Electromagnetically induced transparency with a standing-wave drive in the frequency up-conversion regime

Transcription

1 PHYSICAL REVIEW A, VOLUME 64, Electromagnetically indced transparency with a standing-wave drive in the freqency p-conversion regime F. Silva Departament d Òptica, Universitat de València, E Brjassot, Spain J. Mompart, V. Ahfinger, and R. Corbalán Departament de Física, Universitat Atònoma de Barcelona, E Bellaterra, Spain Received 29 Jly 1999; revised manscript received 9 Febrary 2001; pblished 7 Agst 2001 We stdy electromagnetically indced transparency for a probe traveling-wave TW laser field in closed Doppler-broadened three-level systems driven by a standing-wave SW laser field of moderate intensity its Rabi freqencies are smaller than the Doppler width of the driven transition. We show that probe windows of transparency occr for vales of the probe to drive field freqency ratio R close to half-integer vales. For optical transitions and typical vales of Doppler broadening for atoms in a vapor cell, we show that for R 1 a SW drive field is appreciably more efficient than a TW drive in indcing probe transparency. As examples, we consider parameters for real cascade schemes in barim atoms with R 1.5 and in beryllim atoms with R 3.5 showing that probe transmission vales well above 50% are possible for conditions in which it is almost negligible either withot driving field or with only one of the TW components of the drive. We show that a strongly asymmetric drive having two TW components with neqal intensities is even more eficient than a symmetric SW drive in indcing probe transparency. The case of arbitrary probe intensity is also considered. DOI: /PhysRevA PACS nmber s : Gy, Hz, Qk I. INTRODUCTION Linear and nonlinear optical properties of an atomic medim can be controlled and modified by applying a strong drive laser field that indces atomic coherence. Atomic coherence manifests itself in a rich variety of phenomena that are a sbject of crrent research interest, sch as electromagnetically indced transparency EIT for reviews see, for instance, Refs. 1 4 ; amplification and lasing withot inversion for a review, see Ref. 5 ; high refractive index withot absorption 6 ; steep dispersion and either ltraslow grop velocity 7 or gain-assisted sperlminal light propagation 8 ; and enhanced nonlinear optical ssceptibility 9. Here we consider EIT in some cases it is also called coherent poplation trapping or dark resonance 2, which is among the most basic of these phenomena and has many sefl applications. For instance, it is at the heart of velocity selective coherent poplation trapping, a laser cooling method for free atoms which achieves sbrecoil temperatres 10. It is also at the basis of another very recent laser cooling method for trapped atoms which achieves efficient grond-state cooling 11. EIT in a time-dependent regime has been sed recently for stopping light 12, which cold have important applications in the brgeoning field of qantm information. In EIT, an otherwise absorbing medim is made transparent to a coherent field on resonance with a certain atomic transition by applying an intense coherent drive field to an adjacent transition. There has been a large nmber of theoretical contribtions 1 4 and EIT was demonstrated experimentally in many systems Up to now, most theoretical papers dealing with atomic coherence effects in gas media have considered only the case of a traveling-wave TW drive field. In this paper we stdy EIT in Doppler-broadened three-level media with standingwave SW drive i.e., composed of two eqal-freqency conterpropagating TW fields and focs or analysis on the advantages of sing a SW drive configration instead of a TW one in the so-called freqency p-conversion regime where the probe wavelength is sbstantially shorter than that of the drive laser. A short accont of some of the reslts presented below was pblished recently 20. The possibility of inversionless amplification in the freqency p-conversion regime with an intense, detned SW drive has been recently pointed ot 21. In the opposite regime of freqency downconversion, which appears in coherently pmped far-infrared lasers, it has been demonstrated both theoretically 22 and experimentally 23 that SW pmping of these lasers leads to gain vales larger than those obtained for a TW pmping. The case of a SW drive intitively appears as promising in the freqency p-conversion regime. Let s consider, for instance, a Doppler-broadened three-level ladder scheme Fig. 1 driven on transition 2 3 by a SW laser field and probed on transition 1 3 by a weak collinear TW laser field. The wave nmber and the detning from atomic resonance are (k d, d ) and (k p, p ) for drive and probe fields, respectively. The resonance conditions for the processes contribting to probe absorption can be easily traced if the driveindced ac Stark effect is neglected. Since the probe is weak, only processes involving one probe photon mst be considered. Velocity-tned mltiphoton processes 24 involving one probe photon and several drive photons see Fig. 2, with alternating absorption and emission from the two conterpropagating drive components, occr for velocities satisfying p k p v qk d v, q 0, 2,..., 1a p k p v d qk d v, q 1, 3,..., 1b /2001/64 3 / /$ The American Physical Society

2 SILVA, MOMPART, AHUFINGER, AND CORBALÁN PHYSICAL REVIEW A FIG. 1. Closed cascade three-level system nder investigation. where v is the atomic velocity along the probe beam and q is the nmber of drive photons involved in the process 25. q is positive negative if the drive photons absorbed in the mltiphoton process are copropagating conterpropagating with the probe while those emitted are conterpropagating copropagating. For a TW drive conterpropagating with the probe, only processes 1a with q 0 and 1b with q 1 occr. The resonance condition for this second process can be made v independent only if k p k d. This is the well-known reslt that with TW fields Doppler-free absorption resonances only occr for eqal-freqency drive and probe fields. In contrast, a SW drive allows also Doppler-free absorption resonances at p 0 with k p /k d q 2,4,..., and at p d 0 with k p /k d q 1,3,...,anditistherefore interesting in the freqency p-conversion regime. Note that there is also a Doppler-free absorption resonance at p 0 with k p /k d q 0; this case is relevant for coherently pmped long-wavelength far infrared lasers 22. For the cascade scheme of Fig. 1 there will be a strong probe absorption nder conditions of Doppler-free absorption resonances, since all the atoms, irrespective of their velocities, contribte to the absorption. Therefore, one expects to find windows of transparency for k p /k d q 1 2, i.e., approximately midway between consective Doppler-free mltiphoton absorption resonances. On the other hand, one mst be aware that in a tre SW drive i.e., when its two TW components have the same intensity there are nodes, where the total drive field vanishes. Therefore, the most abndant v 0 atoms placed at the nodes do not interact with the drive and will prodce a strong resonant probe absorption. This ndesirable featre from the perspective of EIT can in principle be avoided by sing a drive field with neqal intensities for its two TW components. All these main conclsions will be confirmed by the detailed analysis presented below. Section II is devoted to the semiclassical density matrix approach to the problem at hand. In Sec. III we briefly review the dressed-atom analysis for the SW excitation of a Doppler-broadened transition. Nmerical reslts for EIT in two real cascade systems in Ba and in Be atoms are presented in Sec. IV. Section V smmarizes or main conclsions. II. SEMICLASSICAL DENSITY MATRIX ANALYSIS To be specific, let s consider the closed cascade threelevel system shown in Fig. 1, althogh similar reslts can be obtained for the folded (V and ) schemes. The Hamiltonian H 0 that describes the internal energy of the three-level atoms is written as H 0 i 1,2,3 E i i i, 2 and ij E i E j is the energy separation between levels i and j. A TW laser field E p (z,t) 1 2 A p exp i( p t k p z) c.c., to which transparency will be indced probes the lower transition 3 1, while the pper transition 2 3 is driven by a collinear strong SW laser field E d (z,t) A d exp i( d t k d z) c.c. The copling of the atoms with the laser fields is characterized by the detnings p p 31 and d d 23 from their respective resonance, and by the three Rabi freqencies 2 31 A p / and 2 23 A d /, where 31 and 23 are electric dipole moments between the corresponding states. Withot loss of generality the three Rabi freqencies will be taken to be real. De to electric dipole selection rles the states 2 and 1 mst have the same parity, therefore The Hamiltonian V that describes the atom-laser copling is given by FIG. 2. Lowest-order processes participating in the probe absorption, with q being the nmber of drive photons involved. Similar processes, interchanging by and by correspond to q V exp i d t k d z 3 2 exp i p t k p z 1 3 H.c., 3

3 ELECTROMAGNETICALLY INDUCED TRANSPARENCY... PHYSICAL REVIEW A where the standard rotating-wave approximation has been sed to eliminate the nonresonant terms. The three-level system is also sbjected to relaxation processes, which will be described by means of a Lioville operator R acting on the density matrix (v,z,t), whose evoltion is given by i v H, R, 4 t z where H H 0 V, and v is the atomic velocity along the z axis. For simplicity, we consider that the relaxation operator R only contains terms 31 and 23 describing the poplation decay rates of the middle and pper levels, respectively, and terms ij describing the dephasing rates of the ij offdiagonal elements. In the so-called radiative limit when there are no dephasing collisions one has /2; ( )/2; and /2. It is convenient to work in the interaction pictre to eliminate the fast dependences in the time evoltion eqations. Ths, we define new density matrix elements ij by and 23 v,z,t 23 v,z,t exp i d t, 31 v,z,t 31 v,z,t exp i p t, 21 v,z,t 21 v,z,t exp i d p t. The reslting eqations of motion are v t z i e ik d z e ik d z 32 c.c., v t z i e ik p z 13 c.c., v t z 23 R id 23 e ik d z e ik d z i e ik p z 21, v t z 31 R id 13 e ik p z i e ik d z e ik d z 21, v t z 21 R i e ik d z e ik d z 31 i e ik p z 23, 5a 5b 5c 5d 5e where d 13 ( ), d 23 ( ), R 23 ( 23 i d ), R 31 ( 31 i p ), and R i( p d ). Since we are considering a closed system, the normalization condition mst be added to Eqs. 5. Notice that there is an incoherent copling between drive and probe fields throgh the poplation of level 3, and also a coherent copling throgh the atomic coherence 21 see Eqs. 5c and 5d. All qantm interference phenomena mentioned in the introdction are based on this coherent copling. Even in the steady state, to which we restrict in the following, these eqations do not have a general analytical soltion de to the simltaneos presence in the right-hand sides of the explicit z dependences exp( ik d z), obviosly linked to the SW natre of the drive field. The soltion mst therefore be obtained nmerically, in contrast with the wellknown case of a TW drive which can be treated analytically. In the case of weak probe and strong drive fields i.e., 31, 23 ) one first solves Eqs. 5 to zeroth order in the probe field and to all orders in the drive field. This soltion is well known from the theory of a high-intensity gas laser 26,27 : The density matrix elements are expanded in spatial Forier components with a fndamental freqency k d. One obtains a set of copled difference eqations in the Forier coefficients c n which may be solved sbject to appropriate bondary conditions on c n for large n. The soltion can be expressed in terms of contined fractions 26,27. This zeroth-order soltion is then sed to solve Eqs. 5 to first order in the probe field 28. In some cases of practical interest it may be necessary to consider also a probe field of arbitrary intensity e.g., in applications of EIT to enhance nonlinear processes the probe beam is not necessarily weak. The steady-state soltion of Eqs. 5 for this case can be obtained introdcing a two-index Forier expansion for the density matrix elements with k d and k p fndamental spatial freqencies. However, this expansion can be more easily handled if we define, withot loss of generality, two integers a and b sch that a/b is a rational nmber and k d ak, k p bk, where k is a redced wave nmber 29. Then the steady-state soltion of Eqs. 5 becomes the one-index expansion: jj q y q jj v exp iqkz, jk y jk v exp iqkz, j,k 1,2,3. 6 q The set of copled difference eqations in the Forier coefficients y q jj and y q jk can be recast in a form identical to that derived in Ref. 26 for complex nmbers, bt involving now vectors and matrices of dimension The soltion can therefore be obtained in terms of contined fractions of 4 4 matrices 29. The contined fractions are evalated nmerically by trncation to a finite nmber of terms in the denominator and it is a standard procedre in the calclations to check if the addition of one term in the denominator modifies the nmerical vale within the reqired accracy. We have also checked that for 31 the matrix continedfraction soltion reprodces the Feldman and Feld s soltion 28. The absorption and dispersion characteristics of the atomic medim with the ladder configration shown in Fig. 1 are derived in the sal way from the components of the polarization P(z,t) N(v)dv (v,z,t)

4 SILVA, MOMPART, AHUFINGER, AND CORBALÁN PHYSICAL REVIEW A (v,z,t) c.c., whose temporal and spatial oscillation matches the copled field. Here N(v)dv is the nmber of atoms per nit volme with velocity v and we take for N(v) the sal Maxwellian distribtion with a most probable velocity. Ths, for instance, the intensity absorption coefficient for the probe is given by a p (2 p /I p ) dvn(v)imy b 31 (v), where I p 0 ca 2 p /2 is the probe intensity. We will consider here the following drive field configrations: i a symmetric SW drive i.e., ), ii a TW drive copropagating with the probe i.e., and 0), iii a conterpropagating TW drive i.e., and 0), and iv an asymmetric SW drive i.e., ). In all cases the drive will be strong enogh to satrate the driven transition, i.e., 23, althogh with a Rabi freqency smaller than the Doppler broadening D of the driven transition, i.e., D 2 ln 2(/c) d. Let s emphasize that Doppler broadening is one of the main difficlties in observing and sing atomic coherence effects in atomic gases, in particlar in the case of a large freqency p-conversion ratio R p / d k p /k d. It is also well known that a way to mitigate the negative inflence of Doppler broadening is to se a high-intensity TW drive with D, which sally reqires plsed lasers. In contrast, we here explore the opposite regime, i.e., D, bt se a SW drive. III. DRESSED-ATOM ANALYSIS The dressed-atom analysis 30 is very sefl for the physical interpretation of nmerical reslts obtained by solving the density matrix Eqs. 5 for the case of a weak probe field. Consider again the ladder scheme of Fig. 1, althogh generalization to any other three-level configration is straightforward. We se a qantm description of the laser fields and denote by i;n,n ;n p the states of the ncopled atom laser fields, with the atom in state i 1,2,3, in the presence of n p probe photons and N (N ) photons of the TW drive component copropagating conterpropagating with the probe field. For a TW drive it is either N 0 orn 0. In the standard rotating-wave approximation, and considering only processes involving one probe photon, de to the weakness of the probe field, one has the following infinite chain of coplings again de to the SW natre of the drive field between the npertrbed states: 2;N,N 1;n p 3;N,N ;n p 2;N 1,N ;n p 1;N,N ;n p 1, 7 where we have indicated the matrix elements of the atomlaser interaction Hamiltonian between the levels connected by the arrows, in the semiclassical approximation. In this approximation, the choice of N, N, and n p is nrelevant, therefore we introdce the following shortened label: 3;N m,n m;n p 3;q 2m,0, 2;N m,n m 1;n p 2;q 2m 1,0, 1;N,N ;n p 1 1;0,1, 8a 8b 8c with m 0, 1, 2,... If the energy of the state 3;0,0 is taken as the origin, the energies of the above npertrbed states become with 1) E 3;q qk d v, q even, 9a E 2;q d qk d v, q odd, 9b E probe p k p v, 9c that represented as a fnction of k d v see Fig. 3 a for the case d 0 appear as two manifolds of straight lines, one of them formed by lines with the even slope passing throgh the origin, and the other by lines with the odd slope passing throgh the point with coordinates (0, d ). The probe level is represented in Fig. 3 a by a straight dotted-dashed line with ordinate at the origin p and slope k p /k d R. Notice that the crossing points between the probe level and the other levels determine the npertrbed resonant velocities at which varios mltiphoton absorption resonances occr. Ths, the crossings of E probe with the levels of slopes q 0,1,2, and 3 determine the resonant velocities for the for processes shown in Fig. 2. For a TW drive copropagating with the probe, only the following three states are involved: 3;q 0,0, 2;q 1,0, 1;0,1. If the fields are conterpropagating the levels involved are 3;q 0,0, 2;q 1,0, 1;0,1. The laser-atom copling can be considered in two steps owing to the weak-probe assmption. First one considers the copling with the strong SW drive field. By diagonalizing

5 ELECTROMAGNETICALLY INDUCED TRANSPARENCY... PHYSICAL REVIEW A FIG. 3. a Energy diagram of the atom dressed by a resonat drive field solid lines with d 0 and. Dotted lines correspond to the case of a TW drive. The dotted-dashed line represents the probe level for R 2.5, p 10. b The black domains represent the space-inhomogeneos continos energy spectrm for v 0 atoms as a fnction of the asymmetry factor ( ) 2 /( ) 2 ( ) 2 of the drive field for a constant total intensity ( ) 2 ( ) 2. the infinite Hamiltonian associated with the first row in Eq. 7 one obtains the pertrbed energies E n and eigenstates x n of the so-called SW-dressed two-level atom 31. The x n states can be written in the form x n q C q,n j;q,0, with j 3 2 for q even odd. These dressed states are labeled by the integer n, whose choice is completely arbitrary. The copling with the probe field is treated as a pertrbation on the basis formed by the x n states and the probe state 1;0,1 and its copling elements depend on the 3;0,0 term in the expansion of x n. Within the weak-probe assmption the inflence of the copling with the probe on the energies E n and E probe is negligible. This copling prodces a transition rate P 1;0,1 xn 2 C 0,n 2, with the probe absorption resonance occrring at the velocity where E n and E probe are degenerate. The TW drive case is mch simpler. One obtains the two pertrbed energies E n and eigenstates x n by diagonalizing the 2 2 matrix corresponding to the copling of 3;q 0,0 and 2;q 1,0 or of 3;q 0,0 and 2;q 1,0 ). The two E n are the branches of a hyperbola whose asymptotes are the npertrbed energy levels see the dotted lines in Fig. 3 a. For v 0, an exact soltion of the SW-dressed two-level atom problem can be obtained analytically only when the drive is on resonance and, while nmerical treatment is reqired in other cases 31. For d 0 and Fig. 3 a, atv 0 the energy levels are identical to the npertrbed ones, however, the associated eigenstates are linear combinations of the npertrbed states. The case v 0 can always be treated analytically. In fact, a stationary atom experiences the SW drive as a monochromatic field which is not the case for moving atoms, de to the opposite Doppler shifts sffered by the two TW components of the drive. This case can therefore be described by the TW soltion with a space-dependent Rabi freqency. For d 0 and, v 0 atoms at some places see both field components in phase and experience a maximm ac Stark splitting 2( ); in other places, where both fields have opposite phases, atoms experience a minimm ac Stark splitting 2 ; all intermediate splitting vales correspond to atoms placed between maximm and minimm field positions. Ths, de to this inhomogeneos ac Stark splitting sffered by the stationary atoms, for a symmetric SW ( ) drive energies at v 0 cover a finite interval between 2 and 2 Fig. 3 a 31. For d 0 and, the black domains in Fig. 3 b represent the spatially inhomogeneos continos energy spectrm of v 0 atoms as a fnction of the asymmetry factor ( ) 2 /( ) 2 ( ) 2 for a constant total intensity ( ) 2 ( ) 2. Since E(v 0) 0 occrs only for a symmetric SW drive ( ) stationary atoms contribte to resonant ( p 0) probe absorption only in this case. Notice that in the resonant and symmetric SW drive case Fig. 3 a, for p 0 and R k p /k d 1,2,3,...,there is a complete sperposition between the probe level and a dressed-state level, i.e., a complete cancellation of the Doppler effect, since all the atomic velocity classes contribte simltaneosly to the probe absorption resonance. IV. NUMERICAL RESULTS AND DISCUSSION For the nmerical reslts shown in this section, we have considered parameters for two real cascade systems Fig. 1 in barim and in beryllim atoms. Let s discss first the case of barim driven by a resonant SW field with 350 MHz copling the pper transition at d 821 nm and a weak field with MHz probing the lower transition at p 554 nm. Note that the probe-to-drive freqency ratio is in this case R 1.48 which, as pointed ot in the introdction, makes this configration very convenient to observe EIT with a SW drive. With 1, 2, and 3 being the levels 1 S 0 m J 0, 1 P 1 m J 1, and 1 D 2 m J 2, respectively, and with all the laser fields circlarly polarized with the same helicity, this three-level cascade system in 138 Ba is almost perfectly closed 32. The spontaneos emission rates of lower and pper transitions read, respectively, MHz and MHz. For a Maxwellian atomic velocity distribtion corresponding to barim atoms in a vapor cell at T 500 K one has k d 930 MHz. For a resonant drive field ( d 0), we plot in Fig. 4 a contor map of N(v)Imy b 31 (v) in the parameter plane p (v) p k p v

6 SILVA, MOMPART, AHUFINGER, AND CORBALÁN PHYSICAL REVIEW A FIG. 4. Probe absorption contor map for Ba atoms in the parameter plane k d v, p (v)] for a resonant SW drive and b resonant TW drive. For parameters and discssion, see the text. p Rk d v vs k d v for a a SW drive ( 350 MHz) and b a copropagating TW drive ( 350 MHz, 0). Regions with dark gray light gray correspond to high low probe absorption. A comparison of Figs. 4 a and 4 b reveals that the absorption characteristics of the atomic medim are dramatically different for SW and TW driving. While for a TW drive see Fig. 4 b the regions of high probe absorption occr close to the two branches of a hyperbola whose asymptotes have slopes 0 and 1, for a SW drive high probe absorption occrs abot straight lines passing throgh the origin with slopes 0, 1, 2,...InFig. 4 a the regions with slopes 0 and 1 are clearly visible, those with slopes 2 can be appreciated clearly only far from the origin. Regions with higher slopes overlap and cannot be seen individally. These reslts clearly reflect the atom s dressed-level strctre discssed in the previos section. To show this strctre most clearly we have assmed in Fig. 4 a flat velocity distribtion i.e., k d ). The probe absorption domains in Fig. 4 are de to probe resonances involving state 1;0,1 and the dressed states bilt by the drive field. The discssion in Sec. III then clarifies the physical origin of the probe absorption shown in Fig. 4 a and shows that the main featres in this figre are generic, i.e., with a resonant SW drive, probe absorption will always occr abot straight lines throgh the origin, with the integer slope, in diagrams of the type shown in Fig. 4. One can already read in Fig. 4 the advantages of sing the resonant SW drive instead of the TW drive when R 1. To obtain the probe absorption coefficient a p one mst perform an integration over the velocity distribtion N(v) in order to obtain the contribtions of all the atoms. If, also, p 0, the points representing different atomic velocity classes in the diagrams of Fig. 4 lie on a straight dashed line throgh the origin with slope R or R in the case of the conterpropagating probe and TW drive. To get a p one mst add the contribtions to probe absorption fond along this dashed line. In Fig. 4 b, with R 1 there is always partial overlap between the dashed lines and the two absorbing regions, whose slope changes continosly from 0 to 1. Notice, however, that with conterpropagating drive and probe fields there is no absorption, i.e., complete transparency, for 0 R 1, which corresponds to freqency downconversion. For the SW drive case, we see in Fig. 4 a that one gets small absorption not only when 0 R 1 bt also for R n 1/2 with n 1,2,3... These featres of the probe absorption coefficient are clearly seen in Fig. 5, where we plot the probe absorption coefficient normalized to its vale in the absence of the drive field, a p /a p ( 0), as a fnction of R and other parameters as mentioned at the beginning of this section for both resonant SW drive and TW drive. Althogh Fig. 5 is not very interesting from a physical point of view, since one cannot explore in practice all the R vales with the same set of parameters selected here for a cascade system in barim atoms with R 1.48), we plot it becase it clearly exhibits the advantages of sing resonant SW drive instead of TW drive in the freqency pconversion regime, i.e., when R 1. For SW drive we observe strong absorption resonances at integer vales of the probe-to-drive freqency ratio, i.e., at R 1,2,3... Between these absorption resonances there are probe windows of electromagnetically indced transparency at R n 1/2 (n 1,2,3...) whose depth decreases as R increases. It is also appreciated that p to R 1 a conterpropagating TW drive is the most convenient configration in order to prodce probe transparency. Nevertheless, for R 1 sch that FIG. 5. Normalized probe absorption coefficient as a fnction of the probe-to-drive freqency ratio R for p d 0, and other parameters as in Fig. 4. Three different cases are represented: i SW drive solid line, ii copropagating TW drive dotted line, and iii conterpropagating TW drive dashed line

7 ELECTROMAGNETICALLY INDUCED TRANSPARENCY... PHYSICAL REVIEW A FIG. 6. a Normalized probe absorption coefficient as a fnction of the probe detning p for i SW drive solid line, ii copropagating TW drive dotted line, and iii conterpropagating TW drive dashed line. b Corresponding probe transmission throgh a vapor cell of barim atoms of length L 4 cm and atomic density of N atoms/cm 3. Other parameters as in Fig. 4. R n 1/2 (n 1,2,3...), the resonant SW drive configration is the most advantageos configration to observe EIT or even the niqe one in which it occrs, e.g., in Fig. 5 there is almost no EIT with a TW drive for R 2 while there are appreciable EIT windows with a SW drive p to R 9.5. Again, for the parameters of the real cascade system in atomic barim we have obtained the probe spectra shown in Fig. 6 a for both SW and TW drive. Clearly, the maximm indced probe transparency is observed for a SW drive configration and, as a characteristic featre of this case, three different peaks of transparency appear. In particlar, at the probe field line center, i.e., p 0, a p redces to abot 0.1a p ( 0) i.e., it redces by 90%). Finally, we have considered the probe propagation throgh a vapor cell of length L 4 cm with an atomic density of N atoms/cm 3. Assming an electric dipole moment for the probed transition of C cm and neglecting drive field depletion throgh the vapor cell, we have obtained the probe transmission spectrm shown in Fig. 6 b. Probe transmission at the line center increases from the vale exp( 10.6) in the absence of drive field to vales above 50% with the SW drive, bt remains almost negligible with a co- or a conterpropagating TW drive field. To obtain the same resonant probe transmission peak as in the SW case, FIG. 7. Normalized probe absorption coefficient a and probe transmission b vs / D for p 0, , and other parameters as in Fig. 6 for i SW drive solid line, ii copropagating TW drive dotted line, and iii conterpropagating TW drive dashed line. with a conterpropagating TW drive one shold se 1680 MHz, which means an intensity 23 times that of one of the TW components of the SW drive sed here. Additional information in this respect can be obtained from Fig. 7 where we plot, for p 0 and , the normalized probe absorption coefficient Fig. 7 a and the probe transmission Fig. 7 b verss / D and other parameters as in Fig. 6. One clearly sees that for / D 2.2 a SW drive prodces larger probe transmissions than if only one of its TW components were sed for driving. Above this vale of / D a conterpropagating TW drive is more efficient than a SW one in indcing probe transparency. We have compted also transmission spectra for increasing probe strength. The peak at p 0 does not change appreciably for p to 3.5 MHz For 2 31 there is still a resonant peak above 50% bt differences between the cases of TW and SW drive are less marked. Finally, we have compted transmission spectra for the same conditions as in Fig. 6 bt for different drive detnings from atomic resonance. Reslts are qalitatively similar to those shown in Fig. 6 b for drive detnings p to d 350 MHz, which indicates that it is not critical to tne the drive field exactly to resonance. Up to now we have considered a drive field whose two components have the same intensity ( ). In spite of

8 SILVA, MOMPART, AHUFINGER, AND CORBALÁN PHYSICAL REVIEW A FIG. 8. Same as in Fig. 4 a bt for 383 MHz, 913 MHz. the previos discssion, a SW field has a clear drawback for indcing transparency as compared to a TW field. In fact, while the latter field acts on all the atoms, the former does not cople to the most abndant v 0 atoms placed at the nodes of the SW where the drive field vanishes. These stationary atoms therefore contribte to probe absorption in the SW case. This drawback can be mitigated in part sing a drive configration with. In fact, this case can be considered in a first approximation as consisting in a SW drive convenient for v 0 atoms pls an extra conterpropagating TW drive convenient for v 0 atoms. This is confirmed by the nmerical reslts shown in Figs. 8 and 9. In Fig. 8 we represent a contor map of N(v)Imy b 31 (v) for 383 MHz, 913 MHz, and other parameters as for Fig. 4 a. The map is no more symmetric abot v 0, and the dashed line representing different atomic velocity classes for p 0 meets close to v 0 domains of smaller absorption than in Fig. 4 a. The resonant peak transmission as a FIG. 10. The same as in Fig. 9 bt now for Be atoms. From top to bottom crves correspond to ( ) 2 ( ) 2 / , ,3 10 4, and fnction of ( ) 2 / ( ) 2 ( ) 2 for several total drive intensities ( ) 2 ( ) 2 and other parameters as for Fig. 6 b is plotted in Fig. 9. For each fixed total drive intensity there is a different optimm vale of the asymmetry between and at which probe transmission is maximm. Inspection of atomic data tables reveals that there are many probe transitions involving the grond state and having an adjacent transition to be driven sch that R n 1/2 (n 1,2,3,...). Examples inclde Ba (R 1.48), C (R 2.49), Be (R 3.51), and Hg (R 5.48). To show that the advantages of sing in EIT a SW drive instead of a TW one does not restrict to the particlar case of barim atoms, we discss briefly next the case of beryllim. Neglecting hyperfine strctre and sing drive and probe fields linearly polarized along the qantization axis, the levels 2s 1 S 0,2p 1 P 1, and 3s 1 S 0 in beryllim form a cascade three-level system with p nm, d nm i.e., R 3.51), MHz, and MHz. Taking other parameters as for Ba, we plot in Fig. 10 transmission crves for Be analoges to those in Fig. 8. In both cases peak transmissions above 85% are possible with a strongly asymmetric drive. V. CONCLUSIONS FIG. 9. Resonant peak transmission as a fnction of ( ) 2 / ( ) 2 ( ) 2 for several total drive intensities ( ) 2 ( ) 2 and other parameters as for Fig. 6 b. From top to bottom crves correspond to ( ) 2 ( ) 2 / , , and In this paper we have analyzed, in the framework of standard semiclassical density matrix formalism, EIT for a probe TW field in closed Doppler-broadened three-level systems driven by a SW field. Electromagnetically indced probe transparency windows appear close to half integer vales of the probe to drive field freqency ratio R. These reslts have been compared to those obtained for a TW drive proving that for R 1 a SW drive field is appreciably more efficient than a TW drive in indcing probe transparency. In particlar, we have considered parameters for a real closed cascade scheme in barim atoms with R 1.5 showing probe transmission resonances with vales above 50% when the transmission is almost negligible either withot drive field or with only one of the TW components of the SW drive. We have shown that a drive field with two conterpropagating components of dif

9 ELECTROMAGNETICALLY INDUCED TRANSPARENCY... PHYSICAL REVIEW A ferent intensity is even more efficient than a SW one in indcing probe transparency. In ladder schemes, to achieve optimm probe transmission with a given total drive intensity, the drive component conterpropagating with the probe shold be stronger. Many real three-level systems exist with R n 1/2 (n 1,2,3,...). In addition to the Ba case, another system in beryllim atoms with R 3.51 has been considered explicitly. Detailed contor maps of the probe field absorption in the parameter plane k d v, p (v)] have been presented and discssed with the help of the dressed-atom approach. ACKNOWLEDGMENTS Spport from the DGESIC Spanish Government nder Contract No. PB C03 and from the DGR Catalan Government nder Contract No. 1999SGR00096 is acknowledged. 1 M. O. Sclly and M. S. Zbairy, Qantm Optics Cambridge University Press, Cambridge, U.K., 1997, Chap. 7, pp E. Arimondo, in Progress in Optics XXXV, edited by E. Wolf Elsevier, Amsterdam, 1996, Chap. V, pp S. E. Harris, Phys. Today 50, J. P. Marangos, J. Mod. Opt. 45, J. Mompart and R. Corbalán, J. Opt. B: Qantm Semiclassical Opt. 2, R M. O. Sclly, Phys. Rev. Lett. 67, ; A. S. Zibrov et al., ibid. 76, L. V. Ha, S. E. Harris, Z. Dtton, and C. H. Behroozi, Natre London 397, ; M. M. Kash, V. A. Satenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lkin, Y. Rostovtsev, E. S. Fry, and M. O. Sclly, Phys. Rev. Lett. 82, ; D. Bdker, D. F. Kimball, S. M. Rochester, and V. V. Yashchk, ibid. 83, ; S. Inoye, R. F. Löw, S. Gpta, T. Pfa, A. Görlitz, T. L. Gstavson, D. E. Pritchard, and W. Keterlee, ibid. 85, L. J. Wang, A. Kzmich, and A. Dogari, Natre London 406, K. Hakta, L. Marmet, and B. P. Stoichef, Phys. Rev. Lett. 66, ; G. Z. Zhang, K. Hakta, and B. P. Stoichef, ibid. 71, ; Y. Li and M. Xiao, Opt. Lett. 21, A. Aspect et al., Phys. Rev. Lett. 61, G. Morigi, J. Eschner, and C. H. Keitel, Phys. Rev. Lett. 85, ; C. F. Roos, D. Leibfried, A. Mndt, F. Schmidt- Kaler, J. Eschner, and R. Blatt, ibid. 85, Ch. Li, Z. Dtton, C. H. Behroozi, and L. V. Ha, Natre London 409, ; D. F. Phillips, A. Fleischhaer, A. Mair, R. L. Walsworth, and M. D. Lkin, Phys. Rev. Lett. 86, G. Alzetta, A. Gozzini, L. Moi, and G. Orriols, Lett. Novo Cimento Soc. Ital. Fis. 17, H. R. Gray, R. M. Whitley, and C. R. Strod, Jr., Opt. Lett. 3, K.-J. Boller, A. Imamoğl, and S. E. Harris, Phys. Rev. Lett. 66, ; J. E. Field, K. H. Hahn, and S. E. Harris, ibid. 67, K. Hakta, L. Marmet, and B. Stoicheff, Phys. Rev. Lett. 66, M. Xiao, Y.-q. Li, S.-z. Jin, and J. Gea-Banacloche, Phys. Rev. Lett. 74, J. R. Boon, E. Zeko, D. J. Flton, and M. H. Dnn, Phys. Rev. A 57, S. Wielandy and A. L. Gaeta, Phys. Rev. A 58, F. Silva, J. Mompart, V. Ahfinger, and R. Corbalán, Erophys. Lett. 51, F. Silva, V. Ahfinger, J. Mompart, and R. Corbalán, Laser Phys. 9, F. Silva, R. Corbalán, and R. Vilaseca, Opt. Commn. 114, R. Corbalán, A. N. Pisarchik, V. N. Chizhevsky, and R. Vilaseca, Opt. Commn. 133, J. Reid and T. Oka, Phys. Rev. Lett. 38, For any of the processes of the type shown in Fig. 2 there is a family of processes obtained by the addition to any of the original,,or steps of an arbitrary even nmber of and/or absorption-emission steps. These processes do not prodce new atomic resonances bt reslt in a modification of the resonance conditions given in Eq. 1 de to the driveindced ac Stark effect, which is flly taken into accont in the rest of the paper. 26 See, e.g., S. Stenholm and W. E. Lamb, Jr., Phys. Rev. 181, B. J. Feldman and M. S. Feld, Phys. Rev. A 1, See, e.g., B. J. Feldman and M. S. Feld, Phys. Rev. A 5, R. Vilaseca, G. Orriols, L. Roso, R. Corbalán, and E. Arimondo, Appl. Phys. B: Photophys. Laser Chem. B34, C. Cohen-Tannodji, J. Dpont-Roc, and G. Grynberg, Atom- Photon Interactions Wiley, New York, L. Roso, R. Corbalán, G. Orriols, R. Vilaseca, and E. Arimondo, Appl. Phys. B: Photophys. Laser Chem. B31, P. B. Sellin, G. A. Wilson, K. K. Medri, and T. W. Mossberg, Phys. Rev. A 54,