Fano models---consisting of a Hamiltonian with a discrete-continuous spectrum---are one of the basic toy models in spectroscopy. They have been successful in explaining the line shape of experiments in atomic physics and condensed matter. These models, however, have largely been beyond the scope of dissipative dynamics, with only a handful of works considering the effect of a thermal bath. Yet in nanostructures and condensed-matter systems, dissipation strongly modulates the dynamics. We present an overview of the theory of Fano interferences coupled to a thermal bath and compare them to the scattering formalism. We provide the solution to any discrete-continuous Hamiltonian structure within the wideband approximation coupled to a Markovian bath. In doing so, we update the toy models that have been available for unitary evolution since the 1960s. We find that the Fano line shape is preserved as long as we allow a rescaling of the parameters, and an additional Lorentzian contribution that reflects the destruction of the interference by dephasings. The universality of the line shape can be traced back to specific properties of the effective Liouvillian.
Fano models---consisting of a Hamiltonian with a discrete-continuous spectrum---are one of the basic toy models in spectroscopy. They have been successful in explaining the line shape of experiments in atomic physics and condensed matter. These models, however, have largely been beyond the scope of dissipative dynamics, with only a handful of works considering the effect of a thermal bath. Yet in nanostructures and condensed-matter systems, dissipation strongly modulates the dynamics. We present an overview of the theory of Fano interferences coupled to a thermal bath and compare them to the scattering formalism. We provide the solution to any discrete-continuous Hamiltonian structure within the wideband approximation coupled to a Markovian bath. In doing so, we update the toy models that have been available for unitary evolution since the 1960s. We find that the Fano line shape is preserved as long as we allow a rescaling of the parameters, and an additional Lorentzian contribution that reflects the destruction of the interference by dephasings. The universality of the line shape can be traced back to specific properties of the effective Liouvillian.
The scattering amplitude from a set of discrete states coupled to a continuum became known as the Fano profile, characteristic for its asymmetric line shape and originally investigated in the context of photoionization. The generality of the model and the proliferation of engineered nanostructures with confined states gives immense success to the Fano line shape, which is invoked whenever an asymmetric line shape is encountered. However, many of these systems do not conform to the initial model worked out by Fano in that (i) they are subject to dissipative processes and (ii) the observables are not entirely analogous to the ones measured in the original photoionization experiments. In this Letter, we work out the full optical response of a Fano model with dissipation. We find that the exact result for the excited population, Raman, Rayleigh, and fluorescence emission is a modified Fano profile where the typical line shape has an additional Lorentzian contribution. Expressions to extract model parameters from a set of relevant observables are given.
We obtain an explicit solution for the stationary state populations of a dissipative Fano model, where a discrete excited state is coupled to a continumm set of states; both excited set of states are reachable by photo-excitation from the ground state. The dissipative dynamic is described by a Liouville equation in Lindblad form and the field intensity can take arbitrary values. We show that the continuum states population as a function of laser frequency can always be expressed as a Fano profile plus a Lorentzian function with effective parameters whose explicit expressions are given in the case of a closed system coupled to a bath as well as for the original Fano scattering framework. Although the solution is intricate, it can be elegantly expressed as a linear transformation of the kernel of a 4 × 4 matrix which has the meaning of an effective Liouvillian. We unveil key notable processes related to the optical nonlinearity and which had not been reported to date: electromagnetic induced-transparency, population inversions, power narrowing and broadening, as well as an effective reduction of the Fano asymmetry parameter. The original Fano model was introduced by U. Fano in 1961 [1] to explain the asymmetric profile that was obtained when monitoring the photo-current as a function of the laser frequency used to ionize a gaz of Helium-like atoms. In parallel, Friederichs [2] developed the mathematical formalism to describe the essential feature of the Fano model: a discrete state coupled to a continuum set of states; both sets of states being reachable by photo-excitation from the ground state. The resulting photo-current, which is proportional to the population of the continuum set of states, as a function of the excitation laser frequency ω L is known as the Beutler-Fano or Fano profile: f (q) = (q + 2 /(2 + 1), where q is the ratio of the transition dipole moment of the ground-discrete and ground-continuum transitions, and = (ω L − ω e)/γ where e is the energy of the discrete state relative to the ground state, ω L is te radiation field frequency, and = nπV 2 is the linewidth of the excited state, induced by its coupling (per unit of energy) nV 2 to the continuum set of states, n being the density of states. Two extensions of the original model have been considered: inclusion of incoherent relaxation and dephasing processes and high field intensities. The motivation to include incoherent processes was first to describe the pressure broadening [3] due to elastic collisions, laser phase fluctuations [4] and spontaneous emission [5, 6]. Nowadays , Fano profiles in nanoscale structures are standard [7, 8], for example in plasmonic nanostructures [9–11], quantum dots, decorated nanoparticles [12] and spin filters [13]. The coherent coupling with the incident light induces large Rabi frequencies which in turn compete with the relaxation rates in order to modify the stationary state. The ability to predict the lineshape, and in general to investigate non linear optical phenomena in
We obtain an explicit solution for the stationary-state populations of a dissipative Fano model, where a discrete excited state is coupled to a continuum set of states; both excited sets of states are reachable by photoexcitation from the ground state. The dissipative dynamic is described by a Liouville equation in Lindblad form and the field intensity can take arbitrary values within the model. We show that the population of the continuum states as a function of laser frequency can always be expressed as a Fano profile plus a Lorentzian function with effective parameters whose explicit expressions are given in the case of a closed system coupled to a bath as well as for the original Fano scattering framework. Although the solution is intricate, it can be elegantly expressed as a linear transformation of the kernel of a $4\ifmmode\times\else\texttimes\fi{}4$ matrix which has the meaning of an effective Liouvillian. We unveil key notable processes related to the optical nonlinearity and which had not been reported to date: electromagnetic-induced transparency, population inversions, power narrowing and broadening, as well as an effective reduction of the Fano asymmetry parameter.
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.
We study the Fano effect in dissipative cavity quantum electrodynamics (QED), which originates from the interference between the emitter's direct radiation and that mediated by a cavity mode. Starting from a two-level system coupled to a structured reservoir, we show that a quantum master equation previously derived within the Born-Markov approximation can be rederived by introducing a single auxiliary mode through the pseudomode approach. We identify the corresponding spectral function of the system--environment interaction and show that it consists of a constant contribution and a non-Lorentzian contribution, whose interplay gives rise to a spectral profile of the Fano form. The constant contribution represents a Markovian background and is essential for obtaining a Lindblad master equation. Furthermore, by applying Fano diagonalization to an atom--cavity system coupled to common and independent reservoirs, we independently derive the same spectral function and clarify its physical origin. Our results provide a unified description of the Fano effect in single-mode cavity QED systems and reveal its non-Markovian origin encoded in the spectral function of the structured reservoir.
The Fano line shape arises from the interference of two excitation pathways to reach a continuum. Its generality has resulted in a tremendous success in explaining the line shapes of many one-dimensional spectroscopies---absorption, emission, scattering, conductance, photofragmentation---applied to very varied systems---atoms, molecules, semiconductors, and metals. Unraveling a spectroscopy into a second dimension reveals the relationship between states in addition to decongesting the spectra. Femtosecond-resolved two-dimensional electronic spectroscopy (2DES) is a four-wave mixing technique that measures the time evolution of the populations and coherences of excited states. It has been applied extensively to the dynamics of photosynthetic units, and more recently to materials with extended band structures. In this paper, we solve the full time-dependent third-order response, measured in 2DES, of a Fano model and give the system parameters that become accessible.