  We investigate quantum synchronization phenomena in electrical circuits that
incorporate specifically designed nonconservative elements. A dissipative
theory of classical and quantized electrical circuits is developed based on the
Rayleigh dissipation function. The introduction of this framework enables the
formulation of a generalized version of classical Poisson brackets, which are
termed Poisson-Rayleigh brackets. By using these brackets, we are able to
derive the equations of motion for a given circuit. Remarkably, these equations
are found to correspond to Kirchhoff's current laws when Kirchhoff's voltage
laws are employed to impose topological constraints, and vice versa. In the
quantum setting, the equations of motion are referred to as the
Kirchhoff-Heisenberg equations, as they represent Kirchhoff's laws within the
Heisenberg picture. These Kirchhoff-Heisenberg equations, serving as the native
equations for an electrical circuit, can be used in place of the more abstract
master equations in Lindblad form. To validate our theoretical framework, we
examine three distinct circuits. The first circuit consists of two resonators
coupled via a nonconservative element. The second circuit extends the first to
incorporate weakly nonlinear resonators, such as transmons. Lastly, we
investigate a circuit involving two resonators connected through an inductor in
series with a resistor. This last circuit, which incidentally represents a
realistic implementation, allows for the study of a singular system, where the
absence of a coordinate leads to an ill-defined system of Hamilton's equations.
To analyze such a pathological circuit, we introduce the concept of auxiliary
circuit element. After resolving the singularity, we demonstrate that this
element can be effectively eliminated at the conclusion of the analysis,
recuperating the original circuit.
