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Chinese Control Conference (CCC)

Quantum modeling and control of Josephson junction by quantum Hamilton mechanics

Cian-Dong Yang, Chung-Hsuan Kuo

Published Jul 1, 2014• 0 citations
Publisher

Abstract

In this paper, we apply quantum Hamilton mechanics to describe the dynamical motion of an electronic Cooper pair tunneling through a Josephson junction. Quantum Hamilton equations provide us a set of canonical equations q̇ = f(q, p) and ṗ = g(q, p) to model the tunneling dynamics of a Cooper pair. Instead of using the conventional probabilistic description, we solve complex quantum trajectory q(t) = qR+ qI·i from the Hamilton equations to demonstrate the tunneling dynamics on a geometrical phase plane. In order to control the dynamics of a cooper pair, we add a gate voltage parameter ngto the quantum Hamiltonian. By adjusting the magnitude of ng, we successfully control the tunneling dynamic of the Cooper pair such that the predicted current-voltage relation is in excellent agreement with the experimental measurements.

DOI
10.1109/chicc.2014.6895956

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In this paper, we apply quantum Hamilton mechanics to describe the dynamical motion of an electronic Cooper pair tunneling through a Josephson junction. Quantum Hamilton equations provide us a set of canonical equations q̇ = f(q, p) and ṗ = g(q, p) to model the tunneling dynamics of a Cooper pair. Instead of using the conventional probabilistic description, we solve complex quantum trajectory q(t) = qR+ qI·i from the Hamilton equations to demonstrate the tunneling dynamics on a geometrical phase plane. In order to control the dynamics of a cooper pair, we add a gate voltage parameter ngto the quantum Hamiltonian. By adjusting the magnitude of ng, we successfully control the tunneling dynamic of the Cooper pair such that the predicted current-voltage relation is in excellent agreement with the experimental measurements.
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.
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.
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.
We have analyzed and measured the quantum coherent dynamics of a circuit containing two coupled superconducting charge qubits. Each qubit is based on a Cooper pair box connected to a reservoir electrode through a Josephson junction. Two qubits are coupled electrostatically by a small island overlapping both Cooper pair boxes. Quantum state manipulation ofthe qubit circuit is done by applying non-adiabatic voltage pulses to the common gate. We read out each qubit by means of probe electrodes connected to Cooper pair boxes through high-Ohmic tunnel junctions. With such a setup the measured pulse-induced probe currents are proportional to the probability for each qubit to have an extra Cooper pai1r after the manipulation. As expected from theory and observed experimentally the measured pulse-induced current in each probe has two frequency components whose position on the frequency axis can be externally controlled. This is a result ofthe inter-qubit coupling which is also responsible for the avoided level crossing that we observed in the qubits' spectra. Our simulations show that in the absence of decoherence and with a rectangular pulse shape the system remains entangled most ofthe time reaching maximally entangled states at certain instances.
Within a semiclassical approximation and at the weak-coupling limit, where the charging energy dominates the Josephson coupling energy, the dynamics of small-capacitance superconducting tunnel junctions can be described by a stochastic master equation for the probability distribution of junction charges. We develop a method of solving this equation and obtain an analytic solution when Cooper-pair, quasiparticle, and Zener tunneling are all present under an arbitrary Ohmic conductance at zero temperature. We use this solution to systematically investigate the static and dynamic properties of small Josephson junctions. The charge distributions, current-voltage characteristics, and voltage spectra are analytically calculated and numerically evaluated for wide ranges of parameters. The separate contributions from quasiparticle and Cooper-pair tunneling to voltage spectra are evaluated and compared based on their respective sum rules. The linewidths of the resonance peaks for single-electron tunneling and Bloch oscillations are also discussed.
Electrons in a superconductor form Cooper pairs and thus condense into a ground state which can be described by a macroscopic quantum wavefunction. The phase of the wavefunction represents a collective movement of millions of Cooper pairs. Using devices based on Josephson tunnel junctions we are able to control and detect the phase and number of Cooper pairs in a superconductor. Josephson devices are therefore excellent systems for demonstrating macroscopic quantum phenomena. We present a short review of the macroscopic quantum phenomena in superconducting Josephson devices, as well as their applications and prospects in quantum computationl.