5 0 10 20 30 40 50 0 2 4 6 8 t GLYPH<c121> m (t) 2 1 3 4 FIG. 3: (color online) Saturation of secular Bragg scattering for a Gaussian wave packet that excites the complex lattice of Fig.2 at normal incidence. The curves show the evolution of the maximum wave packet amplitude ψ m ( t ) versus t for a few values of input wave packet spot size w . Curve 1: w = 40; curve 2: w = 80; curve 3: w = 150; curve 4: w = 300. FIG. 4: (color online) (a) Evolution of a Gaussian wave packet [modulus of ψ ( x, t )] in the complex lattice (5) for V 0 = 0 . 2, a = 1, w = 80, and below the PT symmetry breaking point ( λ = 0 . 9). In (b), the corresponding evolution of the maxi- mum wave packet amplitude ψ m ( t ) is shown. packet input spot size w . Note that in the early stage of the dynamics the peak amplitude linearly increases with t , as expected for a plane wave according to Theorem 3. The linear growth then saturates to a steady state value and, as Fig.3 clearly shows, the saturation process occurs earlier for wave packets with smaller input size w . It should be pointed out that saturation of ψ$_{m}$ ( t ) to a steady-state value, as predicted by Eq.(7) and confirmed by numerical simulations depicted in Figs.2 and 3, is indeed a signature of spectral singularities of the underlying Hamiltonian that arise at λ = λ$_{c}$ = 1. This is clearly shown in Figs.4 and 5, where a typical wave packet evolution is reported for the complex crystal defined by Eq.(5) either below ( λ = 0 . 9 λ$_{c}$ = 0 . 9, | ξ | > 1. Equation (7) shows that, owing to wave packet broadening in momentum space, the secular growth with t of the diffracted beam saturates to the value ∼ ( π/k$_{B}$ ) | F ( − k$_{B}$/ 2) | while the beam assumes a square shape whose width w$_{s}$ spreads in space with a con- stant speed v = dw$_{s}$/dt = k$_{B}$ . Note that v equals the translation speed of the freely-diffracting wave packet ψ$_{0}$ . This behavior is confirmed by direct numerical simulations of Eq.(1), as shown in Fig.2. The figure depicts the evolution of | ψ ( x, t ) | for a Gaussian wave packet ψ ( x, 0) = exp[ − ( x/w ) 2 − ik$_{B}$x/ 2] with spectrum F ( k ) = [ w/ $_{(2}$√$_{π}$$_{)] exp[}$ − ( k + k$_{B}$/ 2) $^{2}$w$^{2}$/ 4] that excites the crystal at t = 0. The evolution of the freely diffract- ing wave packet | ψ$_{0}$ | and of the asymptotic behavior of | ψ$_{1}$ | predicted by Eq.(6) are also shown for comparison. Note the formation of interference fringes on the left side FIG. 2: (color online) Evolution of a Gaussian wave packet in the complex lattice V ( x ) = V 0 exp( ik B x ) for V 0 = 0 . 2, a = 1 and w = 80. In (a) a snapshot of | ψ ( x, t ) | is shown, whereas in (b) the profiles of | ψ ( x, t ) | at a few values of t are reported. In (b), the dashed lines correspond to the wave packet evolution in absence of the lattice, whereas the dotted curves correspond to | ψ 1 ( x, t ) | as predicted by Eq.(7). t=10 t=30 t=90 0 3 -800 -400 0 400 800 0 3 0 3 0 3 t=60 -800 800 0 0 90 30 60 t x x (a) (b) w$_{s}$ -1200 1200 0 t x (a) (b) t GLYPH<c121> m (t) 0 160 0 50 100 150 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 is depicted for a few decreasing values of the wave ψ$_{m}$ ( t ) = max$_{x}$ | ψ ( x, t ) | , (8) of the wave packet, which arise from the interference of ψ$_{0}$ and ψ$_{1}$ . The spectral-broadening-induced saturation of the secular growth of the wave packet is clearly shown in Fig.3, where the behavior of the maximum amplitude ψ$_{m}$ ( t ) of the wave packet versus t , defined by where ψ$_{n}$ ( x, t ) = ∫ dkF ( k ) c$_{n}$ ( k, t ) exp[ i ( k + nk$_{B}$ ) x ] with c$_{0}$ ( k, t ) = exp( − ik $^{2}$t ) and c$_{n}$ ( k, t ) = − i ∫ t 0 dξc$_{n}$$_{−}$$_{1}$ ( k, ξ ) exp[ i ( k + nk$_{B}$ ) $^{2}$( ξ − t )] for n ≥ 1. For a shallow lattice, i.e. for | V$_{0}$ | ≪ 1, we can limit to consider the first two terms on the right hand side of Eq.(6). The leading term, ψ$_{0}$ ( x, t ) = ∫ dkF ( k ) exp( − ik $^{2}$t + ikx ), is simply the freely-diffracting wave packet that one would observe in the absence of the crystal and that propagates with a constant speed v = dx/dt = k$_{B}$ . The expression of ψ$_{1}$ ( x, t ) is more involved, however its asymptotic behavior for t ≫ π/ ( k$_{B}$ Δ k ) assumes a rather simple and physically interesting form, namely ψ ( x, t ) = ψ$_{0}$ ( x, t ) + V$_{0}$ψ$_{1}$ ( x, t ) + V 2 0 ψ$_{2}$ ( x, t ) + ... (6) one-side diffraction at various orders, namely ψ$_{1}$ ∼ − iπ k$_{B}$ F ($_{−}$ k$_{B}$ 2 ) exp ( i k$_{B}$x 2 − i k 2 $_{B}$t 4 ) Φ ( x k$_{B}$t ) (7) where Φ( ξ ) = 1 for | ξ | < 1 and Φ( ξ ) = 0 for