Abstract
In the third lesson, we dealt with the formalism of the quantum Langevin equation, which we applied to establish the evolution equations of the non-degenerate three-wave mixer introduced in the previous lesson. For an electrical oscillator, damping and excitation can be physically described by the Nyquist model, which replaces resistance and source by a semi-infinite transmission line in which two propagating fields, the incoming and outgoing fields, flow in opposite directions. The outgoing field accounts for dissipation and corresponds to the field radiated by the oscillator in the line. The incoming field provides the excitation and noise associated with dissipation. The quantum Langevin equation can be seen quite naturally as resulting from Kirchhoff's law of conservation of current at a node : the total current supplied per line must be equal to the sum of the currents sent through the capacitance and inductance. The quantum Langevin equation thus establishes a relationship between the incoming field and the stationary field inside the resonator. The undamped nature of the resonators in the problem allows us to invoke the so-called rotating wave approximation, transforming the Langevin equation into a series of linear equations with time-independent coefficients.
