How to interpret the spectral density of the Keldysh nonequilibrium Green's function

TL;DR

This paper investigates the spectral function of the Keldysh nonequilibrium Green's function, revealing its limitations as a probability density and proposing a convoluted spectral function for better interpretation, demonstrated through quantum dot charge dynamics.

Contribution

It introduces a convoluted spectral function incorporating the time-energy uncertainty principle to enable a probability interpretation of the spectral density.

Findings

Spectral function can take negative values, challenging probability interpretation.

A convoluted spectral function allows for a probabilistic interpretation.

Application to quantum dot charge dynamics demonstrates the method's usefulness.

Abstract

This paper is devoted to the study and interpretation of the spectral function $\mathbf{A}(\omega, T)$ of the Keldysh nonequilibrium Green's function. The spatial diagonal of the spectral function is often interpreted as a time-dependent local density of states. We show that this object can take negative values implying that a simple probability interpretation as a time-dependent density of states is not possible. The same issue also occurs for the Wigner function $P(x,p)$ where it is solved by taking the uncertainty principle into account. We follow the same path and incorporate the time-energy uncertainty relation to define a convoluted spectral function that allows for a probability interpretation. The usefulness of this quantity as a interpretative tool is demonstrated by visualizing the charge dynamics in a quantum dot coupled to superconducting leads.