Mesoscopic Hamiltonian for Josephson travelling-wave parametric amplifiers

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. [Phys. Rev. Lett. $113:157001$] is shortly reviewed. Then a derivation of the full quantum theory is given using mesoscopic quantisation 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.

21 pages, 11 figures (excluding subfigures). Equation (18) has been found erroneous in the previous version and has been updated. This error, however, do not influence any of the final results or conclusions

References in corpus (12)

• QuTiP 2: A Python framework for the dynamics of open quantum systems
• Microwave photonics with superconducting quantum circuits
• Phase preserving amplification near the quantum limit with a Josephson Ring Modulator
• A widely tunable parametric amplifier based on a SQUID array resonator
• Introduction to Quantum Electromagnetic Circuits
• Resonantly phase-matched Josephson junction traveling wave parametric amplifier
• Quantum limited amplification and entanglement in coupled nonlinear resonators
• Broadband parametric amplification with impedance engineering: Beyond the gain-bandwidth product
• Widely tunable, non-degenerate three-wave mixing microwave device operating near the quantum limit
• Low-noise kinetic inductance traveling-wave amplifier using three-wave mixing
• Squeezing and quantum state engineering with Josephson traveling wave amplifiers
• Broadband parametric amplifiers based on nonlinear kinetic inductance artificial transmission lines

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• Continuous measurement of a qudit using dispersively coupled radiation
• Shock wave in series connected Josephson transmission line: Theoretical foundations and effects of resistive elements
• On Parametric Amplification In Discrete Josephson Transmission Line