We experimentally study the group time delay for a light pulse propagating
through hot Rb vapor in the presence of a strong coupling field in a $\Lambda$
configuration. We demonstrate that the ultra-slow pulse propagation is
transformed into superluminal propagation as the one-photon detuning of the
light increases due to the change in the transmission resonance lineshape.
Negative group velocity as low as -c/10^6=-80 m/s is recorded. We also find
that the advance time in the regime of the superluminal propagation grows
linearly with increasing laser field power.Comment: 5 pages, 6 figure

A. B. Matsko

A. M. Akulshin

E. E. Mikhailov

Eugeniy E. Mikhailov

F. L. Kien

George R. Welch

I. Novikova

Irina Novikova

J. Vanier

M. Merimaa

R. Y. Chiao

Vladimir A. Sautenkov

English

arXiv

Mikhailov, Eugeniy E.

Sautenkov, Vladimir A.

Novikova, Irina

Welch, George R.

OAKTrust Digital Repository (Texas A&M Univ)

arXiv:quant-ph/0402120v1  18 Feb 2004Large negative and positive delay of optical pulses in coherently prepared dense Rbvapor with buffer gas.Eugeniy E. Mikhailov,1, 2, ∗ Vladimir A. Sautenkov,1, 3 Irina Novikova,1, 4 and George R. Welch11 Department of Physics and Institute for Quantum Studies,Texas A&M University, College Station, TX 778432 Center for Space Research, Massachusetts Institute of Technology, Cambridge, MA 021393 Lebedev Institute of Physics, 119991 Moscow, Russia4 Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138(Dated: August 23, 2018)We experimentally study the group time delay for a light pulse propagating through hot 87Rbvapor in the presence of a strong coupling field in a Λ configuration. We demonstrate that the ultra-slow pulse propagation is transformed into superluminal propagation as the one-photon detuningof the light increases due to the change in the transmission resonance lineshape. Negative groupvelocity as low as −c/106 = −80 m/s is recorded. We also find that the advance time in the regimeof the superluminal propagation grows linearly with increasing laser field power.PACS numbers: 42.50.Gy, 42.50.-p, 42.25.BsI. INTRODUCTIONCoherent population trapping (CPT) occurs whenthree-level atoms interact with two coherent electromag-netic fields in a Λ configuration. In this case, atoms areoptically pumped into the non-interacting superpositionof the two ground states (into the “dark” state), and thelinear absorption of the optical fields is greatly reduced.In addition to the absorption cancellation, such atomicmedia exhibit steep nonlinear dispersion. This steep dis-persion can result in significant reduction of the groupvelocity vg of a pulse propagating in a coherent medium,even though the refractive index n is still very close tounity:vg =cn+ ω ∂n∂ω. (1)Group velocity as low as a few meters per second has beendemonstrated experimentally in alkali atoms in Bose-Einstein condensates [1], thermal atomic vapor [2, 3],ruby crystal at room temperature [4] and in Pr dopedY2SiO5 crystals [5]. See Ref. [6] for a comprehensive re-view of slow light-related research.The amplitude and width of the transmission and dis-persion resonances, and therefore the nonlinear proper-ties of the medium, are largely determined by the timeatoms stay in the dark state. In thermal alkali vaporsthis time is limited by the interaction time of the atomswith the laser fields, which in turn is usually limited bythe average time-of-flight of an atom through the laserbeam. To prolong the interaction time, an inert buffergas is often added to the atomic vapor to slow the diffu-sion of the coherently prepared atoms through the laserbeam [7, 8, 9]. However, it has been recently demon-∗Electronic address: evmik@leona.physics.tamu.edustrated that significant modification of the dark reso-nance is possible in atomic vapor in the presence of buffergas [10, 11]. Namely, the initially symmetric narrowtransmission resonance is transformed into an asymmet-ric dispersion-like lineshape and then into a (potentiallynarrower) nearly symmetric absorption resonance as theone-photon detuning of the laser fields from the excitedstate is increased. Therefore a far-detuned Λ system mayexhibit superluminal propagation of the probe pulse dueto the steep anomalous dispersion associated with a nar-row absorption line.Although the superluminal propagation of a light pulsenear resonant absorption or gain lines has been knownand discussed for more than a century [12], interest inthis effect has been revived recently by the observationof ultra-low negative group velocity [13, 14, 15, 16, 17,18, 19, 20, 21]. Here we present an extensive study of thetime delay of probe pulses propagating in a Λ schemeunder the conditions of CPT. We trace the modificationof positive delay for zero one-photon detuning to negativedelay for large one-photon detuning. Also we present thedependence of the delay time on total laser power.II. EXPERIMENTAL SETUPA schematic of the experimental setup is shown inFig. 1b. An extended cavity diode laser is tuned to52S1/2F = 2 → 52P1/2F′ = 2 transition of 87Rb(λ = 795 nm). The laser output is phase modulatedusing an electro-optic modulator (EOM) at a frequencyclose to the 87Rb ground-state splitting (6.835 GHz), sothat one of the modulation sidebands is resonant with52S1/2F = 1 → 52P1/2F = 2 transition, forming theΛ system shown in Fig. 1a. The other sideband is far-detuned from all 87Rb transitions, and has no effect onthe phenomena described below. Approximately 7% ofthe total laser power is transferred to the probe field.After the EOM the probe and drive fields pass through2EdEpEdEp/4λEdEEdpF=2F=1∆δF’=2PDRbEOMMagnetic shieldPulse generator SpectrumAnalyzerPBSComputerτExternal cavitydiode laser(b)(a)FIG. 1: (a) 87Rb level diagram; (b) Schematic of the experi-mental apparatus.a single mode optical fiber to ensure a Gaussian spatialintensity distribution, and their polarization is made cir-cular by a high-quality polarizer followed by a quarterwave-plate. The overall light intensity is controlled bythe rotation of an additional polarizer placed after thefiber. The diameter of the laser beam at the entrance of87Rb cell is approximately 7 mm.The experiment is conducted using a 2.5 cm long cylin-drical glass cell filled with a mixture of isotopically en-hanced 87Rb and 30 Torr of Ne buffer gas. The cell isplaced inside a 3 layer magnetic shield to screen outthe Earth’s magnetic field, and maintained at 68o C,which results in an atomic density of 87Rb vapor N =4.7×1011 cm−3. After the cell the optical fields are mixedwith an additional optical field shifted by 60 MHz withrespect to the original laser frequency, and the amplitudeof the beat signal between this field and and the probefield is used to monitor the changes in the transmittedprobe field intensity.III. EXPERIMENTAL RESULTSTo observe the modification of the dark resonance line-shape we change the one-photon detuning ∆ by chang-ing the laser frequency, and then record the probe fieldtransmission as a function of two-photon detuning byscanning the microwave frequency driving the EOM. Inthe case of a nearly resonant Λ system (∆ = 0), a nar-row transmission resonance is observed due to coherentpopulation trapping. An example of such a dark reso-nance is shown in Fig. 2a. As the one-photon detuningincreases, the shape of the resonance changes, and forlarge detuning it becomes almost purely absorptive (seeFig. 2b). Note that the widths of both resonances are 0 0.2 0.4 0.6 0.8 1 1.2-20 -10  0 10 20Transmission Two-photon detuning δ (kHz)(a)(b)FIG. 2: The probe field transmission resonance for (a) zeroone-photon detuning ∆ = 0 and (b) for ∆ = 1.45 GHz. Theoutput signals are shown on the same scale and normalized tothe non-absorbed transmission. The total power of the laserbeam is 400 µW.similar (≈ 2.5 kHz) and are determined by the decay rateof the ground-state atomic coherence.The corresponding changes in the atomic dispersionfrom normal (∂n/∂ω > 0) to anomalous (∂n/∂ω < 0)produces a modification of the pulse delay from posi-tive (slow light) to negative (superluminal regime, or fastlight). Examples of such pulses are shown in Fig. 3. Asexpected the transmitted pulse for the resonant probefield is delayed with respect to the reference for τ =370 µs, whereas for the far-detuned Λ system the maxi-mum of the output pulse leaves the cell noticeably earlierthan the input. For the laser detuning of ∆ = 1.45 GHzthe advance time is 300 µs, which corresponds to a groupindex ng ≃ −4×106.It is easy to see that only minimal reshaping of bothretarded and advanced probe pulses is observed. In ei-ther case, the output pulses are still very close to theoriginal Gaussian waveform, although their widths areslightly reduced. For a 1 ms duration input probe pulsethe duration of the transmitted pulse is 0.94 ms in thecase of slow light propagation and 0.81 ms in the case ofsuperluminal propagation.It is interesting to track the variation of the group ve-locity as the shape of the dark resonance changes fromtransmission to absorption with increasing laser detun-ing. This dependence of the delay time on the laser fre-quency is shown in Fig. 4. This dependence is measuredin the following way: for each one photon detuning weadjust the the two-photon tuning of the probe field to thetransmission peak (maximum or minimum) by adjustingthe EOM modulation frequency. Then we propagate aGaussian temporal probe field pulse through the mediumand measure its relative delay with respect to the refer-ence pulse. Close to atomic resonance (∆ < 1 GHz) theEIT transmission peak is observed, although it becomesasymmetric as the laser detuning increases. The gap in3-3 -2 -1 0 1 2 3Signal (a.u.)time  (mS)(a)(b)(c)FIG. 3: Examples of transmitted pulses: (a) reference pulse(no atoms); (b) advanced pulse recorded at one-photon de-tuning ∆ = 1.45 GHz (total laser power was 700 µW); (c)retarded pulse recorded at ∆ = 0 (total laser power 145 µW).Solid lines represents the Gaussian lineshape. The amplitudesof all pulses are normalized for easier comparison.-400-300-200-100  0100200 0  500  1000  1500  2000  2500  3000Delay time τ (µs)Detuning ∆ (MHz)FIG. 4: Dependence of the positive and negative pulse delayson one photon laser detuning. Total power of the laser beamis 400 µW.measurements between 1 GHz and 1.4 GHz correspondsto the range where the dark resonance lineshape is toodispersion-like to find a meaningful peak. Although the-oretically both slow and superluminal group velocitiesmight be measured in this case for two slightly differenttwo-photon detunings, in practice it is rather difficult toperform such measurements. In this regime the measuredgroup velocity is extremely sensitive to any changes in theone-photon detuning, and therefore the instability of thelaser frequency creates a huge variation of the measuredpulse delays. Once the absorption resonance begins todominate (for ∆ > 1.4 GHz), we follow the minimumof the probe transmission by again adjusting the two-photon detuning appropriately.We have also studied the dependence of the group ve-locity on the drive field power. We measure the pulse de-lay both for near-resonant (EIT) and far-detuned fields(enhanced absorption resonance) as the laser intensity ischanged. The results are shown in Fig. 2. One can seethat in the case of slow pulse propagation the group de-lay is inversely proportional to the laser intensity. This-400-300-200-100  0100200300400 0  100  200  300  400  500  600  700Delay time τ, (µs)Laser power (µW)FIG. 5: The dependence of the positive and negative pulsedelays on the laser power. The two graphs are for ∆ = 0(x) and ∆ = 1.44 GHz (+). Solid lines represent fits to theexperimental data for the two cases: if the lasers are tuned tothe atomic resonances, τ ∝ 1/|Ω|2 , and for the far-detuned Λsystem τ ∝ −|Ω|2.dependence is well described by the power broadening ofthe dark resonance [2]:τ ≃38piNλ2Lγr|Ω|2. (2)This assumes the usual EIT conditions, namely |Ω|2 ≫γ0γ and |Ω|2 ≫√γ0/γWD, where Ω is the drive fieldRabi frequency, γr and γ are the radiative and total de-cay rates of the excited states, WD is the width of theDoppler-broadened absorption line, and γ0 is the darkstate decoherence rate.Very different behavior is observed for the superlumi-nal pulse propagation. Namely, the advance time lin-early increases with the laser power, which means thatthe steepness of the anomalous dispersion increases aswell. This behavior may be explained by the fact thatfor the far-detuned Λ system the width of the absorp-tion resonance is much less affected by power broaden-ing. The asymptotic behavior may be deduced from thegeneral expression for the dark resonance width [11]γEIT ≃ γ0 +γ2|Ω|42γ0∆4(3)The experimental measurements of the resonance widthvs. laser power are shown in Fig. 6 and are in good agree-ment with this expression. At the same time, the ampli-tude of the resonance is directly proportional to the laserpower. The combination of these two effects providesthe linear growth of the atomic dispersion and the probepulse advance time.We can check this conclusion in the following way. Thenonlinear dispersion of the atomic medium may be cal-culated analytically using the well-known density matrixequations for a three-level Λ system, if the motion of the40.00.51.01.52.02.53.03.5 0  100  200  300  400  500  600 0 5 10 15 20 25 30 35Resonance width (kHz)Resonance amplitude (Arb.units)Laser power (µW)FIG. 6: The width (+) and the amplitude (x) of the absorp-tion resonance as a function of the laser power. The data aretaken for one photon detuning ∆ = 1.44 GHz.atoms is neglected. Following the calculations presentedin [11] we find the time delay is given by:τ =38piNλ2Lγrγ2 +∆22γ0∆2 + γ|Ω|2×γ|Ω|4 − 2|Ω|2∆2γ2 +∆2(γ0∆2 + γ|Ω|2)γ20∆2(γ2 +∆2) + γ2|Ω|4(4)Although this expression is very cumbersome it reducesto Eq. (2) at one-photon resonance (∆ = 0). A simpleasymptotic behavior of the time delay may be also foundfor large detunings ∆2 ≫ γ/γ0|Ω|2 we findτ ≃ −38piNλ2Lγr|Ω|2γ20∆2. (5)This linear increase in pulse advance time with laserpower is just what is seen in Fig. 5.IV. DISCUSSIONAs previously mentioned, both large positive and neg-ative group delay are the manifestation of steep atomicdispersion, either normal or anomalous. This meansthat the accumulated phase of an electromagnetic fieldtraversing such a medium is very sensitive to even smallvariations in its frequency. Therefore, it may be mea-sured with high precision if phase-sensitive measure-ments of the output field are made. The cancellationof the linear absorption is undoubtedly the biggest ad-vantage of the steep dispersion associated with EIT, andhas been widely exploited for precision magnetic fieldmeasurements [8, 22, 23, 24, 25, 26] and atomic stan-dards [27, 28, 29, 30].In this light, our experimental data suggest thatthe narrow absorption resonances observed in bufferedatomic cells in a far-detuned Λ system are also good can-didates for precision measurements. We can estimate thephase ∆ϕ acquired by the probe field in this regime as:∆ϕ ≈ωcL∂n∂ωδ ≈ τγEIT (6)where we assume that the detuning of the probe fieldfrom the center of the absorption resonance δ is on theorder of the resonance width γEIT . Substituting the ex-perimental value for the group advance time τ ≃ 0.3 msand for the resonance width γ ≃ 2.5 kHz, we find themaximum phase difference to be about 5 rad, which ison the same order of magnitude as the large polarizationrotation angle in nonlinear Faraday rotation [31] and thelarge phase shifts induced by gradient magnetic fields instored light experiments [32].V. CONCLUSIONWe have studied the propagation of a weak probe pulsein an atomic cell filled with optically thick 87Rb va-por and Ne buffer gas for various one-photon detuningsfrom the upper atomic state of the Λ system. The pulsepropagation undergoes a transition from an ultra-slow toa superluminal regime as the dark resonance lineshapechanges from a narrow transmission to a narrow absorp-tion resonance. In the latter case, advance time as longas 300µs has been observed, which is equivalent to a neg-ative group velocity of vg ≃ −80 m/s.We have also studied the dependence of the group de-lay on the laser power. While the group velocity underEIT conditions increases linearly, in agreement with thetheoretical predictions, we see a linear growth of negativegroup delay with the laser power. The main reason is thatfor large one-photon detuning the contrast of the absorp-tion resonance increases with laser intensity, whereas thechange in the resonance width is negligible.The large anomalous dispersion associated with the re-ported absorption resonances is associated with high sen-sitivity of the phase of the probe field to the frequencyvariations, which may be find application in precisionmeasurements.AcknowledgmentsThe authors thank Yuri V. Rostovtsev, Andrey B.Matsko, A. Zhang and M. O. Scully for useful discus-sions. This work was supported by the the Office of NavalResearch, AFOSR, and the DARPA.5[1] L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi,Nature 397, 594 (1999).[2] M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg,G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, andM. O. Scully, Phys. Rev. Lett. 82, 5229 (1999).[3] D. Budker, D. F. Kimball, S. M. Rochester, and V. V.Yashchuk, Phys. Rev. Lett. 83, 1767 (1999).[4] M. Bigelow, N. Lepeshkin, and R. Boyd, Phys. Rev. Lett.90, 113903 (2003).[5] A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A.Musser, B. S. Ham, and P. R. Hemmer, Phys. Rev. Lett.88, 023602 (2002).[6] A. B. Matsko, O. Kocharovskaya, Y. Rostovtsev, G. R.Welch, A. S. Zibrov, and M. O. Scully, Advan Atom MolOpt Phys 46, 191 (2001).[7] S. Brandt, A. Nagel, R. Wynands, and D. Meschede,Phys. Rev. A 56, R1063 (1997).[8] R. Wynands and A. Nagel, Appl. Phys. B 68, 1 (1998).[9] M. Erhard and H. Helm, Phys. Rev. A 63, 043813 (2001).[10] E. E. Mikhailov, V. A. Sautenkov, Y. V. Rostovtsev,and G. R. Welch, LANL e-Print archive (2003), URLhttp://arxiv.org/abs/quant-ph/0309151.[11] E. E. Mikhailov, I. Novikova, Y. V. Rostovtsev, andG. R. Welch, LANL e-Print archive (2003), URLhttp://arxiv.org/abs/quant-ph/0309171.[12] R. Y. Chiao, Amazing Light (Springer, New York, 1996).[13] L. J. Wang, A. Kuzmich, and A. Dogariu, Nature 406,277 (2000).[14] A. Dogariu, A. Kuzmich, and L. J. Wang, Phys. Rev. A63, 053806 (2001).[15] A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, andR. Y. Chiao, Phys. Rev. Lett. 86, 3925 (2001).[16] A. Godone, F. Levi, and S. Micalizio, Phys. Rev. A 66,043804 (2002).[17] A. M. Akulshin, A. Cimmino, A. I. Sidorov, P. Han-naford, and G. I. Opat, Phys. Rev. A 67, 011801(R)(2003).[18] A. M. Akulshin, A. Cimmino, A. I. Sidorov, R. McLean,and P. Hannaford, J. Opt. B: Quantum Semiclass. Opt.5, S479 (2003).[19] M. Bigelow, N. Lepeshkin, and R. Boyd, Science 301,200 (2003).[20] M. Stenner, D. Gauthier, and M. Neifeld, Nature 425,695 (2003).[21] K. Kim, H. Moon, C. Lee, S. Kim, and J. Kim, Phys.Rev. A 68, 013810 (2003).[22] M. O. Scully and M. Fleischhauer, Phys. Rev. Lett. 69,1360 (1992).[23] M. Fleischhauer and M. O. Scully, Phys. Rev. A 49, 1973(1994).[24] M. Stahler, S. Knappe, C. Affolderbach, W. Kemp, andR. Wynands, Europhysics Lett. 54, 323 (2001).[25] D. Budker, D. F. Kimball, S. M. Rochester, V. V.Yashchuk, and M. Zolotorev, Phys. Rev. A 62, 043403(2000).[26] I. Novikova and G. R. Welch, J. Mod. Opt. 49, 349(2002).[27] J. Vanier, A. Godone, and F. Levi, Phys. Rev. A 58,2345 (1998).[28] J. Kitching, S. Knappe, and L. Hollberg, Appl. Phys.Lett. 81, 553 (2002).[29] M. Merimaa, T. Lindvall, I. Tittonen, and E. Ikonen, J.Opt. Soc. Am. B 20, 273 (2003).[30] J. Vanier, M. W. Levine, D. Janssen, and M. J. Delaney,IEEE Trans. Instrum. Meas. 52, 822 (2003).[31] I. Novikova, A. B. Matsko, and G. R. Welch, Opt. Lett.26, 1016 (2001).[32] A. Mair, J. Hager, D. F. Phillips, R. L. Walsworth, andM. D. Lukin, Phys. Rev. A 65, 031802 (2002).

Physical Review A

Large negative and positive delay of optical pulses in coherently prepared dense Rb vapor with buffer gas

Journals published by the American Physical Society can be found at http://publish.aps.org/We experimentally study the group time delay for a light pulse propagating through hot Rb-87 vapor in the presence of a strong coupling field in a Lambda configuration. We demonstrate that the ultraslow pulse propagation is transformed into superluminal propagation as the one-photon detuning of the light increases due to the change in the transmission resonance line shape. Negative group velocity as low as -c/10(6)=-80 m/s is recorded. We also find that the advance time in the regime of the superluminal propagation grows linearly with increasing laser field power

Mikhailov, E. E.

Sautenkov, V. A.

Novikova, I.

Large negative and positive delay of optical pulses in coherently prepared dense Rb vapor with buffer gas RID B-9041-2008

Texas A&M University

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Large negative and positive delay of optical pulses in coherently prepared dense Rb
vapor with buffer gas.
Eugeniy E. Mikhailov,1, 2, ∗ Vladimir A. Sautenkov,1, 3 Irina Novikova,1, 4 and George R. Welch1
1 Department of Physics and Institute for Quantum Studies,
Texas A&M University, College Station, TX 77843
2 Center for Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139
3 Lebedev Institute of Physics, 119991 Moscow, Russia
4 Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138
(Dated: August 23, 2018)
We experimentally study the group time delay for a light pulse propagating through hot 87Rb
vapor in the presence of a strong coupling field in a Λ configuration. We demonstrate that the ultra-
slow pulse propagation is transformed into superluminal propagation as the one-photon detuning
of the light increases due to the change in the transmission resonance lineshape. Negative group
velocity as low as −c/106 = −80 m/s is recorded. We also find that the advance time in the regime
of the superluminal propagation grows linearly with increasing laser field power.
PACS numbers: 42.50.Gy, 42.50.-p, 42.25.Bs
I. INTRODUCTION
Coherent population trapping (CPT) occurs when
three-level atoms interact with two coherent electromag-
netic fields in a Λ configuration. In this case, atoms are
optically pumped into the non-interacting superposition
of the two ground states (into the “dark” state), and the
linear absorption of the optical fields is greatly reduced.
In addition to the absorption cancellation, such atomic
media exhibit steep nonlinear dispersion. This steep dis-
persion can result in significant reduction of the group
velocity vg of a pulse propagating in a coherent medium,
even though the refractive index n is still very close to
unity:
vg =
c
n+ ω ∂n∂ω
. (1)
Group velocity as low as a few meters per second has been
demonstrated experimentally in alkali atoms in Bose-
Einstein condensates [1], thermal atomic vapor [2, 3],
ruby crystal at room temperature [4] and in Pr doped
Y2SiO5 crystals [5]. See Ref. [6] for a comprehensive re-
view of slow light-related research.
The amplitude and width of the transmission and dis-
persion resonances, and therefore the nonlinear proper-
ties of the medium, are largely determined by the time
atoms stay in the dark state. In thermal alkali vapors
this time is limited by the interaction time of the atoms
with the laser fields, which in turn is usually limited by
the average time-of-flight of an atom through the laser
beam. To prolong the interaction time, an inert buffer
gas is often added to the atomic vapor to slow the diffu-
sion of the coherently prepared atoms through the laser
beam [7, 8, 9]. However, it has been recently demon-
∗Electronic address: evmik@leona.physics.tamu.edu
strated that significant modification of the dark reso-
nance is possible in atomic vapor in the presence of buffer
gas [10, 11]. Namely, the initially symmetric narrow
transmission resonance is transformed into an asymmet-
ric dispersion-like lineshape and then into a (potentially
narrower) nearly symmetric absorption resonance as the
one-photon detuning of the laser fields from the excited
state is increased. Therefore a far-detuned Λ system may
exhibit superluminal propagation of the probe pulse due
to the steep anomalous dispersion associated with a nar-
row absorption line.
Although the superluminal propagation of a light pulse
near resonant absorption or gain lines has been known
and discussed for more than a century [12], interest in
this effect has been revived recently by the observation
of ultra-low negative group velocity [13, 14, 15, 16, 17,
18, 19, 20, 21]. Here we present an extensive study of the
time delay of probe pulses propagating in a Λ scheme
under the conditions of CPT. We trace the modification
of positive delay for zero one-photon detuning to negative
delay for large one-photon detuning. Also we present the
dependence of the delay time on total laser power.
II. EXPERIMENTAL SETUP
A schematic of the experimental setup is shown in
Fig. 1b. An extended cavity diode laser is tuned to
52S1/2F = 2 → 5
2P1/2F
′ = 2 transition of 87Rb
(λ = 795 nm). The laser output is phase modulated
using an electro-optic modulator (EOM) at a frequency
close to the 87Rb ground-state splitting (6.835 GHz), so
that one of the modulation sidebands is resonant with
52S1/2F = 1 → 5
2P1/2F = 2 transition, forming the
Λ system shown in Fig. 1a. The other sideband is far-
detuned from all 87Rb transitions, and has no effect on
the phenomena described below. Approximately 7% of
the total laser power is transferred to the probe field.
After the EOM the probe and drive fields pass through
2Ed
Ep
Ed
Ep
/4λ
Ed
E
Ed
p
F=2
F=1
∆
δ
F’=2
PD
Rb
EOM
Magnetic shield
Pulse generator Spectrum
Analyzer
PBS
Computer
τ
External cavity
diode laser
(b)
(a)
FIG. 1: (a) 87Rb level diagram; (b) Schematic of the experi-
mental apparatus.
a single mode optical fiber to ensure a Gaussian spatial
intensity distribution, and their polarization is made cir-
cular by a high-quality polarizer followed by a quarter
wave-plate. The overall light intensity is controlled by
the rotation of an additional polarizer placed after the
fiber. The diameter of the laser beam at the entrance of
87Rb cell is approximately 7 mm.
The experiment is conducted using a 2.5 cm long cylin-
drical glass cell filled with a mixture of isotopically en-
hanced 87Rb and 30 Torr of Ne buffer gas. The cell is
placed inside a 3 layer magnetic shield to screen out
the Earth’s magnetic field, and maintained at 68o C,
which results in an atomic density of 87Rb vapor N =
4.7×1011 cm−3. After the cell the optical fields are mixed
with an additional optical field shifted by 60 MHz with
respect to the original laser frequency, and the amplitude
of the beat signal between this field and and the probe
field is used to monitor the changes in the transmitted
probe field intensity.
III. EXPERIMENTAL RESULTS
To observe the modification of the dark resonance line-
shape we change the one-photon detuning ∆ by chang-
ing the laser frequency, and then record the probe field
transmission as a function of two-photon detuning by
scanning the microwave frequency driving the EOM. In
the case of a nearly resonant Λ system (∆ = 0), a nar-
row transmission resonance is observed due to coherent
population trapping. An example of such a dark reso-
nance is shown in Fig. 2a. As the one-photon detuning
increases, the shape of the resonance changes, and for
large detuning it becomes almost purely absorptive (see
Fig. 2b). Note that the widths of both resonances are
 0
 0.2
 0.4
 0.6
 0.8
 1
 1.2
-20 -10  0 10 20
Tr
an
sm
is
si
on
 
Two-photon detuning δ (kHz)
(a)
(b)
FIG. 2: The probe field transmission resonance for (a) zero
one-photon detuning ∆ = 0 and (b) for ∆ = 1.45 GHz. The
output signals are shown on the same scale and normalized to
the non-absorbed transmission. The total power of the laser
beam is 400 µW.
similar (≈ 2.5 kHz) and are determined by the decay rate
of the ground-state atomic coherence.
The corresponding changes in the atomic dispersion
from normal (∂n/∂ω > 0) to anomalous (∂n/∂ω < 0)
produces a modification of the pulse delay from posi-
tive (slow light) to negative (superluminal regime, or fast
light). Examples of such pulses are shown in Fig. 3. As
expected the transmitted pulse for the resonant probe
field is delayed with respect to the reference for τ =
370 µs, whereas for the far-detuned Λ system the maxi-
mum of the output pulse leaves the cell noticeably earlier
than the input. For the laser detuning of ∆ = 1.45 GHz
the advance time is 300 µs, which corresponds to a group
index ng ≃ −4×10
6.
It is easy to see that only minimal reshaping of both
retarded and advanced probe pulses is observed. In ei-
ther case, the output pulses are still very close to the
original Gaussian waveform, although their widths are
slightly reduced. For a 1 ms duration input probe pulse
the duration of the transmitted pulse is 0.94 ms in the
case of slow light propagation and 0.81 ms in the case of
superluminal propagation.
It is interesting to track the variation of the group ve-
locity as the shape of the dark resonance changes from
transmission to absorption with increasing laser detun-
ing. This dependence of the delay time on the laser fre-
quency is shown in Fig. 4. This dependence is measured
in the following way: for each one photon detuning we
adjust the the two-photon tuning of the probe field to the
transmission peak (maximum or minimum) by adjusting
the EOM modulation frequency. Then we propagate a
Gaussian temporal probe field pulse through the medium
and measure its relative delay with respect to the refer-
ence pulse. Close to atomic resonance (∆ < 1 GHz) the
EIT transmission peak is observed, although it becomes
asymmetric as the laser detuning increases. The gap in
3-3 -2 -1 0 1 2 3
Si
gn
al
 (a
.u.
)
time  (mS)
(a)
(b)
(c)
FIG. 3: Examples of transmitted pulses: (a) reference pulse
(no atoms); (b) advanced pulse recorded at one-photon de-
tuning ∆ = 1.45 GHz (total laser power was 700 µW); (c)
retarded pulse recorded at ∆ = 0 (total laser power 145 µW).
Solid lines represents the Gaussian lineshape. The amplitudes
of all pulses are normalized for easier comparison.
-400
-300
-200
-100
  0
100
200
 0  500  1000  1500  2000  2500  3000
D
el
ay
 ti
m
e 
τ 
(µs
)
Detuning ∆ (MHz)
FIG. 4: Dependence of the positive and negative pulse delays
on one photon laser detuning. Total power of the laser beam
is 400 µW.
measurements between 1 GHz and 1.4 GHz corresponds
to the range where the dark resonance lineshape is too
dispersion-like to find a meaningful peak. Although the-
oretically both slow and superluminal group velocities
might be measured in this case for two slightly different
two-photon detunings, in practice it is rather difficult to
perform such measurements. In this regime the measured
group velocity is extremely sensitive to any changes in the
one-photon detuning, and therefore the instability of the
laser frequency creates a huge variation of the measured
pulse delays. Once the absorption resonance begins to
dominate (for ∆ > 1.4 GHz), we follow the minimum
of the probe transmission by again adjusting the two-
photon detuning appropriately.
We have also studied the dependence of the group ve-
locity on the drive field power. We measure the pulse de-
lay both for near-resonant (EIT) and far-detuned fields
(enhanced absorption resonance) as the laser intensity is
changed. The results are shown in Fig. 2. One can see
that in the case of slow pulse propagation the group de-
lay is inversely proportional to the laser intensity. This
-400
-300
-200
-100
  0
100
200
300
400
 0  100  200  300  400  500  600  700
D
el
ay
 ti
m
e 
τ,
 
(µs
)
Laser power (µW)
FIG. 5: The dependence of the positive and negative pulse
delays on the laser power. The two graphs are for ∆ = 0
(x) and ∆ = 1.44 GHz (+). Solid lines represent fits to the
experimental data for the two cases: if the lasers are tuned to
the atomic resonances, τ ∝ 1/|Ω|2 , and for the far-detuned Λ
system τ ∝ −|Ω|2.
dependence is well described by the power broadening of
the dark resonance [2]:
τ ≃
3
8pi
Nλ2L
γr
|Ω|2
. (2)
This assumes the usual EIT conditions, namely |Ω|2 ≫
γ0γ and |Ω|
2 ≫
√
γ0/γWD, where Ω is the drive field
Rabi frequency, γr and γ are the radiative and total de-
cay rates of the excited states, WD is the width of the
Doppler-broadened absorption line, and γ0 is the dark
state decoherence rate.
Very different behavior is observed for the superlumi-
nal pulse propagation. Namely, the advance time lin-
early increases with the laser power, which means that
the steepness of the anomalous dispersion increases as
well. This behavior may be explained by the fact that
for the far-detuned Λ system the width of the absorp-
tion resonance is much less affected by power broaden-
ing. The asymptotic behavior may be deduced from the
general expression for the dark resonance width [11]
γEIT ≃ γ0 +
γ2|Ω|4
2γ0∆4
(3)
The experimental measurements of the resonance width
vs. laser power are shown in Fig. 6 and are in good agree-
ment with this expression. At the same time, the ampli-
tude of the resonance is directly proportional to the laser
power. The combination of these two effects provides
the linear growth of the atomic dispersion and the probe
pulse advance time.
We can check this conclusion in the following way. The
nonlinear dispersion of the atomic medium may be cal-
culated analytically using the well-known density matrix
equations for a three-level Λ system, if the motion of the
40.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
 0  100  200  300  400  500  600
 0
 5
 10
 15
 20
 25
 30
 35
R
es
on
an
ce
 w
id
th
 (k
Hz
)
R
es
on
an
ce
 a
m
pl
itu
de
 (A
rb.
un
its
)
Laser power (µW)
FIG. 6: The width (+) and the amplitude (x) of the absorp-
tion resonance as a function of the laser power. The data are
taken for one photon detuning ∆ = 1.44 GHz.
atoms is neglected. Following the calculations presented
in [11] we find the time delay is given by:
τ =
3
8pi
Nλ2Lγr
γ2 +∆2
2γ0∆2 + γ|Ω|2
×
γ|Ω|4 − 2|Ω|2
∆2
γ2 +∆2
(γ0∆
2 + γ|Ω|2)
γ20∆
2(γ2 +∆2) + γ2|Ω|4
(4)
Although this expression is very cumbersome it reduces
to Eq. (2) at one-photon resonance (∆ = 0). A simple
asymptotic behavior of the time delay may be also found
for large detunings ∆2 ≫ γ/γ0|Ω|
2 we find
τ ≃ −
3
8pi
Nλ2L
γr|Ω|
2
γ20∆
2
. (5)
This linear increase in pulse advance time with laser
power is just what is seen in Fig. 5.
IV. DISCUSSION
As previously mentioned, both large positive and neg-
ative group delay are the manifestation of steep atomic
dispersion, either normal or anomalous. This means
that the accumulated phase of an electromagnetic field
traversing such a medium is very sensitive to even small
variations in its frequency. Therefore, it may be mea-
sured with high precision if phase-sensitive measure-
ments of the output field are made. The cancellation
of the linear absorption is undoubtedly the biggest ad-
vantage of the steep dispersion associated with EIT, and
has been widely exploited for precision magnetic field
measurements [8, 22, 23, 24, 25, 26] and atomic stan-
dards [27, 28, 29, 30].
In this light, our experimental data suggest that
the narrow absorption resonances observed in buffered
atomic cells in a far-detuned Λ system are also good can-
didates for precision measurements. We can estimate the
phase ∆ϕ acquired by the probe field in this regime as:
∆ϕ ≈
ω
c
L
∂n
∂ω
δ ≈ τγEIT (6)
where we assume that the detuning of the probe field
from the center of the absorption resonance δ is on the
order of the resonance width γEIT . Substituting the ex-
perimental value for the group advance time τ ≃ 0.3 ms
and for the resonance width γ ≃ 2.5 kHz, we find the
maximum phase difference to be about 5 rad, which is
on the same order of magnitude as the large polarization
rotation angle in nonlinear Faraday rotation [31] and the
large phase shifts induced by gradient magnetic fields in
stored light experiments [32].
V. CONCLUSION
We have studied the propagation of a weak probe pulse
in an atomic cell filled with optically thick 87Rb va-
por and Ne buffer gas for various one-photon detunings
from the upper atomic state of the Λ system. The pulse
propagation undergoes a transition from an ultra-slow to
a superluminal regime as the dark resonance lineshape
changes from a narrow transmission to a narrow absorp-
tion resonance. In the latter case, advance time as long
as 300µs has been observed, which is equivalent to a neg-
ative group velocity of vg ≃ −80 m/s.
We have also studied the dependence of the group de-
lay on the laser power. While the group velocity under
EIT conditions increases linearly, in agreement with the
theoretical predictions, we see a linear growth of negative
group delay with the laser power. The main reason is that
for large one-photon detuning the contrast of the absorp-
tion resonance increases with laser intensity, whereas the
change in the resonance width is negligible.
The large anomalous dispersion associated with the re-
ported absorption resonances is associated with high sen-
sitivity of the phase of the probe field to the frequency
variations, which may be find application in precision
measurements.
Acknowledgments
The authors thank Yuri V. Rostovtsev, Andrey B.
Matsko, A. Zhang and M. O. Scully for useful discus-
sions. This work was supported by the the Office of Naval
Research, AFOSR, and the DARPA.
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Large negative and positive delay of optical pulses in coherently
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Large negative and positive delay of optical pulses in coherently prepared dense Rb vapor with buffer gas

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