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Chinese Journal of Quantum Electronics

Number-phase quantization scheme for LC circuit and Josephson junction's Cooper pairs

Liang Bao-Long

Published Jan 1, 2007• 3 citations
OpenAlex

Abstract

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.

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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.
For a mesoscopic L-C circuit, besides the Louisell's quantization scheme in which electric charge q and electric current I are respectively quantized as the coordinate operator Q and momentum operator P , in this paper we propose a new quantization scheme in the context of number-phase quantization through the standard Lagrangian formalism. The comparison between this number-phase quantization with the Josephson junction's Cooper pair number-phase-difference quantization scheme is made.
By introducing the entangled state representation and Feynman assumption that `electron pairs are bosons, ..., a bound pair acts as a Bose particle', we construct an operator Hamiltonian for a mesoscopic inductance-capacitance (LC) circuit including a Josephson junction, then we use the Heisenberg equation of motion to derive the current equation and the voltage equation across the inductance as well as across the Josephson junction. The result manifestly shows how the junction voltage is affected by the capacitance coupling. In this way the Cooper-pair number-phase quantization for this system is completed.
Based on the entangled state representation and Feynman's idea that ‘electron pairs are bosons, …, a bound pair acts as a Bose particle’, we present a cooper-pair number-phase quantization scheme for the mesoscopic LC circuit including two coupled Josephson junctions (JJs). Then we use the Heisenberg equation of motion to obtain the modified current equation and voltage equation across each JJ, as well as the equation for realizing quantum control. Besides, we investigate how the phases in two JJs are affected mutually through the capacitor coupling and the coupled JJs.
This paper examines the quantization of mesoscopic circuit including Josephson junctions. Following Feynman's assumption, via the Hamilton dynamic approach and by virtue of the entangled state representation, it constructs Hamiltonian operator for the double-Josephson-junction mesoscopic circuit coupled by a capacitor. Then it uses the Heisenberg equation of motion to derive the induction voltage across each Josephson junction. The result manifestly shows how the voltage is affected by the capacitance coupling.