Dibyendu Roy

Heat Transport in Harmonic Lattices

We work out the non-equilibrium steady state properties of a harmonic lattice which is connected to heat reservoirs at different temperatures. The heat reservoirs are themselves modeled as harmonic systems. Our approach is to write quantum Langevin equations for the system and solve these to obtain steady state properties such as currents and other second moments involving the position and momentum operators. The resulting expressions will be seen to be similar in form to results obtained for electronic transport using the non-equilibrium Green’s function formalism. As an application of the formalism we discuss heat conduction in a harmonic chain connected to self-consistent reservoirs. We obtain a temperature dependent thermal conductivity which, in the high-temperature classical limit, reproduces the exact result on this model obtained recently by Bonetto, Lebowitz and Lukkarinen.

Few-photon optical diode

We propose a scheme of realizing an optical diode at the few-photon level. The system consists of a one-dimensional waveguide coupled asymmetrically to a two-level system. The two or multiphoton transport in this system is strongly correlated. We derive exactly the single and two-photon current and show that the two-photon current is asymmetric for the asymmetric coupling. Thus the system serves as an optical diode which allows transmission of photons in one direction much more efficiently than the opposite.

Colloquium: Strongly interacting photons in one-dimensional continuum

Photons, the particles of light, are in most conditions very weakly interacting. Nevertheless, it is possible to make them interact by altering environmental conditions, for instance, in the interior of certain materials or by squeezing them in confined geometries. In this Colloquium the topic of photons interacting strongly when confined to a one-dimensional geometry is discussed from experimental and theoretical perspectives.

Heat transport and phonon localization in mass-disordered harmonic crystals

We investigate the steady-state heat current in two- and three-dimensional disordered harmonic crystals in a slab geometry connected at the boundaries to stochastic white-noise heat baths at different temperatures. The disorder causes short-wavelength phonon modes to be localized so the heat current in this system is carried by the extended phonon modes which can be either diffusive or ballistic. Using ideas both from localization theory and from kinetic theory we estimate the contribution of various modes to the heat current and from this we obtain the asymptotic system size dependence of the current. These estimates are compared with results obtained from a numerical evaluation of an exact formula for the current, given in terms of a frequency-transmission function, as well as from direct nonequilibrium simulations. These yield a strong dependence of the heat flux on boundary conditions. Our analytical arguments show that for realistic boundary conditions the conductivity is finite in three dimensions but we are not able to verify this numerically, except in the case where the system is subjected to an external pinning potential. This case is closely related to the problem of localization of electrons in a random potential and here we numerically verify that the pinned three-dimensional system satisfies Fouriers law while the two-dimensional system is a heat insulator. We also investigate the inverse participation ratio of different normal modes.

Electron transport in a one dimensional conductor with inelastic scattering by self-consistent reservoirs

equilibrium Green’s functions. In the linear-response limit, we are able to solve the model exactly and obtain expressions for various transport coefficients. Standard linear-response relations are shown to be valid. We also explicitly compute the heat dissipation and show that for wires of length N, where is a coherence length scale, dissipation takes place uniformly along the wire. For N, when transport is ballistic, dissipation is mostly at the contacts. In the intermediate range between Ohmic and ballistic transport, we find that the chemical-potential profile is linear in the bulk with sharp jumps at the boundaries. These are explained using a simple model where the left and right moving electrons behave as persistent random walkers.

Crossover from ballistic to diffusive thermal transport in quantum Langevin dynamics study of a harmonic chain connected to self-consistent reservoirs.

Through an exact analysis using quantum Langevin dynamics, we demonstrate the crossover from ballistic to diffusive thermal transport in a harmonic chain with each site connected to Ohmic heat reservoirs. The temperatures of the two heat baths at the boundaries are specified from the beginning, whereas the temperatures of the interior heat reservoirs are determined self-consistently by demanding that in the steady state, on average, there is no heat current between any such (self-consistent) reservoir and the harmonic chain. The essence of our study is that the effective mean free path separating the ballistic regime of transport from the diffusive one emerges naturally.

Heat conduction and phonon localization in disordered harmonic crystals

We investigate the steady-state heat current in two- and three-dimensional isotopically disordered harmonic lattices. Using localization theory as well as kinetic theory we estimate the system size dependence of the current. These estimates are compared with numerical results obtained using an exact formula for the current given in terms of a phonon transmission function, as well as by direct nonequilibrium simulations. We find that heat conduction by high frequency modes is suppressed by localization while low frequency modes are strongly affected by boundary conditions. Our heuristic arguments show that Fouriers law is valid in a three-dimensional disordered solid except for special boundary conditions. We also study the pinned case relevant to localization in quantum systems and often used as a model system to study the validity of Fouriers law. Here we provide the first numerical verification of Fouriers law in three dimensions. In the two-dimensional pinned case we find that localization of phonon modes leads to a heat insulator.

Majorana fermions in a topological superconducting wire out of equilibrium: Exact microscopic transport analysis of ap-wave open chain coupled to normal leads

Topological superconductors are prime candidates for the implementation of topological-quantum-computation ideas because they can support non-Abelian excitations like Majorana fermions. We go beyond the low-energy effective-model descriptions of Majorana bound states (MBSs), to derive non-equilibrium transport properties of wire geometries of these systems in the presence of arbitrarily large applied voltages. Our approach involves quantum Langevin equations and non-equilibrium Greens functions. By virtue of a full microscopic calculation we are able to model the tunnel coupling between the superconducting wire and the metallic leads realistically; study the role of high-energy non-topological excitations; predict how the behavior compares for increasing number of odd vs. even number of sites; and study the evolution across the topological quantum phase transition (QPT). We find that the normalized spectral weight in the MBSs can be remarkably large and goes to zero continuously at the topological QPT. Our results have concrete implications for the experimental search and study of MBSs.

Scattering of electrons from an interacting region

We address the problem of transmission of electrons between two noninteracting leads through a region where they interact (quantum dot). We use a model of spinless electrons hopping on a one-dimensional lattice and with an interaction on a single bond. We show that all two-particle scattering states can be found exactly. Comparisons are made with numerical results on the time evolution of a two-particle wave packet, and several interesting features are found. For N particles, the scattering state is obtained within a two-particle scattering approximation. For a dot connected to Fermi seas at different chemical potentials, we find an expression for the change in the Landauer current resulting from the interactions on the dot. We end with some comments on the case of spin-1/2 electrons.

Heat Transport in Ordered Harmonic Lattices

We consider heat conduction across an ordered oscillator chain with harmonic interparticle interactions and also onsite harmonic potentials. The onsite spring constant is the same for all sites excepting the boundary sites. The chain is connected to Ohmic heat reservoirs at different temperatures. We use an approach following from a direct solution of the Langevin equations of motion. This works both in the classical and quantum regimes. In the classical case we obtain an exact formula for the heat current in the limit of system size N→∞. In special cases this reduces to earlier results obtained by Rieder, Lebowitz and Lieb and by Nakazawa. We also obtain results for the quantum mechanical case where we study the temperature dependence of the heat current. We briefly discuss results in higher dimensions.