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arXiv:cond-mat/9908012

Comment on "quantum theory for mesosocopic electric circuits". Cond-mat/9907171 and cond-mat/9606206

Juan Cesar Flores

Published Aug 1, 1999• 0 citations
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Abstract

In references cond-mat/9907171 and cond-mat/9606206 (Phys.Rev.B.53, 4927 (1996)) by You-Quan Li and Bin Chen, was considered a mesoscopic LC circuit with charge discreteness. So, it was proposed a finite difference Schroedinger equation for the charge time behavior. In this comment, we generalize the corresponding mesoscopic Hamiltonian in order to taken into account the dissipative effects (resistance R). Namely, a quantum term RI, proportional to the current, is added to the mesoscopic LC circuit equation. This is carried-out in analogy with the theory of Caldirola-Kanai for quantum one particle damping.

DOI
10.48550/arxiv.cond-mat/9908012

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In references cond-mat/9907171 and cond-mat/9606206 (Phys.Rev.B.53, 4927 (1996)) by You-Quan Li and Bin Chen, was considered a mesoscopic LC circuit with charge discreteness. So, it was proposed a finite difference Schroedinger equation for the charge time behavior. In this comment, we generalize the corresponding mesoscopic Hamiltonian in order to taken into account the dissipative effects (resistance R). Namely, a quantum term RI, proportional to the current, is added to the mesoscopic LC circuit equation. This is carried-out in analogy with the theory of Caldirola-Kanai for quantum one particle damping.
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.
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.
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.
In this paper one deals with the quantization of mesoscopic LC-circuits under the influence of an external time dependent voltage. The canonically conjugated variables, such as given by the electric charge and the magnetic flux get established by resorting to the hamiltonian equations of motion. This time the discretization of the electric charge is accounted for by working in combination with both right-hand and left-hand discrete derivatives. This results in the onset of related magnetic flux operators. Besides the discrete Schrödinger equation with nearest-neighbor hoppings, a nontrivial next nearest neighbor generalization has also been established. Handling the time dependent voltage within the nearest neighbor description opens the way to the derivation of dynamic localization effects in terms of the approach discussed before by Dunlap and Kenkre, L-ring configurations included.
Proofs are given that the quantum-mechanical description of the LC-circuit with a time dependent external source can be readily established by starting from a more general discretization rule of the electric charge. For this purpose one resorts to an arbitrary but integer-dependent real function F(n) instead of n. This results in a nontrivial generalization of the discrete time dependent Schrödingerequation established before via F(n) = n, as well as to modified charge conservation laws. However, selected descriptions can also be done by looking for a unique derivation of the effective inductance. This leads to site independent inductances, but site dependent ones get implied by accounting for periodic solutions to F(n) in terms of Jacobian elliptic functions. Many-charge generalizations of quantum circuits, including the modified continuity equation for total charge and current densities, have also been discussed. Keywords: Quantum LC-circuits; Charge discretization; Discrete Schrödinger equations; Many-charge generalizations I.