Optical measurements of strong microwave fields with Rydberg atoms in a vapor cell

2.1 MB·

by  David A. Anderson

Optical measurements of strong microwave fields with Rydberg atoms in a vapor cell
D. A. Anderson, S. A. Miller, and G. Raithel
Department of Physics, University of Michigan, Ann Arbor, MI 48109
J. A. Gordon, M. L. Butler, and C. L. Holloway
National Institute for Standards and Technology,
U.S. Department of Commerce, Boulder, CO 80305
(Dated: January 12, 2016)
WepresentaspectralanalysisofRydbergatomsinstrongmicrowavefieldsusingelectromagnet-
ically induced transparency (EIT) as an all-optical readout. The measured spectroscopic response
6 enables optical, atom-based electric field measurements of high-power microwaves. In our experi-
1 ments,microwavesareirradiatedintoaroom-temperaturerubidiumvaporcell. Themicrowavesare
0 tuned near the two-photon 65D−66D Rydberg transition and reach an electric field strength of
2 230V/m,about20%ofthemicrowaveionizationthresholdoftheseatoms. AFloquettreatmentis
usedtomodeltheRydberglevelenergiesandtheirexcitationrates. Wearriveatanempiricalmodel
n
a forthefield-strengthdistributioninsidethespectroscopiccellthatyieldsexcellentoverallagreement
J between the measuredandcalculatedRydbergEIT-Floquet spectra. Usingspectralfeaturesinthe
Floquet maps we achieve an absolute strong-field measurement precision of 6%.
1
1

] I. INTRODUCTION a new approach is necessary to accurately describe the
h atomic system. Extending the atom-based measurement
p
approachtoahighpowerregimecouldenable,forexam-
- Significant progress has been made in recent years to-
m wards establishing atomic measurement standards for ple, sub-wavelength characterization of antennas radiat-
inghigh-powermicrowavesamongotherapplications(for
o field quantities [1–3]. Rydberg atoms hold particu-
anoverviewofhigh-powerRFtechnologiesseeRef.[16]).
t lar appeal for applications in electrometry due to their
a In this work, we study the response of a Rydberg EIT
large transition electric dipole moments, which lead to a
.
s strong atomic response to electric (E) fields [4, 5]. Ryd- systemtostrongmicrowavefieldsuptoabout20%ofthe
c microwaveionizationlimitofthechosenatomicRydberg
i berg electromagnetically induced transparency (EIT) in
s states [4, 17]. In this strong-field regime, the atomic re-
atomic vapors [6] has recently been demonstrated as a
y
sponsebecomeshighlynonlinear,andanon-perturbative
h practical approach to absolute measurements of radio-
Floquet treatment must be employed for an accurate
p frequency (RF) electric (E) fields over a broad frequency
model of the Rydberg level shifts and excitation rates.
[ range (1 to 500 GHz) suitable for the development of
The Floquet model allows us to obtain quantitative in-
calibration-free broadband RF sensors [7]. The utility
1 formation on the E-field of the high-power RF radiation
of the Rydberg EIT technique in characterizing RF E-
v
source of interest.
5 fields has been demonstrated in a number of applica-
This paper is organized as follows. We begin in Sec-
3 tions. These include microwave polarization measure-
5 ments [8], millimeter-wave (mm-wave) sensing [9], and tion II with a description of our experimental setup. To
2 sub-wavelength imaging [10, 11]. The approach has also illustratethecontrastbetweentheweak-fieldandstrong-
0 fieldregimesinaRydbergEITexperiment,inSectionIII
been employed in room-temperature studies of multi-
1. photon transitions in Rydberg atoms [12], as well as in we compare Rydberg EIT spectra for weak mm-waves,
0 measurements of static E-fields [13] for precise determi- tuned to the one-photon 26D5/2−27P3/2 Rydberg tran-
6 sition, with spectra for strong microwaves, tuned near
nations of quantum defects [14].
1 the two-photon 65D−66D Rydberg transition. In Sec-
Todate,theRydbergEITmeasurementtechniquehas
v: beenemployedinmeasurementsofweakRFfields. Inthe tion IV we outline the Floquet theory required for an
i accuratemodeloftheatomicresponseinthestrong-field
X weak-field regime, the atom-field interaction strength is
regime. The theory is then employed in Section V to
small compared to the Rydberg energy-level structure,
r extract quantitative information on the microwave field
a and the level shifts of the relevant coupled atom-field
anditsdistributionwithinthespectroscopiccellusedfor
states are well described using perturbation theory [15].
the strong-field sample spectra in Section III. We con-
By exploiting near-resonant and resonant dipole tran-
clude with a discussion on how our results may be used
sitions between high-lying Rydberg levels, which elicit
in atomic high-power RF measurement methods.
a maximal atomic response, RF fields from as small
as ∼ 1 mV/m to a few tens of V/m have been mea-
sured [3, 12]. For measurements of strong RF E-fields,
the atom-field interaction cannot be modeled using per- II. EXPERIMENTAL SETUP
turbative methods, precluding the utility of the atom-
basedtechniqueasdemonstratedthusfarforE-fieldmea- The Rydberg EIT three-level system in 85Rb relevant
surements of high-power RF sources. In strong fields, to this work is illustrated in Fig 1(a). It includes a weak
2
probe laser resonant with the 5S (F = 3) → 5P using high n-values, strong RF fields, or a combination
1/2 3/2
transition and a strong coupling laser resonant with the of both, leading to considerably richer atom-field inter-
5P →Rydbergtransition. Applicationofthecoupling action phenomena. In Section III we show two sample
3/2
laser results in an increased transmission of the probe spectra to highlight the qualitatively different nature of
fieldwhena5S →5P →Rydbergdoubleresonance the atomic response in weak- and strong-field regimes.
1/2 3/2
conditionismet. Inourexperiment,theprobelaserwith Theweak-fieldmeasurementisperformedinthefar-field
λ =780.24nmhasapowerof100µWandisfocusedto of a mm-wave horn in the WR-6 band (injection power
p
a beam width of 94 µm FWHM; the coupling laser with −18.7 to 2.4 dBm). The emitted field has a linear polar-
λ ≈ 480 nm has a power of 37 mW and is focused to a ization that is parallel to the optical-beam polarizations.
c
beam width of 144 µm FWHM. Forthestrong-fieldmeasurement,microwavesintheK -
u
band are emitted from a horn placed to within about
1 cm of the optical beam foci, which is in the near-field
of the horn (injection power 13 to 24 dBm).
III. WEAK-FIELD AND STRONG-FIELD
MEASUREMENTS
In weak RF fields, the Rydberg levels are dynami-
cally (AC) Stark-shifted and, in the case of near- or on-
resonant drive of a Rydberg transition, exhibit Autler-
Townessplittings[15]. Forsingle-photontransitions, the
FIG. 1. (a) Energy-level diagram for Rydberg EIT and the RF E-field strength is obtained directly from the Autler-
Rydberg states coupled by RF fields. (b) Illustration of Townes splitting of the Rydberg EIT line, which is given
the experimental setup showing the vapor cell, 480 nm and by the Rabi frequency Ω = d · E/¯h, where d is the
780nmlaserbeams(dashedlines),photodiode(PD),bluelens Rydberg transition dipole moment and E is the RF ra-
(LB), red lens (LR), dichroic mirror (DM), and microwave diation E-field vector. Figure 2(a) shows experimental
horn.
spectra for the on-resonant one-photon 26D −27P
5/2 3/2
mm-wave (132.6495 GHz) transition as a function of
The experimental setup is similar to one described in the square root of mm-wave power [18]. Here, the EIT
previouswork[12];aschematicisshowninFig.1(b). The coupling-laser frequency is resonant with the mm-wave-
probe and coupling beams are counter-propagating and free 85Rb 5P (F(cid:48) = 4) to 26D transition, where F(cid:48)
3/2 5/2
overlapped through a room-temperature isotopic 85Rb denotes the intermediate-state hyperfine component. As
vapor cell and have parallel linear polarizations along z. expected in weak RF fields, Ω is a linear function of the
The probe laser frequency is scanned over the Doppler- square root of power (which is proportional to E). The
broadened 5S → 5P transition at a rate of about faint level pairs centered at about -70 and -110 MHz
1/2 3/2
1 Hz, while the coupling laser frequency is fixed near the correspond to spectra associated with the intermediate
5P → Rydberg transition. The coupling laser power 5P (F(cid:48) = 2,3) hyperfine components [19]. From the
3/2 3/2
is modulated by a 30 kHz square pulse with a 50% duty measured spitting, using Ω = d E /¯h and the known
z z
cycle, while the transmission of the probe beam through values of the dipole moments for a z-polarized field d
z
the cell is detected on a photodiode. The photodiode (405 ea for magnetic quantum number m = 1/2 and
0 j
signal is processed with a lock-in amplifier and recorded 331 ea for m = 3/2) the E-field is obtained from the
0 j
on an oscilloscope. EITspectra. Itisfoundthatthemaximumfieldreached
Microwaves are produced by a signal generator and inFig.2(a)is16V/m,about0.02%ofthemicrowaveion-
emitted from a horn aimed at the vapor cell perpendic- izationfieldoftheseatomsandwellwithintheweak-field
ular to the propagation direction of the optical beams, limit.
as shown in Fig. 1(b). The dipole moments and frequen- Tostudytheatomicresponseinstrongfields,wedrive
cies of dominant RF transitions between Rydberg levels Rydberg atoms at the zero-field 65D −66D two-
5/2 5/2
typically scale as ∼n2 and ∼n−3, respectively (n is the photon resonance frequency (12.4611548 GHz). This
Rydberg atom principal quantum number). Weak-field two-photon Rydberg transition was chosen to accommo-
atomic response can be observed over a wide RF fre- datehigh-powermicrowavesintheK band. Figure2(b)
u
quencyrangebymatchingtheRFfrequencywiththatof shows experimental EIT spectra centered on the 65D
a strong Rydberg transition, at suitable quantum num- level for microwave powers ranging from 13 to 24 dBm
bers, and by keeping the RF field low enough that the in steps of 1 dBm. The twelve data sets are plotted as a
Rabi frequency of the Rydberg transition remains much functionofpower. Forthishigh-powermeasurement,the
smaller than the RF frequency and smaller than typical microwave-inducedshiftsareintherangeofseveralhun-
level separations in the unperturbed spectrum. Nonlin- dredMHz,i.e. aboutafactoroftenlargerthantheshifts
ear response and high-order state mixing occurs when of the low-power measurement in Fig. 2(a). As detailed
3
in Section V, the maximum field reached in Fig. 2(b) is inmicrowavepowerthepeaksignalstrengthsoftheindi-
230 V/m, about 20% of the microwave field-ionization vidual spectral lines are reduced by more than an order
limitforthatcase. Wenotethatthefield-freefinestruc- of magnitude. As the microwave power is increased fur-
tureofthe65D and66D states(∼40MHz)isbrokenup ther, the shifts of the spectral lines become non-linear
in the strong-field regime because it is small compared in power, reflecting substantial state-mixing and higher-
to the microwave-induced shifts. This aspect heralds the order couplings. The transition from linear to non-linear
greater complexity of high-power Rydberg EIT spectra behavior occurs gradually as a function of power and
compared to low-power spectra, evident from a qualita- depends on the level. As seen by close inspection of
tive comparison of Figs. 2(a) and (b). Fig. 2(b), even at the lowest powers, most levels exhibit
some degree of non-linearity. A quantitative model of
the complex level structure in the strong-field regime is
(a ) E le c tric F ie ld (V /m ) (b ) E le c tric F ie ld (V /m )
2 4 6 8 1 0 1 2 1 4 1 6 1 0 0 1 5 0 2 0 0 described in detail in Section IV.
8 0 4 0 0
Inhomogeneous fields within the measurement volume
6 0
3 0 0 contribute to the background and additional spectral
4 0
lines observed in Fig. 2(b). The field inhomogeneity is
z) 2 0 2 0 0
H attributed in part to the presence of the dielectric cell,
y (M 0 1 0 0 aswasobservedinpreviouswork[10,12],andtothefact
c -2 0
uen -4 0 0 thatthemeasurementisdoneinthenear-fieldofthemi-
q
Fre -6 0 -1 0 0 crowave horn. The effects of the field inhomogeneity are
discussed in detail in Section V.
e -8 0
tiv -2 0 0
la-1 0 0
e
R-1 2 0 -3 0 0
IV. FLOQUET ANALYSIS
-1 4 0
-4 0 0
-1 6 0
-1 8 0 -5 0 0 In strong fields, where typical Rabi frequencies ap-
0 .2 0 .4 0 .6 0 .8 1 .0 1 .2 5 0 1 0 0 1 5 0 2 0 0 2 5 0 proach or exceed atomic transition frequencies, high-
P o w e r/m W P o w e r/m W order couplings become significant and perturbative ap-
proaches are no longer valid. To model the strong-field
FIG.2. Experimentalspectra: (a)Weak-fieldmeasurementof
experimental spectra we use a non-purturbative Floquet
132.6495GHzmm-wavesonthe26D −27P one-photon
√ 5/2 3/2 method. FollowingtheFloquettheorem,thesolutionsto
transitionversus power. Eachspectrumisanaverageof25
Schr¨odinger’s equation for a time-periodic Hamiltonian
traces and the signal is represented on a linear gray scale in
arbitrary units. A calculated spectrum is overlaid with rela- Hˆ(t)=Hˆ(t+T), where T is the period of the RF field,
tive excitation rates from 5P , given by the dot areas, and are of the form
3/2
E-field values on the top axis. (b) Strong-field measurement
Ψ (t)=e−iWνt/h¯ψ (t). (1)
of12.4611548GHzmicrowavesonthe65D−66Dtwo-photon ν ν
transition versus power. Each spectrum is an average of 20
Here, ψ (t)=ψ (t+T) are the periodic Floquet modes
ν ν
and the signal is represented on a linear gray scale. The Flo-
andW theirquasienergies,withanarbitrarymodelabel
quet calculation is overlaid with excitation rates from 5P , ν
3/2 ν. Fortheatom-fieldinteractionstrengthofinteresthere,
given by the dot areas, and E-field values on the top axis.
the Floquet modes can be represented using standard
Background signal and additional features in the experimen-
basis states |n,(cid:96),j,m (cid:105)=|k(cid:105), i.e.
tal strong-field spectrum are due to the microwave E-field j
inhomogeneity within the cell (see Section V). (cid:88)
Ψ (t)=e−iWνt/h¯ C (t)|k(cid:105), (2)
ν ν,k
k
At the lowest microwave power in Fig. 2(b), the mi-
crowave interaction broadens the 65D EIT resonance to with time-periodic (complex) coefficient functions
a FWHM width of 2π×50±1 MHz, which is a factor of Cν,k(t) = Cν,k(t + T). The Floquet energies Wν and
2 larger than that of the microwave-free EIT resonance states Ψν(t=0) are determined by finding the eigenval-
(not shown). For increasing microwave power, the EIT uesandvectorsofthetime-evolutionoperatorUˆ(t,T+t).
signal splits into multiple distinguishable spectral lines. ThecoefficientfunctionsC (t)arethenobtainedbyin-
ν,k
For two-photon transitions Ω ∼ E2 and the lines are tegrating Ψ (t) over one field period, t∈[0,T].
ν
expected to shift linearly as a function of power. Most InthelaserexcitationofFloquetstatesfromtheinter-
levels in Fig. 2(b) exhibit linear shifts up to microwave mediate 5P state, multi-photon processes are impor-
3/2
powers of ∼70 mW. tant because the atom may emit or absorb a number of
In strong fields, higher-order couplings lead to a re- microwave photons together with an optical photon. To
distribution of oscillator strength between many field- compute excitation line strengths, the functions C (t)
ν,k
perturbed Rydberg states, resulting in smaller signal are Fourier-expanded,
strengths compared to those in weak fields. This is re-
∞
flected in Fig. 2(b), where we observe a rapid initial de- Ψ (t)=e−iWνt/h¯(cid:88) (cid:88) C˜ e−iNωRFt|k(cid:105),
ν ν,k,N
crease in signal strength; over the first 30 mW increase
k N=−∞
4
6 0 0
1 : 1 4 0 V /m
4 0 0 2000
1500
)2 0 0
z 1000
H
(M 0 500
cy )0
en-2 0 0 z (a 0
u
Freq-4 0 0 3 -1-050000
e
lativ-6 0 0 1 2 -1500
e -2000
R
-8 0 0 -2000 -1500 -1000 -500 0 500 1000 1500 2000
2 : 1 6 5 V /m
5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0
2000
E le c tric F ie ld (V /m )
1500
FIG.3. Calculated|m |=1/2(black)and3/2(gray)Floquet
j 1000
quasienergies and their relative excitation rates (dot area)
from 5P . The E-field axis is scaled such that linear dis- 500
tanceisp3/r2oportionaltointensity,allowingacomparisonwith )0 0
the experimental map in Fig. 4. The region inside the black z (a
-500
box corresponds to the parameter space covered in the ex-
periment. The Floquet wave-packet dynamics for the levels -1000
marked 1, 2, and 3 are shown in Fig. 4. These calculations
-1500
useabasisofallstateswitheffectiveprincipalquantumnum-
ber 56≤ n∗ ≤ 73 and orbital quantum number (cid:96) ≤ 10. The -2000
Floquet series are terminated at N =±8 photon numbers. -2000 -1500 -1000 -500 0 500 1000 1500 2000
3 : 1 8 0 V /m
2000
1 (cid:90) T
C˜ν,k,N = T Cν,k(t)eiNωRFtdt. (3) 1500
0
1000
The integer N is interpreted as a number of microwave
500
photons with frequency ω associated with the bare
atomicstate|k(cid:105). ThelaserfRrFequenciesωL,whereFloquet (a)0 0
levelsareresonantlyexcitedfromthe5P3/2 level,andthe z-500
corresponding line strengths S are then given by
ν,N -1000
¯hωL =Wν +N¯hωRF -1500
(cid:12) (cid:12)2
Sν,N =(eEL/¯h)2(cid:12)(cid:12)(cid:12)(cid:88)C˜ν,k,N(cid:15)ˆ·(cid:104)k|ˆr|5P3/2,mj(cid:105)(cid:12)(cid:12)(cid:12) , (4) -2000 -2000 -1500 -1000 -500 0 500 1000 1500 2000
(cid:12) (cid:12) x (a )
k 0
where E is the amplitude of the laser E-field, (cid:15)ˆ is
L FIG. 4. Videos 1, 2, and 3 showing the Floquet wave-packet
the laser polarization vector, and (cid:104)k|ˆr|5P ,m (cid:105) are the
3/2 j dynamicsthroughonefullmicrowavecycleforthethreelevels
electric-dipole matrix elements of the basis states with
marked1,2,and3,respectively,inFig.3. Theinitialframeof
|5P ,m (cid:105). EachFloquetlevelW leadstomultipleres-
3/2 j ν each(shown)istakenatthetimewhenthemicrowaveE-field
onances, which are associated with the microwave pho- is maximal. The E-field points along +z.
ton number N. This procedure is similar to one used
by Yoshida et al. [20]. Due to parity, in the absence of
additional static fields a Floquet level W may generate
ν
displayedinFig.3itisalwaysN =0. Furtherinspection
resonances either for even N or odd N, but not both.
of the calculated Floquet energies and excitation rates,
In Fig. 3 we show calculated Floquet energies and ex-
not presented in detail here, shows that for fields above
citation rates S in the vicinity of the 65D Rydberg
ν,N
∼ 150 V/m several Floquet levels W visible in Fig. 3
level for a microwave frequency of 12.4611548 GHz and ν
have copies with high excitation rates for even N-values
field strengths ranging from 0 to 350 V/m. The field is
between about -8 and +8.
displayed on a quadratic scale to show the dependence
of the atomic level shifts on power, and for direct com- The Floquet modes in strong fields exhibit non-trivial
parison with Fig. 2(b). Over the short frequency range wave-packet motion, and their optical excitation rates
5
have to be calculated according to Eq. (4). It would, for is reached. The uncertainty is given by half the exper-
instance, be incorrect to associate the excitation rates of imental step size (±0.5 dB, corresponding to ±6% in
the Floquet modes in Figs. 3 and 5 with the 65D prob- field).
abilities the modes carry. To qualitatively explain this, The experimental spectrum plotted in Figs. 2(b) and
we note that in weak fields the dressed-state coefficients 5(a) contains information on the continuous distribution
and dipole moments have fixed amplitudes and phases of the microwave E-field strength along the length of
relativetothefield(inthefieldpictureandusingthero- the EIT probe and coupling beams passing through the
tating wave approximation). In strong fields, wave func- spectroscopic cell. The microwave boundary conditions
tion coefficients and dipole moments vary significantly are symmetric about the xy-plane, with the incident mi-
throughout the microwave-field cycle. Specifically, the crowave E-field being z-polarized and the optical beams
Floquet modes are time-periodic wave-packets that are propagating along the x axis (see Fig. 1(b)). Since the
synchronizedwiththedrivingRFfield. Tovisualizetyp- microwavefieldisprimarilyz-polarizedalongtheoptical
ical Floquet wave-packet dynamics, see the videos of the beams, it drives ∆m = 0 Rydberg transitions within
j
wave-packet evolution of the wave functions shown in the m =±1/2 and m =±3/2 manifolds of states. The
j j
Fig. 4. We have also investigated the wave-packet dis- field distribution is a result of superpositions of reflec-
tribution over the quantum numbers n and (cid:96) of the ba- tions from the cell walls. Also, the cell is placed within
sis states. The distributions over n and (cid:96) oscillate syn- thenearfieldofthesource,leadingtoadditionalvariabil-
chronously with the driving RF field, as expected, and ity of the microwave field along the probe beam. Hence,
have significant populations over the quantum-number onemaypicturetheRFfieldasaspecklepattern,akinto
ranges 60 <∼ n <∼ 73 and 0 ≤ (cid:96) <∼ 5. The wave-packets specklepattersseenforgeneral,non-idealcoherentfields.
therefore cover a basis-state energy range equivalent to Here, one should expect the number of speckles to be on
the absorption or emission of up to about ten RF pho- the order of the cell length divided by the wavelength,
tons. which in our case is on the order of three. Also, since
therearenostructureswithinthecellthatareveryclose
to the optical probe beam path, we do not expect any
sharp spatial features in the microwave E-field (features
V. ANALYSIS OF EXPERIMENTAL SPECTRA
whichmightotherwisearisefromsharpmetallicordielec-
tric edges and the like). Therefore, for each theoretical
In Fig. 2(b) we find an excellent overall agreement be-
line in Fig. 3 the measured EIT spectra are expected to
tween dominant features in the experimental and the-
exhibit a small number of spectral features that corre-
oretical Floquet maps. Additional features evident in
spondtothelocalmaximaandminimaofthemicrowave
theexperimentalmapareduetoE-fieldinhomogeneities,
field along the length of the probe beam.
which are discussed in detail below. Both experimental
Based on the observation of five downward spectral
andtheoreticalmapsexhibitseveralarchedlinesatposi-
lines [labeled in Fig. 5(a)], and the fact that the cal-
tivefrequenciesand,atnegativefrequencies,severallines
culated spectrum has only one corresponding downward
that shift approximately linearly in power. The domi-
line (feature 1 in Fig. 3), we model the spectrum con-
nant down-shifting line suddenly terminates at close to
sidering populations of atoms located within a set of
−400 MHz. It is evident from Fig. 3 that the sudden
five dominant microwave E-field regions. In our model
termination of the down-shifting lines is due to a wide
the probability distribution for intensity on a dB scale is
Floquet avoided crossing. Avoided crossings in Floquet
given by
maps, such as those in Fig. 3, provide convenient mark-
ers for spectroscopic determination of the RF E-field 3/2 5
(cid:88) (cid:88)
on an absolute scale. In the present case, the promi- P (s)= w (|m |)w (k)P (s+∆s ).
dBi mj j k dBi0 k
nent avoided crossing at 165 V/m (see label 2 in Fig. 3) |mj|=1/2k=1
has several matching locations in the experimental spec- (5)
trum shown in Fig. 2(b). These are seen more clearly in Here, k is an index for the five microwave field domains,
Fig. 5(a), which shows the experimental data on a dBm w (k)istheprobabilitythatanatomcontributingtothe
k
scale. The appearance of multiple copies of the calcu- signal resides within domain k, w (|m |) is the proba-
mj j
lated avoided crossing in the experiment points to the bilitythatanatomcontributingtothesignalhasamag-
fact that the microwave field within the cell must have netic quantum number |m |, and P is a Gaussian
j dBi0
multiple dominant domains, each of which produces its point-spread function that accounts for inhomogeneous
own rendering of the avoided crossing. The rendering at spectral broadening within the five domains. The values
the lowest injected microwave power corresponds to the ∆s indicate by what amount (in dB) the central mi-
k
E-field domain with the highest field at a given injected crowave intensity within the k-th microwave field region
power (calculation and E-field axis shown in Fig. 2(b)). isshiftedrelativetotheintensityinthehighest-intensity
In Figure 2(b) the avoided crossing is observed first at (k = 1) domain. For P we assume a Gaussian that
dBi0
130 mW, at which point the domain that has the high- is the same for all k. The fit parameters in the model
estfieldreaches165V/m. Itfollowsthatatamicrowave are ∆s , w , w , and the standard deviation, σ , for
k mj k dBi
powerof250mWamaximumRFE-fieldof230±14V/m P . We account for the optical EIT line broadening
dBi0
6
andlaserlinedriftsbyaGaussianspreadfunctioninfre- trum in Fig. 4 shows that a strong-field Floquet analysis
quency, P (ν), which has a standard deviation σ . From of the atomic physics of Rydberg atoms in microwave
ν ν
the theoretical spectrum, S (s,ν), the model spectrum, fields, combined with a straightforward empirical model
T
S (s ,ν ), is then given by the convolution of the microwave intensity distribution and the |m |-
E 0 0 j
weightinginthesample,leadstoremarkablygoodagree-
(cid:90)
S (s ,ν )= S (s −s(cid:48),ν −ν(cid:48))P (ν(cid:48))P (s(cid:48))dν(cid:48)ds(cid:48) , ment between spectra with rather complex features.
E 0 0 T 0 0 ν dBi
VI. DISCUSSION AND CONCLUSION
(6)
where the intensities in the arguments of S and S are
E T Utilizing a combination of resonant, strong electric-
measuredindBi,definedas10log [I/(W/m2)],whereI
10 dipoletransitionsasinFig.2(a)andhigher-ordertransi-
is the RF field intensity.
tionssuchasthetwo-photontransitionin2(b), itispos-
Figures 5(b) shows the model spectrum S (s ,ν )
E 0 0 sible to observe level shifts in Rydberg EIT spectra over
in dBi for the data in Fig. 5(a). In the model spec-
a wide dynamic range of the applied RF intensity. Since
trum, the field domains have empirically fitted inten-
the spectroscopic response is well described by the Flo-
sity shifts of ∆s = 0.0,−2.0,−4.17,−6.0,−8.0 dB for
k quet theory laid out in Section IV, the measured spectra
k = 1,...,5. The corresponding fitted weighting factors
can be used to directly determine the RF E-field caus-
are w = 0.39,0.21,0.27,0.09,0.04. The intensity shifts
k ing the observed spectral features in a calibration-free
are significant to better than about 0.5 dB, while the
manner. Specifically, there are no antenna systems and
weighting factors are significant to better than about
readout instruments that need to be calibrated to trans-
±0.04. The weighting factors w for the |m | = 1/2
mj j late a reading into a field, because spectral features such
and 3/2 states are 0.7 and 0.3, respectively (significance
aslineshiftsandavoidedcrossingsfollowfromtheinvari-
level better than about 0.1). The larger |m | = 1/2-
j able nature of the underlying atomic physics. The field
weight likely results from optical pumping by the EIT
measurement precision is given by how well the spectral
probe field. Further, σ =1 dBi and σ =30 MHz.
dBi ν features are resolved. For instance, in the present work
the avoided crossing pointed out in Figs. 2(b), 3 and 4
canberesolvedwith±0.5dBiuncertainty,corresponding
to an absolute field uncertainty of ±6%.
In the weak-field domain, an example of which is seen
in Fig. 2(a), the E-field strength is obtained by a mea-
surement of the Autler-Townes splitting of the Rydberg
EITline. Inthestrong-fielddomain,anexampleofwhich
is seen in Fig. 2(b) and analyzed in depth through most
ofthispaper,acomparisonofexperimentalRydbergEIT
spectra with calculated Floquet maps allows us to deter-
mine the E-field strength. Our analysis of the spectrum
also reveals information about the E-field distribution
within the active measurement volume.
The extension of the Rydberg EIT method to higher
fields, explored in this work, paves the way to extend
the dynamic range of the Rydberg EIT RF-field sensing
methodology into the realm of strong microwave intensi-
FIG. 5. (a) Experimental spectra of the 65D −66D two-
ties. With the present measurement we have reached a
photontransitionversusmicrowavepower. Thesignalisrep-
field of 230 V/m, corresponding to a six-fold increase in
resented on a linear gray scale in arbitrary units. The ar-
fieldanda40-foldincreaseinintensityrelativetoaprevi-
rows indicate coordinates that both correspond to the same
ousmeasurement[12]. Anextensionofthemeasurement
avoidedcrossinginthetheoreticalFloquetmap,marked2in
Fig.3. Thesolidarrowcorrespondstothespatialregionwith rangetoevenhigherfieldswillbereadilyachievableusing
the highest microwave intensity along the probe beam. The more powerful RF testing sources. Fundamental limita-
labels 1-5 mark the renderings of the same level in Fig. 3, tions due to the underlying atomic physics of Rydberg
observedinthefivedifferentintensitydomains(seetext). (b) EIT will eventually occur when the Rydberg atoms ex-
Composite Floquet map model of (a). perience substantial microwave ionization rates [4, 17].
This work was supported by the AFOSR (FA9550-10-
A comparison of measured spectrum and model spec- 1-0453) and DARPA’s QuASAR program.
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