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Chinese Physics B

Number-phase quantization of a mesoscopic RLC circuit

Chenglin Xu

Published Feb 1, 2012• 5 citations
Publisher

Abstract

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.

DOI
10.1088/1674-1056/21/2/020402

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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.
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.
Traditionally, the quantization scheme for a mesoscopic LC circuit lies in that electric charge q and electric current L×I are respectively quantized as the coordinate operator Q and momentum operator P. A new quantization scheme is proposed in which number n of the electric charge q(q=en) is quantized as the charge number operator and the relation between electric current and phase operatorθcan be derived, in this way the quantization of mesoscopic LC circuit in the context of number-phase is realized and the corresponding Hamiltonian is obtained. The results show that the Cooper pair number-phase-difference quantization scheme is also available for Josephson junction after introducing the entangled state representation.
The mesoscopic RLC series circuit are quantized by five different ways which are the forced damp harmonic vibration,the construct the Hamiltonian and associated canonical transformation,the damped harmonic vibration canonical transformation,the construct the Hamiltonian and associated unitary transformation and the inner construction of the resistance are summarized.The quantum fluctuations of the charge,the current and their products are analyzed when the system in the vacuum state.The different results from the same circuit quantized by different methods are due to the different chice of the Hamiltonion of the system.