Bloch oscillations in complex crystals with PT symmetry S. Longhi Dipartimento di Fisica and Istituto di Fotonica e Nanotecnologie del CNR, Politecnico di Milano, Piazza L. da Vinci 32, I-20133 Milano, Italy Bloch oscillations (BO) in complex lattices with PT symmetry are theoretically investigated with specific reference to optical BO in photonic lattices with gain/loss regions. Novel dynamical phe- nomena with no counterpart in ordinary lattices, such as non-reciprocal BO related to violation of the Friedel’s law of Bragg scattering in complex potentials, are highlighted. PACS numbers: 42.50.Xa, 42.70.Qs, 11.30.Er, 72.10.Bg Bloch oscillations (BO), i.e. the coherent oscillatory motion of a quantum particle in a periodic potential driven by an external dc force, represent one of the most striking predictions of wave mechanics of periodic sys- tems. In the quantum realm, BO have been experimen- tally observed for electrons in semiconductor superlat- tices [1] and for ultracold atoms or Bose-Einstein con- densates in tilted optical lattices [2]. Classical analogues of BO have been also proposed and observed for optical [3] and acoustical [4] waves. The phenomenon of BO is related to the transition of energy spectrum from contin- uous (energy bands) to (nearly) discrete, with the forma- tion of Wannier Stark (WS) ladders when the dc force is applied. In physical space, BO are generally explained in terms of wave Bragg scattering off the periodic poten- tial, causing a wave packet to oscillate rather than trans- late through the lattice. Previous studies on BO have considered particle motion in real-valued potentials, may be in presence of nonlinearity, lattice disorder or parti- cle interactions [5]. However, particle or wave dynamics in complex potentials is not uncommon in physics [6– 9]. Complex potentials for matter waves emerge in the near resonant interaction of light with open two-level sys- tems [7], whereas in optics wave propagation in lossy or active waveguides is described by a complex refractive index [8]. Complex potentials have received in recent years an enormous interest since the discovery that non- Hermitian Hamiltonians with parity-time ( PT ) symme- try may possess an entirely real-valued energy spectrum below a phase transition (symmetry-breaking) point [10]. PT Hamiltonians belong to the more general class of pseudo-Hermitian systems [11]. Optical realizations of PT -symmetric potentials, based on light propagation in coupled waveguides with a complex refractive index, have been recently proposed [12–14], whereas energy band for- mation and spontaneous symmetry-breaking thresholds have been investigated in a few PT potential models [15] and in presence of nonlinearity [16] or lattice disorder [17]. It is the aim of this Letter to investigate the onset of BO in periodic lattices with PT symmetry, highlight- ing some unique features arising from the complex nature of the scattering potential. We consider here BO for light waves in photonic lattices [3], however the analysis can be applied to other physical systems, such as to matter waves in accelerating complex optical lattices. Let us first consider the case α < α$_{c}$ , and indi- cate by φ$_{n}$ ( x, κ ) = u$_{n}$ ( x, κ ) exp( iκx ) the Bloch-Floquet eigenfunction of H$_{0}$ , corresponding to the n -th lattice band and to wave number κ in the first Brillouin zone ( − π/a ≤ κ < π/a ) with periodic part u$_{n}$ ( x, κ ). Thus H$_{0}$ φ$_{n}$ ( x, κ ) = E$_{n}$ ( κ ) φ ( x, κ ), where E$_{n}$ ( κ ) is the real- valued band dispersion curve. As for real potentials, one can show that E$_{n}$ ( − κ ) = E$_{n}$ ( κ ). Additionally, Re( u$_{n}$ ( κ, x )) and Im( u$_{n}$ ( κ, x )) have well defined and op- posite parity under the inversion x → − x , whereas φ † $_{n}$( x, κ ) = φ$_{n}$ ( − x, − κ ) are eigenfunctions of the ad- joint H † 0 with same eigenvalues E$_{n}$ ( κ ). Assuming that I$_{n}$ ≡ ∫ a 0 dxu ∗ $_{n}$( − x, − κ ) u$_{n}$ ( x, κ ) does not vanish, i.e. in absence of spectral singularities, the orthonormal condi- tions 〈 φ † $_{l}$( x, κ $^{′}$) | φ$_{n}$ ( x, κ ) 〉 ≡ ∫ ∞ −∞ dxφ †∗ l ( x, κ $^{′}$) φ$_{n}$ ( x, κ ) = D$_{n}$ δ$_{n,l}$δ ( κ − κ $^{′}$) hold, where D$_{n}$ = ± 1 is given by the one has α$_{c}$ = 1 [13]. where λ = λ/ (2 π ) is the reduced wavelength, n$_{s}$ is the substrate refractive index, V ( x ) = n$_{s}$ − n ( x ) is the po- tential, n ( x ) = n ( x + a ) is the refractive index profile of the lattice (spatial period a ), and F is the refractive index gradient which mimics an external dc force. For F = 0, PT symmetry requires V ( − x ) = V $^{∗}$( x ), i.e. suit- able combinations of optical gain and loss regions in the lattice, as discussed in [12–14]. The real and imaginary parts of the potential are denoted by V$_{R}$ ( x ) and αV$_{I}$ ( x ), respectively, where α ≥ 0 is a dimensionless parameter that measures the anti-Hermitian strength of H$_{0}$ . For given profiles of V$_{R}$ and V$_{I}$ , the spectrum of H$_{0}$ turns out to be real-valued and composed by bands separated by gaps for α < α$_{c}$ , where α$_{c}$ ≥ 0 corresponds to the spontaneous symmetry breaking point. For α ≥ α$_{c}$ , band merging and appearance of pairs of complex-conjugates eigenvalues is observed [13, 15]. For instance, for the potential Paraxial propagation of light at wavelength λ in a weakling-guiding one-dimensional waveguide lattice with a superimposed transverse refractive index gradient is de- scribed by the Schr¨ odinger-type equation [13, 18] iλ∂$_{z}$ψ = − λ 2 2 n$_{s}$ ∂ 2 $_{x}$ψ + V ( x ) ψ + F xψ ≡ ( H$_{0}$ + F x ) ψ (1) V$_{R}$ ( x ) = V$_{0}$ cos(2 πx/a ) , V$_{I}$ ( x ) = V$_{0}$ sin(2 πx/a ) (2) arXiv:1001.0981v1 [cond-mat.other] 6 Jan 2010