Analytical solution of driven time-dependent mesoscopic circuits

When a time-dependent harmonic system is further driven by an external source, it presents significant challenges to obtaining analytical solutions for the system. This paper applies the Lewis-Riesenfeld invariant method to a mesoscopic time-dependent dissipative inductor-capacitor (RLC) circuit with an external source. By constructing quantum invariant Hermitian operators and introducing auxiliary equations that can describe the effects of external sources and resistance, we analytically solve the Schrödinger equation of the system. Taking an alternating current (AC) voltage source as an example, we derive the generalized coherent states of a mesoscopic time-dependent RLC circuit with an external source, calculate the quantum fluctuations of charge and current, and obtain their corresponding uncertainty relations. The results indicate that the time-dependent inductance and resistance disrupt the minimum uncertainty relation of the Glauble coherent states in driven time-dependent mesoscopic RLC circuits. This research contributes to the establishment of a quantum theory for mesoscopic circuits.