Quantum interference in the rate of two-photon excitation of the 6S1/2 → 6P3/2 → 6D5/2 transition in atomic cesium is exploited to demonstrate phase-sensitive frequency demodulation for an optical interval of 612.5 THz. By thus using atoms as ultrafast nonlinear mixing elements, we suggest and analyze a new avenue for absolute comparisons of a dense set of frequencies over the range of 200–2000 nm

Georgiades, N. Ph.

Kimble, H. J.

Polzik, E. S.

English

Caltech Authors - Main

1688 OPTICS LETTERS / Vol. 21, No. 20 / October 15, 1996Frequency metrology by use of quantum interference
N. Ph. Georgiades, E. S. Polzik,* and H. J. Kimble
Norman Bridge Laboratory of Physics, 12-33 California Institute of Technology, Pasadena, California 91125Received May 31, 1996
Quantum interference in the rate of two-photon excitation of the 6S1/2 ! 6P3/2 ! 6D5/2 transition in atomic
cesium is exploited to demonstrate phase-sensitive frequency demodulation for an optical interval of 612.5 THz.
By thus using atoms as ultrafast nonlinear mixing elements, we suggest and analyze a new avenue for absolute
comparisons of a dense set of frequencies over the range of 200–2000 nm.  1996 Optical Society of AmericaQuantum interference (QI) of multiple excitation path-
ways is ubiquitous in atomic, molecular, and optical
physics, with applications to the coherent control of
products in a chemical reaction,1 phase-sensitive yields
in above-threshold ionization,2 and the synthesis of
atomic wave functions3 being but three recent exam-
ples. In this Letter we present observations of QI for
two-photon excitation whereby interfering paths lead
to modulation of excited-state population as a function
of the relative phase of three excitation beams with
widely separated frequencies (corresponding to 852,
884, and 917 nm). By using an atom as an ultrafast
nonlinear mixing element, we demonstrate optical de-
modulation for an interval of 612.5 THz. In an exten-
sion of this capability, we suggest a novel metrological
technique that should allow for absolute comparisons
with dense coverage over the region 200–2000 nm.
We begin by considering two-photon excitation of an
atomic ladder of the form 1 ! 2 ! 3 with eigenfrequen-
cies v12, v23, and v13 ­ v12 1 v23 by three lasers at
frequencies va ø v12, vb ø v23, and vc ø 1/2 v13 (dot-
ted box in Fig. 1). QI in this system arises from two
possible pathways of excitation, the stepwise dipole
absorption of two photons at frequencies sva, vbd and
the quadrupole two-photon absorption at vc. Devi-
ations of the excitation frequencies from the trian-
gle equality va 1 vb ­ 2vc result in a modulation
of the excited-state population of the form r33std ,
Af1 1 V cossDvt 1 fdg at frequency Dv ­ va 1 vb 2
2vc with visibility V and phase f ­ fa 1 fb 2 2fc,
where fi is the phase of laser i si ­ a, b, cd. In this
scheme the atom acts as a nonlinear mixer that com-
pares va 1 vb with 2vc; the process corresponds to
demodulation of sidebands at frequency offsets Dn ­
jva 2 vcjy2p ø jvc 2 vbjy2p with respect to a local
oscillator at vc. Note that r33std depends on the exci-
tation scheme and is sinusoidal only in the absence of
saturation. Explicit expressions for A, V, and f fol-
low from theoretical analyses based on either pertur-
bative or master equation approaches.4
Independently of these more complex treatments, the
origin of QI for two-photon excitation can be under-
stood by consideration of Mollow’s classic expression,5
r33 ~ kEˆyEˆyEˆEˆl, where now Eˆ ­ Eˆa 1 Eˆb 1 Eˆc is
the total excitation field and Eˆa, b, c are taken to be of
the form Eˆi ­ Eˆi exps2iFiˆd with Fi ­ vit 1 fi. Ex-
panding the fourth-order correlation function of the to-
tal field under the assumption of a coherent state for
the field Eˆc of eigenvalue Ec, and retaining only those0146-9592/96/201688-03$10.00/0terms that correspond to small two-photon detuning for
the transition amplitudes, we find that r33 . a1jEcj4 1
a2kEˆbyEˆayEˆa Eˆbl 1 a3sEc2kEˆayEˆbyl 1 c.c.d. Note that
the first term represents the usual quadrupole two-
photon excitation that is proportional to the square
of the intensity and the second term corresponds to
stepwise dipole absorption at frequencies va and vb.
In the case of nonclassical squeezed fields this second
term gives rise to a linear rate of two-photon excita-
tion.6 The last term is the focus of the present study
and results from interference in two-photon excitation
as a function of the overall phase F ­ Fa 1 Fb 2 2Fc
for the fields sEˆa, Eˆb, Eˆcd.
For an initial proof-of-principle experiment, we ex-
cite the 6S1/2, F ­ 4 ! 6P3/2, F 0 ­ 5 ! 6D5/2, F 00 ­ 6
transition s1 ! 2 ! 3d in magneto-optically trapped
atomic Cs by three laser beams at wavelengths
la ø 852, lb ø 917, and lc ø 884 nm, as shown in
Fig. 1.7 The field at 884 nm originates from a tunable
Ti:sapphire laser externally locked to a reference cavity
that is in turn locked to the Doppler-free two-photon
transition 1 ! 3 of Cs in an auxiliary vapor cell. The
field at 852 nm is provided by a diode laser operated
with an external grating and stabilized to the 1 ! 2
transition.8 Finally, the field at 917 nm is derived by
nonlinear mixing of the 852- and 442-nm (double of
884) beams in a parametric amplifier composed of a
subthreshold optical parametric oscillator.6
The experiment is conducted continuously in two al-
ternating stages that are switched at a rate of 4 kHz.
Fig. 1. Experimental setup: PZT, piezoelectric trans-
ducer; OPO, optical parametric oscillator; MOT, magneto-
optical trap. 1996 Optical Society of America
October 15, 1996 / Vol. 21, No. 20 / OPTICS LETTERS 1689In the first stage atoms are maintained in a MOT.7 In
the second stage the trapping beams are mechanically
chopped off, and the atoms are illuminated with the
three lasers at la, b, c with powers of approximately
0.5 nW, 0.5 nW, and 5 mW, respectively. We monitor
the excited-state population r33 by observing the f lu-
orescent decay 3 ! 2 at 917 nm, using an avalanche
photodiode.
The phase f (and thereby the excited-state popu-
lation r33) is externally modulated with a sawtooth
ramp applied to a moving mirror in the path of the
884-nm laser. During the long linear segment of the
ramp, we have fc ! fcstd . vmt (with vmy2p ,10–
20 Hz) and hence jfstdj ­ 2vmt. Note that Dv ­ 0
by virtue of the parametric process used to generate
vb from sva, vcd, although the same experiment could
be carried out with three totally independent lasers.
As illustrated in Fig. 2, a modulation of r33 at fre-
quency Vm ­ 2vm is observed with visibility V . 0.3,
thus demonstrating QI in the two-photon excitation
process. We stress that the modulation displayed in
Fig. 2 corresponds to demodulation of the optical fre-
quency difference jva, b 2 svc 1 vmdjy2p ø 12.5 THz.
Because the response of the nonlinear mixer in this
particular demonstration is centered around the spe-
cif ic frequencies shown in Fig. 1, its bandwidth is
necessarily quite narrow (being intrinsically associ-
ated with atomic linewidths), although one can achieve
an appreciable response for terahertz detunings for
sva, vbd by increasing the incident power to the mil-
liwatt level. However, our scheme of obtaining QI in
two-photon excitation is easily generalized to other
transitions of the form 1 ! 2 ! 3. Indeed, by expand-
ing our viewpoint to the alkali elements Li, Na, K, Rb,
and Cs but constraining consideration to only ladder
transitions for which each of the three wavelengths
involved is less than 2 mm,9 we find that there are
,6800 possible triangles of the form v13 ­ v12 1 v23,
each of which can be viewed as a different nonlinear
mixer for obtaining QI and hence phase-sensitive de-
tection of Dv ­ va 1 vb 2 2vc for fields of frequen-
cies va . v12, vb . v23, and vc . 1/2 v13.10 Note that
each of these triangles is characterized by its cen-
tral frequency 1/2 v13 and sideband separation Dn ­
1/2p jv12 2 1/2 v13j ø 1/2p j1/2v13 2 v23j. To illustrate
the density of coverage, we display in Fig. 3 Dn versus
lc ­ 2pcyvc for the transitions in the database, with
Dn typically 10–100 THz.
The capability for optical demodulation (Fig. 2), to-
gether with the numerous possibilities for nonlinear
mixing (Fig. 3), suggests that QI can be exploited
in two-photon excitation for applications in frequency
metrology. In particular, to obtain an absolute mea-
surement of a given target frequency in terms of a
handful of reference frequencies, one faces the daunting
challenge of bridging intervals that are often compara-
ble to the optical frequency itself. Numerous schemes
to conquer these large frequency intervals have been
investigated,11 – 14 including resonant multiwave mixing
in atomic vapors.15
Within this context, we propose a new method for ab-
solute comparison of an (arbitrary) target frequency vt
one or more reference frequencies vr1, vr2, . . . ,14 by ex-ploiting atoms as ultrafast nonlinear mixing elements.
As for the actual implementation, we would first con-
struct from the set of available frequencies V0 ­
hvt, vr1, vr2, . . .j a new set V1 ­ hw10, v2 0, . . . , vN1 0j,
where one allows for combinations of the form vi 6 vj
with hvi, vj j # V0. This process can be repeated for
as many as k stages to generate a final set of fre-
quencies Vk, with the number of resulting frequen-
cies growing exponentially in k. For example, with
two reference and one target lasers, k ­ 2 stages
of nonlinear conversion produces ,150 distinct fre-
quencies. Of course, finally, we need only three of
these frequencies to mix with the aid of an atom.
To decide which three must be generated, we con-
sider all possible triplets sva 0, vb 0vc 0d from the set
V ­ V0 < V1 . . . < Vk that can be traced back to
vt and at least one of the vpi ’s and keep only those
Fig. 2. Fluorescence from the 3 ! 2 transition at as
a function of time for illumination of the MOT optical
trap with three fields at la, b, c ­ s852, 917, 884d nm. Here
the relative phase fstd ­ 2f884std 2 f852 2 f917, where
f884std ­ vmt and vmy2p ­ 11 Hz. The solid curve is a
fit of a constant plus a sinusoidal variation with visibility
n ø 0.3.
Fig. 3. Nonlinear mixer characteristics for the sideband-
to-carrier separation Dn versus carrier wavelength lc from
our database of 6800 transitions in the alkali elements.
The circled point corresponds to the current experiment,
for which lc ­ 884 nm and Dn ­ 12.5 THz.
1690 OPTICS LETTERS / Vol. 21, No. 20 / October 15, 1996triplets that satisfy va0 1 vb 0 2 2vc 0 , Dv0, where
Dv0 is the maximum offset that can be either directly
detected or compensated by electro-optic modulation
sDv0 # 500 GHzd. These triplets are then cross refer-
enced to our database of two-photon transitions with a
search algorithm that minimizes the sum
P
of absolute
detunings,
P
­ jva 2 v12j 1 jvb 2 v23j 1 jvc 2 1/2 v13j,
and thereby chooses the particular atomic transition
for the measurement. Alternative criteria can also by
considered.
The complexity of possibilities that arise from this
database of transitions10 and from the set of possible
reference wavelengths14 does not allow for an explicit
algorithm to be presented here. However, a computer
program has been developed that analyzes the problem
as described above and finds optimized measurement
strategies for any target wavelength in the range 200–
2000 nm for a given set of reference wavelengths.
Particular examples of optimized solutions for the
measurement of several wavelengths have been found
by our program (including the 121.6-nm, Lyman-a
transition in atomic hydrogen) and will be discussed
elsewhere.16
Another possible application of our scheme is in
the establishment of frequency standards in the
1.3- and 1.5-mm bands relevant to optical commu-
nications.17 Toward this end we have constructed
measurement strategies for wavelengths between 1300
and 1500 nm in increments of 1 nm.16 Yet another po-
tential investigation is a test of the internal consistency
of existing frequency standards by employing some
as targets and others as references and then making
direct comparisons with atoms serving as nonlinear
mixers.
Turning finally to the issue of the conversion effi-
ciency for our scheme, we introduce an effective non-
linear coefficient ENL for the conversion of the powers
Pa, b, c into an equivalent power P
f
HET for the f luores-
cence carried at frequency d, where 2vc ­ va 1 vb 1 d.
Explicitly, we find P
f
HET ­ ENLsPc2PaPbd1/2, where
ENL ­ sE cNLE abNLd, with sE cNL, E abNLd denoting the inde-
pendent conversion efficiencies from the incident pow-
ers sPa, Pb , Pcd into f luorescent powers from the 3 ! 2
decay, including the collection efficiency e for a finite
solid angle. Note that the form for P
f
HET follows from
an interference of three field amplitudes equivalent
to the a3 term in the previous expression for r33 and
that a completely analogous expression can be derived
for Ha¨nsch’s divide-and-conquer scheme.12 Although
explicit forms for E ab, cNL will be presented elsewhere,4
we find for the case of our experiment that E abNL ,
4 3 106yW and E cNL , 1028yW, so that the effective
nonlinear coefficient ENL , 0.2yW (for a trap density of
109ycm3 and e , 0.08), which is apparently quite com-
petitive with more conventional mixing schemes even
given our (nonoptimal) setup. On the other hand, note
that for our initial experiment the saturation powers
are P sata, b , 0.5 nW and P
sat
a, b , 10 mW, so P
f
HET # 1 pW,
which, however, could be signif icantly increased with a
new trap design.In conclusion, by exploiting QI based on parallel ex-
citation paths for two-photon transitions, we have ob-
served the beat note arising from a small frequency
offset sø10 Hzd between three fields at wavelengths
of 852, 884, and 917 nm, demonstrating in this way
demodulation of optical offsets of 612.5 THz. By ex-
tending the idea of using atoms as nonlinear mixing
elements to more general two-photon transitions, we
have proposed a scheme for absolute-frequency com-
parisons between an arbitrary target frequency and a
small set of reference frequencies.
This research was supported by the Division of
Chemical Science, Office of Basic Energy Science, Of-
fice of Energy Research, U.S. Department of Energy.
We gratefully acknowledge discussions with J. L. Hall.
*Permanent address, Institute of Physics and
Astronomy, Aarhus University, Aarhus C, DK-8000,
Denmark.
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Frequency metrology by use of quantum interference

Details will appear on the Internet at http:// www.cco.caltech.edu/˜qoptics.

In principle V and L two-photon transitions will also exhibit QI. We could likewise include atoms other than the alkalis.

Our database of possible triangles was constructed from atomic energy levels for Li, Rb, and Cs Vols.

https://authors.library.caltech.edu/2935/1/GEOol96.pdf

Frequency metrology by use of quantum interference

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