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Chinese Journal of Quantum Electronics

Application of Heisenberg correspondence principle in the mesoscopic time-dependent coupled circuits

Zhaoxia Sheng

Published Jan 1, 2006• 0 citations
OpenAlex

Abstract

For the general time-dependent inductance-capacitance coupled circuits, quantum-classical correspondence and the quantum fluctuations are studied by using the Heisenberg correspondence principle. Through the wave functions and exact solution of equations of motion in Heisenberg picture, the classical solution is derived from quantum solution in the limit of large quantum numbers carrying out the average over the initial phase in the matrix elements, quantum fluctuations of the charges and magnetic fluxes are obtained. When the inductances in the circuit increase and the capacitances decrease exponentially with time, quantum fluctuations of the charge and current decrease exponentially with time; When the inductances and capacitances in the two component circuits are equal to each other and time-independent, it is found that the coupling capacitance tends to reduce the quantum fluctuations of the electric current and the coupling inductance tends to reduce the charge quantum fluctuations.

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For the general time-dependent inductance-capacitance coupled circuits, quantum-classical correspondence and the quantum fluctuations are studied by using the Heisenberg correspondence principle. Through the wave functions and exact solution of equations of motion in Heisenberg picture, the classical solution is derived from quantum solution in the limit of large quantum numbers carrying out the average over the initial phase in the matrix elements, quantum fluctuations of the charges and magnetic fluxes are obtained. When the inductances in the circuit increase and the capacitances decrease exponentially with time, quantum fluctuations of the charge and current decrease exponentially with time; When the inductances and capacitances in the two component circuits are equal to each other and time-independent, it is found that the coupling capacitance tends to reduce the quantum fluctuations of the electric current and the coupling inductance tends to reduce the charge quantum fluctuations.
For the damped inductive coupled electric circuit with time-dependent inductances and capacitances,the system is quantized with the electron-phonon interaction included and the quantum fluctuations are calculated.In the studied coupled circuit,there are resistances both in the component circuits and in the coupling part.When the inductances and the capacitances are independent of time,the physical quantities will first decrease with oscillations and then reach a stable state.At the stable state,if the inductances and capacitances in the two component circuits are the same and the resistances go to zero,the quantum fluctuations will tend to that of the undamped circuit.
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Starting from Kirchhoff's equation for each mesoscopic coupled circuit, we investigated the quantum fluctuations of a dissipative mesoscopic capacitance–resistance–inductance coupled circuit using canonical transformation and unitary transformation. It is shown that the quantum fluctuations of charges and currents are not only related to the parameters of components in the self-circuit and coupled part, but also influenced by the parameters of other circuit. The uncertainty relations of the charges and currents are analyzed briefly.
The quantum fluctuations of charges and currents of mesoscopic capacitance coupled circuit with mutual inductance and power source were investigated by using canonical transformation and unitary transformation. The results show that the quantum fluctuations of charges and currents are independent of power source. When the components of the circuit are given, if the value of L_1/L_2 is very small or very large, the effect of coupling coefficient on quantum fluctuations of charges and currents are great. From no mutual inductance to having mutual inductance, the quantum fluctuation of charge of loop 2 and the quantum fluctuation of current of loop 1 are obvious. We can conclude that it play a very important role whether the mutual inductance exists or not.