5 computed as louin zone. Note, however, that in Eqs.(7) and (11) n is a positive (but not a negative) integer number: this circumstance is a clear signature that Bragg scattering is highly non-reciprocal, a feature which is peculiar to the complex nature of the crystal [12, 14]. If the cross- ings of the stationary points are fast enough, a$_{n}$ ( z ) is basically constant far from the stationary points, with abrupt changes at z = z 0 , namely | R | ≃ V 0 λ ∣ ∫ ∞ −∞ dξ exp ( i 2 d $^{2}$ϕ 1 dz 2 ξ 2 )∣ ∣ = V 0 λ √ 2 πn$_{s}$ k$_{B}$ √ F 2 0 − F 2 c . (16) In deriving Eq.(16), we have taken into account that ( d $^{2}$ϕ 1 /dz $^{2}$) = F ( z 0 ) k$_{B}$/n$_{s}$ , | F ( z 0 ) | = ( F 2 0 − F 2 c ) 1 / 2 and ∫ ∞ $_{−∞}$dξ exp( iξ $^{2}$) = √$_{iπ}$$_{. For parameter values used in the}$ simulations of Fig.3(c), from Eq.(16) one has | R | ≃ 0 . 95, which is in excellent agreement with the staircase be- havior of beam power P ( z ) shown in the right panel of Fig.3(c). Equation (16) fails to predict the correct amplitude factor R when F 0 → F + c , i.e. when the two crossing points in each ac oscillation cycle coalesce. For F 0 = F$_{c}$ , the crossing is parabolic [see Fig.3(b), central panel], and ϕ 1 ( z ) in Eq.(13) should now be approximated as ϕ 1 ( z ) = ϕ 1 ( z 0 ) + (1 / 6)( d $^{3}$ϕ 1 /dz $^{3}$)( z − z 0 ) $^{3}$, where the derivative ( d $^{3}$ϕ 1 /dz $^{3}$) is calculated at the crossing point z = z 0 . In this case, in place of Eq.(16) one has where Ai( ξ ) is the Airy function. In deriving Eq.(17), we have taken into account that ( d $^{3}$ϕ 1 /dz $^{3}$) = ( k$_{B}$/n$_{s}$ )( dF/dz ) = λk 2 $_{B}$ω $^{2}$/ (2 n$_{s}$ ) at z = z 0 , and ∫ ∞ $_{−∞}$dξ exp( iξ $^{3}$/ 3) = 2 π Ai(0). For parameter values used in the simulations of Fig.3(b), from Eq.(17) one has | R | ≃ 2 . 245. Note that, with this amplitude factor, the staircase behavior of beam power P ( z ) shown in the right panel of Fig.3(b) is reproduced with excellent accuracy. | R | ≃ V 0 λ ∣ ∫ ∞ −∞ dξ exp ( i 6 d $^{3}$ϕ 1 dz 3 ξ 3 )∣ ∣ = V 0 λ 2 π Ai(0) ( 4 n$_{s}$ λω $^{2}$k 2 B ) 1 / 3 (17) and the integral on the right hand side of Eq.(13) is ex- tended to the neighborhood of z 0 . From these results one can readily explain the abrupt changes of the total beam power P ( z ) observed in Fig.3 (right panels) at the crossing points shown in the central panels of Fig.3, the existence of a critical forcing F$_{c}$ below which no bifur- cating wave packets appear, as well as the occurrence of higher-order bifurcating wave packets and a dynamical scenario with increasing complexity as the amplitude of forcing is increased. In fact, according to Eqs. (8) and (11) the stationary points which generate the bifurcating wave packets of order n are obtained from the equation The coherent motion of a Bloch wave packet in a tight- binding lattice driven by an ac electric field is known to show a self-imaging effect that arises from quasi-energy band collapse of the time-periodic Hamiltonian. This phenomenon, referred to as dynamic localization [1, 2], has been recently observed for both matter and optical waves as a suppression of wave packet broadening in the lattice. In this work, we investigated theoretically the be- havior of a Bloch particle in a complex crystal with P T symmetry subjected to a sinusoidal ac-like force. As com- pared to ordinary crystals, complex crystals exhibit some unique properties, such as violation of the Friedel’s law of Bragg scattering and nonreciprocal diffraction. For an unbroken P T symmetry and in the single-band ap- proximation, it has been shown that the quasi-energy spectrum of the time-periodic non-Hermitian Hamilto- nian remains real-valued. In this regime, like in an or- dinary crystal exact band collapse is possible within the IV. CONCLUSIONS is the critical forcing amplitude. Therefore, for F 0 < F$_{c}$ no diffracted wave packets of any order are generated [Fig.3(a)], for F$_{c}$ ≤ F 0 < 3 F$_{c}$ first-order diffracted wave packets are generated, for 3 F$_{c}$ ≤ F 0 < 5 F$_{c}$ first-order and second-order wave packets are generated, and so on. The graphical determination of the stationary points is depicted in the central panels of Fig.3 as the crossing of the sinusoidal curve k ( z ) /k$_{B}$ with the vertical dashed lines k/k$_{B}$ = − 0 . 5 (for first-order Bragg diffraction) and k/k$_{B}$ = − 1 . 5 [for second-order Bragg diffraction, shown solely in Fig.3(d)]. Let ut discuss in some detail the gen- eration of first-order diffracted wave packets ( n = 1). For F$_{c}$ < F 0 < 3 F$_{c}$ [Fig.3(c)], in each ac oscillation cycle there are two stationary points, and the crossing is linear. Near each of the stationary points, ϕ 1 ( z ) can be thus ap- proximated as ϕ 1 ( z ) = ϕ 1 ( z 0 )+(1 / 2)( d $^{2}$ϕ 1 /dz $^{2}$)( z − z 0 ) $^{2}$, where the derivative ( d $^{2}$ϕ 1 /dz $^{2}$) is calculated at the sta- tionary point z = z 0 . From Eq.(13), it follows that the amplitude factor | R | of the generated diffracted beam after each stationary phase point can be approximately which can be satisfied provided that the amplitude F 0 of forcing is larger than F$_{c}$ (2 n − 1), where F 0 λω sin( ωz 0 ) = − k$_{B}$ 2 (2 n − 1) , (14) F$_{c}$ = 1 2 λωk$_{B}$ (15) a$_{n}$ ( z + 0 ) ≃ a$_{n}$ ( z − 0 ) + Ra$_{n}$ − 1 ( z 0 ) (12) R = − i V 0 λ ∫ dξ exp[ iϕ$_{n}$ ( ξ )] (13) where