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  • Open Access

Path integral approach to input-output theory

Aaron Daniel1,*, Matteo Brunelli1,2, Aashish A. Clerk3, and Patrick P. Potts1

  • 1Department of Physics and Swiss Nanoscience Institute, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
  • 2JEIP, UAR 3573 CNRS, Collège de France, PSL Research University, 11 Place Marcelin Berthelot, 75321 Paris Cedex 05, France
  • 3Pritzker School of Molecular Engineering, University of Chicago, Chicago, Illinois 60637, USA

  • *Contact author: aaron.daniel@unibas.ch
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Phys. Rev. Research 8, 023012 – Published 2 April, 2026

DOI: https://doi.org/10.1103/tds9-2tyz

Abstract

Input-output theory is a well-known tool in quantum optics and ubiquitous in the description of quantum systems probed by light. Owing to the generality of the setup it describes, the theory finds application in a wide variety of experiments in circuit and cavity QED. We present an approach to input-output theory using the Schwinger-Keldysh path integral formalism that gives us direct access to the full output field statistics such as the first- and second-order coherence functions. By making the rich toolbox of nonequilibrium quantum field theory accessible, our formalism greatly simplifies the treatment of nonlinear systems and provides a uniform way of obtaining perturbative results. We showcase this particular strength by computing the output field statistics of a Kerr nonlinear oscillator at finite temperatures through the use of diagrams and diagram summation techniques. We find a reduction in reflection that is not due to photon leakage but rather associated to the squeezing of the output light.

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Physics Subject Headings (PhySH)

Corrections

21 September, 2026

Correction: The omission of a factor in the minus quadrature of Eq. (80) has been fixed. A corresponding error in Eq. (81) has also been corrected.

Article Text

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Phys. Rev. Research 8, 023012– Published 2 April, 2026
  • Received 12 September 2025
  • Accepted 27 February 2026
  • Corrected 21 September 2026
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