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arXiv:cond-mat/0605141

Discrete-charge quantum circuits in semiclassical approach

Constantino A. Utreras-Díaz, Juan Cesar Flores

Published May 5, 2006• 0 citations
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Abstract

We discuss a new approach to describe mesoscopic systems, based on the ideas of quantum electrical circuits with charge discreteness. This approach has allowed us to propose a simple alternative descriptions of some mesoscopic systems, with interesting results for some mesoscopic systems. In his work, we show that the application of the Bohr-Sommerfeld quantization rules to the Quantum $LC$ circuit with discrete charge allows us to easily reproduce previous results.

DOI
10.48550/arxiv.cond-mat/0605141

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We discuss a new approach to describe mesoscopic systems, based on the ideas of quantum electrical circuits with charge discreteness. This approach has allowed us to propose a simple alternative descriptions of some mesoscopic systems, with interesting results for some mesoscopic systems. In his work, we show that the application of the Bohr-Sommerfeld quantization rules to the Quantum LCLCLC circuit with discrete charge allows us to easily reproduce previous results.
In the talk, I briefly demonstrate the quantum theory for mesoscopic electric circuits and its applications. In the theory, the importance of the charge discreteness in a mesoscopic electric circuit is addressed. As a result, a new kind of commutation relation for electric charge and current occurred inevitably. The charge representation, canonical current representation and pseudo-current representation are discussed extensively. It not only provides a concrete realization of mathematical models which discuss the space quantization in high energy physics and quantum gravity but also presents a sequence of applications in condensed matter physics from a different point of view. A possible generalization to coupled circuits is also proposed.
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 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.
Based on energy conservation considerations we study the nonlinear dynamic behavior of a quantum mesoscopic circuit, which is characterized through parameters of inductance and capacitance.Nonlinearity is given by the initial conditions of magnetic flux and discreteness charge which oscillate in the interval [-0 α , + 0 α ], being 0 α the magnetic flux normalized by e 0/ q = φ h .This LC circuit with quantized electric charge is excited by energy battery that can produce an electrical discreteness charge on the capacitor.The dynamics of the mesoscopic circuit is highly nonlinear.Our results show for the magnetic flux a nearly square wave with an elongated period when compared with the linear case and a train of narrow pulses for the discrete charge.
A quantum mesoscopic electrical LC-circuit with charge discreteness including a Josephson junction is considered and a nonlinear Hamiltonian that describing the dynamic of such circuit is introduced. The quantum dynamical behavior (persistent current probability) is studied in the charge and phase regimes by numerical solution approaches. The time evolution of charge and current, number-difference and the bosonic phase and also the energy spectrum of a quantum mesoscopic electric LC-circuit with charge discreteness that coupled with a Josephson junction device are investigated. We show the role of the coupling energy and the electrostatic Coulomb energy of the Josephson junction in description of the quantum behavior and the spectral properties of a quantum mesoscopic electrical LC-circuits with charge discreteness.