Vibrations of glass-like disordered chains

Propagation of vibrational modes in classical harmonic lattice with correlated disorder

The vibrational modes with nonzero frequency are localized in harmonic lattice with disordered masses. In our work, we investigated numerically the propagation of vibrational energy in harmonic lattice with long-range correlated disordered masses, which are randomly distributed with power law spectrum S ( k ) ∝ k - α . For α = 0, a standard uncorrelated disordered mass distribution was observed and for α > 0 its distribution exhibits intrinsic long-range correlations. Our procedure was done by the numerical solution of the classical equations for the mass displacement and velocities. Energy flow was investigated after injection of an initial wave-packet with energy E0 and the dynamics of the vibrational energy wave-packet was analyzed. We also investigated the dynamics of a pulse pumped at one side of the lattice. Our calculations suggest that vibrational modes with nonzero frequency propagate within harmonic lattice with correlated disordered masses distribution.

Enhanced localization, energy anomalous diffusion and resonant mode in harmonic chains with correlated mass-spring disorder

In this work, we study the vibrational modes and energy spreading in a harmonic chain model with diluted second-neighbors couplings and correlated mass-spring disorder. While all nearest neighbor masses are coupled by an elastic spring, second neighbors springs are introduced with a probability pD. The masses are randomly distributed according to the site connectivity mi = m0 (1 + 1/n(α)(I), where ni is the connectivity of the site i and α is a tunable exponent. We show that maximum localization of the vibrational modes is achieved for α ≃ 3/4. The time-evolution of the energy wave-packet is followed after an initial localized excitation. While the participation number remains finite, the energy spread is shown to be sub-diffusive after a displacement and super-diffusive after an impulse excitation. These features are related to the development of a power-law tail in the wave-packet distribution. Further, we unveil that the spring dilution leads to the emergence of a resonant localized state which is signaled by a van Hove singularity in the density of states.

Energy transport in a one-dimensional harmonic ternary chain with Ornstein-Uhlenbeck disorder

In this paper we study a one-dimensional ternary harmonic chain with the mass distribution constructed from an Ornstein-Uhlenbeck process. We generate a ternary mass disordered distribution by generating the correlated Ornstein-Uhlenbeck process and mapping it into a sequence of three different values. The probability of each value is controlled by a fixed parameter b. We analyze the localization aspect of the above model by numerical solution of the Hamilton equations and by the transfer matrix formalism. Our results indicate that the correlated ternary mass distribution does not promote the appearance of new extended modes. In good agreement with previous work, we obtain extended modes for b → ∞; however, we explain in detail the main issue behind this apparent localization- delocalization transition. In addition, we obtain the energy dynamics for this classical chain.

Geometrical and Anderson transitions in harmonic chains with constrained long-range couplings

Low-dimensional systems with long-range couplings usually present phase transitions which are absent in the short-ranged counterpart model. In this work, we show that a harmonic chain with long-range couplings restricted by a cost function proportional to the chain length N exhibits two distinct phase transitions. In the present model, two sites at a distance r>1 are connected by a spring with probability 1/r(α) with the constraint that the total length of the non-nearest-neighbor couplings is limited to λN, where λ is a cost parameter. A geometrical phase transition is found at α=1.5 between a phase with a finite number of long-range couplings and a phase on which the number of long-range couplings is proportional to the system size. Further, the normal vibrational modes of this chain display a phase transition from delocalized to localized modes at a smaller value of α. Maximum effective disorder is reached at α=2 for which the frequency of the lowest vibrational mode exhibits a pronounced peak.

Vibrational modes in a two-dimensional aperiodic harmonic lattice

We study the nature of collective excitations in classical harmonic lattices with aperiodic and pseudo-random mass distributions. Using a matrix recursive reformulation of the mass displacement equation, we compute the localization length within the band of allowed frequencies. Our numerical calculations indicate that, for aperiodic arrays of masses, a new phase of extended states appears in this model. Solving numerically the Hamilton equations for momentum and displacement along the chain, we compute the spreading of an initially localized energy excitation. We find that for sufficient aperiodicity, there is a ballistic propagation of the energy pulse.

Effect of disorder on slow light velocity in optical slow-wave structures

Slow-wave optical structures such as coupled photonic crystal cavities, coupled microresonators, and similar coupled-resonator optical waveguides are being proposed for slowing light because of the nature of their dispersion relationship. Since the group velocity becomes small, slow light and enhanced light-matter interaction may be observed at the edges of the waveguiding band. We derive a model of the effects of disorder on slow light in such structures, obtaining a relationship between the root-mean-square variation in the coupling coefficients and how slow the light is at the band edge.

Vibrons in a one-dimensional lattice of hydrogen-bonded peptide units: influence of the inhomogeneous mass distribution in the amino acid sequence

The influence of the inhomogeneous mass distribution in the amino acid sequence on amide-I vibrons in alpha-helices is studied within the small polaron approach in a one-dimensional (1D) model. It is shown that inhomogeneities in the sequence favor a randomness in the polaron Hamiltonian via the dressing mechanism. The polaron dynamics is thus described by a 1D tight binding model with correlated off-diagonal disorder. At low temperature, the polaron hopping constants exhibit small fluctuations around a large average value so that the polaron Hamiltonian appears weakly disordered. Extended states occur over a wide range of energies around the band center whereas the states close to the band edges appear localized. By contrast, at biological temperature, a stronger disorder takes place which originates in a drastic decrease of the average hopping constant. The number of localized states increases but few states close to the band center exhibit a localization length about to or greater than the lattice size. The extended nature of these latter states is attributed to the occurrence of short range correlations in the random hopping constants.