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sign of I$_{n}$ in each band [10, 13]. To investigate WS lo- calization, like in ordinary crystals we expand the wave packet ψ as a superposition of Bloch-Floquet states [19], ψ ( x, z ) = ∑ n ∫ dκc$_{n}$ ( κ, z ) φ$_{n}$ ( κ, z ). The scalar product of both sides of Eq.(1) with 〈 φ † $_{n}$( x, κ ) | yields the following evolution equations for the spectral coefficients c$_{n}$ ( κ, z ) 

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where X$_{n,l}$ ( κ ) ≡ (2 πi/a ) ∫ a 0 dxu ∗ $_{n}$( − x, − κ ) ∂$_{κ}$u$_{l}$ ( x, κ ). The off-diagonal elements X$_{n,l}$ ( n ̸ = l ) in Eq.(3) are responsible for interband transitions, i.e. Zener tunneling (ZT), and can be written in the equivalent form X$_{n,l}$ = (2 πλ $^{2}$/an$_{s}$ )[ E$_{n}$ ( κ ) − E$_{l}$ ( κ )] − 2 ∫ a 0 u ∗ $_{n}$( − x, − κ ) ∂$_{x}$V ( x ) u$_{l}$ ( x, κ ). If bands n and l are separate by a large gap and the force F is small enough such that F | X$_{n,l}$ ( κ ) | ≪ | E$_{n}$ ( κ ) − E$_{l}$ ( κ ) | in the entire Brillouin zone , ZT is negligible as in ordinary lattices [19]. Taking X$_{n,l}$ ≃ 0 for n ̸ = l (single-band approximation), the WS spectrum of the n -th band is then obtained by letting in Eq.(3) c$_{n}$ ( κ, z ) = a$_{n}$ ( κ ) exp( − iEz/λ ) and imposing the boundary condition a$_{n}$ (2 π/a ) = a$_{n}$ (0). This yields 

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where we have set 〈 E$_{n}$ 〉 = ( a/ 2 π ) ∫ π/a − $_{π/a}$dκE$_{n}$ ( κ ) and γ$_{n}$ = ( D$_{n}$ F a/ 2 π ) ∫ π/a − $_{π/a}$dκ Im( X$_{n,n}$ ( κ )). As compared to an ordinary crystal [19], the main difference here is that the WS ladder spectrum is complex-valued, even though α < α$_{c}$ . This is simply due to the fact that the external force F breaks the P T symmetry of the Hamiltonian. In each ladder, the imaginary parts of the eigenvalues have the same value (i.e., independent of l ), equal to γ$_{n}$ . Since Im( X$_{n,n}$ ( κ )) is an even function of κ , γ$_{n}$ is usually non- vanishing, and its sign is reversed as the direction of the force, i.e. the sign of F , changes. Therefore, in complex lattices below the phase transition point, BO with spa- tial period z$_{B}$ = λ/ ( F a ) are observed, however they are either amplified or damped depending on the sign of F . This behavior is illustrated in Fig.1, which shows the on- set of amplified or damped BO as obtained by numerical simulations of Eq.(1) for the complex lattice of Eq.(2) with α = 0 . 3 and for an input excitation of the lattice with a broad Gaussian beam at normal incidence. In the figures, the evolution of the total beam power P = P ( z ) is also shown. Note that, in spite all WS eigenvalues have the same imaginary part, P ( z ) shows a non-exponential behavior, with abrupt changes of P ( z ) in correspondence of Bragg scattering of the beam from the lattice. This behavior is ascribable to the non-orthogonality of WS states in complex potentials. 

As the non-Hermitian parameter α is increased to cross the phase transition value α$_{c}$ , gap narrowing till band merging, associated to the appearance of pairs of complex-conjugate eigenvalues, is observed [13], making 

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the previous analysis inapplicable. Here we will limit to investigate the onset of BO in the α = α$_{c}$ and α > α$_{c}$ regimes by considering two specific examples which en- able an analytical treatment. 

The first example is given by the potential (2) at 

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the phase transition point α = α$_{c}$ = 1, i.e. V ( x ) = V$_{0}$ exp(2 πix/a ) [20], for which band gaps disappear and the energy dispersion is described by the free-particle parabola E ( κ ) = ( λ $^{2}$/ 2 n$_{s}$ ) κ $^{2}$, periodically folded inside the first Brillouin zone. In presence of an external force F , numerical simulations indicate that the wave packet evolution strongly depends on the sign of F , as shown in Fig.2. Remarkably, BO disappear for F < 0, and the wave packet propagates following a parabolic path as if the complex lattice were absent [Fig.2(a)]. This result in- dicates that a net drift along the lattice can be achieved by the application of a dc force, an effect which is not pos- sible in ordinary crystals without collisions. Conversely, for F > 0 an oscillatory dynamics survives, with spatial period equal to z$_{B}$ = λ/ ( F a ) [Fig.2(b)]. In this case, 

