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Chinese Physics B

Investigating mesoscopic circuits with entangled state representations

Bao‐Long Liang, Ji‐Suo Wang, Xiang‐Guo Meng, Qin-Ying Yang

Published Jan 1, 2013• 2 citations
Publisher

Abstract

By employing the continuous parameter entangled state representations, we investigate the energy level and the wave function for a capacitively and mutual-inductively coupled LC mesoscopic circuit. It is found that investigating the mesoscopic circuit in such representations can bring us the following conveniences. Firstly, the dynamical equation is naturally transformed into a single-variable differential equation. Secondly, the center-of-mass kinetic energy is included in the energy level of the system. Thus it is instructive to introduce the entangled state representation into the investigation of mesoscopic circuits.

DOI
10.1088/1674-1056/22/1/016804

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By employing the continuous parameter entangled state representations, we investigate the energy level and the wave function for a capacitively and mutual-inductively coupled LC mesoscopic circuit. It is found that investigating the mesoscopic circuit in such representations can bring us the following conveniences. Firstly, the dynamical equation is naturally transformed into a single-variable differential equation. Secondly, the center-of-mass kinetic energy is included in the energy level of the system. Thus it is instructive to introduce the entangled state representation into the investigation of mesoscopic circuits.
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.
Since the birth of quantum mechanics, its classical correspondence (or analogy) has been a hot topic for physicists. In this paper, we first discuss whether there is a classical correspondence of quantum entanglement. We give a positive answer through the following examples: in the framework of quantization of mesoscopic circuits, two mesoscopic capacitance inductance (LC) circuits with mutual inductance are proved to be the source of quantum entanglement by using the integration within an ordered product, and then the formula of their characteristic frequency is obtained, It is found that it is similar to the expression of the small oscillating frequency of a classical system described below. The classical system is shown in Fig. 1 . Two walls are connected with the same spring. And between the two springs a sliding trolley can move on a smooth table. The trolley is hung with a simple pendulum, The small oscillating frequency of the system is calculated by analytical mechanics. It is found that the swing of the simple pendulum will cause the trolley to oscillate back and forth. The mutual restraint effect of the pendulum, the trolley and the spring reflects the “entanglement” between them.
A mesoscopic circuit, where field quantization (e.g., coherent states and squeezed states) can be exhibited, is one of the ideal quantum systems for manifesting diverse quantum field theoretical phenomena and thermally statistical characteristics. Because of inevitable electromagnetic coupling (e.g., mutual capacitor coupling and mutual inductor coupling) in two neighboring mesoscopic electric circuits, quantum effects relevant to mesoscopic circuit coupling may play a key role in some devices of quantum information processing. Such a circuit coupling can lead to circuit entangled states, where the Fock number states of energy quanta in different circuits are entangled with each other. Since the time dependence in parameters of coupled-circuit Hamiltonian would result from environmental perturbations (e.g., temperature fluctuations in devices) or from some controllable manipulation (e.g., an incident ultrasonic wave), the circuit entangled states would evolve due to these external conditions. Time evolution of entangled states in coupled mesoscopic circuits is considered within a framework of time-dependent quantum theoretical formalism. Note that an applied photon field can also be coupled to such quantum entangled circuits. Then it is possible that an effect of entanglement transfer (i.e., transportation of superposition coefficients in the entangled state) from the quantum mesoscopic coupled LC circuits to the applied external photon field would also occur under some proper parameter conditions. The present characteristics of both quantized coupled mesoscopic circuits and entanglement transfer can also be found in some alternative microscopic- and mesoscopic-scale devices such as the systems in cavity quantum electrodynamics (QED).
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.