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In circuit QED, we construct the hamiltonian of a circuit following Devoret's procedure.

We start from a classical treatment, in which we impose Kirchhoff's voltage laws on fundamental loops and construct the lagrangian in such a way that it returns Kirchhoff's current laws on independent nodes as equations of motion. Once the lagrangian has been constructed, we can move on to the hamiltonian and then quantize the system.

Therefore, the first step is to build the classical theory, and then quantize it (as it is done for the LC circuit, for example).

However, is there a classical theory, if there is a Josephson junction in the circuit? Is it correct to start with a classical treatment, if there is a junction in the circuit, as if it were a classical circuit element, or is it a formal extension of the procedure?

Typically, we do this way even in this case. Now, does the classical lagrangian, that we obtain in such a case, describe a real classical system, at least in some regime, that we want to quantize? Shouldn't current and phase be operators? Is there a regime in which all Kirchhoff's laws hold classically, that justify the first step of the procedure even in this case?

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I’ll respond with what I know. The answer won’t be 100% complete, but it’s a good start. To analyze circuits containing junctions, generally, as you mentioned, we start by considering a classical Hamiltonian. In this initial phase, since the circuit is analyzed CLASSICALLY, the junction is replaced with a linear inductance. This circuit is not identical to the original one, but it is very similar. So, the classical Lagrangian describes a system where instead of junctions, you have simple inductances. Why do we do this? Because it allows us to diagonalize the "CLASSICAL" system and subsequently express the nonlinearities introduced by the junction using the \phi operators. As you might have understood, in the case of junctions, we always start with the standard classical procedure and then add the nonlinearities later. When you move to the quantum Hamiltonian, all physical quantities are redefined as operators, so yes, the phase is an operator. As for the current, it can certainly be redefined as operators, but I’ve never seen it done because, for the purpose of writing the Hamiltonian, it is not necessary. To recap the steps:

  1. Look at the circuit; if there are junctions, transform them into inductances. Write the Lagrangian of the classical circuit.
  2. Find the Hamiltonian.
  3. Transition to the quantum Hamiltonian by transforming the variables with which the Hamiltonian was written (e.g., charge and flux) into operators.
  4. Add the nonlinearities of the junctions.
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