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Physical review. B, Condensed matter

Nonequilibrium quantum noise in chiral Luttinger liquids

Claudio Chamon, Denise E. Freed, Xiao-Gang Wen

Published Feb 15, 1996• 83 citations
Publisher

Abstract

We study nonequilibrium noise in chiral Luttinger liquids using the Landauer-Buttiker scattering approach, obtaining the current and voltage noise spectrum for a four-terminal measurement scheme. Experimental consequences of the tunneling of charges are present in the four-terminal measurement of both the low-frequency shot noise (\ensuremath{\omega} near 0), and the high-frequency Josephson noise (\ensuremath{\omega} near ${\mathrm{\ensuremath{\omega}}}_{\mathit{J}}$=${\mathit{e}}^{\mathrm{*}}$V/\ensuremath{\Elzxh}). Within perturbation theory, an algebraic singularity is present at the Josephson frequency ${\mathrm{\ensuremath{\omega}}}_{\mathit{J}}$=${\mathit{e}}^{\mathrm{*}}$V/\ensuremath{\Elzxh}, whose position depends on the charge ${\mathit{e}}^{\mathrm{*}}$ of the tunneling particles, either electrons or fractionally charged quasiparticles. These two types of tunneling are related by a strong-weak--coupling duality transformation. We show in a nonperturbative calculation for an exactly solvable point that the singularity at the quasiparticle frequency exists only in the limit of vanishing coupling, whereas the singularity at the electron frequency is present for all coupling strengths. The vanishing coupling limit corresponds to perfectly quantized Hall conductance in the case of quasiparticle tunneling between edge states in the fractional quantum Hall regime, and thus tunneling destroys the singularity at the quasiparticle frequency concomitantly with the quantized current. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.53.4033

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We study nonequilibrium noise in chiral Luttinger liquids using the Landauer-Buttiker scattering approach, obtaining the current and voltage noise spectrum for a four-terminal measurement scheme. Experimental consequences of the tunneling of charges are present in the four-terminal measurement of both the low-frequency shot noise (\ensuremath{\omega} near 0), and the high-frequency Josephson noise (\ensuremath{\omega} near \ensuremathωJ{\mathrm{\ensuremath{\omega}}}_{\mathit{J}}\ensuremathωJ​=e∗{\mathit{e}}^{\mathrm{*}}e∗V/\ensuremath{\Elzxh}). Within perturbation theory, an algebraic singularity is present at the Josephson frequency \ensuremathωJ{\mathrm{\ensuremath{\omega}}}_{\mathit{J}}\ensuremathωJ​=e∗{\mathit{e}}^{\mathrm{*}}e∗V/\ensuremath{\Elzxh}, whose position depends on the charge e∗{\mathit{e}}^{\mathrm{*}}e∗ of the tunneling particles, either electrons or fractionally charged quasiparticles. These two types of tunneling are related by a strong-weak--coupling duality transformation. We show in a nonperturbative calculation for an exactly solvable point that the singularity at the quasiparticle frequency exists only in the limit of vanishing coupling, whereas the singularity at the electron frequency is present for all coupling strengths. The vanishing coupling limit corresponds to perfectly quantized Hall conductance in the case of quasiparticle tunneling between edge states in the fractional quantum Hall regime, and thus tunneling destroys the singularity at the quasiparticle frequency concomitantly with the quantized current. \textcopyright{} 1996 The American Physical Society.
We study non-equilibrium noise in Chiral Luttinger Liquids using the Landauer-Buttiker Scattering approach, obtaining the current/voltage noise spectrum for a four-terminal measurement scheme. Experimental consequences of the tunneling of charges are present in the four-terminal measurement of both the low-frequency shot noise (ωωω near 0), and the high-frequency Josephson noise (ωωω near ωJ=e∗V/ℏω_J=e^*V/\hbarωJ​=e∗V/ℏ). Within perturbation theory, an algebraic singularity is present (to all orders) at the Josephson frequency ωJ=e∗V/ℏω_J=e^*V/\hbarωJ​=e∗V/ℏ, whose position depends on the charge e∗e^*e∗ of the tunneling particles, either electrons or fractionally charged quasiparticles. We show in a non-perturbative calculation for an exactly solvable point that the singularity at the quasiparticle frequency exists only in the limit of vanishing coupling, whereas the singularity at the electron frequency is present for all coupling strengths. The vanishing coupling limit corresponds to perfectly quantized Hall conductance in the case of quasiparticle tunneling between edge states in the fractional quantum Hall regime, and thus tunneling destroys the singularity at the quasiparticle frequency concomitantly with the quantized current.
We study nonequilibrium noise in the transmission current through barriers in one-dimensional Luttinger liquids and in the tunneling current between edges of fractional quantum Hall liquids. The distribution of tunneling events through narrow barriers can be described by a Coulomb gas lying in the time axis along a Keldysh (or nonequilibrium) contour. We show that the charges tend to reorganize as a dipole gas,which we use to describe the tunneling statistics. The dipole-gas picture allows us to have a unified description of the low-frequency shot noise and the high-frequency Josephson noise. The correlation between the charges within a dipole (intradipole) contributes to the high-frequency Josephson noise, which has an algebraic singularity at \ensuremath{\omega}=e∗{\mathit{e}}^{\mathrm{*}}e∗V/\ensuremath{\Elzxh}, whereas the correlations between dipoles (interdipole) are responsible for the low-frequency noise. We show that an independent or noninteracting dipole approximation gives a Poisson distribution for the locations of the dipole centers of mass, which gives a flat noise spectrum at low frequencies and corresponds to uncorrelated shot noise. Including interdipole interactions gives an additional 1/t2{\mathit{t}}^{2}t2 correlation between the tunneling events that results in an \ensuremath{\Vert}\ensuremath{\omega}\ensuremath{\Vert} singularity in the noise spectrum. We present a diagrammatic technique to calculate the correlations in perturbation theory, and show that contributions from terms of order higher than the dipole-dipole interaction should only affect the strength of the \ensuremath{\Vert}\ensuremath{\omega}\ensuremath{\Vert} singularity, but its form should remain \ensuremath{\sim}\ensuremath{\Vert}\ensuremath{\omega}\ensuremath{\Vert} to all orders in perturbation theory. A counting argument also suggests that the leading algebraic singularity at ${\mathrm{\ensuremath{\omega}}}_{\mathit{J}}$ should be \ensuremath{\propto}\ensuremath{\Vert}\ensuremath{\omega}-${\mathrm{\ensuremath{\omega}}}_{\mathit{J}}$${\mathrm{\ensuremath{\Vert}}}^{2\mathit{g}\mathrm{\ensuremath{-}}1}$ to all orders in perturbation theory.
A point contact in a Luttinger liquid couples left- and right-moving channels, producing shot noise. We calculate exactly the dc shot noise at zero temperature in the out-of-equilibrium steady state where current is flowing. Integrability ensures the existence of a quasiparticle basis where scattering is ``one by one'' off the point contact, enabling us to generalize the Landauer approach to shot noise to this interacting model. Experiments in a fractional quantum Hall effect device should test our results, providing a clear signal of the fractional charge of the Laughlin quasiparticles.
We compute exactly the nonequilibrium dc noise in a Luttinger liquid with an impurity and an applied voltage. By generalizing Landauer transport theory for Fermi liquids to interacting, integrable systems, we relate this noise to the density fluctuations of quasiparticles. We then show how to compute these fluctuations using the Bethe ansatz. The nontrivial density correlations from the interactions result in a substantial part of the nonequilibrium noise. The final result for the noise is a scaling function of the voltage, temperature, and impurity coupling. It may eventually be observable in tunneling between edges of a fractional quantum Hall effect device. \textcopyright{} 1996 The American Physical Society.
The current noise of a voltage‐biased interacting quantum wire connected to leads is computed in presence of an impurity in the wire. We find that in the weak backscattering limit the Fano factor characterizing the ratio between noise and backscattered current crucially depends on the noise frequency ω relative to the ballistic frequency vF/gL, where vF is the Fermi velocity, g the electron interaction strength, and L the length of the wire. In contrast to chiral Luttinger liquids, the noise is not only due to the Poissonian backscattering of fractionally charged quasiparticles at the impurity, but also to Andreev‐type reflections at the contacts. The frequency dependence of the noise needs to be analyzed to extract the fractional charge e* = eg of the bulk excitations.
We have recently calculated exact nonequilibrium quantum transport properties through a point contact in a Luttinger liquid. Using a particular quasiparticle basis of the Hilbert space dictated by integrability, we here compute explicitly the exact I(V) characteristic and conductance out of equilibrium as a function of driving voltage V and temperature T. These are described by universal scaling functions of two variables, the scaled point-contact interaction strength, and V/T. The differential-conductance curve as a function of the interaction strength broadens significantly as V/T is increased, and develops a pronounced maximum at a (universal) critical value (eV/kB{\mathit{k}}_{\mathit{B}}kB​T)=7.188 68. . .. In addition, we derive an exact duality between strong and weak backscattering. The theory presented here has recently been realized experimentally in resonant tunneling-transport experiments between edge states in fractional quantum Hall effect devices. In this context the exact duality is between electron tunneling and Laughlin-quasiparticle tunneling.