Assessment of Rydberg Atoms for Wideband Electric Field Sensing

David H. Meyer,∗ Zachary A. Castillo,† Kevin C. Cox, and Paul D. KunzU.S. Army Research Laboratory, 2800 Powder Mill Rd, Adelphi MD 20783, USA

(Dated: January 13, 2020)

Rydberg atoms have attracted significant interest recently as electric field sensors. In orderto assess potential applications, detailed understanding of relevant figures of merit is necessary,particularly in relation to other, more mature, sensor technologies. Here we present a quantitativeanalysis of the Rydberg sensor’s sensitivity to oscillating electric fields with frequencies between1 kHz and 1 THz. Sensitivity is calculated using a combination of analytical and semi-classicalFloquet models. Using these models, optimal sensitivity at arbitrary field frequency is determined.We validate the numeric Floquet model via experimental Rydberg sensor measurements over arange of 1–20 GHz. Using analytical models, we compare with two prominent electric field sensortechnologies: electro-optic crystals and dipole antenna-coupled passive electronics.

I. INTRODUCTION

Vapors of alkali Rydberg atoms, i.e. where each atom’svalence electron is highly excited, have recently gainedattention as a promising candidate for electric field sen-sors thanks to some distinct characteristics. 1) Theyare identical quantum particles with known response di-rectly tied to fundamental constants. 2) They exhibit alarge polarizability and sensitivity over an ultra-wide fre-quency range. 3) They are small and broadly available.And 4) they are compatible with optical/laser technology.Explicit demonstrations include electric field sensitivitydown to less than 1 (µV/cm)/

√Hz [1] with record ab-

solute accuracy [2], detection of fields from 10 kHz [3]up to 1 THz [4], sub-wavelength imaging [5], commu-nication bandwidths of over 1 MHz [6, 7], and effectiveoperation in the extreme electrically small regime [8].These demonstrations provide validation for Rydberg-based sensors as a useful technology platform.

Unsurprisingly the technology space related to electricfield sensing is large and varied, given the wide spec-trum of frequencies and dynamic ranges that are of in-terest. More commercially mature technologies, includ-ing plasmonic sensors [9, 10], electro-optic crystals [11],and traditional electronic circuits coupled to antennashave found value in a broad array of marketable applica-tions. Other notable technologies, which can be consid-ered quantum sensors like the Rydberg sensor, include su-perconducting transition edge bolometers [12] that haveenabled cutting edge scientific results such as character-izing cosmic microwave background radiation, trappedions [13], and NV diamond color centers [14, 15]. Identi-fying applications where the Rydberg sensor can providea significant advantage over these technologies is an openquestion.

The benefits of sub-wavelength, resonant, non-destructive, precise measurements afforded by Rydberg

∗ Corresponding author: david.h.meyer3.civ@mail.mil† Also at: Department of Physics, University of Maryland, CollegePark MD 20742, USA

vapors have merited applications in calibration andmetrology of radio-frequency (RF) fields where currentstandards rely on manufactured off-resonant dipole an-tennas coupled to a diode rectifier [2]. The possibility ofRF communications has been investigated recently as an-other potential application for Rydberg sensors [6–8, 16–19]. However, no work so far has presented a quantitativeanalysis of the Rydberg sensor’s sensitivity over its widespectral range, particularly in comparison with existingelectric field sensing technology.

In this work we perform such an analysis by calculat-ing the Rydberg sensor’s field sensitivity across its oper-ational frequency spectrum and compare with sensors ofsimilar size (∼1 cm) based on electro-optic crystals andtraditional passive electronic elements. We begin in Sec-tion II with a discussion of the fundamental differencesin operation of these systems, followed in Section III byanalytic derivations for the sensitivity in the electrically-small, low frequency regime. The focus is fundamentalsensitivity limits of representative model systems whilehighlighting various distinctive characteristics. In Sec-tion IV we present a more generalized, numerical treat-ment for the Rydberg sensor in order to calculate thesensitivity for fields of arbitrary frequency up to 1 THz.We experimentally confirm our model’s calculations forfrequencies between 1–20 GHz.

II. BACKGROUND

For any electric field sensor, the measurement processcan be divided into three stages: 1) state preparation,including mode shaping of sensor to the incident fieldand/or sensor initialization, 2) field–sensor interaction,often parameterized using macroscopic susceptibility (χ)or microscopic polarizability (α), and 3) sensor readout.Each step impacts the various figures of merit and overallperformance of a given sensor. Each step also has a fun-damental limitation that depends on the type of sensor.

When comparing disparate technologies terminologycan present challenges. In particular, the notions ofbandwidth and sensitivity are often used inconsistentlyacross different communities.

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To be explicit about our terminology regarding band-width, “carrier spectral range” signifies the system’srange of operational carrier frequencies, while “instanta-neous bandwidth” signifies the maximum rate of changeof the carrier to which the system is sensitive. Ryd-berg atoms and electro-optic crystals have a large carrierspectral range, as discussed below, while dipole-coupledpassive electronics sensors are typically more restricteddue to challenges of impedance matching the dipole tothe readout load. In contrast, the instantaneous band-width for passive electronic sensors is often equal to thecarrier spectral range – resulting in little distinction be-tween the two for this technology. The instantaneousbandwidth for electro-optic and Rydberg sensors is typ-ically limited by the readout process. For electro-opticsthis corresponds to the bandwidth of the photodetector.For Rydberg sensors that rely on the electromagnetically-induced-transparency (EIT) probing method, the probephoton scattering rate of the intermediate atomic reso-nance (of order 10 MHz) is the limiting bandwidth [6, 20].

We determine a sensor’s sensitivity by deriving the fun-damental signal-to-noise ratio (SNR) of the measurementprocess, in standard deviations of the field amplitude.The sensitivity is then defined as the incident signal fieldamplitude (E) [21] that results in SNR = 1 as measuredin a one second integration time. If the SNR ∝ E and thenoise is white, this definition is equivalent to the standardfield sensitivity unit of (V/m)/

√Hz and can be straight-

forwardly scaled to other measurement times and fieldamplitudes. However, this is not universally true for theRydberg sensor, where SNR ∝ Eβ with β ranging be-tween 1 and 2 (as discussed in Section IV A), so we useour more general definition of sensitivity to avoid mis-interpretation. Note that often, especially outside of alaboratory context, environmental noise dominates theoverall noise profile of the measurement result. Here wechoose to set aside these external noise sources in or-der to characterize the basic sensor technologies them-selves. When evaluating the sensitivity requirement foran E-field sensor in a particular application, these exter-nal noise sources are important to consider.

Figure 1 illustrates the three primary electric field sen-sors discussed in this work: the Rydberg sensor and twocomparison sensors based on dipole-coupled passive elec-tronics or electro-optic crystals. This figure also diagramsa simple, conceptual model that governs each underlyingsensor: a two-level atom, an equivalent circuit, and aphasor diagram, respectively.

Atoms are best described in the language of quantummechanics, and Rydberg sensors can rightfully be consid-ered “quantum sensors”, particularly as they have per-formed at the standard quantum (shot noise) limit [8, 22].Their sensitivity to electric fields relies on large electricdipole moments and the corresponding energy shifts tothe atomic spectroscopy that are detected optically [23].Although not essential, Rydberg sensors to date havegenerally relied on the EIT method for state prepara-tion and sensor readout [24]. The EIT dark state, which

FIG. 1. Examples of electric field sensors: Rydberg Sen-sor) A dilute vapor of highly excited Rydberg atoms are per-turbed by an incident RF field. These perturbations shiftthe atomic energy levels and are detected using optical spec-troscopy. Passive Electronic Sensor) The incident RF fieldis coupled to a passive sensing load using a center-fed dipoleantenna. The strength of this coupling depends on the di-mensions of the antenna relative to the field wavelength andthe impedance matching to the load. Electro-Optic Sen-sor) The incident RF field induces a change in the refractiveindices of the crystal. A probing optical field is then used tomeasure this change, typically using a Mach-Zehnder inter-ferometer.

is a coherent quantum superposition of ground and Ry-dberg states, results in a narrow spectral resonance wellsuited for precision measurement. In a broader contextRydberg atoms have been used to create exotic quantumentangled states [25], and shown promise in the field ofquantum information science [26, 27]. Though quantumproperties are not the primary focus of the present work,it is worth highlighting that quantum sensors bring im-portant general features such as the ability to achievesub-shot noise level measurements.

Electro-optic (EO) crystal-based sensors, in whichchanges of the indices of refraction due to the presenceof electric-fields are detected with lasers, are similar inmany ways to Rydberg sensors. Both are dielectric andcan be made without any conductive material near thesensing volume. They are therefore transparent over awide range of electric-field frequencies and this enables

3

a non-destructive sensing interaction. Additionally, bothdevices work by transducing the RF information ontoan optical field, lending to highly effective interferomet-ric phase readout. The interaction strength between thefield and sensing element can be characterized by thematerial’s susceptibility, χ. One difference between EOcrystals and a Rydberg vapor is that crystals typicallyuse a second order χ(2) nonlinearity while Rydberg va-pors rely on a third order χ(3) due to the vapor’s spatiallycentrosymmetric nature nullifying its χ(2) response.

Traditional electronics represent the most common andhighly developed forms of electric field sensors due totheir long history, low cost, and familiar implementation.In this work we restrict our consideration to a center-feddipole antenna of length 1 cm, similar in size to the Ry-dberg sensing volume, with electronic readout using anideal rectifier and a load resistor. This system is readilymodeled as a voltage divider connected to an ideal volt-age source. While simple, this model reasonably char-acterizes the nominal performance of room-temperatureelectronic readout, which is fundamentally limited bythermal noise. It does not account for non-Foster cir-cuit elements that can allow for higher sensitivities [28].

We recognize that there is a wide array of electric fieldsensors that we do not consider in this work. Our motiva-tion is to provide a foundation for broader considerationof the application space for Rydberg sensors in the con-text of some common sensor platforms rather than anexhaustive comparison with the entire field.

III. SENSITIVITY COMPARISON IN LOWFREQUENCY REGIME

If we limit the frequency of the incident electric fieldto be much less than a device’s lowest natural reso-nance, one can obtain simple analytic solutions for thesensitivity of Rydberg, passive electronic, and EO sen-sor systems. In the context of antenna engineering,this is known as the electrically small regime where`/λ 1, with ` being the physical size of the sensor,and leads to fundamental effects such as the Wheeler-Chulimit[29, 30]. In this section we derive analytic formulasfor the sensitivity of a Rydberg sensor, a small dipoleelectronic sensor, and an electro-optic sensor in the lowfrequency regime.

A. Rydberg atoms

The common method for implementing a Rydberg elec-tric field sensor involves optical pumping to prepare theatoms into a sensitive Rydberg superposition state, in-teraction of that state with the incident electric field viaStark shifts, then optical readout of the collective phaseshift of the initial state; see simplified diagram in Figure1 and more detailed diagram in Figure 5. As describedin our previous work [8], the SNR of this process is ul-

timately limited by the phase resolution of the readoutstage due to the finite number of participating Rydbergatoms and the standard quantum limit. Here we outlinethat derivation.

While the Rydberg sensor analysis and qualitativetrends in this manuscript are transferable to other speciesof Rydberg sensor, the details and quantitative resultswill change depending on the specific atomic species used.We do not claim that our choice of species (rubidium) isinherently superior. Such a decision would depend on de-tails of the intended use. For example, desires for specificRF resonances, laser colors, or vapor density/operatingtemperature will influence the choice of species.

We begin by defining the SNR as ϕ/∆ϕ where ϕ = Ωτis the accumulated phase between two quantum states inan evolution time τ due to the atomic frequency shift Ω.The phase noise ∆ϕ is assumed to be at the standardquantum limit, i.e., ∆ϕSQL = 1/

√N , with N being the

number of atoms.The finite coherence time of our atomic sensor, Tc,

gives an effective measurement/evolution time, τ , thatdepends on whether the measurement time, t, is greateror less than Tc.

τ =

t t < Tc√

Tct t t > Tc

(1)

When t > Tc an optically-pumped superposition statewill, on average, collapse before readout and be re-pumped. This reset leads to a smaller observed phaseshift from an ensemble of atoms by a factor of

√Tc/t

[31]. The coherence time Tc is influenced by many ex-perimental details including transit broadening from thethermal motion of the atoms. In this work we assume aconservative Tc = 52 ns.

For electric field frequencies much lower than anyatomic resonance considered (i.e. which is ∼2 GHz con-sidering the n = 100 D state of Rb), the frequency shiftdue to the incident field can be estimated using the DCStark shift [32],

Ω = −1

2αE2 ≈ −1

2

(a0e

~

)2 ~n7

R∞E2 (2)

where a0 is the Bohr radius, ~ is the reduced Planck’sconstant, e is the charge of the election, R∞ is the Ryd-berg constant and n is the principal quantum number ofthe Rydberg state. The polarizability α of the Rydbergstate can be approximated as shown under the rotatingwave approximation. Finding the field, ERydberg, whichmakes the SNR equal to one yields

ERydberg = N−14

√2

ατ≈ N− 1

4

√2~R∞a2

0e2n7τ

(3)

We see that the strength of a Rydberg sensor, in termsof sensitivity, lies in the scaling of the polarizibility withprinciple quantum number, n, and the potential to use

4

FIG. 2. Minimum detectable field in a 1 second measurement versus RF frequency for 1 cm systems. a) Quasi-DC regime:The solid(dashed) red lines show the minimum field using a |100D5/2,mJ=1/2〉 target state with 103(104) Rb Rydberg atoms.The solid(dashed) black lines show the minimum field for a ` = 1 cm passive dipole electronic sensor, optimized for operationat 1 MHz with a resistive 2.1 MΩ (tuned inductor with 50 Ω) load. The green line shows the minimum field for a 1 cm ZnTeelectro-optic sensor with 150 µW of optical probe power. b) AC-regime: Each data point represents the minimum detectablefield for a Rb-based Rydberg sensor, allowing for optimal choice of n denoted by color. Square, circle, and star points representthe scaling of the SNR with E , β = 1, 2 or in between, respectively. The gray line shows the minimum field for a ` = 1 cmcenter-fed dipole antenna terminated with a 50 Ω load.

many identical atoms, N , which can be packed withinone electric field wavelength thanks to their small rela-tive size. However, because the SNR scales with E2, thesensitivity’s scaling with n and N is suppressed by theadditional square root. For example, to reduce ERydberg

by a factor of 10 would require a factor of 104 more atoms.

The accuracy of the approximation of α in Eq. 2 de-pends on how many nearby atomic resonances are takeninto account and the validity of the rotating wave approx-imation. For example, estimating the polarizability dueto a low frequency field considering only the next neareststate from |100D5/2〉 yields −45.4 GHz (cm/V)2, whereasaccounting for all nearby Rydberg states (calculated nu-merically [33]) yields −8.6 GHz (cm/V)2. The impact ofeach subsequent state diminishes as the respective de-tunings get larger, but in our particular case the secondnearest state plays a significant role since D states sitrather symmetrically between P and F states, meaningthat the second nearest state is almost equally detunedas the first. It happens that the second state contributesa counteracting shift, which reduces the effective electricfield sensitivity for the given target state (as in the ex-ample given). In Figure 2a we account for all states andplot the low frequency Rydberg sensor sensitivity usingthe numerically obtained polarizability with atom num-bers N = 103 and N = 104 shown as solid and dashed redtraces respectively. These numbers represent optimisticvalues for a Rydberg sensor using EIT readout wherevelocity selective probing significantly reduces the num-ber of participating atoms [34, 35]. High Rydberg atomdensities can also lead to complicating ion formation andRydberg-Rydberg interactions.

While the quadratic signal scaling is a disadvantagewhen sensing weak fields directly, it opens the possibilityof superposing a known strong field, Ebias E , to am-plify the effect of the weak field under test (i.e. hetero-dyning). Under this assumption the sensor’s SNR scaleslinearly with E . Assuming the uncertainty and noise ofEbias is less than the SQL, this technique improves theminimum detectable field to

ERyd-bias ≈1

ατ√NEbias

(4)

Using this method, a sensitivity better than1 (mV/m)/

√Hz has been recently observed using

n = 100 Rydberg atoms and a non-zero Ebias forωRF < 10 kHz [36].

B. Passive Electronics

As size constraints generally affect the performance ofany sensor, we consider a short dipole antenna that iscomparable in size to our Rydberg vapor sensing volumeand connected to passive readout electronics. For low fre-quencies this means that the antenna will be electricallysmall (i.e. λ `/10). Along with loop antennas, dipoleantennas form the majority of electrically small antennasin use today [37].

To determine this sensor’s fundamental SNR, we esti-mate the signal strength by modeling the short dipoleantenna as an ideal voltage source, with an intrinsicimpedance dependent on the geometry, coupled to aread out resistor through an ideal full-wave rectifier (see

5

Fig. 1). We assume the dominant noise source to be thesense resistor’s thermal rms Johnson noise,

√4kbTR∆f .

Here kB is Boltzmann’s constant, T = 300 K is roomtemperature, R is resistance, and ∆f = 1/t is the mea-surement bandwidth.

The magnitude of the voltage source signal is given bythe product of the electric field and the full length, `, ofthe dipole. The impedance of the antenna, Za, is predom-inantly capacitive and is given to good approximation as[28]:

Za ≈ i[Z0

π

(1− ln

(`

2a

))cot

(ωRF`

2c

)](5)

where a is the radius of the conductor, ωRF is the angularfrequency of the incident radiation, Z0 is the impedanceof free space, and c is the speed of light. The signalstrength will depend on the degree of impedance match-ing between the antenna and load resistor, Rl. The equiv-alent circuit model reduces to that of a simple voltagedivider, and the SNR of the measurement is

SNRDipole =E`√

4kbTRl∆f

Rl|Zl + Za|

(6)

where Zl is the lumped impedance of the load includingthe readout resistor and any matching network.

If no matching network is used (i.e. Zl = Rl) theSNR is maximized at a particular frequency by matchingRl = |Za|. The associated rms minimum detectable fieldin a one second measurement is

EDipole =

√8kBT |Za|

`2(7)

This result, with ` = 1 cm, a = 300 µm, and optimizedRl = 2.1 MΩ at 1 MHz, is shown in Figure 2a as the solidblack trace. While not flat across this portion of thespectrum, the sensitivity is significantly greater than theRydberg sensor. This is to be expected since the dipoleantenna, even in this regime, acts as a superior couplerof the incident field than the free-space atoms. Usingthe dipole coupler with the Rydberg sensor would leadto significantly higher sensitivity as well.

Enhanced sensitivity at a desired frequency can beaccomplished, at the cost of sacrificing carrier spectralrange and instantaneous bandwidth, by the addition ofan impedance matching network in the form of an induc-tor that cancels the capacitance of the antenna to create aresonant dipole with higher Q-factor. An example of thiswith Rl = 50 Ω is shown in Figure 2a as the dashed blacktrace. This line approaches the Chu-Wheeler limit for the1 cm electrically small antenna, and we indeed see highersensitivity, though only over a very small bandwidth.

In this model we have assumed an ideal, passive rec-tifier, since rectification is necessary in order to measurea non-zero rms voltage over the sense resistor. In prac-tice this is implemented using diodes. For small signalinputs (which are implicit when defining minimum de-tectable field) this means the circuit is driving a non-linear load with non-zero forward voltage drop [38, 39].

This presents a significant technical limitation to the re-alizable minimum field for passive electronic readout, onthe order of 1 (V/m)/

√Hz, that we have not included in

our model [40]. This issue can be avoided using activecomponents/measurement techniques such as transistorsor RF heterodyning. While we do not explicitly considerthese detection schemes, both are ultimately limited byJohnson noise and would therefore have similar idealizedperformance to the simple model presented here.

C. Electro-Optic Crystals

The Pockels effect in an electro-optic crystal is an es-tablished mechanism for detecting electric fields [11, 41].In a similar way to the Rydberg sensor, we define thesignal from a Pockels-based EO sensor to be the opticalphase shift on a probing field due to the RF field in thecrystal medium. Measurement of this phase is typicallydone using a Mach-Zehnder interferometer configuration.The noise limit in this case is determined by optical shotnoise.

Assuming proper polarization when entering a 43m or23 crystal, the relative phase shift on the probe light is

ϕ =2πL

λ0∆n =

4πL

λ0

n30rE

1 +√εr

(8)

where L is the length of the crystal (interaction length),λ0 is the wavelength of the probe in vacuum, ∆n is thedifference in the index of refraction for the ordinary andextraordinary axes of the material, n0 is the index ofrefraction of the ordinary axis, r is the EO coefficient, andεr is the dielectric constant of the bulk crystal. The factorof 1+

√εr accounts for the reduction of free-space E inside

the crystal due to its dielectric constant [42]. Variouschoices of crystals exist, as discussed in References [11,42]. For the sake of comparison, we have chosen ZnTewith r ≈ 4.0 pm/V, εr = 10.1, and n0 ≈ 2.8 at a probingwavelength λ0 = 633 nm.

The phase uncertainty due to photon shot noise is

∆ϕ = 1/2√N(t), where N(t) is the average number of

photons from the probe light expected in a measurementtime t. Again taking the minimum detectable E-field,EEOM, to be when SNR = ϕ/∆ϕ = 1, we find

EEOM =λ0(1 +

√εr)

8πn30rL

√N(t)

(9)

This result is shown as the green trace of Figure 2a,taking L = 1 cm and probe power Pprobe = 150 µW,

or N(1 s) = (Pprobet)/(hfProbe) = 4.8 × 1014 (where his Planck’s constant and fProbe is the frequency of theprobe light). Since the SNR is linearly proportional to E ,there is a more favorable scaling with photon number ascompared to the scaling with atom number in the Ryd-berg sensor. Because of this, the EO sensor performs at a

6

similar level to the Rydberg sensor despite comparativelyweak nonlinear susceptibility. While the sensitivity alsoscales favorably with the crystal length and probe power,it is not practical to arbitrarily increase both due to thechallenges of large crystal growth and achieving optimal,shot-noise limited Mach-Zehnder performance for ever in-creasing photon number. Demonstrated performance ofan EO sensor on the order of 1 (mV/m)/

√Hz has been

reported in the literature [43].Another point of comparison is the minimal perturba-

tion to the measured field from the dielectric EO crystal.Similar to the Rydberg sensor, the EO sensor head canbe made without conductors, enabling a relatively non-destructive measurement. The remaining perturbationto the field is due to the step in index of refraction at thecrystal surface, which can be significant for EO crystals[42]. Comparing with the Rydberg sensor, the Rydbergvapor presents a significantly smaller index change andcorrespondingly smaller perturbation. However, the glasscell containing the vapor often presents a significant in-dex change and must also be considered.

Finally, the sensitivity is relatively flat and indepen-dent of ωRF, which is convenient for sensor operation.Resonances that limit this flat response do arise, par-ticularly as the RF wavelength approaches the lengthscale `, and these depend strongly on the mechanical de-sign of the sensor. To reflect these considerations, wehave extended the low-frequency result into Fig. 2b upto ∼20 GHz, an operational range that commercial EOsensors readily achieve.

IV. WIDE SPECTRUM SENSITIVITY OF THERYDBERG SENSOR

In this section we extend the quantitative measure ofthe Rydberg sensor’s sensitivity to cover a wider car-rier spectral range. At frequencies >100 MHz, atoms ex-cited to a Rydberg state provide a structured spectrumof sensitivity to electric fields due to strong resonant andoff-resonant interactions with many dipole-allowed tran-sitions to nearby Rydberg states. As discussed above,these interactions produce Stark shifts with respect tothe target Rydberg state that can be detected via opticalspectroscopy. The scaling of this shift with the appliedelectric field amplitude, E , depends on the frequency ofthe radiation, ωRF, relative to the atomic resonances.Near resonance, the Stark shift takes the form of Autler-Townes splitting (a special case of the AC Stark effect)and is proportional to n2E . Off-resonance, the shift takesthe form of a general AC Stark shift and is proportionalto n7E2. Using both regimes, sensitivity to fields withfrequencies ranging from 10 kHz[3, 8] to 1 THz[4] havealready been demonstrated.

Here we develop a theoretical treatment based on semi-classical Floquet theory to estimate the minimum de-tectable field of the Rydberg sensor for arbitrary carrierfrequencies that is valid for sub-ionizing field strengths.

We also confirm the theoretical model via comparisonwith experimental data obtained using a commercially-available wideband antenna operating over 1–20 GHz forthree particular Rydberg states.

A. Modeling

If we first limit consideration to relatively weak fieldstrengths common in communication or remote sensingapplications and frequencies near atomic resonances, astandard textbook model of the AC Stark shift usingthe Rotating Wave Approximation (RWA) is valid. Thismodel assumes a two-level system with a strong couplingfield detuned much less than the transition frequency be-tween the two levels. Going to the rotating frame of theRF field and ignoring the counter-rotating term, the ACStark shifted energies of the two levels take the form of

ΩAC =1

2

(∆±

√∆2 + Ω2

)(10)

where ∆ is the detuning of the incident RF field fromresonance, Ω = d · E/~ is the resonant Rabi frequency ofthe RF field and d is the dipole moment of the transi-tion. Which shift corresponds to the lower energy statedepends on the sign of ∆: if ∆ > 0 corresponding to ablue detuning the minus sign is used, ∆ < 0 uses the posi-tive sign. At ∆ = 0, both roots have the same magnitude(Ω/2) resulting in the common-mode splitting known asAutler-Townes splitting.

The total Stark shift from multiple nearby levels isfound by summing together the contribution of eachtwo-level system calculated separately. Figure 3 showsthis model’s estimate of the absolute Stark shift of the|50D5/2,mJ=1/2〉 Rydberg state due to a E = 100 mV/mfield versus frequency in comparison with the more com-plete Floquet models developed below. Near atomic res-onances this simple model has good agreement. Furtherfrom resonances where the detuning is on order with thetransition frequency the counter-rotating term cannotbe ignored and the validity of the approximation breaksdown. While less accurate in these far-detuned regimes,this model is very fast to calculate numerically comparedwith the Floquet model and therefore can be useful forrough sensitivity estimates.

A more complete model is derived using semi-classicalFloquet theory, outlined in detail in Ref. [44]. Floquettheory is capable of modeling Stark shifts for arbitraryfield amplitude and frequency, and represents a morecomplete solution when determining the Rydberg sen-sor’s sensitivity [45]. Here we briefly outline the Starkshift calculation procedure using this theory.

We start with the time-dependent Schrodinger equa-tion

[H0 + V (t)] Ψ(t) = i~∂

∂tΨ(t) (11)

where perturbation potential V from the RF field is peri-odic in time such that V (t+ τ) = V (t) (ωRFτ = 2π) and

7

the bare atomic Hamiltonian H0 has eigenfunctions suchthat H0 |α〉 = E0

α |α〉, 〈β|α〉 = δαβ . The precise numeri-cal values for the energy levels (H0) and dipole moments(V ) are found using numerical integration as providedby the Alkali-Rydberg-Calculator (ARC) Python pack-age [46].

The Floquet theorem states that the periodic natureof the perturbation potential implies the solutions to theSchrodinger equation should also be periodic such thatΨ(t) = e−iεt/~Φ(t) where ε, known as the quasi-energies,is a diagonal matrix of unique, real numbers, εβ , up tointeger multiples of 2π/τ and Φ(t+ τ) = Φ(t) is a matrixof corresponding periodic functions. The time-periodicHamiltonian H(t) = H0+V (t) and Ψ(t) can be expandedinto the Floquet-state basis |αn〉 = |α〉 |n〉 where |n〉 areFourier vectors corresponding to harmonics of ωRF suchthat 〈t|n〉 = einωRFt.

〈α|Ψ(t) |β〉 =∞∑

n=−∞Φ

(n)αβ e

inωRFte−iεβt/~ (12)

〈α|H(t) |β〉 =∞∑

n=−∞H

[n]αβe

inωRFt (13)

Here H[n]αβ represents the nth-order component of the

Fourier expansion of the Hamiltonian (i.e. H [0] = H0

and H [±1] = V ).Solving for ε and Φ(t) then gives the energy shifts to

the bare atomic states and the transition probabilitiesbetween states. These solutions typically must be foundnumerically since the large dipole moments between themany nearby Rydberg states lead to significant, compet-ing interactions that all must be taken into account. Atypical basis for H0 includes Rydberg states with n± 10relative to the target state and orbital quantum number` ≤ 20 resulting in over 800 states. Moreover, multi-photon resonances are possible for relatively small ap-plied fields meaning the infinite sums of Eqs. 12 & 13can only be truncated to n ' 10, each order adding amultiple of two of the nominal atomic basis to the Flo-quet basis.

Semi-classical Floquet theory has already been appliedin the context of Rydberg electrometers for large fieldamplitudes [45, 47, 48], where the solve takes the form ofnumerically integrating the the time-evolution operatorU(t + τ, t). Here we are not interested in the transi-tion probabilities and can thus choose to use the simplerShirley’s time-independent Floquet Hamiltonian method[49]. This is done by substituting Eqs. 12 & 13 into Eq. 11to obtain an infinite dimension eigenvalue equation

∑

γm

〈αn|HF |γm〉Φ(m)γβ = εβΦ

(n)αβ (14)

where HF is a block tri-diagonal matrix with elements

〈αn|HF |βm〉 = H[n−m]αβ + n~ωRFδαβδnm (15)

FIG. 3. Comparison of Rydberg Models: a) The black lineis the full Floquet model calculated for |50D5/2,mJ=1/2〉 statewith E = 100 mV/m. The red line is the reduced Floquetmodel. The blue line is the perturbative model. b) The nor-malized residuals between the two approximate models andthe full Floquet theory are shown above in dB; note 0 dBrepresents an error of 100 %.

In this case, because we are focused on weak fields, wecan truncate HF to n = ±1 while also avoiding the in-tegration of the time-evolution operator. Furthermore,we can reduce the basis of HF to only include those Ry-dberg states that have direct dipole-allowed transitionswith the target state. This reduces the basis from ∼ 800to ∼ 40, significantly improving the speed of computa-tion. We will refer to this reduced basis solution as thereduced Floquet model.

In Figure 3 we choose an applied field of 100 mV/m anda single target state, |50D5/2,mJ=1/2〉, and show compar-isons of predicted Stark shifts for the full Floquet model(black trace), the reduced Floquet model (red trace),and the RWA model (blue trace). The magnitude ofthe normalized residuals between the two Floquet models(shown red in part b) is mostly less than −10 dB, exceptfor small regions around intermediate detunings wherethe differences are somewhat larger. For example, thisdiscrepancy is most visible for the set of features around6–9× 1010 Hz, where the shifts from nearby states con-spire to significantly suppress the response. The RWAmodel, where the Stark shift from each dipole-allowedtransition is added together to produce an average shiftof the target state, shows larger discrepancies with thefull Floquet model except on atomic resonances whereagreement is quite good. It may seem surprising thatthis model is as effective as it is (within a factor 2 formost of the frequency range), but this is due to the rela-tive weakness of the applied RF field, which reduces theinfluence of the far-detuned resonances that violate theRWA assumptions.

Each peak in Figure 3a is actually a pair of two

8

FIG. 4. Scaling of minimum detectable field (blue) and β(red) versus detuning from RF transition. The square, circle,and star symbols match those used in Fig. 2b and correspondto β = 1, 2 (±1%) or somewhere in between, respectively.

nearby resonances (the lowest couplet near 17 GHz, isvisibly resolved) since the D5/2 states sit nearly symmet-rically between the P3/2 and F7/2 states (illustrated inFig. 5b). Peaks at increasing RF frequency are coupletswith increasing ∆n, i.e. |50D5/2〉 → |(50±∆n)P3/2〉 &|(50∓∆n)F7/2〉.

The structure of the frequency response shown in Fig-ure 3 has important implications for a wideband sen-sor. While the sensor has some measurable response atall frequencies, the discrete resonances (each .10 MHzwide) provide amplified response at specific frequencies.This behavior is reminiscent of the harmonics of a dipoleantenna (as seen in Figure 2b and described below),however the Rydberg sensor resonances are not relatedby harmonics. The implications are similar: the Ryd-berg sensor can preferentially detect many RF frequen-cies spread across its carrier spectral range without mod-ification while effectively rejecting large portions wherethe atom response is significantly weaker. One impor-tant distinction is that the Rydberg atomic resonancesare absolutely well known, and each atom is identical(a quantum advantage). Another distinction is that theRydberg sensor signal depends primarily on the detuningof the RF field to the nearest resonance which does notconvey the RF frequency directly. This can make deter-mination of unknown RF frequencies more challengingand methods for addressing this will be the subject offuture work.

While Figure 3 shows the Stark shift of a single Ry-dberg target state over the full spectrum of consid-ered frequencies, there are many Rydberg states thatcan be taken advantage of by simply tuning the Ryd-berg laser. Restricting the target Rydberg state to be|nD5/2,mJ=1/2〉 the optimal target state for maximizingsensitivity to a given RF frequency is shown above, inFigure 2b, calculated using the reduced Floquet model.A comparison to |nP3/2,mJ=1/2〉 and |nS1/2,mJ=1/2〉 tar-get Rydberg states is located in the Supplemental Mate-rials.

To keep Figure 2b legible, we restricted the numberof data points to include all dipole-allowed resonances as

well as 300 more points distributed on a log scale pereach decade of frequency. Each point is calculated byfirst comparing the absolute Stark shift for a fixed E toidentify the optimal n for each frequency. We then usenumerical optimization to find the SNR = 1 point forthat optimal state.

Numerical optimization is necessary because the scal-ing of the minimum detectable field is not known a-prioriat every frequency, rather it depends on the detuningfrom nearby atomic resonances. As an example, Fig-ure 4 shows how the sensitivity and the scaling of theSNR, β (as in SNR ∝ Eβ), vary with detuning from the|50D5/2〉 → |51P3/2〉 transition. The precise width andtransition point of the β transition from 1 to 2 dependson the strength of the applied field, however the generalshape is consistent for any Rydberg resonance. The cor-responding value of β for each point in Figure 2b is notedby the shape of the point: a square for β = 1, a circlefor β = 2, and a star for an intermediate value. Knowingthe value of β allows one to use the results for any sensorin Figure 2 to determine the SNR for any E in a 1 secondmeasurement. For example, in a region where β = 1, ifE1 s = 1 µV/m then the expected SNR in standard devi-ation for a 1 second measurement of a 100 µV/m field of

the same frequency is (E/E1 s)β

= 100. Scaling the mini-mum detectable field to other measurement bandwidthsrequires understanding of how the SNR scales with t.With the exception of the Rydberg sensor for t < Tc,this scaling takes the form of Et = E1 st

− 1/2β, assumingwhite noise sources. If E1 s = 10 µV/m and β = 2, theminimum detectable field in a t = 1 ms measurement is56 µV/m.

Figure 2b reveals a few basic patterns. First, the gen-eral rule for picking a target Rydberg level to use for agiven field frequency is to choose the n which allows thenearest to resonance. If there are multiple equally closeresonances, use the n with the smallest ∆n to the tar-get level. Second, it is interesting to observe that thereare two clusters of resonant transitions, one with smallerdetectable field values and one with larger. The moresensitive set of transitions are made up of couplings to|(n+ ∆n)P3/2〉 and |(n−∆n)F7/2〉. The couplings withopposite sign are significantly weaker due to mismatch inthe overlap of the radial wavefunctions with that of thetarget |nD5/2〉 state, rendering them less sensitive i.e.larger minimum detectable field values. For more detailssee the Supplemental Materials.

The choice to cut off consideration of Rydberg levelsgreater than n = 100 is somewhat arbitrary, though suchRydberg levels have recently been used for low-frequencyE-field measurements [36]. There are complicating fac-tors not included in the model that cause concern as theRydberg levels increase. These include the challenge ofgetting good EIT contrast and SNR, the need for morelaser power to couple the same number of atoms, and var-ious atomic interactions, particularly Rydberg-Rydberginteractions.

For reference, current state of the art Rydberg sensor

9

FIG. 5. Experimental setup: a) Optical and electronic configuration for the homodyne/heterodyne measurement. Experimentalcontrol and analysis is done using the labscript suite [50]. b) Rubidium level diagram showing Rydberg excitation path andRF coupling to manifold of nearby Rydberg states. c) Homodyne/Heterodyne frequency spectrum. The solid red lines showthe probe spectrum with EOM modulation applied. The solid orange line shows the frequency of the local oscillator (LO)reference. The laser frequency is set such that the LO and lower sideband of the probe are resonant with the D2 probingtransition. Detuning the laser (dashed lines) moves both spectra in unison.

performance is ∼100 (µV/m)/√

Hz using n = 50 [1, 6].Ideally this value would be near the bottom line of pointsin Fig. 2b, but low quantum efficiency of detection, re-sulting in a reduced effective Rydberg atom number, haslimited the minimum detectable field to date [8].

Electrically Large Dipole Sensor

As a point of reference, the gray line of Figure 2b showsthe sensitivity of the same ` = 1 cm center-fed dipolefrom Section III B, but in an electrically-large regime(`/λ & 1) with a 50 Ω load sense resistor. This responseis calculated using the induced-emf method to determinethe intrinsic impedance and directivity gain of the dipoleantenna as a function of frequency [37, 51]. The sensitiv-ity is then found using the same equivalent circuit modelas the low-frequency passive electronic sensor, but withthe length ` of Eq. 5 replaced with an effective length,

`eff =

√Raλ2G(λ, `)

πηZ0(16)

where G(λ, `) is the antenna gain and η is the radiationefficiency. The resulting minimum detectable field is thengiven as

EDipole =

√4kbTRl∆f

K=|Za + Zl|

√πηZ0

√4kbTRl∆f

Rl√G(λ, `)Raλ2

(17)where K is commonly known as the antenna factor. Thehalf-wave frequency is ∼15 GHz and corresponds to thefirst high sensitivity dip. Higher frequency dips corre-spond to the ` = (m + 1/2)λ frequencies. The low sen-sitivity peaks correspond to frequencies of integer wave-length multiples (` = mλ) where the center-feed pointof the wire then sits at a node of the field, or in other

words, becomes a high impedance point. We again notethat this calculation is an idealized model. In practice itis difficult to accurately back out the absolute incidentfield strength due to the importance of parasitic elementsat these frequencies.

Comparing with the Rydberg sensor, we observe thatthe passive electronics sensor has lower minimum de-tectable field at its optimal half-wave frequencies. How-ever, coverage of the entire spectrum at this level mayrequire difficult design and optimization of both the an-tenna and sensing system at each frequency. In contrast,the Rydberg system can be tuned to any of its optimalsensitivity points by simply tuning a laser frequency.

B. Experiment

We confirm the validity of our theoretical model byexperimentally measuring the sensor response over a fre-quency range of 1–20 GHz using three different Rydbergtarget levels: |50D5/2〉, |60D5/2〉, and |70D5/2〉. Thefrequency range of this measurement was dictated bythe microwave source system; specifically the operationalrange of the widest-band transmission antenna readilyavailable [52].

The experimental setup and level diagram are shownin Figure 5a-b, and largely follow the standard Ryd-berg electrometer configuration: linearly polarized probeand coupling beams counter-propagate in a rubidium va-por cell establishing ladder Electromagnetically-InducedTransparency (EIT) spectroscopy of the Rydberg level,which is shifted by the presence of an RF field (for de-tails see the Supplemental Materials). The transmit-ted probe light is measured using an optical homodynemethod similar to those in Refs. [3, 53] which allows forprecise, photon-shot-noise limited measurements in boththe phase and amplitude quadratures. Our implemen-

10

FIG. 6. Atomic response versus RF frequency for |nD5/2〉target states. Ranging from top to bottom is n = 50, 60, 70.The RF set power is 16 dBm, 12 dBm and 3 dBm respectively.The black lines show the expected level shifts from the Flo-quet theory. The vertical dashed lines indicate where whichexample sweep traces are shown to the right of each contourplot. The black dashed line shows the atomic response at farRF detuning, the red trace near the lowest couplet of reso-nances, and blue between the couplet resonances.

tation follows a modification used in Ref. [54]. A singlelaser sent through acousto-optic modulators (AOMs) cre-ates both the probe and reference beams; a subsequentelectro-optic modulator (EOM) places sidebands on theprobe beam such that the lower sideband is equal in fre-quency to the reference. An optical heterodyne signalbetween the carrier and the reference is used to activelystabilize the relative beam path phase, see Fig. 5c. Thismethod allows us to easily change measurements betweenamplitude and phase quadratures without the need for asecond reference laser.

Figure 6 shows a contour plot of experimental data for1–20 GHz fields and target Rydberg levels n = 50, 60, 70,where the spectrum amplitudes are normalized to thebare EIT peak (i.e. no RF present). In order to maintainsimilar-order Stark shifts for each level, the RF power wasdecreased with increasing n (Pset = 16, 9, 3 dBm, respec-tively). To the right of each contour plot we show threeslices for probe sweeps at RF frequencies that are far fromresonance (black), near the lower resonant doublet (red),and inside the nearest resonant doublet (blue). One no-tices the red and cyan trace peaks are broadened relativeto the far-detuned black traces, due to the influence of thevarious mJ = 1/2, 3/2, 5/2 transitions. As the appliedfield strength is increased, these sublevels become re-

FIG. 7. Atomic response versus RF frequency in the ACStark regime. The black, red, and blue data correspond ton = 50, 60, 70 |nD5/2〉 target states, respectively. The linesrepresent the corresponding Floquet model predictions. Thecolored regions show the corresponding error in model esti-mate from field calibration error while the error bars showthe corresponding error in experimental peak extraction.

solved. The n = 60 blue trace reveals some of this behav-ior as the nearby |60D3/2〉 → |61P1/2〉, which has slightlyhigher resonance frequency, experiences Stark shifts thatoverlap with those of the |60D5/2〉 target state.

For direct comparison with Floquet theory, we cali-brate the applied RF electric field amplitude througha resonant Autler-Townes splitting measurement of the|50D5/2〉 → |51P3/2〉 transition at 17.0415 GHz [2, 24].We then use the manufacturer-specified antenna gainprofile and measured cable losses to extrapolate over themeasured range, 1–20 GHz. The overlaid solid black linesshow the Floquet-predicted shifts as a function of RF fre-quency, which shows good agreement with the measure-ments.

We performed narrower probe sweeps with a fixed RFset power of Pset = 16 dBm for each level in order to makea more detailed comparison with theory. In Figure 7 weshow the extracted Stark shift of the EIT peak relativeto no applied field for n = 50, 60, 70 as black circles, redsquares, and blue diamonds, respectively. The error barsrepresent the sweep-to-sweep jitter of the measured EITresonance. As expected, higher n leads to larger Starkshifts for the same applied field, and as the frequency ap-proaches a Rydberg resonance the Stark shift increases.The solid lines represent the Floquet predictions, and theshaded regions correspond to ±3 dB changes in the ap-plied RF field power, which accounts for fluctuations ofenvironmental reflections/scatter, horn calibration error,and RF etalons within the vapor cell [35]. The difficultyin calibrating the wideband horn antenna versus the ab-solute atomic measurement uncertainty is demonstratedin this figure: while the accuracy of the Floquet predic-tions is limited by the horn calibration errors, the atomicmeasurements are significantly more accurate by natureand could be used to improve the calibration. Note thatthe EIT resonance has a linewidth of Γ ∼ 5 MHz andStark shifts less than this width (Γ/100 or 50 kHz) are

11

difficult to resolve accurately; this is particularly rele-vant for the n = 50 data. Similar to the heterodyning inthe low-frequency regime mentioned earlier, the additionof a biasing RF field can be helpful in addressing thischallenge [16, 55].

These results reinforce confidence in the Floquet modelas an effective predictor of Stark shifts due to arbitraryRF frequencies and amplitudes of interest. This allowsus to not only determine optimal target Rydberg statesfor a given frequency and field, but also could enable theidentification of unknown frequency fields by comparingStark shifts on multiple target states.

V. CONCLUSION

With the current interest in Rydberg-based electricfield sensors, there have been numerous creative pro-posals identifying potential application spaces. Rydbergsensors’ wide spectral coverage and sensitivity have beentouted as strengths, and are important figures of meritfor many applications. We have presented multiple the-oretical models of varying accuracy and computationalcomplexity that predict the Rydberg sensor’s spectralsensitivity over a wide range of field frequencies and am-plitudes. We validated these models experimentally us-ing a simultaneous homodyne/heterodyne measurement

technique for three Rydberg levels over a frequency rangeof 1–20 GHz.

In this work we have also compared the Rydberg sensorto prominent, established electric field sensors; namelyelectro-optic crystals and dipole-coupled passive electron-ics. We used relatively simple models and assumed fun-damental noise sources in order to be as general andbroadly applicable as possible. We find the Rydberg sen-sor to be competitive with these technologies and havehighlighted some of their unique aspects. In particular,being atomic sensors, they hold special appeal as calibra-tion tools since they can be linked directly to fundamen-tal constants and well calculable models. They also onlyvery weakly perturb the measured field which addition-ally lends to their capabilities as precision sensors. As arelatively new technology, active research is steadily im-proving their sensitivity and performance with respect toother metrics of interest. While the exact, high-impactapplication has yet to be conclusively identified for theRydberg sensor, this work should aid in identifying it.

ACKNOWLEDGMENTS

We thank Fredrik Fatemi and Yuan-Yu Jau for usefuldiscussions. This work was partially supported by theDefense Advanced Research Projects Agency (DARPA).

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1

Supplemental Materials for

Assessment of Rydberg Atoms for Wideband Electric Field Sensing

I. OTHER TARGET RYDBERG STATES

As mentioned in Section IV of the main text, the|nD5/2〉 series of Rydberg states are not the only targetstates that can be used for the Rydberg sensor. Usingthe same EIT excitation/readout scheme the |nS1/2〉 se-ries of states are also accessible via appropriate selectionof laser detuning. Due to selection rules, direct opti-cal coupling to the P states requires either single photonexcitation [1] or three photon excitation [2]. While theoverall strength of the optical coupling to the D-series ofRydberg states is the strongest, leading to the highestSNR signals, these other series provide RF resonances atdifferent frequencies. Incorporating these series into theRydberg sensor therefore provides a greater range of cov-erage for the highly sensitive, dipole-allowed transitionsthan the subset shown in Figure 2b.

In Figure S1 we reproduce the Rydberg sensor min-imum detectable field using the |nD5/2,mJ=1/2〉 seriesof Rydberg target states along with the correspondingminimum fields for |nP3/2,mJ=1/2〉 (middle panel) and|nS1/2,mJ=1/2〉 (lower panel) target states, using the samemethod described in Section IV A of the main text. Theshape and color of each point follows the same conven-tion as that of Figure 2b: the color represents the n thatproduces the lowest minimum field at that frequency andthe shape denotes the SNR scaling of that point with E .

Added to each figure are regional colorings that high-light the different types of RF couplings to the targetRydberg state. Circles in the pink regions representoff-resonant, AC Stark couplings. These points indicatethere are no nearby dipole-allowed Rydberg transitionsfor that particular series of target states. Squares in thegreen regions indicate the RF frequency is very near oron resonance with a dipole-allowed transition. This typeof resonant coupling is very strong and leads to the lowestminimum detectable fields for a particular series. Bothof these regions are discussed in detail in Section IV A ofthe main text. The square points in the purple regionsrepresent dipole-allowed transitions where the couplingis suppressed by ∼ 2.5 orders of magnitude. For the Dseries target states, these points correspond to couplingswith |(n−∆n)P3/2〉 and |(n+ ∆n)F7/2〉. For the P se-ries target states, these points correspond to couplingswith |(n+ ∆n)D5/2〉. The mechanism that describes thissuppressed coupling is described in the next appendix.

In analyzing these figures, one must first recall thatsymmetry in the coupling of the Rydberg levels leads tothe same transitions on multiple plots. For example, allof the resonant S→P transitions shown in the bottompanel are present in the middle panel as P→S transitionswith the same magnitude. This property helps in iden-tifying which couplings lead to which lines of resonantminimum field points. Again considering the bottom two

FIG. S1. Minimum detectable field in 1 second using othertarget rubidium Rydberg series. The top panel is the samedata for the |nD5/2,mJ=1/2〉 from Figure 2b. The mid-dle and lower panels show the predicted minimum field for|nP3/2,mJ=1/2〉 and |nS1/2,mJ=1/2〉. The green, purple, andpink regions represent different types of couplings as discussedin the text.

panels, the S→P transitions of the bottom panel are vis-ible in the middle panel as the second series of pointsin the primary resonance line, slightly offset in mini-mum field and minimum frequency. The other pointsof that line are then due to P→D couplings, which arealso present in the top panel.

In general, the minimum detectable field for each tar-get series has similar trends and absolute values, thoughthe exact location of resonances vary from series to se-ries. There are two important difference to note. 1)The S↔P transitions generally have smaller dipole mo-ments than the other series and are thus less sensi-tive by a near-unity factor (the minimum field for the|100P3/2〉 → |101S1/2〉 transition is ×1.3 larger than thatof |100D5/2〉 → |101P3/2〉). 2) The P and S series havesmaller minimum detectable field than the D series inthe far-detuned, low frequency regime by factors of 3and 1.5, respectively. This is because the next nearesttransition that would contribute a Stark shift of oppositesign is much further from the lowest n = 100 resonance

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for these series.

II. PREDICTING RESONANT RYDBERGSENSITIVITIES

The strength of the coupling of an RF field to any Ryd-berg transition is ultimately related to the dipole momentof that transition.

d = 〈nLJ,mJ| er |n′L′J′,m′

J〉 (S1)

This matrix element represents a measure of the amountof overlap of the electron wavefunctions of the two statesand can be reduced, using the Wigner-Eckart theorem,to radial and angular terms. The angular terms for alltransitions considered in this work are of order 0.5. Sig-nificant differences in the dipole moments, observed inthe calculations of Section IV of the main text and Sec-tion I of this supplement are due to overlap of the ra-dial wavefunctions. Predicting which transitions will besuppressed (or enhanced) requires some detailed under-standing of the radial wavefunction for a Rydberg state.

The radial wavefunction for a Hydrogenic atom can befound by solving the radial portion of the Schrodingerequation [3]

∂2ρ(r)

∂r2+

[2

r− 1

n∗2− l(l + 1)

r2

]ρ(r) = 0 (S2)

where n∗ = n − δnlj is the effective n, δnlj is thequantum defect, and the wavefunction Ψnlm(r, θ, φ) =Ylm(θ, φ)ρ(r)/r.

For a Rydberg state with large n the expectation valueof the electron radial position is 〈r〉 = (3n∗2− l(l+1))/2,and gives the portion of the wavefunction with strongestcontribution to the wavefunction overlap between twostates. The approximate solution of Equation S2 nearthis point is an Airy function of the first kind,

ρ(〈r〉+ δr) ∝ Ai(−n∗4/3

(n∗2 − δr2 − l(l + 1)

))(S3)

which has an oscillatory nature with a large final peakbefore trending to 0 for r →∞. The location of this finalpeak roughly corresponds to 〈r〉 and the dipole momentfor these Rydberg transitions strongly depends on theoverlap of the final peaks for the wavefunctions of thetwo states of a transition. To lowest order, the shift ofthis peak from an |nLJ〉 target state is

∆r ≈ 3n−1/3(∆n−∆δ) (S4)

where ∆n is the difference in principle quantum numbersand ∆δ is the difference in quantum defects between thetarget state and the coupled state. If ∆r falls withinthe full-width half-max (FWHM) of the final peak of thesquared Airy function (∼ 1.6), a large dipole moment forthe transition results.

For the S↔P, P↔D, and D↔F transitions consideredin this work, ∆δ ≈±0.48, ±1.30 and ±1.33 respectively,

with the sign chosen opposite that of ∆l relative to thetarget state. Therefore, ∆r is minimized and the dipolemoment increased if the signs of ∆n and ∆l for the tran-sition are opposite. This trend is clear in the differencebetween resonant sensitivities in the green and purple re-gions of Figure S1, where transitions to states with thesame magnitude ∆n, ∆l can differ by nearly three ordersof magnitude in sensitivity depending on relative signs.

For the S↔P transitions this trend is less clear becausethe absolute value of ∆δ is small enough that ∆r for∆n, ∆l of the same sign is closer to the FWHM value.However, these transitions are still an order of magnitudeweaker than when ∆n, ∆l have opposite sign.

This analysis can also explain why the S↔P transitionsare generally weaker than the other two series. Limitingourselves to the most prominent ∆n = −1 transitions,|∆r| ∝ 0.52 for S→P but only 0.3 for P→D and D→F.The increased overlap leads to the larger dipole momentsand corresponding lower resonant mimimum detectablefields.

Finally, this analysis highlights how details of theatomic species used (the quantum defects in this case)can significantly alter the realized sensitivity of the Ry-dberg sensor. As an example, consider another commonspecies used in Rydberg electrometery: caesium. Thevalues of ∆δ for the S↔P, P↔D, and D↔F transitionsare approximately ±0.46, ±1.12 and ±2.44 respectively.Comparing with the rubidium values described above,we can expect a caesium Rydberg sensor to have similartrends due to wavefunction overlap for the S↔P transi-tions, slightly improved overlap for the P↔D transitions,and somewhat degraded overlap for the D↔F transitionswith the strongest resonance corresponding to one of the|∆n| = 2 states.

III. EXPERIMENTAL METHODS

The experimental configuration is shown in Figure 5of the main text, and it largely follows the standard Ry-dberg electrometer configuration found in the literature.The probing light is near resonance with the 85Rb D2transition at 780.24 nm and has a power of 13 µW in a1/e2 beam diameter of 410 µm. Its frequency is controlledvia a Direct-Digital Synthesis (DDS) tunable beat-notelock to a master laser that is frequency stabilized viasaturated absorption spectroscopy. The ∼480 nm Ryd-berg coupling light, with a 1/e2 beam diameter of 380 µmand typical power of ∼500 mW, is frequency stabilized toan ultra-low expansion (ULE) reference cavity. The RFfields are applied using a waveguide horn antenna, withpolarization parallel to that of the light. The atomicresponse is measured by sweeping the probe light fre-quency through resonance sufficiently slowly (typically67 MHz/s) to ensure a steady-state probing regime. Ex-perimental timing and control is performed using thelabscript suite [4].

The transmitted probe light is measured using an opti-

3

cal homodyne method similar to that used in Refs. [5, 6]which allows for precise, photon-shot-noise limited mea-surements in both the phase and amplitude quadratures.Our implementation follows that of Ref. [7] where we si-multaneously measure an optical heterodyne with a side-band of the probe to stabilize the relative beam pathphase. This method has the advantage of an easy changebetween measurement quadratures without the need fora second reference laser. In this work, all data is takenin the amplitude quadrature.

In our implementation the 780 nm laser light is initiallyseparated into probe and local oscillator (LO) paths, andfrequency shifted up and down 78.6 MHz, respectively,by separate acousto-optic modulators (AOMs). TheseAOMs are also used to stabilize the power in the opticalbeams. An electro-optic phase modulator (EOM) thenimparts 157.2 MHz sidebands on the probe beam, suchthat the lower sideband is at the same frequency as theLO beam, facilitating homodyne detection, see Fig. 5cof the main text. The relative beam path phase is stabi-lized using the simultaneous 157.2 MHz heterodyne signalbetween LO and probe carrier frequency. Thus, the bal-anced photodetector output has two signals of interest, aDC-coupled homodyne signal carrying the spectroscopicinformation, and a heterodyne beat-note at 157.2 MHz,which provides the correction signal that is fed back to

the EOM for path stabilization. Note that the drivepower of the EOM is relatively low, meaning most of theprobe power is in the carrier. This minimizes the influ-ence of the upper sideband and affords precise control ofthe probe power in the lower sideband while maintaininghigher carrier optical power for the heterodyne signal.

This system requires some care in the details of its con-figuration. First, the probe carrier and upper sidebandshould be blue detuned from atomic resonance to avoidtheir interaction with the EIT signal of interest. Second,the path lengths should be passively balanced and theRF signals to the AOMs, EOM, and heterodyne mixer’sLO should be phase coherent in order to reduce the im-pact of the RF source’s phase noise. We obtained phasecoherence by deriving all RF signals from the same DDSsynthesizer (that was externally clocked to a low-noise100 MHz reference oscillator, which was in turn stabilizedto a rubidium reference), and matching the cable delaysfor the AOM and EOM drives. Due to the slow acous-tic velocity in the AOMs (4 mm/µs), particular attentionmust be taken to ensure the probe and reference beamsare the same distance from the transducer. The opticalpath length between the LO and probe should also bebalanced to limit the influence of phase noise from theprobe laser itself.

[1] P. Thoumany, T. Hansch, G. Stania, L. Urbonas, andTh Becker, “Optical spectroscopy of rubidium Rydbergatoms with a 297 nm frequency-doubled dye laser,” OpticsLetters 34, 1621–1623 (2009).

[2] Christopher Carr, Monsit Tanasittikosol, ArmenSargsyan, David Sarkisyan, Charles S. Adams, andKevin J. Weatherill, “Three-photon electromagneticallyinduced transparency using Rydberg states,” OpticsLetters 37, 3858 (2012).

[3] Thomas Gallagher, Rydberg Atoms, 1st ed., CambridgeMonographs on Atomic, Molecular and Chemical PhysicsNo. 3 (Cambridge University Press, Cambridge, 2005).

[4] P. T. Starkey, C. J. Billington, S. P. Johnstone,M. Jasperse, K. Helmerson, L. D. Turner, and R. P.Anderson, “A scripted control system for autonomoushardware-timed experiments,” Review of Scientific Instru-

ments 84, 085111 (2013).[5] Ashok K. Mohapatra, Mark G. Bason, Bjorn Butscher,

Kevin J. Weatherill, and Charles S. Adams, “A giantelectro-optic effect using polarizable dark states,” NaturePhysics 4, 890–894 (2008).

[6] Santosh Kumar, Haoquan Fan, Harald Kubler, JitengSheng, and James P. Shaffer, “Atom-Based Sensing ofWeak Radio Frequency Electric Fields Using HomodyneReadout,” Scientific Reports 7, 42981 (2017).

[7] K. C. Cox, J. M. Weiner, G. P. Greve, and J. K. Thomp-son, “Generating entanglement between atomic spins withlow-noise probing of an optical cavity,” in 2015 Joint Con-ference of the IEEE International Frequency Control Sym-posium the European Frequency and Time Forum (2015)pp. 351–356.