volume could, in principle, weakly perturb the 58S_{1/2} and 59S_{1/2} levels by adding some Padmixture to these levels. Any microwaveradiation leakage into the atomfield interaction region under the simultaneous presence of a DC electric field would drive the transition as an electricdipole (E1) transition between the weakly perturbed 58S_{1/2} and 59S_{1/2} levels. To verify that our spectral line is not due to such a coincidence, we look for a spectral line while leaving all microwave equipment fully powered but blocking the weak, intensitymodulated optical beam in the MachZehnder interferometer (beam block ‘B1’ in Fig. 1). In this case, the atoms are still trapped via the highpower, unmodulated beam. A stray DC electric field and leakage of microwave radiation into the chamber could then drive the transition as a singlestep E1 transition. In Fig. 3a, we scan the microwave frequency in a manner identical to the procedure used for Fig. 2a. No spectral line is evident in Fig. 3a. This establishes that lattice light modulation, rather than microwave leakage in the presence of a stray DC electric field, drives the observed transition.
Next, we aim to distinguish between the A·A transition mechanism and a possible twostep A·p transition mechanism (equation (1)). As detailed in the next paragraph, either mechanism may, in principle, drive the observed 58S_{1/2}→59S_{1/2} transition via lattice modulation. However, these fundamentally distinct 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 58S_{1/2}→59S_{1/2} transition). The presence of multiple frequencies allows, in principle, for the A·p term to couple 58S_{1/2} to 59S_{1/2} in a twostep (secondorder) process via a stimulated Raman transition through one or more states that have an energy separation ≈ℏω from the 58S_{1/2} and 59S_{1/2} levels. Following E1 selection rules, this mechanism would involve optical S→P and P→S electricdipole transitions through distant intermediate Pstates. The Raman coupling mechanism would be effective in both runningwave or standingwave laser fields. In contrast, the coupling due to the A·A term is proportional to ‹59S_{1/2}I(z)58S_{1/2}›sin(Ωt), where I (z) is the light intensity. For this coupling to be effective in first order, two conditions must be simultaneously fulfilled: the light intensity must substantially vary as a function of position z within the volume of the atom, and the modulation frequency Ω must correspond to the energylevel difference. Without spatial variation of I(z), the coupling vanishes due to the orthogonality of the atomic states. Hence, the A·A coupling is present in the intensitymodulated standing wave, while it is absent in an intensitymodulated running wave.
To test whether it is a secondorder A·p or a firstorder A·A interaction that causes our spectroscopic signal, in Fig. 3b we exchange the intensitymodulated standingwave optical lattice for an intensitymodulated runningwave beam by blocking the retroreflected lattice beam with beam block ‘B2’ in Fig. 1. We then scan the microwave frequency in a manner identical to the procedure used for Fig. 2a. No spectral line is evident in Fig. 3b. Therefore, the transition mechanism responsible for the spectral line observed in Fig. 2a is indeed a singlestep atomfield interaction arising from the modulated optical standingwave intensity (via a firstorder A·A interaction), not a twostep electricdipole Raman coupling process arising from the frequency sidebands in the light field (via a secondorder A·p interaction).
Dependence on experimental parameters
Here we characterize the dependence of ponderomotive spectroscopy on several experimental parameters. In Fig. 4 we show the scaling behaviour of the spectral line width and amplitude on the atomfield interaction time. Experimentally, the interaction time t_{int} is defined as the time between when the atoms are excited to the 58S_{1/2} state and when the atoms are ionized for detection (Methods), typically t_{int}=6 μs. During the atomfield interaction time, the transitions are driven by the intensitymodulated lattice (which is always on), and the atoms undergo a squarepulse coupling to state 59S_{1/2} of duration t_{int}. In the limit of weak saturation and assuming a Fourierlimited spectral profile of the driving field, the fullwidthathalf maximum (FWHM) of the spectral line is expected to decrease with increased interaction time as ≈0.9/t_{int}. This agrees with the trend observed in Fig. 4. A doubleGaussian fit of the 3μs spectral line and tripleGaussian fits of the 6and9μs spectral lines (red solid curves in Fig. 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 Fig. 4 further, we also find that the maximum signal height approximately doubles between 3 and 6 μs; the additional increase between 6 and 9 μs is relatively minor. This observation is consistent with a Rabi frequency within the range of 50–100 kHz. This result is in qualitative agreement with our calculated Rabi frequency presented in Supplementary Note 2.
In Fig. 5 we summarize the dependence of the spectralline height on additional experimental parameters. In Fig. 5a we investigate the dependence of spectralline height on modulation 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 (2), the Rabi frequency χ has a firstorder Bessel function (J_{1}) dependence on V_{IM}. As the height of the spectral line indicates the peak fraction of population in 59S_{1/2}, we expect the height to scale as , for fixed t_{int} and in the limit of weak saturation. In Fig. 5a, we plot the spectral line height as a function of the Bessel function argument, πV_{IM}/V_{π}, as we vary V_{IM}. Fitting the data to a function proportional to yields good agreement.
The spectralline height also depends on the distance of the retroreflector from the atoms. Both the incident and the retroreflected intensitymodulated lattice beams can be viewed as periodic sequences of pulses with a repetition frequency Ω. The spectralline height is maximal if the pulse trains of the incident and retroreflected lattice beams arrive synchronously at the atoms’ location. Considering the time delay between the retroreflected and incident pulses, it is seen that the spectralline height should sinusoidally vary with the position of the retroreflector mirror with a period of 4 mm. In Fig. 5b we plot the spectralline height as a function of retroreflector displacement over a range of a few millimetres. A sinusoidal curve with a period of 4 mm has also been plotted for reference. We observe good qualitative agreement, with deviations attributed to alignment drift of the excitation beams during acquisition of multiple data scans. This test provides another verification of our interpretation of the transition mechanism.
Discussion
In the analysis of our experimental evidence above, we have presented a classicalfield picture. Spectroscopy using the ponderomotive interaction can also be interpreted using quantized fields. Using quantized fields, the lattice beams carry photons of three different frequencies: the optical carrier frequency, ω, and the optical carrier frequency with microwave sidebands, ω±Ω. The transitions described in this work are due to absorption of a photon from one lattice beam and (simultaneous) reemission of a photon into the other lattice beam, with the frequency difference between the two photons being Ω. It is necessary to have two counterpropagating (or noncopropagating) fields to retain the spatial coupling between Rydberg levels.
In the standard description of the interaction between an atom and a quantized electromagnetic field, the A·p term in equation (1) leads to wellknown multipole transitions characterized as E1, M1, E2 and so on. The terms arise from an expansion of the exponential phase factor e^{ikz} of the field, assuming a plane running wave propagating in the zdirection (without spatial amplitude modulation). The leading term of the expansion, E1, describes electric dipole transitions. These are often dominant, but higherorder terms such as the electricquadrupole term, E2, are also used in precision spectroscopy^{17}. All electromagnetic multipole transitions can, in principle, be driven in first order and are associated with wellknown selection rules^{10}. One common way to visualize firstorder E1, M1, E2, etc., A·p transitions is via a threebranch Feynman diagram^{10} (one vertex with three branches), in which an atom in an initial state transitions into a final state by absorption or emission of a photon.
In contrast, the common way in which to visualize firstorder A·A interactions is via a fourbranch Feynman diagram^{10} (one vertex with four branches), in which an atom in an initial state interacts simultaneously with an incoming and an outgoing photon, resulting in an atom in a final state. When a plane running wave is assumed, the atomic state and the photon energy are unchanged (in the nonrelativistic case, relevant to this work). The process is an elastic scattering process that gives rise to Thomson scattering. Adding the secondorder A·p interaction leads to the KramersHeisenberg formula, which describes Rayleigh scattering^{10}, also an elastic scattering process.
However, if the field amplitude substantially varies within the volume of the atom, as in the present work, then the field is not a plane running wave, and the atomic state and the photon energy may indeed be changed (an inelastic process). Such a statechanging A·A interaction is typically not considered in spectroscopy because the field intensity is typically constant within the volume of the atom. In our work we have demonstrated that a field inhomogeneity can be intentionally introduced into an atomfield system so that the A·A interaction results in statechanging transitions. Explicitly, since A·A is proportional to field intensity I, the matrix element ‹R2A·AR1› representing a transition between two (different) Rydberg states R1› and R2› is different from zero only if ‹R2IR1›≠0, that is, if the intensity I depends on position within the atom. For fixed intensity, the matrix element vanishes due to the orthogonality of the atomic states. Therefore, to drive transitions the light field must contain different spatial modes, to retain the spatial coupling between the Rydberg levels, and different optical frequencies, with a frequency difference corresponding to the energy difference between states R1› and R2›. An optical lattice that is amplitudemodulated at the desired atomic transition frequency, such as the lattice used in this work, satisfies these conditions. The transitions can be further described in terms of the fourbranch Feynman diagram^{10} referenced above: instantaneous exchange of a photon between two different field modes of the lattice yields different initial and final Rydberg states R1› and R2›. Finally, it should be noted that this process is not the same as stimulated Raman scattering because the latter is a secondorder A·p process, which can be visualized by a sequence of two threebranch, A·ptype Feynman diagrams.
Although transitions driven by the A·A term and electricmultipole transitions (E1, E2, E3, and so on, in first and higherorder) may connect the same atomic states, we have explained above that the physical origins of these mechanisms are quite different. This results in different scaling laws. For instance, the Rabi frequencies of firstorder multipole transitions scale proportional to the driving field, while Rabi frequencies in ponderomotive spectroscopy scale proportional to the intensity (see equation (2) below).
One advantage of ponderomotive spectroscopy over established spectroscopic methods is flexible transition rules. Ponderomotive spectroscopy affords singlestep access to atomic transitions that are forbidden by electricdipole selection rules, an example of which is the 58S_{1/2}→59S_{1/2} transition demonstrated in this report. Atomic states with large differences in angular momentum can be coupled by choosing an appropriate atom size relative to the lattice period^{14}. Accurate measurements of transitions for a wide range of angularmomentum separations may improve the determination of quantum defects and other atomic constants.
As with any highprecision measurement method, lightinduced level shifts must be considered. In spectroscopic studies of lowlying atomic states, E2 transitions in the optical spectral range between longlived states can be driven as a singlephoton process in lowintensity fields^{17}. In contrast, in microwave spectroscopy of Rydberg levels, similar transitions are usually driven as E1 transitions in higher order (see, for instance, our Fig. 2b), necessitating relatively strong microwave fields. These typically result in strong AC Stark shifts of the measured transition (the small shift seen in Fig. 2b is an anomoly of the specific transition in rubidium). In contrast, in ponderomotive spectroscopy lightinduced shifts originate solely from the optical trapping fields acting on the Rydberg levels via the nonstatechanging part of the A·A interaction. While in the present work these shifts are relatively large (on the order of 100 kHz), stateoftheart laser cooling techniques will allow users to reduce the atom temperature from ~100 μK (our work) to ≲1 μK. This will allow a reduction of the trap depth and associated light shifts by at least a factor of 100. Furthermore, the differential light shift between selected upper and lower Rydberg levels cancels under certain ‘magic wavelength’ conditions, leading to at least one more orderofmagnitude reduction in the shift of the measured transition. We therefore believe that ponderomotive spectroscopy has immediate potential as a tool for highprecision spectroscopy of Rydberg atoms.
Furthermore, another powerful innovation afforded by ponderomotive spectroscopy is a spatial resolution orders of magnitude better than the frequency of the transition would suggest. Electronic transitions in a particle are typically driven by applying radiation resonant with the transition frequency. The best possible spatial resolution will be at the diffraction limit of the applied radiation, 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 ponderomotive spectroscopy, the frequency of the applied radiation is very different from the frequency of the transition being driven. The applied radiation is a standing wave with a wavelength on the order of the particle size. The frequency resonant with the desired transition is introduced by intensitymodulating the standing wave. In this report, a microwavefrequency atomic transition (typical resolution: centimetrescale) has been driven by intensitymodulating a standingwave optical lattice (typical resolution: micrometrescale). This spatial addressability could streamline certain gate schemes in quantum computing, especially those schemes involving controlled electrostatic interactions between Rydberg atoms^{7,8}. For example, by using siteselective transitions driven by ponderomotive spectroscopy, a recently proposed controlledz gate scheme^{18} could be modified such that the same Rydberg state is used for both control and target qubits (and no microwave field is needed). Ponderomotive spectroscopy may enable other advances in quantum computing, where singlesite addressability plays a central role.
In conclusion, we have successfully demonstrated ponderomotive spectroscopy. We have demonstrated major advantages over standard spectroscopy, including improved spatial addressability and flexible transition rules, which are relevant in a broad range of applications. Using a temporally and spatially modulated ponderomotive interaction, we have driven an atomic microwave transition forbidden by established electricdipole selection rules, with a spatial resolution in the micrometre range. In our specific case, we have demonstrated ponderomotive spectroscopy using cold Rydberg atoms and an intensitymodulated standingwave laser beam. One application of the method is in quantum computing^{8,19}, where singlesite addressability plays a central role. Another application is in precision measurement of atomic characteristics^{20} and physical constants (for example, the Rydberg constant^{21}, leading to the proton size^{22}); thereby, flexible spectroscopic transition rules will be very convenient. In the future, one may also explore the possibility of extending ponderomotive spectroscopy to smallersized atoms or molecules trapped in shorterwavelength optical lattices.
Methods
Preparation of cold Rydberg atoms in an optical lattice
Initially, ^{85}Rb atoms are cooled to a temperature of about 150 μK in a magnetooptical trap (MOT). A onedimensional, 1,064nm optical lattice is applied to the MOT region. The optical lattice is formed by focusing an incident laser beam into the MOT region, retroreflecting and refocusing it. The incident beam interferes with the return beam and forms a standing wave. Pointing stability is greatly improved by using a retroreflector rather than a plane mirror. See Fig. 1 in the main text for an illustration. The intensity ratio between the modulated and unmodulated portions of the incident lattice beam is about 1:100. The lattice has 880 mW of incident power at the MOT region with an 11micrometre beam waist radius, and 490 mW of return power with a ~37micrometre beam waist radius. See Supplementary Note 2 for a discussion of these numbers. Both MOT magnetic field and lattice light remain on throughout a data scan, while the MOT light is turned off during Rydberg atom excitation and probing.
The Rydberg state is excited via a twostage excitation 5S_{1/2}→5P_{3/2}→58S_{1/2} using 780nm and 480nm lasers, detuned from the intermediate 5P_{3/2} level by ≈1 GHz. Immediately after Rydbergatom excitation, the lattice is inverted using an electrooptic component, required to efficiently trap Rydberg atoms at intensity minima. The Rydberg atoms are trapped in the optical field using the energy shift of the quasifree Rydberg electron^{11}. During a data scan, the Rydberg atom excitation laser wavelength is tuned so as to produce Rydberg atoms in the optical lattice at an average of 0.5 Rydberg atoms per experimental cycle. This ensures a negligible probability of atom–atom interactions.
Zeroing the fields
The electric field is zeroed before a series of data runs by performing Stark spectroscopy^{23} of the 58D_{3/2} and 58D_{5/2} finestructure levels, while scanning the potentials applied to a set of electricfield compensation electrodes. The residual field is <60 mV cm^{−1}. The MOT magnetic field, which is on during the experiment, does not cause Zeeman shifts of the 58S_{1/2}→59S_{1/2} transition.
Lattice modulation
While the 58S_{1/2} Rydberg atoms are trapped in optical lattice wells, the intensity of the lattice is modulated at the expected 58S_{1/2}→59S_{1/2} transition frequency. The electrooptic modulator used to perform the modulation is a fibrecoupled, polarizationmaintaining, zcut lithium niobate modulator, tunable from DC to 40 GHz. The modulator has an input optical power limit of 200 mW, which is, in the present experiment, insufficient for Rydbergatom trapping. In a MachZehndertype interferometric setup, we split the 1,064nm laser into a highpower (3.9 W), unmodulated beam and a lowpower (190 mW), intensitymodulated beam, which is passed through the fibre modulator. The two beams are coherently recombined at the exit beamsplitter of the MachZehnder interferometer. The recombined beam incident at the MOT region has ~1 W of average power, which is sufficient for Rydberg atom trapping. See Supplementary Note 2.
Operating point of the fibre modulator
The fibre modulator has two voltage inputs, one for the microwave voltage signal and another for a DC bias voltage. As described by equations (1 and 2) in Supplementary Note 2, the intensity transmitted through the fibre modulator depends on the values of these voltage inputs. For most of the work, the amplitude of the microwave signal is set to V_{IM}=V_{π}/2, which yields maximal intensity modulation when the DC bias voltage is set for a timeaveraged transmission through the modulator that equals 50% of the maximum possible. Owing to thermal drifts in the fibre modulator, the DC bias voltage must be actively regulated to maintain this operating point. The lock circuit utilizes a photodetector (‘PD1’ in Fig. 1) and a PID regulator.
Locking the interferometer
Owing to drifts in the optical path length difference between the arms of the MachZehnder interferometer, the path length of one of the interferometer arms must be actively regulated to maintain a fixed phase difference at the recombination beamsplitter. This phase difference is locked so that the intensity sent to the experiment is at a maximum (by maintaining an intensity minimum at the unused output of the recombination beamsplitter). The lock circuit utilizes a photodetector (‘PD2’ in Fig. 1), a mirror mounted on a piezoelectric transducer (‘Piezo’ in Fig. 1), and a PID regulator.
Rabi frequency estimate
To initialize certain experimental parameters (such as atomfield interaction time), we must make an estimate of the expected Rabi frequency. Because of the coherent mixing of the lowpower, modulated lattice beam with a highpower, unmodulated beam, the timeaveraged lattice depth is large enough that most atoms remain trapped in the lattice while the 58S_{1/2}→59S_{1/2} transition is probed. The coherent mixing of the two beams provides the additional benefit that it enhances the modulation in the atomfield interaction region, resulting in a much larger 58S_{1/2}→59S_{1/2} coupling than would be possible with the weak modulated beam alone. The Rabi frequency for a transition n,l,m›→n′,l′,m′› between two Rydberg states that are resonantly coupled by the intensitymodulated lattice is
where V_{IM} is the amplitude of the microwave voltage signal that drives the fibre modulator, V_{π} is the voltage difference between minimum and maximum intensity transmission through the modulator (a fixed modulator property), ε is the intensity ratio at the atom location between the return and incident beams forming the lattice, e is electron charge, m_{e} is electron mass, ω is the angular frequency of the opticallattice light, and and are the incident intensities of the modulated and unmodulated lattice beams at the atom location, respectively. The transition matrix element (unitless) has been derived in previous work^{14} and is 0.215 for the 58S_{1/2}→59S_{1/2} transition for an atom located at a minimum in our lattice. This value is large compared with those of other possible transitions due to a favourable ratio between atom size and lattice period (which is on the order of one). A detailed derivation of equation (2) is given in Supplementary Note 2 along with values for the parameters listed above, yielding a typical Rabi frequency estimate on the order of 100 kHz. The inphase addition of the fields corresponding to intensities and , using the MachZehnder beam combination setup, leads to the enhancement term in square brackets in equation (2). In our experiment, the Rabi frequency is enhanced by a factor of ≈20, which aids significantly in observing the transition. While in our experiment the enhancement afforded by the interferometric setup is critical for a successful demonstration, different laser wavelengths, modulation schemes, subDoppler and evaporative cooling techniques or the use of separate trapping and modulation beams could make the interferometric setup unnecessary.
Detection
The spectral line is detected through stateselective field ionization^{12}, in which Rydberg atoms are ionized by a ramped electric field. Freed electrons are detected by a microchannel plate, and detections on the microchannel plate are registered by a pulse counter. Counting gates are synchronized with the field ionization ramp to enable stateselective detection of the 58S_{1/2} and 59S_{1/2} Rydberg levels.
Readout protocol
During a data scan, the microwave frequency of the intensitymodulation is stepped across the expected 58S_{1/2}→59S_{1/2} resonance frequency. At each microwave frequency step, the pulse counter registers counts for 200 experimental cycles. Average counts per cycle are recorded before advancing to the next frequency step. The interferometer lock status is queried before and after each set of 200 experimental cycles. If either query indicates an unlocked interferometer, the data for that frequency step are ignored and retaken.
Additional information
How to cite this article: Moore, K. R. et al. Forbidden atomic transitions driven by an intensitymodulated laser trap. Nat. Commun. 6:6090 doi: 10.1038/ncomms7090 (2015).
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Acknowledgements
S.E.A. acknowledges support from DOE SCGF. This work was supported by NSF Grant No. PHY1205559 and NIST Grant No. 60NANB12D268.
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All authors contributed extensively to the work presented in this paper.
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Correspondence to Kaitlin R. Moore.
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Moore, K., Anderson, S. & Raithel, G. Forbidden atomic transitions driven by an intensitymodulated laser trap. Nat Commun 6, 6090 (2015). https://doi.org/10.1038/ncomms7090
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