Spectral singularities and Bragg scattering in complex crystals arXiv:1001.0962v1 [quant-ph] 6 Jan 2010 I. INTRODUCTION 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 Spectral singularities that spoil the completeness of Bloch-Floquet states may occur in non- Hermitian Hamiltonians with complex periodic potentials. Here an equivalence is established be- tween spectral singularities in complex crystals and secularities that arise in Bragg diffraction pat- terns. Signatures of spectral singularities in a scattering process with wave packets are elucidated for a PT -symmetric complex crystal. PACS numbers: 03.65.-w, 11.30.Er, 42.50.Xa The Dirac-von Neumann formulation of quantum me- chanics prescribes that the Hamiltonian H of a physical system must be Hermitian. This requirement ensures a real-valued energy spectrum and a unitary (probability- preserving) temporal evolution. In the last decade, complex extensions of quantum mechanics which relax the Hermiticity constraint have been proposed (see e.g. [1, 2]). Indeed, a diagonalizable Non-Hermitian Hamil- tonian (NHH) having a real spectrum is enough to con- struct a unitary quantum system, provided that the inner product of the Hilbert space is properly modified [2]. In particular, Bender and collaborators showed that a NHH possessing parity-time ( PT ) symmetry may serve to de- velop a complex extension of quantum mechanics below a phase transition (symmetry-breaking) point [3, 4]. Ex- perimental realizations of PT Hamiltonians have been recently proposed in optical media with a complex re- fractive index [5–8], and the first observation of PT - symmetry breaking has been reported in a passive optical waveguide coupler [9]. For a NHH, the reality of the spectrum does not ensure diagonalizability, which may be prevented by the pres- ence of exceptional points in the point-spectrum of H [10], or of spectral singularities in the continuous part of the spectrum [11–13]. Exceptional points, also re- ferred to as Hermitian degeneracies (see [10]), correspond to degeneracies where both eigenvalues and eigenvectors coalesce as a system parameter is varied. Other singu- lar points may also occur in low-dimensional (e.g. ma- trix) non-Hermitian Hamiltonians, such as branch points in the complex plane in which eigenvalue degeneracy do not correspond to lack of completeness of the spectrum [14]. Such singularities have various physical implica- tions, which have been investigated in different physi- cal fields (see for instance [7, 9, 10, 15] and references therein). They play an important role in the study of open quantum systems, particularly in relation with the resonance states (for a recent review see [16]). But they must not be confused with spectral singularities as de- fined in [11–13]. Unlike exceptional and branch points that can be present for non-Hermitian operators with a discrete spectrum, spectral singularities are exclusive features of certain non-Hermitian operators having a continuous part in their spectrum [13]. Following [11– 13], spectral singularities refer to divergences (poles) of the resolvent on the continuous spectrum of H , which do not correspond to square integrable eigenfunctions. The physical implications of such spectral singularities have been investigated so far for wave scattering from complex potential barriers [12, 13, 17, 18] and shown -notably by Mostafazadeh- to correspond to resonance states with vanishing spectral width [17, 18]. Another physically relevant class of NHHs with continuous spec- trum is provided by complex periodic potentials [22–24]. Complex crystals have been investigated in different ar- eas of physics, ranging from matter waves [19–21] to op- tics [6, 8, 20, 21, 25]. As compared to ordinary crys- tals, complex crystals exhibit some unique properties, such as violation of the Friedel’s law of Bragg scattering, double refraction, nonreciprocal diffraction, and anoma- lous transport [6, 19–21, 25], which make them a rather unique class of synthetic materials. Spectral singularities for periodic non-self-adjoint Schr¨ odinger operators have been studied by mathematicians [27], however their phys- ical implications in connection to complex crystals have not been explored yet. It is the aim of this work to investigate the onset of spec- tral singularities in complex crystals and to show their physical impact in Bragg scattering processes. In par- ticular, we prove rather generally that spectral singular- ities are associated to a secular growth of plane waves diffracted off the crystal when the incident angle is an integer multiple of the Bragg angle. With reference to a specific PT -symmetric complex crystal, previously con- sidered in Ref.[6], we show that spectral singularities occur at the PT symmetry breaking point and can be revealed in a diffraction experiment using a spatially- confined wave packet that excites the crystal at normal incidence. Unlike plane wave excitation, broadening of the angular spectrum for a wave packet is shown to lead to a saturation of the secular growth of the wave ampli- tude.