Exact nonlinear response theory for lossy superconducting quantum circuits

Quantum systems inherently interact with their environments, typically consisting of a macroscopic number of degrees of freedom [1]. These interactions give rise

to a diverse range of phenomena, among which the most important are dissipation and decoherence [2]. In addition to the inevitable coupling to the natural environment, the study of open quantum systems is also motivated by the need to carry out measurements on real systems [3]. In such cases, the measurement apparatus itself acts as an environment, leading to decoherence. In our work we provide a formalism for calculation of electromagnetic field emitted by the superconducting circuits, which can be measured experimentally [4]. Starting from the Feynman–Vernon path integral formalism for the lossy circuits, we eliminate the influence functional by introducing auxiliary harmonic modes with complex-valued frequencies coupled to the non-linear degrees of freedom of the circuit. This results in a many-body Liouville equation for the state of the system. We propose a concept of time-averaged observables, inspired by experiment, and provide an explicit formula for producing their quasiprobability distribution. Finally, we demonstrate the applicability of our formalism through a study on the dispersive readout of a superconducting qubit.

[1] U. Weiss, Quantum dissipative systems (World Scientific, 2012)

[2] A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. Fisher, A. Garg, and W. Zwerger, Reviews of Modern Physics 59, 1 (1987).

[3] K. Jacobs, Quantum measurement theory and its applications (Cambridge University Press, 2014).

[4] A. Blais, A. L. Grimsmo, S. Girvin, and A. Wallraff, Reviews of Modern Physics 93, 025005 (2021)