4 F = 1 . 618 m − $^{1}$, corresponding to point A ( g/f ≃ 0 . 91, Δ /f ≃ 8 . 72) on the third resonance curve of Fig.3(c). Photonic PT lattices may provide an accessible labo- ratory to experimentally observe BO in complex crys- tals. Complex refractive indices could be realized in en- gineered arrays of active semiconductor waveguides with selective pumping to achieve gain and loss regions. At optical or near-infrared wavelengths, a modulation depth of the imaginary refractive index Δ n$_{I}$ ∼ 2 × 10 − 4 (as in Fig.2) requires a gain/absorption coefficient of the order ∼ 10-20 cm − $^{1}$, which is available with current semicon- ductor optical amplifier technology. Smaller gain/loss coefficients are needed for the example of Fig.3. Non- reciprocal BO could be also observed in passive (i.e. with- out gain) waveguide arrays with loss modulation, in a geometrical setting similar to the one recently realized to demonstrate PT symmetry breaking in a passive Al- GaAs optical coupler [22]. The transverse index gradient can be realized by either chirping the waveguide width [3] or by waveguide axis bending [18]. In conclusion, BO in complex crystals with PT symmetry behave differently than in ordinary lattices owing to non- reciprocity of Bragg scattering. Our results indicate that the conventional wisdom of BO and transport for classi- cal or matter waves needs to be modified when dealing with complex crystals. ( l = 0 , ± 1 , ± 2 , ... ) where y$_{1}$$_{,}$$_{2}$ ≡ | y$_{1}$$_{,}$$_{2}$ | exp( iϕ$_{1}$$_{,}$$_{2}$ ) = Re( σ ) ± [1 + Re( σ ) $^{2}$]1 / $^{2}$, σ ≡ S$_{1}$$_{,}$$_{2}$ , and the 2 × 2 matrix S is the propagator for the differential system dx$_{1}$$_{,}$$_{2}$/dκ = ± i (Δ / 4 f ) cos( κ/ 2) x$_{1}$$_{,}$$_{2}$ + ( g/ 4 f ) x$_{2}$$_{,}$$_{1}$ from κ = 0 to κ = 2 π . Note that the two WS ladders are generally complex- valued with opposite sign of their imaginary part. How- ever, if the condition Re( σ ) = 0 is satisfied, y$_{1}$$_{,}$$_{2}$ = ± 1 and the spectrum becomes real-valued and given by a single WS ladder, namely E$_{l}$ = F al ( l = 0 , ± 1 , ± 2 , ... ). Figure 3(c) depicts the numerically-computed locus in the (Δ /f, g/f ) plane where Re( σ ) = 0, showing the exis- tence of a sequence of resonance curves. A perturbative calculation of the propagator S in the g → 0 limit shows that the resonance curves depart from the points of the Δ /f axis that are the roots of Bessel function J$_{0}$ [21]. Along such resonance curves, a forced Bloch particle un- dergoes undamped nor amplified BO, with spatial period z$_{B}$ = λ/ ( F a ). This is shown in Fig.3(d), which depicts an example of a BO breathing mode, obtained by nu- merical analysis of Eq.(1) for single waveguide excitation of the complex binary lattice of Fig.3(a), with a forcing E (1 , 2) l = 2 F al − F aϕ$_{1}$$_{,}$$_{2}$ π + i F a ln | y$_{1}$$_{,}$$_{2}$ | π (9) purely discrete and composed by two interleaved WS lad- ders E (1) l and E (2) l , as in a binary lattice with real po- tential [21]. Specifically, one can show that [1] C. Waschke, H. G. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. K¨ ohler, Phys. Rev. Lett. 70 , 3319 (1993). [2] M. BenDahan, E. Peik, J. Reichel, Y. Castin, and C. Sa- lomon, Phys. Rev. Lett. 76 , 4508 (1996); S. R. Wilkin- son, C. F. Bharucha, K. W. Madison, Q. Niu, and M. G. Raizen, Phys. Rev. Lett. 76 , 4512 (1996); B.P. Anderson and M. A. Kasevich, Science 282 , 1686 (1998). [3] R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisen- berg, and Y. Silberberg, Phys. Rev. Lett. 83 , 4756 (1999); T. Pertsch, P. Dannberg, W. Elflein, A. Br¨ auer, and F. Lederer, Phys. Rev. Lett. 83 , 4752 (1999); H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Br¨ auer, and U. Peschel, Phys. Rev. Lett. 96 , 023901 (2006). [4] H. Sanchis-Alepuz, Y.A. Kosevich, and J. Sanchez- Dehesa, Phys. Rev. Lett. 98 , 134301 (2007). [5] D. Cai, A.R. Bishop, N. Grønbech-Jensen, and M. Salerno, Phys. Rev. Lett. 74 , 1186 (1995); F. Dominguez- Adame, V.A. Malyshev, F.A.B.F. de Moura, and M.L. Lyra, Phys. Rev. Lett. 91 , 197402 (2003); A. Buchleitner and A.R. Kolovsky, Phys. Rev. Lett. 91 , 253002 (2003). [6] N. Hatano and D.R. Nelson, Phys. Rev. Lett. 77 , 570 (1996). [7] C. Keller, M.K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, and A. Zeilinger, Phys. Rev. Lett. 79 , 3327 (1997). [8] A. Kostenbauder, Y. Sun, and A.E. Siegman, J. Opt. Soc. Am. A 14 , 1780 (1997). [9] M.V. Berry, Czech. J. Phys. 54 , 1039 (2004). [22] A. Guo, G.J. Salamo, D. Duchesne, R. Morandotti, M. [10] C.M. Bender and S. Boettcher, Phys. Rev. Lett. 80 , 5243 (1998); C.M. Bender, Rep. Prog. Phys. 70 , 947 (2007). [11] A. Mostafazadeh, J. Math. Phys. 43 , 2814 (2002). [12] R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, Opt. Lett. 32 , 2632 (2007). [13] K.G. Makris, R. El-Ganainy, D.N. Christodoulides, and Z.H. Musslimani, Phys. Rev. Lett. 100 , 103904 (2008). [14] S. Klaiman, U. G¨ unther, and N. Moiseyev, Phys. Rev. Lett. 101 , 080402 (2008). [15] C.M. Bender, G.V. Dunne, P.N. Meisinger, Phys. Lett. A 252 , 272 (1999); H.F. Jones, Phys. Lett. A 262 , 242 (1999); Z. Ahmed, Phys. Lett. A 286 , 231 (2001); A. Khare and U. Sukhatme, J. Math. Phys. 47 , 062103 (2006); T. Curtright and L. Mezincescu, J. Math. Phys. 48 , 092106 (2007). [16] Z.H. Musslimani, K.G. Makris, R.El-Ganainy, and D.N. Christodoulides, J. Phys. A 41 , 244019 (2008). [17] O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, Phys. Rev. Lett. 103 , 030402 (2009). [18] S. Longhi, Laser & Photon. Rev. 3 , 243 (2009). [19] J. Callaway, Quantum Theory of the Solid State (Aca- demic Press, New York, 1974), pp. 465-478; V. Grecchi and A. Sacchetti, arXiv:quant-ph/0506057. [20] M.V. Berry, J. Phys. A 31 , 3493 (1998). [21] B.M. Breid, D.Witthaut, and H.J. Korsch, New J. Phys. 8 , 110 (2006); F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. T¨ unnermann, and S. Longhi, Phys. Rev. Lett. 102 , 076802 (2009).