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Chinese Physics Letters

Number-Phase Quantization and Deriving Energy-Level Gap of Two LC Circuits with Mutual-Inductance

Meng Xiang-Guo, Ji‐Suo Wang, Zhai Yun, Fan Hong-Yi

Published Apr 1, 2008• 6 citations
Publisher

Abstract

For two LC circuits with mutual-inductance, we introduce a new quantization scheme in the context of number-phase quantization through the standard Lagrangian formalism. The commutative relation between the charge operator and the magnetic flux operator is derived. Then we use the Heisenberg equation of motion to obtain the current and voltage equation across the inductance and capacity. The results clearly show how the current and voltage in a single LC circuit are affected by the circuit parameters and inductance coupling coefficient. In addition, adopting invariant eigen-operator method the energy-level gap of the dynamic Hamiltonian which describes two LC circuits with mutual-inductance is obtained.

DOI
10.1088/0256-307x/25/4/012

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For two LC circuits with mutual-inductance, we introduce a new quantization scheme in the context of number-phase quantization through the standard Lagrangian formalism. The commutative relation between the charge operator and the magnetic flux operator is derived. Then we use the Heisenberg equation of motion to obtain the current and voltage equation across the inductance and capacity. The results clearly show how the current and voltage in a single LC circuit are affected by the circuit parameters and inductance coupling coefficient. In addition, adopting invariant eigen-operator method the energy-level gap of the dynamic Hamiltonian which describes two LC circuits with mutual-inductance is obtained.
为二个 LC 电路与相互电感，我们通过标准 Lagrangian 形式主义在数字阶段量子化的上下文介绍一个新量子化计划。在费用操作员和磁性的流动操作员之间的可交换的关系被导出。然后，我们使用运动的海森堡方程越过电感和电容获得电流和电压方程。结果清楚地显示出在一个单个 LC 电路的电流和电压怎么被电路参数和电感联合系数影响。另外，采用不变的 eigen 操作员方法描述二个 LC 电路与的动态 Hamiltonian 的精力级的差距相互电感被获得。
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.
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.
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.
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. This time the discretization of the electric charge is accounted for by working in combination with both right-hand and left-hand discrete derivatives. This results in the onset of related magnetic flux operators. Besides the discrete Schrödinger equation with nearest-neighbor hoppings, a nontrivial next nearest neighbor generalization has also been established. Handling the time dependent voltage within the nearest neighbor description opens the way to the derivation of dynamic localization effects in terms of the approach discussed before by Dunlap and Kenkre, L-ring configurations included.
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.