ELECTROMAGNETICALLY INDUCED TRANSPARENCY

Transcription

1 14 ELECTROMAGNETICALLY INDUCED TRANSPARENCY J.P. Marangos Quantum Optics and Laser Science Group Blackett Laboratory, Imperial College London, United Kingdom T. Halfmann Institute of Applied Physics Technical University of Darmstadt Germany 14.1 GLOSSARY Terms and Acronyms EIT electromagnetically induced transparency CPT coherent population trapping STIRAP stimulated Raman adiabatic passage RAP rapid adiabatic passage SCRAP Stark chirped rapid adiabatic passage CPR coherent population return LWI lasing without inversion lambda (Λ) scheme three coupled atomic levels with the initial and final states at lower energy than the intermediate state ladder scheme three coupled atomic levels with the energy of the initial state below the inter mediate state, and the energy of the intermediate state below the final state vee (V) scheme three coupled atomic levels with the initial and final states at higher energy than the intermediate state cw continuous wave Symbols a quantum state of atom E a energy of quantum state a w ab angular transition frequency between states a and b (rad s 1 ) Δ ab detuning of a light field from an atomic transition at w ab (rad s 1 ) m ab transition dipole moment between two states a and b (Cm) _Bass_V4_Ch14_p indd /25/09 6:22:38 PM

2 14.2 NONLINEAR OPTICS Re c (1) real part of linear susceptibility (dispersion) Im c (1) imaginary part of linear susceptibility (absorption) c (3) nonlinear susceptibility of third order (m 2 V 2 ) N A number of atoms n A atomic density (cm 3 ) E electric field strength (Vm 1 ) w angular frequency of a radiation field (rad s 1 ) l wavelength of a radiation field (nm) P macroscopic polarization (Cm 2 ) Ω Rabi frequency (rad s 1 ) r ab density matrix element Γ a decay rate of state a (rad s 1 ) g ab dephasing rate of the coherence r ab (rad s 1 ) g Doppler Doppler (inhomogeneous) linewidth (rad s 1 ) g laser laser bandwidth (rad s 1 ) 14.2 INTRODUCTION Electromagnetically induced transparency (EIT) is a quantum interference phenomenon that arises when coherent optical fields couple to the states of a material quantum system. In EIT the interference occurs between alternative transition pathways, driven by radiation fields within the internal states of the quantum system. Interference effects arise, because in quantum mechanics the probability amplitudes (which may be positive or negative in sign), rather than probabilities, must be summed and squared to obtain the total transition probability between the relevant quantum states. Interference between the amplitudes may lead to either an enhancement (constructive interference) or a complete cancellation (destructive interference) in the total transition probability. As a consequence, interference effects can lead to profound modification of the optical and nonlinear optical properties of a medium. Thus, control of optical or nonlinear optical properties and processes becomes possible. Interference effects of this kind are well known in physics. These occur naturally if there are two transition pathways available to the same final state. Fano interferences exhibit an example of interference between two transition pathways. In this case the two pathways are direct photoionization from a quantum state to the ionization continuum and photoionization from the quantum state via an intermediate autoionizing state. 1,2 The interference between these two pathways leads to asymmetric resonances in the photoionization spectrum. The photoionization cross section vanishes at certain excitation frequencies, that is, complete destructive interference occurs. This process is well known for radiative transitions to autoionizing states in atoms or to predissociating states in molecules. It was also predicted to occur in semiconductor quantum wells. 3 Interfering transition pathways can also be deliberately induced by application of resonant laser fields to multilevel atomic systems. Perhaps the most striking example for this type of interference is the cancellation of absorption for a probe radiation field, tuned in resonance to an atomic transition. Usually the resonant excitation would lead to strong absorption of the probe field. However, if the atoms are prepared by EIT, 4 6 the absorption is essentially switched off. EIT exhibits a laserinduced interference effect between the quantum coherences in the atom, which renders an initially highly opaque medium into an almost transparent medium. Similarly the refractive properties of the medium may be greatly modified. 5,7,8 For instance the usual correlation of high refractive index with high absorption can be broken, leading to the creation of media with unique optical properties. There has been a considerable research effort devoted to EIT and related topics over the last few years. This has been motivated by the recognition of a number of new potential applications, for example, lasers without inversion, highly efficient nonlinear optical processes, storage of light pulses 14_Bass_V4_Ch14_p indd /25/09 6:22:38 PM

3 ELECTROMAGNETICALLY INDUCED TRANSPARENCY 14.3 and quantum information, lossless propagation of laser beams through optically thick media, and highly efficient and selective population transfer via coherent adiabatic processes. EIT is one of an interrelated group of processes, for example, including coherent population trapping (CPT) and coherent adiabatic population transfer, that result from externally induced quantum mechanical coherence and interference. In contrast, the earlier ideas associated with CPT (first observed in 1976) 9 had found application mostly as a tool of high-resolution spectroscopy, rather than as a new direction in nonlinear optics. Therefore, the concept of EIT has contributed a distinctive new thrust to work on atomic coherence and its applications a thrust, which is of direct interest to optical scientists and engineers. To explain the basic idea and applications of EIT, an equivalent picture to interfering transition pathways is provided by the concept of laser-dressed states. In terms of quantum mechanics, the dressed states are the eigenstates of the quantum system, including strong interaction with driving radiation fields. The dressed states are coherent superpositions of the bare states, that is, the eigenstates of the quantum system without external interaction. The coherent superpositions have well-defined amplitudes and phases that describe the relationship between the atomic states in the superposition. The reader is referred to The Theory of Coherent Atomic Excitation by Shore 10 for a complete account of these ideas. An important feature of EIT is the preparation of large populations of these coherently driven, uniformly phased atoms. Such media are termed phasesonium by Scully, 7 to convey the basic idea. The (both linear and nonlinear) optical properties of the coherent medium are very different from those of a normal, incoherently driven medium. In the dressed medium the terms of linear and nonlinear susceptibilities can be retained only to the extent that it is recognized that all these resonant processes are highly nonpeturbative. As we will discuss later, even the so-called linear processes now involve the coupling of atoms with many photons. An important consequence of this is that the magnitudes of linear and nonlinear susceptibilities can reach equality in a phasecoherent medium. This is in marked contrast to the normal situation. Usually the nonlinear susceptibility would give rise to nonlinear optical processes, which are many orders of magnitude weaker than those arising from the linear susceptibility. The exceptionally high efficiencies of nonlinear optical processes in gas phase media, prepared by EIT, therefore constitute an important feature of EIT. The large conversion efficiencies in gas phase media, driven to EIT, become comparable to nonlinear optical processes in optical crystals. Thus a renewed interest in gas phase nonlinear optical devices possessing unique capabilities [e.g., high conversion efficiencies into the spectral regions of vacuum-ultraviolet (VUV) and far-infrared (IR) radiation] has occurred. There have been a number of notable recent demonstrations of EIT, applied to frequency conversion. A near-unity frequency conversion into the far-ultraviolet spectral region was reported for a four-wave mixing scheme in lead vapor. The lead atoms were prepared in the state of maximal coherence, that is, maximal polarization, by EIT. The uniquely high conversion efficiency arises since the nonlinear terms become equal in magnitude to the linear terms. Besides applications in dense gas phase media with thermal velocity distribution, large optical nonlinearities, induced by EIT were also studied in laser-cooled atoms. In such media, a successful implementation of EIT requires only quite weak laser couplings. Thus, at maximum transparency there is an extremely steep dispersion as a function of the driving laser frequency, that is, the detuning from the atomic resonance. The consequence of this steep slope is a very slow group velocity for optical pulses propagating through the medium. 11 Massive optical nonlinearities accompany the steep dispersion. These are manifested as very large nonlinear refractive indices that are many orders of magnitude larger than any previously observed. These huge nonlinearities are the subject of current research activity. They offer the likelihood of efficient nonlinear optical processes at the few photon level. In addition to the applications, described above, the coherence and interference effects related to EIT may also permit new possibilities to build short wavelength lasers, that is, lasers in the x-ray spectral range. As the Einstein coefficient for spontaneous emission scales with the cube of the transition frequency w, 3 it is usually hard to achieve population inversion by optical pumping in an x-ray laser. In contrast, the laser concept based on EIT does not rely on population inversion in the atomic laser medium anymore. 12,13 Lasing without inversion (LWI) has been demonstrated in sodium atoms and rubidium atoms in the visible range. 14,15 The prospects that this might lead to the construction of lasers which are able to circumvent the usual constraints of achieving inversion in short wavelength 14_Bass_V4_Ch14_p indd /25/09 6:22:38 PM

4 14.4 NONLINEAR OPTICS lasers has been much discussed. 16,17 Related effects on LWI and EIT in semiconductor quantum wells have also been theoretically explored using laser-induced processes 18 or bandgap engineering 19 to create the necessary coherences. Moreover, atomic and molecular coherences, driven by EIT, were also exploited for efficient frequency conversion and the generation of ultrashort laser pulses ,28 39 It is the aim of this chapter to provide an accessible summary of EIT and to present some of the main results obtained in recent research. It is not possible in the space available to cover all work in this field, and we apologize to authors whose important contributions are not directly mentioned. An extensive review of all theoretical and experimental work on related atomic coherence phenomena is also beyond the scope of the present article. The reader is advised to look at a number of reviews on LWI, 16,17 coherent population trapping, 10,40 43 coherent adiabatic population transfer, 44 efficient frequency conversion in coherently prepared media 45 and laser-induced continuum structure to find these topics presented in detail. The theory pertinent to EIT is sketched in the text, but again the reader is referred to the more detailed treatments published in the literature. 5 References to relevant literature will be given as they arise. The purpose of the chapter is (1) to communicate the underlying physical principles of EIT and related effects, (2) to describe the manifestations of EIT and to summarize the conditions required to create EIT, and (3) to introduce some potential applications in optical technology. A a f 14.3 COHERENCE IN TWO- AND THREE-LEVEL ATOMIC SYSTEMS The first experimental work on laser-induced atomic coherence was carried out in the 1970s. Earlier relevant work includes the investigation of dressing two-level systems by strong microwave fields. This led to the observation of splittings between dressed states, that is, Autler-Townes splittings, 51 and photon echoes in two-level systems. 52 Mollow 53,54 reported novel features, subsequently termed the Mollow triplet, of resonance fluorescence in a two-level system driven by a strong resonant laser. Much work on two-level systems has been carried out since. 10,44,55,56 Although two-level systems remain a subject of considerable interest, our concern here is primarily with three-level systems (and in some cases four-level systems). Atomic coherence and interference in three-level systems was first observed experimentally by Alzetta, Arimondo et al., 9,57 and by Gray et al. 58 The first group of authors 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 field Alzetta, Arimondo et al. observed a series of spatially separated dark lines. These resonances correspond to the locations in the magnetic field where the Zeeman splitting matched the frequency difference between modes of the coupling laser. This situation corresponds to a two-photon resonant lambda-type level scheme. Thus, this was the first experimental observation of CPT. The experiments of Gray et al. 58 involved the preparation of coherence between the hyperfine lower levels of sodium atoms. In these experiments two coincident laser fields are coupled to a three-level lambda scheme of states to create superpositions of the two lower states 1 and 2 (see Fig. l). One of the superpositions, that is, the coupled state or bright state C interacts with the fields [see Eq. (l)]. For the other superposition, that is, the noncoupled or dark state NC, interference causes cancellation of the two driven dipoles. Once the coherent states are formed, the population in the system will all be optically pumped into the dark state through spontaneous emission from the intermediate state. The optical pumping process occurs on the timescale of a few times the radiative decay time. Once in the dark state there is no process to remove the population. Thus the population is trapped in the dark state. The basic idea of CPT has been extended to systems, driven by time-varying optical fields to yield very efficient excitation of atomic and molecular states. 44,59 61 In stimulated Raman adiabatic passage (STIRAP), the noncoupled state NC exhibits a specific evolution in time. Initially NC is prepared such that it is composed purely of the lowest state 1. For intermediate times NC evolves as a superposition of the two lower states 1 and 2, with no contribution from the intermediate state 3. Finally NC is composed purely of state 2. Thus, population is transferred completely 14_Bass_V4_Ch14_p indd /25/09 6:22:39 PM

5 ELECTROMAGNETICALLY INDUCED TRANSPARENCY 14.5 Δ 13 Δ 23 3 Ω 1 Ω FIGURE 1 CPT in a three-level lambda configuration. Two radiation fields, that is, couplings Ω 1 and Ω 2, are applied with frequencies close to the single-photon resonances at w 13 and w 23. between the lower states 1 and 2, with no intermediate storage in state 3. The bright state is not populated during the process. This specific preparation of a dark state in STIRAP is achieved by employing counterintuitive pulse sequences, that is, the couplings (or the laser pulses) Ω 1 and Ω 2 still partly overlap in time, but Ω 2 reaches its peak value prior to the peak of Ω 1. The timescale for STIRAP is determined by the evolution of the laser pulses. This contrasts to conventional CPT, that is, driven with coincident radiation fields, when the time evolution is determined by spontaneous emission. We also note that recently extended work has been performed to utilize CPT also in laser cooling and manipulation of trapped atoms. The technique, used in these experiments was termed velocity-selective coherent population trapping (VSCPT) In experiments, dedicated to CPT the primary concern is focused on the manipulation of state populations, essentially of individual atoms. In contrast, for EIT the main interest is the optical response, rather than simply the populations, of the entire medium. The optical response is determined by the coherences rather than the populations. In terms of density matrix elements, in CPT the pertinent quantities are the on-diagonal density matrix elements, that is, the populations; while in EIT they are the off-diagonal density matrix elements, that is, the coherences. Most important, in the limit of a strong coupling field Ω 2 and population initially in the ground state, the coherences, driven by EIT are almost instantaneously established. The timescale of the evolution in EIT is determined by l/ω 2. For strong excitation, even driven by quite long nanosecond (ns) pulses, the timescale of l/ω 2 easily reaches the regime of picoseconds, that is, well below the pulse duration. For a successful implementation of population trapping a timescale of several radiative lifetimes is required, that is, typically in the regime of many nanoseconds. From these considerations we see, that though EIT and CPT are closely related, some of their important features as well as their aim are very different THE BASIC PHYSICAL CONCEPT OF ELECTROMAGNETICALLY INDUCED TRANSPARENCY As discussed in the preceding section, there is a close link between EIT and other phenomena, relying on atomic coherence, that is, adiabatic population transfer processes. 9,43,44,57 61,65 67 In all these processes, three-level atomic systems are involved that is, systems that can be adequately reduced to three levels when interaction with the pertinent electromagnetic fields are considered. The atomic dipole selection rules require that two pairs of levels are dipole-coupled, while the transition between the third pair is dipole-forbidden. In Fig. 2, we show the basic three-level schemes. All of the level schemes, discussed in this paper can be reduced to one or other of these schemes. Following 14_Bass_V4_Ch14_p indd /25/09 6:22:39 PM

6 14.6 NONLINEAR OPTICS (a) (b) (c) FIGURE 2 Basic three-level schemes: (a) ladder (or cascade) scheme with E 1 < E 3 < E 2, (b) lambda (Λ) scheme with E 1 < E 2 < E 3, (c) vee scheme with E 3 < E 1 and E 3 < E 2. 2 the nomenclature of Harris et al, 6 we label the levels 1, 2, and 3. The dipole-allowed transitions are between states 1 and 3 and between states 2 and 3. Classification of the schemes then depends upon the relative energies of the three states: (1) ladder (or cascade) scheme with E 1 < E 3 < E 2 (2) lambda (Λ) scheme with E 1 < E 2 < E 3, and (3) vee (V) scheme with E 3 < E 1 and E 3 < E 2. EIT has been extensively studied in all three of these configurations. In a lambda or ladder scheme state 1 is normally the ground state of the atom. This is, where initially the majority of the population resides. In EIT there is no need for significant population transfer. Thus states 2 and 3 remain essentially unpopulated throughout the process. It should be noted that the lambda scheme has a special importance due to the metastability of state 2. As a consequence, very long-lived coherences can be established between states 1 and 2. This leads to near-perfect conditions for EIT. To understand in more detail, how laser fields interact with a three-level atom to create coherent superpositions of the atomic bare states, we will consider now CPT in a lambda scheme. A threelevel lambda system (see Fig. l) is coupled by two near-resonant laser fields. The interaction strength is defined by the Rabi frequencies Ω 1 = m 13 E 1 / at frequency w 1 and Ω 2 = m 23 E 2 / at frequency w 2 with the dipole transition moments m 13 and m 23, and the electric fields E 1 and E 2. The transition frequencies are defined as w 12 and w 23. The one-photon detunings of the radiation fields from the atomic resonances are Δ 13 = w 13 w 1, Δ 23 = w 23 w 2.The two-photon (Raman) detuning is Δ = [(w 13 w 23 ) (w 1 w 2 )]. The Hamiltonian of the bare atom H 0 must be modified to include the interactions due to the two couplings. Thus H = H 0 + V 1 + V 2, with the interactions V j = ħ Ω j. The eigenstates of this new Hamiltonian will be linear superpositions of the bare atomic states 1, 2, and 3 (see, Refs. 10, 42, 68). For exact two-photon resonance and, that is, Δ = 0, two of the three eigenstates of the total Hamiltonian H turn out to be symmetric and antisymmetric coherent superpositions of the two lower bare states. These superpositions read 1 C = ( ) 1 2 Ω Ω Ω 1 NC = ( ) 2 1 Ω Ω Ω (1a) (1b) where Ω = [Ω Ω 22 ] 1/2. It is important to note that no component of the bare state 3 appears in these superpositions. The superposition state C is coupled to the intermediate state 3 via electric dipole interaction, that is, C is a bright state. In contrast, the other state NC is not coupled to state 3, that is, NC is a dark state or trapped state. This is obvious from the total dipole moment for a transition from state NC to the bare state 3. If the magnitudes of the coupling fields Ω 1 and Ω 2 are appropriately balanced, the negative sign in the superposition of 1 and 2, which forms the state NC, causes the transition moment NC m 3 to vanish. In effect, the two terms 14_Bass_V4_Ch14_p indd /25/09 6:22:39 PM

7 ELECTROMAGNETICALLY INDUCED TRANSPARENCY 14.7 that are summed to give the transition amplitude between NC and 3 are of equal and opposite magnitude, and hence the total amplitude will vanish. In a classical picture, this corresponds to the electron driven by two fields, both of which may be strong, but which exert forces of exactly equal magnitude and opposite directions. This interaction leads to a zero net force and hence the electron stays at rest. In conventional CPT and assuming steady-state conditions, the superposition state NC will acquire all of the population of the system through optical pumping. Thus spontaneous emission from state 3 populates state NC, but absorption losses from state NC back to state 3 are not possible. The noncoupled state NC also serves as the key component for coherent population transfer by STIRAP, 44 which was already briefly discussed above. We assume, that initially all population is in state 1. State 2 is assumed to be metastable, for example, if the level scheme is of lambda-type configuration. If at early times the Rabi frequencies (i.e., the corresponding radiation fields) are applied such that Ω 2 >> Ω 1, state NC is equal to state 1 [see (Eq. 1)]. All population of the system is prepared in the dark state NC. No population is in the bright state C. If at the end of the interaction Ω 1 >> Ω 2, state NC aligns now parallel to the target state 2. As Ω 1 >> Ω 2, the contribution of state 1 is negligible, that is, all population is transferred to the target state 2. The sequence of an initially strong coupling between the states 2 and 3 and a finally strong coupling between the states 1 and 2 exhibits a counterintuitive laser pulse order, which is the typical feature of STIRAP. We note that in the previous description we ignored a fast time oscillation of the bare states 1 and 2 in the superpositions in Eq. (1). The oscillation occurs at frequencies e 1 /ħ and e 3 /ħ, with the energy of the bare states e i. In fact these terms disappear when the dipole moments are formed. In typical implementations of CPT, for example, STIRAP, the couplings are of comparable strength, that is, Ω 1 Ω 2, and the two-photon transition is strongly driven. The transition is saturated if the terminology of incoherent excitation is used. We note an interesting feature of CPT with respect to the coherences for the case of the laser frequencies tuned to exact two-photon resonance, but with large single-photon detunings. In this case, state 3 can be adiabatically eliminated from the level scheme. 10 Thus state 3 does not enter into the consideration of the coupling between atoms and fields any more. However, also in this case, that is, even far detuned from the single-photon resonances, the two-photon resonance condition alone is sufficient to drive large coherences between states 1 and 2. We also stress the point that in general in CPT the initial population may be distributed between the lower states 1 and 2. This is usually the case if the lower states are provided by sublevels of an atomic ground state, for example, for Zeeman or hyperfine split states. As the energy difference of the lower states 1 and 2 is very small in this case, the initial thermal populations will be almost the same. In this case, a careful analysis of states NC and dark state C is required to determine the population dynamics for the specific CPT process under consideration, for example, STIRAP. However, also in the case of an initial population in both lower states, the state NC is still a dark state. In contrast, in implementations of EIT, the population is initially and for all times completely stored in state 1. In CPT, interference effects arise from both coupling fields, since they are of comparable strength. If only one of the fields is strong, that is, Ω 1 << Ω 2, only interference effects due to processes driven by Ω 2 will be important. This is the situation in many implementations of EIT and is discussed by a number of authors (see, e.g., Ref. 5 and references therein). In such EIT experiments, Ω 2 is usually called the coupling field and labeled Ω C, and Ω 1 is a weaker probe field, labeled Ω P. In the following we will use the notation Ω C and Ω P, whenever appropriate. Based on the considerations, presented above, we will now discuss some simple and straightforward approaches alternative approaches to understand the basic concept of EIT. First, let us consider the basis, formed by the coupled state C and the noncoupled state NC [see Eq. (1)]. We can write the bare state 1 in this basis: 1 1 = ( + ) Ω Ω NC Ω C (2) C P Very obviously, for the case Ω P << Ω C state 1 is almost equivalent to NC, that is, the dark state. Thus absorption to state 3 vanishes. The population remains in the ground state 1 throughout 14_Bass_V4_Ch14_p indd /25/09 6:22:40 PM

8 14.8 NONLINEAR OPTICS 3 + Ω C Ω P Ω P 2 (a) 1 (b) 1 FIGURE 3 EIT in a lambda scheme (compare Fig. 4) viewed in terms of (a) bare atomic states, driven by a weak probe field Ω P and a strong coupling field Ω C, (b) dressed states, generated by the strong coupling Ω C. The probe field is still at the bare state resonance frequency w 13. the interaction with the two radiation fields. The probe laser propagates through the medium without any absorption losses, that is, the medium is driven to EIT. Alternatively, as Ω P << Ω C, we can treat the three-level system of states 1, 2, and 3 as a composition of a strongly coupled two-level subsystem of states 2 and 3, with the weakly coupled state 1 attached to the subsystem. Thus it is straightforward to describe the subsystem in terms of the dressed states, which arise due to the strong interaction 5 (see Fig. 3). For a strong resonant coupling at the single-photon resonance Δ 23 = 0 the dressed states of the subsystem are 1 + = ( + ) = ( ) (3b) The transition amplitude at the (undressed) resonant frequency w 13 from the ground state 1 to the dressed states will be the sum of the contributions to the dressed states + and, that is, [ 1 m m ] [ 1 m m m 2 1 m 3 ] = m 12 + m 13 + m 12 m 13, with the transition moments m ij. As we assumed in the definition of our three-level system, the transition between states 1 and 2 is forbidden, that is, m 12 = 0. The transition moments m 13 enter the sum with opposite signs. Thus the transition amplitude, that is, the absorption, reduces exactly to zero. The system is driven now to EIT. Finally, another approach to an understanding of EIT is based on the concept of quantum interferences. 13,69 71 Interference, associated with EIT, arises because the transition amplitude between states 1 and 3 includes different pathways from the ground state 1 to the excited state 3 : One term, which is due to excitation by the resonant field Ω P only, that is, a direct path from state 1 to state 3 ; an additional term, which is due to the presence of the second field Ω C (see Fig. 4), that is, a indirect path from state 1 to 3 further on to state 2 and back to 3. The additional term and similar higher-order terms have a negative sign with respect to the direct path. Hence the higherorder terms cancel completely the direct path. This situation is closely related to interferences, mediated at Fano-type resonances, 1 for example, via autoionizing states, or to laser-induced continuum structure Equivalently within the picture of EIT in terms of the interfering pathways between the bare atomic states, the coherences are the quantities pertinent to the interference. Coherences can be thought of, in a semiclassical picture, as associated with the oscillating electric dipoles driven by the coupling fields applied between pairs of quantum states of the system, for example, between states i and j. Strong excitation of these dipoles occurs whenever electromagnetic fields are applied close to resonance with an electric dipole transition between two states. If there are several pathways (3a) 14_Bass_V4_Ch14_p indd /25/09 6:22:40 PM

9 ELECTROMAGNETICALLY INDUCED TRANSPARENCY Ω P Ω P Ω C 2 (a) 1 1 (b) FIGURE 4 EIT in a lambda scheme, viewed in terms of Fano-type interference (compare Fig. 3), (a) shows the direct channel for the excitation 1 3 by the probe field Ω P, (b) shows the lowest-order multiphoton channel induced by the coupling field Ω C, that is, the sequence Interference between pathway (a) and (b) (also including higher-order terms) occurs. to excite the oscillating dipole at frequency w ij, then interference arises between the various contributions to this dipole. These contributions must be summed to give the total amplitude to the electric dipole oscillation (see Fig. 4). Mathematically, coherences are identified with the off-diagonal density matrix elements r ij. These are formed by taking bilinear combinations of probability amplitudes of two quantum states, that is, by the weighting factors associated with the outer products such as i j. 72 Off-diagonal elements in the density matrix play a critical role in the evolution of an atom coupled to electro-magnetic fields. 73 Many calculations of atomic coherence effects 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. The magnitudes of the relevant density matrix elements are computed from the basic coupled evolution equations, that is, the Liouville equation, 10,72 and are found to depend upon experimental parameters (e.g., detunings and laser intensities). 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). 67 EIT serves to control the absorption of a probe laser on the transition between the states 1 and 3 in the three-level systems, defined and discussed above. Thus EIT will manifest itself in the density matrix element r 13. The real and imaginary part of r 13 both vanish at zero detuning, that is, the coherence is cancelled by interference of the excitation pathways. A set of coupled equations connecting the density matrix elements (e.g., r 12, r 23, and r 13 ) and their temporal derivatives is deduced from the Liouville equation. These coupled equations are then solved for various sets of conditions by either analytical or numerical means. Interference that leads to EIT arises from the coherences r 23 and r 12, which are coupled to r 13. The coherence r 12 between the ground state 1 and state 2 is present only due to the additional laser coupling. The contribution to the coherence r 13 from the coherences r 23 and r 12 (driven by both laser fields) cancels with the direct contribution (driven by the probe laser field alone). Although this use of density matrix elements is convenient, it is by no means essential, and many theoretical treatments that give clear physical insight have been performed in terms of probability amplitudes. Additional physical insights have been obtained by adopting alternative approaches, for instance by a careful consideration of the Feynman diagrams representing the various processes involved that lead to interference, or by applying a quantum jump approach. 74 In all cases the predictions are essentially identical. Analytical solutions are generally only possible for steady-state conditions [corresponding to continuous wave (cw) laser fields]. A time-dependent calculation of the density matrix is appropriate, if laser pulses rather than cw radiation drives the atom. A time-dependent calculation is also vital to account for transient effects or pulse propagation. Some analytical solutions also for timedependent calculations of the density matrix have been obtained, but unless restrictive simplifying assumptions are applied, 75 time-dependent calculations must be performed numerically. In many 14_Bass_V4_Ch14_p indd /25/09 6:22:40 PM

10 14.10 NONLINEAR OPTICS cases the results of a full time-dependent treatment will be comparable to the results, obtained in a steady-state approach. This holds true at least in so far, as qualitative trends are concerned. To calculate the propagation of laser pulses through an extended medium the time-dependent equations for the density matrix elements must be coupled to Maxwell s equations. 10 This is necessary, for example, to compute the propagation of matched pulses, the efficiency of frequency conversion processes in coherently prepared media, 20 27,28 39 to account for losses in the driving laser fields or to model pulse shape modifications MANIPULATION OF OPTICAL PROPERTIES BY ELECTROMAGNETICALLY INDUCED TRANSPARENCY Any optical process in a medium, driven by radiation pulses, is determined by the polarization, that is induced be the light fields. The macroscopic polarization P at the transition frequency w 13 can be related to the microscopic coherence r 13 via the expression P = n µ ρ (4) 13 A where n A is the number of atoms per unit volume in the ground state within the medium, and m 13 is the dipole matrix element associated with the (undressed) transition. 73 In this way imaginary and real parts of the linear susceptibility at frequency w can be directly related to r 13 via the macroscopic polarization. 55 The latter is defined as P ( ω) = ε χ( ω) E (5) 13 0 introducing the susceptibility c(w). In this paper, the microscopic coherences are treated quantum mechanically, while the electromagnetic fields are treated classically (i.e., using Maxwell s equations and the susceptibilities). This semiclassical approach is not essential, and fully quantum mechanical treatments for CPT (see, Ref. 41) and EIT have been developed. These fully quantum approaches are appropriate for cases such as the coupling of atoms to modes in cavities, 80,81 or when the statistical properties of the light fields are of interest. The latter is the case, for example, in proposals to generate squeezed light using EIT. 79 For relatively strong light fields, present in most laser experiments, a semiclassical treatment, with spontaneous decay added as a phenomenological damping process, proves adequate. The real and imaginary parts of the (dressed) linear susceptibility, associated with the dispersion and absorption of the medium respectively, are given by 5,6 2 μ n 2 2 ( ) () Re χ ( ω, ω ) A 4Δ Ω C 4Δ Δ + 4 Δ Γ = D P P ε ħ ( 4Δ Δ Γ Γ Ω ) + 4( Γ Δ + Γ Δ ) C μ n ( () 1 13 Im χ ( ω, ω ) = A 8Δ Γ 2Γ Ω C + ΓΓ) 3 2 D P P ε ħ ( 4Δ Δ Γ Γ Ω ) + 4( Γ Δ + Γ Δ ) C (6a) (6b) To deduce the linear susceptibility, monochromatic fields and negligible collisional or Doppler broadening were assumed. The real and imaginary part of the linear susceptibility, along with the nonlinear susceptibility, are plotted in Fig. 5 (see figure caption for explanation of the labeling) as a function of the detuning Δ 13, at Δ 23 = 0, that is, for resonant excitation by the coupling field Ω C. A striking result, the absorption for the probe field vanishes at exact resonance, if the coupling field Ω C is switched on and state 2 is perfectly metastable (i.e., Γ 2 = 0). Simultaneously the dispersion is significantly modified. The dispersion is still zero at line center, that is, the same value as in the case of the coupling field Ω C switched off. However, the group velocity (dependent upon the slope of [Re c (1) ]) becomes anomalously low 5,82 when absorption has vanished. This offers the possibility of 14_Bass_V4_Ch14_p indd /25/09 6:22:41 PM

11 ELECTROMAGNETICALLY INDUCED TRANSPARENCY (a) Ω 23 = 0 Ω 23 = 2Γ 3 (1) D Normalized Im c^ (b) Ω 23 = 2Γ 3 (1) D Normalized Re c^ (c) Ω 23 = 2Γ 3 (3) (arbitrary units) D c^ (w d w 3 /(Γ 3 /2) 2 4 FIGURE 5 The dressed susceptibilities in terms of the normalized detuning (w d w 3 )/(Γ 3 /2) [in our notation this corresponds to the detuning of the probe laser, scaled to the decay rate, i.e. Δ 13 /(Γ 3 /2)] for a value of the coupling Rabi field Ω 23 = 2 Γ 3 [in our notation Ω C = 2 Γ 3 ]. (a) [Im c (1) ], i.e. absorption; (b) [Re c (1) ], i.e. dispersion; (c) c (3), i.e., nonlinear response. [Reprinted figure with permission from Ref 6. Copyright (1990) by the American Physical Society.] 14_Bass_V4_Ch14_p indd /25/09 6:22:41 PM

12 14.12 NONLINEAR OPTICS slowing down the speed of light by EIT, that is, a most prominent example for the striking features of EIT, which is discussed elsewhere in this book. Inclusion of finite laser linewidths, collisional and Doppler broadening in these deductions is straightforward. 6 Including these effect, the dressed susceptibilities still retain the key features, provided some limits for the experimental parameters are kept in view. The medium that would in the absence of the coupling field be optically thick is now rendered transparent. The reduction in absorption is not merely that caused by the effective detuning induced by the Autler-Townes splitting of the dressed state absorption peaks (see Fig. 3), that is, the absorption that would be measured if the probe field were interrogating the absorption coefficient of the medium in the wing of the absorption lines of the dressed states. Additionally, there is destructive interference at the transition frequency w 13 that leads to complete cancellation of all absorption, provided there are no additional dephasing processes in the system. Even if the transition dipole moment m 12 is not zero (i.e., if the spontaneous decay rate is Γ 2 0), the absorption due to EIT will be reduced compared to the weak field absorption by a factor of Γ 2 /Γ 3. 6 In the preceding considerations it was implicitly assumed that the probability amplitudes of state 3 remain close to zero (i.e., the probe field is very weak). If there is an incoherent population pump process into the upper states 2 or 3, such that these populations no longer remain negligible, then gain on the transition between the states 3 and 1 can result (see Sec. 14.8). The remarkable feature of this gain is that under the circumstances in which EIT occurs, that is, 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 amplification without inversion. The process has successfully been implemented in a vee-type scheme in Rubidium atoms 15 and a lambda scheme in sodium atoms. 83 Much theoretical work 12 14, has been reported on LWI. In early work, LWI was predicted by Arkhipkin and Heller, 85 and then further elucidated by Harris, 86 Kocharovskaya and Khanin, 12 and Scully et al. 84 A long-term objective in this work is the prospect of overcoming the familiar difficulties of constructing short wavelength lasers, that is, the requirement of very high energy pump fields in order to compensate for the small transition moment at far-infrared wavelengths. In addition to gain without population inversion, any incoherent pumping of population into the upper states also modifies the dispersion in the medium. In particular, it is then possible to obtain a very large refractive index for specific wavelength regimes. The refractive index can reach values comparable to those normally encountered very close to an absorption line, while here the absorption now vanishes. 8 The prospects for engineering the refractive properties of media to give novel combinations of absorption, gain, and dispersion have been explored in a number of theoretical and experimental studies (see Sec. 14.9). The successful implementation of EIT depends upon a number of critical parameters, both inherent to the quantum system and the experimental setup, for example, the driving laser pulses. A correct choice of the atomic energy level configuration is essential. The configuration must satisfy the conditions, already discussed above, that is, dipole allowed transitions 1 3 and 2 3, while the transition 1 2 is dipole forbidden. Radiative couplings to other energy levels outside of these states, that lead to an open three-level system, and additional level substructure must also be considered. Collisions with other species in the medium or photoionization must be minimized in order to prohibit perturbing decay or dephasing of the coherence r 12, which is essential to EIT. The couplings may be either driven by cw or pulsed lasers. In both cases the couplings must be sufficiently strong to overcome the inhomogeneous broadening. Morover, sufficiently monochromatic or radiation with transform-limited bandwidth in the pulsed case is required, in order not to dephase the coherence r 12. These critical parameters are summarized in Table 1. In the following section we discuss in more detail the most crucial parameters for a successful implementation of EIT. Intrinsic Dephasing of Atomic Coherence in Gas Phase Media For processes of laser-induced atomic coherence in a realistic medium the maintenance of the phase of the coherence during the interaction is essential for effective quantum interference. Any dephasing of the coherence will wash out and eventually nullify the interference effects. Dephasing can 14_Bass_V4_Ch14_p indd /25/09 6:22:41 PM

13 ELECTROMAGNETICALLY INDUCED TRANSPARENCY TABLE 1 Summary of Critical Experimental Parameters for a Successful Implementation of EIT Physical Parameter Constraint Typical Values Radiative decay rate Γ 3 of state MHz Radiative decay rate Γ 2 of state 2 Γ 2 << Γ 3 <1 MHz Photoionization rate Γ ion Γ ion >> Γ 3 Depends upon the laser intensity Coherence dephasing g ij g 12 < g 13, g MHz (gases), GHz (solids) Laser linewidth g laser Transform-limited <1 MHz (cw) or 1/t pulse (pulsed) Doppler linewidth g Doppler g Doppler < Ω C 1 GHz Rabi frequency Ω C Ω C > g 12 Ω C = m 23 E C / Laser pulse energy E C [see Eq. (8)] E C > f 13 /f 23 w N A arise from a variety of different sources, for example, the excitation of a multitude of closely spaced hyperfine or Zeeman components (see, e.g., Refs. 10,15,94), radiative decay of state 2, photoionization channels, 95 and collisions. 6,96 Following these arguments, it is obvious, that in a ladder scheme perfect EIT is usually not possible, because state 2 is not metastable, but undergoes spontaneous emission to state 3 as well as to other states outside the three-state system. Collisional broadening with atoms of the same and other species is also critical and places strict limits on sample purity, otherwise leading to foreign gas broadening. Possibly also limits are imposed on the sample density. Moreover, at large atomic density the local field effects due to dipole-dipole couplings between the atoms may be important. 96,97 In this case the simple relationship between macroscopic polarization and the coherences in Eq. (5) may break down. Dephasing due to Phase Fluctuations in the Laser Fields If cw lasers with narrow bandwidth Δν < 1 MHz (i.e., negligible phase fluctuations) are used, the quality of an implementation of EIT usually approaches the expectations, based on steady-state conditions and strictly monochromatic excitation. Due to phase-diffusion processes laser line broadening gives rise to linewidths above the limit of the allowed radiative decay rate. This will destroy EIT. EIT may also be implemented with pulsed laser, provided the laser pulses exhibits transform limited bandwidth. Though these transform-limited bandwidth is inevitably larger than those of cw lasers, EIT is not reduced at all. A single-mode transform-limited laser pulse (i.e., without excess phase fluctuation) will introduce insufficient dephasing during the interaction time (i.e., the pulse duration t pulse ) to disturb the phases of the atomic coherences. It should also be appreciated that hyperfine sublevels will in general cause dephasing of coherences on a timescale, which is given by the inverse of their frequency separation Δw HFS. 10 In a pulsed excitation the dephasing due to hyperfine splitting within the laser bandwidth will therefore be negligible, provided t pulse < l/δw HFS (i.e., if the hyperfine splittings are sufficiently small). Dephasing Processes in Solids A few experiments on EIT have also been performed in solid-state media In contrast to implementations of EIT in gaseous media, coherent interactions in solids suffer from additional dephasings. These dephasings are induced by quantum processes in the crystal lattice (e.g., by photon-phonon interactions). The dephasings lead to significant additional homogeneous broadening in spectral lines of optical transitions. At room temperature the broadening may reach the regime of many gigahertz. In the gas phase, homogeneous broadening (e.g., as observed in the natural linewidth of a transitions) is usually much smaller than inhomogeneous broadening [e.g., induced by Doppler shifts (see below)]. In contrast, in solids it is usually the homogeneous broadening which dominates. This homogeneous broadening (i.e., the dephasing) will wash out any quantum interferences. In principle, there are two ways to deal with the fast dephasing processes in solids: (1) to freeze phonon processes, that is, to cool 14_Bass_V4_Ch14_p indd /25/09 6:22:41 PM

14 14.14 NONLINEAR OPTICS the medium to cryogenic temperatures such that phonon interactions may be neglected; (2) to drive the medium with ultrafast laser pulses, that is, pulse durations below the timescale of the dephasing processes. However, ultrashort laser pulses usually do not provide sufficient pulse energy to saturate the transitions in EIT. Thus, in most experiments on EIT in solids, cryogenic cooling of the medium is a necessary requirement. Inhomogeneous Broadening In many experiments (inhomogeneous) Doppler broadening presents a serious limit since it introduces a randomization in the effective laser detunings over the ensemble of atoms in the sample. 6,116,117 Various methods have been employed to eliminate this effect, for example, Doppler-free excitation 118,119 or using cooled atoms in a magneto-optical trap. 120,121 Alternatively, at coupling Rabi frequencies larger than the Doppler width the influence of inhomogeneous broadening can be overcome. 6 In spectroscopic terms, the laser-induced power broadening beats the Doppler broadening in this case. This concept requires Rabi frequencies exceeding the Doppler width, that is, typically in the regime of 1 GHz. Such Rabi frequencies can be provided by a cw laser, with a typical power of 1 W or less, only under conditions of tight focusing. This may lead to undesirable effects 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. 119 For pulsed lasers, with intrinsically high peak power, it is usually not necessary to focus the laser to reach the required intensity. Thus defocusing effects are negligible and the interaction volume (i.e., the atom number) will be much larger than under conditions of focusing. Moreover, for short- or ultrashort laser pulses the laser bandwidth already exceeds the Doppler width. However, short- or ultrashort laser pulses are usually no good choice to drive EIT (see below). Inhomogeneous broadening also plays a major role for applications of EIT in solid-state systems. Such inhomogeneous broadenings are induced, for example, when doped atoms in a crystal experience different electric fields in the background of the host crystal. This leads to an inhomogeneous distribution of transition frequencies. In general, the implementation of EIT in inhomogeneously broadened solids requires large laser intensities. This increases the risk of optical damage to the sample. An exception to this is the work on solids under conditions of cryogenic cooling. In such media, exceptionally small inhomogeneous widths are encountered. 122 Moreover, inhomogeneous broadening in doped solids can be overcome by appropriate optical preparation (e.g., spectral hole burning). In this case, a specific ensemble of atoms in the inhomogeneously broadened medium is prepared to exhibit spectrally narrow transitions. Coupling Laser Power In addition to the conditions, described above, there are some more constraints for the coupling laser Rabi frequency: (1) The coupling laser Rabi frequency must be sufficiently large to induce a transparency with a spectral width exceeding the linewidth of the Raman (two-photon) transition. For Raman transitions involving large detunings from the single-photon resonances, this may require large laser powers. (2) Harris and Luo 79 derived a condition for the laser pulse energy, which demands a sufficient number of photons in the laser pulse to match the number of atoms, weighted by the transition oscillator strength, in the laser path [see Eq. (8)]. Essentially, the number of photons in the coupling laser pulse must exceed the number of atoms in the medium. (3) In the adiabatic limit, the pulse durations must exceed the time evolution of the transparency, which is in the order of 1/Ω C. Thus, the product Ω t pulse must be large, that is, Ω t pulse >> 1. In terms of incoherent excitation, the laser must saturate the transition. This adiabaticity criterion can be derived in a very similar form also for other adiabatic processes. 44 As Ω t pulse is a combination of the electric field and the pulse duration, laser pulses with large intensity and/or large interaction time are a good choice to fulfill the adiabaticity criterion. An analysis of typical laser systems with specific pulse duration reveals that laser pulses with medium pulse duration (i.e., in the regime of short nanosecond (ns) 14_Bass_V4_Ch14_p indd /25/09 6:22:42 PM