Perturbative fluctuation dissipation relation for nonequilibrium finite-frequency noise in quantum circuits

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.

Figure 1

A general two-terminal conductor coupled arbitrarily an environment, which includes an impedance $Z\left(\omega \right)$ as well as other quantum conductors. Here, the conductor is a spatially extended tunnel junction with capacitive couplings, and $q$ denotes the tunneling charge.

Figure 2

Various regimes for the noise as a function of $\omega$ and ${\omega }_{\mathrm{dc}}$. The $(\omega ,{\omega }_{\mathrm{dc}})$ plane is partitioned into four quadrants separated by diagonal bands of width $k_{B}T/\hslash$ (orange) corresponding to thermal excitation of electron/hole pairs. The quantum regime corresponds to $\hslash \left|\omega \right|\gg k_{B}T$ or $\hslash |{\omega }_{\mathrm{dc}}|\gg k_{B}T$ whereas the thermal regime corresponds to both $\hslash \omega$ and $\hslash {\omega }_{\mathrm{dc}}$ much smaller than $k_{B}T$ (center).

Figure 3

A good conductor coupled to an arbitrary resistance $R$: density plots of the ratio ${\mathcal{S}}_R(\omega ,{\omega }_{\mathrm{dc}})$ of the backscattering current noise for $R\ne 0$ to its value at $R=0$ (no environment), as a function of $\hslash \omega /E_B$ and $\hslash {\omega }_{\mathrm{dc}}/E_B$, for $R=2R_q$ ($K=1/3$) and $R=R_q/2$ ($K=2/3$). Here, $k_{B}T=5E_B$ and $\hslash {\omega }_{\mathrm{RC}}=40E_B$.

Figure 4

Weakly transmitting conductor or low energy regime (below $E_B$) in a good conductor coupled to a resistance $R$: density plot of the ratio ${\mathcal{S}}_R(\omega ,{\omega }_{\mathrm{dc}})$ of the finite-frequency noise at $R\ne 0$ to its value at $R=0$ (no environment) as a function of $\hslash \omega /E_B$ and $\hslash {\omega }_{\mathrm{dc}}/E_B$, for $R=2R_q$ ($K=1/3$, left) and for $R=R_q/2$ ($K=2/3$, right). Here, $\hslash {\omega }_{\text{RC}}=40E_B$ and $k_{B}T=E_B/10$.

Figure 5

Infrared divergences for a good conductor at low temperature: density plots of the ratio ${\mathcal{S}}_R(\omega ,{\omega }_{\mathrm{dc}})$ of the backscattering current noise for $R\ne 0$ to the same quantity at $R=0$ (no environment) as function of $\hslash \omega /E_B$ and $\hslash {\omega }_{\mathrm{dc}}/E_B$ for $R=2R_q$ ($K=1/3$) and various values of the temperature: (a) high temperature, $k_{B}T=2E_B$, (b) lower temperature, $k_{B}T=0.5E_B$, and (c) the lowest temperature, $k_{B}T=0.1E_B$. Frequencies $\omega$ and ${\omega }_{\mathrm{dc}}$ are kept below the high-energy cutoff $\hslash {\omega }_{\mathrm{RC}}=40E_B$. Note the different color scales of each graph.