Effects of thermal motion on electromagnetically induced absorption
aa r X i v : . [ qu a n t - ph ] J u l Effects of thermal motion on electromagnetically induced absorption
E. Tilchin and A. D. Wilson-Gordon
Department of Chemistry, Bar-Ilan University, Ramat Gan 52900, Israel
O. Firstenberg
Department of Physics, Technion-Israel Institute of Technology, Haifa 32000, Israel
We describe the effect of thermal motion and buffer-gas collisions on a four-level closed N systeminteracting with strong pump(s) and a weak probe. This is the simplest system that experienceselectromagnetically induced absorption (EIA) due to transfer of coherence via spontaneous emissionfrom the excited to ground state. We investigate the influence of Doppler broadening, velocity-changing collisions (VCC), and phase-changing collisions (PCC) with a buffer gas on the EIA spec-trum of optically active atoms. In addition to exact expressions, we present an approximate solutionfor the probe absorption spectrum, which provides physical insight into the behavior of the EIApeak due to VCC, PCC, and wave-vector difference between the pump and probe beams. VCC areshown to produce a wide pedestal at the base of the EIA peak, which is scarcely affected by thepump-probe angular deviation, whereas the sharp central EIA peak becomes weaker and broaderdue to the residual Doppler-Dicke effect. Using diffusion-like equations for the atomic coherencesand populations, we construct a spatial-frequency filter for a spatially structured probe beam andshow that Ramsey narrowing of the EIA peak is obtained for beams of finite width. PACS numbers: 42.50.Gy, 32.70.Jz
I. INTRODUCTION
The absorption spectrum of a weak probe, interact-ing with a pumped nearly-degenerate two-level transi-tion, can exhibit either a sharp subnatural dip or peakat line center [1], depending on the degeneracy of thelevels, the polarizations of the fields, and the absence orpresence of a weak magnetic field. The phenomenon istermed electromagnetically-induced transparency (EIT)[2, 3] when there is a dip in the probe spectrum and elec-tromagnetically induced absorption (EIA) [4] when thereis a peak.In the case of orthogonal polarizations of the pump andprobe, both EIT and EIA are related to the ground-levelZeeman coherence, which is induced by the simultane-ous action of both fields. The simplest model systemthat exhibits EIT is the three-level Λ system, where thetwo lower states g , are Zeeman sublevels of the groundhyperfine level F g . In a Λ system, quantum coherencecan lead to the destructive interference between the twopossible paths of excitation. As a result, if the pumpfield is tuned to resonance, the narrow dip in the probeabsorption spectrum at the two-photon resonance can beinterpreted as EIT caused by a coherent population trap-ping [5] in the lower levels. The simplest system thatexhibits EIA is the four-level N system [6, 7] (Fig. 1,top), consisting of states g , and e , which are Zeemansublevels of the ground ( F g ) and excited ( F e ) hyperfinelevels, where the g i ↔ e i , i = 1 ,
2, transitions interactwith non-saturating pump(s), and the g ↔ e transi-tion interacts with a weak probe. The N system givessimilar results to those obtained for a closed alkali-metal F g → F e = F g + 1 transition interacting with a σ ± po-larized pump, and a weak π polarized probe [7, 8]. Ithas been shown [6, 7, 9], that the EIA peak is due to FIG. 1: Top: The N − configuration atom. The light-inducedtransitions are marked by solid (pump field) and dashed(probe field) lines, and the wavy arrows are spontaneous decaypaths. The thick arrow illustrates the spontaneous transfer ofcoherence (TOC). Bottom: probe and pump beam(s), of pos-sibly a finite size, propagating through the vapor cell. Theoptical axis is parallel to ˆ z , while ˆ x and ˆ y form the opticaltransverse plane. transfer of coherence (TOC) from the excited state tothe ground state, via spontaneous emission. The excited-state coherence only exists in systems where the coherentpopulation trapping is incomplete so that there is somepopulation in the excited state [9, 10]. The transfer ofthis coherence to the ground state leads to a peak in thecontribution of the ground-state two-photon coherenceto the probe absorption at line center, instead of the dipthat occurs in its absence (for example, in a Λ system ora non-degenerate N system) [11].In this paper, we investigate the effect of the thermalmotion of the alkali-metal gas on the EIA spectrum, inthe presence of a buffer gas. In a previous paper [12], wediscussed the effect of phase-changing collisions (PCC)with the buffer gas on an N system and showed that theylead to considerable narrowing of the EIA peak in boththe presence and absence of Doppler broadening. Thesecollisions increase the transverse decay rate of the opticaltransitions, resulting in the so-called pressure broadeningof the optical spectral line, and are thus easily incorpo-rated in the Bloch equations. However, in order to de-scribe the overall effect of buffer-gas collisions, it is neces-sary to include both velocity-changing collisions (VCC)as well as PCC [13, 14], which is a much greater chal-lenge. Due to the complexity of the problem, we limitour discussion to a four-level N system, and to buffer-gas pressures that are sufficiently low so that collisionaldecoherence of the excited state [15] can be neglected.The Doppler effect occurs in the limit of ballistic atomic motion, when the mean free-path between VCCis much larger than the radiation wavelength. Due totheir narrow spectral response, Raman processes suchas EIT and EIA are much more sensitive to the “resid-ual” Doppler effect, arising when there is a differencebetween the wavevectors of the Raman fields. In manycases however, the Raman wavelength can become muchlarger than the typical free-path between collisions. Forexample, an angular deviation of a milliradian betweenthe two optical beams yields a superposition pattern witha wavelength in the order of a millimeter. In this limit,the atoms effectively perform a diffusion motion throughthe spatial oscillations of the superposition field, leadingto the Dicke narrowing of the residual Doppler width.While the residual Doppler broadening is linearly propor-tional to the Raman wavevector, Dicke narrowing shows aquadratic dependence. This behavior was demonstratedin EIT with non-collimated pump and probe [16, 17].Recently, a model describing thermal motion and col-lisions for EIT was presented [17–19], utilizing the den-sity matrix distribution in space and velocity with aBoltzmann relaxation formalism. The model describesa range of motional phenomena, including Dicke nar-rowing, and diffusion in the presence of electromagneticfields and during storage of light. This diffusion modelwas used to describe a spatial frequency filter for a spa-tially structured probe [19] and also Ramsey narrowing[20, 21]. Here, we utilize a similar formalism to estimatethe influence of the atomic thermal motion in a buffer-gasenvironment, including VCC and PCC, on the spectralshape of EIA in a four-level N system, with collimatedor non-collimated light beams. In Sec. II, the Dopplerbroadening and Dicke narrowing effects are studied forplane-wave fields. As the full mathematical treatment islengthy, it appears in Appendix A. However, an approx-imate equation which describes the main features of thespectra is presented in Sec. II. Diffusion-like equationsfor the ground and excited state coherences and popula-tions are derived in Appendix B. Two main phenomena are described using this model: (i) a spatial-frequency fil-ter for structured probe fields which is presented in Sec.III, and (ii) atomic diffusion through a finite-sized beamresulting in Ramsey narrowing of the EIA peak, which isdiscussed in Sec. IV. Finally, conclusions are drawn inSec. V. II. THE DOPPLER-DICKE LINE SHAPES OFEIA
Consider the near-resonant interaction of a four-stateatom in an N configuration, depicted in Fig. 1. The twolower states g and g are degenerate and belong to theground level with zero energy, and the excited states e and e are degenerate with energy ~ ω . The light fieldconsists of three beams, each with a carrier frequency ω j and wavevector q j , where j = 1 , j = p the weak probe, ˘E ( r , t ) = X j =1 , ,p E j ( r , t ) e − iω j t + i q j · r + c.c. (1)Here, E j ( r , t ) are the slowly varying envelopes in spaceand time. The pumps drive the g ↔ e and g ↔ e transitions, and the probe is coupled to the g ↔ e transition.Our model will incorporate four relaxation rates: Γ , the spontaneous emission rate from the each of the ex-cited states to all the ground states; Γ pcc , the pressurebroadening of the optical transitions resulting from PCC; γ vcc , the velocity autocorrelation relaxation rate (1 /γ vcc is the time it takes the velocity vector to vary substan-tially) [22], which is proportional to the rate of VCC;and γ is the homogenous decoherence rate within theground and excited state manifolds due, for example, tospin-exchange and spin-destruction collisions [26]. In themodel, the transition g ↔ e is forbidden (due to someselection rule such as angular momentum).To focus the discussion, we assume that all threebeams are continuous waves, namely E j ( r , t ) = E j ( r ).We then obtain stationary Rabi frequencies, given by V j = V j ( r ) = µ j E j ( r ) / ~ , where µ j is the transitiondipole moment. The complete set of Bloch equations forthe four-level N system consists of sixteen equations [7].In order to simplify the application of the theory to EIA,we assume that V p ≪ V , < Γ and that the pump tran-sitions are well below saturation, so that in the absenceof the probe, the population concentrates in the g state,the g ↔ e dipole is excited, and the e state is emptyup to second order in the pump field [7]. The equationscan then be written up to the first order in the probe field V p [6], which reduces the number of Bloch equations tofive.The complete analytical development is presented inAppendix A, and an example of the calculated probeabsorption spectrum for collinear and degenerate beams( q = q = q ) is given in Fig. 2(a) (blue line). For the −100 −50 0 50 10000.05 ∆ p / Γ I m ( R e ) Γ / V p (a) −0.1 −0.05 0 0.05 0.10.05 ∆ p / Γ exact solutionapproximate equation −0.1 0 0.10.030.04 ∆ p / Γ I m ( R e ) Γ / V p (b) FIG. 2: (a) The probe absorption, calculated from the ex-act solution for the density matrix (blue line) and from theapproximate solution Eq. (7) (red line), for collinear plane-wave beams ( q = q = q p ), Γ pcc = 5Γ , γ vcc = 0 . ,A = 0 . , V = 0 . , V = AV , V p = γ = 0 . , and∆ = ∆ = 0. The inset depicts a zoom on the EIA peakwith its wide pedestal. (b) The EIA spectrum (red line) isthe sum of three contributions in Eq. (7): the one-photonabsorption (black dashed line), a pedestal at the base of thepeak (brown dotted line), and the sharp peak (green dash-dotted line). For the clarity of presentation, the one-photonabsorption is added to the pedestal and to the sharp peakcurves. numerical calculations, we have considered the D lineof Rb (wavelength 780 nm) at room temperature, witha total spontaneous emission rate Γ = 2 π × ξ = (∆ p − ∆ ) − ( q p − q ) · v + i ( γ + γ vcc ) , (2a) ξ = ∆ p − q p · v + i (˜Γ + γ vcc ) , (2b) ξ = (∆ p − ∆ ) − ( q p − q ) · v + i (Γ + γ + γ vcc ) , (2c) ξ = (∆ p − ∆ − ∆ ) − ( q p − q − q ) · v + i (˜Γ + γ vcc ) , (2d)with the one-photon detunings ∆ j = ω j − ω e j g j ( j = 1 , p = ω p − ω e g , and ˜Γ = Γ / pcc + γ . The fre-quency ξ is related to the probe transition and includesthe one-photon Doppler shift q p · v . ξ and ξ relate tothe slowly varying ground and excited state coherencesand include the residual Doppler shift ( q p − q i ) · v and the Raman (two-photon) detuning. ξ relates to the three-photon transition (whose direct optical-dipole is forbid-den), required for the EIA process. Note that the fastoptical decay rates (Γ or ˜Γ) is absent only from ξ .In EIA, in contrast to EIT, a strong optical-dipoletransition ( g ↔ e ) is excited even in the absence ofthe probe. Its excitation depends on its resonance withthe pump field, and is thus affected by Doppler broad-ening. This leads to velocity-dependent equations evenin zero-order in the probe field, and introduces the addi-tional complex frequency ξ = − ∆ + q · v + i (˜Γ + γ vcc ) , (3)with the one-photon Doppler shift q · v . The overalldynamics is thus governed by the five equations (A9a)-(A9e).We start by calculating the probe absorption spectrumfor uniform pump and probe fields (plane waves) by solv-ing the equations analytically. The spectrum depends on18 different integrals over velocity, of the form G i = Z d v ξ α · · · ξ β ξ ξ d F ( v ) , (4)where F ( v ) = (cid:0) πv (cid:1) − / e − v / v is the Boltzmannvelocity distribution, and v = k b T /m is the mean ther-mal velocity. The determinant ξ d , ξ d = ξ ξ ξ ξ − ξ ( ξ V + ξ V ) + iV V bA Γ ( ξ + ξ ) , (5)introduces the power broadening effect (first and secondterms), i.e. the dependence of the Raman spectral widthon the pump powers, and the spontaneous TOC from theexcited state to the ground state (last term). The lastterm is associated with the TOC due to its dependenceon the parameter b , which sets the amount of TOC inthe original dynamic equations (A1), and can take eitherthe value 0 (no TOC) or 1 [6]. The spontaneous decaybranching ratio is given by A = µ e g / (cid:0) µ e g + µ e g (cid:1) [7]. The TOC term in Eq. (5) depends on the complexfrequency ξ + ξ = (2∆ p − ∆ − ∆ ) − (2 q p − q − q ) · v +2 i (Γ / pcc + γ + γ vcc ) . (6)It is important to note that, although each of the individ-ual frequencies ξ and ξ is affected by a Doppler shift(either one- or three-photon), the sum ξ + ξ exhibits only a residual Doppler shift (assuming nearly collinearpumps, q ≈ q ). Nevertheless the relaxation rate(Γ / pcc + γ + γ vcc ) is the same as that characterizingthe decay of the optical transitions. As a consequence,even when Γ pcc is much smaller than the optical Dopplerwidth, it plays a significant role in determining the in-tensity of the EIA spectrum. This is in contrast to one-and two-photon processes (such as EIT), in which Γ pcc is irrelevant when it is much smaller than the Dopplerwidth. It can also be seen that when q ≈ q , the var-ious residual Doppler shifts are negligible compared tothe relaxation rates in the determinant ξ d , so that ξ d isonly weakly dependent on these shifts.Examining the absorption spectrum in Fig. 2(a), weobserve the narrow absorption peak on top of the broadone-photon curve. Moreover, as can be seen in the inset,the EIA resonance consists of two independent features:a “pedestal” at the base and a sharp absorption peakat the center. In order to obtain physical insight intothese features, we have derived an approximate solutionfor the probe absorption which incorporates the maincontributions to the EIA, namely the underlying EITmechanism plus the spontaneous TOC. The approximateFourier transform of the nondiagonal density-matrix ele-ment for the probe is R e g = n (cid:20) − G + V G + iV V bA Γ iG G γ vcc − iG γ vcc (cid:21) V p , (7)where G = R d v ξ ξ ξ F ( v ) ξ d , G = R d v ξ ξ F ( v ) ξ d , G = R d v ξ ξ F ( v ) ξ ξ d , G = R d v ξ ξ ξ F ( v ) ξ d , G = R d v ξ F ( v ) ξ d ,and n is the number density of the active atoms. It canbe shown that Eq. (7) is valid provided γ vcc ≪ Γ pcc +Γ / . For an atom at rest and in the absence of collisions, sothat γ vcc = 0 , v th → , and Γ pcc = 0 , Eq. (7) is identicalto the expression obtained by Taichenachev et al . [6](with b = 1), R rest e g = in V p Γ / − i ∆ p " A | V | / Γ2 (1 − A ) | V | / Γ − i ∆ p . (8)The first term in the square brackets in Eqs. (7) and (8)describes the one-photon (background) absorption, andthe other terms are the EIA peak.For a moving atom, the spectrum resulting from Eq.(7) is shown in Fig. 2(a) (red dashed line) and is com-pared with the exact solution; evidently, there is a goodagreement between the spectra. Despite the small dis-crepancy in the intensity of the sharp peak, the approx-imate solution preserves the main features in the reso-nance. When plotted separately in Fig. 2(b), the threeterms in Eq. (7) can be identified with the different spec-tral features: − G (black dashed line) describes the back-ground absorption; V G (brown dotted line), which con-stitutes the total peak in the absence of VCC, describesthe wide pedestal; and iG G γ vcc / (1 − G γ vcc ) (greendashed-dotted line) describes the sharp EIA peak, in-duced by VCC.Fig. 3 shows the effect of varying the VCC rate, for afixed PCC rate (Γ pcc = 5Γ) and zero pump-probe angulardeviation. The width of the pedestal feature depends onthe VCC rate and is given by γ vcc + γ, while the widthof the narrow peak shows only a very weak dependenceon γ vcc . Increasing the VCC rate leads to a decreasein the overall EIA intensity, but to an increase in the −0.05 0 0.050.030.060.09 ∆ p / Γ I m ( R e ) Γ / V p γ vcc =0 γ vcc =0.01 Γγ vcc =0.03 Γγ vcc =0.05 Γ FIG. 3: The EIA peak for different γ vcc rates and Γ pcc = 5Γat zero pump-probe angular deviation; other parameters asin Fig. 2 ratio between the amplitude of the narrow peak and thepedestal baseline.We now turn to explore the residual (two-photon andfour-photon) Doppler and Dicke effects due to wave-vector mismatch between the pump fields and the probe,introduced in principle either by a frequency detuningbetween the fields, | q p | 6 = | q , | , or due to an angulardeviation between them, q p ∦ q , . We mainly focuson the latter, which may be found in a nearly degen-erate level scheme, and we further take the two pumpfields to be the same, namely q = q . Figure 4 presentsthe probe absorption spectrum for different values of thewave-vector difference, δ q = q p − q , , when γ vcc = 0 . pcc = Γ. As can be seen, increasing δ q broad-ens the EIA spectrum (see inset). This is analogous tothe broadening of an EIT transmission peak in a sim-ilar configuration [17]. However, the wide collisionally-broadened pedestal remains unaffected by the changes in δ q , indicating that it mostly originates from homogenousdecay processes. Figure 5(a) summarizes the full-widthat half-maximum (FWHM) of the EIA peak for Γ pcc = Γand for various values of γ vcc , as a function of δ q . Be-cause of the difficulty of separating the sharp peak fromthe background in the calculated spectra [27], the widthsof the sharp EIA peak were obtained only from the thirdterm in Eq. (7). In contrast to an EIT peak, whichdoes not depend on γ vcc when δ q = (collinear degener-ate beams) [19], the FWHM of the EIA peak at δ q = depends weakly on the VCC rate (although barely no-ticeable in the figure). This difference derives from theeffect of collisions on the pump absorption in the case ofEIA, as described earlier.For δ q = the FWHM of the peak in the Dicke limit(high γ vcc ) depends on γ vcc and is proportional to theresidual Doppler-Dicke width, 2 v th δ q /γ vcc . In this limit, −0.5 0 0.50.060.090.120.03 ∆ p / Γ I m ( R e ) Γ / V p δ q=0 δ q=2 mm −1 δ q=5 mm −1 −0.01 0 0.010.060.120 ∆ p / Γ FIG. 4: Calculated probe absorption spectra with γ vcc = 0 . pcc = Γ, for different pump-probe angular deviations.Other parameters as in Fig. 2. the results are well approximated by the analytic expres-sion [19] [dotted lines in Fig. 5(a)]:FWHM = 2 × a γ vcc H (cid:18) a v th δqγ vcc (cid:19) , (9)where H ( x ) = e − x − x and a = 2 / ln 2. Increasingthe pump-probe angular deviation reduces the efficiencyof the EIA process and thus results in a decrease in theprobe absorption [Fig. 5(b)]. This is of course the oppo-site trend to that of EIT (blue stars), where the depthof the dip decreases (the absorption increases) with in-creasing δq [19]. III. SPATIAL-FREQUENCY FILTER
We now to turn to discuss the results of our model fromthe viewpoint of a spatial-frequency filter for a structuredprobe beam. When non-uniform beams are considered,the different spatial frequencies that comprise the beamsresult in different Doppler and Dicke widths. Conse-quently, the various spatial-frequency components expe-rience different absorption and refraction in the medium.Specifically, the dependence of the absorption on thetransverse wave-vectors of the probe beam manifests afilter for the probe in Fourier space.We assume an optical configuration of two collinearuniform pumps (plane waves with V and V constant)and a spatially varying propagating probe, V p = V p ( r , t ).Since the medium exhibits a non-local response due theatomic motion, the evolution of the probe is more natu-rally described in the Fourier space V p ( k , ω ) where k and ω are the spatial and temporal frequencies of the envelopeof the probe. Under these assumptions, the model re-sults in a Diffusion-like equations for the populations and δ q (mm −1 ) I m ( R e ) Γ / V p (b) γ vcc =0.1 Γ , Γ pcc =10 Γ EIT0 1 2 3 4 50255075100125 δ q (mm −1 ) P ea k F W H M ( KH z ) (a) γ vcc =0.025 Γγ vcc =0.1 Γγ vcc =0.25 Γ FIG. 5: Calculated EIA FWHM (a) and absorption (b) forΓ pcc = Γ and various values of γ vcc , as a function of thepump-probe wave-vector difference δ q . The blue starts arethe EIT absorption on the Raman resonance conditions, withΓ pcc = Γ and γ vcc = 0 . coherences of the atomic medium, derived in AppendixB. To simplify the general dynamics of Eqs. (B8a) and(B9), we take the stationary case [ ω = 0 , V p = V p ( k )]and assume that the carrier wave-vector of the probeis the same as that of the pumps, q p = q = q , sothat δ q = δ q = 0. Taking the Fourier transform [seeEq. (A11)], we obtain a set of steady-state equationsfor the spatially-dependent atomic coherences, R g g ( k ), R e e ( k ), and R e g ( k ) , h i (∆ p − ∆ ) − γ − K | V | − K | V | − Dk i R g g = ( Dk − bA Γ) R e e + K V ∗ V p n , (10a) (cid:2) i (∆ p − ∆ ) − Γ − γ − Dk (cid:3) R e e = − V V ∗ ( K + K ) R g g − V ∗ ( K + K pump ) V p n , (10b) R e g = iK ( V R g g + V p n ) (10c)where K = iG / (1 − iG γ vcc ) is the one-photon ab-sorption spectrum with G = R F ( v ) /ξ d v ; K = iG / (1 − iG γ vcc ) is the three-photon absorption spec-trum with G = R F ( v ) /ξ d v ; and K pump = iG pump / (1 − iG pump γ vcc ) is the one-photon (pump) ab-sorption spectrum with G pump = R F ( v ) /ξ d v , as de-scribed in Appendix B. Solving Eq. (10) for R e g ( k , ω ),substituting the result into the expression for the linear-susceptibility [Eq. (A14)], assuming that V = ηV (0 < η ≤ / Γ, we obtain −2 −1 0 1 200.050.10.150.2 k (units of (D/ Γ hom ) ) R e ( L ) ∆ p =0 ∆ p = Γ hom ∆ p =2 Γ hom FIG. 6: The EIA spatial-frequency filter, given in Eq. (11b),as a function of k , with γ vcc = 0 . pcc = 10Γ. Redcurve is plotted for ∆ p on resonance, blue-dashed and black-dotted curves demonstrate the behavior at nonzero Ramandetuning. χ e g ( k ) = gc iKn (1 + L) , (11a)L = η (2 bA − η ) Γ p − i ∆ p + γ + ( η + 1 − bAη ) Γ p + Dk , (11b)where D = v th /γ vcc is the diffusion coeffi-cient, Γ p = K | V | is the power broaden-ing, and K ≈ K ≈ K pump = K = R F ( v ) / [ q p · v + i (Γ / pcc + γ + γ vcc )] d v for∆ p ≪ Γ hom = γ + Γ p . In the case where η = A, Eqs.(11) is similar to Eq. (8) obtained by Taichenachev etal. [6], except for the diffusion term Dk , which vanishesfor an atom at rest.The imaginary part of the susceptibility in Eq. (11)yields the absorption of the probe for various values of k . The first term in the brackets in Eq. (11a) is thelinear one-photon absorption, and the second term is the k -dependent EIA contribution. Thus, the real part of L in Eq. (11b) describes an “absorbing” spatial-frequencyfilter, the same way as was done for EIT [24, 25]. Fig.6 summarizes several examples of the EIA spatial filterbehavior as a function of k for ∆ p = 0 , ∆ p = ± Γ hom , and ∆ p = ± hom . At ∆ p = 0 , the curve is a Lorentzianand maximum absorption is achieved. When ∆ p = 0 thefilter becomes more transparent. IV. RAMSEY NARROWING
We now consider the N system interacting withcollinear probe and pump beams that have finite widths.Due to thermal motion, the alkali atoms spend a period −0.01 0 0.010.0330.036 ∆ p / Γ I m ( R e ) Γ / V p (b) −0.01 0 0.010.03060.03090.0312 ∆ p / Γ I m ( R e ) Γ / V p (a) FIG. 7: Calculated probe absorption spectra (blue line) forone-dimensional stepwise beam with finite thickness: (a) 2 a =100 µm and (b) 2 a = 10 mm , and fitted Lorentzian (reddashed line). All other parameters are the same as in Fig.2 of time in the interaction region and then leave the lightbeams, evolve ‘in the dark’, and diffuse back inside. Sucha random periodic motion was described recently by Xiao et al. [20, 21] for an EIT system, and was shown to resultin a cusp-like spectrum. Near its center, the line is muchnarrower than that expected from time-of-flight broaden-ing and power broadening, and the effect, resulting fromthe contribution of bright-dark-bright atomic trajectoriesof random durations, was named Ramsey narrowing.Ramsey-narrowed spectra can be calculated analyti-cally from the diffusion equations of the atomic coher-ences when the light fields of both the probe and pumpbeams have finite widths [19]. The EIA spectrum result-ing from a one-dimensional uniform light-sheet of thick-ness 2 a in the x − direction is derived analytically in Ap-pendix B [Eq. (B12)]. In Fig. 7, we show the spectrumfor two different thicknesses and the fitted Lorentziancurves. Near the resonance, the EIA line for the 100 µm sheet is spectrally sharper than the fitted Lorentzian– the characteristic signature of Ramsey narrowing. Incontrast, the EIA peak calculated for a 10 mm beam iswell fitted by the Lorentzian. In addition, the EIA con-trast deteriorates as the beam becomes narrower, sincethe interaction area decreases and fewer atoms interactwith the fields. V. CONCLUSIONS
In this paper, we extended the theory that describesthe effect of buffer-gas collisions on three-level Λ systemsin an EIT configuration [17–19] to the case of a four-level closed N system which is the simplest system thatexperiences EIA due to TOC. Using this formalism, weinvestigated the influence of collisions of optically activeatoms with a buffer gas on the EIA peak. In additionto the exact expressions, we presented an approximatesolution for the probe absorption spectrum, which pro-vides a physical insight into the behavior of the EIA peakdue to VCC, PCC, and wave-vector difference betweenthe pump and probe beams. VCC were shown to pro-duce a wide pedestal at the base of the EIA peak; in-creasing the pump-probe angular deviation scarcely af-fects the pedestal whereas the sharp central EIA peakbecomes weaker and broader due to the residual Doppler-Dicke effect. Using diffusion-like equations for the atomiccoherences and populations, the spatial-frequency filterand the Ramsey-narrowed spectrum were analytically ob-tained.In extending the description from the Λ to the N schemes, we have considered several elements that arelikely to be important in other four-level systems. Theseinclude the diffusion of excited-state coherences and theinfluence of the thermal motion on the optical dipole inthe absence of the probe. The latter introduces a Dopplercontribution into the pumping terms and consequentlyaffects the power broadening of the narrow resonances. Appendix A: Reduced density matrix
Consider the near-resonant interaction of a light fieldconsisting of one or two moderately strong pumps anda weak probe, as given in Eq. (1), with the four-leveldegenerate N system of Fig. 1(a). We use the first-orderapproximation in the probe amplitude, V p , and assumethat V < Γ , V V , V p < V , . Since the pump transi-tions are assumed non-saturated, the atomic populationin the absence of the probe concentrates in the g state,and the population in other states can be neglected. The g ↔ e dipole, excited in the absence of the probe, is ofimportance and is thus considered. The resulting Blochequations are [6]˙˘ ρ (1) ,ig g ( ω p − ω ) = − [ i ( ω e g − ω e g ) + γ ] ˘ ρ (1) ,ig g + i ˘ V ∗ ˘ ρ (1) ,ie g − i ˘ V ˘ ρ (1) ,ig e + bA Γ˘ ρ (1) ,ie e , (A1a)˙˘ ρ (1) ,ie g ( ω p ) = − [ iω e g + Γ / pcc ] ˘ ρ (1) ,ie g + i ˘ V p ˘ ρ (0) ,ig g + i ˘ V ˘ ρ (1) ,ig g , (A1b)˙˘ ρ (1) ,ie e ( ω p − ω ) = − [ i ( ω e g − ω e g ) + Γ + γ ] ˘ ρ (1) ,ie e + i ˘ V p ˘ ρ (0) ,ig e + i ˘ V ˘ ρ (1) ,ig e − i ˘ V ∗ ˘ ρ (1) ,ie g , (A1c)˙˘ ρ (1) ,ig e ( ω p − ω − ω ) = − [ i ( ω e g − ω e g − ω e g )+ Γ / pcc ] ˘ ρ (1) ,ig g − i ˘ V ∗ ˘ ρ (1) ,ig g , (A1d)˙˘ ρ (0) ,ig e ( − ω ) = − [Γ / pcc − iω e g ] ˘ ρ (0) ,ig e ( − ω )+ i ˘ V ∗ (cid:16) ˘ ρ (0) ,ie e − ˘ ρ (0) ,ig g (cid:17) . (A1e) Here, ˘ ρ ( j ) ,iss ′ is the density-matrix element of the i − thatom (one of many identical particles) to the j − th or-der in the probe, and apart from ˘ ρ (0) ,ig g ≈ , ˘ ρ (0) ,iss = 0.We also consider the envelopes of the pumps to be con-stant in time so that V , is shorthand for V , ( r ). Thewave equation for the probe field is (cid:18) ∇ − c ∂ ∂t (cid:19) ˘E p ( r , t ) = 4 πc ∂ ∂t ˘P e g ( r , t ) , (A2)where ˘P e g ( r , t ) = P e g ( r , t ) e − iω p t e − i q p · t is the con-tribution of the e ↔ g transition to the expectationvalue of the polarization, P e g is the slowly varying po-larization, and ∇ is the three-dimensional Laplacian op-erator. With Eq. (1), and assuming without loss of gen-erality that ˆq p = ˆz q p , as shown in Fig. 1(b), Eq. (A2)can be written in the paraxial approximation as (cid:18) ∂∂t + c ∂∂z − i c q p ∇ ⊥ (cid:19) V p ( r , t ) = i gµ ∗ e g P e g ( r , t ) , (A3)where ∇ ⊥ is the transverse Laplacian operator, and g =2 πω p | µ e g | / ~ is a coupling constant .Following [19], we introduce a density-matrix distribu-tion function in space and velocity,˘ ρ ss ′ = ˘ ρ ss ′ ( r , v , t ) = X i ˘ ρ iss ′ ( t ) δ ( r − r i ( t )) δ ( v − v i ( t )) , (A4)where the time dependence of ˘ ρ iss ′ ( t ) is determined byEqs. (A1). Differentiating Eq. (A4) with respect totime, we arrive at ∂∂t ˘ ρ ss ′ + v · ∂∂ r ˘ ρ ss ′ + (cid:20) ∂∂t ˘ ρ ss ′ (cid:21) col = X i ∂∂t ˘ ρ iss ′ ( t ) δ ( r − r i ( t )) δ ( v − v i ( t )) , (A5)where the effect of velocity-changing collisions is takenin the strong collision limit in the form of a Boltzmannrelaxation term [22], (cid:20) ∂∂t ˘ ρ ss ′ (cid:21) col = − γ vcc h ˘ ρ ss ′ ( r , v , t ) − ˘ R ss ′ ( r , t ) F ( v ) i , (A6)with ˘ R ss ′ = ˘ R ss ′ ( r , t ) = R d v ˘ ρ ss ′ ( r , v , t ) being thedensity-number of atoms per unit volume, near r inspace, and F = F ( v ) = (2 πv th ) − / e − v / v th , v th = k b Tm (A7)is the Boltzmann distribution.Before writing the coupled dynamics of the internaland motional degrees of freedom, we introduce the slowlyvarying envelopes of the density-matrix elements, ρ ss ′ = ρ ss ′ ( r , v , t ), as˘ ρ g g = ρ g g e − i ( ω p − ω ) t e i ( q p − q ) · r , ˘ ρ e g = ρ e g e − iω p t e i q p · r , ˘ ρ e e = ρ e e e i ( q p − q ) · r , ˘ ρ g e = ρ g e e − i ( ω p − ω − ω ) t e i ( q p − q − q ) · r , ˘ ρ g e = ρ g e e iω t e − i q · r , (A8)and similarly the slowly varying densities R ss ′ = R d v ρ ss ′ . Eqs. (A1) then become (cid:20) ∂∂t + v · ∂∂ r − iξ (cid:21) ρ g g − γ vcc R g g F = i ( V ∗ ρ e g − V ρ g e ) + bA Γ ρ e e , (A9a) (cid:20) ∂∂t + v · ∂∂ r − iξ (cid:21) ρ e g − γ vcc R e g F = i [ V p n F + V ρ g g ] , (A9b) (cid:20) ∂∂t + v · ∂∂ r − iξ (cid:21) ρ e e − γ vcc R e e F = i ( V ρ g e − V ∗ ρ e g ) + iV p ρ g e , (A9c) (cid:20) ∂∂t + v · ∂∂ r − iξ (cid:21) ρ g e − γ vcc R g e F = − iV ∗ ρ g g , (A9d) (cid:20) ∂∂t + v · ∂∂ r − iξ (cid:21) ρ g e − γ vcc R g e F = − iV ∗ n F, (A9e)where ξ i ( i = 1 −
5) are given in Eq. (2). The ex-pectation value of the polarization density P e g ( r , t ) interms of the number density R e g ( r , t ) is P e g ( r , t ) = µ ∗ e g R e g ( r , t ), and Eq. (A3) becomes (cid:18) ∂∂t + c ∂∂z − i c q p ∇ ⊥ (cid:19) V p ( r , t ) = igR e g ( r , t ) . (A10)We now consider the case of stationary plane-wavepumps. For this case, it is convenient to introduce theFourier transform f ( r , t ) = + ∞ Z −∞ d k π e i kr + ∞ Z −∞ dω π e − iωt f ( k , ω ) , (A11)and write Eqs. (A9) as [ ω − k · v + ξ ] ρ g g − iγ vcc R g g ( k , ω ) F = ( V ρ g e − V ∗ ρ e g ) + ibA Γ ρ e e , (A12a)[ ω − k · v + ξ ] ρ e g − iγ vcc R e g ( k , ω ) F = − ( V p n F + V ρ g g ) , (A12b)[ ω − k · v + ξ ] ρ e e − iγ vcc R e e ( k , ω ) F = ( V ∗ ρ e g − V ρ g e ) − V p ρ g e , (A12c)[ ω − k · v + ξ ] ρ g e − iγ vcc R g e ( k , ω ) F = V ∗ ρ g g , (A12d)[ ω − k · v + ξ ] ρ g e − iγ vcc R g e ( k , ω ) F = V ∗ n F, (A12e)and Eq. (A10) as (cid:18) ik z − i ωc + i k q p (cid:19) V p ( k , ω ) = i gc R e g ( k , ω ) . (A13)The linear susceptibility χ e g ( k , ω ) is defined by R e g ( k , ω ) = χ e g ( k , ω ) cg V p ( k , ω ) . (A14)In order to find the probe absorption spectrum, wesolve Eqs. (A12) analytically, obtain an expression for ρ ss ′ , and formally integrate it over velocity. This leadsto an expression for R ss ′ in terms of integrals over veloc-ity, in the form of Eq. (4), such as G = R d v ξ ξ ξ F ( v ) ξ d ,which can be evaluated numerically. In the general case,the resulting expression for R ss ′ is very complicated andis not reproduced here. In order to explore the under-lying physics, we developed an approximate expressionfor the Fourier transform of the density-matrix elementthat refers to the probe transition, namely, R e g [see Eq.(7)].One can verify that in the absence of the pumps( V = V = 0), the resulting one-photon complex spec-trum simplifies to the well known result for the strongcollision regime, K = iG/ (1 − iγ vcc G ), where G = R d v F/ ( ω − k · v + ξ ) [22]. Appendix B: Diffusion in the presence of fields
In order to obtain diffusion-like equations for thedensity-matrix elements and the probe fields, we beginby integrating Eqs. (A9a) and (A9c) over velocity andobtain (cid:20) ∂∂ r + iδ q (cid:21) · J g g + (cid:20) ∂∂t − i (∆ p − ∆ ) + γ (cid:21) R g g = i ( V ∗ R e g − V R g e ) + bA Γ R e e , (B1a) (cid:20) ∂∂ r + iδ q (cid:21) · J e e + " ∂∂t − i (∆ p − ∆ )+ Γ + γ R e e = i ( V R g e − V ∗ R e g + V p R g e ) , (B1b)where J ss ′ = J ss ′ ( r , t ) = R d v v ρ ss ′ is the envelope ofthe current density. Expanding ρ g g and ρ e e in Eqs.(A9a) and (A9c) as ρ ss ′ = R ss ′ F +1 /γ vcc ρ (1) ss ′ , multiplyingEqs. (A9a) and (A9c) by v , integrating the resultingequations over velocity using Z d v j v i ∂∂x i R ss ′ F = δ ij v th ∂∂x i R ss ′ , (B2)defining the current density of the density matrix by γ vcc J ss ′ = Z d v j ρ (1) ss ′ , (B3)and retaining the leading terms in 1 /γ vcc , we obtain J g g + D (cid:20) ∂∂ r + iδ q (cid:21) R g g = iγ vcc ( V ∗ J e g − V J g g ) − bA Γ γ vcc J e e , (B4a) J e e + D (cid:20) ∂∂ r + iδ q (cid:21) R e e = iγ vcc (cid:16) V J g e − V ∗ J e g + ˜ V p J g e (cid:17) , (B4b)where D = v th /γ vcc . Substituting J g g , J e e from Eq.(B4) into Eq. (B1), we get " ∂∂t − i (∆ p − ∆ ) + γ − D (cid:18) ∂∂ r + iδ q (cid:19) R g g = i ( V ∗ R e g − V R g e ) + bA Γ R e e − D (cid:18) ∂∂ r + iδ q (cid:19) × (cid:20) iγ vcc ( V ∗ J e g − V J g e ) − bA Γ γ vcc J e e (cid:21) , (B5a) " ∂∂t − i (∆ p − ∆ ) + Γ + γ − D (cid:18) ∂∂ r + iδ q (cid:19) R e e = i ( V R g e − V ∗ R e g + V p R g e ) − D (cid:18) ∂∂ r + iδ q (cid:19) × (cid:20) iγ vcc ( V J g e − V ∗ J e g + V p J g e ) (cid:21) . (B5b) In order to calculate R e g , R g e , R g e , and J e g , J g e , J g e , we assume in Eqs. (A9b), (A9d), and(A9e) that the envelopes change slowly enough such that | ∂/∂t + v · ∂/∂ r | ≪ | ξ , , | , and get − iξ ρ e g = γ vcc R e g F + i ( V p n F + V ρ g g ) , (B6a) − iξ ρ g e = γ vcc R g e F − iV ∗ ρ g g , (B6b) − iξ ρ g e = γ vcc R g e F − iV ∗ n F. (B6c)Solving Eq. (B6) formally for ρ e g , ρ g e , ρ g e and sub-stituting only their leading parts, i.e. ρ ss ′ = R ss ′ F , wefind ρ e g = [ γ vcc R e g − V R g g ) − V p n ] F/ξ , (B7a) ρ g e = [ γ vcc R g e + V ∗ R g g ] F/ξ , (B7b) ρ g e = [ γ vcc R g e + V ∗ n ] F/ξ . (B7c)Integrating Eqs. (B7) over velocity we get R e g = iK [ V R g g + V p n ] , (B8a) R g e = − iK V ∗ R g g , (B8b) R g e = − iK pump V ∗ n , (B8c)where K = iG / (1 − G γ vcc ) is the one-photonabsorption spectrum with G = R F/ξ d v , K = iG / (1 − G γ vcc ) is the three-photon absorptionspectrum with G = R F/ξ d v and K pump = iG pump / (1 − G pump γ vcc ) is the one-photon (pump) ab-sorption spectrum with G pump = R F/ξ d v . In thecase of collinear pump and probe beams δ q = δ q , = q p − q , = δq b z , Eqs. (B5) and (B8) form a closed setwhen (cid:18) ∂∂ r + iδ q , (cid:19) · iV , ( r ) γ vcc J e g , (cid:18) ∂∂ r + iδ q , (cid:19) · iV , ( r ) γ vcc J g e , (cid:18) ∂∂ r + iδ q (cid:19) · iV p ( r ,t ) γ vcc J g e can be neglected in Eq. (B5). These terms vanish com-pletely in the special case of pump and probe which areplane waves ( ∂/∂ r = ), and also collinear and degen-erate ( δ q = ). They can also be neglected whenever | V , ,p | ≪ γ vcc as is the case in many realistic situations.However, the term ( ∂/∂ r + iδ q ) · bA Γ /γ vcc J e e in Eq.(B5a) cannot be neglected in the case of collinear pumpand probe beams since bA Γ /γ vcc does not go to zero.Substituting Eq. (B4b) into Eq. (B5a), and Eq. (B8)into Eq. (B5), we find:0 (cid:26) ∂∂t − i (∆ p − ∆ ) + γ + K | V | + K | V | (cid:27) R g g = D (cid:18) ∂∂ r + iδ q (cid:19) R g g + D (cid:18) ∂∂ r + iδ q (cid:19) R e e + bA Γ R e e − K V ∗ V p n , (B9a) (cid:26) ∂∂t − i (∆ p − ∆ ) + Γ + γ (cid:27) R e e = D (cid:18) ∂∂ r + iδ q (cid:19) R e e + V V ∗ ( K + K ) R g g + V ∗ ( K + K pump ) V p n . (B9b)These are the final diffusion-like coupled equations forthe ground- and excited-state coherences.In order to investigate the Ramsey narrowing of theEIA peak, we consider finite probe and pump beamsand restrict the discussion to collinear EIA. We assumethat the fields are stationary and overlap in their crosssections with negligible variation along the z − direction, V p ( r , t ) = V p w ( r ⊥ ), V ( r ) = V w ( r ⊥ ), V ( r , t ) = V w ( r ⊥ ) , where w ( r ⊥ ) is the transverse profile of thefields. We further take δq = 0 and ∆ = ∆ = 0 forbrevity. In the diffusion regime, we rewrite Eqs. (B8)and (B9) as h i ∆ p + γ + (cid:16) K | V | + K | V | (cid:17) w ( r ⊥ ) i R g g = bA Γ (cid:18) Dγ vcc ∇ ⊥ (cid:19) R e e − K V ∗ V p n w ( r ⊥ ) , (B10a) R e g = iK ( V R g g + V p n ) w ( r ⊥ ) , (B10b) (cid:0) i ∆ p + Γ + γ − D ∇ ⊥ (cid:1) R e e = V ( K + K ) R g g V ∗ w ( r ⊥ ) + V p ( K + K pump ) n V ∗ w ( r ⊥ ) , (B10c) R g e = − iK V ∗ R g g w ( r ⊥ ) , (B10d) R g e = − iK pump V ∗ n w ( r ⊥ ) . (B10e)We further consider a probe and pump beams with auniform intensity and phase within a sheet of thickness2 a in the x − direction (one-dimensional stepwise beams): w ( x, y ) = (cid:26) | x | ≤ a | x | > a . The solution for R g g , symmetric in x and decaying as | x | → ∞ , is given by R g g ( | x | ≤ a ) = C cosh ( k x ) + C cosh ( k x ) + bA Γ β + Dα β ( Dα α ) + bA Γ β , (B11a) R e e ( | x | ≤ a ) = C (cid:0) k − α (cid:1) Dγ vcc bA Γ ( Dα + γ vcc ) cosh ( k x )+ C (cid:0) k − α (cid:1) Dγ vcc bA Γ ( Dα + γ vcc ) cosh ( k x ) + β β − β α Dβ bA Γ − ( Dα α ) , (B11b) R g g ( | x | > a ) = C bA Γ (cid:0) Dα + γ vcc (cid:1) ( α − α ) Dγ vcc e − α ( | x |− a ) + C e − α ( | x |− a ) , (B11c) R e e ( | x | > a ) = C e − α ( | x |− a ) , (B11d)where α = ( − i ∆ p + γ + K | V | + K | V | ) /D, α = α + Γ /D, α = ( − i ∆ p + γ ) /D, and β = V ∗ V p K n ,β = V V ∗ ( K + K ) , β = V ∗ V p ( K + K pump ) n . Thecomplex diffusion wave-numbers are obtained from2 Dγ vcc k , = Dγ vcc α + β bA Γ ∓ (cid:2) ( Dγ vcc ) α − + β bA Γ (cid:0) Dγ vcc α + β bA Γ (cid:1)(cid:3) / , with α ± = α ± α . The coefficients C i ( i = 1 − R ss ′ and( ∂/∂x ) R ss ′ at | x | = a . From Eq. 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0, the third term in Eq. (7) graduallyvanishes, and the second term ( V G ) is responsible forthe EIA peak, as indicated by the brown-dotted line inFig. 2(b). Its width is limited by homogenous broadeningmechanisms and determined by γ and Γ pccpcc