Introducing time-dependent drive into the Hamiltonian of quantised electric circuits Suppose I have the schematic of a superconducting electric circuit composed of (quasi) lossless linear inductances and capacitances and some non-linear inductances, eg. Josephson junctions. The Hamiltonian of the circuit is derived according to Devoret, 1997.
However this article considers the case only of a classical DC drive (so calles bias). Questions:
(1) Do you have any idea how to introduce a time-dependent drive into the system and how to consctruct the Hamiltonian of the given system? The driving may be considered classical and to begin with sinusoidal is enough.
(2) Do you have any idea how to treat the a non-classical drive, ie. the energy from the generators into the cicrcuit is coming in the form of single photons well separated in time?
 A: A single photon mode is described by a creation operator. The problem is not complicated and really generic in bosonic problems. 
How to include the driving is just a matter of taste. Suppose your circuit is described by the degrees of freedom $x$ and $p$ (say position and momentum). A force $f$ would be added as a term $x\cdot f$ in the Hamiltonian for instance (modulo all the required integral / most certainly in time for lumped elements). 
For a circuit it's the same, except $p$ and $x$ are usually flux/charge and/or current/voltage or something else, but certainly less intuitive than position and momentum. It means you can find a coupling with either $p$ and/or $x$, so either something like $f\cdot x$ or $f\cdot p$ (modulo problems of dimensions). 
To make it time-dependent is trivial, say $f\left(t\right)\cdot x$ is the coupling term in the Hamiltonian.
To quantise the coupling, you should impose a dynamics of this extra degree of freedom. For instance, if it's a photon mode, you can discuss this degrees of freedom as an oscillator, say $H_{\text{photon}}=p^{2}/2m+\omega^{2}x^{2}=\hbar\omega a^{\dagger}a$ and you make a coupling position-position, momentum-position, etc... depending on the nature of the coupling (inductive / capacitive / mixed ...)
So for instance suppose your system is a simple LC circuit, so it corresponds to a harmonic oscillator, say with creation/annihilation operators $b$ and $b^{\dagger}$. Further suppose that the photon mode is harmonic as well, and that the coupling is position-position like, then you have something like 
$$H\sim\hbar\omega a^{\dagger}a+\hbar\Omega b^{\dagger}b+g\left(b+b^{\dagger}\right)\left(a+a^{\dagger}\right)$$
with $\Omega$ the LC resonance-frequency and $g$ a coupling constant (to simplify).
