Rydberg atoms for radio-frequency communications and sensing: atomic receivers for pulsed RF field and phase detection
David Alexander Anderson, Rachel Elizabeth Sapiro, Georg Raithel
RRydberg atoms for radio-frequency communications and sensing: atomic receivers forpulsed RF field and phase detection
D. A. Anderson (email: [email protected]), R.E. Sapiro, and G. Raithel
Rydberg Technologies Inc., Ann Arbor, MI 48103 USA (Dated: October 18, 2019)
I. INTRODUCTION
The emergence of atomic sensor technologies is driving a paradigm shift in modern sensing and measurement byexploiting quantum phenomena to realize fundamentally new detection capabilities unmatched by their classicalcounterparts [1]. Atomic sensing of radio-frequency (RF) electric fields using Rydberg electromagnetically-inducedtransparency (EIT) in atomic vapors has been the subject of growing scientific interest [2–4]. This has beenmotivated in part by a drive at National Metrology Institutes to replace century-old antennas as RF standardswith absolute (atomic) standards for RF electric fields [3], and has recently been established as a novel quantumtechnology platform with broad capabilities [5, 6] that has matured into commercial RF detection and measurementinstrumentation [7, 8]. A notable advance in atomic RF devices and measurement tools is the recent realization ofthe first Rydberg RF field probe (RFP) and measurement system (RFMS) for self-calibrated SI-traceable broadbandRF measurement and imaging of continuous, pulsed, or modulated fields. Relevant developments include therealization of compact atomic sensing elements capable of broadband RF electric-field measurement from MHz to >
100 GHz [9], fiber-coupled atomic vapor-cell RF field probes [6, 10], the demonstration of ultra-wide dynamic fieldranges spanning sub-10 mV/m up to >
10 kV/m (dynamic range >
120 dB) [11, 12], and all-optical circuit-free RFsensors for EMP/EMI-tolerant detection and operational integrity in high-intensity RF environments [13]. Hybridatomic RF technology that combines atom-based optical sensing with traditional RF circuitry and resonators hasalso been developed realizing hybrid sensors with augmented performance capabilities such as resonator-enhancedultra-high sensitivity polarization-selective RF detectors [14], waveguide-embedded atomic RF E-field measurementfor SI-traceable RF power standards [15], and atom-mediated optical RF-power/voltage transducers and receivers [16].Recently, Rydberg atom-based field sensing has also been adapted to modulated RF field detection promising newpossibilities in RF communications, with demonstrations including a Rydberg-atom transmission system for digitalcommunication [17], atom radio-over-fiber [18], and “Atomic Radio” [19] using a multi-band atomic AM and FMradio receiver based on direct atom-mediated RF-to-optical conversion of baseband signals picked up from modulatedRF carriers [20].In this article we describe the basic principles of the atomic RF sensing method and present the development ofatomic pulsed RF detection and RF phase sensing establishing capabilities pertinent to applications in communicationsand sensing. To date advances in Rydberg atom-based RF field sensors have been rooted in a method in which thefundamental physical quantity being detected and measured is the electric field amplitude, E , of the incident RF elec-tromagnetic wave. Sections III and IV are focused on using atom-based E -field measurement for RF field-sensing andcommunications applications. With established phase-sensitive technologies, such as synthetic aperture radar (SAR)as well as emerging trends in phased-array antennas in 5G, a method is desired that allows robust, optical retrievalof the RF phase using an enhanced atom-based field sensor. Section V is focused on a fundamentally new atomicRF sensor and measurement method for the phase of the RF electromagnetic wave that affords all the performanceadvantages exhibited by the atomic sensor [6]. The presented phase-sensitive RF field detection capability opensatomic RF sensor technology to a wide array of application areas including phase-modulated signal communicationsystems, radar, and field amplitude and phase mapping for near-field/far-field antenna characterizations. II. ATOMIC-PHYSICS AND FIELD/PHASE-SENSING BACKGROUND
Our atom-based field sensors use Rydberg atoms as an RF-receiver medium. Classically, a Rydberg state is a stateof an atom in which a valence electron resides in an orbit far from the atomic core. The weakly-bound, quasi-freeelectron of a Rydberg atom affords the atom a unique set of physical properties including a high sensitivity toexternal electric and magnetic fields. The atomic-physics principles of one- and two-electron systems are describedby Bethe and Salpeter [21]. Rydberg atoms of alkali, earth alkali and a variety of other species fall within this classof atomic systems. Several textbooks that are specifically focused on the physics of Rydberg atoms include theworks by Gallagher [22] and Stebbings and Dunning [23]. For the present purpose Rydberg atoms may be viewed as a r X i v : . [ qu a n t - ph ] O c t quantum oscillators that are fairly easy to prepare via laser excitation, and that are perfectly frequency-matched toa selection of incident RF frequencies. This is because the orbital frequencies of the Rydberg valence electron can betuned into resonance with RF radiation. The set of highly responsive frequencies is different for every Rydberg state.Since there is a wide variety of different Rydberg states that are accessible by tuning the Rydberg-atom excitationlasers, Rydberg atoms offer broadband RF coverage from the MHz into the THz regime.A single Rydberg-atom receiver consists of a valence electron of a single atom that has been laser-excited into aRydberg state, whose orbital frequency allows a (near-)resonant, RF-driven transition into another Rydberg state.The frequency match affords a combination of very small receiver size, and high electric-field sensitivity. A singlereceiver Rydberg atom has a size on the order of a µ m, while an atomic ensemble that is large enough for theconstruction of a technically viable and robust receiver instrument can range between hundreds of µ m and a fewcm in size. The response of the atomic ensemble to an incident RF field amounts to quantum-mechanical energylevel splittings and level shifts that are observed by the means of EIT laser beams, which present an all-optical,robust tool to measure the atomic response, and to thereby determine the RF field. As the measurement is basedon invariable atomic properties that are well known, this method of RF field determination is atom-based andintrinsically calibration-free. In Sections III and IV we employ a Rydberg-atom field sensor to measure RF fieldamplitudes and to receive modulated RF signals.To achieve phase sensitivity in an atom-based Rydberg receiver, we employ elements of holographic phase-sensingmethodologies. The phase of the signal wave, φ , is defined relative to the phase of a reference oscillator or referencewave, φ ref . To enable phase measurement of the signal wave, the signal electromagnetic field has to be brought into aninterferometric relationship with the reference field. In practice, the phase reference is often mediated via a referencewave that is physically superimposed with the signal wave on top of a detector that measures field amplitude. Due tothe superposition principle, which is common to all wave phenomena that follow the (linear) wave equation, the phasedifference, φ − φ ref − φ ofs , is obtained from an interference measurement. In its most basic implementation, the netsignal is given by the sum of two sine waves with the same frequency, A sin( ωt + φ ) + A ref sin( ωt + φ ref + φ ofs ), withamplitudes A and a controllable offset phase φ ofs that is used to tune the interference pattern from constructive todestructive, and to thereby find a value for φ − φ ref . A measurement of the net wave amplitude versus φ ofs yieldsthe phase difference between the wave to be tested and the reference wave, φ − φ ref . This usually sums up thetask of phase measurement. The principle of differential-phase measurement by the means of superposition of objectand reference waves is widely used in holography. There, phase- and amplitude-sensitive recordings of interferencepatterns of signal- and reference waves on a planar recording medium with sub-wavelength spatial resolution allowaccurate, three-dimensional reconstruction of the signal wave field. This holography concept can be translated fromthe optical into the RF domain. In Section V we describe a method of atom-based RF phase detection, measurementand enhanced receiving that we have recently devised. The method is not limited to signal and reference waves ofthe same RF frequency. Reference waves that are offset in frequency enable heterodyne and superheterodyne signalamplitude and phase detection. III. ATOMIC RF ELECTRIC FIELD SENSING
Atomic RF receiver technology employs EIT as a quantum-optical readout of Rydberg states of atoms in avapor [2–4, 24]. Figure 1(a) shows a picture of a miniature atomic vapor cell sensing element containing a purecesium gas next to a standard K a -band horn antenna. To sense and measure parameters of an incident RF field,optical beams are passed through the vapor cell to interrogate field-sensitive Rydberg states of the atoms exposed tothe RF field. Detected changes in the transmission of an optical probing beam through the atomic vapor provide adirect RF-to-optical readout and information on the incident RF signal field. Under typical operating conditions, theatomic vapor has an optical density for the EIT probe laser beam propagated through the cell that is sufficiently highto obtain a robust EIT signal with high signal to noise, as required for RF field detection. Further, the atomic vaporin the cell is dilute enough so that interactions of the Rydberg atoms can be neglected. Therefore, the spectroscopicresponse of the medium to the fields can be modeled based on a quantum-mechanical picture of a single, isolated atom.Figure 1(b) shows an atomic energy-level diagram illustrating a two-photon Rydberg EIT readout scheme for acesium vapor. In this basic scheme, two optical laser fields couple atomic states to a high-lying Rydberg state (30D inFig. 1(b)), with a weak optical probe beam resonant with the first atomic transition between ground and intermediatestates, and a relatively stronger optical coupler beam tuned into resonance with a second atomic transition betweenintermediate and Rydberg state. When the coupler laser-frequency is in resonance with the Rydberg state, an EITwindow opens for the probe beam through the vapor [25, 26]. Owing to the sensitivity of the atomic Rydberg levels tothe RF electric field, the Rydberg EIT signal provides an optical readout for the RF field. An example Rydberg EITresonance is shown in Figure 1(c) (black curve). In the presence of a moderate RF field at a frequency near-resonantwith an allowed transition between the optically excited Rydberg level and a second Rydberg level of the atom, theEIT-detected atomic Rydberg line splits into a pair of Autler-Townes (AT) lines whose splitting is proportional to theRF electric-field amplitude (Figure 1(c) (magneta curve)). In this linear AC Stark effect regime, the E-field is givenby [26] E = (cid:126) Ω /d, (1)where Ω is the Rabi frequency of the RF-coupled atomic Rydberg transition (near-identical to the AT splittingmeasured optically in units 2 π × Hz), d is the electric dipole moment of the Rydberg transition in units Cm, and (cid:126) = 6 . × − Js / (2 π ) is Planck’s constant.FIG. 1: (a) An atomic vapor-cell RF sensing element in front of a K a -band horn antenna. Photo courtesy ofRydberg Technologies Inc. [8], (b) atomic energy-level diagram for RF sensing and measurement using a two-photonRydberg EIT scheme in cesium, (c) Rydberg EIT signal readout from Rydberg state without RF (black), withon-resonant weak RF field (teal) and moderate RF field exhibiting Autler-Townes splitting (magenta).From Eq. 1 one obtains a direct, optical measurement of the electric field amplitude E of the RF wave in absolute(atomic) units traceable to fundamental constants. Generally, for low RF field levels, the sensitivity of the atomicreceivers is dictated by (1) the electric dipole moment d of the Rydberg-Rydberg transition resonant with the incidentRF field and (2) the spectroscopic EIT linewidth in the optical readout which determines the achievable resolutionfor measuring Ω. For RF-field frequencies in the range of 100 MHz to 500 GHz, resonant dipole moments in alkaliatoms typically range from 10 to 10 ea , where e is the elementary charge and a is the Bohr radius, with theprincipal quantum number n ranging from about 10 to 300, while Rydberg EIT linewidths are typically about 1 MHzor more. Equation 1 provides a useful approach to RF E-field sensing and measurement with EIT in Rydberg atomvapors, but serves to a large extent as a didactic model because it is valid only within a relatively limited E-fieldrange and for a discrete, albeit large, set of RF field frequencies near-resonant with Rydberg transitions, therebyrendering it impractical in many real-world E-field measurement scenarios. This is addressed by a well-developedmeasurement method and approach using EIT and exploiting the full quantum response of the Rydberg atominteraction with RF fields that includes off-resonance AC Stark shift readouts [9], enabling direct E RMS measurementof continuous-frequency RF field frequencies over tens of GHz with a >
60 dB dynamic range. A full non-perturbativeFloquet treatment allows measurement of the electric-field values and frequencies of even stronger (coherent) RFfields [4, 5].
IV. COMMUNICATIONS AND MODULATED RF FIELD SENSING WITH ATOMIC RECEIVERS
The adaptation of the Rydberg atom-based RF E-field sensing and measurement approach to the detection ofmodulated and time-varying RF fields promises to enable new capabilities in RF sensing and communications [6].Recent laboratory work has been performed demonstrating modulated RF E-field detection and baseband signalreception with Rydberg EIT in atomic vapors. Highlights include a Rydberg atom-based transmission system fordigital communications [17], atom radio-over-fiber [18], and a multi-band atomic AM and FM receiver for radiocommunications [20] recently adapted to two-channel reception using two atomic species [27]. Atomic receivers forcommunications are a nascent technology prime for advanced development and adaptation to real-world systems.The basic operating principle of an atomic RF receiver based on Rydberg EIT in vapor cells exploits the largedifferential dipole moments of Rydberg states of atoms. With an RF carrier wave applied to the atomic sensingvolume, the coupler-laser frequency is set to an operating point on one of the inflection points of the EIT spectral line(see, for example, Figs. 1(b) and (c)). As the incident modulated RF wave impinges on the atoms, the atoms respondsynchronously to the time-varying RF electric field leading to a change in the probe light transmission through thevapor. This realizes a direct Rydberg-atom-mediated optical pick-up and demodulation of the baseband-modulatedRF carrier signal, where the demodulation occurs in the atomic vapor cell without need for any demodulation orsignal-processing electronics required by traditional antenna-receiver technology.For the general case of a transmitted AM signal and differential dipole moment d of the target Rydberg statesin the atomic receiver, a typical range in AM depth δE/E is given by δE/E ∼ h × δ Γ / ( Ed ), where δ Γ is the EITlinewidth. Figure 2(a) shows the real-time optical readout from an atomic rubidium vapor-cell receiver detectingand demodulating 1 kHz baseband signals transmitted in free space on an AM-modulated 37.4065 GHz RF carrierwave. Received signals are shown for three different AM modulation depths of the carrier. The modulationdepths can typically range from several 10% down to below 1%, depending on exact operating conditions andreceiver sensitivity requirements. In addition to being sensitive to changes in RF field amplitude, Rydberg statesare also sensitive to changes in RF field frequency, allowing receiver pick-up and demodulation of FM RF carriersignals using a similar approach. This basic approach has been implemented in the reception of both AM andFM radio communications on RF carrier waves over a wide range of carrier bands, with wide-band operation ofa single atomic receiver demonstrated for carrier frequencies spanning more than four octaves, from C-band to Q-band.In addition to radio and digital communications, pulse-modulated RF field detection and measurement withRydberg atom receivers promises to expand atom-based RF technology for enhanced performance capabilitiesin application areas including high-intensity pulsed-RF measurement and electromagnetic testing, pulsed radar,surveillance, and electronic support measures (ESM) systems. To this end, in the following we discuss the directdetection of pulsed RF fields with an atomic receiver in the time-domain and investigate the behavior and responsetimes of the atomic detector to both pulsed RF field detection and pulsed Rydberg EIT readout without RF toisolate the atom-optical interaction from the atom-RF interaction under typical EIT operating conditions.Figure 2(b) shows the time-domain detection and measurement of a 1 µ s-long 36.2-GHz narrow-band RF fieldpulse by a rubidium Rydberg-based atomic detector. The pulse-modulated RF pulse measurement is performed inthe weak-field regime where the RF is resonant with rubidium 47S / to 47P / transition and AT-splits the EIT linefollowing Eq. 1. One observes in the measured data that the AT-splitting is well-resolved in time for 1 µ s-long RFpulses. The temporal resolution in the detection in Fig. 2(b) approaches the ∼
10 ns level and is limited primarily bythe response time of the photo-detector used in the measurements. An extension to shorter RF pulse-width detectionis readily achievable and corresponding larger RF detection bandwidths. Development in this area is on-going.In a closely related study of time-dependent effects, we have investigated the time-dependence of the underlying EITreadout from the atomic vapor for pulsed Rydberg EIT alone, without application of external RF fields. This allowsus to distinguish between atom-optical and atom-RF interaction effects contributing to the detection process, and toshed light on the short time-scale response of the Rydberg EIT pulse in a thermal atomic vapor for typical moderateoptical Rabi frequencies. Figure 2(c) shows the EIT probe transmission (gray-scale) for a 5 µ s-long coupler-lightpulse as a function of time (vertical axis) and coupler laser frequency (horizontal axis) near the field-free Rb 5P / to 30D / Rydberg state resonance. Here, the coupler pulse is switched on at 11.7 µ s and off again at 16.7 µ s with aprecision <
100 ns. When the coupler pulse turns on a sudden decrease in transmission is observed, or equivalently anincrease in probe absorption, over a period of about 20 ns (white horizontal stripe in the data, labelled I in Fig. 2(c)).This is followed by an increase in transmission until reaching a steady-state value over a period of 1 to 2 µ s. At theturn-off of the coupler pulse, a sudden increase (gain) in optical transmission is observed, also over about 20 ns (blackFIG. 2: (a)Real-time optical readout from the atomic receiver for an AM 1 kHz baseband signal transmitted on a37.406 GHz RF carrier resonantly driving the cesium 47S / to 47P / Rydberg transition. The received signals forthree AM modulation depths of 5% (blue), 25% (purple), and 45% (black) are shown for the coupler laser-frequencyoperating point set to the field-free cesium 47S / Rydberg line [20] (b) time-domain detection and measurement ofa 36.2-GHz 1 µ s-long RF field pulse using a rubidium Rydberg-based atomic detector as a function of time andcoupler-laser frequency. The time evolution is along the y-axis. The coupler laser beam is switched on at 11.7 µ s andleft on, and the RF pulse incident on the sensing element is switched on at 21.7 µ s. The RF frequency is resonantwith the rubidium 47S / to 47P / Rydberg transition (dipole moment d = 745 ea ) and produces a splitting(double-sided arrows) of the EIT line proportional to the pulsed RF field amplitude of about 5 V/m. (c) RelativeEIT probe transmission for a 5 µ s-long coupler laser square-pulse (probe laser on continuously) as a function of timeand coupler-laser frequency near the rubidium 30D / Rydberg state. No RF is applied. The coupler pulse is on at11.7 µ s and off again at 16.7 µ s with an uncertainty <
100 ns. The probe transmission is in gray-scale; coupler-freeabsorption background is at a level of 0.236; with relative increasing transmission from white to black.horizontal stripe in the data, labelled II in Fig. 2(c)), followed by a decay of the signal to zero over several microseconds.The transients measured at both the beginning and the end of the Rydberg EIT coupler pulse in an atomic vaporhave to our knowledge not been observed before. The observed process appears akin to - but distinct from - photonstorage and retrieval via EIT-mediated Rydberg polaritons in cold-atom systems, where a probe photon is storedas a collective Rydberg excitation in the medium in presence of the coupler beam and released/retrieved when thecoupler is turned off [28–30]. In our presented case, a certain excess amount of probe-pulse energy (contained inthe probe light incident on the medium) is stored and released. In this interpretation, the ’stored’ 780 nm lightis ’retrieved’ after an extremely long time ( > µ s-long pulses in other experiments), exceeding the < µ s transittime of atoms through the EIT beams used. The concept of collective Rydberg-polaritons propagating along thelaser-beam direction, through a medium of atoms that are frozen in place (a picture commonly used in cold-atomEIT experiments [28–30]), is not directly applicable to our situation. However, during the short, ∼ p = 2 π ×
18 MHz for the probe andΩ c = 2 π × . p = 2 π ×
44 MHz and Ω c = 2 π ×
10 MHz. In that case, the large probe Rabi frequency causes alarger amount of saturation broadening, leading to EIT lines that are about 20 MHz wide.
V. ATOMIC RF PHASE DETECTORS
RF electric-field sensing and measurement based on EIT readout of field-sensitive Rydberg states of atoms inthermal vapor cells has made rapid progress towards establishing atomic RF E -field standards. Here we describe anatomic RF phase, amplitude, and polarization sensor that employs a novel quantum-optical readout scheme from anRF field-sensitive Rydberg vapor to achieve RF phase sensitivity [6].The holography concept outlined in Sec. II can be translated from the optical into the RF domain. Measurementshave been performed by combining RF signal and reference waves in or close to Rydberg-EIT vapor cells [31–33].The magnitude of the coherent electric-field sum of the object and reference RF or microwave fields is measuredusing vapor-cell Rydbrg-EIT methods within the atomic vapor cell or hybrid atom-cavity cell structure, as describedin Sec. III. According to principles of holography, this allows measurement of amplitude and phase of the signal wave,with the reference wave providing the phase reference as well as amplification [31–33]. Towards practical applications,a phase-sensitive recording of a coherent electromagnetic field on a surface allows the reconstruction of the field inall space. RF-applications of this reconstruction principle are abound and include radars based on interferometricschemes, such as SAR and InSAR, and far-field characterization of antenna radiation patterns based on near-fieldmeasurements of amplitude and phase of the field emitted by the antenna under test. In the last application listed,the measurement has to be performed on a surface, and a near-field to far-field transformation is applied to calculatethe field in all space.To achieve phase sensitivity in the holographic RF field measurement, the reference wave can be interfered with thewaves emitted by or reflected from an object. The generation of a clean RF reference wave presents a considerableproblem. In optical holography, the reference wave typically is an expanded, near-perfect plane-wave laser beamthat interferes with the object scatter within a layer of photographic emulsion (or an equivalent substance). Thepurity of the reference wave is important, i.e. it should be free of diffraction rings caused by dust particles andother imperfections. Interference from specular reflections of the reference wave from planar surfaces should alsobe avoided. In quantitative work, it would also be important that the reference wave has a fixed amplitude or, atleast, a well-known, slowly varying amplitude function. In holographic measurements in the RF domain, equivalentconditions are hard to meet. The preparation of a defect-free RF reference wave that has a smooth amplitudebehavior over a large surface presents a great challenge. In some cases, it will be fundamentally impossible to preparea stationary reference wave. This applies, for instance, to SAR radar applications, where the detector is mountedon a moving platform, like an airplane or a satellite, or in cases where a mm-wave or microwave field needs to befully characterized over a large surface in space. In another class of applications, the object waves are located withinclose quarters where multiple reflecting surfaces cannot be covered with anechoic material (“urban radar”); there,reflections from unknown surfaces spoil the reference wave.The cited previous implementations of holographic RF phase detection with atoms have required an antenna orsimilar for the generation of the reference RF wave, precluding the approach from providing a stand-alone atomicdetector solution for RF waves propagating in free space [31–33]. In the following we present the holographic schemein which an RF reference signal is provided via phase modulation of one of the EIT laser beams [6]. Our presentedapproach removes the need for RF reference waves, and therefore eliminates the aforementioned shortcomings of RFreference waves.For RF phase measurement using RF-modulated optical beams, we consider a phase modulation imprinted on anoptical coupling laser beam via an electro-optic modulation technique. Using a fiber-optic high-frequency modulator,which is commercially available, the coupler beam is frequency- or amplitude-modulated with a signal at frequency ω RF that is near the frequency of the RF field to be measured, and that is phase-coherent with the RF field to bemeasured. For the purpose of describing the basic concept, in the following we consider a rubidium atom and a casewhere the (optical) coupler field is phase-modulated at a frequency that is identical with the RF signal frequency ω RF .Here, ω RF also approximately equals half the separation between two neighboring Rydberg levels of rubidium, nS / and ( n + 1) S / , as shown in Fig. 3(a). The carrier frequency of the coupler laser beam is resonant with the forbiddentransition 5 P / → nP / . Due to the quantum defects in rubidium, the nP / level is approximately at the midpointbetween the nS / and ( n +1) S / levels, and the electric-dipole matrix elements for the allowed microwave transitions, d A and d B , are about the same. Also, the detunings of ω RF from the respective atomic transition frequencies, ∆ A andFIG. 3: (a) Quantum mechanical level scheme and optical/RF excitation pathways used in an implementation of thephase-sensitive RF electric-field measurement method [6]. (b) Setup illustration of the phase-sensitive measurementimplementation. The microwave horn (MW) stands for any antenna under test or other object wave of interest. Thefiber modulator phase-coherently imprints an RF reference beat onto the coupler beam sent to the atoms in thevapor cell. The RF reference beat replaces the reference beam that is normally used in phase sensitive (holographic)field measurement. The vapor cell in the atom-based RF sensing element can be very small ( ∼ B , are approximately equal in magnitude and opposite in sign (see Fig. 3(a)). For the simplified discussion presentedhere, we assume that the detunings ∆ A and ∆ B have magnitudes on the order of 100 MHz, which is much largerthan the Rabi frequencies of any of the involved transitions. Hence, the two-photon Rabi frequencies that describethe transitions from 5 P / into nP / via the absorption of one coupling-laser sideband photon and the absorption(channel B in Fig. 3(a)) or the stimulated emission (channel A in Fig. 3(a)) of an RF photon are given byΩ A = Ω P ( n +1) S Ω ( n +1) SnP A exp(i( φ P ( n +1) S − φ RF ))Ω B = Ω P nS Ω nSnP B exp(i( φ P nS + φ RF )) (2)There, Ω P nS and Ω P ( n +1) S are the Rabi frequencies of the optical coupler-laser transitions into the S Rydberglevels, Ω nSnP and Ω ( n +1) SnP are the Rabi frequencies of the RF transitions from the S Rydberg levels into the nP / Rydberg level, and φ RF is the phase of the RF field. Also, φ P nS and φ P ( n +1) S are the phases of the modulationsidebands of the coupling laser. Note there is an important difference in sign in front of the φ RF in the above equations.Further, the RF field amplitude, E , is included in the RF Rabi frequencies. It is, for instance, Ω P nS = Ed B / (cid:126) . Thenet coupling, Ω C , due to the coupler lasers is given by the coherent sum of Ω A and Ω B . Noting that Ω P nS ≈ Ω P ( n +1) S and Ω nSnP ≈ Ω ( n +1) SnP , and noting that a suitable choice of levels allows us to set ∆ B = − ∆ A =: ∆, forthe present simplified discussion we haveΩ C = Ω P nS Ω nSnP
2∆ (exp(i( φ P ( n +1) S − φ RF )) − exp(i( φ P nS + φ RF ))) (3)The approximations made to arrive at this expression are not crucial; they serve to simplify the math to betterelucidate the important aspects of the method. The optical phases φ P nS and φ P ( n +1) S are well-defined and arenot prone to drift, because all frequency components of the modulated coupling laser beam follow the exact samegeometrical path. An optical delay line in the beam path of the FM-modulated coupler laser is used to control thedifference between the optical phases φ P nS and φ P ( n +1) S . A translation by amount L (see Fig. 3(b)) causes a phaseshift of Lc ω RF . For RF frequencies in the 10-GHz range, a translation of about 1 cm will scan the optical-phasedifference φ P nS − φ P ( n +1) S over a range of 2 π . It is seen from the previous equation that the net EIT coupling takesthe form Ω c = Ω c cos( φ RF + φ opt ) , (4)with a (complex) factor Ω c that neither depends on φ RF nor on the delay-line-controlled optical phase φ opt . It isthus seen that net EIT coupling Rabi frequency Ω c can be tuned between zero and ± Ω c by adjusting the opticalphase φ opt with the coupler-beam delay line (see Fig. 3(b)). The presented analysis shows that the optical phase φ opt is equivalent with the tunable offset phase φ ofs in the introductory discussion II. Also, φ RF corresponds with thefrequency difference φ − φref that is to be measured.Since the strengths of the Rydberg-EIT lines observed in the spectra are generally proportional to | Ω c | , the EITline strength is proportional to cos ( φ RF + φ opt ). The EIT line strength, measured as a function of the optical phase, φ opt , allows one to measure the phase φ RF . The microwave phase φ RF can therefore be retrieved as long as it remainsstable over the time scale needed to scan the optical delay line over a range of 2 π . Using mechanical delay lines, thedynamic range of this RF phase measurement method will be at about 10 s − , while electro-optic phase shifters willallow a phase measurement bandwidth ranging into the MHz-range.We note that in the presented scheme the 5 P / to nP / transition is forbidden; therefore, the coupler-beamcarrier (thin blue line in Fig. 3(a)) does not introduce an additional term in the analysis. In more general cases,such a term could, of course, be included. Further, the magnitude of the pre-factor Ω c can be determined by findingthe peak EIT line strength while varying φ opt . The obtained peak value for Ω c then reveals the RF electric field, E , via the known electric dipole moments of the RF transitions. In this way, both E and the phase φ RF can bemeasured. This capability enables the aforementioned applications in antenna characterization, phase-sensitive radar,communications, and sensing. VI. CONCLUSION
In this work we have demonstrated the capability of pulsed RF field detection and measurement with an atomicreceiver. Pulsed RF field detection was performed in the time-domain with a temporal resolution at the 10 ns-level, limited by photodetector bandwidths. The behavior and response times of the atomic detector to both pulsedRF field detection and pulsed Rydberg EIT readout without RF have been investigated to isolate the atom-opticalinteraction from the atom-RF interaction under typical EIT operating conditions. In pulsed Rydberg EIT readoutfrom the atomic vapor, transient behavior was experimentally observed resulting in a drop in optical transmissionat the onset of the coupler pulse and gain in optical transmission at the turn-off of the coupler pulse with dynamicson a 10 ns timescale, also limited by photodetector bandwidth. Modeling of these system dynamics has separatelybeen performed reproducing the observed transient behavior in great detail and affirming the physical existence ofthe phenomenon, with underlying physics distinct from the interpretation of collective Rydberg-excitation polaritonspropagating in the medium [28–30]. Fast quantum-optical transient dynamics in Rydberg EIT readout at time-scales on the sub-10 ns level have been studied, and their implementation in RF field sensing has been proposed toenable, for example, reception of modulated RF communications signals approaching 100 MHz bandwidth, short RFpulse detection, and high-frequency RF noise measurements. In the present work, we also describe a new methodfor atomic RF phase sensing and measurement to realize atomic sensors for phase-sensitive detection of RF fields [6]critical to a wide range of application areas such as antenna near-field characterizations, radar based on interferometricschemes, and phase-modulated signal transmission and telecommunications. The atomic RF phase sensor developmentenables the realization of atomic sensors, receivers and measurement tools capable of RF phase, amplitude, andpolarization detection with a single, vapor-cell sensing element. Atomic RF sensors and receivers based on Rydbergatom-mediated RF-to-optical transduction hold promise as a basic technology platform to realize advanced passiveradar and electronic support measures (ESM) systems. Implementation of coherent conversion between microwaveoptical photons in Rydberg gases [34], for example, may be implemented in the Rydberg atom-based detector platformto realize coherent RF-to-optical transducers in quantum communications schemes and radar.
VII. ACKNOWLEDGEMENT
This work was supported by Rydberg Technologies Inc. Part of the presented material is based upon work sup-ported by the Defense Advanced Research Projects Agency (DARPA) and the Army Contracting Command-AberdeenProving Grounds (ACC-APG) under Contract Number W911NF-17-C-0007. The views, opinions and/or findings ex-pressed are those of the author and should not be interpreted as representing the official views or policies of theDepartment of Defense or the U.S. Government. [1] A. G. J. MacFarlane, J. P. Dowling, and G. J. Milburn, “Quantum technology: the second quantum revolution,”
Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences , vol.361, no. 1809, pp. 1655–1674, 2003. [Online]. Available: https://royalsocietypublishing.org/doi/abs/10.1098/rsta.2003.1227[2] J. A. Sedlacek, A. Schwettmann, H. K¨ubler, R. L¨ow, T. Pfau, and J. P. Shaffer, “Microwave electrometry with rydbergatoms in a vapour cell using bright atomic resonances,”
Nat. Phys. , vol. 8, pp. 819–824, November 2012.[3] C. Holloway, J. Gordon, S. Jefferts, A. Schwarzkopf, D. Anderson, S. Miller, N. Thaicharoen, and G. Raithel, “Broadbandrydberg atom-based electric-field probe for si-traceable, self-calibrated measurements,”
IEEE Transactions on Antennasand Propagation , vol. 62, no. 12, pp. 6169–6182, Dec 2014.[4] D. A. Anderson, S. A. Miller, G. Raithel, J. A. Gordon, M. L. Butler, and C. L. Holloway, “Optical measurements ofstrong microwave fields with rydberg atoms in a vapor cell,”
Phys. Rev. Applied , vol. 5, p. 034003, Mar 2016. [Online].Available: http://link.aps.org/doi/10.1103/PhysRevApplied.5.034003[5] D. A. Anderson, G. Raithel, T. Nithiwadee, S. A. Miller, and A. Schwarzkopf, “Atom-based electromagnetic radiationelectric-field and power sensor,” Patent US 9,970,973 B2, 5 15, 2018.[6] D. A. Anderson, G. Raithel, E. G. Paradis, and R. E. Sapiro, “Atom-based electromagnetic field sensing element andmeasurement system,” Patent US-2019-0 187 198-A1, 6 20, 2019.[7] D. A. Anderson, R. E. Sapiro, and G. Raithel, “A self-calibrating si-traceable broadband rydberg atom-basedradio-frequency electric field probe and measurement instrument,” arXiv:1910.07107 [physics.atom-ph] , May 2018, pp. 1–3. [Online]. Available: https://ieeexplore.ieee.org/document/8439437/[10] M. T. Simons, J. A. Gordon, and C. L. Holloway, “Fiber-coupled vapor cell for a portable rydberg atom-basedradio frequency electric field sensor,”
Appl. Opt. , vol. 57, no. 22, pp. 6456–6460, Aug 2018. [Online]. Available:http://ao.osa.org/abstract.cfm?URI=ao-57-22-6456[11] H. K¨ubler, J. Keaveney, C. Lui, J. Ramirez-Serrano, H. Amarloo, J. Erskine, G. Gillet, and J. P. Shaffer, “Atom-basedsensing of microwave electric fields using highly excited atoms: mechanisms affecting sensitivity,” vol. 10934, 2019.[Online]. Available: https://doi.org/10.1117/12.2515587[12] E. Paradis, G. Raithel, and D. A. Anderson, “Atomic measurements of high-intensity vhf-band radio-frequencyfields with a rydberg vapor-cell detector,”
Phys. Rev. A , vol. 100, p. 013420, Jul 2019. [Online]. Available:https://link.aps.org/doi/10.1103/PhysRevA.100.013420[13] D. A. Anderson and G. Raithel, “Continuous-frequency measurements of high-intensity microwave electric fieldswith atomic vapor cells,”
Applied Physics Letters , vol. 111, no. 5, p. 053504, 2017. [Online]. Available:http://dx.doi.org/10.1063/1.4996234[14] D. A. Anderson, E. G. Paradis, and G. Raithel, “A vapor-cell atomic sensor for radio-frequency field detection using apolarization-selective field enhancement resonator,”
Applied Physics Letters , vol. 113, no. 7, p. 073501, 2018. [Online].Available: https://doi.org/10.1063/1.5038550[15] C. L. Holloway, M. T. Simons, M. D. Kautz, A. H. Haddab, J. A. Gordon, and T. P. Crowley, “Aquantum-based power standard: Using rydberg atoms for a si-traceable radio-frequency power measurementtechnique in rectangular waveguides,”
Applied Physics Letters , vol. 113, no. 9, p. 094101, 2018. [Online]. Available:https://doi.org/10.1063/1.5045212[16] R. E. Sapiro, G. Raithel, and D. A. Anderson, “Atom-based optical rf-power/voltage transducer and sensor,” vol. 64,no. 4, 2019. [Online]. Available: http://meetings.aps.org/Meeting/DAMOP19/Session/L01.31[17] D. H. Meyer, K. C. Cox, F. K. Fatemi, and P. D. Kunz, “Digital communication with rydberg atoms andamplitude-modulated microwave fields,”
Applied Physics Letters , vol. 112, no. 21, p. 211108, 2018. [Online]. Available:https://doi.org/10.1063/1.5028357[18] A. B. Deb and N. Kjrgaard, “Radio-over-fiber using an optical antenna based on rydberg states of atoms,”
AppliedPhysics Letters arXiv:1808.08589 , Aug 2018. [Online]. Available: https://arxiv.org/abs/1808.08589[21] H. Bethe and E. Salpeter,
Quantum mechanics of one- and two-electron atoms . Springer, 1957. [Online]. Available:https://books.google.com/books?id=1ZUuAAAAIAAJ[22] T. F. Gallagher,
Rydberg Atoms . Cambridge University Press, Cambridge, 1994.[23] R. F. Stebbings and F. B. Dunning,
Rydberg States of Atoms and Molecules . Cambridge University Press, Cambridge,1983.[24] A. K. Mohapatra, T. R. Jackson, and C. S. Adams, “Coherent optical detection of highly excited rydberg statesusing electromagnetically induced transparency,”
Phys. Rev. Lett. , vol. 98, p. 113003, Mar 2007. [Online]. Available:http://link.aps.org/doi/10.1103/PhysRevLett.98.113003[25] M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,”
Rev. Mod. Phys. , vol. 77, pp. 633–673, Jul 2005. [Online]. Available: http://link.aps.org/doi/10.1103/RevModPhys.77.633[26] P. R. Berman and V. S. Malinovsky,
Principles of Laser Spectroscopy and Quantum Optics . Princeton, NJ, USA: PrincetonUniversity Press, 2011.[27] C. L. Holloway, M. T. Simons, A. H. Haddab, J. A. Gordon, and S. D. Voran, “A multiple-band rydberg-atombased receiver/antenna: Am/fm stereo reception,” arXiv:1903.00786 [physics.atom-ph] , Mar 2019. [Online]. Available:https://arxiv.org/abs/1903.00786[28] Y. O. Dudin and A. Kuzmich, “Strongly interacting rydberg excitations of a cold atomic gas,”
Science
Nature , vol. 488, pp. 57–60, August 2012.[30] D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones,and C. S. Adams, “Storage and control of optical photons using rydberg polaritons,”
Phys. Rev. Lett. , vol. 110, p. 103001,Mar 2013. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevLett.110.103001[31] M. Jing, Y. Hu, J. Ma, H. Zhang, L. Zhang, L. Xiao, and S. Jia, “Quantum superhet based on microwave-dressed rydbergatoms,” arXiv:1902.11063 [physics.atom-ph] , Feb 2019. [Online]. Available: https://arxiv.org/pdf/1902.11063.pdf[32] M. T. Simons, A. H. Haddab, J. A. Gordon, and C. L. Holloway, “A rydberg atom-based mixer: Measuring thephase of a radio frequency wave,”
Applied Physics Letters , vol. 114, no. 11, p. 114101, 2019. [Online]. Available:https://doi.org/10.1063/1.5088821[33] J. A. Gordon, M. T. Simons, A. H. Haddab, and C. L. Holloway, “Weak electric-field detection with sub-1 hz resolutionat radio frequencies using a rydberg atom-based mixer,”
AIP Advances , vol. 9, no. 4, p. 045030, 2019. [Online]. Available:https://doi.org/10.1063/1.5095633[34] J. Han, T. Vogt, C. Gross, D. Jaksch, M. Kiffner, and W. Li, “Coherent microwave-to-optical conversionvia six-wave mixing in rydberg atoms,”