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egundo, California 

Atomic Stabilization of Electromagnetic Field 
Strength Using Rabi Resonances 

10 March 2000 

Prepared by 

Electronics and Photonics Laboratory 
Laboratory Operations 

Prepared for 

2430 E. El Segundo Boulevard 
Los Angeles Air Force Base, CA 90245 

Engineering and Technology Group 



20000425 116 

This report was submitted by The Aerospace Corporation, El Segundo, CA 90245-4691, under 
Contract No. F04701-93-C-0094 with the Space and Missile Systems Center, 2430 E. El Segundo 
Blvd., Los Angeles Air Force Base, CA 90245. It was reviewed and approved for The Aerospace 
Corporation by B. Jaduszliwer, Principal Director, Electronics and Photonics Laboratory. Michael 
Zambrana was the project officer for the Mission-Oriented Investigation and Experimentation 
(MOIE) program. 

This report has been reviewed by the Public Affairs Office (PAS) and is releasable to the National 
Technical Information Service (NTIS). At NTIS, it will be available to the general public, including 
foreign nationals. 

This technical report has been reviewed and is approved for publication. Publication of this report 
does not constitute Air Force approval of the report's findings or conclusions. It is published only for 
the exchange and stimulation of ideas. 

Michael Zambrana 


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1 . AGENCY USE ONLY ( Leave blank) 


10 March 2000 

3. REPOR1 



Atomic Stabilization of Electromagnetic Field 

Strength Using Rabi Resonances 




J. C. Camparo 


The Aerospace Corporation 

Technology Operations 

El Segundo, CA 90245-4691 




Space and Missile Systems Center 

Air Force Materiel Command 

2430 E. El Segundo Boulevard 

Los Angeles Air Force Base, CA 90245 






Approved for public release; distribution unlimited 

13. ABSTRACT (Maximum 200 words) 

For nearly fifty years, atomic resonances have been employed to stabilize electromagnetic 
field frequency. Here, a dynamical Rabi resonance, associated with an alkali atom's 
ground-state hyperfine transition, is used to stabilize electromagnetic field strength : A 7% 
sinusoidal variation was superimposed on a microwave field's strength, and through atomic 
stabilization, this variation was reduced to 0.1 %. Although the method is demonstrated 
with microwave fields, the technique could also be employed to stabilize optical fields via 
Rabi resonances associated with electronic transistions in atoms or molecules. 


Atomic clock 











NSN 7540-01-280-5500 

Standard Form 298 (Rev. 2-89) 
Prescribed by ANSI Std. Z39-18 


1. Experimental arrangement. 2 

2. (a) Rabi resonance associated with the ground state hyperfine transition 

of Rb 87 as observed via the atoms’ second harmonic response to the modulated 
microwave field, (b) Rapid change in the phase of the atoms second harmonic 
response as a function of microwave attenuator setting. 3 

3. Field-strength discriminator obtained from the output of lock-in #2; 

the time constant of lock-in #1 was set to 1 msec. 3 

4. Demonstration of the efficacy of field-strength stabilization using a Rabi resonance. 3 



Transitions between eigenstates in atoms (or molecules) 
have great utility for stabilizing electromagnetic field fre¬ 
quency. For example, it is now possible to realize the sec¬ 
ond to fourteen significant figures by locking the frequency 
of a microwave field to the ground state hyperfine transi¬ 
tion of Cs 133 [1]. To no small extent, the utility of atoms 
in this regard is achieved because the transition probability 
between eigenstates is a resonant function of energy, and 
because the transition frequency is (in principle) a constant 
of nature. This, of course, is well known, and quantum 
mechanical transitions have been employed for frequency 
stabilization since 1948 [2]. 

The stabilization of electromagnetic field strength , how¬ 
ever, is another matter, since the transition probability 
between two atomic eigenstates is, in general, a mono- 
tonically increasing function of intensity. Consequently, 
resonances between energy eigenstates are not particularly 
useful for field-strength stabilization (except in the case 
of dc fields, where the Zeeman and Stark effects can be 
exploited [3]), and other methods must be found. In the 
case of lasers, where field stability can be of importance for 
precision spectroscopy, field strength is typically stabilized 
by comparing a voltage measurement of laser intensity to 
some reference voltage [4]. A deviation of the laser inten¬ 
sity from the reference produces an error signal that can be 
exploited in a feedback control loop to stabilize the laser’s 
field strength. Identical techniques are employed for the 
stabilization of microwave field strengths [5], Of course, 
one disadvantage with this approach is that the reference 
voltage is arbitrary, sometimes making it a cumbersome 
process to reference the field strength to an absolute value, 
for example in ac Stark shift measurements. Additionally, 
the reference voltage may drift over time, giving rise to 
long term field-strength instability. It should be noted that 
the achievement of stable microwave field strength over 
long times has technological implications, as it is known 
that microwave power variations in atomic clocks degrade 
timekeeping ability [6,7]. 

In the present paper, field-strength stabilization is 
achieved by “locking” the field strength to an atomic Rabi 

resonance [8-12]. In simplest terms, when an atom or 
molecule is subjected to a phase modulated resonant field, 
a Fourier component of the resulting population variations 
shows a resonant increase when the corresponding Fourier 
frequency matches the Rabi frequency (i.e., the electro¬ 
magnetic field strength); hence the term '‘Rabi resonance.” 
It is to be noted, however, that even though the Rabi 
resonance appears in a quantum system it is not a typical 
resonance between energy eigenstates. (We are referring 
to the so-called “bare-atom” eigenstates as opposed to 
the quantum system’s “dressed-atom” eigenstates [13].) 
Rather, it is a dynamical resonance associated with a 
frequency match between the rate of a perturbation’s 
induced atomic variations (at harmonics of the phase 
modulation frequency, ^ m od) and an atom’s internal rate 
of response to that perturbation (i.e., the Rabi frequency, 
O). Here, a microwave field (resonant with the ground 
state (F = 2, m? — 0) — (1,0) hyperfine transition of 
Rb 87 , A^hfs) was phase modulated, and the resulting Rabi 
resonance was observed as an enhancement in the atoms’ 
second harmonic response to the modulated field. Using 
standard heterodyne detection methods, the change in the 
atoms’ response with microwave field strength was used to 
generate an error signal for field-strength stabilization. As 
a test of this atomic stabilization technique, a slow 1.2 dB 
(peak-to-peak) sinusoidal power variation (i.e., “drift") 
was superimposed on the microwave field. By locking 
the microwave field strength to the Rabi resonance, this 
drift was reduced by nearly two orders of magnitude. 

Figure 1 is a block diagram of the experimental ar¬ 
rangement. A Coming 7070 glass resonance cell contain¬ 
ing isotopically pure Rb 87 and 10 Torr of N 2 was placed 
in a microwave cavity whose TEon mode was resonant 
with Ai'hfs at 6834.7 MHz. The cyclindrical cavity had 
a radius of 2.8 cm and a length of 5 cm, and the reso¬ 
nance cell filled the cavity volume. Braided windings, 
wrapped around the cavity, heated the resonance cell to 
approximately 40 °C, and the entire assembly was cen¬ 
trally located in a set of three perpendicular Helmholtz 
coil pairs: Two pairs zeroed out the Earth’s magnetic field 


FIG. 1. Experimental arrangement. 

while the third pair (—300 mG) provided a quantization 
axis for the atoms parallel to the cavity’s cylindrical axis. 
An AlGaAs diode laser (—3 mW) was tuned to the Rb 
5 2 Pi /2 - 5 2 S\/2(F = 2) transition [14] and was attenu¬ 
ated by a 2.0 neutral density filter. The transmission of 
the laser light through the vapor was monitored with a Si 
photodiode, and the propagation direction of the laser was 
parallel to the cavity axis. In addition to optical pump¬ 
ing [15], the laser light monitored the F = 2 population 
density: In the absence of resonant microwaves, optical 
pumping reduced the F = 2 atom density and resulted in 
an increased level of transmitted light; when resonant mi¬ 
crowaves were present, atoms returned to the 5 2 S\/2(F = 
2) state from the F = 1 state, thereby reducing the amount 
of transmitted light. As a result, any microwave field in¬ 
duced oscillation of the atomic population could be ob¬ 
served as oscillations in the transmitted light intensity. 

The microwaves were derived from a voltage-controlled 
crystal oscillator (VCXO) which had a modulation band¬ 
width of 10 kHz, and its output frequency at —107 MHz 
was multiplied up to A rvs* The microwaves were attenu¬ 
ated by the combination of a voltage-controlled attenuator 
(VCA) and a fixed attenuator (labeled as — dB in the figure) 
before being amplified by a +30 dB solid state amplifier. 
The attenuators were calibrated to Rabi frequency by 
measuring the hyperfine transition linewidth [16]. The 
output from a frequency synthesizer at 357 Hz and a dc 
voltage were added, and these provided the VCXO’s con¬ 
trol voltage, V c . The dc level of V c tuned the average mi¬ 
crowave frequency to Az>hfs> while the sine wave provided 
microwave phase modulation. The amplitude of the phase 
modulation was 2.28 radians, and this was chosen by maxi¬ 
mizing the amplitude of the atoms’ Rabi resonance. 

The Rabi resonance was manifested in the atoms’ second 
harmonic response to the phase-modulated microwave 
field, and thus occurred for fl = 2z/ mo d- The output of 
the photodiode was sent to a spectrum analyzer (25 kHz 

bandwidth) and a lock-in amplifier (labeled as #1 in Fig. 1) 
referenced to 2v mod . The spectrum analyzer was used 
primarily to tune the microwave frequency to A^ h fs by 
zeroing the atoms’ fundamental response to the phase- 
modulated microwaves. The photodiode-preamp-lock- 
in combination acted as a low-pass detector for the atoms’ 
second harmonic response. (The bandwidth of the preamp 
was 1 MHz, and the time constant of the lock-in amplifier 
depended on the experiment.) 

Figure 2(a) shows the amplitude of the atoms’ second 
harmonic response as a function of the fixed attenu¬ 
ator setting. Two sets of data are shown corresponding 
to measurements made by the spectrum analyzer (open 
circles) and measurements obtained at the output of lock-in 
#1 (filled circles). Both measures of the second harmonic 
response show a resonance centered at about 31 dB (fl = 
850 Hz) with a linewidth of about 10 dB; this is the 
Rabi resonance. Figure 2(b) shows that the phase of the 
second harmonic signal varies rapidly in the vicinity of 
the Rabi resonance [11], and explains the slight difference 
in width between the two resonances shown in Fig. 2(a). 

To generate an error signal for field-strength stabiliza¬ 
tion, the fixed attenuator was set to 31 dB, and the mi¬ 
crowave power was modulated by applying a sinusoidal 
signal to the VCA’s control voltage (i.e., 0.8 dB peak-to- 
peak modulation at 47 Hz). The field-strength modulation 
resulted in a modulation of the atoms’ second harmonic re¬ 
sponse, and this could be observed in the output of lock-in 
#1. The amplitude of atomic modulation was monitored 
in a heterodyne fashion with the aid of lock-in #2, whose 
output thus became a “field-strength discriminator.” This 
is illustrated in Fig. 3, where the output of lock-in #2 and 
the spectrum analyzer’s indication of second harmonic re¬ 
sponse are plotted as a function of fixed attenuator setting. 

A field-strength feedback control loop was closed by 
adding the field-strength discriminator voltage to the VC A 
control voltage. (No efforts were made to optimize the 
















Microwave attenuator [dB] 

Microwave attenuator [dB] 

Microwave attenuator [dB] 

FIG. 2. (a) Rabi resonance associated with the ground state 

hyperfine transition of Rb 87 as observed via the atoms’ 
second harmonic response to the modulated microwave field. 
Open circles correspond to spectrum analyzer measurements, 
while closed circles correspond to lock-in #1 measurements, 
(b) Rapid change in the phase of the atoms’ second harmonic 
response as a function of microwave attenuator setting. Since 
a lock-in amplifier’s output voltage depends on the relative 
phase between the reference and the signal, this rapid change in 
phase explains the slight difference in width between the two 
resonances shown in (a). 

control loop parameters.) The efficacy of this atomic 
stabilization technique was tested by adding a slowly 
varying voltage to the VC A control voltage, and was 
accomplished with the aid of the “slow-drift” function 
generator shown in Fig. 1. 

Referring to Fig. 1, relative field-strength variations 
were measured by splitting the microwave signal just prior 
to the VCA and just after the VCA, and then combining 
the two signals in a microwave mixer. As illustrated by 
the solid line in Fig. 4, the slow-drift function generator 
produced a 7.2% variation of the microwave field strength 
when the feedback control loop was open. The filled cir¬ 
cles in Fig. 4 show the microwave field strength with the 
feedback control loop closed: The amplitude of the field- 
strength variations was considerably reduced, and the av¬ 
erage value of the field strength was changed. The change 
in average field-strength value corresponds to a —0.12 dB 
change in microwave power attenuation, and arose because 

FIG. 3. Field-strength discriminator obtained from the output 
of lock-in #2 (closed circles); the time constant of lock-in #1 
was set to 1 msec. For comparison, the open diamonds show 
the Rabi resonance obtained by measuring the second harmonic 
amplitude with the spectrum analyzer. 

the fixed attenuator (having units of integer dB) could not 
be set to the exact center of the Rabi resonance. Examin¬ 
ing the cumulative probability distribution of filled-circle 
deviations from their average, 90% of the microwave field- 
strength variations were within ±0.12% of the average 
field strength with the feedback control loop closed. Thus, 
by stabilizing the field strength to an atomic Rabi reso¬ 
nance, there was factor of 60 reduction in the field-strength 

While we have demonstrated atomic field-strength sta¬ 
bilization with microwaves, nothing precludes its use with 
optical fields. Rabi resonances have been observed in the 
optical regime [10], and one could employ an electro¬ 
optic modulator as a VCA to stabilize laser intensity. 

FIG. 4. Demonstration of the efficacy of field-strength sta¬ 
bilization using a Rabi resonance. Relative microwave field- 
strength variations were measured via the IF output port of the 
mixer shown in Fig. 1. The solid line corresponds to the mi¬ 
crowave field-strength variations that were externally imposed 
by the slow-drift function generator shown in Fig. 1. The filled 
circles correspond to the field-strength variations after atomic 
stabilization. Percentages are referenced to the average mi¬ 
crowave field strength. 


Moreover, if one employed square-wave phase modula¬ 
tion instead of sine-wave phase modulation, a spectrum of 
Rabi resonances would be generated [9], and each Rabi 
resonance in the spectrum could be employed to stabilize 
laser intensity at a different value. 

With regard to microwave stabilization, the Rabi reso¬ 
nance method may have relevance to atomic clocks, where 
microwave power variations are known to give rise to 
atomic clock frequency variations [6,7]. In such an ap¬ 
plication, the atoms’ fundamental response to a phase- 
modulated microwave field would be used to stabilize 
field frequency (as is presently done in atomic clocks), 
while the atoms’ second harmonic response could be used 
to stabilize field strength. In the case of the gas-cell 
atomic clock, where different atoms in a vapor contribute 
to the clock signal to varying degrees, an attractive feature 
of the Rabi resonance method would be that those atoms 
dominating the frequency stabilization signal would also 
dominate the field-strength stabilization signal. 

It should be noted that the Rabi resonance condition 
depends on the exact value of t'mod '■ Should the phase- 
modulation frequency change, the Rabi resonance con¬ 
dition will occur for a different value of field strength. 
Consequently, in this atomic stabilization technique, the 
field-strength stability can only be as good as the stability 
of r'mod- However, not only is the output frequency of a 
synthesizer typically quite stable, but the synthesizer can 
be referenced to an external atomic, frequency standard. 
Hence, the Rabi resonance condition (via ^ mo d) has the 
potential to be as stable as the output of an atomic clock. 

Referencing z’ moc i to a cesium atomic clock is an in¬ 
triguing notion, as it suggests a direct link between field 
strength and a fundamental definition of time interval. 
Note that the Rabi frequency has a well-defined rela¬ 
tionship with field strength, and that the Rabi resonance 
condition defines Cl in terms of r'mod- Since the phase- 
modulation frequency would be traceable to a defini¬ 
tion of the second via the cesium atomic clock, the field 
strength would also be traceable to a definition of the sec¬ 
ond. In other words, Rabi resonance stabilization of field 
strength could provide a means of defining the units of 
field strength in terms of the second at the atomic level. 

The author would like to thank J. Coffer for his assis¬ 
tance in performing some of the experiments. Addition¬ 

ally, I would like to thank R. Frueholz and B. Jaduszli- 
wer for several stimulating discussions regarding the Rabi 
resonance phenomena and a critical reading of the manu¬ 
script. This work was supported the U.S. Air Force Space 
Division under Contract No. F04701-93-C-0094. 

[1] R.E. Drullinger, J.H. Shirley, J.P. Lowe, and D. J. Glaze, 
IEEE Trans. Instrum. Meas. 42, 453 (1993). 

[2] W.D. Hershberger and L.E. Norton, RCA Rev. 9, 38 

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[8] U. Cappeller and H. Mueller, Ann. Phys. (Leipzig) 42, 250 

[9] J. C. Camparo and R. P. Frueholz, Phys. Rev. A 38, 6143 

[10] S. Papademetriou, S. Chakmakjian, and C. R. Stroud, 
J. Opt. Soc. Am. 9, 1182 (1992). 

[11] J.C. Camparo and R. P. Frueholz, in Proceedings of the 
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(IEEE Press, New York, 1993), pp. 114-119. 

[12] J.C. Camparo, J.G. Coffer, and R.P. Frueholz, Phys. Rev. 
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[13] S. Reynaud and C. Cohen-Tannoudji, J. Phys. 43, 1021 

[14] J.C. Camparo, Contemp. Phys. 26, 443 (1985); C.E. 
Wieman and L. Hollberg, Rev. Sci. Instrum. 62, 1 

[15] W. Happer, Rev. Mod. Phys. 44, 169 (1972). 

[16] J.C. Camparo and R.P. Frueholz, Phys. Rev. A 31, 1440 
(1985); J.C. Camparo and R.P. Frueholz, Phys. Rev. A 
32, 1888 (1985).