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arXiv:cond-mat/0211681

Quantum noise, detailed balance and Kubo formula in nonequilibrium quantum systems

Uri Gavish, Y. Imry, Yehoshua B. Levinson

Published Nov 29, 2002• 4 citations
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Abstract

Current quantum noise can be pictured as a sum over transitions through which the electronic system exchanges energy with its environment. We formulate this picture and use it to show which type of current correlators are measurable, and in what measurement the zero point fluctuations will play a role (the answer to the latter is as expected: only if the detector excites the system.) Using the above picture, we calculate and give physical interpretation of the finite-frequency finite-temperature current noise in a noninteracting Landauer-type system, where the chemical potentials of terminals 1 and 2 are $μ+eV/2$ and $μ-eV/2$ respectively, and derive a detailed-balance condition for this nonequilibrium system. Finally, we derive a generalized form of the Kubo formula for a wide class of interacting nonequilibrium systems, relating the differential conductivity to the current noise.

DOI
10.48550/arxiv.cond-mat/0211681

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Current quantum noise can be pictured as a sum over transitions through which the electronic system exchanges energy with its environment. We formulate this picture and use it to show which type of current correlators are measurable, and in what measurement the zero point fluctuations will play a role (the answer to the latter is as expected: only if the detector excites the system.) Using the above picture, we calculate and give physical interpretation of the finite-frequency finite-temperature current noise in a noninteracting Landauer-type system, where the chemical potentials of terminals 1 and 2 are μ+eV/2μ+eV/2μ+eV/2 and μ−eV/2μ-eV/2μ−eV/2 respectively, and derive a detailed-balance condition for this nonequilibrium system. Finally, we derive a generalized form of the Kubo formula for a wide class of interacting nonequilibrium systems, relating the differential conductivity to the current noise.
The conductance of one-dimensional interacting electron systems is calculated in a manner similar to Landauer's argument for non-interacting systems. Unlike in previous studies in which the Kubo formula was used, the conductance is directly evaluated as the ratio of current JJJ to the chemical potential difference ΔμΔμΔμ between right-going and left-going particles. It is shown that both JJJ and ΔμΔμΔμ are renormalized by electron-electron (e-e) interactions, but their ratio, the conductance, is not renormalized at all if the e-e interactions are the only scattering mechanism. It is also shown that nonequilibrium current fluctuation at low frequency is absent in such a case. These conclusions are drawn for both Fermi liquids (in which quasi-particles are accompanied with the backflow) and Tomonaga-Luttinger liquids.
The Kubo fluctuation-dissipation theorem relates the current fluctuations of a system in an equilibrium state with the linear AC-conductance. This theorem holds also out of equilibrium provided that the system is in a stationary state and that the linear conductance is replaced by the (dynamic) conductance with respect to the nonequilibrium state. We provide a simple proof for that statement and then apply it in two cases. We first show that in an excess noise measurement at zero temperature, in which the impedance matching is maintained while driving a mesoscopic sample out of equilibrium, it is the nonsymmetrized noise power spectrum which is measured, even if the bare measurement, i.e. without extracting the excess part of the noise, obtains the symmetrized noise. As a second application we derive a commutation relation for the two components of fermionic or bosonic currents which holds in every stationary state and which is a generalization of the one valid only for bosonic currents. As is usually the case, such a commutation relation can be used e.g. to derive Heisenberg uncertainty relationships among these current components.
The conductance of one-dimensional interacting electron systems is calculated in a manner similar to Landauer's argument for non-interacting systems.Unlike in previous studies in which the Kubo formula was used, the conductance is directly evaluated as the ratio of current J to the chemical potential difference ∆µ between right-going and left-going particles.It is shown that both J and ∆µ are renormalized by electron-electron (e-e) interactions, but their ratio, the conductance, is not renormalized at all if the e-e interactions are the only scattering mechanism.It is also shown that nonequilibrium current fluctuation at low frequency is absent in such a case.These conclusions are drawn for both Fermi liquids (in which quasi-particles are accompanied with the backflow) and Tomonaga-Luttinger liquids.
We develop a general perturbative computation of finite-frequency quantum noise that applies, in particular, to both good or weakly transmitting strongly correlated conductors coupled to a generic environment. Under a minimal set of hypotheses, we show that the noise can be expressed through the nonequilibrium dc current only, generalizing a nonequilibrium fluctuation dissipation relation. We use this relation to derive explicit predictions for the nonequilibrium finite-frequency noise for a single channel conductor connected to an arbitrary Ohmic environment.
We calculate the finite-frequency current noise of a nonequilibrium resonance-level quantum dot close to a dissipative quantum phase transition of the Kosterlitz-Thouless (KT) type between a de-localized phase for weak dissipation and a localized phase for strong dissipation. The resonance-level is coupled to two spinless fermionic baths with a finite bias voltage and an Ohmic boson bath representing the dissipative environment. The system is equivalent to an effective anisotropic Kondo model out of equilibrium. To compute the finite-frequency noise, we combine two recently developed Functional Renormalization Group (FRG) approaches in Refs.[17,22] and in Ref.[23]. The nonequilibrium current noise at zero-temperature and finite frequencies shows a singular dip in the de-localized phase for the magnitude of frequencies equal to the bias voltage; while the dip is smeared out as the system moves to the localized phase. The corresponding peak-to-dip crossover is found in the AC conductance for the magnitude of frequencies equal to the bias voltage. The relevance and applications of our results for the experiments and for tunnelings between Fractional Quantum Hall Edge (FQHE) states and chiral Luttinger liquids are discussed.