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arXiv:cond-mat/9709261

Mesoscopic quantum circuit theory to the persistent current and Coulomb blockade

You‐Quan Li

Published Sep 24, 1997• 2 citations
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Abstract

The quantum theory for mesoscopic electric circuit is briefly described. The uncertainty relation for electric charge and current modifies the tranditional Heisenberg uncertainty relation. The mesoscopic ring is regarded as a pure L-design, and the persistent current is obtained explicitly. The Coulomb blockade phenomenon appears when applying to the pure C-design.

DOI
10.48550/arxiv.cond-mat/9709261

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The quantum theory for mesoscopic electric circuit is briefly described. The uncertainty relation for electric charge and current modifies the tranditional Heisenberg uncertainty relation. The mesoscopic ring is regarded as a pure L-design, and the persistent current is obtained explicitly. The Coulomb blockade phenomenon appears when applying to the pure C-design.
This talk is assumed to exhibit an overview of the quantum theory for mesoscopic electric circuits and some of its further developments. In the theory the importance of the discreteness of electronic charge in mesoscopic electric circuit is addressed. The mesoscopic LC-design is quantized in accord with the charge discreteness. The uncertainty relation for electric charge and current is given. Because the stationary Schrödinger equation is turned to be Mathieu equation in ppp-representation, the wave function and energy spectrum is formally solved. As further applications, the persistent current is obtained by considering the mesoscopic ring as a pure L-design. The Coulomb blockade phenomenon occurs when applying the theory to the pure C-design. Concerning the time evolution of the states for mesoscopic electric circuit, we are able to study it by the method of characteristics. In order to study the dissipative effect in the circuit, we use density-matrix formulation. In this formulation, several type of ``off diagonal'' dissipations are expected to be discussed.
The quantum theory for mesoscopic electric LC circuits with charge discreteness is briefly described. We take into account a resistance element (R) as an environment of the discrete-charge mesoscopic quantum LC circuit which is modeled by a Hamiltonian consisting of oscillators with continuous range of frequencies. Using a minimal coupling method, we investigate the quantum dynamics of this system. Hereby, the persistent current on a quantum damped L-design under the external potential source is obtained. Then, we write Heisenberg equations for a driven mesoscopic quantum RLC circuit with a dissipative term proportional to Ohmic damping and obtain persistent current on such a mesoscopic electric circuit.
Using the quantum theory for a mesoscopic circuit based on the discreteness of electric charges, the finite-differenced Schrodinger equation of the non-dissipative mesoscopic capacity coupling circuit are achieved. The quantum effect not only comes from the quanumnization of circuit but also that of charge. Only after the properties of charge quanumnization is considered can the Coulomb blockade effect of a mesoscopic circuit be discussed. The Schrodinger equation can be divided into two Mathieu equations in p-representation. With the WKBJ method, the currents quantum fluctuations in the ground states of the two circuits are calculated. And the fluctuations are the main noise sources of quantum in the mesoscopic circuit.
Using the quantum theory for a mesoscopic circuit based on the discretenes of electric charges, the finite-difference Schrödinger equation of the non-dissipative mesoscopic inductance and capacity coupling circuit is achieved. The Coulomb blockade effect, which is caused by the discreteness of electric charges, is studied. Appropriately choose the components in the circuits, the finite-difference Schrödinger equation can be divided into two Mathieu equations in representation. With the WKBJ method, the currents quantum fluctuations in the ground states of the two circuits are calculated. The results show that the currents quantum zero-point fluctuations of the two circuits are exist and correlated.