Low-loss negative refraction by laser induced magneto-electric cross-coupling
Jürgen Kästel, Michael Fleischhauer, Susanne F. Yelin, Ronald L. Walsworth
aa r X i v : . [ qu a n t - ph ] J a n Low-loss negative refraction by laser induced magneto-electric cross-coupling
J¨urgen K¨astel and Michael Fleischhauer
Fachbereich Physik, Technische Universit¨at Kaiserslautern, D-67663 Kaiserslautern, Germany
Susanne F. Yelin
Department Of Physics, University of Connecticut, Storrs, Connecticut 06269, USA andITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02138, USA
Ronald L. Walsworth
Harvard-Smithsonian Center for Astrophysics and Department of Physics,Harvard University, Cambridge, Massachusetts 02138, USA (Dated: October 30, 2018)We discuss the feasibility of negative refraction with reduced absorption in coherently drivenatomic media. Coherent coupling of an electric and a magnetic dipole transition by laser fieldsinduces magneto-electric cross-coupling and negative refraction at dipole densities which are consid-erably smaller than necessary to achieve a negative permeability. At the same time the absorptiongets minimized due to destructive quantum interference and the ratio of negative refraction indexto absorption becomes orders of magnitude larger than in systems without coherent cross-coupling.The proposed scheme allows for a fine-tuning of the refractive index. We derive a generalizedexpression for the impedance of a medium with magneto-electric cross coupling and show thatimpedance matching to vacuum can easily be achieved. Finally we discuss the tensorial propertiesof the medium response and derive expressions for the dependence of the refractive index on thepropagation direction.
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I. INTRODUCTION
Negative refraction of light, first predicted to occur inmaterials with simultaneous negative permittivity andpermeability in the late 60’s [1], has become one of themost active fields of research in photonics in the lastdecade. Since the theoretical proposal for its realiza-tion in meta-materials [2, 3] and its first experimentaldemonstration [4] for RF radiation, substantial techno-logical progress has been made towards negative refrac-tion for shorter and shorter wavelengths [5, 6]. This in-cludes various approaches based on split-ring resonatormeta-materials [4, 7, 8, 9], photonic crystals [10, 11, 12]as well as more unconventional designs like double rod[13, 14, 15] or fishnet structures [16, 17].Despite the wide variety of implementations a ma-jor challenge is the large loss rate of these materials[5, 18]. Especially for potential applications such as sub-diffraction limit imaging [19] or electromagnetic cloaking[20, 21, 22] the suppression of absorption proofs to becrucial [23, 24]. The usually adopted figure of merit
FoM = − Re[ n ]Im[ n ] (1)reaches only values on the order of unity in all currentmeta-material implementations [5] with a record value of FoM = 3 [25]. This means that the absorption length ofthese materials is only on the order of the wavelength.Recently we have proposed a scheme in which coherentcross-coupling of an electric and a magnetic dipole reso-nance with the same transition frequencies in an atomic system [26] leads to negative refraction with strongly sup-pressed absorption [27] due to quantum interference ef-fects similar to electromagnetically induced transparency[28, 29]. Furthermore, the value of the refractive indexcan be fine-tuned by the strength of the coherent cou-pling. In the present paper we provide a more detailedanalysis of this scheme. In particular we will discuss un-der what conditions a magneto-electric cross coupling caninduce negative refraction in atomic media without re-quiring Re[ µ ] <
0. The model level scheme introducedin [27] will be analyzed in detail. Explicit expressions forthe susceptibilities and cross-coupling coefficients will bederived and the limits of linear response theory explored.The important issue of non-radiative broadenings andlocal field corrections to the response will be discussed.Furthermore we will derive an explicit expression for theimpedance of a medium with magneto-electric cross cou-pling and show that it can be matched to vacuum viaexternal laser fields such that reflection losses at inter-faces can be avoided. Finally we will give full account ofthe tensorial properties of the induced magneto-electriccross coupling and the resulting refractive index in themodel system. We will show that in the model systemdiscussed it is in general only possible to obtain isotropicnegative refraction in 2D.
II. FUNDAMENTAL CONCEPTS
In the following we discuss the prospects of nega-tive refraction in media with magneto-electric cross-coupling. The electromagnetic constitutive relations be-tween medium polarization P or magnetization M andthe electromagnetic fields E and H are usually expressedin terms on permittivity and permeability only. Impor-tant aspects of linear optical systems such as optical ac-tivity, which describes the rotation of linear polarizationin chiral media cannot be described in this way however.The most general linear relations that also include theseeffects read P = ¯ χ e E + ¯ ξ EH π HM = ¯ ξ HE π E + ¯ χ m H . (2)Eqs.(2) describe media with magneto-electric cross cou-pling which are also known as bianisotropic media [30].Here the polarization P gets an additional contributioninduced by the magnetic field strength H and likewise themagnetization is coupled to the electric field component E . ¯ ξ EH and ¯ ξ HE denote tensorial coupling coefficientsbetween the electric and magnetic degrees of freedom,while ¯ ε = 1 + 4 π ¯ χ e and ¯ µ = 1 + 4 π ¯ χ m are the complex-valued permittivity and permeability tensors. As we useGaussian units the coefficients ¯ χ e , ¯ χ m , ¯ ξ EH , and ¯ ξ HE areunitless.The propagation properties of electromagnetic wavesin such media are governed by the Helmholtz equation h ¯ ε + ( ¯ ξ EH + cω k × )¯ µ − ( cω k × − ¯ ξ HE ) i E = 0 . (3)A general solution of this equation for the wave vec-tor k is very tedious [31] and a comprehensive discus-sion of the most general case almost impossible. Forthe sake of simplicity we therefore assume the permit-tivity ¯ ε and the permeability ¯ µ to be isotropic ¯ ε = ε ,¯ µ = µ . We furthermore restrict ourselves to a one-dimensional theory by choosing the wave to propagatein the z -direction which leaves only the upper left 2 × ξ EH and ¯ ξ HE relevant. Atthis point we restrict the discussion to media which al-low for conservation of the photonic angular momen-tum at their interfaces. In particular, we assume theresponse matrices ¯ ξ EH and ¯ ξ HE to be diagonal in thebasis { e + , e − , e z } . Here e ± denote circular polarizationbasis vectors e ± = ( e x ± i e y ) / √
2. This leads, e.g., for¯ ξ EH , to¯ ξ EH = ξ + EH e + ⊗ e ∗ + + ξ − EH e − ⊗ e ∗− + ξ zEH e z ⊗ e z = ( ξ + EH + ξ − EH ) / − i ( ξ + EH − ξ − EH ) / i ( ξ + EH − ξ − EH ) / ξ + EH + ξ − EH ) / ξ zEH . (4)Note that the tensor (4) includes also biisotropic mediafirst discussed by Pendry [32] for negative refraction asa special case. Such biisotropic media display a chiral behavior, i.e., isotropic refractive indices which are dif-ferent for the two circular eigen-polarizations. In sectionVI we will give an example which implements a polariza-tion independent but anisotropic index of refraction.Using (4) the propagation equation for a left circularpolarized wave traveling in the z -direction can be ex-pressed as εµ − (cid:16) ξ + EH + i cω k + z (cid:17) (cid:16) ξ + HE − i cω k + z (cid:17) = 0 (5)which can be solved for k + z . As k + z is related to thecorresponding refractive index via n + = k + z c/ω we find n + = s εµ − (cid:0) ξ + EH + ξ + HE (cid:1) i (cid:0) ξ + EH − ξ + HE (cid:1) . (6)Note that for a vanishing cross-coupling ( ξ EH = ξ HE =0) this simplifies to the well known expression n = √ εµ .Equation (6) has more degrees of freedom than thenon-chiral version and allows for negative refraction with-out requiring a negative permeability. For example if weset ξ EH = − ξ HE = iξ , with ξ > n = √ εµ − ξ (7)Here and in the following we drop the superscript + fornotational simplicity.Magnetic dipole transitions in atomic systems are arelativistic effect and thus magnetic dipoles are typicallysmaller than electric ones by factor given by the finestructure constant α ≈ / χ m per dipole is typically lessthan the electric susceptibility χ e by a factor given by thefine structure constant squared α . On the other hand, aswe will show later on, the cross-coupling coefficients ξ EH and ξ HE scale only with one factor of α . Thus Eq. (7)represents an improvement compared to non-chiral ap-proaches, since a negative index can be achieved at den-sities where µ is still positive but ξ > √ εµ . Negativerefraction with Re[ µ ] > ξ EH and ξ HE and additionally to control their phases in order toget close enough to ξ EH = − ξ HE = iξ necessary for neg-ative refraction.As in [27] we will analyze these fundamental conceptsin more detail and consider a modified V-type three-levelsystem (Fig. 1). It consists of an electric dipole transi-tion which couples the ground state | i and the excitedstate | i as well as a magnetic dipole transition between | i and the upper state | i . We assume states | i and | i to be energetically degenerate such that the electric ( E )and magnetic ( B ) field components of the probe field cancouple efficiently to the transitions | i − | i and | i − | i ,respectively. In order to couple the electric and mag-netic dipoles, i.e., to induce a cross-coupling in the senseof equation (2) we add a strong resonant coherent cou-pling field between the two upper states | i and | i with a B E Ω c γ γ | i| i | i FIG. 1: A simplistic three-level system which employs electro-magnetically-induced cross-coupling. E , B are the electricand magnetic components of the probe field, γ and γ thedecay rates out of levels | i and | i . Ω c is an applied fieldwhich couples levels | i and | i . Rabi-frequency Ω c . Note that the configuration of Fig. 1complies with the requirements of parity rules. The mag-netization of the system at the probe field frequency isgiven by the coherence of the | i − | i transition whichalso gets a contribution induced by the electric field E and likewise the polarization, connected to | i − | i , isnot only induced by E but also by the magnetic field B .Following the discussion above the level scheme of Fig. 1should therefore show a negative refractive index withoutrequiring Re[ µ ] < n stems from the per-mittivity ε . The radiative decay rate γ of the magneticdipole transition | i − | i is typically a factor α smallerthan γ , the decay rate of the electric dipole transition.As a consequence the strong field Ω c couples state | i strongly to the metastable state | i which, on two-photonresonance, is the condition for destructive quantum inter-ference for the imaginary part of the permittivity, knownas electromagnetically induced transparency (EIT) [28].Additionally, for closed loop schemes (Fig. 1) it is knownfrom resonant nonlinear optics based on EIT [33, 34],that the dispersive cross-coupling, which in our case is themagneto-electric cross coupling, experiences constructiveinterference.In essence the coupling of an electric to a magneticdipole transition should lead to negative refraction forsignificantly smaller densities of scatterers compared tonon-chiral proposals [35, 36, 37]. We additionally expectthe imaginary part of ε , which represents the major con-tribution to absorption, to be strongly suppressed dueto quantum interference effects while simultaneously thecross-coupling terms should be further enhanced.Though conceptually easy the scheme of Fig. 1 has sev-eral drawbacks which demand a modification of the levelstructure. (i) As stated above the phase of the chiral-ity coefficients ξ EH , ξ HE must be adjustable to controlthe sign of the refractive index and induce Re[ n ] < c is a dc -field in the scheme of Fig. 1 the phases of B E B E Ω Ω Ω c Ω c | i| i| i | i| i | i| i | i FIG. 2: (color online) Modification of the level scheme ofFig. 1. The ground state | i is substituted by the dark state | D i = (Ω | i − Ω | i ) / p Ω + Ω of the 3-level Λ-type sub-system formed by levels {| i , | i , | i} . ξ EH , ξ HE are solely given by the intrinsic phase of thetransition moments and therefore can not be controlled.(ii) To suppress the absorption efficiently there must behigh-contrast EIT for the probe field. The critical pa-rameter for this effect is the dephasing rate γ of thecoherence ρ between the two EIT “ground”-states | i and | i . Since state | i has an energy difference to state | i on the order of the probe field frequency, the coher-ence ρ is highly susceptible to additional homogeneousor inhomogeneous broadenings which ultimately can de-stroy EIT. (iii) The level structure must be appropriatefor media of interest (atoms, molecules, excitons, etc.).Although the scheme of Fig. 1 is not forbidden on fun-damental grounds it is very restricting to require thatelectric and magnetic transitions be energetically degen-erate while having a common ground state.One possible alternative level structure which solvesthese problems is shown in Fig. 2. The former groundstate | i is now substituted by the dark state | D i ofthe 3-level Λ-type subsystem formed by the new lev-els {| i , | i , | i} as shown in Fig. 2. | D i is determinedby the two coupling field Rabi frequencies Ω and Ω : | D i = (Ω | i − Ω | i ) / p Ω + Ω . This simple manipu-lation indeed solves the above mentioned problems: Theupper states | i and | i are no longer degenerate, i.e.the coupling Rabi frequency Ω c is now given by an ac -field. By adjustment of Ω c the phase of ξ EH and ξ HE can be controlled. The critical parameter of EIT for thisscheme is the dephasing rate γ of the coherence ρ between states | i and | i . By assuming levels | i and | i of Fig. 2 to be close to degenerate ρ can be takento be insensitive to additional broadenings. Comparedto the 3-level system of Fig. 1 the electric and magnetictransitions here do not share a common state while thetransition frequencies are still degenerate. This leavesmuch more freedom regarding a realization in real sys-tems.In the next section we will analyze the 5-level-systemin detail. | i| i | i| i | i EB ∆ E δ c ∆ B Ω Ω c Ω FIG. 3: 5-level scheme for the implementation of negative re-fraction via electromagnetically induced cross-coupling. Themagnetic dipole transition | i − | i and the electric dipoletransition | i − | i are coupled by Ω c to induce chirality.The “ground”-state of the system is formed by the darkstate | D i = (Ω | i − Ω | i ) / p Ω + Ω of the subsystem {| i , | i , | i} . III. 5-LEVEL SCHEMEA. Linear response
The level scheme in question is shown in greater detailin Fig. 3. The transition | i − | i is magnetic dipoleby nature, all other transitions are electric dipole ones.Note that this complies with the demands of parity. Itis assumed that for reasons of selection rules or absenceof resonance no other transitions than the ones sketchedin Fig. 3 are allowed. The Hamiltonian H = H atom − d · E ( t ) − µµµ · B ( t ) of the system can be written explicitly as H = X n =1 ~ ω An | n i h n | + (cid:26) − d Ee − iω p t | i h | − µ Be − iω p t | i h |− ~ e − iω t | i h | − ~ e − iω t | i h |− ~ c e − iω c t | i h | + h.c. (cid:27) . (8)Here d = h | e r · ˆ e E | i and µ = h | µµµ · ˆ e B | i are theelectric and magnetic dipole moments, E and B the elec-tric and magnetic components of the weak probe fieldwhich oscillates at a frequency ω p . The Rabi frequenciesΩ , Ω and Ω c belong to strong coupling lasers whichoscillate at frequencies ω , ω and ω c , respectively. Wechoose d and µ as well as Ω and Ω to be real whereasthe strong coupling Rabi frequency Ω c has to stay com-plex for the closed loop scheme. To include losses in our description we solve for thesteady state solutions of the Liouville equation of thedensity matrix. In doing so we introduce populationdecay rates γ i , i ∈ { , , , , } . As we focus on thelinear response we treat the probe field amplitudes E and B as weak fields which allows to neglect the upperstate populations ρ and ρ . In contrast the subsystem {| i , | i , | i} contains strong fields with Rabi frequenciesΩ and Ω and should be treated non-perturbatively.We first solve the 3-level subsystem {| i , | i , | i} undis-turbed by the probe field. Under the assumption that | i is meta-stable and therefore γ ≪ γ holds, the exact so-lution is: ρ (0)11 = | Ω | | Ω | + | Ω | ρ (0)44 = | Ω | | Ω | + | Ω | ˜ ρ (0)41 = − Ω Ω | Ω | + | Ω | ρ (0)55 = ˜ ρ (0)51 = ˜ ρ (0)54 = 0 . (9)This solution for the Λ-type subsystem indeed corre-sponds to the pure state | D i = (Ω | i− Ω | i ) / p Ω + Ω via ρ subsys . = | D i h D | . Note that density matrix compo-nents with tildas denote slowly varying quantities.We proceed by solving for the polarizabilities of thecomplete 5-level-scheme up to first order in the probefield amplitudes E and B . Since the induced Polarization P is proportional to the coherence of the electric dipoletransition ˜ ρ whereas the induced Magnetization M isproportional to the density matrix element ˜ ρ we arriveat P = ̺ d ˜ ρ := ̺ α EE E + ̺ α EB BM = ̺ µ ˜ ρ := ̺ α BE E + ̺ α BB B. (10)Here ̺ is the number density of scatterers, α EE , α EB , α BE and α BB are the direct and the cross-coupling po-larizabilities. They are given by: α EE = i ~ d ρ (0)44 ( γ + i (∆ E − δ c ))( γ + i (∆ E − δ c ))( γ + i ∆ E ) + | Ω c | / , (11) α BB = i ~ µ ρ (0)11 ( γ + i (∆ B + δ c ))( γ + i (∆ B + δ c ))( γ + i ∆ B ) + | Ω c | / α EB = − ~ d µ ˜ ρ (0)41 Ω c ( γ + i (∆ E − δ c ))( γ + i ∆ E ) + | Ω c | / , (13) α BE = − ~ d µ ˜ ρ (0)41 Ω ∗ c ( γ + i (∆ B + δ c ))( γ + i ∆ B ) + | Ω c | / . (14)Here the definitions γ ij = ( γ i + γ j ) /
2, ∆ E = ω − ω p ,∆ B = ω − ω p and δ c = ω − ω c apply, with ω µν = ω Aµ − ω Aν being the transition frequencies between levels | µ i and | ν i . Note that these solutions are only valid - - - - - - - - - - - - - - - - * - α EE α BB α EB α BE ∆ /γ ∆ /γ ∆ /γ ∆ /γ FIG. 4: (color online) Real (solid lines) and imaginary (dashedlines) parts of the electric ( α EE ) and magnetic ( α BB ) polariz-abilities as well as the cross-coupling parameters ( α EB , α BE )for arbitrary but the same units for Ω c = 10 γ e iπ/ . for ∆ E = ∆ B + δ c which corresponds to the resonancecondition ω c = ω − ω (15)which ensures the total frequency in the closed loopscheme to sum up to zero.In order to visualize the polarizabilities we set themagnetic dipole decay rate to a typical radiative valuefor optical frequencies [38] of γ = 1kHz, the electricdipole decay rates correspondingly to γ = γ = 137 γ and γ = γ = 0 for the (meta-)stable states | i and | i . The electric and magnetic dipole matrix elements d and µ are determined from the radiative decayrates via the Wigner-Weisskopf result [39] d ( µ ) = p γ ( γ ) ~ c / (4 ω ) for an optical frequency correspond-ing to λ = 600 nm. The Rabi frequencies of the Λ-type subsystem attain real values Ω = Ω = 10 γ while the coupling Rabi frequency Ω c is chosen com-plex, Ω c = | Ω c | e iφ . We furthermore specialize to δ c = 0which implies ∆ E = ∆ B . In order to have increas-ing photon energy from left to right in figures presentedhere all following spectra are plotted as a function of∆ = − ∆ E = − ∆ B .For Ω c = 0 the cross-coupling coefficients α EB and α BE vanish exactly whereas the electric as well as themagnetic polarizability both show a simple Lorentzianresonance. Introducing a non-zero coupling strength Ω c changes the response dramatically. As shown in Fig. 4for Ω c = 10 γ e iπ/ the electric polarizability α EE in-deed displays electromagnetically induced transparency(EIT). The value of | Ω c | is optimized as will be dis-cussed in section III C. As long as the coupling field Ω c ispresent Im[ α EE ] on resonance is proportional to the de-coherence rate γ of the two EIT ground states | i and | i which can become very small. Thus the prominentfeature of EIT emerges: suppression of absorption. Incontrast the magnetic polarizability α BB still shows anordinary Lorentzian resonance since γ is always large. - - - - - - - - - - - - - - - - - - - - ∆ /γ ∆ /γ ∆ /γ ∆ /γ α EE α BB α EB α BE FIG. 5: (color online) Deviation (16) of real and imaginaryparts of the exact polarizabilities α IJ ( E, B ) compared to thelinear response results for 2 different probe field Rabi frequen-cies: Ω E = 137 Ω B = γ (real: solid lines, imaginary: dashedlines) and Ω E = 137 Ω B = 10 · γ (real: dash-dotted lines,imaginary: dotted lines). Due to the coupling to the strong electric dipole transi-tion the magnetic resonance is broadened as comparedto its radiative linewidth which is accompanied by a sig-nificant decrease of the magnetic response susceptibilityon resonance.For a non-vanishing Ω c the two cross-couplings α EB and α BE show strongly peaked spectra. Note that thephase φ = π/ α EB = − α BE ∝ i holds, as demanded in section II.Note furthermore that we verified numerically that allpolarizabilities and cross-coupling terms are causal andthus fulfill Kramers-Kronig relations. B. Limits of linear response theory
For radiatively broadened systems γ ( B )21 ≈ α γ ( E )21 holds. Thus magnetic transitions saturate at much lowerfield amplitudes than corresponding electric dipole tran-sitions. For this reason we have to analyze the saturationbehavior of the system.To rule out saturation effects and validate the use oflinear response theory we solve the Liouvillian equationfor the 5-level system of Fig. 3 to all orders in the electricand magnetic field amplitudes E and B which can onlybe done numerically. We determine the polarizabilities α EE ( E, B ), α BE ( E, B ), α BB ( E, B ) and α EB ( E, B ) fromthe numerically accessible density matrix elements ˜ ρ and ˜ ρ (see Appendix A) for the worst case scenario ofpurely radiatively broadened transitions and compare tothe results of the linear response theory. Fig. 5 showsthe deviation of the exact solution to the linear responseresults log (cid:12)(cid:12)(cid:12)(cid:12) − α IJ ( E, B ) α IJ (cid:12)(cid:12)(cid:12)(cid:12) (16)for all polarizabilities. The solid lines correspond toΩ E = 137 Ω B = γ , the dashed lines to Ω E = 137 Ω B =10 · γ . Here Ω E = d E/ ~ and Ω B = µ B/ ~ denote theelectric and magnetic probe field Rabi frequencies for the5-level system.For a radiatively broadened electric dipole 2-level atomwe can estimate the probe-field Rabi frequency requiredto lead to a 1% upper state population to be ˜Ω E = √ . γ ≈ . γ . ¿From Fig. 5 we note that even forΩ E = 137 Ω B = 10 × γ , i.e., the same order of magnitudeas ˜Ω E , the deviation of the exact result from the spec-trum obtained in linear response approximation neverexceeds 10 − . As a result we conclude that the 5-levelscheme is not significantly more sensitive to saturationeffects than any ordinary electric dipole transition.This behavior is a result of the coupling Rabi frequencyΩ c due to which the | i − | i transition experiences anadditional broadening which makes it less susceptible tosaturation. C. Non-radiative broadenings
As noted in section I additional broadenings are essen-tial for the description of spectral properties as soon asmagnetic transitions are involved.To add an additional homogeneous dephasing rate weformally add to every static detuning ∆ E , ∆ B , δ c anextra random term x E , x B , x c (e.g. ∆ E −→ ∆ E + x E )which has to be convoluted with a Lorentzian distribution L ( x ) = 1 π γ p γ p + x . (17)For, e.g., α EE this amounts to˜ α EE = Z dx E dx B dx c α EE ( x E , x B , x c ) L ( x E ) L ( x B ) L ( x c ) . (18)Levels | i and | i are approximately degenerate and thusassumed to experience correlated phase fluctuations. Asa consequence γ , which is relevant for EIT, remainsunchanged while the same width γ p applies for both ∆ E and δ c . For simplicity we choose the same width for ∆ B .The convolution integral can be solved analytically whichresults in the substitution of the off-diagonal decay rates γ ij , i = j (11)-(14) according to γ −→ γ γ −→ γ + γ p γ −→ γ + γ p γ −→ γ + 2 γ p . (19)In contrast γ and γ encounter a broadening γ p , like-wise γ experiences a broadening 2 γ p .We choose the value γ p = 10 γ which is typical forrare-earth doped crystals at cryogenic temperatures. Fora given broadening γ p the cross-coupling terms reach amaximum for the coupling Rabi frequency attaining theoptimal values 137 γ and 1370 √ γ , respectively. Fig. 6shows the polarizabilities for an intermediate value of - - - - - - - - - - - - - - - α EE ∆ /γ ∆ /γ ∆ /γ ∆ /γ α BB α EB α BE × × × FIG. 6: (color online) Real (solid lines) and imaginary (dashedlines) parts of the electric ( α EE ) and magnetic ( α BB ) polariz-abilities as well as the cross-coupling parameters ( α EB , α BE )for arbitrary but the same units for Ω c = 10 γ e iπ/ . In con-trast to Fig. 4 additional homogeneous broadenings accordingto Eq. (19) with γ p = 10 γ apply. | Ω c | = 10 γ . Since γ remains unbroadened the elec-tric polarizability α EE still shows EIT while the spec-trum of α BB shows a simple but broadened resonancewith α EE ≈ α BB . Similarly α EE ≈ α EB and α EE ≈ α BE hold approximately.To incorporate the effect of an inhomogeneous Dopplerbroadening mechanism on the spectrum the same formal-ism as for the homogeneous case, but with a Gaussianinstead of a Lorentzian distribution, can be used. D. Local field correction
So far we have dealt with a single individual radia-tor (e.g., atom) responding to the locally acting fields.For large responses these local fields are known to differfrom the averaged Maxwell field. We therefore have tocorrect the results of section III A by the use of Clausius-Mossotti type local field corrections. Note that due tothe cross-coupling the influence of the magnetic proper-ties is enhanced by a factor of approximately 137. Wetherefore also include magnetic local field corrections inthe treatment.To add local field corrections to the response the fields E and B Eq. (10) are interpreted as local or microscopicones: P = ̺ α EE E micro + ̺ α EB B micro M = ̺ α BE E micro + ̺ α BB B micro (20)( ̺ : number density of scatterers). The relations betweenthe local and the corresponding macroscopic field ampli-tudes can be obtained from phenomenological considera-tions [40, 41] which read E micro = E + 4 π P, H micro = H + 4 π M (21) - - - ∆ /γ n (a) ̺ = 5 · cm − - - - - - ∆ /γ p n (b) ̺ = 5 · cm − FIG. 7: (color online) Real (solid lines) and imaginary (dashedlines) parts of the refractive index, including local field effectsfor two different densities. for the electric and the magnetic field, respectively. Notethat we need to replace B by H to find the permittiv-ity ε , permeability µ , and coefficients ξ EH and ξ HE ofequation (2) in terms of the polarizabilities of equations(11) - (14). This can be done most easily for the localmicroscopic fields for which B micro = H micro holds [41].Solving Eq. (20) together with (21) for P and M in termsof the macroscopic field amplitudes E and H yields ε =1 + 4 π ̺ L loc × (cid:26) α EE + 4 π ̺ (cid:16) α EB α BE − α EE α BB (cid:17)(cid:27) ,µ =1 + 4 π ̺ L loc × (cid:26) α BB + 4 π ̺ (cid:16) α EB α BE − α EE α BB (cid:17)(cid:27) , (22) ξ EH =4 π ̺ L loc α EB ,ξ HE =4 π ̺ L loc α BE , (23)with the denominator L loc =1 − π ̺ α EE − π ̺ α BB − (cid:18) π (cid:19) ̺ (cid:16) α EB α BE − α EE α BB (cid:17) . Note that for media without a magneto-electric cross-coupling a rigorous microscopic derivation [42] validatesthe phenomenological procedure adopted here.
E. Negative refraction with low absorption
With the permittivity ε and the permeability µ givenby Eq. (22) and the parameters ξ EH and ξ HE [Eq. (23)]we determine the index of refraction from Eq. (6). As anexample, Fig. 7(a) shows the calculated real and imag-inary parts of the refractive index as a function of theprobe field detuning ∆ for a density of ̺ = 5 · cm − and otherwise using the parameter values defined in sec-tions III A and III C. The shape of the spectrum is gov-erned by the permittivity ε with the prominent features - - n ∆ /γ p (a) Π€€€€€ Π Π€€€€€€€€€ Π- - - n φ (b) FIG. 8: (color online) (a) Real (solid line) and imaginary(dashed line) parts of the refractive index for ̺ = 5 · cm − .Compared to Fig. 7(b) here ˜ ρ (0)41 = 0 applies. (b) Real (solidline) and imaginary (dashed line) parts of the refractive indexas a function of the phase φ of the coupling Rabi frequencyΩ c . of EIT: suppression of absorption and steep slope of thedispersion on resonance. Clearly for this density there isno negative refraction yet.In Fig. 7(b) the spectrum of n for an increased den-sity of ̺ = 5 · cm − is shown. Note that in contrastto Fig. 7(a) the frequency axis is scaled in units of thebroadening γ p as local field effects start to influence theshape of the spectral line at this density. We find sub-stantial negative refraction and minimal absorption forthis density. The density is about a factor 10 smallerthan the density needed without taking chirality into ac-count [35].In order to validate that the negative index results fromthe cross-coupling we compare the spectrum of Fig. 7(b)to a non-chiral version. As setting Ω c = 0 influences thepermittivity and permeability as well, we set ˜ ρ (0)41 = 0by hand such that the cross-couplings vanish identically.The resulting index of refraction is shown in Fig. 8(a)for a density ̺ = 5 · cm − . We find that withoutcross-coupling no negative refraction occurs. Thus thenegative index at this density is clearly a consequence ofthe cross-coupling.Following the qualitative discussion in section II wehave set the phase φ of the coupling Rabi frequency Ω c to φ = π/
2. Fig. 8(b) shows the dependence of the refractiveindex on φ taken at the spectral position ∆ = − , γ p at which Re[ n ] reaches its minimum for ̺ = 5 · cm − .As expected the refractive index is strongly phase de-pendent. A change of the phase by δφ = π for examplereverses the influence of the cross-coupling and gives apositive index of refraction Re[ n ] >
0. Note that the sym-metry Re[ n ( φ )] = − Re[ n (2 π − φ )] is coincidental since forthe chosen parameters ε ≈ ̺ further theoptical response of the medium increases. For a spec-tral position slightly below resonance (∆ = − , γ p )where negative refraction is obtained most effectively weshow Re[ ε ] as well as Im[ ξ EH ] and Im[ ξ HE ] as functionsof the density ̺ [Fig. 9(a)]. Due to local field correctionsthe permittivity is of the same order of magnitude as thecross-coupling terms. The imaginary parts of the param-eters ξ EH and ξ HE increase strongly with opposite signs
16 16.2 16.4 16.6 16.8 17 - log ̺ (a)
16 16.2 16.4 16.6 16.8 17 - log ̺ × (b) FIG. 9: (color online) Real part of (a) the permittivity ε (solid line) as well as the imaginary parts of ξ EH (dashed line)and ξ HE (dotted line) and (b) real (solid line) and imaginary(dashed line, ×
50) parts of the refractive index, as well as thereal part of the permeability (dotted line) as a function of thelogarithm of the density log ̺ . - - - - - n ∆ /γ p FIG. 10: (color online) Real (solid line) and imaginary(dashed line) parts of the refractive index, including local fieldeffects for a density of ̺ = 5 · cm − . causing the refractive index to become negative. Thecorresponding density dependence of the refractive indexis shown in Fig. 9(b). We find that Re[ n ] becomes nega-tive while the absorption Im[ n ] stays small (note that inFig. 9(b) Im[ n ] is amplified by a factor of 50). Addition-ally Re[ µ ] is positive and becomes larger for increasingdensity as a consequence of operating on the red detunedside of the resonance (∆ < n isshown for ̺ = 5 · cm − in Fig. 10. Compared to thecase of ̺ = 5 · cm − [Fig. 7(b)] Im[ n ] did not changemuch qualitatively while Re[ n ] reaches larger negativevalues.Remarkably, in Fig. 9(b) we find that the absorptionIm[ n ] reaches a maximum and then decreases with in-creasing density of scatterers. This peculiar behavior isdue to local field effects which invariably get importantat such high values of the response. As a consequence thespectral band with minimal absorption broadens with in-creasing density due to local field effects. Hence the cho-sen spectral position moves from the tail of the band edgeto the middle of the minimal absorption band.As a consequence of the low absorption and corre-sponding increasing values of Re[ n ] the FoM continuesto increase with density and reaches rather large val- - - FoM ∆ /γ p FIG. 11: (color online) The Figure of Merit
FoM for densities ̺ = 5 · cm − (solid line) and ̺ = 5 · cm − (dashedline) as a function of the detuning ∆. ues. In Fig. 11 we show the FoM as a function of ∆for ̺ = 5 · cm − and ̺ = 5 · cm − . Whilethe FoM reaches for ̺ = 5 · cm − values of ≈ ̺ = 5 · cm − up to FoM ≈ IV. TUNABILITY
As first noted by Smith et al. [23] and Merlin [24]sub-wavelength imaging using a flat lens of thickness d requires not only some negative refractive index but anextreme control of the absolute value of n . For an in-tended resolution ∆ x the accuracy with which the value n = − | ∆ n | = exp (cid:26) − πd ∆ x (cid:27) . (24)For a metamaterial with Re[ n ] < d ≫ λ ) as it demandsan extreme fine-tuning of the refractive index in order toachieve a resolution beyond the diffraction limit.Our scheme allows to achieve such a fine tuning. InFig. 12 we show the real and imaginary parts of n asa function of log[ | Ω c | /γ ] for a density of ̺ = 1 , · cm − . As the coupling Rabi frequency approaches γ we find small Im[ n ] while Re[ n ] attains negative val-ues. The dispersion then changes only slightly with asmall slope and values around n = −
1. Therefore therefractive index can be fine tuned by relatively coarse ad-justments of the strength of the coupling field Ω c . Notethat the value and slope of n for | Ω c | ≈ γ can be cho-sen by adjusting the density ̺ of the medium and thespectral position of the probe light frequency.Apart from potential imaging applications the 5-levelquantum interference scheme allows for devices operating - - - - - - - - - n log [ | Ω c | /γ ] FIG. 12: (color online) Real (solid line) and imaginary(dashed line) parts of the refractive index as a function ofthe coupling field Rabi-frequency | Ω c | relative to the radia-tive decay rate γ , for ̺ = 1 , · cm − . in a wide range of positive and negative refractive indiceswith simultaneously small losses. V. IMPEDANCE MATCHING
When considering the applicability of optical devicesreflection at boundaries between different media play animportant role. Impedance matching at these bound-aries is often essential for the performance of the device.For sub-wavelength resolution imaging the elimination ofreflection losses is particularly important, see Eq. (24).In this section we thus derive conditions under which theboundary between non-chiral and chiral, negative refract-ing media is non or little reflecting.We consider a boundary between a non-chiral medium1 ( z <
0) with ε , µ and medium 2 ( z >
0) whichemploys a cross-coupling ( ε , µ , ξ EH , ξ HE ). We assumeagain a wave propagating in z -direction in a medium cor-responding to the tensor structure (4) such that we canrestrict to an effectively scalar theory for, e.g., left circu-lar polarization.We decompose the e + field component in region 1 intoan incoming E i and a reflected part E r E ( r ) = ( E i e ik z + E r e − ik z ) e + (25)( k = | k i | = | k r | ). In medium 2 ( z >
0) only a trans-mitted wave E t exists due to the boundary condition atinfinity E ( r ) = E t e ik z e + . (26)The connection of these modes at the interfaces andsimilar ones for the magnetic field H ( r ) is establishedby the boundary conditions n × ( E − E ) = 0 and n × ( H − H ) = 0. At z = 0 we find E i + E r = E t H i + H r = H t . (27)Moreover an independent set of equations is found fromMaxwell’s equations in Fourier space together with the material equations (2). For medium 1 we get k i × e + E i e ik z + k r × e + E r e − ik z = ωc µ (cid:0) H i e ik z + H r e − ik z (cid:1) e + . (28)Noting that e z × e ± = ± i e ± holds, this simplifies for z = 0 to the scalar equation ik ( E i − E r ) = ωc µ ( H i + H r ) (29)where k i = − k r = k e z has been applied. Similarly weobtain for medium 2 ik E t = ωc ( ξ HE E t + µ H t ) . (30)We eliminate the magnetic field amplitudes from (27),(29), and (30) and solve for the ratio of reflected andincoming electric field amplitudes which yields E r E i = 1 − r µ ε n + iξ HE µ r µ ε n + iξ HE µ . (31)Here the wave numbers k and k have been replaced by k = n ω/c = √ ε µ ω/c and k = n ω/c . Equation (31)is a generalization of the well-known Fresnel formulas fornormal incidence to a cross-coupled medium. Impedancematching is defined as the vanishing of the reflected wave E r = 0, i.e., a complete transfer of the incoming field intomedium 2: r µ ε n + iξ HE µ = 1 . (32)Using the explicit form of n for the particular polariza-tion mode (6) we find the more convenient expression r ε µ = r ε µ s − (cid:18) ξ EH + ξ HE √ ε µ (cid:19) + i ξ EH + ξ HE √ ε µ (33)which obviously simplifies to the well known result p ε /µ = p ε /µ for the non-chiral case ξ EH = ξ HE =0. The right hand side of (33) is the inverse impedance Z − of the cross-coupled medium. For causality reasonsthe square root in Z for passive media has to be takensuch that Re[ Z ] ≥ Z − and of the index of re-fraction. For the case ε = µ = 1 the impedances ofthe two media are matched at the interface as soon as Z − = 1 + i ̺ = 1 . · cm − we find Z − = 1 .
003 + i . . · − γ p while the index of refractionattains n = − . i . FoM ≈ - - - - ∆ /γ p FIG. 13: (color online) Real (solid line) and imaginary(dashed line) parts of the refractive index as well as the real(dash-dotted line) and imaginary (dotted line) parts of theimpedance Z − for ̺ = 1 , · cm − . VI. BEYOND 1D: ANGLE DEPENDENCE
In section II we specialized our discussion to an effec-tively scalar theory by restricting to a particular direc-tion of propagation and left circular polarization ( e + ).We now want to analyze the dependence of the refractiveindex on the propagation direction of the light, whichrequires to take into account the tensor properties of alllinear response coefficients. To this end we consider ageneralization of the 5-level scheme Fig. 3 that includesthe full Zeeman sublevel structure shown in Fig. 14. Thecoupling field E c is assumed to be linear polarized in the z -direction. As the quantization axis we choose the prop-agation direction of the probe light. In this scheme therequirements of section II are fulfilled.Let us first consider the case when the probe light prop-agates along the z axis. Due to Clebsch-Gordon rulesthis solely leads to couplings Ω ++ c and Ω −− c between thetransitions | , + i − | , + i and | , −i − | , −i , respectively.As a result the two circular polarizations of a probe fieldtraveling in z -direction are eigenmodes and the scalartreatment from section II is valid. We therefore get n ± = s εµ − (cid:0) ξ ± EH + ξ ± HE (cid:1) ± i (cid:0) ξ ± EH − ξ ± HE (cid:1) (34)for the left [cf. Eq. (6))] and right circular polarizations.The Wigner-Eckart theorem [38] implies that the electricdipole moments of the | , + i − | i and the | , −i − | i transition coincide: d +34 = d − . Similarly the matrix ele-ments of the magnetic dipole transitions are independentof the polarization state: µ +21 = µ − . In contrast we find d ++32 = − d −− . Thus the coupling Rabi frequencies of theleft and right circular branches Ω ±± c = d ±± | E c | / ~ havea relative sign Ω ++ c = − Ω −− c . (35)From (11) - (14) together with the results obtained in Ω c B + E + B − E − Ω | i| i | , −i| , −i Ω | , i| , + i | , i| , + i | i FIG. 14: (color online) For a direction of incidence other thanthe direction of the coupling field vector E c numerous addi-tional angle dependent couplings occur. section III C we find the relations ε + = ε − µ + = µ − ξ + EH = − ξ − EH ξ + HE = − ξ − HE (36)One recognizes that the refractive indices of e + - and e − -polarizations are identical n + = n − . (37)Hence the index of refraction of the scheme from Fig. 14 isindependent of the polarization state of probe light prop-agating in z -direction. For this reason the electromagnet-ically induced cross-coupling in the scheme of Fig. 14 doesnot correspond to a chiral medium for which the circularcomponents should have different refractive indices.Next we allow for an angle between the propagationdirection of the probe field and the direction of the cou-pling field vector E c . In particular we employ a frameof reference whose z -direction is fixed to the k -vector: k ∼ e z . The direction of the coupling field which is fixedwith regard to the laboratory frame is then given by polarangles θ , φ :( E c,x , E c,y , E c,z ) = | E c | (sin θ cos φ, sin θ sin φ, cos θ ) . (38)As the atomic quantization axis is assumed to be given bythe e z -axis of the k -frame, the probe field will encounteran unchanged atomic level structure irrespective of thedirection of propagation. As indicated in Fig. 14 the | i − | i transition is assumed to be a J = 0 , M = 0to J = 0 , M = 0 transition and thus the dark state isspherically symmetric and does not depend on the polarangles θ and φ .In this framework angle-dependent propagation is tak-en into account by means of angle dependent coupling1Rabi frequenciesΩ ++ c = Ω c, cos θ = − Ω −− c , (39)Ω +0 c = Ω c, √ θe − iφ = Ω − c , (40)Ω − c = Ω c, √ θe iφ = Ω c (41)according to (38). Here Ω c, = h || d || i · | E c | / ( √ ~ ) isfound from the Wigner-Eckart theorem where h || d || i denotes the reduced dipole matrix element.The angle-dependent cross coupling tensor for the elec-trically induced magnetization reads¯ α BE = α BE cos θ θ e − iφ √ − cos θ sin θ e + iφ √ θ e + iφ √ θ e − iφ √ (42)with α BE given by (14). Note that the tensor is expressedin the { + , − , z } -basis. For example the coefficient α EB + z which describes the e + -polarized electric field induced bya e z -polarized magnetic field (in the k -frame) is given bythe upper right entry. For ¯ α EB we find (42) as well butwith α BE replaced by α EB from equation (13). On theother hand the electric polarizability is given by¯ α EE = α EE + α EE | Ω c | D D ×× sin θ − sin θ e iφ − sin θ cos θ e iφ √ − sin θ e − iφ θ θ e − iφ √ − sin θ cos θ e − iφ √ θ e iφ √ θ (43)with α EE determined by Eq. (11) and D = ( γ + i (∆ E − δ c )), D = ( γ + i ∆ E ). For the magnetic polar-izability ¯ α BB the same tensor structure applies. Again α EE has to be replaced by (12) and D D is substitutedby D D = ( γ + i (∆ B + δ c ))( γ + i ∆ B ).For incidence in the z -direction the tensors simplifysignificantly. The cross-couplings reduce to a tensor pro-portional to e + ⊗ e ∗ + − e − ⊗ e ∗− which identically corre-sponds to (4) for ξ z = 0 and ξ + = − ξ − . In the same limit( θ = 0) the electric and magnetic polarizabilities becomediagonal. In particular α EE ++ and α EE −− are given by (11)and therefore potentially display EIT while the α EEzz en-try simplifies to a simple Lorentzian resonance structure.In contrast the diagonal elements of ¯ α BB always displaya Lorentzian resonance with ( α BB ++ , α BB −− ) and without( α BBzz ) coherent coupling. Π€€€€€€ Π€€€€€€
Π€€€€€€€€€€€ Π- θn FIG. 15: (color online) Real (solid line) and imaginary(dashed line) parts of the refractive index (44) as a functionof θ . From the angle dependent response tensors we find anangle dependent index of refraction. The true index ofrefraction which takes the full form of (43) into account,i.e., the angle dependent correction to ε and µ , gets verycomplicated. We here note that under the assumption ofisotropic permittivity and permeability we find the fairlysimple result n + = n − = r ǫµ − ξ EH ξ HE − ( ξ EH − ξ HE ) cos ( θ )4+ i ξ EH − ξ HE ) cos( θ ) . (44)independent of the polarization state. We conclude thateven the idealized case ¯ ε ∼ ∼ ¯ µ does not give anisotropic index of refraction.In Fig. 15 we show the index of refraction (44) as afunction of the polar angle θ . We use values of theresponse functions taken at a spectral position ∆ = − , γ p for a density ̺ = 5 · cm − . We emphasizethat the angle dependence in (44) results in an index ofrefraction which varies over a broad spectrum of positiveand negative values for different angles. VII. CONCLUSION
In conclusion we have shown that coherent magneto-electric cross-coupling improves the prospects to obtainlow-loss negative refraction in several ways. The densi-ties needed to get Re[ n ] < Acknowledgments
M.F. and J.K. thank the Institute for Atomic, Molecu-lar and Optical Physics at the Harvard-Smithsonian Cen-ter for Astrophysics and the Harvard Physics Depart-ment for their hospitality and support. R.W. thanksD. Phillips for useful discussions. J.K. acknowledges fi-nancial support by the Deutsche Forschungsgemeinschaftthrough the GRK 792 “Nichtlineare Optik und Ultra-kurzzeitphysik” and by the DFG grant FL 210/14. S.Y.thanks the NSF for support.
Appendix A. EXACT NUMERICAL SOLUTIONOF THE LIOUVILLE EQUATION
To solve the Liouville equation to all orders in theprobe field amplitudes E and B we first transform toa rotating frame and specialize to steady state solutions.This gives a set of 25 algebraic equations which we castinto a matrix form by arranging the 5 diagonal ρ . . . ρ and 20 off-diagonal density matrix elements ˜ ρ . . . intoa 25-dimensional vector ~ρ . We end up with an inhomo-geneous matrix equation M ~ρ = ~a (A.1)where the inhomogeneity is given by the 25-dimensionalvector ~a = (1 , , , . . . ). The matrix M contains all cou-plings, detunings and decay rates for the system in ques-tion. To solve equation (A.1) for the sought density ma-trix elements ˜ ρ and ˜ ρ we have to invert M which canonly be done numerically after specifying explicit num-bers for all parameters.In general we find that ˜ ρ and ˜ ρ are functions ofboth, the electric and the magnetic field amplitude˜ ρ = f ( E, B ) ˜ ρ = g ( E, B ) . (A.2)We emphasize that the analytical form of the functions f ( E, B ) and g ( E, B ) is unknown. As we want to comparewith the result of linear response theory we have to bring(A.2) in the form of Eq. (10) d ˜ ρ = α EE ( E, B ) E + α EB ( E, B ) B (A.3) µ ˜ ρ = α BE ( E, B ) E + α BB ( E, B ) B. (A.4)In contrast to the linear response theory here we dealwith the exact solution of the Liouville equation andtherefore the polarizabilities are still functions of the fields E and B . At first glance the separation does notseem to be unique. To determine α EE ( E, B ), α EB ( E, B ), α BE ( E, B ), and α BB ( E, B ) numerically we formally ex-pand f and g in a power series in E and Bf ( E, B ) = X n,m f nm E n B m (A.5) g ( E, B ) = X n,m g nm E n B m . (A.6)To separate the electric and magnetic properties uniquelywe use the fact that physically there must be an oddpower of field amplitudes. In fact all but one (fastly rotat-ing) factors e − iω p t must be compensated by factors e iω p t .Otherwise the (untransformed) polarizabilities would notoscillate with the probe field frequency ω p . Since an oddpower of field amplitudes can only be realized by an oddpower in E and an even power in B or vice versa an evenpower in E and an odd power in B we formally split intoeven and odd subseries f ( E, B ) = X n,m f Enm | E | n | B | m E + X n,m f Bnm | E | n | B | m B,g ( E, B ) = X n,m g Enm | E | n | B | m E + X n,m g Bnm | E | n | B | m B. 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