Quantum Theory of a Damped Electrical Oscillator and Noise

Abstract

Field quantization is applied to an electrical oscillating circuit. Damping effects are treated by perturbation theory. Quantum effects occur both in the damping and in the noise, and are discussed in detail. An interpretation is given of the infinite zero point contribution which appears in the theory of Callen and Welton. The average electromagnetic field energy of an oscillator with capacitance $C$, conductance $G$, and natural frequency $\omega$ as a function of time is given by $U=\left[\frac{1}{2}\hslash \omega +\frac{\hslash \omega }{\exp \left(\frac{\hslash \omega }{\mathrm{kT}}\right)-1\mathrm{}}\right][1-e^{-\frac{\mathrm{Gt}}{C}}]+U_0{e}^{-\frac{\mathrm{Gt}}{C}}.$ The mean squared noise voltage which would be measured in an experiment with a damped oscillator is given by ${\overline{V}}^2=\frac{1}{C}\left[\frac{1}{2}\hslash \omega +\frac{\hslash \omega }{\exp \left(\frac{\hslash \omega }{\mathrm{kT}}\right)-1\mathrm{}}\right].$ The maximum noise power which a conductance $G$ at temperature $T$ can transfer to a damped oscillator approaches the value $\frac{G\hslash \omega }{C\left[\exp \right(\frac{\hslash \omega }{\mathrm{kT}})-1]}.$