3 sites at z = 0 [2 w = 5 a , Fig.2(a)], and a narrow Gaussian beam that excites at normal incidence a single well of the lattice [2 w = 2 a/ 3, Fig.2(b)]. In both cases, the lowest band of the array is mainly excited, as discussed e.g. in [8]. After application of the sinusoidal force [Figs.2(c-f)], suppression of beam diffraction and self-imaging effects are clearly observed when period and amplitude of forc- ing satisfy the condition Γ ≡ F 0 a/ ( λω ) = 2 . 405 (first zero of Bessel J 0 function), as shown in Figs.2(e) and (f). As α is increased to reach and cross the PT symmetry- breaking point α$_{c}$ , gap narrowing till band merging, as- sociated to the appearance of pairs of complex-conjugate eigenvalues, is observed [13]. For instance, for the po- tential defined by Eq.(2), at the transition point the band diagram is given by the free-particle energy dis- persion curve E = λ $^{2}$κ$^{2}$/ (2 n$_{s}$ ), periodically folded in- side the first Brillouin zone [13] (see the dotted curve in Fig.1). In this case, as previously noticed for BOs [17], wave packet transport is deeply modified and can not be described by means of the canonical model (3) in- troduced in the previous section. From a physical view- point, this is related to the highly non-reciprocal behav- ior of Bragg scattering in the crystal and violation of Friedel’s law of Bragg diffraction for crystal inversion [12, 14]. As in Ref. [17], we limit here to consider the dynamical behavior of a broad wave packet in the complex lattice defined by Eq.(2) at α = α$_{c}$ = 1, i.e. V ( x ) = V 0 exp( ik$_{B}$x ), which enables a rather simple ana- lytical and physical analysis. As opposed to Ref. [17], we assume here an ac-like force, namely F ( z ) = F 0 cos( ωz ). Figure 3 shows a few typical examples of wave packet dynamics as obtained by a numerical analysis of Eq.(1) when the lattice is excited at z = 0 with a broad Gaus- sian beam ψ ( x, 0) = exp( − x $^{2}$/w$^{2}$) for a fixed value of the ac modulation period Λ = 2 π/ω and for increas- ing values of the amplitude F 0 . For a small amplitude F 0 , it turns out that the wave packet propagates as if the lattice were absent [Fig.3(a)], following an oscillatory path as predicted by the semiclassical analysis of Eq.(1) with V = 0. Owing to the external force, the mean wave packet momentum varies periodically according to λk ( z ) = ∫ z 0 dξF ( ξ ) = ( F 0 /ω ) sin( ωz ) . As the forcing F 0 reaches and crosses the critical value F$_{c}$ = k$_{B}$λω/ 2, new wave packets, arising from first-order Bragg diffrac- tion, periodically bifurcate from the primary beam at propagation distances z satisfying the Bragg condition k ( z ) = − k$_{B}$/ 2 [see Figs.3(b) and 3(c)]. The mean mo- mentum of the first-order diffracted wave packets differs from that of the primary wave packet by an additional term k$_{B}$ , which explains the refraction of the first-order diffracted beams at the angle θ = dx/dz = k$_{B}$λ/n$_{s}$ ob- served in Figs.3(b) and 3(c). At stronger forcing, namely for F ≥ 3 F$_{c}$ , additional wave packets bifurcate from the III. WAVE PACKET DYNAMICS AT THE PT SYMMETRY-BREAKING POINT FIG. 1: (color online) Band diagram of the complex potential defined by Eq.(2) for parameter values λ = 633 nm, a = 8 µ m, n s = 1 . 42, V 0 = 0 . 002, and α = 0 . 3. The dotted curve shows, for comparison, the parabolic dispersion relation E ( κ ) = ( λκ ) $^{2}$/ (2 n s ), folded inside the first Brillouin zone, corresponding to the critical case α = α c = 1. The picture on the right hand side is an enlargement of the lowest band (solid curve), fitted by a sinusoidal curve (dotted curve), almost overlapped with the solid one. 0 firstGLYPH<.notdef>band GLYPH<c45>GLYPH<c112>GLYPH<c47> a GLYPH<c112>GLYPH<c47> a E n ( ) GLYPH<c107> waveGLYPH<.notdef>number GLYPH<c107> 0 -7 -6.8 -6.6 -6.4 -6.2 xGLYPH<.notdef>10 -4 firstGLYPH<.notdef>band GLYPH<c112>GLYPH<c47> a GLYPH<c45>GLYPH<c112>GLYPH<c47> a waveGLYPH<.notdef>number GLYPH<c107> -2 0 2 4 6 8 xGLYPH<.notdef>10 -3 -300 300 0 0 5 4 3 2 1 -300 300 0 0 4 3 2 1 x ( m) GLYPH<c109> x ( m) GLYPH<c109> z (cm) z (cm) -300 300 0 0 -300 300 0 5 4 3 2 1 z (cm) 0 5 4 3 2 1 z (cm) 0 5 4 3 2 1 z (cm) -300 300 0 0 5 4 3 2 1 z (cm) -100 100 0 (a) (b) (d) (f) (e) (c) FIG. 2: (color online) Wave packet broadening (discrete diffraction) and dynamic localization in the lattice of Fig.1 for broad beam excitation at normal incidence (left panels) and for single site excitation at normal incidence (right pan- els). In (a) and (b), the external force is absent ( F 0 = 0). In (c), (d) a sinusoidal force with period Λ = 1 cm and am- plitude F 0 = 13 . 32 µ m $^{$_{−}$1}$, corresponding to Γ = 1 . 684, is applied. In (e) and (f), the forcing amplitude is increased to F 0 = 19 . 03 µ m $^{$_{−}$1}$, corresponding to the DL condition Γ = 2 . 405. band. DL is thus expected to occur provided that the lattice is excited in its lowest order band. Figures 2(a) and (b) show a typical spreading (discrete diffraction) of a Gaussian wave packet ψ ( x, 0) = exp( − x $^{2}$/w$^{2}$) in the absence of the external force, for either a broad Gaus- sian beam that excites at normal incidence a few lattice