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Progress of Theoretical Physics

Quantum Description of a Mesoscopic RLC Circuit

I. A. Pedrosa, Anderson Pereira Pinheiro

Published Jun 1, 2011• 29 citations
Publisher

Abstract

We present a quantum description of a mesoscopic RLC circuit. Based on the Caldirola-Kanai Hamiltonian and quantum invariant method we solve the Schrödinger equation for this circuit and write the corresponding wave functions in terms of a particular solution of the Milne-Pinney equation. In addition, we construct coherent and squeezed states for the quantized RLC circuit and evaluate the quantum fluctuations of the charge and the magnetic flux as well as the uncertainty product. Subject Index: 060, 064 The study of mesoscopic systems has attracted a lot of attention in the litera-ture in the last few years.1)–19) The great interest in these systems is motived partly because of the rapid development of nanophysics and nanoelectronics and partly because in the manufacture of electronics devices the miniaturization of integrated

DOI
10.1143/ptp.125.1133

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We present a quantum description of a mesoscopic RLC circuit. Based on the Caldirola-Kanai Hamiltonian and quantum invariant method we solve the Schrödinger equation for this circuit and write the corresponding wave functions in terms of a particular solution of the Milne-Pinney equation. In addition, we construct coherent and squeezed states for the quantized RLC circuit and evaluate the quantum fluctuations of the charge and the magnetic flux as well as the uncertainty product. Subject Index: 060, 064 The study of mesoscopic systems has attracted a lot of attention in the litera-ture in the last few years.1)–19) The great interest in these systems is motived partly because of the rapid development of nanophysics and nanoelectronics and partly because in the manufacture of electronics devices the miniaturization of integrated
In this work we present a quantum description of a mesoscopic RLC circuit without source. For this purpose, we model this system for that of a damped harmonic oscillator which is described by the Caldirola-Kanai Hamiltonian. Then, with the aid of the quantum invariant method we solve the Schrödinger equation associated with this Hamiltonian and write the corresponding wave functions in terms of a particular solution of the Milne-Pinney equation. We also construct coherent states for the RLC quantized, and evaluate the quantum fluctuations of the charge and the magnetic flux, as well as the corresponding product of uncertainty.
In this paper, we present a comprehensive quantum description of a mesoscopic RLC circuit with time-dependent resistance, inductance and capacitance. Based on the dynamical invariant method and using quadratic invariants, we derive exact nonstationary quantum states for this circuit and write them in terms of solutions of the Milne–Pinney equation. Afterwards, we use quadratic invariants to construct coherent states for this quantized system and employ them to investigate quantum properties of the RLC circuit. In particular, we show that the product of the quantum fluctuations of the charge and the magnetic flux does not satisfy the minimum uncertainty relation.
In this paper, we use Hermitian linear invariants and the Lewis and Riesenfeld invariant method to obtain the general solution of the Schrödinger equation for a mesoscopic RLC circuit with time-dependent resistance, inductance, capacitance and a power source and represent it in terms of an arbitrary weight function. In addition, we construct Gaussian wave packet solutions for this electromagnetic oscillation circuit and employ them to calculate the quantum fluctuations of the charge and the magnetic flux as well as the associated uncertainty product. We also show that the width of the Gaussian packet and the fluctuations do not depend on the external power.
The quantum wavefunctions and the corresponding energy levels of a RLC (Resistance-Inductance-Capacity) electric circuit are obtained by using canonical quantization method and unitary transformation from the classical equation of motion. The quantum fluctuations of charge and current in an arbitrary eigenstate of the system have also been given as well as the uncertainty relation. It is showed that even if at 0 K charge and current in the circuit exhibit quantum fluctuations, which originates from fluctuations of zero point vibrations of the system.