Decay of a nonlinear impurity in a structured continuum from a nonlinear Fano-Anderson model

Stefano Longhi

Phys. Rev. B 75, 184306 – Published 18 May 2007

Abstract

The decay dynamics of a nonlinear impurity mode embedded in a linear structured continuum is theoretically investigated in the framework of a nonlinear Fano-Anderson model. A gradient flow dynamics for the survival probability is derived in the Van Hove $(λ^{2} t)$ limit by a multiple-scale asymptotic analysis, and the role of nonlinearity on the decay law is discussed. In particular, it is shown that the existence of bound states embedded in the continuum acts as transient trapping states which slow down the decay. The dynamical behavior predicted in the $λ^{2} t$ limit is studied in detail for a simple tight-binding one-dimensional lattice model, which may describe electron or photon transport in condensed matter or photonic systems. Numerical simulations of the underlying equations confirm, in particular, the trapping effect in the decay process due to bound states embedded in the continuum.

Received 4 January 2007

DOI:https://doi.org/10.1103/PhysRevB.75.184306

Authors & Affiliations

Stefano Longhi

Dipartimento di Fisica and Istituto di Fotonica e Nanotecnologie del CNR, Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133 Milan, Italy

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Issue

Vol. 75, Iss. 18 — 1 May 2007

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Images

Figure 1
(Color online) Schematic behavior of a reservoir spectral function $G (ω)$ for (a) a structured reservoir showing pointlike gaps in the band and (b) corresponding behavior of the potential $V (P)$ governing the gradient flow decay in the $λ^{2} t$ limit. In (a), the horizontal thick solid line indicates the frequency range $(ω_{a} - γ, ω_{a})$ spanned by the nonlinear instantaneous frequency $ω_{a}^{NL} = ω_{a} - γ P$ of the impurity mode during the decay process, whereas $Ω_{1}, Ω_{2}, \dots$ , are the zeros of $G (ω)$ embedded in this range. In (b), $P_{c} = (ω_{a} - ω_{1}) ∕ γ$ is the critical value of $P$ above which $V (P) = const$ , $P_{1}, P_{2}, \dots$ , are the horizontal flex points of the potential associated with the roots $Ω_{1}, Ω_{2}, \dots$ , of $G (ω)$ , and the shaded area is a forbidden region of the motion. For $P_{c} > 1$ as in the figure, starting from the initial condition $P (0) = 1$ , the globally stable attractor of the flow is $P = 0$ , whereas the fixed points at the flexes $P_{1}, P_{2}, \dots$ , represent unstable trapping states which slow down the decay of the survival probability $P (t)$ .Reuse & Permissions
Figure 2
(Color online) Schematic of a nonlinear site $∣ a ⟩$ side coupled to a semi-infinite tight-binding lattice. $κ_{0}$ is the hopping amplitude between adjacent sites in the lattice, whereas $κ_{a}$ is the hopping amplitude between the localized state $∣ a ⟩$ and the site $∣ n_{0} ⟩$ of the lattice.Reuse & Permissions
Figure 3
(Color online) Decay dynamics of the nonlinear site $∣ a ⟩$ in the lattice model of Fig. 2 for $n_{0} = 1$ and for $ω_{a} = 1.2$ , $κ_{0} = 1$ , $κ_{a} = 0.2$ , and $γ = 2$ . (a) Behavior of the reservoir spectral function $G$ , as given by Eq. (47). (b) Behavior of the potential $V (P)$ [Eq. (36)]. (c) Behavior of the survival probability $P (t) = {∣ c_{a} (t) ∣}^{2}$ versus time as computed by numerical simulations of Eqs. (39, 40, 41) (solid curve), and corresponding behavior as predicted by the gradient flow [Eq. (35)] (dotted line, almost overlapped with the solid curve). The inset shows the behavior of the instantaneous decay rate $α (t) = - [\ln P (t)] ∕ t$ versus time. (d) Grayscale image showing the temporal evolution of the occupation probabilities ${∣ c_{n} (t) ∣}^{2}$ at the lattice sites.Reuse & Permissions
Figure 4
(Color online) Decay dynamics of the survival probability $P (t) = {∣ c_{a} (t) ∣}^{2}$ of state $∣ a ⟩$ in the linear regime. Parameter values are the same as in Fig. 3 except for $γ = 0$ . The inset shows the behavior of the instantaneous decay rate $α (t) = - [\ln P (t)] ∕ t$ versus time. Solid and dotted curves refer to the results as obtained by numerical simulations of Eqs. (39, 40, 41) and by the gradient flow [Eq. (35)], respectively.Reuse & Permissions
Figure 5
(Color online) Same as Fig. 3 but for $n_{0} = 8$ . For the sake of clearness, in the inset in (c), the horizontal time scale is logarithmic. The arrows in the figures correspond to slowing down of the decay associated with the three flex points (1), (2), and (3) of the potential $V$ .Reuse & Permissions

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