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URSI General Assembly and Scientific Symposium

Spatial Evolution of Quantized Discrete-Mode Operators in a Lossy Nonlinear Josephson-Embedded Transmission Line

Yongjie Yuan, Michael Haider, Johannes A. Russer, P. Russer, Christian Jirauschek

Published Aug 28, 2021• 0 citations
Publisher

Abstract

We present a description of the spatial evolution of quantized discrete-mode operators along a lossy nonlinear transmission line. The nonlinearity is formed by hundereds or even thousands of Josephson junctions which are placed periodically along a microwave transmission line. Dissipation is added to the system Hamiltonian by coupling the nonlinear transmission line to an Ohmic bath. Using the Hamiltonian of the open quantum system, Heisenberg equations of motion for the discrete mode operators can be derived in terms of quantum Langevin equations. The temporal equations of motion are then translated to the spatial domain to investigate the performance of a nonlinear four-wave-mixing process, while signals propagate along the transmission line.

DOI
10.23919/ursigass51995.2021.9560410

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We present a description of the spatial evolution of quantized discrete-mode operators along a lossy nonlinear transmission line. The nonlinearity is formed by hundereds or even thousands of Josephson junctions which are placed periodically along a microwave transmission line. Dissipation is added to the system Hamiltonian by coupling the nonlinear transmission line to an Ohmic bath. Using the Hamiltonian of the open quantum system, Heisenberg equations of motion for the discrete mode operators can be derived in terms of quantum Langevin equations. The temporal equations of motion are then translated to the spatial domain to investigate the performance of a nonlinear four-wave-mixing process, while signals propagate along the transmission line.
In this paper, we present a theoretical framework for investigating losses and thermal fluctuations in a Josephson traveling-wave parametric amplifier (JTWPA). Our model is based on a discrete-mode Hamiltonian which includes a four-wave-mixing process and system-reservoir interactions. From this Hamiltonian, we derive a quantum master equation that describes the lossy Josephson-junction-embedded transmission line. The resulting equation of motion for the reduced density operator is applied for evaluating the average photon number in the signal mode. We present an analytic solution for the case of a dispersionless transmission line, where the phase-modulation effects can be neglected, and evaluate the photon number spectrum of a JTWPA structure with parameters from the literature. The quantum master equation is especially advantageous for a direct treatment of the expectation values of the photon number, in comparison to other strategies.
An electronic transmission line that contains an array of nonlinear elements (Josephson junctions) is studied theoretically. A continuous nonlinear wave equation describing the dynamics of the node flux along the transmission line is derived. It is shown that due to the nonlinearity of the system, a mixing process between four waves with different frequencies is possible. The mixing process can be utilized for amplification of weak signals due to the interaction with a strong pump wave. An analytical solution for the spatial evolution of the wave amplitudes is derived, and found to be in excellent agreement with the results of numerical computations. Simulations of realistic parameters show that the power gain can exceed 20 dB over a bandwidth of more than 2 GHz.
We present a theory describing parametric amplification in a Josephson junction embedded transmission line. We will focus on the process of four-wave mixing under the assumption of an undepleted pump. However, the approach taken is quite general, such that a different parametric process or the process under different assumptions is easily derived. First the classical theory of the coupled-mode equations as presented by O'Brien et al. [O'Brien et al., Phys. Rev. Lett. 113, 157001 (2014)] is briefly reviewed. Then a derivation of the full quantum theory is given using mesoscopic quantization techniques in terms of discrete-mode operators. This results in a Hamiltonian that describes the process of parametric amplification. We show that the coupled-mode equations can be derived from this Hamiltonian in the classical limit and elaborate on the validity of the theory.
In this contribution, we investigate different types of Josephson junction arrays that are embedded in coplanar microwave transmission lines, acting as traveling-wave parametric amplifiers. For the realization of traveling-wave parametric amplifiers, different Josephson junction array geometries have been proposed in the literature. Here, we investigate the amplifier’s Josephson nonlinearity along with the gain performance, following a general method to derive the nonlinear wave-mixing coefficients from the current-phase relation of different junction array topologies. These nonlinear coefficients are then incorporated into a corresponding Hamiltonian, which is used in order to solve the Heisenberg equations of motion of the photon number operator, determining the gain of the amplifier.
This thesis studies quantum coherence in macroscopic and mesoscopic dissipative electrical circuits, including LC circuits, microwave resonators, and Josephson junctions. \nFor the LC resonator and the terminated transmission line microwave resonator, second quantization is carried out for the lossless system and dissipation in modeled as the coupling to a bath of harmonic oscillators. Stationary states of the linear and nonlinear resonator circuits as well as the associated energy levels are found, and the time evolution of uncertainty relations for the observables such as flux, charge, current, and voltage are obtained. Coherent states of both the lossless and weakly dissipative circuits are studied within a quantum optical approach based on a Fokker-Plank equation for the P-representation of the density matrix which has been utilized to obtain time-variations of the averages and \nuncertainties of circuit observables. \nMacroscopic quantum tunneling is addressed for a driven dissipative Josephson resonator from its metastable current state to the continuum of stable voltage states. The Caldeira-Leggett method and the instanton path integral technique have been used to find the tunneling rate of a driven Josephson junction from a zero-voltage state to the continuum of the voltage states in the presence of dissipation. Upper and lower bounds are obtained for the tunneling rate at the intermediate loss and approximate closed form expressions are derived for the overdamped and underdamped limits.
In this paper, we are constructing a circuit quantum electrodynamic model describing dissipative effects in a Josephson traveling-wave parametric amplifier (JTWPA). Losses and noise are treated using the quantum Langevin method. The total Hamiltonian of our model describes the coupling of the modes in a parametric four-wave-mixing scheme, as well as the interaction with the environment. Based on the Heisenberg equations of motion, we obtain the temporal evolution of the signal photon number and analyze the signal gain of the JTWPA.