Dynamic localization and transport in complex crystals 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 The behavior of a Bloch particle in a complex crystal with PT symmetry subjected to a sinusoidal ac force is theoretically investigated. For unbroken PT symmetry and in the single-band approxima- tion, it is shown that time reversal symmetry of the ac force preserves the reality of the quasienergy spectrum. Like in ordinary crystals, exact band collapse, corresponding to dynamic localization, is attained for a sinusoidal band shape. The wave packet dynamics turns out to be deeply modified at the PT symmetry breaking point, where band merging occurs and Bragg scattering in the crystal becomes highly non-reciprocal. PACS numbers: 73.23.Ad, 11.30.Er, 72.10.Bg The dynamics of matter or classical waves in peri- odic potentials subjected to external dc or ac forces is strongly influenced by Bragg scattering, which is re- sponsible for such important effects as Bloch oscillations (BOs) and dynamic localization (DL). Dynamic local- ization was originally proposed by Dunlap and Kenkre [1] as a suppression of the broadening of a charged- particle wave packet as it moves along a tight-binding lattice driven by a sinusoidal ac electric field. DL was then explained in terms of quasienergy band collapse [2], and the general conditions of DL beyond the nearest- neighboring tight-binding (NNTB) approximation used by Dunlap and Kenkre were subsequently investigated in Ref.[3]. The interest on DL has been recently renewed since the first experimental observations of DL have been reported for matter waves trapped in dynamical optical lattices [4–6], and for light waves in curved waveguide arrays [7–11]. Matter or light waves may also interact with complex potentials. Complex crystals for matter waves emerge, for instance, in the near resonant inter- action of light with open two-level systems [12], whereas in optics complex crystals are realized by waveguide ar- rays that include gain and/or loss regions [13]. 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 [12–14]. A special class of complex crystals is provided by com- plex potentials possessing parity-time ( PT ) symmetry [13, 15, 16]. An important property of crystals is to admit of an entirely real-valued energy spectrum below a phase transition (symmetry-breaking) point, in spite of the non-Hermiticity of the underlying Hamiltonian [15]. A recent study on BOs in PT crystals [17] has shown that the common wisdom of coherent quantum transport in a crystal is greatly modified when dealing with a complex crystal. It is the aim of this work to investigate the coherent mo- tion of wave packets in a complex PT crystal driven by an ac-like force. In particular, it is shown that in the un- broken PT symmetry region time reversal symmetry of 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 = F ( z ) is a fic- titious refractive index gradient proportional to the lo- cal waveguide axis curvature which mimics the action of a driving force [18]. In particular, a sinusoidal ac-like force is simply mimicked by a sinusoidal bending profile of the waveguides [7]. In a complex lattice, the refrac- tive index is complex, and the PT symmetry require- ment V ( − x ) = V $^{∗}$( x ) corresponds to suitable combina- tions of optical gain and loss regions in the lattice as discussed in [13]. 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 In optics, the coherent motion of charged quantum par- ticles in periodic potentials driven by an ac electric field can be mimicked by the propagation of light waves in a periodically-curved photonic lattice [7, 18]. In the scalar and paraxial approximations, light propagation at wave- length λ in the lattice is described by the Schr¨ odinger- type wave equation the ac-like force preserves the reality of the quasienergy spectrum, and that a full band collapse, corresponding to dynamic localization, occurs within the single-band and NNTB approximations like in ordinary crystals. How- ever, the transport properties of the lattice are deeply modified at the PT symmetry breaking, where Bragg scattering in the crystal becomes highly non-reciprocal. In the following analysis, we will consider specifically wave packets in a photonic lattice system [7, 11, 13, 17], however the results can be applied to other lattice real- izations, such as to matter wave tunneling in dynamic complex optical lattices. II. DYNAMIC LOCALIZATION IN COMPLEX CRYSTALS WITH UNBROKEN PT SYMMETRY I. INTRODUCTION arXiv:1001.0974v1 [quant-ph] 6 Jan 2010 iλ∂$_{z}$ψ = − λ 2 2 n$_{s}$ ∂ 2 $_{x}$ψ + V ( x ) ψ − F xψ ≡ ( H 0 − F x ) ψ. (1)