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Chinese Physics

The quantum Kirchhoff equation and quantum current and energy spectrum of a homogeneous mesoscopic dissipation transmission line

Cui Yuan-shun

Published Oct 1, 2007• 2 citations
Publisher

Abstract

On the basis of quantization of charge, the loop equations of quantum circuits are investigated by using the Heisenberg motion equation for a mesoscopic dissipation transmission line. On the supposition that the system has a symmetry under translation in charge space, the quantum current and the quantum energy spectrum in the mesoscopic transmission line are given by solving their eigenvalue equations. Results show that the quantum current and the quantum energy spectrum are not only related to the parameters of the transmission line, but also dependent on the quantized character of the charge obviously.

DOI
10.1088/1009-1963/16/10/045

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On the basis of quantization of charge, the loop equations of quantum circuits are investigated by using the Heisenberg motion equation for a mesoscopic dissipation transmission line. On the supposition that the system has a symmetry under translation in charge space, the quantum current and the quantum energy spectrum in the mesoscopic transmission line are given by solving their eigenvalue equations. Results show that the quantum current and the quantum energy spectrum are not only related to the parameters of the transmission line, but also dependent on the quantized character of the charge obviously.
On the basis of the charge discreteness, the inductance-coupling rnesoscopic circuit is quantized, the quantum loop equations in coupling form are given, and the energy spectrum is computed in charge representation. The results show that, taking account of the charge discreteness, it will make the quantum loop equations complicated, and the energy spectrum of the circuit is not only related with the circuit parameters, but depended on the quantization character of the charge and the undetermined phases.
For the mesoscopic metal ring system in external magnetic field,supposing the system has a symmetry under translation in charge space,the quantum current and the quantum energy spectrum in mesoscopic metal ring are given by solving thire eigenvalue equations;and the quantum fluctuations of the current and the energy have been calculated by the character of the minimum shift operator.The properties of the quantum current and the quantum energy spectrum have been investigated,and the effects of the parameters on the quantum fluctuations are analysed.The results show that the quantum current and the quantum energy spectrum are not only related with the external magnetic field and the parameters of the mesoscopic metal ring,but depended on the quantization character of the charge evidently.
The quantum theory for the dual LC mesoscopic coupled electric circuit with charge discreteness is briefly described. The quantum Kirchoff's moving equation, and energy spectrum of the mesoscopic coupled circuit is given under the non-interaction Hamilton digenstate vectors. The results show that, taking account of the charge discreteness, a evident change occurred for the equation of mesoscopic circuit, and the energy spectrum of the circuit are not only related with the circuit parameters, but depended the quantization character of the charge.
The quantization of a mesoscopic transmission line in accordance with the discreteness of electric charges is proposed under the Born–Von-Karmann periodic boundary condition. The finite-difference Schrödinger equation of the circuit has been obtained in charge representation, and the Schrödinger equation is transformed into Mathieu equation in current representation. The wavefunction and energy spectrum can be solved by adopting the canonical transformation and unitary transformation method. Our results indicate that in the transmission line, quantum fluctuation of currents, which also have distributed properties, are related to both the circuit parameters and the positions and the mode of signals.
On the basis of charge discreteness,the properties of quantum current in a mesoscopic electron resonator are investigated by the minimum shift operator.Quantum Kirchhoff’s equation,the current relation,and the fluctuations of current are given.The results show that when charge quantization is taken into account,the charge in the resonator has the characteristic of quantum vibration;the quantum current and the fluctuation are related to the charge quantum and Planck constant,respectively,and the size is determined by the self-inductance of the system.