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Modern Physics Letters B

On the dynamics of a time-dependent mesoscopic LC circuit with a negative inductance

I. A. Pedrosa, E. Nogueira, I. Guedes

Published May 10, 2016• 6 citations
Publisher

Abstract

We discuss the problem of a mesoscopic LC circuit with a negative inductance ruled by a time-dependent Hermitian Hamiltonian. Classically, we find unusual expressions for the Faraday’s law and for the inductance of a solenoid. Quantum mechanically, we solve exactly the time-dependent Schrödinger equation through the Lewis and Riesenfeld invariant operator method and construct Gaussian wave packet solutions for this time-dependent LC circuit. We also evaluate the expectation values of the charge and the magnetic flux in these Gaussian states, their quantum fluctuations and the corresponding uncertainty product.

DOI
10.1142/s0217984916501220

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We discuss the problem of a mesoscopic LC circuit with a negative inductance ruled by a time-dependent Hermitian Hamiltonian. Classically, we find unusual expressions for the Faraday’s law and for the inductance of a solenoid. Quantum mechanically, we solve exactly the time-dependent Schrödinger equation through the Lewis and Riesenfeld invariant operator method and construct Gaussian wave packet solutions for this time-dependent LC circuit. We also evaluate the expectation values of the charge and the magnetic flux in these Gaussian states, their quantum fluctuations and the corresponding uncertainty product.
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.
When a time-dependent harmonic system is further driven by an external source, it presents significant challenges to obtaining analytical solutions for the system. This paper applies the Lewis-Riesenfeld invariant method to a mesoscopic time-dependent dissipative inductor-capacitor (RLC) circuit with an external source. By constructing quantum invariant Hermitian operators and introducing auxiliary equations that can describe the effects of external sources and resistance, we analytically solve the Schrödinger equation of the system. Taking an alternating current (AC) voltage source as an example, we derive the generalized coherent states of a mesoscopic time-dependent RLC circuit with an external source, calculate the quantum fluctuations of charge and current, and obtain their corresponding uncertainty relations. The results indicate that the time-dependent inductance and resistance disrupt the minimum uncertainty relation of the Glauble coherent states in driven time-dependent mesoscopic RLC circuits. This research contributes to the establishment of a quantum theory for mesoscopic circuits.
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 work we study the classical and quantum dynamics of a London superconductor and of a time-dependent mesoscopic or nanoscale LC circuit by assuming that the inductance and capacitance vary exponentially with time at constant rate. Surprisingly, we find that the behavior of these two systems are equivalent, both classically and quantum mechanically, and can be mapped into a standard damped harmonic oscillator which is described by the Caldirola-Kanai Hamiltonian. With the aid of the dynamical invariant method and Fock states, we solve the time-dependent Schr\"odinger equation associated with this Hamiltonian and calculate some important physical properties of these systems such as expectation values of the charge and magnetic flux, their variances and the respective uncertainty principle.
The quantum double wave theory is applied to the time-dependent mesoscopic LC electric circuit with time-dependent capacitance and inductance. The averages and quantum fluctuations of the electric charge and magnetic flux (electric current) are given for a single time-dependent LC electric circuit. The obtained averages of the electric charge and magnetic flux satisfy the classical equation of motion. After making statistical average over the initial phases, the average and quantum fluctuation of every quantity reduce to that derived by the standard quantum mechanics. These results show that the double wave theory can be used to describe a single time-dependent mesoscopic electric circuit.
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 provided by both Faraday and Kirchhoff laws. This time the discretization of the electric charge is accounted for, so that magnetic flux operators one looks for should proceed in terms of discrete derivatives. However, the flux operators one deals with are not Hermitian, which means that subsequent symmetrizations are in order. The eigenvalues characterizing such operators can be readily established in terms of twisted boundary conditions. Besides the discrete Schrödinger equation with nearest-neighbor hopping, a nontrivial next nearest neighbor generalization has also been established. Such issues open the way to the derivation of persistent currents in terms of effective k-dependent Hamiltonians. Handling the time dependent voltage within the nearest neighbor description leads to the derivation of dynamic localization effects in L-ring configurations, such as discussed before by Dunlap and Kenkre. The onset of the magnetic flux quantum has also been discussed in some more detail.