Lorentz Meets Fano in Spectral Line Shapes: A Universal Phase and Its Laser Control

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Science  10 May 2013:
Vol. 340, Issue 6133, pp. 716-720
DOI: 10.1126/science.1234407

A Phase for Fano

In spectroscopy, samples placed between a steady light source and a detector are characterized based on the relative intensities of light absorbed at different frequencies. Temporal behavior—the relaxation of a photoexcited state—can be indirectly inferred from the absorption band shapes. The advent of ultrafast laser technology has enabled increasingly sophisticated measurements directly in the time domain. Ott et al. (p. 716; see the Perspective by Lin and Chu) present an analytical framework to account for asymmetric band shapes, termed Fano profiles, on the basis of a phase shift in the temporal dipole response.


Symmetric Lorentzian and asymmetric Fano line shapes are fundamental spectroscopic signatures that quantify the structural and dynamical properties of nuclei, atoms, molecules, and solids. This study introduces a universal temporal-phase formalism, mapping the Fano asymmetry parameter q to a phase ϕ of the time-dependent dipole response function. The formalism is confirmed experimentally by laser-transforming Fano absorption lines of autoionizing helium into Lorentzian lines after attosecond-pulsed excitation. We also demonstrate the inverse, the transformation of a naturally Lorentzian line into a Fano profile. A further application of this formalism uses quantum-phase control to amplify extreme-ultraviolet light resonantly interacting with He atoms. The quantum phase of excited states and its response to interactions can thus be extracted from line-shape analysis, with applications in many branches of spectroscopy.

In spectroscopic detection of electromagnetic radiation, the sample’s temporal dipole response—the time-dependent dipole moment of the system after an infinitesimally short (Dirac delta function) excitation—gives rise to line shapes observed in fluorescence or absorption. If a continuum of states is excited, this temporal dipole response corresponds also to a delta function, which is the superposition of a continuous spectrum of emitting dipoles at all frequencies. The more commonly observed exponential decay of a discrete excited state with a finite lifetime gives rise to the well-known symmetric Lorentzian line shape.

Asymmetric Fano absorption line shapes emerge when discrete excited states are coupled to a continuum of states (1, 2), which is a general phenomenon throughout nuclear (3), atomic (46), and solid-state physics (710), as well as molecular spectroscopy in chemistry (11). As a result of this discrete-continuum coupling mechanism, the temporal dipole response function is not just the sum of the exponentially decaying and deltalike dipole responses of the isolated state and continuum, respectively. The exponential dipole response is shifted in phase with respect to the Lorentzian response, which is the origin of the asymmetric line shape of the Fano resonance. By a mathematical transformation [supplementary text (12) section 1] similar to the one recently conducted for a classical Fano oscillator (13), we mapped this phase shift ϕ in the time domain into the q parameter, which was introduced by Ugo Fano (1, 2) and thereafter used to characterize and quantify the asymmetric Fano line shape. The cross section at photon energy E = ħω is given in terms of q byσFano(E)=σ0(q+ε)21+ε2,ε=EE0(Γ/2) (1)where ε denotes the reduced energy containing E0 and Γ as the position and width of the resonance, respectively, ħ denotes the reduced Planck constant, and σ0 is the cross section far away from the resonance.

In general, the absorption cross section σabs is proportional to the imaginary part of the index of refraction, which in turn is directly related to the polarizability (5) and thus to the frequency-dependent dipole response function d(E):

σabs(E)Im[d(E)] (2)

Via the Fourier transform, d(E) is connected to the time-dependent linear response d˜(t) of the medium after a deltalike excitation pulse. For a Lorentzian spectral line shape of width Γ, d˜Lorentz(t) is an exponentially decaying function of time with time constant τ = 1/Γ. For the Fano resonance, equating σabs=σFano (Eqs. 1 and 2) and using causality results in the following expression for the time-dependent response function [see supplementary text (12) section 1 for the derivation]d˜Fano(t)cqδ(t)+exp{Γ2t+i[E0t+ϕ(q)]} (3)that is the sum of a scaled Dirac delta function δ(t) (with scaling factor cq) and an exponentially decaying dipole moment with a phase given byϕ(q)=2arg(qi)q(ϕ)=i[1+exp(iϕ)1exp(iϕ)]=cot(ϕ2) (4a) and in turn (4b)For the special case of ϕ = 0 = cq, the Lorentzian dipole response function is obtained. This correspondence is illustrated in Figs. 1 and 2.

Fig. 1 Lorentz and Fano line shapes, their temporal dipole response functions, and a universal phase that governs configuration interaction and pulsed (kicklike) laser interaction with a state.

(A) Lorentzian spectral absorption line shape. (B) Lorentzian temporal dipole response function, the well-known exponential decay. (C) Fano spectral absorption line shape. (D) Fano temporal dipole response function, consisting of an instantaneous delta function (continuum absorption) and an exponentially decaying response, governed by autoionization. The asymmetric line shape is understood to be caused by the phase shift ϕFano of the long-lived dipole response as compared to the Lorentzian case. This phase shift is created by configuration interaction (CI) associated with electron-electron interaction in the case of a traditional Fano resonance. (E and F) Alternatively, a Lorentzian decaying state can be laser-dressed impulsively (kicklike) immediately after its excitation, inducing a phase shift ϕLaser governed by the product of duration of the laser pulse Δt and energy shift ΔE. The state will then decay with a phase shift of its dipole moment (F), inducing the same Fano-like absorption line shape (E).

Fig. 2

Mapping of Fano’s q (line-shape asymmetry) parameter to the temporal response-function phase ϕ. A bijective map between the two parameters is obtained in a range from (–π,π), although the function is periodic in 2π. Lorentzian line shapes are obtained for the extreme cases of ϕ→2nπ (integer n), corresponding to q→–∞ and q→+∞, respectively, whereas between these regimes Fano line shapes are obtained, with the special case of a window resonance at ϕ = (2n + 1)π, q = 0. (Insets) The absorption line shapes σ(ε) (as depicted in Fig. 1) for selected values of q(ϕ). The laser interaction creates an additional phase shift (horizontal arrows) that changes the character of the observed resonance line shape. The dots represent the situations measured in the experiment and shown in Fig. 3.

In the context of configuration interaction of electronic states as considered in Fano’s original theory, this phase ϕ can be understood as the consequence of the strong coupling of continuum and discrete quantum states. Originally this coupling was formulated in terms of the q parameter as shown in Eq. 1. The reformulation here provides a physically intuitive picture in terms of the dipole response function and unveils the universal nature of the phase ϕ. In particular, an important implication of this mapping between q and the phase ϕ is the fact that any phenomenon that shifts the dipole evolution of a system out of phase with an initial excitation can be used to modify the q parameter—for example, transforming a Lorentzian into a Fano line-shape profile (14) and vice versa—and thus to control the absorption process in general.

Since their invention, laser light sources have evolved technologically to provide few-cycle light fields in the visible and near-infrared (NIR) and delta-like attosecond pulses (15) in the extreme ultraviolet (XUV) spectral regions. Having these optical precision tools at hand, here we ask the scientific questions: Can we use lasers to quantum-simulate Fano resonances? To what extent can absorption, in general, be controlled? Answers to these questions could help to understand and to formulate new descriptions of electron-electron interaction dynamics on one hand and enable x-ray or even γ-ray laser applications on the other hand.

In the following, we start out by experimentally proving and applying the above Fano-q/dipole-phase correspondence to the paradigmatic two-electron system of helium (to which Fano’s theory was originally applied in his 1961 formulation), switching a Fano line shape into a Lorentzian line shape and then vice versa by introducing an additional phase shift with a laser field.

In the experiment, broadband attosecond-pulsed XUV light [few-pulse train; see supplementary text (12) section 3 for the experimental setup] was sent through a sample of He atoms at a density of ~100 mbar, serving as a deltalike excitation compared to the state lifetime. The transmitted spectrum (exciting XUV pulse plus coherent dipole response) was resolved by a flat-field grating spectrometer in the vicinity of the sp2n+ doubly excited state resonance series and detected with a backside-illuminated x-ray charge-coupled device camera. In addition, a collinearly propagating 7-fs NIR laser pulse of controlled intensity was directed through the sample at a fixed delay of ~5 fs after the XUV pulse. The intensity of the NIR pulse was controlled by opening and closing an iris in the beam path. When the NIR pulse was absent, the coherent dipole response produced the natural absorption spectrum; that is, the commonly known Fano line shapes were observed (Fig. 3A). With a laser field present, the coherent dipole response was modified ~5 fs after its excitation, and we observed strong modifications of the spectral features (16). At a laser intensity of 2.0 × 1012 W/cm2 ± 1.0 × 1012 W/cm2, the asymmetric Fano profiles were replaced by symmetric Lorentzian line shapes in the spectrum (Fig. 3B). Assuming a ponderomotive shift of these states in the laser field during the pulse duration [supplementary text (12) section 2], we calculated a total phase shift of (−0.35 ± 0.18)π = −1.1 rad ± 0.5 rad imprinted by the NIR pulse right after (i.e., short compared to the state lifetime) the excitation. From Eq. 4, the dipole phase ϕ of the original Fano resonance sp24+, exhibiting a q of −2.55 (6), amounts to ϕ = 0.24π (0.75 rad). The two phases add up, and the NIR pulse thus compensates the original (Fano) dipole phase of the state and shifts it back to near zero (−0.11 ± 0.16)π (Fig. 2), producing the Lorentzian absorption line (Fig. 3B). The lifetime of the considered states, and thus their coherent dipole emission times, is on the order of 100 fs or longer and not affected by the much shorter NIR pulse. In this limit, the phase modification right after excitation can be considered kicklike (impulsive). We can thus apply the above (Eq. 4) mapping between ϕ and q as a good approximation. Note that all states shown in Fig. 3A exhibit q = −2.55 ± 0.03. The fact that all of these states assume their Lorentzian profile (Fig. 3B) at the same intensity is thus a further confirmation of the validity of the introduced formalism. The lower-lying states, such as the 2s2p and sp23+, exhibit a complicated mixture of the here-discussed nonresonant (ponderomotive) and resonant coupling via the 2p2 state (16). This results in more complicated and time-dependent dipole phase and amplitude shifts beyond the applicability of the present formalism.

Fig. 3 Transforming asymmetric Fano spectral absorption lines into symmetric Lorentzian absorption peaks in doubly excited He and vice versa, from Lorentz to Fano, in singly excited He.

(A) Field-free (static) absorption spectrum of doubly excited states of the N = 2 series in He. The well-known Fano absorption profiles are observed in the transmitted spectrum of a broad-band attosecond pulse. (B) When a 7-fs laser pulse immediately follows the attosecond-pulsed (deltalike) excitation (time delayed by ~5 fs) at an intensity of 2.0 × 1012 W/cm2, the Fano absorption profiles are converted to Lorentzian profiles. (C) Field-free (static) absorption spectrum of singly excited He states below the first ionization threshold (24.6 eV): Lorentzian line shapes are visible in the attosecond-pulse absorption spectrum. (D) Absorption spectrum of the states in (C), when the attosecond pulse is again followed by the 7-fs laser pulse, at an intensity of 2.1 × 1012 W/cm2. The initially Lorentzian absorption profile has been laser-transformed into an asymmetric Fano profile. The solid black lines are the measurement results; the red lines are generated by using tabulated values in (A) from (6) and (C) from (30), whereas the red line in (B) represents Lorentzians at the resonance positions of the original Fano lines. The red line in (D) shows Fano profiles with expected laser-induced q = 1.49 (Fig. 2) at the resonance positions of the original Lorentzian resonances. [For details of the calculated profiles and the experimental data, see supplementary text (12) sections 4 and 5, respectively.] ΔOD, relative optical density as defined in supplementary text (12) section 5.

To provide additional evidence for the validity of this concept, we experimentally applied the time-domain Fano-phase framework to singly excited states in He. For these states, their natural decay proceeds fully radiatively with no interfering continuum and thus normally results in Lorentzian absorption line shapes (17). Here, we show the inverse transformation to the case above: the transformation of an initially Lorentzian atomic absorption line into a Fano resonance.

We used the same experimental configuration as above, except with the spectrometer set to resolve lower photon energies in the vicinity of 24 eV. When the laser field was absent, we recorded the well-known absorption spectrum of helium in the 1snp series just below its first ionization threshold at 24.6 eV (Fig. 3C). When the laser intensity was set to 2.1 × 1012 W/cm2 ± 1.1 × 1012 W/cm2, the symmetric Lorentzian line shapes were replaced by asymmetric Fano line shapes (Fig. 3D). Again, calculating the ponderomotive phase shift induced by the laser at this intensity results in ϕ = (−0.38 ± 0.19)π = −1.2 rad ± 0.6 rad. According to the time-domain phase formalism (Eq. 4), this implies a Fano q of +1.49 with experimental error as indicated by the red shaded area in Fig. 2, which is in agreement with the measured line shape in Fig. 3D. The line broadening in Fig. 3D is caused by a reduction in the states’ (originally nanosecond) life times resulting from single-photon ionization in the long-duration pedestals typically accompanying few-cycle laser pulses. Small additional line-shape modifications on the 1s6p and lower-lying states are due to the onset of near-resonant couplings to other bound states. Their description is currently beyond the scope of the phase-shift model introduced here. A subcycle time-delay–dependent shift of the 1s3p and 1s4p resonances has recently been observed in transient-absorption measurements (18), whereas here we focus on a global change of the dipole phase ϕ right after excitation (at a constant time delay) and relate it to the transformation of the Fano q parameter.

In addition to the change of the line profile, we can also control the sign of the absorption by varying only the phase ϕ. Thus, gain can be optically induced solely by modifying the phase of the polarization response after perturbative excitation in the absence of additional amplitude control or even population inversion of the excited state. This becomes clear by starting out from the dipole response of a Lorentzian absorption line (with an unperturbed ϕ = 0), where the laser-controlled tunable phase ϕ is again introduced to obtaind˜(t)exp[Γ2t+i(E0t+ϕ)] (5)resulting inσabs(E)Im[d(E)]=Im(ε1+ε2eiϕ+i11+ε2eiϕ)ε=EE0(Γ/2) with (6)which now can become negative when ϕ departs from its original (no laser) value of 0. This is because the negatively valued (for ε < 0) dispersive term ε1+ε2 rotates, in the complex plane, by the angle ϕ and acquires an imaginary part. The phase ϕ can thus be interpreted as a laser-controllable mixing angle between dispersion [the real part of d(E)] and absorption.

To confirm this control and transformation of absorption into gain, we acquired a spectrum spatially resolved across the vertical axis, with and without laser control (Fig. 4). The presence of the laser field enhances the EUV light intensity at the resonance positions of the singly excited states of helium, which are normally absorbing (in the absence of the laser field). The same general mechanism could, in the future, be applied to hard-x-ray or even γ-ray transitions with much longer (>100 ns, for example, 57Fe Mössbauer) life times. There, the ϕ = π phase flip could also be achieved, for example, by nanosecond-pulsed magnetic fields (Zeeman shift) or a physical displacement of a solid-state absorber (19) right after excitation by a femtosecond free-electron laser (FEL) or a subnanosecond synchrotron pulse.

Fig. 4 Laser-controlled amplification of resonant light in the EUV.

(A) Spectrum of transmitted EUV light without control laser: The helium resonant absorption lines can be observed as local minima in an otherwise smoothly varying spectrum centered at a vertical position of 0 mm. (B) Spectrum of transmitted and amplified EUV light in the presence of the control laser. Amplification can be observed exactly at the He resonance positions corresponding to absorption in (A). It is also observed slightly off-axis, which is likely due to imperfect angular alignment of the optical control laser with respect to the EUV beam.

Having this experimental confirmation, Eq. 6 thus allows a general interpretation of effects such as electromagnetically induced transparency (EIT) (20, 21){σabs(E)Im[d(E)]=0} and lasing without inversion (22, 23) [σabs(E)<0] in a unified intuitive picture. It also connects to general dispersion control (14, 23) with short-pulsed fields. The formalism developed here is universal and may also help to provide intuitive physical pictures for the case of nonresonant amplification of light recently experimentally observed in helium atoms (24). The Fano-phase formalism provides a time-domain picture for existing theory (25) and experiments on the laser coupling and control of Fano resonances (26), including recent work using attosecond pulses (27, 28). In a complementary theoretical work, modifications of Fano profiles in transient-absorption spectra for the case of two laser-coupled Fano resonances (including coherent population transfer) have recently been observed and analytically described (29). Going beyond the approximation of the kicklike phase shift near time zero as discussed here could provide enhanced physical understanding of such general time-delay–dependent transient-absorption measurements.

The existence of a direct correspondence between Fano’s q parameter and the dipole phase of an excited state has thus been proved. Because the latter is susceptible to laser fields, Fano absorption profiles can be induced in the absence of effects such as autoionization. This is important because the change of the absorption profile is in turn a measure of the induced phase shift of a complex quantum-mechanical state amplitude in a laser field, with numerous applications in spectroscopy and quantum-state holography. The now well-understood phase-to-q correspondence allows mapping of the coupling of these states to laser fields or other interactions, providing information especially when the coupling mechanisms are more complex than just a ponderomotive coupling, which was considered here to introduce the principle. These additional couplings are already expected in helium atoms for the more strongly bound quantum states, or for states in which both electrons are excited to the same or similar quantum numbers, for which electron-electron interaction effects play a major role while interacting with the laser. There is no reason why the mechanism should not be applicable to molecules or excitons in condensed phase or mesoscopic materials. The Fano-phase formalism also provides an intuitive link between quantum (configuration interaction, energy shifts of quantum states) and classical phenomena (classical light fields, phase shifts of oscillators), which could possibly spawn novel quantum-classical hybrid pictures of multielectron dynamics.

Supplementary Materials

Supplementary Text

Figs. S1 to S3

Tables S1 and S2

References (3034)

References and Notes

  1. Materials and methods are available as supplementary materials on Science Online.
  2. Acknowledgments: We acknowledge helpful discussions with R. Moshammer and J. Ullrich. Financial support from the Max-Planck Research Group program and the Deutsche Forschungsgemeinschaft (grant no. PF 790/1-1) is gratefully acknowledged. The work of C.H.G. is supported in part by the U.S. Department of Energy, Office of Science. The authors declare no competing financial interest.

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