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International Journal of Modern Physics B

THE QUANTIZATION OF A MESOSCOPIC TRANSMISSION LINE WITH THE DISCRETENESS OF ELECTRIC CHARGE

Yinghua Ji

Published Jun 20, 2009• 1 citation
Publisher

Abstract

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.

DOI
10.1142/s0217979209052509

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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.
Under the Born–von-Karmann periodic boundary condition, we propose a quantization scheme for a non-dissipative distributed parameter circuit (a uniform transmission line). Quantum fluctuations of charge and current in the vacuum state are investigated by adopting the canonical transformation and unitary transformation method. Our results indicate that in distributed parameter circuits, quantum fluctuations, which also have distributed properties, are related to both the circuit parameters and the positions and the mode of signals.
The importance of discreteness of electron charge for mesoscopic electric circuit is addressed. A quantization of the circuit in accord with the discreteness of electric charges is proposed, and a Schrödinger equation in ‘charge representation’ is obtained. The Schrödinger equation is turned to be Mathieu equation in ‘current representation’, the wavefunction and energy spectrum can be solved exactly. The WKB solutions of the energy levels for the ground state and the excited states are given.
Under the Born–von-Karmann periodic boundary condition, we propose a quantization scheme for non-dissipative distributed parameter circuits (i.e. a uniform periodic transmission line). We find the unitary operator for diagonalizing the Hamiltonian of the uniform periodic transmission line. The unitary operator is expressed in a coordinate representation that brings convenience to deriving the density matrix ρ( q , q ′,β). The quantum fluctuations of charge and current at a definite temperature have been studied. It is shown that quantum fluctuations of distributed parameter circuits, which also have distributed properties, are related to both the circuit parameters and the positions and the mode of signals and temperature T . The higher the temperature is, the stronger quantum noise the circuit exhibits.
Under the Born-von-Karmann periodic boundary condition, we propose a quantization scheme for nondissipative distributed parameter circuits (i.e. a uniform periodic transmission line). We find the unitary operator for diagonalizing the Hamiltonian of the uniform periodic transmission line. The unitary operator is expressed in a coordinate representation that brings convenience to deriving the density matrix p(q, q',β). The quantum fluctuations of charge and current at a definite temperature have been studied. It is shown that quantum fluctuations of distributed parameter circuits, which also have distributed properties, are related to both the circuit parameters and the positions and the mode of signals and temperature T. The higher the temperature is, the stronger quantum noise the circuit exhibits.
A quantum theory of mesoscopic circuits, based on the discreteness of electric charges, was recently proposed. However, it is not applied widely, mainly because of the difficulty of the mathematical solution to the finite-difference Schrödinger equation. In this paper, we propose an improved perturbation method to calculate Schrödinger equations of the inductance and capacity coupling mesoscopic circuit. With a unitary transformation, the finite differential Schrödinger equation of the system is divided into two equations in generalized momentum representation. The concrete value of the parameter in circuits is important to solving the equation. Both the perturbation method suitable case and the improved perturbation method suitable case, the wave functions, and energy level of the system are achieved. As an application, the current fluctuations are calculated and discussed. The improved mathematical method would be helpful for the popularization of the quantum circuits theory.
The research work on the quantum effects in mesoscopic circuits has undergone a rapid development recently, however the whole quantum theory of the mesoscopic circuits should consider the discreteness of the electric charge. In this paper, based on the fundamental fact that the electric charge takes discrete values, the finite-difference Schrödinger equation of the mesoscopic RLC circuit with a source is achieved. With a unitary transformation, the Schrödinger equation becomes the standard Mathieu equation, then the energy spectrum and the wave functions of the system are obtained. Using the WKBJ method, the average of currents and square of the current are calculated. The results show the existence of the current fluctuation, which causes noise in the circuits. This paper is an application of the whole quantum mesoscopic circuits theory to the fundamental circuits, and the results will shed light on the design of the miniation circuits, especially on the purpose of reducing quantum noise coherent controlling of the mesoscopic quantum states.