Dipole-Induced Electromagnetic Transparency
Raiju Puthumpally-Joseph, Maxim Sukharev, Osman Atabek, Eric Charron
DDipole-Induced Electromagnetic Transparency
Raiju Puthumpally-Joseph, Maxim Sukharev, Osman Atabek, and Eric Charron Universit´e Paris-Sud, Institut des Sciences Mol´eculaires d’Orsay (CNRS), F-91405 Orsay, France Science and Mathematics Faculty, School of Letters and Sciences,Arizona State University, Mesa, Arizona 85212, USA (Dated: August 20, 2018)We determine the optical response of a thin and dense layer of interacting quantum emitters. Weshow that in such a dense system, the Lorentz redshift and the associated interaction broadening canbe used to control the transmission and reflection spectra. In the presence of overlapping resonances,a Dipole-Induced Electromagnetic Transparency (DIET) regime, similar to Electromagnetically In-duced Transparency (EIT), may be achieved. DIET relies on destructive interference between theelectromagnetic waves emitted by quantum emitters. Carefully tuning material parameters allowsto achieve narrow transmission windows in otherwise completely opaque media. We analyze in de-tails this coherent and collective effect using a generalized Lorentz model and show how it can becontrolled. Several potential applications of the phenomenon, such as slow light, are proposed.
Light-matter interaction has been a topic of intense re-search for many decades. It is currently experiencing asignificant growth in the area of nano-optics [1]. Lightscattering by a system of nanometric size is an exam-ple where important applications can be foreseen. Thetheoretical description of light scattering is very well un-derstood when dealing with individual quantum emitterssuch as atoms or molecules [2]. In the case of two (ormore) strongly interacting emitters, and in general forlarge densities, the physics is far more complex since thebehavior of the ensemble of emitters cannot be describedanymore as the sum of their individual response. In thiscase, the field experienced by an emitter depends not onlyon the incident field but also on the one radiated by allits neighbors. The latter are also affected by the emitter,thus leading to a very complex highly coupled dynamicswhich must be described self-consistently.For an oscillating dipole of resonant wavelength λ ,high densities n are achieved when n λ (cid:62) i.e. whenthere is more than one emitter in the volume associatedwith the dipole wavelength [3]. In this situation, strongdipole-dipole couplings come into play, and collective ex-citation modes quickly dominate the optical response ofthe sample. This results usually in an enhancement oflight-matter interaction. This cooperative effect is clearlyobserved in the superradiance or superfluorescence pro-cesses initially discussed by Dicke [4].The topic of strong dipole-dipole interactions has re-cently been the subject of a considerable interest inthe context of quantum information with cold atoms[5], following an early proposal by Jaksch et al to usedipole blockade as a source of quantum entanglement [6].This initial proposal, limited to two interacting dipoles,was soon extended to many-atom ensemble qubits [7].With highly excited Rydberg atoms, this regime can beachieved for atomic densities as low as 10 cm − [8, 9].Higher densities, of the order of 10 cm − , are typicallyrequired for ground state atoms. As an example of coop-erative effects, collective Lamb and Lorentz shifts were z Absorption & DissipationReflection !"
Transmission
Thin slab of quantum emitters ! ( density n ) incident radiation ℓ x y FIG. 1. (Color online) Schematic view of the thin dense vaporof quantum emitters interacting with the incident field. recently measured in a thin thermal atomic vapor layersimilar to the system studied here [10].Following these last developments, the present paperdeals with a theoretical study of the optical responseof a thin dense vapor of quantum emitters, atoms ormolecules. We show that, in such systems, strong dipole-dipole interactions can be used to manipulate the spec-tral properties of the light scattered by the sample. Wealso show that in the presence of overlapping resonances[11], the medium may become partially transparent for aparticular frequency which can be controlled to a certainextent. In addition, the radiation at the neighboring fre-quencies is nearly perfectly reflected, opening the way topotential applications in optics.As shown in Fig. 1, we consider a thin layer of quantumemitters whose transverse dimension ( z -axis) is denotedby (cid:96) . The longitudinal dimensions of the slab in the x and y dimensions are assumed to be much larger than (cid:96) . Thedipoles considered here are two-level emitters, with stateslabeled as | (cid:105) and | (cid:105) . Their associated energies are (cid:126) ω and (cid:126) ω . ω = ω − ω denotes the Bohr frequency. Thedensity matrix ˆ ρ ( z, t ) [12, 13] describing the quantumdynamics satisfies the dissipative Liouville-von Neumann a r X i v : . [ qu a n t - ph ] S e p equation i (cid:126) ∂ t ˆ ρ = [ ˆ H, ˆ ρ ] − i (cid:126) ˆΓˆ ρ, (1)where ˆ H = ˆ H + ˆ V ( z, t ) is the total Hamiltonian and ˆΓ isa superoperator taken in the Lindblad form [14], describ-ing relaxation and dephasing processes under Markov ap-proximation. The field free Hamiltonian readsˆ H = (cid:126) ω | (cid:105)(cid:104) | + (cid:126) ω | (cid:105)(cid:104) | , (2)and the interaction of the two-level system with the elec-tromagnetic radiation is taken in the formˆ V ( z, t ) = (cid:126) Ω( z, t ) (cid:0) | (cid:105)(cid:104) | + | (cid:105)(cid:104) | (cid:1) (3)where Ω( z, t ) is the local instantaneous Rabi frequencyassociated with the transition dipole µ . In Eq. (1), thenon-diagonal elements of the operator ˆΓ include a puredephasing rate γ ∗ , and the diagonal elements of this op-erator consist of the radiationless decay rate Γ of theexcited state. The total decoherence rate is denoted by γ = γ ∗ + Γ /
2. Equations (1)-(3) lead to the well-knownBloch optical equations [15, 16] describing the quantumdynamics of a coupled two-level system. It is assumedthat the system is initially in the ground state | (cid:105) .The incident radiation is normal to the slab and prop-agates in the positive z -direction (see Fig. 1). It is rep-resented by a transverse electric mode with respect tothe propagation axis and is characterized by one in-planeelectric and one out-of-plane magnetic field components,namely E x ( z, t ) and H y ( z, t ). Time-domain Maxwell’sequations in such a geometry read µ ∂ t H y = − ∂ z E x (4) (cid:15) ∂ t E x = − ∂ z H y − ∂ t P x (5)The system of Maxwell’s equations is solved using a gen-eralized finite-difference time-domain technique whereboth the electric and magnetic fields are propagated indiscretized time and space [13, 17]. The macroscopic po-larization P x ( z, t ) = n (cid:104) ˆ µ (cid:105) = n Tr[ˆ ρ ( z, t )ˆ µ ] is takenas the product of the atomic density n with the expecta-tion value of the transition dipole moment operator ˆ µ .The coupled Liouville-Maxwell equations are integratednumerically in a self-consistent manner. The couplingbetween Eqs.(1) and (5) is through the polarization cur-rent ∂ t P x due to the quantum system taken as a sourceterm in Ampere’s law (5) but, as discussed below, this isnot sufficient in the case of high densities.An exact treatment of light scattering in the presenceof strong interactions between a large number of quan-tum emitters is extremely difficult. It has been shownthat an efficient and accurate approach consists in theintroduction of a local field correction to the averagedmacroscopic electric field E x ( z, t ) [18]. In this mean-fieldapproach, the individual quantum emitters are driven bythe corrected local field E local ( z, t ) = E x ( z, t ) + P x ( z, t ) / (3 (cid:15) ) . (6) E x ti n c ti on (a) ∆ / γ = 0.05∆ / γ = 2∆ / γ = 18 T r a n s m i ss i on (b) -100 -75 -50 -25 0 25 50 75 100 Reduced detuning δ R e f l ec ti on (c) FIG. 2. (Color online) Extinction (a), transmission (b) andreflection (c) probabilities as a function of the reduced detun-ing for a thickness (cid:96) = λ / .
55 = 400 nm. The decay andpure dephasing rates are 10 and 10 Hz. The solid blacklines correspond to a weak density [∆ /γ = 0 . /γ = 2], and the dash-dotted brown lines stand for a large density [∆ /γ = 18]. This local field E local ( z, t ) enters the dissipative Liou-ville equation (1) through the Rabi frequency Ω( z, t ) = µ E local ( z, t ) / (cid:126) . It is well known that the replace-ment of E x ( z, t ) with E local ( z, t ) leads to a frequencyshift in the linear response functions of the medium forlarge densities, the so-called Lorentz-Lorenz (LL) shift∆ = n µ / (9 (cid:126) (cid:15) ) [18, 19].Figure 2 shows the calculated one-photon extinction,transmission and reflection spectra as a function of thereduced detuning δ = ( ω − ω ) /γ at three differentdensities. These spectra are obtained via the compu-tation of the normalized Poynting vector on the inputand output sides of the layer [13, 16]. It is importantto note that for weak densities (solid black lines) the ex-tinction spectrum [panel (a)] shows a typical Lorentzianlineshape of half-width γ . Light absorption affects thetransmission spectrum [panel (b)] such that a hole is ob-served, and no reflection is seen in panel (c). An in-creased density (blue dashed lines) leads to a splitting ofthe extinction signal into two lines: the red-shifted linecorresponds to a configuration where the dipoles oscil-late in-phase with the incident field, whereas the blue-shifted line corresponds to an anti-parallel configurationwhere the induced dipoles oscillate out-of-phase with thisfield. In addition, the hole seen in the transmissionspectrum broadens significantly and looses its Lorentzianshape. Concurrently, a strong reflection signal shows upat the transition frequency. The optical response of themedium changes dramatically at high densities (dash- -40-2002040 R e [ χ e ] -100 -75 -50 -25 0 25 50 75 100 Reduced detuning δ R e f l ec ti on ∆ ∆ (a)(b) FIG. 3. (Color online) (a) Real part of the electric susceptibil-ity χ e calculated from our extended Lorentz model as a func-tion of the reduced detuning, for ∆ /γ = 18 (large density).The reflection probability is shown in panel (b). The two dotsin panel (a) indicate the frequencies at which Re [ χ e ] = − dotted brown lines). The medium is then characterizedby a collective dipole excitation which cancels out trans-mission over a very large window around the transitionfrequency. In this frequency range, almost total reflec-tion is observed. This collective effect can be understoodfrom the extended Lorentz model we introduce below.In this model, the dipoles, driven by the electric field,experience linear restoring and classical damping forces.The time evolution of the macroscopic polarization canthen be written as [18] ∂ tt P x + γ ∂ t P x + ω P x = (cid:15) ω p [ E x + P x / (3 (cid:15) )] (7)where ω p denotes the plasma frequency. Compared tothe usual formulation of the classical Lorentz model, wehave added here the local field correction P x / (3 (cid:15) ). Inaddition, with the assumption that the maximum am-plitude of oscillation of the dipoles in the absence of adriving field is given by the quantum harmonic oscillatorlength, we obtain the plasma frequency ω p = √ ω ∆,where ∆ is the LL shift. Finally, in the particular case ofa monochromatic excitation, Eq. (7) is easily solved, andthe electric susceptibility χ e = P x / ( (cid:15) E x ) is obtainedas χ e ( ω ) = 6 ω /f ( ω ), where f ( ω ) = ( ω − ω ∆ − ω + iγω ) / ∆. Compared to the standard Lorentz model,we observe here a frequency shift ω → ω − ω ∆.When ∆ /ω (cid:28)
1, we see that the resonance frequency ω is simply red-shifted by the LL shift − ∆, as expected[19]. We see clearly in panel (a) of Fig. 3 such a strongredshift of the resonance. The broad reflection windowseen in panel (b) for large densities, which was alreadypredicted by Glauber et al [20], can now be explainedusing simple considerations. Assuming a non-absorbingmedium, and therefore γ = 0, the reflectance R ( ω ) atthe interface is given by R = | (1 − n ) / (1 + n ) | , where n ( ω ) = Re [ (cid:112) χ e ( ω )] is the real part of the refrac-tive index of the slab. We see that total reflection is -2 -1 | χ e | P r ob a b ilit y (a)(b)(c) -150 -100 -50 0 50 100 150 200 Reduced detuning δ I nd e x o fr e fr ac ti on Re[ n ]Im[ n ]
20 25 30 0.00.51.01.5
Lorentzian Fano ω * FIG. 4. (Color online) Transmission (solid brown line) andreflection (dashed green line) probabilities (a) as a functionof the detuning in the case of a dense mixture of two differentquantum emitters (see text for details). The thin blue line isthe transmission probability when the coupling between thetwo dipoles is neglected. The electric susceptibility and therefractive index are shown in panels (b) and (c). obtained when n = 0 and therefore when χ e ( ω ) (cid:54) − ω − ∆ , ω + 2∆]. The width of the re-flection window is therefore 3∆ = n µ / (3 (cid:126) (cid:15) ), as shownin Fig. 3. Panels (a) and (b) show that the out-of-phaseoscillations of the induced dipoles correspond to a nearlyopaque overall sample. Here, the strongly coupled oscil-lating dipoles emit a radiation which efficiently cancelsout the incident field inside the sample, thus leading tohigh reflection. This phenomenon dominates in case ofhigh densities, where the dipoles coherently cooperate toprevent penetration of the incident radiation in the slab.Let us now use the in-phase vs out-of-phase dipoles tomanipulate both reflection and transmission. We con-sider a mixture of two different quantum emitters withdensities n and n (cid:48) and transition dipoles µ and µ (cid:48) .Fig. 4 shows the case of two dipoles with the same deco-herence rate γ = γ (cid:48) and the same LL shift ∆ = 18 γ = ∆ (cid:48) .The two transitions differ by ( ω (cid:48) − ω ) /γ = 50 and thereduced detuning is still defined with respect to the firsttransition. The two vertical red dotted lines indicate thefrequencies of the two transitions. A very peculiar fea-ture shows up in the reflection (dashed green line) andtransmission (solid brown line) spectra of Fig. 4. At theintermediate detuning δ = 25, a minimum appears in thereflection spectrum. Concurrently, a sharp transmissionpeak appears at the same frequency. The thin blue lineshows the transmission when the coupling between thetwo dipoles is neglected. Clearly, the strong coupling be-tween the two types of dipole renders the medium trans-parent in an otherwise opaque region. We note that theposition and width of the transparency window are con-trolled by material parameters, as discussed below.Let us introduce the coupling between the two dipolesin our extended Lorentz model. The time evolution ofthe polarization P associated with the first dipole reads ∂ tt P + γ∂ t P + ω P = (cid:15) ω p [ E x + ( P + P (cid:48) ) / (3 (cid:15) )] (8)with an equivalent equation describing the polarizationof the second dipole P (cid:48) . The total polarization is thenwritten as P x = P + P (cid:48) . These two coupled equationscan be solved analytically in the case of a monochromaticdriving field, yielding χ e ( ω ) = 6 ω (cid:48) [ f ( ω ) + 2 ω ] + 6 ω [ f (cid:48) ( ω ) + 2 ω (cid:48) ] f ( ω ) f (cid:48) ( ω ) − ω ω (cid:48) (9)where f (cid:48) ( ω ) = ( ω (cid:48) − ω (cid:48) ∆ (cid:48) − ω + iγ (cid:48) ω ) / ∆ (cid:48) . The squaremodulus of this electric susceptibility is shown in panel(b) of Fig. 4. The resonance observed at the detuning δ = −
35 presents the usual Lorentzian profile and marksthe frequency at which reflectance reaches a plateau. An-other resonance is seen in the region δ = 20 - 40. Thisresonance has a Fano profile, characteristic of a quantuminterference effect between two indistinguishable excita-tion pathways [21]. Indeed, in the frequency range be-tween ω and ω (cid:48) , the two transitions overlap and thecontributions from the two types of dipoles add up in themacroscopic polarization of the medium. In addition, inthis frequency range the two dipoles oscillate in oppositedirections (out-of-phase) and one can find a particularfrequency ω ∗ at which they cancel each other. This comesfrom the fact that one type of emitters is blue-detunedwhile the other is red-detuned, leading to opposite signsof their susceptibilities. In the limit γ (cid:28) ∆ we obtain ω ∗ = ( ω ∆) ω (cid:48) + ( ω (cid:48) ∆ (cid:48) ) ω ( ω ∆) + ( ω (cid:48) ∆ (cid:48) ) . (10)At this intermediate frequency ω ∗ (cid:39) ω + 25 γ the sus-ceptibility reaches almost zero and the medium becomestransparent. Knowing that ∆ ∝ n and that ∆ (cid:48) ∝ n (cid:48) ,it appears that the value of ω ∗ can be controlled in therange ω (cid:54) ω ∗ (cid:54) ω (cid:48) by simply adjusting the densities ofthe two emitters. This transparency phenomenon is remi-niscent of EIT [22–25] and we therefore name it “Dipole-Induced Electromagnetic Transparency” (DIET). Com-pared to EIT, the strong coupling induced by the pumplaser is replaced by strong dipole-dipole interactions.An example of application is shown Fig. 5, where theincident field is a weak 50 fs pulse of carrier frequency ω + 35 γ . We show in Fig. 5 the power spectrum ofthe incident, transmitted and reflected fields. Clearly,DIET is imprinted in both the reflected and transmit-ted pulses. Furthermore, as seen in Fig. 4(c), at ω = ω ∗ -75 -50 -25 0 25 50 75 100 125 150 Reduced detuning δ P u l s e s p ec t r u m IncidentTransmittedReflected ω * FIG. 5. (Color online) Incident, transmitted and reflectedpulse spectrum in the case of a dense vapor constituted by amixture of two different quantum emitters. the medium is characterized by a steep dispersion of thereal part of the refractive index, whereas its imaginarypart, and therefore absorption, is negligible. At this fre-quency the group velocity v g = c/ [ n + ω ( dn/dω )] reaches v g (cid:39) c/ (cid:96)/v g , a proof that DIET can in-duce slow light [26–30]. In addition, the slow-down factorcan be controlled by changing the material parameters.In terms of experimental implementations, we believethat DIET could be observed in an atomic vapor confinedin a cell whose thickness is of the order of the opticalwavelength [10, 31]. Such systems suffer from inhomo-geneous Doppler broadening [32]. At room temperature,the induced dephasing may wash out the coherence ofthe system and low temperature atomic vapors wouldbe necessary, or, alternatively, sub-Doppler spectroscopictechniques [33, 34] could be used. Another envisionedexperimental system for DIET is ultra-cold dense atomicclouds [35]. Such systems are inherently free of inho-mogeneous broadening and homogeneous dipole-inducedline broadening effects have been observed very recently[36]. The total number of trapped atoms is however stilltoo limited [36, 37] to observe the coalescence of two sep-arate resonances. In addition, DIET requires a constantatomic density, and therefore the use of a (quasi) uniformatomic trap [38, 39].In the case of multi-level systems, a series of trans-parency frequencies is expected as a result of Fano-typeinterferences between closely-spaced energy levels [40]. Itis therefore anticipated that DIET can be observed in re-alistic multi-level systems. In this case, each system canbe considered as a quantum oscillating dipole with manyallowed transitions such as different electronic and/or ro-vibrational levels. Since the prediction and observationof EIT has offered a number of very exciting applicationssuch as slow light [41] or even stopped light [42–46], wecan envision similar applications with DIET in the nearfuture. We have also verified that DIET survives withstrong incident laser pulses. One may therefore also ex-pect various applications in strong field and attosecondphysics, for instance for the generation of high harmonicsin dense atomic or molecular gases [47, 48].The authors acknowledge support from the EU (ITN-2010-264951 CORINF) and from the Air Force Office ofScientific Research (Summer Faculty Fellowship 2013). [1] L. Novotny and B. Hecht, Principles of nano-optics ,Cambridge University Press, Cambridge, 2012.[2] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg,
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