Interferometric phase detection at x-ray energies via Fano resonance control
K. P. Heeg, C. Ott, D. Schumacher, H.-C. Wille, R. Röhlsberger, T. Pfeifer, J. Evers
IInterferometric phase detection at x-ray energies via Fano resonance control
K. P. Heeg, C. Ott, D. Schumacher, H.-C. Wille, R. R¨ohlsberger, T. Pfeifer, and J. Evers Max-Planck-Institut f¨ur Kernphysik, Saupfercheckweg 1, D-69117 Heidelberg, Germany Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85, 22607 Hamburg, Germany (Dated: September 11, 2018)Modern x-ray light sources promise access to structure and dynamics of matter in largely unex-plored spectral regions. However, the desired information is encoded in the light intensity and phase,whereas detectors register only the intensity. This phase problem is ubiquitous in crystallographyand imaging, and impedes the exploration of quantum effects at x-ray energies. Here, we demon-strate phase-sensitive measurements characterizing the quantum state of a nuclear two-level systemat hard x-ray energies. The nuclei are initially prepared in a superposition state. Subsequently,the relative phase of this superposition is interferometrically reconstructed from the emitted x-rays.Our results form a first step towards x-ray quantum state tomography, and provide new avenues forstructure determination and precision metrology via x-ray Fano interference.
PACS numbers: 78.70.Ck, 03.65.Wj, 32.70.Jz, 07.60.Ly
The phase of electromagnetic fields is the key to inter-ferometry. This technique of superimposing electromag-netic waves is an important method with applicationsacross all the natural sciences [1]. But standard detec-tors for optical or higher-frequency fields are sensitiveto the field intensity only, masking the phase requiredfor many applications [2]. In x-ray science, prominentexamples are crystallography [3] and coherent imaging.In the latter, photons are elastically scattered off of anobject to characterize it [4]. Although the relation be-tween the scattered light and the original object is wellknown, the lack of phase information prevents a straight-forward reconstruction of the original object [5]. Next tostructure determination, also the reconstruction of quan-tum states requires phase-sensitive measurements [6, 7].This quantum state tomography has been successfullydemonstrated at optical frequencies [8], but remains anopen challenge for the emerging field of x-ray quantumoptics [9–17].In this work, we demonstrate phase-sensitive measure-ments on an archetype quantum mechanical two-levelsystem (TLS) at hard x-ray frequencies, represented bya nuclear resonance of a M¨ossbauer isotope. The TLSis realized in a large ensemble of identical nuclei, oper-ated in such a way that the incident x-rays couple theground state to a single collective excited state. Thephase-sensitivity is gained by a cavity-based x-ray inter-ferometer with the TLS in one of its arms. Tuning thephase of the non-TLS path enables us to determine thephase of the light emitted by the TLS via the intensityat one of the output ports of the interferometer. Thex-ray pulse is near-instantaneous on the time scale of thenuclei, and prepares the TLS in an initial coherent su-perposition of its two states. After this, we determinethe phase of the light emitted by the TLS, which allowsus to characterize the initial state prepared by the x-raypulse.The state of the TLS is described by its density matrix, and the phase reconstructed here can be identified withthe phase of the off-diagonal density matrix elements.Our measurement therefore forms an important step to-wards the full tomography of the TLS quantum state,which is a crucial tool for the exploration of quantumeffects at x-ray energies. The interferometric measure-ment technique combined with the precise mapping ofthe spectroscopic lineshape of the TLS in turn providenew avenues for structure determination and precisionmetrology.The experimental scheme and setup are illustrated inFig. 1. A nm-sized thin-film structure of materials withalternating index of refraction is used to form an x-raycavity. The guiding layer of the cavity contains a sheetof Fe nuclei with a M¨ossbauer resonance at 14 . θ , and werecord the spectrum of the reflected light. In this set-ting, we interpret the cavity as an interferometer, com-prising two pathways that contribute to the x-ray re-flectance as depicted in Fig. 1. The first path ( r C e − iφ )consists of light reflected by the cavity alone. In thesecond path ( r N e iφ N ), the x-rays interact with the near-resonant TLS. The total recorded reflectance arises fromthe interference between the two paths, governed by theirrelative phase. As we will show, the phase φ depends onthe incidence angle θ , and thus can externally be con-trolled.We have experimentally explored this phase controlat the Dynamics Beamline P01 of the PETRA III syn-chrotron radiation source (DESY, Hamburg). We em-ployed nuclear resonant scattering, where a short broad-band incident pulse excites the nuclei, and subsequentlythe delayed scattered photons are detected in a time win-dow 40 −
190 ns after excitation. The cavity is formedby a Pd(4 nm)/C(36 nm)/Pd(14 nm) (top to bottom)layer system with the Pd layers acting as the mirrorsand the C as guiding layer. A 1.2 nm thick active layer a r X i v : . [ qu a n t - ph ] N ov ϵ (b) (c) | r C | | r N | φ=0φ=− /4 π φ= − π /4 ϵ (a) θ φ N +φ Fe r N e iφ ħωħω ħω C Pd C Pd q=−1 q=1q=∞ N r C e -iφ r N e iφ N r C e -iφ | r C e - i φ + r N e i φ | N FIG. 1. (Color online) Schematic setup of the x-ray interferometer and origin of the Fano interference. (a) X-ray light isreflected by a thin-film cavity under a grazing angle θ . The Pd layers act as cavity mirrors and C as the guiding layer. Thephases of the reflection by the empty cavity ( r C e − iφ , blue) and from the embedded Fe nuclei in the center ( r N e iφ N , red) canbe controlled via the incidence angle θ and the energy (cid:15) , respectively. (b) The empty-cavity reflection forms a broad spectralcontinuum, whereas the isolated “bound state” nuclear response r N has a Lorentzian shape. (c) The interference of these twopaths leads to asymmetric line shapes controlled by the phase φ , which we observe experimentally. of Fe was placed in the center of the carbon layer. Far-off-resonant background photons are suppressed using ahigh-resolution monochromator for the incident light. Torecord the spectrum of the scattered light, we use a spec-trally narrow resonant absorber foil (consisting of a 6 µ mthick stainless steel foil enriched to 95% in Fe), whichwe scan in energy across the nuclear resonance with thehelp of a Doppler drive [10]. The M¨ossbauer nucleus Fefeatures a transition at (cid:126) ω = 14 . (cid:126) γ = 4 . θ around θ = θ min , where thecavity reflectance assumes a deep minimum at frequen-cies far off the nuclear resonance. Spectra covering awider range of angles are summarized in the Supplemen-tal Material [18]. Clearly, the incidence angle acts as aknob to control the spectral response from a Lorentzianshape for θ = θ min to strongly asymmetric line shapes,demonstrating the interferometric nature of our setup.To interpret these results, we follow a recently devel-oped quantum-optical framework for the description ofnuclei in x-ray waveguides, and obtain at critical cou-pling [19] for the experimentally observed reflectance | R | = (cid:12)(cid:12)(cid:12)(cid:12) √ σ e − iφ + γ SR Γ 1 (cid:15) + i (cid:12)(cid:12)(cid:12)(cid:12) . (1)Here, the first term corresponds to the interferometerpath due to the cavity alone, with empty-cavity re-sponse σ = 1 / (1 + κ / ∆ C ), with cavity loss rate κ ,and ∆ C is the detuning between cavity eigenmode fre-quency and the frequency of the probing x-ray field.The second path is represented by a complex-valued Lorentzian bound state amplitude typical of a TLS inthe second term. The dimensionless energy (cid:15) = ( ω − ω − ∆ LS ) / (Γ /
2) is modified by cooperative phenom-ena: The Lorentzian is not centered on the nuclearresonance at ω , but slightly shifted by ∆ LS due to acollective Lamb shift [10]. Moreover, its width Γ issuperradiantly broadened from the natural line width (cid:126) γ = 4 . Fe to Γ = γ + γ SR . These cooperativemodifications to the resonance position and width aregiven by ∆ LS = − (2 / | g | N ∆ C / ( κ + ∆ C ) and γ SR =(4 / | g | N κ/ ( κ + ∆ C ), respectively [19]. Here, g is thecavity-nucleus coupling constant, and N the number ofnuclei.The experimentally observed line shapes can then beunderstood by noting that the TLS response featuresthe narrow spectral width typical of M¨ossbauer reso-nances, whereas the cavity modes have orders of mag-nitude higher spectral width and therefore act as contin-uum channels. The interference of a narrow bound statewith a continuum is known to give rise to asymmetricFano resonances [20]. Relating the relative phase φ andthe Fano q parameter as φ = arg( q − i ) [21], we can indeedrewrite Eq. (1) as a Fano profile | R | = | (cid:15) + q | (cid:15) σ , q = γ SR Γ κ ∆ C + i γ Γ . (2)We find that q has an imaginary component, which cor-responds to an incoherent loss channel [22]. However,in the strongly superradiant case Γ (cid:29) γ , and the losschannels can be neglected. Then, q ≈ κ/ ∆ C , and | R | ≈ ( (cid:15) + q ) / [(1 + (cid:15) )(1 + q )], such that 0 ≤ | R | ≤ ≈ γ , we find | R | ≈ σ without any spectral signatures. It is therefore the col-lectively enhanced decay rate which enables the Fano im-plementation with full visibility of the reflectance modu-lation despite the low Q factor of the cavity. Note thatasymmetric line shapes for nuclear resonances at x-ray C o un t s [ ] ∆ θ = – µ rad ∆ θ = – µ rad −10 0 10 0.5 1 1.5 C o un t s [ ] Energy ǫ ∆ θ =29 µ rad
0 10
Energy ǫ ∆ θ =51 µ rad FIG. 2. (Color online) Fano line shape control with nuclei.The different panels show the reflected intensity measured atdifferent relative incidence angles ∆ θ = θ − θ min . Experimen-tal raw data is shown without baseline subtraction. Thereforethe intensities in the different panels cannot directly be com-pared. energies have previously been predicted or observed [23–28], though not interpreted as Fano resonances. Closeto the cavity resonance at ∆ C = 0, one can linearize∆ C ≈ δ C ( θ − θ min ), where θ min is the incidence anglewhere the reflectance of the cavity alone vanishes. Thus,the incidence angle θ can be used to control ∆ C , andthereby q and the interferometer phase φ .For a quantitative analysis of the experimental data,we fitted a generic Fano line shape to the experimentallyrecorded spectra. Each fit was repeated multiple timeswith randomly modified initial parameters to avoid bias,and the respective results are indistinguishable withintheir error bars. This procedure enables us to determinethe superradiant decay width Γ, the cooperative Lambshift ∆ LS , as well as the Fano q parameter as a functionof the incidence angle θ independent of our theoreticalmodel. As can be seen from Fig 2, we found good quan-titative agreement of model and data. Using the suchobtained superradiant enhancement γ SR and the cooper-ative Lamb shift ∆ LS , we normalized the experimentalspectra to the dimensionless energy (cid:15) for further anal-ysis (see Supplemental Material [18] for other recordedspectra).A full state tomography requires a measurement of theTLS density matrix [6]. Due to a lack of intensity cali-bration, we can determine the off-diagonal density matrixelements up to a global scaling factor in the present ex-periment. The x-ray photons are coherently scattered,preserving their energy. Selecting all detection events ofa particular photon energy (cid:15) therefore provides access toa large number of identically prepared TLS states. Re-peated measurements on the light emitted by identically − − − π − π ǫ N u c l e a r ph a s e φ N FIG. 3. (Color online) Nuclear phase reconstructed from theexperimental data. The red line shows the phase obtained viaEq. (5) as a function of the scaled energy (cid:15) . Error ranges aredepicted in light red color. The theoretically expected phaseof the Lorentzian typical for a TLS is shown in blue. prepared TLS states then enables us to determine thecharacteristics of the off-diagonal density matrix element ρ eg . Up to a global scaling factor, ρ eg ∼ σ eg ( (cid:15) ) · e iφ N , (3)where σ eg ( (cid:15) ) contains the spectral shape and φ N is thephase of the density matrix element. The shape is di-rectly obtained from the pure nuclear spectrum shown inthe top left panel of Fig. 2, where the empty-cavity re-sponse vanishes at θ = θ min . Further, our interferometricmeasurements provide a handle to determine the desiredphase of the off-diagonal density matrix elements, since ρ eg is directly proportional to the light amplitude emit-ted by the TLS (see Supplemental Material [18]). Hence,the TLS phase can be identified with the phase φ N of theoff-diagonal density matrix elements.For the reconstruction of the phase of the nuclear con-tribution, i.e. the phase of the TLS, we employed a gen-eral ansatz for the reflectance | R (∆ θ, (cid:15) ) | ∼ | r C (∆ θ ) e − iφ + r N ( (cid:15) ) e iφ N | , (4)without assumptions on the shape of the nuclear response r N except that it only depends on (cid:15) and that it vanishesat large detunings r N ( (cid:15) → ±∞ ) = 0. To extract thenuclear phase φ N , we define the experimentally accessiblequantity ξ (∆ θ, (cid:15) ) = | R (∆ θ, (cid:15) ) | − | R (0 , (cid:15) ) | − | R (∆ θ, ±∞ ) | | R (0 , (cid:15) ) || R (∆ θ, ±∞ ) | = cos ( φ + φ N ) . (5)As the relation between the incidence angle and the cav-ity phase φ (∆ θ ) = arg [ q (∆ θ ) − i ] is known from thequantum optical model, the phase of the nuclear con-tribution φ N can then be determined as function of (cid:15) via Eq. (5) by fitting the cosine to the measured ξ forall available ∆ θ values. To evaluate ξ (∆ θ, (cid:15) ) from thedata without referring to the line shape to be recon-structed, we fitted a general rational function R rat =( a + a (cid:15) + a (cid:15) ) / ( b + b (cid:15) + b (cid:15) ) to the data, normal-ized it to 0 ≤ R rat ≤
1, and evaluated it at the accordingvalues for ∆ θ and (cid:15) . Since the angle ∆ θ = 0 was not mea-sured, we obtained | R (0 , (cid:15) ) | from the mean of the resultsfor ∆ θ = ± µ rad.Results of this phase retrieval are shown in Fig. 3.It can be seen that the reconstructed phase of the off-diagonal density matrix element agrees well to the ex-pected Lorentzian shape as function of the dimension-less energy (cid:15) , in particular close to the resonance energy.The main cause of the discrepancy with respect to theexpected phase at large (cid:15) can be traced back to an un-certainty of | R (0 , (cid:15) ) | in the denominator of Eq. (5). Here,the measured values are tiny and already small absolutedeviations result in large relative errors.The phase-sensitive interferometric measurement ofthe optical response of a TLS demonstrated here opensa number of promising research directions. On the onehand, combination of the techniques developed here withmeasurements of the magnitude of the density matrix el-ements, either via detecting light intensity or conversionelectrons [29], could lead to the development of completequantum state tomography at x-ray energies. Impor-tantly, the method demonstrated here does not dependon the Lorentzian line shape, but can be used to recon-struct the phase of arbitrary nuclear line shapes. There-fore, also more advanced setups, e.g., involving multiplemagnetic hyperfine states with selectively coupled reso-nances can be addressed [11, 12]. On the other hand,the nuclear resonances are of primary significance in pre-cision spectroscopy and metrology at x-ray frequenciesdue to their narrow line width [30]. Using our approach,tiny phase changes can be extracted from the measureddata with high precision, assisted by the discovery ofthe mechanism behind the asymmetric line shapes. Con-versely, the interferometric phase can be used to manip-ulate light–matter interactions, as demonstrated by theFano line shape control which enables us to continuouslyadjust between Lorentz and Fano line shapes in the x-ray optical response. The high sensitivity of the Fanoline shape on the arrangement of scatterers allows for amultitude of applications ranging from structure deter-mination with unprecedented accuracy to precision sta-bilization of interferometers. Furthermore, Fano interfer-ences are ubiquitous features in light–matter interaction,and our phase control concept provides access to suchlarge application potential of Fano processes in the x-rayregion [31, 32]. These concepts can also be generalized to-wards active and dynamical control of spectroscopic lineshapes [21], further fueling the emerging field of x-rayquantum optics.We are grateful to T. Guryeva for support during sam- ple preparation and to F. U. Dill and K. 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Supplemental Material to: Interfero-metric phase detection at x-ray energiesvia Fano resonance control
REFLECTION SPECTRA
Spectra covering a wider range of angles are summa-rized in Fig. S1. The red shaded areas show spectralregions overlayed with artefacts from the measurementprocedure. These regions are excluded from the dataanalysis.
DERIVATION OF THE CAVITY REFLECTANCE
For the derivation of Eqs. (1) and (2) in the maintext, we followed a recently developed quantum opticalframework for the description of nuclei in x-ray waveg-uides [19]. We start with the full model including thelarge ensemble of nuclei coupled to the quantized cavityfield. As the magnetic field experienced by the nucleiis approximately zero, polarization and magnetizationeffects are insignificant, and we consider a single inci-dent field mode a in . The corresponding mode withinthe cavity is modelled by field operator a , the reflectedmode observed by the detector by operator a out . The re-flectance accessible experimentally can then be evaluatedas | R | = |(cid:104) a out (cid:105) / (cid:104) a in (cid:105)| . Making use of the large cavityloss rate κ typical in current experiments, we adiabati-cally eliminate the quantized cavity modes to derive aneffective master equation for the nuclei alone. κ is a fixedparameter determined by the material composition andthe geometric structure of the cavity. Next, motivatedby the low resonant intensity of the incident x-ray beam,we evaluate the linear response of the nuclei. By intro-ducing a suitable many-body basis, the nuclear ensembleof N atoms can be modelled as an effective two-level sys-tem with parameters modified by cooperative effects. Us-ing standard input-output formalism for cavities [33], theoutput field can be evaluated as a out = − a in + √ κ R a .Here, κ R is a coupling parameter quantifying the in- andout-coupling of cavity light. As shown in [19], the reflec-tivity of the single unmagnetized iron layer then followsas R = R C + R N with R C = − κ R κ + i ∆ C , (S1) R N = − iκ R ( κ + i ∆ C ) | g | N (∆ − ∆ LS ) + i Γ , (S2)∆ LS = − | g | N ∆ C κ + ∆ C , (S3)Γ = γ + γ SR , (S4) γ SR = 43 | g | N κκ + ∆ C . (S5) R C is the electronic reflection from the cavity alone, as itwould be observed in the absence of resonant nuclei. R N is the contribution of the nuclei, with atom-cavity cou-pling g . The Fano line shapes arise from the interferenceof these two contributions.Finally, we specialize to critical coupling κ = 2 κ R . Inthis case, if the empty cavity is driven by the externalfield on resonance (cavity detuning ∆ C = 0), then thereflection from the cavity alone is zero, R C = 0. Exper-imentally, this condition is achieved, e.g., by choosing asuitable thickness of the top layer of the waveguide. Ex-ploiting critical coupling, the cavity reflectance can berewritten without further approximations as in Eqs. (1)and (2). RELATION BETWEEN THE PHASE φ AND THEFANO q PARAMETER
As explained in the main text the Fano asymmetryparameter q can be mapped to a relative phase φ be-tween the continuum channel and the bound state am-plitude [21] via the relation φ = arg ( q − i ) . (S6)Using this, the Fano formula [Eq. (2) in the main text]can be rewritten to Eq. (1). A similar mapping was foundin Ref. [21], where absorption lines of auto-ionizing he-lium have been studied. However, it is different by afactor 2 from the relation given in Eq. (S6). The reasonfor this is that, in contrast to Ref. [21], our continuum isnot the free space vacuum, but a cavity. The cavity vac-uum also undergoes a phase shift upon a change of theincidence angle, and thus the relative phase is reduced,as we will show in the following.We consider the phases of the different channels inmore detail. To this end we specialize to the caseΓ ≈ γ SR (cid:29) γ such that q = κ/ ∆ C ∈ IR and κ = 2 κ R .From Eqs. (S1) and (S2) we obtain R C = − iq + i = − i (cid:112) q e iφ , (S7) R N = − i (cid:15) + i q − iq + i = − i (cid:15) + i e iφ . (S8)The phase of the cavity reflection changes as φ in depen-dence of q . Furthermore, from Eq. (S8) we find that thephase shift is 2 φ for the nuclear amplitude. Therefore,the relative phase between the continuum R C and thebound state R N is (2 − φ = φ . To interpret the cav-ity as an interferometer, we attribute this relative phaseto the cavity amplitude, such that the phase of the nu-clear contribution to the reflectance depends only on theenergy (cid:15) . Neglecting a global phase, we find R = R C + R N ∝ r C e − iφ + r N e iφ N , (S9)
0 1 2 ∆ θ = − µ rad C o un t s [ ] ∆ θ = − µ rad ∆ θ = − µ rad ∆ θ = − µ rad ∆ θ = − µ rad
0 1 2 ∆ θ = − µ rad C o un t s [ ] ∆ θ = − µ rad ∆ θ = 1 µ radscale × . ∆ θ = 4 µ rad ∆ θ = 9 µ rad −10 0 10 0 1 2 ∆ θ = 19 µ rad C o un t s [ ] Energy ǫ −10 0 10 ∆ θ = 29 µ rad Energy ǫ −10 0 10 ∆ θ = 39 µ rad Energy ǫ −10 0 10 ∆ θ = 49 µ rad Energy ǫ −10 0 10 ∆ θ = 51 µ rad Energy ǫ FIG. S1. Cavity reflectance for several relative incidence angles ∆ θ of the probing x-ray field. Black line shows experimentaldata, overlayed red curves are theory fits. The narrow spikes at the red shaded areas visible in all panels are artefacts of themeasurement procedure and have been excluded from the data analysis. The blue horizontal lines at ∆ θ = ± µ rad are a guideto the eye and indicate the slight asymmetry of the line shapes. with r C = 1 (cid:112) q , (S10) r N = 1 √ (cid:15) , (S11) φ N = arg (cid:18) (cid:15) + i (cid:19) . (S12) RELATION OF THE PHASE OF OFF-DIAGONALNUCLEAR DENSITY MATRIX ELEMENTSWITH THE PHASE OF THE DETECTED LIGHT
The nuclear phase φ N , which we determined in ourexperiment, can be identified with the phase φ ρ of theoff-diagonal elements of the two-level system (TLS) den-sity matrix. This can already be seen by noting thatthe nuclear contribution to the reflected field ampli-tude R N ∝ r N exp ( iφ N ) is directly proportional to off-diagonal density matrix element ρ eg = | ρ eg | exp ( iφ ρ ) asshown in Ref. [19].Here, we briefly outline this accordance using a dif-ferent method. We found in the previous sectionthat the reflected field amplitude can be written as r C exp ( − iφ ) + r N exp ( iφ N ), such that an interferenceterm ∝ cos ( φ + φ N ) is obtained in the intensity. Forsimplicity, we neglect the cavity environment, such that φ = 0, and describe the field amplitude by an alternatefirst principle approach. To this end, we consider thescattering of a plane wave field with positive frequencypart (cid:126)E (+)in = E in (cid:126)(cid:15)e i ( (cid:126)k(cid:126)r − ωt ) (S13)from a TLS. Here, E in is the field amplitude, (cid:126)(cid:15) the polar-ization vector, (cid:126)k = k ˆ k the wavevector, (cid:126)r = r ˆ r the nucleusposition, and ω the field frequency. The interaction withthe TLS leads to a total field (cid:126)E (+) = (cid:126)E (+)in + (cid:126)E (+)scat [34]with (cid:126)E (+)scat = k π(cid:15) r e i ( (cid:126)k(cid:126)r − ωt ) ˆ S − (ˆ r × (cid:126)d ) × (cid:126)r , (S14)where (cid:126)d is the TLS dipole moment assumed parallel tothe polarization, and ˆ S − the transition operator from theupper to the lower state. The total intensity registered bythe detector is the given by I ∝ (cid:104) (cid:126)E ( − ) (cid:126)E (+) (cid:105) , and decom-poses into a constant part ξ = (cid:104) (cid:126)E ( − )in (cid:126)E (+)in (cid:105) + (cid:104) (cid:126)E ( − )scat (cid:126)E (+)scat (cid:105) and an interference part (cid:104) (cid:126)E ( − )in (cid:126)E (+)scat (cid:105) + (cid:104) (cid:126)E ( − )scat (cid:126)E (+)in (cid:105) . Eval-uated in forward direction, one finds I ∝ ξ + E in d k π(cid:15) (cid:16) (cid:104) ˆ S − (cid:105) + (cid:104) ˆ S + (cid:105) (cid:17) (S15)with ˆ S + = ˆ S †− . Using (cid:104) ˆ S − (cid:105) = ρ eg = | ρ eg | e iφ ρ , (S16)where ρ eg is the off-diagonal TLS density matrix element,we obtain I ∝ ξ + E in d k π(cid:15) | ρ eg | cos( φ ρ ) . (S17)As a result, we find that evaluated in forward direction, the scattered field (cid:126)E scat has a phase relative to the inci-dent field given by the phase of ρ eg , which using Eq. (S17)can most easily be observed in the scattered light inten-sity in the interference term ∝ cos( φ ρ ). Comparing thisresult with the previously obtained interference term, wethus find the correspondence of the phases φ N = φ ρρ