Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1812.05907 · Physical Review A

Mesoscopic Hamiltonian for Josephson traveling-wave parametric amplifiers

Thomas H. A. van der Reep

Published Jun 28, 2019• 26 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.99.063838

Related papers

Parametric amplification in Josephson junction embedded transmission lines

Oded Yaakobi, L. Frièdland, Chris Macklin +1 more

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.

Physical Review B•Apr 1, 2013•118 citations

Quantum Master Equation for a Lossy Josephson Traveling-Wave Parametric Amplifier

Yongjie Yuan, Michael Haider, Christian Jirauschek

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.

International Topical Meeting on Microwave Photonics•Oct 1, 2022•3 citations

Comment on "Parametric amplification in Josephson junction embedded transmission lines"

Saptarshi Chaudhuri, Jiansong Gao

Recently Yaakobi and co-workers [Phys. Rev. B 87, 144301 (2013)] theoretically studied four-wave mixing and parametric amplification in a nonlinear transmission line consisting of capacitively shunted Josephson junctions. By deriving and solving the coupled-mode equations, they have arrived at the conclusion that in a wide frequency range around the pump frequency exponential parametric gain (in which the signal grows exponentially with distance) can be achieved. However, we have found a mathematical error in their derivation of the coupled-mode equations (Equation (A13)), which leads to the wrong expression of the gain factor and invalidates their conclusions on the gain and bandwidth. In this comment, we present the correct expression for the parametric gain. We show that for a transmission line with weak dispersion or positive dispersion ($Δk>0 $), as is the case discussed by Yaakobi et al, while quadratic (power) gain can occur around the pump frequency, exponential gain is impossible. Furthermore, for a transmission line with proper intrinsic or engineered dispersion, exponential gain occurs at frequencies where the phase matching condition is met, while around the pump frequency the gain is still quadratic.

arXiv:1308.2951•Aug 13, 2013•1 citation

Superconducting Josephson-Based Metamaterials for Quantum-Limited Parametric Amplification: A Review

Luca Fasolo, Angelo Greco, Emanuele Enrico

In the last few years, several groups have proposed and developed their own platforms demonstrating quantum-limited linear parametric amplification, with evident applications in quantum information and computation, electrical and optical metrology, radio astronomy, and basic physics concerning axion detection. Here, we propose a short review on the physics behind parametric amplification via metamaterials composed by coplanar waveguides embedding several Josephson junctions. We present and compare different schemes that exploit the nonlinearity of the Josephson current-phase relation to mix the so-called signal, idler, and pump tones. The chapter then presents and compares three different theoretical models, developed in the last few years, to predict the dynamics of these nonlinear systems in the particular case of a 4-wave mixing process and under the degenerate undepleted pump assumption. We will demonstrate that, under the same assumption, all the results are comparable in terms of amplification of the output fields.

arXiv:1908.06889•IntechOpen eBooks•Oct 24, 2019•9 citations

The Lagrangian approach to a Josephson traveling-wave parametric amplifier

Bogdan A. Kochetov, Arkady Fedorov

Future progress and development in ultrasensitive detection and quantum-limited amplification of microwave signals in a wide frequency band is expected to be achieved by using new type architectures of parametric amplifiers based on chains of coupled Josephson junctions or superconducting quantum interference devices (SQUID). Over the past years the classical theory of the amplifiers in the framework of a nonlinear wave equation governing the node flux evolution along an array of coupled Josephson oscillators has been developed under continuum approximation. Although such amplifiers imply the quantum nature utilizing the Josephson effects there is no quantum approach to take into account quantum phenomena in Josephson traveling-wave parametric amplifiers (JTWPA). Here we formulate the Lagrangian approach to a one-dimensional chain of coupled Josephson junctions and take the first step to develop the quantum theory of JTWPAs by deriving an approximate quantum Hamiltonian of the chain.

International Conference on Ultrawideband and Ultrashort Impulse Signals•Sep 1, 2016•5 citations

Quantum Models for Flux-Driven Superconducting Traveling-Wave Parametric Amplifiers with Different Nonlinear Junction Topologies

Michael Haider, Yongjie Yuan, Johannes A. Russer +1 more

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.

IEEE MTT-S International Microwave Symposium digest•Jun 11, 2023•5 citations
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.
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.
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.
Recently Yaakobi and co-workers [Phys. Rev. B 87, 144301 (2013)] theoretically studied four-wave mixing and parametric amplification in a nonlinear transmission line consisting of capacitively shunted Josephson junctions. By deriving and solving the coupled-mode equations, they have arrived at the conclusion that in a wide frequency range around the pump frequency exponential parametric gain (in which the signal grows exponentially with distance) can be achieved. However, we have found a mathematical error in their derivation of the coupled-mode equations (Equation (A13)), which leads to the wrong expression of the gain factor and invalidates their conclusions on the gain and bandwidth. In this comment, we present the correct expression for the parametric gain. We show that for a transmission line with weak dispersion or positive dispersion ($Δk>0 $), as is the case discussed by Yaakobi et al, while quadratic (power) gain can occur around the pump frequency, exponential gain is impossible. Furthermore, for a transmission line with proper intrinsic or engineered dispersion, exponential gain occurs at frequencies where the phase matching condition is met, while around the pump frequency the gain is still quadratic.
In the last few years, several groups have proposed and developed their own platforms demonstrating quantum-limited linear parametric amplification, with evident applications in quantum information and computation, electrical and optical metrology, radio astronomy, and basic physics concerning axion detection. Here, we propose a short review on the physics behind parametric amplification via metamaterials composed by coplanar waveguides embedding several Josephson junctions. We present and compare different schemes that exploit the nonlinearity of the Josephson current-phase relation to mix the so-called signal, idler, and pump tones. The chapter then presents and compares three different theoretical models, developed in the last few years, to predict the dynamics of these nonlinear systems in the particular case of a 4-wave mixing process and under the degenerate undepleted pump assumption. We will demonstrate that, under the same assumption, all the results are comparable in terms of amplification of the output fields.
Future progress and development in ultrasensitive detection and quantum-limited amplification of microwave signals in a wide frequency band is expected to be achieved by using new type architectures of parametric amplifiers based on chains of coupled Josephson junctions or superconducting quantum interference devices (SQUID). Over the past years the classical theory of the amplifiers in the framework of a nonlinear wave equation governing the node flux evolution along an array of coupled Josephson oscillators has been developed under continuum approximation. Although such amplifiers imply the quantum nature utilizing the Josephson effects there is no quantum approach to take into account quantum phenomena in Josephson traveling-wave parametric amplifiers (JTWPA). Here we formulate the Lagrangian approach to a one-dimensional chain of coupled Josephson junctions and take the first step to develop the quantum theory of JTWPAs by deriving an approximate quantum Hamiltonian of the chain.
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.