# Anatomy of fluorescence: quantum trajectory statistics from continuously measuring spontaneous emission

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## Abstract

We investigate the continuous quantum measurement of a superconducting qubit undergoing fluorescence. The fluorescence of the qubit is detected via a phase preserving heterodyne measurement, giving the fluorescence quadrature signals as two continuous qubit readout results. Using the stochastic path integral approach to the measurement physics, we derive most likely paths between boundary conditions on the state, and compute approximate time correlation functions between all stochastic variables via diagrammatic perturbation theory. We focus on paths that increase in energy during the continuous measurement. Our results are compared to Monte Carlo numerical simulation of the trajectories, and we find close agreement between direct simulation and theory. We generalize this analysis to arbitrary diffusive quantum systems that are continuously monitored.

### Keywords

Quantum trajectories Quantum measurement Spontaneous emission Heterodyne measurement Stochastic path integral## Notes

### Acknowledgments

This work was supported by US Army Research Office Grants No. W911NF-09-0-01417 and No. W911NF-15-1-0496, by National Science Foundation grant DMR-1506081, by John Templeton Foundation grant ID 58558, and by Development and Promotion of Science and Technology Talents Project Thailand. This work was partly supported by the EMERGENCES grant QUMOTEL of Ville de Paris. Thanks in particular to the COST Action MP1209 for supporting the Third Conference on Quantum Thermodynamics, in Porquerolles, France, where this work was begun. We thank Mark Dykman, Alexander Korotkov, Kater Murch, and Alain Sarlette for discussions and helpful comments on the work.

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