Showing papers on "Dispersion relation published in 2009"
TL;DR: The program phon calculates force constant matrices and phonon frequencies in crystals and from the frequencies it also calculates various thermodynamic quantities, like the Helmholtz free energy, the entropy, the specific heat and the internal energy of the harmonic crystal.
643 citations
TL;DR: In this paper, the phononic properties of a chiral cellular structure were investigated, and the influence of unit cell geometry on dispersion, band gap occurrence and wave directionality.
304 citations
TL;DR: In this article, the experimental observation of band gaps in a synthetic nanostructured magnonic crystal composed of two different magnetic materials was reported, in the form of a one-dimensional periodic array comprising alternating Permalloy and cobalt nanostripes.
Abstract: We report the experimental observation of band gaps in a synthetic nanostructured magnonic crystal composed of two different magnetic materials. The sample, in the form of a one-dimensional periodic array comprising alternating Permalloy and cobalt nanostripes, has been fabricated using advanced lithographic techniques. Dispersion relations of spin waves in the magnonic crystal have been mapped by Brillouin spectroscopy. The center frequency and width of the band gaps observed are tunable by an applied magnetic field. Dispersion relations calculated based on the finite element method accord with the measured data.
243 citations
TL;DR: In this article, an analytic, kinetic model with one-, two-, and three-dimensional particle-in-cell simulations was used to confirm the existence of the cosmic ray current-driven instability in the kinetic regime and determine its saturation mechanisms.
Abstract: The cosmic ray current-driven (CRCD) instability, predicted by Bell, consists of nonresonant, growing plasma waves driven by the electric current of cosmic rays (CRs) that stream along the magnetic field ahead of both relativistic and nonrelativistic shocks. Combining an analytic, kinetic model with one-, two-, and three-dimensional particle-in-cell simulations, we confirm the existence of this instability in the kinetic regime and determine its saturation mechanisms. In the linear regime, we show that, if the background plasma is well magnetized, the CRCD waves grow exponentially at the rates and wavelengths predicted by the analytic dispersion relation. The magnetization condition implies that the growth rate of the instability is much smaller than the ion cyclotron frequency. As the instability becomes nonlinear, significant turbulence forms in the plasma. This turbulence reduces the growth rate of the field and damps the shortest wavelength modes, making the dominant wavelength, λ d , grow proportional to the square of the field. At constant CR current, we find that plasma acceleration along the motion of CRs saturates the instability at the magnetic field level such that vA ~ v d,cr, where vA is the Alfven velocity in the amplified field, and v d,cr is the drift velocity of CRs. The instability can also saturate earlier if CRs get strongly deflected by the amplified field, which happens when their Larmor radii get close to λ d . We apply these results to the case of CRs propagating in the upstream medium of the forward shock in supernova remnants. If we consider only the most energetic CRs that escape from the shock, we obtain that the field amplification factor of ~10 can be reached. This confirms the CRCD instability as a potentially important component of magnetic amplification process in astrophysical shock environments.
206 citations
TL;DR: In this article, the authors derived the dynamical matrix for a monolayer crystal of hexagonal boron nitride (h-BN) and analyzed the phonon dispersion relations.
Abstract: Starting from an empirical force constant model of valence interactions and calculating by Ewald's method the ion-ion force constants, we derive the dynamical matrix for a monolayer crystal of hexagonal boron nitride (h-BN). The phonon dispersion relations are calculated. The interplay between valence and Coulomb forces is discussed. It is shown by analytical methods that the longitudinal and the transverse optical (LO and TO) phonon branches for in-plane motion are degenerate at the $\ensuremath{\Gamma}$ point of the Brillouin zone. Away from $\ensuremath{\Gamma}$, the LO branch exhibits pronounced overbending. It is found that the nonanalytic Coulomb contribution to the dynamical matrix causes a linear increase of the LO branch with increasing wave vector starting at $\ensuremath{\Gamma}$. This effect is general for two-dimensional (2D) ionic crystals. Performing a long-wavelength expansion of the dynamical matrix, we use Born's perturbation method to calculate the elastic constants (tension coefficients). Since the crystal is noncentrosymmetric, internal displacements due to relative shifts between the two sublattices (B and N) contribute to the elastic constants. These internal displacements are responsible for piezoelectric and dielectric phenomena. The piezoelectric stress constant and the dielectric susceptibility of 2D h-BN are calculated.
188 citations
TL;DR: In this paper, a purely kinetic approach to the excitation of waves by cosmic rays in the vicinity of a shock front leads to predict the appearance of a non-Alfvenic fast-growing mode which has the same dispersion relation as that previously found by Bell in 2004 by treating the plasma in the magnetohydrodynamic approximation.
Abstract: We show that a purely kinetic approach to the excitation of waves by cosmic rays in the vicinity of a shock front leads to predict the appearance of a non-Alfvenic fast-growing mode which has the same dispersion relation as that previously found by Bell in 2004 by treating the plasma in the magnetohydrodynamic approximation. The kinetic approach allows us to investigate the dependence of the dispersion relation of these waves on the microphysics of the current which compensates the cosmic ray flow. We also show that a resonant and a non-resonant mode may appear at the same time and one of the two may become dominant on the other depending on the conditions in the acceleration region. We discuss the role of the unstable modes for magnetic field amplification and particle acceleration in supernova remnants at different stages of the remnant evolution.
182 citations
TL;DR: This Letter discusses how linear coupled plasmon particle arrays inspired by radio frequency Yagi-Uda antennas can be used to construct both efficient unidirectional single photon sources and efficient directional single plAsmon sources.
Abstract: This Letter discusses how linear coupled plasmon particle arrays inspired by radio frequency Yagi-Uda antennas can be used to construct both efficient unidirectional single photon sources and efficient directional single plasmon sources. Calculations using an exact multipole expansion method are presented of the spontaneous emission directivity, efficiency, and spontaneous emission decay rates, taking into account material loss in real noble metals. An analysis of the emission properties in terms of the dispersion relation of infinite arrays reveals how one can use guided mode dispersion to achieve desirable figures of merit. The key ingredient is to couple the source to array eigenmodes that are just beyond the light line but still wave vector matched to propagating modes to within the momentum uncertainty set by the inverse antenna length. Finally, this Letter shows that the emission decay rates can be controlled independently of the directionality and without penalty in quantum efficiency.
175 citations
TL;DR: In this paper, it was shown that the radiation energy density exhibits a peculiar dependence on the scale factor of the dispersion relation, where the energy density decreases proportional to a −6.
163 citations
TL;DR: In this article, a nice fitting procedure is exploited to obtain the dispersion relations from which the effective permittivities of 3D metallic nanowire media can be retrieved, and it is shown that the hyperbolic dispersion relation of the 3D wire medium can be valid even for evanescent modes.
Abstract: We design three-dimensional (3D) metallic nanowire media with different structures and numerically demonstrate that they can be homogeneous effective indefinite anisotropic media by showing that their dispersion relations are hyperbolic. For a finite slab, a nice fitting procedure is exploited to obtain the dispersion relations from which we retrieve the effective permittivities. The pseudo focusing for the real 3D wire medium agrees very well with the homogeneous medium having the effective permittivity tensor of the wire medium. Studies also show that in the long-wavelength limit, the hyperbolic dispersion relation of the 3D wire medium can be valid even for evanescent modes.
160 citations
TL;DR: The spectra exhibit a deep minimum that is shown to be independent of the laser intensity, and is thus a clear measure of the electronic structure of the atom, suggesting that electronic structure can be accurately determined in high-harmonic experiments despite the presence of the strong laser field.
Abstract: We report detailed measurements of the high-harmonic spectra generated from argon atoms. The spectra exhibit a deep minimum that is shown to be independent of the laser intensity, and is thus a clear measure of the electronic structure of the atom. We show that exact field-free continuum wave functions reproduce the minimum, but plane wave and Coulomb wave functions do not. This remarkable observation suggests that electronic structure can be accurately determined in high-harmonic experiments despite the presence of the strong laser field. Our results clarify the relation between high-harmonic generation and photoelectron spectroscopy. The use of exact continuum functions also resolves the ambiguity associated with the choice of the dispersion relation.
157 citations
TL;DR: In this article, the authors derived two equivalent vectorial representations of a surface plasmon field using an expansion over surface waves with either a complex wave vector or a complex frequency.
Abstract: Surface plasmons are usually described as surface waves with either a complex wave vector or a complex frequency. When discussing their merits in terms of field confinement or enhancement of the local density of states, controversies have arisen as the results depend on the choice of a complex wave vector or a complex frequency. In particular, the shape of the dispersion curves depends on this choice. In this work, we derive two equivalent vectorial representations of a surface plasmon field using an expansion over surface waves with either a complex wave vector or a complex frequency. These representations can be used to discuss the issue of field confinement and local density of states as they have a nonambiguous relation with the two dispersion relations. They can also be used to account for propagation and diffraction of surface waves. They generalize the scalar approximation often used when discussing surface plasmon diffraction.
TL;DR: In this article, the relevance of the acoustic band gap on the transformation of single and multiple pulses in linear, nonlinear and strongly nonlinear regimes is investigated with numerical calculations and experiments.
Abstract: One-dimensional nonlinear phononic crystals have been assembled from periodic diatomic chains of stainless steel cylinders alternated with Polytetrafluoroethylene spheres. This system allows dramatic changes of behavior (from linear to strongly nonlinear) by application of compressive forces practically without changes of geometry of the system. The relevance of classical acoustic band-gap, characteristic for chain with linear interaction forces and derived from the dispersion relation of the linearized system, on the transformation of single and multiple pulses in linear, nonlinear and strongly nonlinear regimes are investigated with numerical calculations and experiments. The limiting frequencies of the acoustic band-gap for investigated system with given precompression force are within the audible frequency range (20–20,000 Hz) and can be tuned by varying the particle’s material properties, mass and initial compression. In the linear elastic chain the presence of the acoustic band-gap was apparent through fast transformation of incoming pulses within very short distances from the chain entrance. It is interesting that pulses with relatively large amplitude (nonlinear elastic chain) exhibit qualitatively similar behavior indicating relevance of the acoustic band gap also for transformation of nonlinear signals. The effects of an in situ band-gap created by a mean dynamic compression are observed in the strongly nonlinear wave regime.
TL;DR: In this paper, the photonic band gap structures of obliquely incident electromagnetic waves propagating in a one-dimensional plasma photonic crystal with collision have been studied on the basis of electromagnetic theory and transfer matrix approach.
Abstract: The photonic band gap structures of obliquely incident electromagnetic waves propagating in a one-dimension plasma photonic crystal with collision have been studied on the basis of electromagnetic theory and transfer matrix approach. The dispersion relations for both the transverse electric wave case and the transverse magnetic wave case are deduced. And the photonic band gap structures, with their function dependence on the microplasma layer density, microplasma width, collision frequency, background material dielectric constant, and incident angle, are computed. The results show that there exist two photonic band gap structures in an adsorptive plasma photonic crystal: one is a normal photonic band gap structure and the other is an absorption photonic band gap structure. Parameter dependence of the effects is calculated and discussed.
TL;DR: In this paper, a theoretical analysis of wave propagation in a dynamic bulk is presented, based on the concept of a temporal photonic crystal, and the Poynting vectors of the transmitted and reflected fields are analyzed in detail.
Abstract: We present a theoretical description of the response of a dynamic slab, with time-periodic dielectric function $ϵ(t)$, to a normally incident monochromatic plane wave of frequency ${\ensuremath{\omega}}_{o}$. As a consequence of the interaction of this incoming wave with the dynamic slab, the reflected and transmitted waves contain harmonics of the modulating frequency $\ensuremath{\Omega}$, namely, the slab itself becomes a polychromatic source of frequencies ${\ensuremath{\omega}}_{o}\ensuremath{-}n\ensuremath{\Omega}(n=0,\ifmmode\pm\else\textpm\fi{}1,\ifmmode\pm\else\textpm\fi{}2,\dots{})$. We establish a general formalism to quantify the reflected and transmitted fields for any periodic variation in the dielectric function. To achieve this, a description of wave propagation in a dynamic bulk is needed. A theoretical framework to treat such propagation, based on the concept of a temporal photonic crystal, is developed. As a consequence of the Bloch-Floquet theorem, the dispersion relation is a band structure that is periodic with frequency and exhibits forbidden wave vector gaps. The Poynting vectors of the transmitted and reflected fields are analyzed in detail. Our theory is applied to the case in which the dielectric function is modulated sinusoidally. We calculate numerically the magnitudes and phases of the reflection and transmission coefficients for several harmonics ${\ensuremath{\omega}}_{o}\ensuremath{-}n\ensuremath{\Omega}$ and slab thicknesses. Three modulation regimes are considered: weak, moderate, and strong. The response in the weak regime is similar to that of a Fabry-Perot etalon---the strengths of the harmonics are weak. For the strong-modulation regime, the strengths of reflection and transmission coefficients of the harmonics become large; they can even exceed one due to the openness of the system in which an external modulating agent can provide part of the invested energy. Dynamic variation in the dielectric properties of materials can give rise to new effects in wave propagation and to novel optical applications and is readily attainable with present-day technology.
11 May 2009
TL;DR: In this paper, the Lattice-Boltzmann method is applied to the propagation of planar acoustic waves to determine the resolution dependence of numerical dissipation and dispersion.
Abstract: Numerical simulations are performed to investigate the fundamental acoustics properties of the Lattice–Boltzmann method. The propagation of planar acoustic waves is studied to determine the resolution dependence of numerical dissipation and dispersion. The two setups considered correspond to the temporal decay of a standing plane wave in a periodic domain, and the spatial decay of a propagating planar acoustic pulse of Gaussian shape. Theoretical dispersion relations are obtained from the corresponding temporal and spatial analyses of the linearized Navier–Stokes equations. Comparison of theoretical and numerical predictions show good agreement and demonstrate the low dispersive and dissipative capabilities the Lattice–Boltzmann method. The analysis is performed with and without turbulence modeling, and the changes in dissipation and dispersion are discussed. Overall, the results show that the Lattice–Boltzmann method can accurately reproduce time-explicit acoustic phenomena.
TL;DR: Overall good agreement with experiments is obtained, which allows us to assign the Raman and IR peaks to specific phonons in density functional perturbation theory.
Abstract: Phonon dispersion relations and infrared and Raman spectra of crystalline Sb2Te3 were computed within density functional perturbation theory. Overall good agreement with experiments is obtained, which allows us to assign the Raman and IR peaks to specific phonons.
TL;DR: This phenomenon was investigated assuming a cylindrically symmetrical column and using the general dispersion theory of Aris, which relates the height equivalent to the theoretical plate (HETP) contribution due to a radial heterogeneity of the column to the radial distribution of the linear velocities of a compound peak.
Abstract: When columns packed with very fine particles are operated at high mobile phase velocities, the friction of the mobile phase percolating through the column bed generates heat. This heat dissipates along and across the column and axial and radial temperature gradients appear. The wall region of the column tends to be cooler than its center, and due to the influence of temperature on the mobile phase viscosity and on the equilibrium constant of analytes, the band velocity is not constant across the column. This radial heterogeneity of the temperature distribution across the column contributes to band broadening. This phenomenon was investigated assuming a cylindrically symmetrical column and using the general dispersion theory of Aris, which relates the height equivalent to the theoretical plate (HETP) contribution due to a radial heterogeneity of the column to the radial distribution of the linear velocities of a compound peak and to the radial distribution of its apparent dispersion coefficients in the col...
TL;DR: In this paper, the authors consider the problem of wave propagation in highly anisotropic elastic composites and derive two-scale limiting elastodynamic equations for their band-gap structure.
TL;DR: In this paper, an ensemble Monte Carlo method was used to study high field electron and hole transport in wurtzite phase GaN using a nonpolar carrier-phonon interaction, which was treated within the framework of the rigid pseudoion approximation using ab initio techniques to determine the phonon dispersion relation.
Abstract: High field electron and hole transport in wurtzite phase GaN is studied using an ensemble Monte Carlo method. The model includes the details of the full band structure derived from nonlocal empirical pseudopotential calculations. The nonpolar carrier-phonon interaction is treated within the framework of the rigid pseudoion approximation using ab initio techniques to determine the phonon dispersion relation. The calculated carrier-phonon scattering rates are consistent with the electronic structure and the phonon dispersion relation thus removing adjustable parameters such as deformation potential coefficients. The impact ionization transition rate is computed based on the calculated electronic structure and the corresponding wave-vector dependent dielectric function. The complex band structure of wurtzite GaN requires the inclusion of band-to-band tunneling effects that are critical at high electric fields. The electric-field-induced interband transitions are investigated by the direct solution of the time dependent multiband Schrodinger equation. The multiband description of the transport predicts a considerable increase in the impact ionization coefficients compared to the case in which tunneling is not considered. In the second part of this work it will be shown that the proposed numerical model correctly predicts the carrier multiplication gain and breakdown voltage of a variety of GaN avalanche photodetectors that have been recently fabricated by several research groups.
TL;DR: In this article, the dispersion relation of a metal-dielectric-gap optical waveguide was investigated, and it was shown that this mode of propagation has a cutoff at zero wavenumber, and that it is hollow-waveguide-like for small wavenumbers, while it approaches a surface-plasmon-like mode for large wenumbers.
Abstract: We have investigated the dispersion relation of a novel metal-dielectric-gap optical waveguide. This structure confines the optical field strongly in the gap region between metals and dielectric materials, and its size can be reduced to less than the wavelength of the transmitted light. In addition, the propagation length of light extends much greater than that of the surface plasmon modes on metal surfaces. We show that this mode of propagation has a cut-off at zero wavenumber, and that it is hollow-waveguide-like for small wavenumbers, while it approaches a surface-plasmon-like mode for large wavenumbers. A typical propagation length at around the communication wavelength is 10-20 mum, and optical fields are confined into an approximately 100 times 200 nm2 cross section.
TL;DR: The experimental results agree well with an analytical model for plasmon dispersion in coaxial waveguides and can be tuned by changing the coax geometry, composition, and surrounding dielectric index, enabling coaxial cavities with ultrasmall mode volumes.
Abstract: We determine the plasmon dispersion relation in coaxial waveguides composed of a circular channel separating a metallic core and cladding. Optical transmission measurements are performed on isolated coaxial nanoapertures fabricated on a Ag film using focused ion-beam lithography. The dispersion depends strongly on the dielectric material and layer thickness. Our experimental results agree well with an analytical model for plasmon dispersion in coaxial waveguides. We observe large phase shifts at reflection from the end facets of the coaxial cavity, which strongly affect the waveguide resonances and can be tuned by changing the coax geometry, composition, and surrounding dielectric index, enabling coaxial cavities with ultrasmall mode volumes.
TL;DR: In this article, the strong coupling regime occurring between a Tamm plasmon (TP) mode and an exciton from inorganic quantum wells (QWs) was observed.
Abstract: We report on the observation of the strong coupling regime occurring between a Tamm plasmon (TP) mode and an exciton from inorganic quantum wells (QWs). The sample is formed by a silver thin film deposited onto an AlAs/GaAlAs Bragg reflector containing InGaAs QWs located in the high refractive index layers. Angular resolved reflectometry experiments evidence a clear anticrossing in the dispersion relations, a signature of the strong coupling regime. The Rabi splitting energy is 11.5 meV. The experimental data are in very good agreement with simple transfer matrix calculations. The emission from low and high energy TP/exciton polaritons is also demonstrated.
TL;DR: In this paper, a theory for Bloch wave propagation in damped elastic media is presented, where the authors expand the eigenvalue problem governing the dispersion relation using a set of Bloch mode eigenvectors at each wave-vector point.
Abstract: We present a theory for Bloch wave propagation in damped elastic media. We expand the eigenvalue problem governing the dispersion relation using a set of Bloch mode eigenvectors at each wave-vector point. With the assumption of Rayleigh damping, this decomposition allows us to derive the band structure in the Brillouin zone. The damping ratio corresponding to each Bloch mode is also generated. We show that damping qualitatively alters the shape of the dispersion curves. Damping also results in a branch-overtaking phenomenon that has a significant effect on band gaps. As the damping is increased, a band-gap size can drop abruptly.
TL;DR: In this paper, the first experimental observations of small-scale electron density fluctuations inside the plasma volume were obtained, and observed in the expected ranges of spatial and time scales, with the help of a specially designed collective scattering device.
Abstract: Kinetic models and numerical simulations of E×B plasma discharges predict microfluctuations at the scales of the electron cyclotron drift radius and the ion plasma frequency. With the help of a specially designed collective scattering device, the first experimental observations of small-scale electron density fluctuations inside the plasma volume are obtained, and observed in the expected ranges of spatial and time scales. The anisotropy, dispersion relations, form factor, amplitude, and spatial distribution of these electron density fluctuations are described and compared to theoretical expectations.
TL;DR: A robust iterative technique for solving complex transcendental dispersion equations routinely encountered in integrated optics, which befits the multilayer dielectric and plasmonic waveguides forming the basis structures for a host of contemporary nanophotonic devices.
Abstract: We present a robust iterative technique for solving complex transcendental dispersion equations routinely encountered in integrated optics. Our method especially befits the multilayer dielectric and plasmonic waveguides forming the basis structures for a host of contemporary nanophotonic devices. The solution algorithm ports seamlessly from the real to the complex domain--i.e., no extra complexity results when dealing with leaky structures or those with material/metal loss. Unlike several existing numerical approaches, our algorithm exhibits markedly-reduced sensitivity to the initial guess and allows for straightforward implementation on a pocket calculator.
TL;DR: In this paper, the generation of primordial perturbations with a modified dispersion relation in various cosmological evolution scenarios has been studied and the authors stress that the formation of the power spectrum is strongly dependent on the background.
Abstract: In this paper we study the generation of primordial perturbations with a modified dispersion relation in various cosmological evolution scenarios. We stress that the formation of the power spectrum is strongly dependent on the background. By parameterizing a modified dispersion relation, we have listed all potential conditions on background evolution in order to seed scale-invariant primordial spectra. Working in a bounce model with a matterlike contracting phase, we obtain a red-tilted spectrum due to the modified dispersion relation.
TL;DR: For the leading correction in the nonrelativistic limit the exceptional sensitivity of cold-atom-recoil experiments remarkably allows us to set a limit within a single order of magnitude of the desired Planck-scale level, thereby providing the first example of Plankscale sensitivity in the study of the dispersion relation in controlled laboratory experiments.
Abstract: We use the results of ultraprecise cold-atom-recoil experiments to constrain the form of the energy-momentum dispersion relation, a structure that is expected to be modified in several quantum-gravity approaches. Our strategy of analysis applies to the nonrelativistic (small speeds) limit of the dispersion relation, and is therefore complementary to an analogous ongoing effort of investigation of the dispersion relation in the ultrarelativistic regime using observations in astrophysics. For the leading correction in the nonrelativistic limit the exceptional sensitivity of cold-atom-recoil experiments remarkably allows us to set a limit within a single order of magnitude of the desired Planck-scale level, thereby providing the first example of Planck-scale sensitivity in the study of the dispersion relation in controlled laboratory experiments.
TL;DR: In this article, the authors further extend the investigation of holographic gauge theories in external magnetic fields, continuing earlier work, and study the phenomenon of magnetic catalysis of mass generation in 1+3 and 1+2 dimensions.
Abstract: In this work we further extend the investigation of holographic gauge theories in external magnetic fields, continuing earlier work. We study the phenomenon of magnetic catalysis of mass generation in 1+3 and 1+2 dimensions, using D3/D7- and D3/D5-brane systems, respectively. We obtain the low energy effective actions of the corresponding pseudo Goldstone bosons and study their dispersion relations. The D3/D7 system exhibits the usual Gell-Mann–Oakes–Renner (GMOR) relation and a relativistic dispersion relation, while the D3/D5 system exhibits a quadratic non-relativistic dispersion relation and a modified linear GMOR relation. The low energy effective action of the D3/D5 system is related to that describing magnon excitations in a ferromagnet. We also study properties of general Dp/Dq systems in an external magnetic field and verify the universality of the magnetic catalysis of dynamical symmetry breaking.
TL;DR: In this article, a solution to the water-wave interaction with a submerged elastic plate of negligible thickness by the eigenfunction-matching method is presented, which depends on the solution of a special dispersion equation for a submerged plate and this is discussed in detail.
Abstract: We present a solution to the water-wave interaction with a submerged elastic plate of negligible thickness by the eigenfunction-matching method. The eigenfunction expansion depends on the solution of a special dispersion equation for a submerged elastic plate and this is discussed in detail. We show how the solution can be calculated for the case of normal incidence on a semi-infinite plate in two spatial dimensions and then extend this solution to obliquely incident waves, to a plate of finite length and to a circular finite plate in three dimensions. Numerical calculations showing various properties of the solutions are presented and a near-orthogonality relation for the eigenfunctions is used to derive an energy-balance relation.
TL;DR: The complexity of representation of operators in quantum mechanics can be characterized by the operator space entanglement entropy (OSEE), which grows at most logarithmically with time as discussed by the authors.
Abstract: The complexity of representation of operators in quantum mechanics can be characterized by the operator space entanglement entropy (OSEE). We show that in the homogeneous Heisenberg $XY$ spin 1/2 chains the OSEE for initial local operators grows at most logarithmically with time. The prefactor in front of the logarithm generally depends only on the number of stationary points of the quasiparticle dispersion relation and for the $XY$ model changes from 1/3 to 2/3 exactly at the point of quantum phase transition to long-range magnetic correlations in the nonequilibrium steady state. In addition, we show that the presence of a small disorder triggers a saturation of the OSEE.