Superluminal propagation of an optical pulse in a Doppler broadened three-state, single channel active Raman gain medium

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by  K. J. Jiang

Superluminal propagation of an optical pulse in a Doppler broadened three-state,
single channel active Raman gain medium
K.J Jiang,1,2 L. Deng,1 and M.G. Payne1
1Electron & Optical Physics Division, NIST, Gaithersburg, Maryland 20899
2Center for Cold Atom Physics, Wuhan Institute of Physics and Mathematics, CAS, Wuhan, 200062, China
(Dated: February 9, 2008)
Using a single channel active Raman gain medium we show a (220±20)ns advance time for
an optical pulse of τFWHM = 15.4µs propagating through a 10 cm medium, a lead time that is
comparable to what was reported previously. In addition, we have verified experimentally all the
featuresassociatedwiththissinglechannelRamangainsystem. Ourresultsshowthatthereported
gain-assistedsuperluminalpropagationshouldnotbeattributedtotheinterferencebetweenthetwo
frequencies of the pumpfield.
7
0
PACSnumbers: 42.50.Gy, 92.60.Ta,42.65.-k
0
2

n In a recent work [1] in room-temperature Cs atom and gain line, and Steinburg et al. [9, 10] investigated single
a withtwonearbyRamanresonancescreatedbytwopump photon tunneling time and superluminal propagation in
J
frequencies, a 3.7µs optical pulse is observed to travel a medium with a gain doublet. Stenner et al. [11] have

2 superluminally with about 62 ns lead time, correspond- studied the information speed in superluminal propaga-
2
ing to about 1.6 % of the full width at half maximum tion.

(FWHM) of the probe pulse used in the experiment. In
1
theirconclusiontheauthorshaveattributedtheobserved
v
4 superluminal propagation to the interference effect be- We use 85Rb to demonstrate the superluminal propa-
6 tween different frequency components of the pump field gation of an optical pulse in a single frequency chan-
1 [2] (in the anomalous dispersion region provided by the nel active Raman gain medium (Fig. 1). An exter-
1 two resonances) and also to the gain that occurs at the nal cavity diode laser system produces nearly 400 mW
0 frequencyoftheprobepulse. Payneetal. [3]haveshown power and is locked to a rubidium reference cell. The
7 analytically, however, that in this case the superluminal output of the laser passes through a 3 GHz acousto-
0
/ propagation is not based on any interference effect. In- optical modulator (AOM) to generate two laser beams
h deed, they have shown that with a pump field of single with frequency difference of the ground state hyper-fine
p frequency, the superluminal propagation should still be levels |1i = |5S1/2,F = 2i and |3i = |5S1/2,F = 3i
t- present. In this work, we demonstrate experimentally of 85Rb atoms. The AOM is driven by a RF synthe-
n
that an active Raman gain medium enabled by a single sizer which allows accurate adjustment so that the two-
a
u frequency pump field can indeed bring about the super- photon resonance can be precisely maintained. A fre-
q luminalpropagation,aspredictedinRef.[3]. Thus,while quency shifter is then added to the path of the probe
: the attribution of the observed superluminal propaga- field, allowing accurate and independent tuning of the
v
tionisassistedbyRamangainiscorrectitisincorrectto small two-photon detuning. Figure 1 shows the en-
i
X attribute the origin of the superluminal propagation to ergy level diagram and relevant laser couplings for the
r aninterferencebetweenthe twopump frequencycompo- present experiment. The probe and the pump fields
a nents. It is the gain nature of the anomalous dispersion couple the |1i = |5S1/2,F = 2i → |2i = |5P1/2i and
that is at the root of the apparent superluminal propa- |3i = |5S1/2,F = 3i → |2i = |5P1/2i transitions, re-
gation observed. spectively. Both lasers are detuned, however, on the
high energy side of the 5P hyper fine manifold by
1/2
a large (≃3GHz) one-photon detuning. In this experi-
Before describing our experiments we first cite several ment the strong pump field serves two purposes: (1) it
pioneer works on superluminal propagations of optical is on the |1i = |5S ,F = 2i → |2i = |5P i res-
1/2 1/2
pulses in anomalous dispersion regions of optical me- onance (see discussion below), and (2) it also pumps
dia. This include the early work of Garrett and Mc- the |3i = |5S ,F = 3i → |2i = |5P i transitions
1/2 1/2
Cumber [4] on the propagationof a Gaussian light pulse with a large one-photon detuning. The former serves
through an anomalous dispersive medium , the study by as the optical pumping field that keeps nearly all pop-
Casperson and Yariv [5] on pulse proration in a high- ulation in the state |3i, whereas the latter provides the
gain medium, the work by Chu and Wong [6]on the lin- active Raman gain to a weak probe field tuned near the
ear pulse propagation in an absorbing medium. More |1i = |5S ,F = 2i → |2i = |5P i transition. It is in
1/2 1/2
recently, Chiao [7] investigated the superluminal propa- thisactiveRamangainregionwherewestudythesuper-
gationofwavepacketsintransparentmediawithinverted luminal propagation of the weak probe pulse. Contrary
atomic population, Bolda et al. [8] studied optical pulse to the two-frequency pump field scheme used in the ref.
propagationwithnegativegroupvelocityduetoanearby [1], there is no second pump frequency component that
2
can lead to the interference effect discussed in [1]. Fi- addition,κ =2πN ω |D |2/cwhereN andω arethe
12 0 p 21 0 p
nally, the cell for the medium is 100 mm in length and atom number density and frequency of the probe field,
25 mm in diameter with anti-reflection coating on both respectively. We note that the last term in the denom-
ends,andisplacedinatemperaturecontrolledenclosure. inator in Eq. (2b) leads to a small shift of the Raman
Experiments were carried out in the temperature region gain peak and a shift of the minimum group velocity.
of T =60oC to T =90oC. This term has the origin in the small ac Stark shift due
to the pump field.
Theoretically, the system described above can be un-
derstood using a simple life time broadened three-state Payne et al. have carried out a non-steady state calcu-
model where nearly all population remains in the initial lation and shown the group velocity of a probe pulse as
state |3i. This is because under our experimental con- (for the two-photon detuning |δ |=|∆ −∆ |≫γ ),
2ph p c 31
ditions the one-photon detuning is sufficiently large in
comparisonwith the Doppler broadened line widths and ∆2 δ − |Ωc|2 2
the ac Stark shift produced in the 5P hyperfine states V ≈− c(cid:16) 2ph ∆c (cid:17) , (3)
1/2 g κ |Ω |2
by the optical pumping field. Consequently,the Doppler 12 c
broadeningsofthehyperfinestatesareun-importantand where |∆ |,|∆ | ≫ γ (assuming γ ≈ γ ), and
p c 21 21 23
the contributionsby the two hyperfinestates canbe eas- |∆ |,|∆ |≫|Ω | have been used. We note that the neg-
p c c
ily included. Thus, we neglectthe opticalpumping field, ative sign indicates the superluminal propagation, thus
the Dopplerbroadening,andthehyperfinesplitting, and provingthe interferencebetweenthetwo-frequencycom-
assume that nearly all population is maintained in the ponents of a pump field is not the cause of the superlu-
state |3i. Since the probe field is very weak [12, 13], minal propagation. It is the active Raman gain that is
there is never appreciable population can be transfered responsible for apparent superluminal propagation.
to the state |1i.
Our experiment is aimed at demonstrating three predic-
The equations of motion for the relevant density matrix tions given in Eq. (3): (1) the existence of the non-
elements are given as (assuming ρ33 ≈1) distorted apparent superluminal propagation in a single
pump frequency channel, active Raman gain medium;
∂ρ
12 ≈ −iΩ∗ei∆ctρ −γ ρ , (1a) (2)thecharacteristicinverseparabolicdependenceofthe
∂t c 13 12 12 groupvelocityonthepumpfieldRabifrequency;and(3)
∂ρ
13 ≈ iΩ∗ei∆ptρ −iΩ e−i∆ctρ −γ ρ , (1b) quadratic dependence of the group velocity on the two-
∂t p 23 c 12 13 13 photon detuning (neglecting the small ac Stark shift).
∂ρ
23 ≈ iΩ e−i∆ptρ +iΩ e−i∆ct−γ ρ , (1c)
∂t p 13 c 23 23 In Fig. 2 we show a typical data of the advanced prop-
agation of a Gaussian probe pulse with FWHM pulse
where Ω =D E /(2~) and Ω =D E /(2~) are the
p 21 p0 c 23 c0 width of τ = 15.4µs. In order to show the ad-
FWHM
halfRabifrequenciesofthe probe(E )andpump(E )
p0 c0 vancedtime clearlywe haveplotted only a smallportion
fields for the respective transitions, D and γ are the
ij ij oftheprobepulseprofile. Thelower-leftpanelshowsthe
electric dipole moment and the decoherence rate of the
advance of the frontedge (solidline) in comparisonwith
respective transitions. In addition, ∆ and ∆ are one-
p c a reference Gaussian pulse (dashed line) that traverses
photondetuningstothestate|2ibytheprobeandpump
throughtheair,whereasthelower-rightpaneldepictsthe
fields.
advanceoftherareedgeincomparisonwiththesameref-
erence pulse. We have observed excellent S/N ratio [14]
Equation (1) must be solved self-consistently together
and the fitting of the data to a Gaussian using a stan-
with the wave equation for the pulsed probe field in or-
dard statisticalroutine has yielded the advanced time of
der to correctly predict the propagation characteristics
δt = (220±20) ns, or about 1.4 % of the FWHM pulse
of the probe field. Payne et al. have shown that the
widthoftheGaussianprobepulse. Thisiscomparableto
positive frequency part of the probe field amplitude, in
the advance time ofthe two-pump-frequencyexperiment
the case of un-focused plane wave and within the slowly
where 1.6 % advanced time has been reported [1].
varying amplitude and adiabatic approximations, must
satisfythewaveequationwhich,intheFouriertransform
In Fig. 3a we show the group velocity of a Gaussian
space, is given by
probe pulse as a function of the single frequency pump
fieldRabifrequency. Theinversequadraticbehavior(no-
∂Λ ω iκ |Ω |2W(ω)
p 12 c
−i Λ = Λ , (2a) tice the negative sign) of the group velocity as the func-
p p
∂z c (∆ −iγ )(∆ +iγ )
c 23 c 21 tion of the pump field Rabi frequency is in accordto the
1
prediction based on Eq. (3) when the pump field Rabi
W(ω)= , (2b)
ω−(∆ −∆ −iγ )− |Ωc|2 frequencyisrelativesmallthereforetheacStarkshiftcan
c p 31 (∆c+iγ21) beneglected. AsthepumpfieldRabifrequencyisstrong,
whereΛ isthe FouriertransformoftheprobefieldRabi theacStarkshiftterminEq. (3)mustbeconsideredto-
p
frequency Ω andω is the Fouriertransformvariable. In gether with the ground state population depletion. In
p
3
particular,a sizableac Starkshift leads to the transition becauseofthepresenceofthe“accidental”opticalpump-
from the inverse quadratic behavior to the quadratic in- ing field. In our simple three-state treatment given be-
crease of magnitude of the group velocity as can be seen fore, the optical pumping field is not included. Conse-
from Eq. (3) at large Ω limit. In this limit, however, quently, the possible generation of a FWM field is also
c
one should re-derive Eqs. (1-3) by including the equa- neglected. We neglect the “accidental” optical pumping
tion of motion for the ground state population and the field and the FWM field in our treatment because (1).
possible effects due to the “accidental” optical pumping theoretically with all four fields included, one could not
and a four-wave mixing (FWM) field generated by the find a clean analytical solution to the problem, there-
allowed transition |2i → |3i due to the “accidental” op- fore the extract of the essential physics is non-trivial.
tical pumping field. This complicated situation shall be In addition, steady-state solution is highly questionable.
studied later. This is because the equations of motion contain fast os-
cillating factorsthatcannotbe removedby simple phase
In Fig. 3b we show the group velocity of a probe transformation; (2). although the ”accidental” optical
pulse as a function of the two-photon detuning δ2ph = pumping field is on resonance with the |1i → |2i transi-
∆p−∆c. Equation (3) predicts that the group velocity tion, and therefore causes perturbation to the dispersion
is a quadratic function of the two-photon detuning. In properties of state |2i, this perturbation is not very sig-
addition, when |δ2ph|≫Max[|Ωc|2/|∆c|,γ31], the group nificant to the gain process and the propagation of the
velocity should be symmetric with respect to the two- probe field simply because the detunings from state |2i
photondetuning. This is indeedwhathas beenobserved are so large. It should be pointed out that the contri-
experimentally. Here,we plotthe redtwo-photondetun- bution by FWM is small in our case even with a sizable
ing portion of the group velocity dispersion curve. It is gain. Theoretically,however,the gain increases with the
worth noting that the vertical dashed line indicates the pump field intensity, and the ground state depletion will
approximatetwo-photondetuningwherethepropagation occur at sufficient gain. In this strong pumping regime,
characteristics of the probe pulse propagation changes the inclusion of the ”accidental” optical pumping field
from subluminal (above the horizontal dashed line) to and the FWM generation becomez necessary in order to
superluminal (below the horizontal dashed line). This accurately predict the propagationdynamics.
behavior is in complete agreement with the predictions
based on Eq. (2b). Indeed, from Eq. (2b) it is seen that
In conclusion we have verified experimentally all predic-
the when δ ≈ |Ω |2/∆ , the iγ term dominates the
2ph c c 31 tions based on Eq. (3) and ref. [3]. We have demon-
denominator[15], resulting in a subluminalpropagation.
strated experimentally that the superluminal propaga-
This point occurs at about, for red detuned two-photon
tionofanopticalpulse inanactiveRamangainmedium
detuning,
isnottheresultoftheinterferencebetweentwofrequency
componentsinthepumpfield. Indeed,withonlyasingle
|Ω |2
δ ≈ c ≈−2π×200kHz. frequency pump field where no second frequency com-
2ph
∆
c ponent exists to allow the possibilty of interference be-
tween the pump frequencies but with nearly the same
This is very close to what can be seen from Fig. 3b.
Raman gain coefficient (G ≃0.05 cm−1 in our ex-
Raman
It is also worth pointing out another observation that is oeriment and G ≃ 0.04 cm−1 reported in Ref. [1]) we
in accordwith the prediction of Eq. (2a,2b) and ref. [3]. have shown comparable advance times as reported be-
Wehaveobservedabout1%narrowingoftheprobepulse fore by other group. In addition, we have demonstrated
for |δ | ≤ 2π× 400 kHz. Superluminal propagation four features of the single channel active Raman gain
2ph
without pulse narrowing, such as those depicted in Fig. medium for superluminal propagation: (1) the inverse
2 are observed for |δ | > 2π× 400 kHz, as expected quadraticdependence ofthe groupvelocityasa function
2ph
from Eq. (3) and ref.[3]. With a shorter probe pulse of the pump field Rabi frequency; (2) the quadratic and
such as τ = 5µs, we have observed nearly 10 % symmetric dependence of the group velocity as a func-
FWHM
pulse narrowingin addition to a (295±20) ns lead time. tion of the two-photon detuning; (3) the group velocity
This lead time is about 5 % of the FWHM width of the dispersion region where the propagation characteristics
probe pulse used. changes from superluminal to subluminal; and (4) probe
pulse narrowing when |δ | < γ . None of these fea-
2ph 31
We now discuss the possible effect of the “accidental” tures has been demonstrated before with a narrowband
optical pumping field and the likely hood of the possible gain meidium.
FWM generationdue to the allowedtransition |2i→|3i
[1] L.J. Wang et al., Nature(London) 406, 277 (2000). [3] M.G. Payne and L. Deng, Phys. Rev. A 64, R031802
[2] Seethe conclusion of Ref.[1]. (2001).
4
[4] C.G.B. Garrett and D.E. McCumber, Phys. Rev. A 1, widthathalfmaximum(FWHM)τFWHM =15.4µs(top
305 (1970). panel, the dashed line is for a reference pulse). Here, we
[5] L. Casperson and A. Yariv, Phys. Rev. Lett. 26, 293 have plotted portions of the front (lower left) and rare
(1971). (lowerright)edgesofthe probepulse(solidline)incom-
[6] S.Chu and S.Wong, Phys. Rev.Lett 48, 738 (1982).
parison with a reference Gaussian pulse that travels in
[7] R.Y.Chiao, Phys. Rev.A 48, R34 (1993).
the air (dashed line). The verticalaxis is the normalized
[8] E.L. Bolda et al., Phys.Rev.A 48, 2938 (1994).
opticalintensity ofthe Gaussianprobepulse. The probe
[9] A.M. Steinberg,P.G. Kwiat and R.Y. Chiao, Phys. Rev.
pulsehasaslightlyhigherintensitybecauseoftheRaman
Lett.71, 708 (1993).
gain. There is, however, no detectable pulse spreading
[10] A.M. Steiberg and R.Y. Chiao, Phys. Rev. A 49, 2071
(1994). or distortion when the probe field amplitude is normal-
[11] M.D. Stenner, D.J. Gauthier, and M.A. Neifeld, Nature ized. Parameters: |Ωc|=2π×25 MHz, δ2ph ≈400 kHz,
(London) 425, 695 (2003). T =60oC, and ∆ =2π×3 GHz.
c
[12] M. Kozuma et al., Phys. Rev.A A66, R031801 (2002).
[13] L. Denget al., Appl.Phys.Lett. 81, 1168 (2002). Figure 3 (a) Plot of the group velocity of a Gaus-
[14] Inourexperiment,wechoosetheRamangain coefficient
sian probe pulse as a function of the pump field Rabi
tthoebgeaainbocuoteffiGcRiaemntanof≈G0.≈050c.0m4−c1m.−T1hirsepisorctoemdpianraRbelfe.[1to] frequency (a). As expected, the group velocity exhibits
the inverse quadratic dependence of the Rabi frequency
[15] Thatis,whenthedenominatorinEq.(2b)changesform
ofthe single channelpump field, as predicted inEq. (3).
amostlyrealtomostlyimaginary,thepropagationchar-
acteristics changes from superluminal to subluminal. For this plot δ2ph ≈ 2π×400 kHz, ∆c = 2π×2.2 GHz,
and temperature T =60oC. (b) Plot of the groupveloc-
Figure captions
ity of a Gaussian probe pulse as a function of the two-
Figure 1 Top panel: Energy level diagram and rele- photon detuning δ2ph. In the superluminal region where
vantlasercouplings. Levelassignment: |1i=|5S1/2,F = |δ2ph| ≫ γ31, the group velocity exhibits the quadratic
2i,|2i=|5P ,and|3i=|5S ,F =3i. Becauseofthe dependence on the two-photon detuning, as predicted
1/2 1/2
largeone-photondetuning, both the hyper fine splitting, fromEq. (3). When|δ2ph−|Ωc|2/∆c|<<γ31 thepropa-
Doppler broadening and smallac. Stark shift induced in gationcharacteristicschanges fromsuperluminal to sub-
thestate|2ihavebeenneglected. Lowerpanel: Schemat- luminal. For this plot |Ωc|=2π×25 MHz, ∆c =2π×3
ics of the experiment set up. GHz,andtemperatureT =60oC.TheRamangaincoef-
ficientof G ≃0.05 cm−1 used in our exoerimentis
Raman
Figure 2 Typical data (solid line) showing the ad- similarto the gaincoefficeintofG≃0.04cm−1 reported
vanced propagation of a Gaussian probe pulse of full in Ref. [1].
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