arX

iv:q

uant

-ph/

0309

171v

2 7

Jun

200

4

Buffer-gas induced absorption resonances in Rb vapor

Eugeniy E. Mikhailov,∗ Irina Novikova, Yuri V. Rostovtsev, and George R. WelchDepartment of Physics and Institute for Quantum Studies,

Texas A&M University, College Station, Texas 77843-4242

(Dated: February 1, 2008)

We observe transformation of the electromagnetically induced transparency (EIT) resonance intothe absorption resonance in a Λ interaction configuration in a cell filled with 87Rb and a buffer gas.This transformation occurs as a one-photon detuning of the coupling fields is varied from the atomictransition. No such absorption resonance is found in the absence of a buffer gas. The width of theabsorption resonance is several times smaller than the width of the EIT resonance, and the changesof absorption near these resonances are about the same. Similar absorption resonances are detectedin the Hanle configuration in a buffered cell.

I. INTRODUCTION

The coherent interaction of atoms with electromag-netic fields has attracted increasing attention recentlyin studies of nonlinear and quantum optics as well asspectroscopy and precision metrology. Under the com-bined action of several resonant laser fields, atoms areoptically pumped into a coherent superposition of theground-state (hyperfine or magnetic sublevels) which isdecoupled from the original electromagnetic fields. Thatis, the atoms are in a so-called “dark” state. Such amedium possesses some unique optical properties, for ex-ample, coherent population trapping (CPT) [1, 2, 3], can-cellation of absorption due to electromagetically inducedtransparency (EIT) [4, 5, 6], and steep nonlinear disper-sion [4, 7, 8, 9]. The characteristic spectral width offeatures occurring due to these nonlinear effects is deter-mined by the inverse lifetime of an atom in the coherentsuperposition of ground states. Since radiative transi-tions are usually forbidden between these states, the co-herence can be preserved for a long time, and in atomiccells its lifetime is usually determined by the interactiontime of the atom with the laser beams [10, 11].

The addition of a buffer gas (inert gases, N2, CO2,CH4, etc.) to the atomic vapor is a common method forobtaining narrow EIT resonances. Because of the ex-tremely low spin-exchange cross-section, the collisions be-tween rubidium and buffer gas atoms or molecules do notdestroy the quantum coherence of the internal states ofthe atoms, but effectively prolong the time they stay in-side the laser beam(s). The processes of decoherence andredistribution of atomic population have been extensivelystudied in optical pumping experiments [12, 13, 14]. Sub-stantial narrowing of the dark resonance linewidth is re-ported in [15, 16, 17, 18].

In this paper we study the transformation of the trans-mission peaks corresponding to EIT into enhanced ab-sorption peaks for proper laser detuning. We present anextensive experimental and theoretical analysis of thiseffect, previously reported in [19, 20]. We observe nar-

∗Electronic address: evmik@leona.physics.tamu.edu

δ

F’=2, m’=0

δ

∆δ

F=2

F=1

F’=1, m’=0

F’=2

F’=1

F=2, m=−1F=2, m=1

∆

EE

EE+ −

E

S

p

Ed

d

(b)(a)

FIG. 1: (a) Three-level interaction scheme of three laser fieldswith 87Rb atoms: the long-lived coherence is created on hy-perfine ground-state sublevels with strong driving field Ed andweak probe (anti-Stokes) field Ep; the probe and Stokes fieldEs are generated by electro-optic modulation. (b) Hanle con-figuration: the ground-state coherence is created on magneticsublevels with two circularly polarized components E± of amonochromatic linearly polarized laser field. In both cases ∆is the one-photon detuning of the laser(s) from atomic res-onance, and δ is the two-photon detuning due to frequencymismatch in case (a) or an external magnetic field in case (b).

row enhanced absorption resonances in two experimen-tal arrangements: in a bi-chrimatic configuration, wherestrong and weak laser fields form a Λ scheme on twoground-state hyperfine sublevels (Fig. 1a), and in the de-generate Hanle configuration (Fig. 1b). In both situationthe EIT resonance is observed for the laser field(s) tunedto the atomic transition(s) due to formation of a non-interacting ground-state atomic coherence; in both casesthe lineshape of this transmission peak persists for anyvalue of the one-photon detuning unless a buffer gas isadded to the Rb vapor.

Before proceeding into the details of the experiment letus highlight the distinctions between the effect describedbelow, and similarly looking coherent effects:

(i) A transformation of enhanced transmission to en-hanced absorption CPT resonance has been reportedby Affolderbach et al. [21] in their experiments with abichromatic standing wave in hot atomic vapor. Be-cause of large Doppler broadening, moving atoms effec-

2

tively interact with a double-Λ level configuration, whichmay result in either suppression or enhancement of ab-sorption, depending on the relative phases of the drivingfields. This explanation, however, cannot be applied tothe present experimental data, since our experiments arecarried out with running waves.

(ii) Narrow absorption resonances in alkali vapors arealso the manifestation of Electromagnetically InducedAbsorption (EIA) [22, 23, 24, 25, 26, 27, 28], or closelyrelated enhanced absorption Hanle effect [29, 30, 31, 32,33]. In those experiments, a narrow absorption peak ap-pears for two laser fields of close frequencies interactingwith a quasi-degenerate two-level system. However, it isa general requirement in both cases that the degeneracyof the ground state must be lower than that of the excitedstate, i.e. F < F ′, which prohibits any dark state forma-tion. Narrow EIA resonances in this case are due to thespontaneous coherence transfer from the excited states ofthe atoms [25, 26, 34]. In addition, the experimental ar-rangements for traditional EIA experiments involve laserfields resonant with the corresponding atomic transitions,whereas the narrow absorption resonances discussed inthis paper appear for far-detuned laser fields.

(iii) Finally, the effect described below cannot be at-tributed to pressure induced extra resonances (PIER)(fora review of the collision-induced coherent effects see[35]). In PIER the extra resonance is connected to adressed state population that is non-vanishing only in thepresence of collisions. A relatively low buffer gas pressureused in our experiment does not allow the realization ofPEIR.

This paper is organized as follows. In the next Sec-tion we describe the experimental apparatus and mea-surement technique. In Section III we present the ex-perimental study of these resonances in the three-level Λscheme based on the hyperfine coherence in atomic cellswith different amounts of a buffer gas. Theoretical anal-ysis in given in Section IV. Resonant four-wave mixingand the lineshape of the associated resonances in a Stokesfield are discussed in Section V. A brief analysis of thecoherent resonances’ width is given in Section VI. InSection VII, we present the experimental results on theenhanced absorption resonances observed in the Hanle-configuration, together with a qualitative discussion ofthe their origin. A brief summary of the work appears inthe final Section.

II. EXPERIMENTAL SETUP

A schematic of the experimental setup is shown inFig. 2. An external cavity diode laser is tuned to the5S1/2 → 5P1/2 (D1) transition of 87Rb. A probe fieldEp (and an additional Stokes field ES) are producedby an electro-optic modulator (EOM) driven by a sta-ble narrow-band tunable microwave generator operatingat 6.835 GHz to match the 87Rb ground-state hyperfinetransition frequency. The probe and Stokes fields have

Ed

AnalyzerSpectrum

/4λ

Ed

ES

Ep

Ed

ES

Ep

87

PDEOMRb

FIG. 2: Schematic of the experimental setup.

equal intensities of approximately 10% of that of the drivefield. After the EOM, all fields pass through a single-mode optical fiber to create a clean spatial mode witha Gaussian radial intensity distribution and to increasethe diameter of the output beam to 7 mm. The fieldsare circularly polarized with a quarter-wave plate placedafter the fiber.

In this experiment we use several glass cells filled withisotopically enhanced 87Rb and various pressures andtypes (Ne, Kr) of a buffer gas. Each cell is placed in-side a 3-layer magnetic shield to screen the laboratorymagnetic field from the system and is heated to 60o Cto increase the density of the 87Rb vapor to approxi-mately 2.5 × 1011 cm−3. After traversing the cell, allthree fields are mixed on a fast photodiode with an addi-tional field shifted by 60 MHz with respect to the drivingfield. The resulting photo-current is measured with aspectrum analyzer to separate the transmission signalsof the probe and Stokes components (see Fig. 2). Thisdetection scheme has been described in [36, 37].

For our studies in the Hanle configuration, the electro-optical modulator is removed so that there is only onelinearly polarized monochromatic electromagnetic fieldpropagating through the cell. The transmitted intensityis recorded directly from a photodetector. A longitudinalmagnetic field for shifting the Zeeman sublevels is createdby a solenoid mounted inside the inner magnetic shield.

III. ENHANCED ABSORPTION DUE TO A

BUFFER GAS IN HYPERFINE Λ SCHEME

We start with the drive field resonant with 5S1/2F =2 → 5P1/2F

′ = 2 transition; simultaneously, the probefield couples the ground-state 5S1/2F = 1 level to thesame excited state. In that resonant Λ system weobserve expected narrow EIT resonance in the probefield transmission due to efficient optical pumping intonon-interacting dark state. The bottom row of Fig. 3shows the observed resonances for the cells with differentamount of a buffer gas. It is easy to see that all EITpeaks are nearly symmetric, which is in agreement withtheoretical predictions [38, 39, 40, 41] and our numeri-cal simulations. The main difference between differentcells is the narrowing the EIT width resulting from theincreased interaction time for higher buffer gas pressure.

3

00.250.5

0.751

-20 -10 0 10 20

100 Torr of Ne 30 Torr of Ne 0.12 Torr of Kr no buffer gas

two photon detuning δ (kHz)

Tra

nspa

renc

y

0 MHz

00.250.5

0.751 103 MHz

00.250.5

0.751 517 MHz

00.250.5

0.751 828 MHz

00.250.5

0.751 1138 MHz

0

0.2

0.4

-20 -10 0 10 20

0 MHz

0

0.2 103 MHz

0

0.2

0.4

0.6 517 MHz

00.20.40.60.8 828 MHz

0.20.40.60.8

1 1138 MHz

0.6

0.8

1

2173 MHz

0

0.1

0.2

-60 -40 -20 0 20 40 60

0 MHz

0

0.1

0.2 103 MHz

0.4

0.6

0.8

1 517 MHz

0.85

0.9

0.95 828 MHz

0.9

0.95

1 1138 MHz

0

0.05

-200 -100 0 100 200

0 MHz

0

0.05

0.1 103 MHz

0.20.40.60.8

1 517 MHz

0.85

0.9

0.95

1 724 MHz

FIG. 3: Transmission of the probe field as a function of two-photon detuning δ for various one-photon detunings ∆. Thevalues of ∆ are shown on the upper right corner of each graph. These data are recorded in the presence of (columns, leftto right) 100 Torr of Ne, 30 Torr of Ne, 0.12 Torr of Kr, and no buffer gas. All signals are normalized to the non-absorbedtransmission. The asymmetry of the resonance curves for ∆ = 828 and 1138 MHz in the cell with 0.12 Torr Kr, and for ∆ = 517and 740 MHz in the vacuum cell are due to the slope of the one-photon absorption contour. The intensities of the laser fieldsare Id ≃ 1.20mW/cm2 and Ip ≃ 0.13mW/cm2 .

As we tune the laser away from the atomic transition(while maintaining near-zero two photon detuning) theshape of the transmission resonance changes dependingon the presence of a buffer gas in a cell. For the vac-uum cell (the rightmost column in Fig. 3) the resonancestays symmetric while its width increases rapidly withone-photon detuning ∆ due to less efficient density nar-rowing.

However, the behavior is very different in cells with abuffer gas. As ∆ increases, the EIT resonance becomesasymmetric, and then gradually transforms into a nar-row absorption resonance. Let us emphasize here someof the important properties of these resonances. For ex-ample, the cell with 30 Torr of Ne (Fig. 3) shows that theamplitude of the enhanced absorption peak observed forlarge detuning (∆ ≈ 1 – 2 GHz) is comparable to, andsometimes larger than, the amplitude of the EIT peak at∆ = 0, and its width is narrower. Second, these narrowabsorption resonances are observed for the laser detun-ing exceeding the Doppler-broadened one-photon reso-nant absorption width (∆ > 1 GHz). Thus, the enhancedabsorption peaks appears on top of 100% transmission ofthe probe field.

The asymmetry in the EIT resonances for nonzero one-photon detuning of the laser fields has been observed byLevi et al. [42] in maser emission in the CPT process aswell as by Knappe et al. [43] . However, the reported

modification of the resonance lineshape is significantlyweaker. Knappe et al. also suggested an empirical ex-pression for the resonance line-shape:

f(δ) = γAγ + B(δ − δ0)

γ2 + (δ − δ0)2+ C (1)

Here δ is a two-photon (Raman) detuning, γ is an effec-tive width of the coherent resonance, and A, B, and C arefitting parameters which are functions of the one-photondetuning ∆. We introduce a shift δ0 of the resonance po-sition from the exact Raman condition to reproduce theexperimental data. One can see that this expression con-sists of symmetric and anti-symmetric Lorentzian func-tions of δ with amplitudes A and B respectively. Pa-rameter C reflects the residual absorption of the elec-tromagnetic field determined by incoherent processes.Taichenachev et al. [44] derived the analytical expressionsfor these coefficients in the limit of weak interaction fields(perturbation approach). The case of the strong (drive)field is discussed in the next Section.

It is convenient to write the coefficients A and B in thefollowing form:

A = D cos(φ) , B = D sin(φ) , (2)

so that the Eq. (1) can be written as:

f(δ) = ℜ

D(∆)eiφ(∆) γ

γ + i(δ − δ0)

+ C . (3)

4

0.5

π1.

0 π

0 500 1000 1500 2000 2500 3000

angl

e (r

adia

n)

one photon detuning ∆ to the blue side (MHz)

30Torr Ne0.12Torr Kr

vacuum

FIG. 4: Angle φ of two photon resonance for 87Rb cells withdifferent amount of a buffer gas.

In this case, the parameter D characterizes the amplitudeof the resonance, and the angle φ expresses the ratio be-tween the symmetric and anti-symmetric components inEq. (1). For example, φ = 0 represents a symmetricpeak of enhanced transmission, φ = π corresponds to asymmetric peak of enhanced absorption, and φ = ±π/2corresponds to a pure dispersion-like lineshape.

These parameters are shown as functions of one-photondetuning in different cells in Figs. 4 and 5. We note thatno deviation from the symmetric Lorentzian form is ob-served for the EIT resonance in a vacuum cell (φ = 0 forall detunings). However, buffered cells show the changefrom a symmetric transmission resonance (for ∆ = 0) toalmost symmetric absorption resonance for ∆ ≈ 700 MHzfor 0.12 Torr buffer gas pressure and ∆ ≈ 1.4 GHz for30 Torr buffer gas pressure. After reaching its maximum,the angle starts to decline again, although we never ob-serve the recovery of the symmetric EIT peak for largerdetunings.

IV. THEORETICAL ANALYSIS

Let us first give a brief pictorial explanation of the ef-fect. The observation of the narrow absorption resonancefor large laser detunings may be qualitatively understoodby using dressed-state picture. It is known that the in-teraction of the drive field with the transition betweenlevels |a〉 and |c〉 can be described as a splitting of theexcited state [45]. Fig. 6 shows the interaction schemefor the probe beam in the case of the far-detuned drivefield (∆ ≫ |Ωd|, γr).

For that conditions the transition between one of thedressed states (|D1〉 in our notation) and level |b〉 corre-sponds to a regular resonant absorption, and the tran-sition between another dressed state |D2〉 and level |b〉corresponds to a two-photon Raman transition. The de-structive interference of two dressed states occurs only

0

5

10

15

20

25

0 500 1000 1500 2000 2500 3000

ampl

itude

(ar

b.un

it)

one photon detuning ∆ to the blue side (MHz)

30Torr Ne0.12Torr Kr

vacuum

FIG. 5: Amplitude D of the two-photon resonance. For easiercomparison, the values of D are normalized to the resonanceamplitude at zero detuning.

in case of zero two-photon detuning between probe anddrive fields and corresponds to the frequency with zeroabsorption (EIT). For the ideal case of γbc = 0 (no groundstate relaxation) and assuming that all population is inlevel |b〉, the magnitudes of absorption and Raman tran-sition are the same while their widths are different: thewidth of the former is determined by the relaxation rateof optical coherence γ, while the width of the Ramantransition is given by γ|Ωd|2/∆2 and can be small forlarge ∆. However, if spin relaxation is not zero, the am-plitude of resonance dramatically decreases once the ab-sorption resonance is narrower than this relaxation, andfor γbc ≫ γ|Ωd|2/∆2 the line disappears. The additionof a buffer gas, therefore, improves the conditions for theobservation of that narrow resonance: higher buffer gaspressure produces additional pressure broadening of theexcited state ( i.e. makes γ bigger) and at the same timerestricts the diffusion of Rb atoms through the interac-tion region (i.e. makes γbc smaller). [74].

The dressed state picture provides a general idea aboutorigin of the narrow absorption resonances. Note, thatthe asymmetric lineshapes observed in the experimentcannot be fitted by the combination of two symmetricLorentzian peaks — one positive due to coherent popu-lation trapping in the dark state and one negative due tothe enhanced absorption by the bright state. The inter-ference between two dressed state has to be taken into ac-count for proper description of the coherent effects [46].

For more rigorous analysis we derive an expression forthe absorption coefficient of a weak probe field in Rbatoms. It has been shown that a closed three-level Λ-scheme as shown in Fig. 7 provides satisfactory descrip-tion of coherent effects in alkali atoms [41]. The propaga-tion equations for the density matrix elements are well-

5

∆

Ωp

D1⟩

D2⟩∆Ωd

2δ

a⟩

b⟩

∆Ωd

2

FIG. 6: Dressed states picture for three level Λ scheme. State

|D1〉 = ξ1

(

|a〉 + Ωd∆|c〉

)

has approximately the same energy

as bare state |a〉, and state |D2〉 = ξ2

(

|c〉 −Ω∗

d

∆|a〉

)

is close

to the two-photon resonance. The normalization coefficients

are ξ1,2 ≈ 1 + o( |Ωd|2

∆2 ). EIT is observed for δ = 0 (maximumquantum interference), and absorption resonance corresponds

to δ = − |Ωd|2

∆(probe is resonant with |D2〉).

|b

|a

|cγbc

Ωp

Ωd

γγ

∆δ

FIG. 7: Three-level Λ system.

known (see, e.g. [41]) and are the following:

ρbb = iΩ∗

pρab − iΩpρba + γrρaa − γbcρbb + γbcρcc (4)

ρcc = iΩ∗

dρac − iΩdρca + γrρaa − γbcρcc + γbcρbb (5)

ρab = −Γabρab + iΩp(ρbb − ρaa) + iΩdρcb (6)

ρca = −Γcaρca + iΩ∗

d(ρaa − ρcc) − iΩ∗

pρcb (7)

ρcb = −Γcbρcb − iΩpρca + iΩ∗

dρab (8)

where Ωd = ℘acEp/~ and Ωp = ℘abEd/~ are the Rabifrequencies of the drive and probe fields. The generalizeddecay rates are defined as:

Γba = γ + i(∆ + δ) , (9)

Γca = γ + i∆ , (10)

Γbc = γbc + iδ . (11)

Here γ = γr + γdeph is the polarization decay rate,γr ≃ 2π ·3 MHz is the radiative decay rate of the excitedstate, γdeph is the dephasing rate of the optical transi-tion due to non-radiative effects (γdeph/p ≃ 5 MHz/Torr[47]); γbc is the inverse lifetime of the coherence betweenground states |b〉 and |c〉, which is determined by thediffusion time of Rb atoms through the interaction re-gion and nonhomogeneity of the magnetic field due toimperfect screening. The presence of the buffer gas af-fects values of both γbc and γ. On one hand, it allows fora longer ground-state coherence lifetime. On the otherhand it broadens the optical transition, since γdeph growslinearly with buffer gas pressure [48].

Solving Eqs.(4)-(8) in a steady state regime and as-suming |Ωp| ≪ |Ωd|, we obtain the following expressionfor the linear susceptibility of the probe field:

χ = iκΓcb(ρ

(0)aa − ρ

(0)bb ) +

|Ωd|2Γca

(ρ(0)aa − ρ(0)

cc )

ΓabΓcb + |Ωd|2, (12)

Where κ = 38π Nλ2γr, N is the 87Rb density and λ is the

wavelength of the probe field.

The atomic population differences in the approxima-tion of strong driving field (|Ωd|2 ≫ γbcγ) are:

(ρ(0)aa − ρ

(0)bb ) ≃ − γbc∆

2 + γ|Ωd|22γbc∆2 + γ|Ωd|2

(13)

(ρ(0)aa − ρ(0)

cc ) ≃ − γbc(∆2 + γ2)

2γbc∆2 + γ|Ωd|2. (14)

By substituting these expressions into Eq. (12), we findthe absorption coefficient α = Imκ as a function oftwo-photon detuning δ:

α =κ

γ2 + ∆2

γbc∆2 + γ|Ωd|2

2γbc∆2 + γ|Ωd|2γbc|Ωd|2 + γδ2

γ2 + (δ − δ0)2, (15)

where

δ0 = |Ωd|2∆

γ2 + ∆2(16)

is the ac-Stark shift of the excited state, and

γ =

√

γ2|Ωd|4 + γ2bc∆

2(γ2 + ∆2)

γ2 + ∆2(17)

is the effective width of the two-photon transmission res-onance.

Using Eq. (15) for the absorption coefficient we cannow find expressions for the coefficients A, B, and Cin Eq. (1) which describe the line-shape of the two-photon resonance for the probe field propagating througha medium of length L. For the moment we restrictourselves to the case of optically thin media, so that

6

-50

0

50

A, a

.u.

-κL

/2γ

0

κL/2γ

κL/γ

A-50

0

50

B, a

.u.

-κL

/2γ

0

κL/2γ

B

0 1 2 3∆, GHz

0

100

C, a

.u.

0 2 4 6 8∆/γ

1-κL

/γ

1

C

-1

0

1

A, a

.u.

-1

0

B, a

.u.

0 200 400 600∆/γr

0

1

C, a

.u.

FIG. 8: Coefficients A, B, and C describing the line-shapeof the coherent resonance Eq. (1): extracted from the exper-imental data for 87Rb cell with 30 Torr (left column), calcu-lated using Eqs. (18–20) (middle column), and obtained bynumerical modelling (right column).

Ip(z) = Ip(0) · e−αz ≃ Ip(0)(1 − αz):

A = κLη|Ωd|2

γ2 + ∆2

γ|Ωd|2(γ2 − ∆2) − γbc(γ2 + ∆2)2

γ2|Ωd|4 + γ2bc∆

2(γ2 + ∆2)

(18)

B = −κLη∆

γ2 + ∆2(19)

C = 1 − κLηγ

γ2 + ∆2(20)

η =γbc∆

2 + γ|Ωd|22γbc∆2 + γ|Ωd|2

. (21)

It is easy to see that the coefficient C is well approxi-mated by the absorption of a weak probe field in a two-level scheme. The only difference is the coefficient ηwhich describes the redistribution of the atomic popula-tion between ground levels due to optical pumping: in thecase of a small one-photon detuning (∆ ≪ |Ωd|

√

γ/γbc)almost all atoms are in the state |b〉 and therefore η = 1,and if the laser is far-detuned, the populations of thelevels |b〉 and |c〉 are almost the same and η = 1/2.

Coefficient A, which describes the symmetric compo-nent of the resonant lineshape is a symmetric functionof the one-photon detuning. Its sign changes as ∆ getslarger: A is positive for small detunings, then it becomesnegative at ∆ ≈ γ − 2γbcγ

2/|Ωd|2 (i.e. the resonance be-comes absorptive). Coefficient B is an odd function of∆, and exactly zero for ∆ = 0.

A comparison of coefficients A, B, and C given byEqs. (18–20), with those obtained by fitting our experi-mental data for the Rb cell with 30 Torr of Ne is presentedin Fig. 8. One can immediately see that the theoreticalformulae qualitatively describe the dependence of the co-efficients as functions of one-photon detuning, althoughthey are not accurate enough for a quantitative analy-

sis. There are several reasons for this. On one hand, wewill show below that the thermal velocity distribution ofRb atoms is very important and has to be taken intoaccount. On the other hand, we assumed for simplicitythat the medium is optically thin which is not the case.One of the consequences of using optically thick mediumis density narrowing of the EIT resonances which will bediscussed in the following Section.

We perform a numerical simulation of the interactionof the drive and probe field with the three-level Λ sys-tem considered above. Our model takes into accountattenuation of both drive and probe fields as they prop-agate through Rb vapor. The thermal motion of atomsalso implies that atoms with different velocities “see” theelectromagnetic fields at shifted frequencies, and the fi-nal susceptibility has to be averaged over the Maxwellvelocity distribution:

χ(∆) =1√πku

∫ +∞

−∞

χ(∆ − kv)e−

(kv)2

(ku)2 d(kv) , (22)

where k = νd/c is the drive field wave-vector, and u =√

2kBT/M is the most probable thermal speed (here Tis the temperature of the vapor, M is the mass of Rbatom and kB is the Boltzmann constant). The residualDoppler broadening of the ground-state transition (∆ku)may be neglected due to Lamb-Dicke effect [18].

We can reproduce experimental spectra very accu-rately (compare the experimental spectra presented inFig. 3 with the the results of numerical simulations inFig. 9) except for the case of cells with small amount ofa buffer gas (< 1 Torr). This limit corresponds to theregime where the mean free path of Rb atoms is com-parable with the size of the interaction region. Underthese conditions the motion of Rb atoms in and out of thelaser beam has to be taken into account when calculatingthe ground state coherence lifetime. Unfortunately, thismechanism is beyond the theoretical model used in ournumeric simulations. For that reason we do not show anycalculated transmission spectra for the cell with 0.12 Torrof Kr. The results of the numerical simulation for the pa-rameters A, B and C are shown in Fig. 8c.

The calculated values of the resonance amplitude Dand the ratio φ are shown in Figs. 10 and 11. Thesedemonstrate that the inversion of the EIT resonance oc-curs in the cell with a buffer gas, whereas no absorp-tion resonances ever appear in the cell without buffergas. Although a noticeable asymmetry of the resonanceis expected for large laser detuning (∆ > 100γr) , theamplitude of the resonances is very small and hardly de-tectable.

It is also easy to see in Fig. 3 that for non-zero de-tuning ∆ the centers of both EIT and buffer-gas inducedabsorption resonances are shifted from zero two-photondetuning. One of the reasons for this effect is light shiftsof the atomic levels, as shown by Eq. (16). However,this expression fails to describe the behavior of the res-onance shift measured experimentally (Fig. 12). A morerealistic resonance shift as a function of laser detuning is

7

00.250.5

0.751

-20 -10 0 10 20

100 Torr of Ne 30 Torr of Ne no buffer gas

two photon detuning (δ) [kHz]

Tra

nspa

renc

y

0 MHz

00.250.5

0.751 103 MHz

00.250.5

0.751 517 MHz

00.250.5

0.751 828 MHz

0.25

0.5

0.75

1 1138 MHz

0

0.2

0.4

-20 -10 0 10 20

0 MHz

0

0.2

0.4 103 MHz

00.20.40.60.8 517 MHz

00.20.40.60.8

1 828 MHz

0.20.40.60.8

1 1138 MHz

0.6

0.8

1

2173 MHz

0

0.05

-200 -100 0 100 200

0 MHz

0

0.05

0.1 103 MHz

00.20.40.60.8

1 517 MHz

0.750.8

0.850.9

0.951 724 MHz

FIG. 9: Calculated probe field transmission spectra for the conditions corresponding the experimental data in Fig. 3. We usethe following parameters: γdeph = 450 MHz, γbc = 0.5 kHz for the cell with 100 Torr of Ne; γdeph = 150 MHz, γbc = 0.7 kHz forthe cell with 30 Torr of Ne; and γdeph = 0 γbc = 30 kHz for the cell without a buffer gas. These values are in good agreementwith published results for collisional dephasing [47, 49]. For all graphs ku = 250 MHz, Ωd = 2.5 MHz, and Ωp = 0.5 MHz.These values are reasonable close to estimated actual Rabi frequencies Ωd = 2.1 MHz and Ωp = 0.7 MHz. The small deviationbetween them can be attributed to the nonuniform spatial distribution of the laser intensities.

0

0.2

0.4

0.6

0.8

1

1.2

0 100 200 300 400 500 600

ampl

itude

(ar

b.un

it)

one photon detuning ∆/γr

buffervacuum

FIG. 10: Numerical calculation of the resonance strength(D) vs one-photon detuning for a probe field propagating ina medium with Ne buffer gas and with vacuum. One-photondetuning and resonance width are given in units of γr .

obtained by numerical simulation if Doppler averaging isperformed (Fig. 12, inset).

We note that both the prediction of the theoreticalmodel and the result of the numerical simulations pro-

0.5

π1.

0 π

0 100 200 300 400 500 600

angl

e (r

adia

n)

one photon detuning ∆/γr

buffervacuum

FIG. 11: Numerical calculation of angle (φ) vs one-photondetuning for a probe field propagating in a medium with Nebuffer gas and with vacuum. One-photon detuning and reso-nance width are given in units of γr .

vide only qualitative agreement with the experimentalresults. There are several major effects which are notconsidered here. For example, neither hyperfine struc-ture of the excited state nor Zeeman substructure of all

8

-10

-5

0

5

10

0 500 1000 1500 2000 2500 3000

shiff

t (kH

z)

one photon detuning ∆ to the blue side (MHz)

30Torr Ne0.12Torr Kr

vacuum

-0.001

0

0.001

0.002

0.003

0 100 200 300 400 500 600

δ 0/γ

r

∆/γr

FIG. 12: Two photon resonance shift (δ0) as a function ofone-photon detuning for the vacuum cell (squares), cell with0.12 Torr of Kr (stars) and with 30 Torr of Ne (crosses). Inset:

the result of the numerical simulation for the 30 Torr cell.

states are taken into consideration, although this mayhave a profound effect on the coherent interaction. Inaddition, no influence of the four-wave mixing processesis taken into account.

For the better understanding of the conditions forwhich the enhanced absorption resonance occurs, we nu-merically compare the resonance lineshape at a large one-photon detuning (∆ = 250γr) for the cases of a cellwith and without a buffer gas. We see that the absorp-tion resonances appear only for large enough drive laserpower Ωd in the cell with a buffer gas. In the cell with-out any buffer gas the resonance remains dispersion-like(φ ≈ π/2) and vanishingly weak.

V. INFLUENCE OF THE BUFFER GAS ON

FOUR-WAVE MIXING

So far we have completely ignored the presence of aStokes field ES in the medium. Many previous publi-cations [36, 50, 51, 52] have shown that dense coherentmedia contribute to significant enhancement of nonlinearprocesses. In our experiment the strong drive field ap-plied to the F = 1 level is scattered by the ground-statecoherence, which leads to the appearance of narrow reso-nances in the Stokes field transmission with the width de-termined by the ground-state coherence relaxation rate.For near resonant probe and drive fields (∆ ≈ 0)the Λsystem formed by the Stokes and the drive field appliedto F = 1 ground state level is far detuned (see Fig. 1a),and the effects of the associated ground-state coherenceon the probe field are negligible. However, all four fieldshave to be taken into account in the case of large laserdetuning, comparable with the ground state hyperfinesplitting.

The amplitude and the angle φ for the Stokes fieldtransmission resonance in the cell with 30 Torr of Ne are

0.5

π1.

0 π

0 500 1000 1500 2000 2500 30000

5

10

15

20

25

30

35

angl

e (r

adia

n)

ampl

itude

(ar

b.un

it)

one photon detuning ∆ to the blue side (MHz)

angleamplitude

FIG. 13: Angle (φ) and amplitude (D) of the two photonresonance for the generated Stokes field.

shown in Fig. 13. Please note that for ∆ = 0 we observea dispersion-like lineshape (φ ≈ 0.5π), which is differ-ent from the previously published experimental resultsfor the Stokes field generation initiated by spontaneousphotons [37]. As the laser detuning ∆ increases the line-shape of the Stokes field transmission changes, and theresonance is transformed into a symmetric transmissionpeak (φ ≈ 0).

VI. WIDTH OF THE PROBE AND STOKES

RESONANCES

The measured widths of the two-photon resonance γas functions of the one-photon detuning ∆ are shown inFig. 14 for cells with different buffer gas pressure. Againwe see the difference between cells with and without abuffer gas: in the latter case the width of the EIT res-onance increases significantly with one-photon detuning.However, for the buffered cells the resonance is broad-ened only near atomic resonance, but for larger detuningits width actually decreases with ∆.

Fig. 15 demonstrate that we may be able observe thenarrowing of resonances for large detuning even in a vac-uum cell. However according to Fig. 10 at the point whennarrowing takes place (∆/γr > 100) the amplitude of theresonance is so small that it is extremely hard to detectin the experiment.

Narrow resonances with good signal-to-noise ratio areimportant for many applications. For example, narrowEIT resonances are used for precision metrology [16, 53,54] and atomic clocks [55, 56]. Our experiments demon-strate that the coherent absorption resonances, observedfor the far-detuned Λ system, may have more attractivecharacteristics in terms of the resonance width and am-plitude than the EIT resonances observed for the zerodetuning. For example, in the cell with 30 Torr of Nethe amplitude of the absorption resonance for ∆ = 1.2–

9

0

5

10

15

20

25

0 500 1000 1500 2000 2500 3000

wid

th (

kHz)

one photon detuning ∆ to the blue side (MHz)

30Torr Ne0.12Torr Kr

0100200300400500600700

0 300 600 900

wid

th (

kHz)

FIG. 14: Width of the two photon resonance γ as a functionof one-photon detuning ∆ for 87Rb cell with 30 Torr of Ne(cross), 0.12 Torr of Kr (x), and without buffer gas (squares,inset). The minimum width of the EIT resonance in the vac-uum cell with no buffer gas is γ(∆ = 0) = 17 kHz.

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0 100 200 300 400 500 600

reso

nanc

e w

idth

in u

nits

of γ

r

one photon detuning ∆/γr

0

0.05

0.1

0.15

0.2

0 50 100 150 200

FIG. 15: Numerical calculation of resonance width (2γ)vs one-photon detuning for a probe field propagating in amedium with Ne buffer gas and with vacuum (inset).

2 GHz is larger than that of the EIT resonance, while itswidth is narrower (for example, γ(0)/γ(2GHz) ≈ 3.3).

According to Eq. (17) in the strong laser field limit(|Ωd|2 ≫ γbcγ) the width of the EIT resonance for small∆ does not depend on the ground-state coherence de-cay rate and is determined by power broadening: γ ≈|Ωd|2/γ, as in previous studies [38]. The resonance widthdecreases with one-photon detuning, and for ∆ ≫ γ itdrops as 1/∆2. Ultimately for ∆ ≫ |Ωd|

√

γ/γbc , thewidth of the resonance is determined by the coherencedecay rate γbc.

To describe the resonance width more carefully weagain have to take into account the velocity distribu-tion of Rb atoms in the cell. Also the variation of thelaser fields intensities as they propagate through the op-tically dense Rb vapor becomes important; for example,

0

5

10

15

20

25

0 500 1000 1500 2000 2500 3000

reso

nanc

e w

idth

(kH

z)

one photon detuning ∆ to the blue side (MHz)

~1/∆2

~e∆2/2(ku)2

probe fieldStokes field

FIG. 16: Width of the two photon resonance (γ) for theprobe and generated Stokes field.

the resonance width is reduced due to the density narrow-ing [57, 58]. We note that both effects are not very impor-tant for the far-detuned optical fields (∆ ≫ γ, ku) sincethe absorption of the medium is small, and the parame-ters describing the resonance lineshape do not change no-ticeably within the Doppler contour. Thus, for large de-tuning the dependence of the resonance width predictedby Eq. (17) is in good agreement with the experimentalpoints, as shown in Fig. 16.

To describe how the resonance width changes for small∆ we should take both propagation and Doppler averag-ing into account. Since it is virtually impossible to dothat exactly, we make a few simplifications that allow usto roughly predict the dependence of the resonance widthon one-photon detuning. First, we assume that the maincontribution to the observed resonance comes from thevelocity subgroup for which ∆−kv ≃ 0, i.e. the couplingfields are resonant with the atomic transitions. Then themain consequence of the laser detuning is the reductionof the number of atoms in this velocity group:

N(∆) ∝ Ne−

∆2

(ku)2 . (23)

Under these conditions the output intensity Iout is givenby

Iout ≃ Iine−κL

γbc|Ωd|2 e

−κL γδ2

|Ωd|4 . (24)

In this expression the first term represents the residualabsorption under EIT conditions, and the second one de-scribes the shape of the peak as a function of two-photondetuning. It is easy to see that the width of the reso-nance in this case is inversely proportional to the atomicdensity [57]:

γD(N) ≃ |Ωd|2√γγr

(

3

8πNλ2L

)−1/2

. (25)

10

Substituting the density of the resonant atoms given byEq. (23) we obtain the resonance width:

γD(∆) ∝ |Ωd|2√γγr

1√NL

e∆2

2(ku)2 . (26)

Fig. 16 shows that this formula works only for a smallfrequency region in the vicinity of one-photon resonancewhere the resonance is nearly symmetric and deviatesfor larger ∆, which is consistent with the assumptionswe made.

The width of the Stokes field resonance is also shownin Fig. 16. It has approximately the same shape as thecurve for the probe field, but it is worth mentioning thatits width can be narrower than that of the probe fieldwhile having a similar resonance amplitude.

VII. ENHANCED ABSORPTION DUE TO

ZEEMAN COHERENCE

Let us now consider another kind of coherence effectoriginating from coherent population trapping, namelynonlinear magneto-optical polarization rotation. An ex-tensive review of this effect is given in Ref. [59]. In thiscase, the long-lived coherence is created among ground-state Zeeman sublevels by a single electromagnetic field(in general elliptically polarized but usually taken as lin-ear in most experiments and analyses) where the Λ link(s)are formed by the two opposite circularly polarized com-ponents E± of the input field (Fig. 1b). In this con-figuration the transmission is very sensitive to the ap-plied magnetic field, which shifts sublevels with differ-ent magnetic quantum number m. An applied magneticfield B therefore creates a two-photon detuning δ pro-portional to the magnetic field: δ = 2µBB/~, whereµB is the Bohr magneton. In particular, the polar-ization of the incident laser field is changed dramati-cally as a result of the steep dispersion associated withCPT [60, 61]. This effect, known as the nonlinear Fara-day effect, or nonlinear magneto-optical polarization ro-tation, has been extensively studied over the last decadesin atomic beams [62, 63] and in glass cells with or withoutanti-relaxation coating [60, 61, 64, 65, 66, 67, 68, 69, 70].

There are a number of publications which demon-strate strong influence of a buffer gas on the ampli-tude and lineshape of polarization rotation and EIT res-onances [71, 72]. Here we show that presence of a buffergas also allows observation of narrow enhanced absorp-tion peaks for buffer gas pressure higher than 3 Torr.

The transmission spectrum of 87Rb 5S1/2F = 2 →5P1/2 F ′ = 1, 2 consists of two transitions, partially re-solved within the Doppler contour as shown in Fig. 1b.Zeeman coherence can be created on both transitions sothat linear absorption is suppressed for a linearly po-larized electromagnetic field even if the laser frequencydiffers somewhat from the exact atomic transition fre-quency. For example, if the laser is tuned exactly between

0

0.5

1

0

0.05

0.1

0.15

0.2

0

0.5

1

Tra

nsm

issi

on

-1×10-3

-5×10-4

0

Res

onan

ce c

ontr

ast

-2 -1 0 1 2 3Laser detuning (GHz)

0

0.5

1

-600 -400 -200 0 200 400 600Magnetic field (mG)

-0.02

-0.01

0

(a)

(b)

(c)

(d)

(e)

(f)

FIG. 17: Left column: transmission through the 87Rb cellswith (a) 1 Torr; (b)3 Torr (c) 10 Torr of Ne buffer gas as thelaser frequency is swept across the F = 2 → F ′ transitions.Solid line: transmission at zero magnetic field. In this casethe ground state coherence is unperturbed and we denote thetransmission as Tcoh. Dashed line: transmission at large mag-netic field (δ ≫ γ0). In this case the coherence is destroyed,and we denote the transmission as Tlin. Right column: Thepeak contrast in transmission as a function of magnetic field(two-photon detuning) for fixed laser frequency (shown as adotted line on the previous graphs) for the same cells. Thenormalization of the output signal is C = (Tcoh −Tlin)/Tcoh.Thus C > 0 is a manifestation of EIT shown in part (d), andC < 0 indicates enhanced absorption as seen in (e) and (f).Because of the difference in geometrical sizes of the cells, theatomic density is adjusted for each case so that Tlin ∼ 20%.Laser power is 2 mW.

the two transitions (∆ = 406 MHz), the transmission isstill enhanced by 40%.

This is true, however, only if there is no buffer gaspresent in the cell. The transmission spectra for differ-ent pressures of Ne buffer gas are shown in Fig. 17a, b,and c. One can easily see that the effect of EIT de-teriorates at higher buffer gas pressures. Moreover, theenhanced absorption appears for the frequency region be-tween the two transitions. When magnetic field is variedthe width of the observed resonances is similar to thatof EIT peak as shown in Fig. 17d, e, and f, which indi-cates that this effect is due to the ground-state coherence.Similar resonances have been observed in the cells withanti-relaxation wall coating [75]

Here we should emphasize that the reason for the en-hanced absorption resonances described in this Sectionis quite different from the effects described in the previ-ous sections, despite of their similarities (both effects areobserved only in buffered Rb cells and for nonzero one-photon detunings). In the case of Zeeman coherence, thenonlinear enhanced absorption may be explained by theinterplay of the matrix elements of the transitions in-volved in the Λ links. As the laser frequency changes,

11

dark states can be created for both F = 2 → F ′ = 1and F = 2 → F ′ = 2 transitions. However, becauseof the difference in relative sign in the transition matrixelements [48], these dark states are orthogonal:

|D〉2→1 =1√2

(|m = 1〉 + |m = −1〉) ; (27)

|D〉2→2 =1√2

(|m = 1〉 − |m = −1〉) ; (28)

Strictly speaking, because of this difference perfect CPTis not possible in this double-Λ system. However, thenatural width of both transitions (γr ≃ 3 MHz) ismuch smaller than their hyperfine splitting, and for thelaser resonant with one transition, the disturbance intro-duced by the other transition may be neglected. For thecell without buffer gas or anti-relaxation coating, atomsdo not change their velocity while moving through thelaser beam and interact with the same atomic transi-tion. Therefore we observe two Gaussian-shaped trans-mission peaks as the laser frequency is swept across theF = 2 → F ′ transitions.

In the presence of a buffer gas the situation is different.Since the hyperfine structure of the excited level is notcompletely resolved under Doppler broadening, there isa finite probability that an atom, optically pumped intothe dark state on one transition, may later change itsvelocity and become resonant with the other transition.Because of the sign difference, the previously prepareddark state becomes a bright state for that transition, andan atom absorbs light more readily than the one with-out coherence. Thus, for the symmetric scheme as shownin Fig. 1b, we would expect complete destruction of co-herence for the laser tuned exactly between two atomictransitions, so that the probabilities of an atom interact-ing with either of them are equal. In reality, however,the F = 2 → F ′ = 1 transition is stronger then theF = 2 → F ′ = 2 transition, and this imbalance resultsin the enhanced absorption for a small region of laserfrequencies.

VIII. SUMMARY

We have demonstrated significant changes of the EITresonance in warm Rb vapor with a buffer gas. We

showed that the resonance lineshape changes from sym-metric transparency peak to a dispersion-like signal andthen to nearly symmetric absorption peak as one-photondetuning of the laser fields from atomic transition in-creases. We also found that for large enough detuning,the resonance width was limited by the ground level co-herence decay rate (γbc), and there is a region of laserfrequencies with higher resonance contranst than for zeroone-photon detuning, while the width of the resonance isnarrower. Observation of this absorption-like resonanceis power dependent and takes place only for high enoughdrive power. We have also developed a simple theory andperformed the numerical stimulations that qualitativelydescribe the observations, and we have accounted for theorigin of the discrepancies between those and the experi-mental data. We compared the behavior of the probe andStokes field for large one-photon detuning, and showedthat the resonance width of these two fields behaves sim-ilarly, while the shapes of the resonances are different.

We have also shown that an induced absorption res-onance appears with one-photon detuning in the Hanleconfiguration, although its origin differes from the bichro-matic case. In the Hanle configuration, the enhancedabsorption results from the destructive interference be-tween the dark states created via different excited statesin the presence of velocity changing collisions.

The effects reported here differ from previously ob-served Electromagnetically Induced Absorption reso-nances and the absorption Hanle effect. Both of thosecases have strict requirements on selection rules that arenot required or met for our observations.

IX. ACKNOWLEDGMENTS

We thank V. A. Sautenkov, D. Budker, A. B. Matsko,A. Zhang, and M. O. Scully for the useful and stimulat-ing discussions, and M. Klein for careful reading of themanuscript. Financial support is provided by Office ofNaval Research.

[1] E. Arimondo and G. Orriols, Nuovo Cimento Lett. 17,333 (1976).

[2] B. D. Agapyev, M. B. Gornyi, B. G. Matisov, andY. V.Rozhdestvenskii, Usp. Fiz. Nauk 163, 1 (1993).

[3] E. Arimondo, Progress in Optics XXXV, 259 (1996).[4] M. O. Scully and M. S. Zubairy, Quantum Optics (Cam-

bridge University Press, Cambridge, UK, 1997).[5] S. E. Harris, Phys. Today 50, 36 (1997).[6] J. P. Marangos, J. Mod. Opt. 45, 471 (1998).

[7] S. E. Harris, J. E. Field, and A. Kasapi, Phys. Rev. A46, R29 (1992).

[8] O. Schmidt, R. Wynands, Z. Hussein, and D. Meschede,Phys. Rev. A 53, R27 (1996).

[9] F. Renzoni and E. Arimondo, Opt. Commun. 178, 345(2000).

[10] E. Arimondo, Phys. Rev. A 54, 2216 (1996).[11] M. Graf, E. Arimondo, E. S. Fry, D. E. Nikonov, G. G.

Padmabandu, M. O. Scully, and S. Y. Zhu, Phys. Rev.

12

A 51, 4030 (1995).[12] R. Bernheim, Optical Pumping (W. A. Benjamin, Inc.,

New York, 1965).[13] J. Vanier, A. Godone, and F. Levi, Phys. Rev. A 58,

2345 (1998).[14] W. Happer, Rev. Mod. Phys. 44, 169 (1972).[15] S. Brandt, A. Nagel, R. Wynands, and D. Meschede,

Phys. Rev. A 56, R1063 (1997).[16] R. Wynands and A. Nagel, Appl. Phys. B 68, 1 (1998).[17] M. Erhard, S. Nußmann, and H. Helm, Phys. Rev. A 62,

061802(R) (2000).[18] M. Erhard and H. Helm, Phys. Rev. A 63, 043813 (2001).[19] E. E. Mikhailov, V. A. Sautenkov, Y. V. Rostovtsev, and

G. R. Welch, JOSA B 21, 425 (2004).[20] E. E. Mikhailov, V. A. Sautenkov, I. Novikova, and

G. R. Welch, LANL e-Print archive (2004), URLhttp://arxiv.org/abs/quant-ph/0402120.

[21] C. Affolderbach, S. Knappe, R. Wynands, A. V.Taichenachev, and V. I. Yudin, Phys. Rev. A 65, 043810(2002).

[22] A. M. Akulshin, S. Barreiro, and A. Lezama, Phys. Rev.A 57, 2996 (1998).

[23] A. Lezama, S. Barreiro, and A. M. Akulshin, Phys. Rev.A 59, 4732 (1999).

[24] A. Lipsich, S. Barreiro, A. M. Akulshin, and A. Lezama,Phys. Rev. A 61, 053803 (2000).

[25] A. V. Taichenachev, A. M. Tumaikin, and V. I. Yudin,JETP Lett. 69, 819 (1999).

[26] A. V. Taichenachev, A. M. Tumaikin, and V. I. Yudin,Phys. Rev. A 61, 011802 (2000).

[27] M. Kwon, K. Kim, H. S. Moon, H. D. Park, and J. B.Kim, J. Phys. B 34, 2951 (2001).

[28] C. Y. Ye, Y. V. Rostovtsev, A. S. Zibrov, and Y. M.Golubev, Opt. Comm. 207, 227 (2002).

[29] Y. Dancheva, G. Alzetta, S. Cartaleva, M. Taslakov, andC. Andreeva, Opt. Comm. 178, 103 (2000).

[30] G. Alzetta, S. Cartaleva, Y. Dancheva, C. Andreeva,S. Gozzini, L. Botti, and A. Rossi, J. of Opt. B 3, 181(2001).

[31] F. Renzoni, C. Zimmermann, P. Verkerk, and E. Ari-mondo, J. Opt. B 3, S7 (2001).

[32] F. Renzoni, S. Cartaleva, G. Alzetta, and E. Arimondo,Phys. Rev. A 63, 065401 (2001).

[33] C. Andreeva, S. Cartaleva, Y. Dancheva, V. Biancalana,A. Burchianti, C. Marinelli, E. Mariotti, L. Moi, andK. Nasyrov, Phys. Rev. A 66, 012502 (2002).

[34] H. Failache, P. Valente, G. Ban, V. Lorent, andA. Lezama, Phys. Rev. A 67, 043810 (2003).

[35] G. S. Agarwal, Advan Atom Mol Opt Phys 29, 113(1991).

[36] 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).

[37] E. E. Mikhailov, Y. Rostovtsev, and G. R. Welch, Journalof Modern Optics 49, 2535 (2002).

[38] A. Javan, O. Kocharovskaya, H. Lee, and M. O. Scully,Phys. Rev. A 66, 013805 (2002).

[39] Y. Rostovtsev, I. Protsenko, H. Lee, and A. Javan, J.Mod. Opt. 49, 2501 (2002).

[40] E. Kuznetsova, O. Kocharovskaya, P. Hemmer, andM. O. Scully, Phys. Rev. A 66, 063802 (2002).

[41] H. Lee, Y. Rostovtsev, C. J. Bednar, and A. Javan, Appl.Phys. B 76, 33 (2003).

[42] F. Levi, A. Godone, J. V. ans S. Micalizio, and G. Mod-

ugno, Eur. Phys. J. D 12, 53 (2000).[43] S. Knappe, M. Stahler, C. Affolderbach,

A. Taichenachev, V. Yudin, and R. Wynands, Appl.Phys. B 76, 57 (2003).

[44] A. V. Taichenachev, V. I. Yudin, R. Wynands,M. Stahler, J. Kitching, and L. Hollberg, Phys. Rev. A67, 033810 (2003).

[45] Y. Li and M. Xiao, Phys. Rev. A 51, 4959 (1995).[46] B. Lounis and C. Cohen-Tannoudji, J. Phys. II 2, 579

(1992).[47] C. Ottinger, R. Scheps, G. W. York, and A. Gallagher,

Phys. Rev. A 11, 1815 (1975).[48] J. Vanier and C. Audoin, The Quantum Phisics of

Atomic Frequency Standards, vol. 1 (Adam Hilger;Philadelphia, 1989).

[49] W. Happer, Rev. Mod. Phys. 44, 169 (1972).[50] M. D. Lukin, M. Fleischhauer, A. S. Zibrov, H. G. Robin-

son, V. L. Velichansky, L. Hollberg, and M. O. Scully,Phys. Rev. Lett. 79, 2959 (1997).

[51] S. E. Harris and L. V. Hau, Phys. Rev. Lett. 82, 4611(1999).

[52] M. T. Johnsson and M. Fleischhauer, Phys. Rev. A 66,043808 (2002).

[53] C. Affolderbach, M. Stahler, S. Knappe, andR. Wynands, Appl. Phys. B 75, 605 (2002).

[54] D. Budker, D. F. Kimball, S. M. Rochester, V. V.Yashchuk, and M. Zolotorev, Phys. Rev. A 62, 043403(2000).

[55] J. Kitching, S. Knappe, and L. Hollberg, Appl. Phys.Lett. 81, 553 (2002).

[56] M. Merimaa, T. Lindvall, I. Tittonen, and E. Ikonen, J.Opt. Soc. Am. B 20, 273 (2003).

[57] M. D. Lukin, M. Fleischhauer, A. S. Zibrov, H. G. Robin-son, V. L. Velichansky, L. Hollberg, and M. O. Scully,Phys. Rev. Lett. 79, 2959 (1997).

[58] V. Sautenkov, M. Kash, V. Velichansky, and G. Welch,Laser Physics 9, 889 (1999).

[59] D. Budker, W. Gawlik, D. Kimball, S. Rochester,V. Yashchuk, and A. Weis, Rev. Mod. Phys. 74, 1153(2002).

[60] V. A. Sautenkov, M. D. Lukin, C. J. Bednar, I. Novikova,E. Mikhailov, M. Fleischhauer, V. L. Velichansky, G. R.Welch, and M. O. Scully, Phys. Rev. A 62, 023810 (2000).

[61] I. Novikova, A. B. Matsko, and G. R. Welch, Opt. Lett.26, 1016 (2001).

[62] G. Theobald, N. Dimarcq, V. Giordano, and P. Cerez,Opt. Comm. 71, 256 (1989).

[63] B. Schuh, S. I. Kanorsky, A. Weis, and T. W. Hansch,Opt. Comm. 100, 451 (1993).

[64] K. H. Drake, W. Lange, and J. Mlynek, Opt. Comm. 66,315 (1988).

[65] L. M. Barkov, D. A. Melik-Pashaev, and M. S. Zolotorev,Opt. Comm. 70, 467 (1989).

[66] P. E. G. Baird, M. Irie, and T. D. Wolfenden, J. Phys. B22, 1733 (1989).

[67] X. Chen, V. L. Telegdi, and A. Weis, Opt. Comm. 78,337 (1990).

[68] S. I. Kanorsky, A. Weis, and J. Skalla, Appl. Phys. B 60,S165 (1995).

[69] D. Budker, V. Yashchuk, and M. Zolotorev, Phys. Rev.Lett. 81, 5788 (1998).

[70] D. Budker, D. J. Orlando, and V. Yashchuk, Am. J. Phys.67, 584 (1999).

[71] A. S. Zibrov, I. Novikova, and A. B. Matsko, Opt. Lett.

13

26, 1311 (2001).[72] I. Novikova, A. B. Matsko, and G. R. Welch, Appl. Phys.

Lett. 81, 193 (2002).[73] U. D. Rapol, A. Wasan, and V. Natarajan, Phys. Rev. A

67, 053802 (2003).

[74] Recently the sub-natural narrowing of the Doppler-freeresonance for far-detuned incoherent fields in a vacuumcell has been demonstrated by U. D. Rapol et al. [73]

[75] D. Budker, private communication.