10 FIG. 6: (color online) Wave packet reflection in a semi-infinite lattice with an amplifying impurity site for κ 0 = κ$_{a}$ = 1, Re( E$_{a}$ ) = 0 and Im( E$_{a}$ ) = 1. The initial wave packet is Gaussian- shaped, with peak amplitude | c$_{n}$ (0) | = 1 at n = n 0 = 30, width Δ n = 12 and momentum k = 3 π/ 10 in (a), and k = π/ 2 in (b) (corresponding to the spectral singularity energy E 0 = 0). In (b)the incident wave packet, indicated by an arrow, is not visible owing to the large amplification of the reflected wave. neously the two conditions Physically, the condition R → ∞ implies the existence of an outgoing wave in the lattice that is sustained by the amplifying complex potential at the impurity site. Such a divergence of R ( k ) is the signature of a spectral singularity of H . In fact, taking into account that the energy of incident/reflected waves is E 0 = − 2 κ$_{0}$ cos k$_{0}$ , it can be easily shown that Eq.(73) is equivalent to Eq.(55), whereas Eq.(72) is equivalent to Eq.(54) with the upper (positive) sign on the right hand side. Therefore, for an amplifying impurity site [Im( E$_{a}$ ) > 0], the condition R → ∞ is equivalent to the appearance of a spectral singularity. This equivalence extends, to our scattering problem on a truncated lattice, the general result shown by Mostafazadeh [9], suggesting to interpret spectral sin- gularities of a non-Hermitian Hamiltonian as resonances with vanishing width. However, for an absorbing poten- tial energy at the impurity site, i.e. for Im( E$_{a}$ ) < 0, our lattice model indicates that the appearance of spectral singularities has a different physical interpretation. In fact, for Im( E$_{a}$ ) < 0 the reflection probability R ( k ) is bounded from above and smaller than one, which pre- vents R ( k ) to diverge. However, in this case R ( k ) can vanish at wave numbers k = k$_{0}$ (with 0 ≤ k$_{0}$ ≤ π ) that satisfy simultaneously the two conditions [see Eq.(71)] κ 2 $_{a}$sin k$_{0}$ = κ$_{0}$ Im( E$_{a}$ ) (72) ( κ 2 a − 2 κ 2 $_{0}$) cos k$_{0}$ = κ$_{o}$ Re( E$_{a}$ ) . (73) κ 2 $_{a}$sin k$_{0}$ = − κ$_{0}$ Im( E$_{a}$ ) (74) ( κ 2 a − 2 κ 2 $_{0}$) cos k$_{0}$ = κ$_{o}$ Re( E$_{a}$ ) . (75) FIG. 7: (color online) Same as Fig.6, but for an absorbing impurity site Im( E$_{a}$ ) = − 1 (other parameters as in Fig.6). siteGLYPH<.notdef>number n (b) (a) 10 30 50 70 90 10 0 20 30 40 10 30 50 70 90 10 0 20 30 40 siteGLYPH<.notdef>number n t siteGLYPH<.notdef>number n (b) (a) 10 30 50 70 90 10 0 20 30 40 10 30 50 70 90 10 0 20 30 40 siteGLYPH<.notdef>number n t incidentGLYPH<.notdef>waveGLYPH<.notdef>packet Gaussian distribution of site occupation amplitudes, i.e. c$_{n}$ (0) = exp[ − ( n − n$_{0}$ ) $^{2}$/ Δ n 2 − ikn ], where Δ n is the wave packet width, k the mean wave packet momen- tum, and n$_{0}$ ≫ Δ n the wave packet center of mass. The large (diverging) amplification of the reflected wave packet in Fig.6(b), and the almost absence of wave packet reflection in Fig.7(b), are clearly visible when the energy E = − 2 κ$_{0}$ cos k of the incoming wave packet attains the spectral singularity point E 0 = 0. The interplay between spectral singularities and de- cay dynamics, discussed in Sec.II.D, is exemplified in Figs.8 and 9. The system of Eqs.(65-67) has been nu- merically integrated with the initial condition c$_{a}$ (0) = 1 and c$_{n}$ (0) = 1. Note that, in the photonic realization of the semi-infinite lattice model of Ref.[37], such an ini- tial condition simply corresponds to initial excitation of the boundary waveguide. The behavior of the site oc- cupation probability P ( t ) = | c$_{a}$ ( t ) | 2 for an amplifying and for an absorbing impurity site is depicted in Figs.8 and 9, respectively. Note that, according to the gen- Figures 6 and 7 show two examples of wave packet reflection from the lattice boundary for an amplifying (Fig.6) and an absorbing (Fig.7) impurity site. The fig- ures show a snapshot of | c$_{n}$ ( t ) | as obtained by numer- ical analysis of Eqs.(65-67) assuming at t = 0 a broad Note that Eq.(75) is equivalent to Eq.(55), whereas Eq.(74) is equivalent to Eq.(54) with the lower (nega- tive) sign on the right hand side. Therefore, for an ab- sorbing impurity site [Im( E$_{a}$ ) < 0], the appearance of a spectral singularity is equivalent to the vanishing of the reflection probability R : an ingoing plane wave with wave number k$_{0}$ is fully absorbed by the impurity site at the lattice edge. An an example, Fig.4 shows the behavior of R ( k ) for an amplifying [Fig.4(a)] and for an absorbing [Fig.4(b)] impurity site for Re( E$_{a}$ ) = 0 and κ$_{a}$/κ$_{0}$ = 1. The different curves in the figures refer to different values of Im( E$_{a}$ ) /κ$_{0}$ . Note that, at the value of Im( E$_{a}$ ) /κ$_{0}$ cor- responding to the appearance of the spectral singularity, a divergence and a zero in the R ( k ) curve are observed in Figs.4(a) and 4(b), respectively. Figure 5 shows the behavior of R ( k ) as in Fig.4, but for Re( E$_{a}$ ) = 0 and κ$_{a}$/κ$_{0}$ = √$_{2. In this case there are two spectral singu-}$ larities at energies given by Eq.(57), which explain the two peaks [Fig.5(a)] or dips [Fig.5(b)] in the reflectance curve R ( k ). By increasing | Im( E$_{a}$ ) | /κ$_{0}$ , the two peaks (or dips) get closer each other, until they coalesce at | Im( E$_{a}$ ) | /κ$_{0}$ = 2 (curve 3 [50]).