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Bulletin of the American Physical Society

Mesoscopic RLC Circuit and its Associated Occupation Number and Berry Phase

Eric Greenwood

Published Mar 7, 2019• 0 citations
OpenAlex

Abstract

In this paper we consider the quantization of the time-dependent harmonic oscillator and its associated Berry phase using the invariant operator method, as well as the occupation number of the induced quasi-particle production. Furthermore, we point out that in the literature there exist different methods for determining the solution to the Milne-Pinney equation, which leads to different results. By measuring the time-dependent occupation number and associated Berry phase, one can, in principle, determine which of these methods leads to physically realized results. As a concrete example, we consider the mesoscopic RLC circuit and derive the occupation number and associated Berry phase for each of these different methods. We find that, the solution to the Ermakov equations leads to a time-dependent occupation number and associated Berry phase, while the particular solution to the Milne-Pinney equation does not.

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In this paper we consider the quantization of the time-dependent harmonic oscillator and its associated Berry phase using the invariant operator method, as well as the occupation number of the induced quasi-particle production. Furthermore, we point out that in the literature there exist different methods for determining the solution to the Milne-Pinney equation, which leads to different results. By measuring the time-dependent occupation number and associated Berry phase, one can, in principle, determine which of these methods leads to physically realized results. As a concrete example, we consider the mesoscopic RLC circuit and derive the occupation number and associated Berry phase for each of these different methods. We find that, the solution to the Ermakov equations leads to a time-dependent occupation number and associated Berry phase, while the particular solution to the Milne-Pinney equation does not.
In this paper we consider the quantization of the time-dependent harmonic oscillator and its associated Berry phase using the invariant operator method, as well as the occupation number of the induced quasi-particle production. Furthermore, we point out that in the literature there exist different methods for determining the solution to the Milne-Pinney equation, which leads to different results. By measuring the time-dependent occupation number and associated Berry phase, one can, in principle, determine which of these methods leads to physically realized results. As a concrete example, we consider the mesoscopic RLC circuit and derive the occupation number and associated Berry phase for each of these different methods. We find that, the solution to the Ermakov equations leads to a time-dependent occupation number and associated Berry phase, while the particular solution to the Milne-Pinney equation does not.
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.
With the help of the time-dependent Lagrangian for a damped harmonic oscillator, the quantization of mesoscopic RLC circuit in the context of a number-phase quantization scheme is realized and the corresponding Hamiltonian operator is obtained. Then the evolution of the charge number and phase difference across the capacity are obtained. It is shown that the number-phase analysis is useful to tackle the quantization of some mesoscopic circuits and dynamical equations of the corresponding operators.
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