Topical review Electromagnetically induced transparency

Transcription

1 journal of modern optics, 1998, vol. 45, no. 3, 471±503 Topical review Electromagnetically induced transparency J. P. MARANGOS Laser Optics and Spectroscopy Group, Blackett Laboratory, Imperial College, London SW7 2BZ, England (Received 11 August 1997) Abstract. The subject of electromagnetically induced transparency (EIT) is reviewed in this paper. Emphasis is placed on the experimental work reported in this eld since Theoretical work is also covered, although it is not intended to review all the very numerous recent theoretical treatments on this topic. The basic physical ideas behind EIT are elucidated. The relation of EIT to other processes involving laser-induced atomic coherence, such as coherent population trapping, coherent adiabatic population transfer and lasing without inversion, is discussed. Experimental work is described covering the following topics: EIT with pulsed and continuous-wave sources, lasing without inversion, pulse propagation in a laser dressed medium and EIT in nonlinear optical processes. A full set of references and a bibliography are included. 1. Introduction Interference between alternative pathways in quantum-mechanical processes is an ubiquitous e ect in physics (for example [1]). This interference is analogous to constructive and destructive interference between classical waves but, whilst with classical wave interference it is the eld amplitudes that are interfering in the quantum case, however, we must invoke intuitively less de nite quantities (e.g. probability amplitudes) to explain quantum interference phenomena. Matter wave interferometry, especially atom beam interference between spatially separated beam paths, has recently received considerable theoretical and experimental attention [2±4], not least because of the potential for the construction of ultrasensitive interferometers that might be used to measure atomic and molecular interactions with extreme sensitivity [5] and to test fundamental aspects of quantum mechanics (see, for example, the proposal in [6, 7]). Another kind of interference phenomenon, between transition pathways induced within the internal quantum states of atoms and molecules coupled to one or more laser elds, is the subject of the present review. In particular, three-level atomic and molecular systems coupled to two laser elds exhibit interference e ects that can result in cancellation of absorption at a resonance transition frequency and other modi cations of the optical response. It has been known for some time [8] that, if states of an atom are coupled via several possible alternative transition processes, interference between the amplitudes of these processes may lead to an enhancement (constructive interference) or a complete cancellation (destructive interference) of the total transition probability. These e ects arise because in quantum mechanics it is probability amplitudes 0950±0340/98 $12 00 # 1998 Taylor & Francis Ltd.

2 472 J. P. Marangos Figure 1. Interference between pathways in autoionization of an atom by ultraviolet radiation. The ground state j1i is coupled to the continuum state je2; ki via two alternative pathways; channel (a) is direct photoionization by absorption of an ultraviolet photon!vuv, channel (b) is an indirect photoionization process in which absorption of the ultraviolet photon!vuv results in excitation into bound state j2i which is followed by a transition to the continuum state via the interelectronic Coulomb interaction. The amplitudes due to these two channels must be summed to give the overall photoionization amplitude; thus interference is seen in the ionization rate. (which may be positive or negative in sign), rather than probabilities, that must be summed to obtain the total transition probability of a process. An example of this in atomic systems is Fano [8, 9], interference seen for radiative transitions to autoionizing states in atoms and also predicted in semiconductor quantum wells [10], leading to asymmetric spectral pro les. Here ( gure 1) a doubly excited state of the atom j2i of energy E 2, with a bound-state character, lies in the continuum and is energetically degenerate with a continuum state je 2; ki (at the same energy E 2) to which it is coupled via the Coulomb interaction between the two electrons. An ultraviolet electromagnetic eld will cause photoexcitation from the ground state j1i to the continuum state je 2; ki. This photoionization can proceed via two possible channels: rstly a direct photoionization j1i! je 2; ki or secondly an indirect process consisting of excitation j1i! j2i followed by a rapid radiationless transition (caused by the interelectronic Coulomb interaction) j2i! je 2; ki. Interference occurs between these two channels with a sign varying from constructive to destructive depending on the frequency of the ultraviolet electromagnetic eld. This type of interference is deliberately induced by an applied laser eld in recent work concerned with the laser control of the optical properties of atomic media. The cancellation of absorption for a probe eld tuned in resonance to an atomic transition, for which strong absorption would normally be expected, is perhaps the most striking example. This process has been termed electromagnetically induced transparency (EIT) [11, 12], the e ect being caused by the interference between the coherences excited in the atom by the electromagnetic elds and leading to an initially highly opaque medium being rendered almost transparent. Similarly the refractive properties of the medium may be greatly modi ed [13, 14]; for instance the usual correlation of high refractive index with high absorption can be broken, leading to the creation of media with very unusual

3 Electromagnetically induced transparency 473 optical properties. Recently classical eld interference of an analogous type has been proposed as a means for the cancellation of absorption of electromagnetic radiation propagating through an ideal plasma at a frequency which would otherwise be below the transmission cut-o [15]. One of the consequences of the coherence and interference e ects related to EIT that has been recognized is the possibility of building a short-wavelength laser operating without the need to achieve population inversion in the atomic medium [16±22]. Lasing without inversion has been demonstrated in Na and Rb in the visible range [23, 24]. The prospects that this might lead to the construction of lasers able to circumvent the usual constraints placed on achieving inversion of short-wavelength lasers (owing to the 3 scaling of the spontaneous emission) has been much discussed (see the reviews in [25, 26]). Related lasing without inversion and EIT e ects in semiconductor quantum wells have also been theoretically explored using laser-induced processes [27] or bandgap engineering [28] to create the necessary coherences. It is the aim of this review to provide a summary of the experimental work that has so far been reported on EIT and to present an overview of the general progress in this topic (including also theory) since An extensive review of all theoretical and experimental work related to atomic coherence phenomena is beyond the scope of the present article and the reader is advised to look at a number of reviews on lasing without inversion [25, 26], coherent population trapping (CPT) [29±31] (a very comprehensive recent review on this subject has been given in [32]) and laser-induced continuum structure [33, 34] to nd these topics presented in detail. The theory pertinent to EIT is sketched in this text, but again the reader is referred to the more detailed treatments published in the literature which will be given as they arise. Although this review is intended to be of use to the specialist reader, perhaps already engaged in research in this or related elds, I hope that it will also be accessible to a wider cross-section of physicists and that it will provide a useful introduction to what is proving to be a fascinating area of research. The review is organized in the following fashion. Section 2 examines the physical basis for EIT and related phenomena. It attempts to provide a physical picture of these e ects; in the process some rigour will be sacri ed but I hope that this is compensated by providing an intuitively accessible picture that clari es some basic issues whilst providing the reader with comprehensive references to more detailed treatments. Also in section 2.2 we shall introduce the most signi cant physical consequences of EIT, namely modi cations to the linear and nonlinear optical response of the medium. The possible importance of this eld to future technological applications will also be touched upon at this point. Finally in section 2.3 some of the very important physical constraints which can reduce or extinguish observable e ects in real systems will be discussed. This provides a link between the predictions of basic theory and the situations likely to be encountered in experiments. Section 3 deals with experimental techniques and results. It is intended to provide a solid overview of the eld but it cannot hope to be comprehensive in its scope owing to the large body of work on related topics (e.g. coherent population transfer, dark resonances, and laser-induced modi cations to spontaneous emission) that must necessarily be omitted or only partially covered. The discussion will be organized along the lines of the scienti c principles being investigated and

4 474 J. P. Marangos Figure 2. The basic energy level schemes for three-level atoms interacting with two near-resonance electromagnetic elds. In all cases, j1i is dipole coupled to j2i, and j2i to j3i, but j3i is not dipole coupled with j1i (metastable). The schemes are de ned by the relative energies of the three states: (a) a ladder or cascade system; (b) a scheme; (c) a V scheme. the experimental techniques used rather than in any strict chronological order. Thus rst (section 3.1) some of the earlier experimental investigations of atomic coherence e ects are brie y reviewed. Then the rst EIT experiments that employed pulsed (section 3.2) and continuous-wave (CW) (section 3.3) lasers are discussed. Related experimental work on ampli cation without inversion and lasing without inversion is presented in section 3.4. Pulse propagation and matched pulse e ects are a development that connects EIT with recent work on coherent adiabatic population transfer, and both theoretical and experimental work will be discussed in section 3.5. Finally, in section 3.6, recent experiments that have used EIT and related processes to enhance nonlinear optical processes (a very promising area) will be presented. The review is completed with a discussion and conclusion in section 4 that overviews the work done so far and the prospects for further developments in the near future. 2. Physics of EIT 2.1. The physical basis of electromagnetically induced transparency There is a close link between EIT and other atomic coherence phenomena such as coherent population trapping [32, 35±37] and coherent adiabatic population transfer processes [38±41]. In all these processes, three-level atomic systems are involved (or at least systems that can be adequately reduced to three levels when interaction with the pertinent electromagnetic elds are considered). The usual atomic dipole selection rules normally require that two pairs of levels are dipole coupled whilst the transition between the third pair is dipole forbidden. In gure 2 we show the three basic level schemes for three-level atoms; all the level schemes involved in the experiments discussed in this paper can be reduced to one or other of these schemes. We label the levels j1i, j2i and j3i, where j1i j2i and j2i j3i are dipole allowed but j1i j3i is not since j3i is a metastable state. Classi cation

5 Electromagnetically induced transparency 475 of the schemes then depends upon the relative energies of these three states [31]: (a) in a ladder (or cascade) scheme E 1 < E 2 < E 3 ; (b) in a scheme E 1 < E 3 < E 2 (although, in a symmetric scheme, E 1 and E 3 are almost degenerate); (c) in a V scheme E 2 < E 1 and E 3 (although again in a symmetric V scheme, E 1 and E 3 are almost degenerate). In a or ladder scheme, j1i is normally the ground state of the atom and is where the majority of the population (initially) resides. EIT is associated with and ladder schemes since in this process population transfer is not required (in contrast with CPT) and states j2i and j3i can remain (essentially) unpopulated throughout the process. To understand more clearly how laser elds may interact with a three-level atom and create coherent superpositions of the atomic basis states we shall rst consider CPT in a scheme. A three-level system ( gure 3) consisting of states j1i, j2i and j3i is coupled by two near-resonance laser elds of strength (de ned in terms of the Rabi frequency [31]) O 1 (at frequency!1) and O 2 (at frequency!2). De ning the frequency of transitions between states as!12 ˆ E 2 E 1 )/ h,!23 ˆ E 2 E 3 = h and!13 ˆ E 3 E 1 = h we can further de ne one- and twophoton (Raman) frequency detunings as D 12 ˆ!12!1, D 23 ˆ!23!2 and D ˆ!12!23!1!2. The Hamiltonian H 0 of the bare atom should be modi ed to include the interactions due to the two couplings, that is H ˆ H 0 V 1 V 2 (where the interaction V j ˆ hoj). The eigenstates of this new Hamiltonian will be linear superpositions of the bare atomic states j1i, j2i and j3i. For the situation of exact two-photon resonance that is D ˆ 0 (or rather exact twophoton resonance taking into account any light shifts) two of the eigenstates of the total Hamiltonian H 0 V 1 V 2 turn out to be symmetric and antisymmetric coherent superpositions of the two lower states of the bare atom basis that are of the form: O 1 O 0 j 1i O 2 O 0 j 3i ; 1 a O 2 O 0 j 1i O 1 O 0 j 3i ; 1 b where O 0 ˆ O 2 1 O 2 2 1=2. Importantly no component of the bare atom state j2i appears in these superpositions. These form eigenstates of the atom± eld system of which one j i is coupled to the intermediate state j2i via the electric dipole interaction whilst the other state j i becomes decoupled (a so-called `dark or `trapped state). This can be seen if we form the dipole moment between j i or j i with the remaining bare atom state j2i [31]. If the magnitudes of the coupling elds O 1 and O 2 are appropriately balanced, the negative sign in the superposition of j1i and j3i which form j i will ensure that the corresponding dipole moment will vanish. In e ect the two terms that are summed to give the transition amplitude between j2i and j i are of equal and opposite magnitude, and hence the total amplitude will vanish. State j i is often referred to as a non-coupled state jnci whilst j i remains coupled to the electromagnetic elds jci. In the above description the situation has been simpli ed by ignoring the fast time development (at frequencies E 1= h and E 3= h) of the states in the superposition in equation (1) (these terms will in fact disappear when the dipole moments are

6 476 J. P. Marangos Figure 3. The basic (Raman) scheme leading to coherent population trapping. The applied elds need not be in single-photon resonance, but the two-photon (Raman) resonance condition should be met in this three-level system. In a CPT experiment the laser elds O 1 and O 2 are of comparable strength that this is su cient to saturate the two-photon transition. formed). On the assumption that a steady-state situation has been reached, the superposition state jnci will acquire all the population of the system through the action of optical pumping (spontaneous emission from j2i will populate jnci but there is no absorption process from jnci to j2i to depopulate it). In the usual CPT scheme, O 1 º O 2, that is both coupling strengths are of comparable magnitude, whilst the elds are also strong enough to reach the twophoton transition saturation condition. In fact this does not necessarily require that the elds are su cient to saturate the single-photon transitions j1i j2i and j2i j3i since, under the two-photon resonance condition, state j2i can be adiabatically eliminated and so does not enter into the consideration of the coupling between atoms and elds. In the CPT system, interference e ects arise from both the coupling elds since they are of comparable strength. If only one of the elds, that is O 2, is strong such that O 1 ½ O 2, then only interference e ects due to processes driven by O 2 will be important. This is the situation in EIT schemes and this close connection between EIT and CPT has been discussed by a number of workers (for example [32, 42]). Typically in a CPT scheme the states j1i and j3i are Zeeman or hyper ne sublevels of the ground state and are thus both initially populated. In contrast, in many EIT schemes, j3i is an excited state and has no population at any time during the process. Unlike the case of CPT, where the time scale for population trapping in the jnci state is several radiative lifetimes, in the case of EIT the e ect is established (within a single atom) in a time on the order of 1=O 2 which is generally much faster. Comparison with coherent adiabatic population transfer schemes (e.g. stimulated Raman adiabatic passage (STIRAP)) [38, 39] also shows that the EIT situation is equivalent to the initial stages in the population transfer process when the counter-intuitively sequenced laser pulses satisfy the condition O 1 ½ O 2. The interference associated with EIT arises because the transition amplitude between j1i and j2i includes, as well as a term due to the resonant eld O 1, an additional amplitude due to the presence of the other eld O 2. This additional term has a negative sign with respect to the rst and hence in an ideal situation will

7 Electromagnetically induced transparency 477 Figure 4. (a) The basic scheme for an EIT experiment, illustrated here in a system in the bare states. The eld coupling j1i to j2i is a relatively weak probe O P (equivalent to O 1), whilst the eld coupling states j2i and j3i called the coupling eld O C (equivalent to O 2) is strong. This makes the adoption of the dressed-state basis logical as illustrated in (b). (b) The dressed states labelled here as j3di and j2di (referred to as jai and jbi in the text); destructive interference between the probe absorption amplitudes due to these two dressed states leads to EIT. cancel the rst term completely. In the case of EIT, since the eld O 2 is large (in EIT experiments, O 2 is usually called the coupling eld and labelled O C, and O 1 is a weaker probe eld labelled O P), it is logical to choose the dressed state basis [21] to analyse this system ( gure 4); on this basis, upper states form a coherent superposition, which for a resonant coupling D 23 ˆ 0 is of the form jai ˆ 1 2 1=2 j 3i j2iš ; 2 a jbi ˆ 1 2 1=2 j 3i j2iš : 2 b The transition amplitude at the (undressed) resonant frequency (E 2 E 1 h from the ground state j1i to the dressed states will be the sum of the contributions to states jai and jbi. If j3i is metastable, then the contributions from the j1i! j2i transition cancel since they enter the sum with opposite signs. This cancellation of absorption on the j1i j2i transition can also be viewed in terms of Fano-type interference [21, 42±44]. The connection between equation (2) and the coupled and non-coupled states of equation (1) can readily be seen. The states jnci and jci (i.e. j i and j i in equations (1)) in the case of two-photon (Raman) resonance are related to the bare atomic states by jci ˆ O1 j 1i O 2 j3i O 0 ; 3 a jnci ˆ O2 j 1i O 1 j3i O 0 ; 3 b or alternatively we can de ne the bare atomic states in terms of jnci and jci. Thus, for example, state j1i (atom ground state) can be written

8 478 J. P. Marangos j1i ˆ O2 j NCi O 1 jci O 0 : 4 For the case O 1 ½ O 2 it is clear from equation (4) that j1i will be almost equivalent to jnci and thus (for two-photon resonance and a metastable state j3i) absorption vanishes. Alternatively, if the EIT process is viewed within the atomic bare-state basis (rather than the dressed states), the so-called `coherences can be seen as being the quantities pertinent to the interference. These coherences can be thought of, in a semiclassical picture, as associated with the oscillating electric dipoles driven by the coupling elds applied between pairs of quantum states of the system, for example jii j ji. Strong excitation of these dipoles occurs whenever electromagnetic elds are applied close to resonance with an electric dipole transition between two states. If there are several ways to excite the oscillating dipole associated with jii j ji, then it is possible for interference to arise between the various contributions to this dipole, and these must be summed to give the total electric dipole oscillation between jii and j ji (see gure 7 later). This is directly analogous to the Fano [8] e ect in autoionization, to the classical interference proposed in ideal plasmas [15] and to e ects discussed in the context of laser-induced continuum structure [34]. Of course, formally coherences are identi ed with the o -diagonal density matrix elements ij formed by taking bilinear combinations of probability amplitudes of two quantum states of the system (i.e. the weighting factors associated with the outer products such as jiih jj) [45]. O -diagonal elements of the density matrix play a critical role in the evolution of an atom coupled to electromagnetic elds [46]. Many calculations of atomic coherence e ects and of EIT (as well as general nonlinear optics and laser action) in three-level systems are therefore developed in terms of the density matrix [14, 47±49]. This approach also naturally lends itself to the inclusion of dampings that cause the decay of populations and coherences (e.g. radiative decay and collisions) [46]. Although convenient it is by no means essential to use a density matrix approach and many theoretical treatments that give clear physical insight, have been performed in terms of probability amplitudes (for example [12, 20, 21]). Additional physical insight has also been obtained by adopting other alternative approaches, for instance by a careful consideration of the Feynman diagrams representing the various processes involved that lead to interference [42, 44], or by applying a quantum jump approach [50]. In all cases the predictions are essentially identical. The magnitudes of the relevant density matrix elements can be computed from the basic coupled evolution equations (the Liouville equation) [31, 45] and are found to depend upon parameters that are controllable within an experiment (e.g. detunings and laser intensities). In gure 5 we show a prototypical EIT scheme (in this case a ladder con guration, but entirely equivalent to the schemes discussed so far if j2i is higher in energy than the metastable state j3i). The detunings between the elds E 1 at!1 and E 2 at!2 with respect to the associated transition frequencies!12 and!23 are de ned by D 12 ˆ!12!1 and D 23 ˆ!23 ˆ!2 as above. The Rabi frequencies are O 1 and O 2 respectively. In addition to dependence on these laser properties the EIT will also depend critically on atomic parameters. For instance since perfect absorption cancellation depends on the metastability of j3i any radiative or collisional decay of this state will lead to nite absorption even at

9 Electromagnetically induced transparency 479 Figure 5. Illustration of a prototypical scheme for EIT, in this case a ladder (cascade) system is shown. The single-photon detunings D 12 and D 23 (de ned in text) are shown. zero detuning!1 ˆ!2 ˆ 0. EIT will manifest itself in the value of the density matrix element 12 whose real and imaginary parts should both vanish at zero detuning (i.e. the coherence is cancelled by the interference of the pathways that can excite it). A set of coupled equations connecting the density matrix elements, for example 12, 23 and 13, and their temporal derivatives can be written down and solved for various sets of conditions by either analytical or numerical means. It should be noted that the interference that leads to EIT arises from the existence of coherences 23 and 13 that are coupled to 12. The coherence 13 between the ground state j1i and metastable state j3i is present only because of the additional laser coupling. The contribution to the coherence 12 from the coherences 23 and 13 cancels with the direct contribution driving this coherence due to the applied probe eld (at frequency!1). If a steady-state limit is assumed, all the derivatives in the density matrix elements vanish; in this case the solution of coupled density matrix equations is greatly simpli ed. This can be solved exactly usually under the additional assumption that only the coupling eld O C is strong and hence is the only eld that must be retained to all orders. This solution is frequently obtained by using an algebraic manipulation programme to invert the super-matrix resulting from the set of equations [51]. These solutions yield the steady-state populations and coherences (on- and o -diagonal density matrices respectively) and in particular give the appropriate value of 12 (the coherence associated with the transition at the probe frequency) that is of direct interest in describing EIT. The expression that is derived for 12 will include a number of terms in the various parameters (e.g. O C and D) that will lead to cancellation of its value (both real and imaginary parts) when in two-photon resonance. A time-dependent calculation is of course appropriate if there are timedependent laser pulses coupling with the atom. This is vital for modelling the results of pulsed laser experiments and to account for transient e ects correctly. Some time-dependent calculations have been made but, unless restrictive simplifying assumptions are applied [52], these calculations must be performed numerically rather than analytically. In most cases the results of the time-dependent

10 480 J. P. Marangos calculation will be comparable with those of the steady-state method, at least in so far as qualitative trends are concerned. In calculating the results of propagation of pulses through an extended ensemble of atoms, the time-dependent density matrix equations must also be coupled to Maxwell s equations. This is necessary for instance in computing the propagation of matched pulses [53±55] and in computing preparation losses and pulse shape modi cations [56] Physical consequences of EIT electromagnetically induced transparency To understand the modi cation of the optical properties (absorption and gain, and dispersion and nonlinear response) of a laser dressed medium we need to examine the linear (and nonlinear) optical susceptibilities rather than the density matrix elements themselves. The macroscopic polarization at the transition frequency!12 can be related to the microscopic coherence 12 via the expression P 12 ˆ N 12 12, where N is the number of equivalent atoms in the ground-state within the medium, and 12 is the dipole matrix element associated with the (undressed) transition [46]. This relation holds for a medium su ciently dilute that dipole±dipole coupling between atoms can be ignored; otherwise these local eld e ects must be incorporated into the treatment which complicates interpretation [57, 58]. In this way imaginary and real parts of the linear susceptibility at frequency! can be directly related to 12 (calculated by the methods outlined in section 2.1), via the macroscopic polarization P 12! ˆ "0À! E [46]. Whilst the microscopic coherences are treated quantum mechanically the electromagnetic elds themselves are treated classically (in terms of Maxwell s equations and susceptibilities). This semiclassical approach is not essential and fully quantum treatments for CPT (for example [30]) and EIT [59±61] have been developed. These fully quantum approaches are appropriate for cases such as the coupling of atoms to modes in cavities [59, 60], or when the statistical properties of the light are of interest as they are in proposals to generate squeezed light using EIT [61]. For the relatively large elds envisaged in most laser experiments a semiclassical treatment (with spontaneous decay added as a phenomenological damping) proves adequate. For the prototypical system introduced above, the real and imaginary parts of the (dressed) linear susceptibility, associated with the dispersion and absorption of the medium respectively, will be given by expressions of the form [12]: Re À 1 D!1;!1 ˆ j 12 j2 N 4D 31 jo 2 j 2 4D 31 D 32 4D 21 G 2 3 "0 h 4D 31 D 21 G 3 G 2 jo 2 j G 3 D 21 G 2 D 31 2 ; Im À 1 D!1;!1 ˆ j j 12 j2 N "0 h 8D 2 31 G 2 2G 3 jo 2 j 2 G 3 G 2 4D 31 D 21 G 3 G 2 jo 2 j G 3 D 21 G 2 D 31 2 ; 5 where 12 is the electric dipole moment of the probe transition, D ij are the detunings de ned in section 2.1, and G 2 and G 3 are the radiative decays of states j2i and j3i respectively. In [12] these were derived using the equations describing the time-dependent atomic bare-state amplitudes in a steady-state situation with monochromatic coupling elds and with no collisional or Doppler broadening included. Of course, identical expressions are obtained using a density matrix approach. These susceptibilities are plotted in gure 6 as a function of detuning

11 Electromagnetically induced transparency 481 Figure 6. The dressed susceptibilities are shown for a three-level ladder with the coupling laser in exact resonance D 23 and with the probe laser tuned over the ranges shown (in units of the decay rate 12 between states j2i and j1i). The coupling eld Rabi frequency O C ˆ in these plots. (a) The imaginary part (Im À 1 D ) of the dressed linear susceptibility that determines the absorption in the medium; note the vanishing value at exact resonance due to destructive interference. (b) Re À 1 D ) that determines the dispersion in the medium. (c) The third-order dressed susceptibility jà 3 D j that characterizes the coupling strength in the four-wave mixing process; note the enhancement of this quantity at exact resonance due to constructive interference. D 12 with D 23 ˆ 0 (coupling eld resonant). This is a striking result when compared with the case when no coupling eld is present O 2 ˆ 0 [12]; the absorption vanishes at exact resonance (if j3i is perfectly metastable). Simultaneously the dispersion is modi ed so that, although still zero at line centre as in the uncoupled case, the group velocity (dependent upon the slope of Re À 1 ) can become anomalously low [62] where absorption has vanished. The consequence of this transparency is that a medium that would otherwise be optically thick is now rendered transparent (or at least the opacity is greatly reduced). The reduction in absorption is not merely that caused by the e ective detuning induced by the Autler±Townes splitting of the dressed-state absorption peaks, that is the absorption that would be measured if the probe eld were interrogating the absorption coe cient of the medium in the wing of the absorp-

12 482 J. P. Marangos tion pro le at a detuning O 2=2. Instead there is a destructive interference at this frequency that leads to complete cancellation of all absorption if there are no additional dephasing channels operating in the system. Even if the j1i! j3i transition dipole moment is not zero (i.e. if there is a spontaneous decay rate G 3) the absorption will be reduced compared with the weak- eld absorption at the detuning O 2=2 in the ratio G 3=G 2 [12]. In the preceding case it was implicitly assumed that the upper state probability amplitudes remain close to zero (i.e. the probe eld is always very weak). If there is an incoherent population pump into the upper states, such that these no longer remain negligible, then gain on the j2i! j1i transition can result. The remarkable feature of this gain is that under the circumstances in which EIT occurs (i.e. when absorption is cancelled) the gain can be present without the requirement of population inversion in the bare atomic states; this is an example of ampli cation without inversion. This process has successfully been incorporated into a laser without inversion in a V scheme in Rb [23] and a scheme in Na [24]. Much theoretical work [16±22] has been reported on this e ect (earlier predicted by Arkhipkin and Heller [19], and then further elucidated by Harris [20], Kocharovskaya and Khanin [16] and Scully et al. [18]). A long-term objective in this work is the prospect of overcoming the familiar di culties of constructing short-wavelength lasers (i.e. very-high-energy-density pump requirements). As well as leading to gain without population inversion, any incoherent pumping of population into the upper states also modi es the dispersive behaviour. In particular, it is then possible to have spectral regions for which the refractive index is very high (with values comparable with those normally encountered at the half-widths of the absorption pro le) whilst the absorption vanishes [14]. The prospects for engineering the refractive properties of media to give novel combinations of absorption or gain, and dispersion have been explored in a number of theoretical [63±69] and experimental studies (see below). An application proposed earlier for refractive index modi cations of this kind was for a high-sensitivity magnetometer [70]. The large dispersion at the point of vanishing absorption could it was suggested be used to detect, with high sensitivity, magnetic level shifts via optical phase measurements in a Mach±Zehnder interferometer. Modi cation of absorption relative to stimulated emission is a key element of the lasing without inversion concept. Laser-induced modi cations of the spontaneous emission have also been discussed as a potential route to reducing the pump energy density requirements to achieve lasing in an atomic medium [18, 20]. These e ects were rst studied some time ago (for example [71]) and more recently theoretical work has predicted spectral-linewidth elimination and spontaneous emission cancellation [72, 73]. These predictions were con rmed by experiments on Na dimers [74]. A full review of this subject is outside the scope of this paper. It is clear that the susceptibilities given by expressions such as equation (5) are no longer the results of rst-order perturbation theory since they implicitly include the coupling eld O 2 to all orders. In this sense they can be thought of as being the dressed linear susceptibility of the system. In the same spirit it is possible to consider a dressed nonlinear susceptibility in the presence of appropriate coupling elds. Consider our prototypical system in gure 5 and imagine that this is now incorporated into a four-wave mixing scheme such that, whilst there is now no probe eld present initially, one is generated by the mixing between the coupling eld and the additional elds (say!a and!b) that are in two-photon resonance with

13 Electromagnetically induced transparency 483 Figure 7. (a), (b) The rst two diagrams for a perturbative expansion of terms associated with the linear susceptibility in the dressed system; there are an in nite number of higher order terms of the same type as, (b) involving the exchange of an even number of photons from the coupling eld with the atom. (c), (d) The rst two terms associated with a perturbative expansion of the nonlinear susceptibility in the four-wave mixing process. Note that in this case there will be an in nite number of terms (like (d)) associated with the exchange of an odd number of photons from the coupling eld with the atom. the j1i j3i transition (this will be referred to as the two-photon coupling). In addition to the dressed linear susceptibility, given by equations (5), which describes the response of the medium to a weak probe at!1, there is now also a dressed nonlinear susceptibility that describes the coupling of the atom to the fourwave mixing process!1 ˆ!a!b!2. This can be computed [12] along similar lines to the nonlinear susceptibility and yields the result 23 12N À 3 D!1;!a;!b;!2 ˆ 6"0 h 3 D 31 jg 3=2 D 21 jg 2=2 jo 2 j 2 =4 X! 1 i1 i2 i!i 1!a!i ; 6!b where the sum on the right-hand side represents the contribution to the nonlinear susceptibility of all the states of the atom; the elds at frequencies!a and!b are close to two-photon resonant with j1i j3i and are necessary to complete a fourwave-mixing scheme with!2 in order to generate a eld at frequency!1. This nonlinear susceptibility is plotted in gure 6 (c); although similar to Im À 1 D ( gure 6 (a)) in the sense that the response displays the familiar Autler±Townes splitting,

14 484 J. P. Marangos there is a fundamental di erence. Paying attention to the centre of the pro le at detuning D 12 ˆ 0 we see that instead of destructive interference there is constructive interference at this point. To get some physical insight into why this should be so we can consider the diagrammatic representation of processes leading to À 1 D and À 3 D ( gure 7). What is clear from the gure is that, whilst in the case of À 1 D all the diagrams representing higher-order interactions are even in the number of photons exchanged with the coupling eld O 2, those for À 3 D involve an odd number of photons being exchanged with this eld. In any four-wave mixing process the phase matching (propagation) of the generated and driving elds is of critical importance. In the limit of an optically deep medium considering plane-wave elds the gure of merit determining the 3 conversion e ciency is given j À j D =À 1 D ; this is physically reasonable since À 1 D characterizes the reabsorption and wave-vector mismatch in the medium, and À 3 D the coupling strength that generates the new eld. Thus we can see the importance of the destructive interference in the value of À 1 D over the frequency range where À 3 D experiences constructive interference. This enhances the generation e ciency (in principle) in a Doppler broadened medium (see below) by many orders of magnitude compared with the case with no coupling eld causing a dressing of the medium. Within the enhancement in À 1 D are both the e ect of reduced absorption and the imposition of essentially perfect phase matching for all the resonant elds (residual mismatch arising only through the dispersion caused by the remaining o -resonance states). These e ects were rst suggested as a means to enhance nonlinear optical frequency conversion e ciencies by Tewari and Agarwal [75] and by Harris et al. [12] and have been further discussed by several workers [51, 76±79]. Propagation of two coupled pulses in a (Raman) type of system has recently been discussed by Harris et al. [53, 54] and Eberly et al. [55]. This Raman scheme is equivalent to EIT in the limit where one of the elds is a weak probe and the other a strong-coupling eld. However, if both elds O 1 and O 2 are strong, the dressed atomic system reacts back on the eld modes in such a way as to result in lossless propagation through the medium for both the elds. A proper insight into this process is best obtained by thinking in terms of CPT. Essentially the coupling laser elds cause the formation in the atomic system of coupled jci and uncoupled jnci coherent superpositions of the bare atom lower states. However, there is additionally a strong coupling between these atomic states and the two driving elds. Hence there are also formed stable normal modes of the driving elds, one of which is uncoupled from the `uncoupled atomic state and the other of which is `uncoupled from the coupled atomic state. These new eld modes result in the lossless propagation of pulses through a normally `lossy medium once a certain preparation energy has been extracted from the laser elds [56]. Other work on matched pulse propagation in a double-v system has also been reported recently [80] Physical constraints to electromagnetically induced transparency For laser-induced atomic coherence processes in a real medium the maintenance of the phase of the coherence during the interaction is essential for e ective interference. Any dephasing of the coherence will reduce and, eventually, nullify the interference e ects. Dephasing can arise from a variety of di erent sources, for example, the excitation of several closely spaced hyper ne or Zeeman components

15 Electromagnetically induced transparency 485 (for example [23, 31, 81]), radiative decays of state j3i (e.g. on the j3i j1i channel), the existence of additional photoionization channels [82], collisions [12, 58] and phase di usion of the laser eld causing deviation from the transform limit. Therefore in experiments these parameters need to be controlled by choosing appropriate atomic systems, density ranges, laser intensities and laser system parameters. In general experiments with CW lasers, in which phase uctuations can be made small with laser bandwidths reduced to the 1 MHz level or less, most fully approach the steady-state monochromatic limit explored in the elementary steady-state theory of these processes. In many experiments, (inhomogeneous) Doppler broadening presents a serious limit since it introduces a randomization in the e ective laser detunings over the ensemble of atoms in the sample [12, 83, 84]. Various methods have been employed to eliminate this e ect, that is working in Doppler-free con gurations [85, 86], or using cooled atoms trapped in a magneto-optical trap [87, 88]. Alternatively by working with a coupling Rabi frequency larger than the width of the (Gaussian) Doppler pro le the in uence of inhomogeneous broadening can be, in e ect, overcome [12]. To generate Rabi couplings of greater than the Doppler width (0.03 cm 1 a CW laser, with a power typically of 1 W or less, must be focused fairly tightly. This may lead to undesirable e ects such as defocusing due to the interplay between the dressed refractive index and the transverse intensity variation across the beam waist in the region of the focus [86]. For pulsed lasers, with intrinsically high peak power, it is not generally necessary to focus the laser to achieve the required intensity and thus the defocusing e ects are not important and there is also the important advantage that large numbers of atoms will be within the dressed ensemble. Pulsed laser EIT e ects are not best modelled by a steady-state treatment. Although pulsed laser bandwidths are inevitably larger than those of CW lasers provided that the laser is transform limited the EIT e ect is not reduced. Some recent calculations [51, 78] have been somewhat arti cial in including relatively large laser linewidths pertinent to the pulsed laser in a steady-state calculation. In fact, provided that the laser is single mode and therefore transform limited (i.e. without excess phase uctuation), the dephasing introduced by the laser bandwidth will not su ciently disturb the phases of the coherences responsible for interference within the interaction time scale (i.e. the duration ½P of the pulse). It should also be appreciated that hyper ne sublevels will in general cause dephasing of coherences on a time scale given by the inverse of their frequency separation!hf [31]. In a pulsed excitation the dephasing due to hyper ne levels within the laser bandwidth will therefore be negligible provided that the interaction time ½P < 1=!hf (i.e. if the hyper ne splittings are su ciently small). In practice, incorporation of laser linewidths is most usually made via the Wiener±Levy phase di usion model (for example [51, 78, 89±91]). In this way the laser phase di usion can be added directly to the relevant density matrix coupled equations. This model, however, predicts Lorentzian laser linewidths, and this may severely overestimate the line wing of the laser [91]. This leads to overly pessimistic predictions about the e ects of laser linewidth in the large O C limit where most pulsed laser EIT experiments need to be performed in order to overcome the Doppler restrictions. More realistic phase di usion models are available [92±94] but are in general harder to implement in a calculation.

16 486 J. P. Marangos 3. Experimental work 3.1. Related atomic coherence experiments The rst experimental work on laser-induced atomic coherence dates back to the 1970s. Evidence for atomic coherence induced by the illumination of atoms by a modulated optical lamp in conjunction with resonant radio-frequency elds was reported still earlier [95]. Other relevant early work includes the investigation of dressing of two-level systems by strong microwave elds (Autler± Townes splitting) [96], and work on photon echoes in two-level systems [97]. Mollow [98, 99] reported novel features, subsequently termed the Mollow triplet, of resonance uorescence in a two-level system. Much work on two-level systems has been carried out since (see for instance the review and books given in [31, 100, 101] but, although two-level systems remain a subject of considerable interest (e.g. in dressed-state lasers and quantum optics experiments), our concern here is primarily with three-level systems (and in some cases with four-level systems). Atomic coherence and interference in three-level systems was rst seen experimentally in the work of Orriols and co-workers [35, 102, 103] in Pisa and Gray et al. [104] in Rochester. The Pisa group performed experiments that established coherence between the Zeeman split lower levels of a sodium atom using a multimode laser. By employing a spatially varying magnetic eld they were able to observe a series of spatially separated dark lines (resonances), corresponding to the locations where the Zeeman splitting matched the frequency di erence between modes of the coupling laser. This situation corresponds to a two-photon resonant (Raman-resonant) scheme and thus this is the rst experimental observation of CPT. The experiments of Gray et al. [104] are similar to this but involved establishing coherence between the hyper ne lower levels of Na. In essence in these experiments the laser elds couple to the atoms to create superpositions of the two ground states. One of these superpositions can interact with the elds (see section 2.1), a so-called `bright state (i.e. the coupled state jci), but the other does not because of cancellation of the two driven dipoles and is thus termed a `dark state (i.e. the non-coupled state jnci). Once formed, the population in the system will all be optically pumped into this dark state, in a matter of a few times the radiative decay rate. Once in the dark state there is, by de nition, no process to remove the population and so there is a `trapping of the population in this state. There has been a considerable body of experimental work carried out on CPT and its applications (see [32] for a full review). Recently the basic idea of CPT has been utilized in systems with time-varying optical elds to yield very e cient excitation of atomic and molecular states [38, 40, 41]. In these coherent adiabatic population transfer schemes (e.g. STIRAP) the non-coupled ground-state superposition jnci of CPT is evolved in time from being a pure state of the lowest state j1i to being a pure state of the upper state j3i as the relative strengths of the (strong) laser couplings vary. This is achieved by employing counter-intuitive pulse sequences, typically with identical Gaussian pulses, in which the coupling O 2 reaches its peak value around the time that O 1 is just switching on and has decreased back to almost zero when the latter is still just passing its peak. In his way the dark state jnci in equation (3) is initially almost purely bare state j1i (the initial state of the system), then evolves through being an admixture of j1i and j3i whilst both elds are strong and nally corresponds to being purely state j3i. The bright state jci remains unpopulated

17 Electromagnetically induced transparency 487 throughout the process. Recently work has been performed which demonstrates the utility of these e ects in laser cooling of trapped atoms in a technique called velocity-selective coherent population trapping (VSCPT) [105±107] Electromagnetically induced transparency experiments with pulsed lasers Most of the experiments on CPT systems have concentrated on the establishment of the population trap state and optical pumping of the population into it. Similarly coherent adiabatic population transfer experiments have concentrated on the e cient movement of the population into an atomic or molecular excited state. In EIT work the main emphasis has been on the linear (and nonlinear) optical response of an extended ensemble of atoms. In these experiments, measurements are made of the transmission of the (weak) probe eld through an otherwise optically deep medium that is made possible by the presence of EIT induced by a strong-coupling eld resonant with a pair of excited states. The rst demonstrations of this were by the group of Harris at Stanford in two atomic systems Sr [108] and Pb [109]. In both these experiments narrow-band pulsed laser radiation was used. In the Sr experiment, the rst to be reported, the atoms are initially pumped into an excited state via a pulsed laser. It is the transition (at nm) between this 5s5p 1 P 1 excited state (j1i) and the 4d5d 1 D 2 autoionizing (j2i) state that was to be rendered transparent (it should be noted that in our notation the state labels j2i and j3i are swapped compared with those used by these workers). A coupling laser (at nm), derived from a single-mode Littman dye laser, was applied between this autoionizing state and a metastable bound state 4d5p 1 D 2 (j3i). As in the prototypical scheme the probe eld excited the system to a state j2i with a large decay, and in the case of a no coupling laser there was a strong absorption; the Sr vapour was completely opaque at resonance, with an inferred transmission of exp ( 20 1). When the coupling laser was applied, the transmission at line centre increased by a large factor to exp 1 0:1. It is pointed out by these workers that, were it not for the interference e ect, the detuning from line centre induced by the Rabi splitting would only account for a reduction in transmission to a value exp 7:0. The latter point illustrates just how important the interference e ect is in ensuring a large reduction in the absorption of the medium. In an experiment in Pb [109] transparency was demonstrated within the bound states of a collisionally broadened medium. The three levels are in a ladder con guration with the probe between the 6s 2 6p 2 3 P 0 ground state j1i and the 6s 2 6p7s 3 P 0 1 excited state j2i, with the coupling eld between j2i and the 6s 2 6p7p 3 D 1 state j3i. This scheme was chosen because of the approximate coincidence between the frequency of the 1064 nm Nd-doped yttrium aluminium garnet laser (which was frequency stabilized by injection seeding and was thus a true transform-limited pulse) and the j2i±j3i transition frequency; there was, however, a 6 cm 1 detuning of the laser eld from exact resonance. An important feature of this experiment was the role of resonance broadening, which was the dominant broadening channel for state j2i (about 40 times larger than the radiative width). Owing to the destructive interference between the components of j2i in the two dressed states these collisions have no e ect on transparency. In contrast the collisions that dephase state j3i, and which will a ect the degree of transparency, are not resonance collisions and hence have a small e ect on this scheme. The