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arXiv:1509.04653 · Physical Review A

Nonlinear Fano interferences in open quantum systems: An exactly solvable model

Daniel Finkelstein‐Shapiro, Mónica Calatayud, Osman Atabek, Vladimiro Mújica, Arne Keller

Published Jun 16, 2016• 12 citations
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Abstract

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.

DOI
10.1103/physreva.93.063414

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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

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We discuss the propagation of a short laser pulse in an autoionizing medium with double Fano structure. By solving the coupled equations for atoms and fields simultaneously, we can consider the influence of double Fano structure on the pulse laser in the transmission process. Full Text: PDF References A. Raczyński, M. Rzepecka, J. Zaremba, and S. Zielińska-Kaniasty, "Electromagnetically induced transparency and light slowdown for Λ-like systems with a structured continuum", Opt. Commun. 266, 552 (2006). CrossRef T. Bui Dinh, W. Leoński, V. Cao Long and J. Perina Jr, "Electromagnetically induced transparency for a double Fano-profile system", Eur. Phys. J. D, 68 (2014) CrossRef T. Bui Dinh, W. Leoński, V. Cao Long and J. Perina Jr, "Electromagnetically induced transparency in systems with degenerate autoionizing levels in Λ-configuration", Optica Applicata 43, 471 (2013). CrossRef Cao Long Van, Structured continuum in various optical phenomena, in Physics and applications I: Quantum Optics" Ed. Wiesław Leoński, University of Zielona Góra Press, p.195 (2012). DirectLink M. Shapiro, "Electromagnetically induced transparency with structured multicontinua", Phys.Rev.A 75, 013424 (2007). CrossRef E. Paspalakis, P.L. Knight, "Population transfer via an autoionizing state with temporally delayed chirped laser pulses", J. Phys. B 31, 2753 (1998). CrossRef S.Y. Zhu and E.E. Fill, "Nonlinear theory of lasers without inversion with two upper lasing levels", Phys. Rev. A 42, 5684 (1990). CrossRef S. E. Harris, "Lasers without inversion: Interference of lifetime-broadened resonances", Phys. Rev. A 62, 1033 (1989). CrossRef E. Paspalakis, N. J. Kylstra, and P. L. Knight, "Propagation dynamics in an autoionization medium", Phys. Rev. A 60, 642 (1999). CrossRef W. Leoński, R. Tanaś and S. Kielich, "Laser-induced autoionization from a double Fano system", J. Opt. Am. B 4, 72 (1987). CrossRef

Photonics Letters of Poland•Sep 30, 2016•0 citations
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×\else\texttimes\fi44\ifmmode\times\else\texttimes\fi{}44\ifmmode×\else\texttimes\fi4 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.
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
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.
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 resonance is a well-celebrated weak-coupling phenomenon whereby a small interaction between a discrete and a continuum of excited states leads to an interference between the respective two transition amplitudes, resulting in an asymmetric line shape in the excitation spectra. We present the strong-coupling phase of the Fano's model where the interaction is localized in a region $\ensuremath{\xi}$ that is smaller than the collective coupling between the discrete state and the continuum. We explore the behavior of the excitation probability $\ensuremath{\sigma}$ as a function of $\ensuremath{\xi}$, the ratio of the bare transition amplitudes, and the energy of the discrete state. In the strong-coupling phase, the spectral function of the discrete state splits up and produces up to three complete destructive interferences or zeros of $\ensuremath{\sigma}$ along with ultrasharp features. We also find that the discrete state does not have to lie within the continuum to produce the well-known weak-coupling profiles of $\ensuremath{\sigma}$ as long as the collective coupling is strong enough. In addition, we consider a broadened state instead of a continuum and present a method that exactly treats such cases by explicitly including the source of the broadening---a weak coupling to the continuum of environment states---in the model.
We discuss the propagation of a short laser pulse in an autoionizing medium with double Fano structure. By solving the coupled equations for atoms and fields simultaneously, we can consider the influence of double Fano structure on the pulse laser in the transmission process. Full Text: PDF References A. Raczyński, M. Rzepecka, J. Zaremba, and S. Zielińska-Kaniasty, "Electromagnetically induced transparency and light slowdown for Λ-like systems with a structured continuum", Opt. Commun. 266, 552 (2006). CrossRef T. Bui Dinh, W. Leoński, V. Cao Long and J. Perina Jr, "Electromagnetically induced transparency for a double Fano-profile system", Eur. Phys. J. D, 68 (2014) CrossRef T. Bui Dinh, W. Leoński, V. Cao Long and J. Perina Jr, "Electromagnetically induced transparency in systems with degenerate autoionizing levels in Λ-configuration", Optica Applicata 43, 471 (2013). CrossRef Cao Long Van, Structured continuum in various optical phenomena, in Physics and applications I: Quantum Optics" Ed. Wiesław Leoński, University of Zielona Góra Press, p.195 (2012). DirectLink M. Shapiro, "Electromagnetically induced transparency with structured multicontinua", Phys.Rev.A 75, 013424 (2007). CrossRef E. Paspalakis, P.L. Knight, "Population transfer via an autoionizing state with temporally delayed chirped laser pulses", J. Phys. B 31, 2753 (1998). CrossRef S.Y. Zhu and E.E. Fill, "Nonlinear theory of lasers without inversion with two upper lasing levels", Phys. Rev. A 42, 5684 (1990). CrossRef S. E. Harris, "Lasers without inversion: Interference of lifetime-broadened resonances", Phys. Rev. A 62, 1033 (1989). CrossRef E. Paspalakis, N. J. Kylstra, and P. L. Knight, "Propagation dynamics in an autoionization medium", Phys. Rev. A 60, 642 (1999). CrossRef W. Leoński, R. Tanaś and S. Kielich, "Laser-induced autoionization from a double Fano system", J. Opt. Am. B 4, 72 (1987). CrossRef