Third Quantization for Order Parameters (II): Local Field Quantization in Superconducting Quantum Circuits

TL;DR

This paper derives a microscopic basis for the quantum behavior of superconducting transmission-line resonators, extending third quantization to local phases, and unifies circuit element quantization with underlying electron-phonon physics.

Contribution

It develops a microscopic derivation of local superconducting phase quantization, linking macroscopic circuit variables to microscopic parameters, and extends third quantization to spatially local order parameters.

Findings

Quantitative relations between current, voltage, and local superconducting phase.

Local phase becomes a genuine quantum variable after low-energy restriction.

Unified microscopic framework for superconducting circuit element quantization.

Abstract

The quantization of superconducting transmission-line resonators is usually introduced phenomenologically by modeling the resonator as an effective LC circuit and imposing canonical commutation relations on macroscopic variables such as charge and flux. Although this approach is highly successful, it leaves open why these macroscopic variables should obey quantum commutation relations and how this behavior emerges from the superconducting state. In this work, starting from the microscopic pairing Hamiltonian underlying BCS superconductivity, we derive the low-energy effective Hamiltonian of a circuit-QED architecture containing a superconducting transmission line with distributed capacitive and inductive elements. We establish quantitative relations between macroscopic observables, including current and voltage, and the spatially local superconducting phase, as well as the microscopic parameters of the electron-phonon system. We then extend the third quantization of the superconducting order parameter, introduced in Paper (I) for the global phase, to the spatially local case. This gives a macroscopic field quantization of the superconducting phase. We show that, after restriction to the low-energy excitation subspace, the local superconducting phase becomes a genuine quantum dynamical variable. Thus, the quantum behavior of transmission-line resonators need not be postulated at the macroscopic level, but follows from the third quantization of the superconducting order parameter. These results suggest that capacitive and inductive superconducting circuit elements share the same microscopic origin, providing a unified framework for superconducting circuit quantization.