Abstract
The fifth lesson began with a reminder of the relationship between the number of photons in a propagative mode and the corresponding root-mean-square values of currents and voltages. From this type of relationship, we can calculate the fluctuations of electrical quantities for an LC circuit, and thereby establish for any impedance the relationship between the real part of the impedance and the spectral density of Johnson noise fluctuations. In the quantum case, this spectral density is asymmetrical : positive frequencies, corresponding to spontaneous and stimulated emission processes from the circuit connected to the impedance, are more intense than negative frequencies, corresponding to absorption processes. We have presented this spectral density calculation both from the Caldeira-Leggett point of view, where the impedance is replaced by an infinite series of harmonic oscillators (stationary modes), and from the Nyquist point of view, which replaces the dissipative part of the impedance by a semi-infinite transmission line (propagative modes), populated by an incident thermal field. The input-output formalism is very useful for passing from the circuit equations with the two dissipation and forcing terms, to the field diffusion equations on the core formed by the reactive part of the circuit. In this way, the quantum fluctuation-dissipation theorem can be seen as a consequence of the symmetry property of the circuit: the latter cannot distinguish, in the process of field diffusion conducted by the transmission line, a deterministic signal from thermal noise.
