Quantum Studies: Mathematics and Foundations

, Volume 3, Issue 3, pp 237–263

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

  • Andrew N. Jordan
  • Areeya Chantasri
  • Pierre Rouchon
  • Benjamin Huard
Regular Paper

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 

References

  1. 1.
    Aharonov, Y., Albert, D.Z., Vaidman, L.: How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100. Phys. Rev. Lett. 60, 1351 (1988)CrossRefGoogle Scholar
  2. 2.
    Alonso, J.J., Lutz, E., Romito, A.: Thermodynamics of weakly measured quantum systems. Phys. Rev. Lett. 116, 080403 (2015)CrossRefGoogle Scholar
  3. 3.
    Barchielli, A., Gregoratti, M.: Quantum trajectories and measurements in continuous time. Springer-Verlag, Berlin Heidelberg (2009). doi:10.1007/978-3-642-01298-3
  4. 4.
    Bolund, A., Mølmer, K.: Stochastic excitation during the decay of a two-level emitter subject to homodyne and heterodyne detection. Phys. Rev. A 89, 023827 (2014)CrossRefGoogle Scholar
  5. 5.
    Campagne-Ibarcq, P., Six, P., Bretheau, L., Sarlette, A., Mirrahimi, M., Rouchon, P., Huard, B.: Observing quantum state diffusion by heterodyne detection of fluorescence. Phys. Rev. X 6, 011002 (2016)Google Scholar
  6. 6.
    Carmichael, H.: An open systems approach to quantum optics: lectures presented at the Université Libre de Bruxelles, October 28 to November 4, 1991. Springer Science and Business Media (2009)Google Scholar
  7. 7.
    Carmichael, H.J., Singh, S., Vyas, R., Rice, P.R.: Photoelectron waiting times and atomic state reduction in resonance fluorescence. Phys. Rev. A 39, 1200 (1989)CrossRefGoogle Scholar
  8. 8.
    Caves, C.M.: Quantum limits on noise in linear amplifiers. Phys. Rev. D 26, 1817 (1982)CrossRefGoogle Scholar
  9. 9.
    Caves, C.M., Combes, J., Jiang, Z., Pandey, S.: Quantum limits on phase-preserving linear amplifiers. Phys. Rev. A 86, 063802 (2012)CrossRefGoogle Scholar
  10. 10.
    Chantasri, A., Dressel, J., Jordan, A.N.: Action principle for continuous quantum measurement. Phys. Rev. A 88, 042110 (2013)CrossRefGoogle Scholar
  11. 11.
    Chantasri, A., Jordan, A.N.: Stochastic path-integral formalism for continuous quantum measurement. Phys. Rev. A 92, 032125 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Dalibard, J., Castin, Y., Mølmer, K.: Wave-function approach to dissipative processes in quantum optics. Phys. Rev. Lett. 68, 580 (1992)CrossRefGoogle Scholar
  13. 13.
    Davies, E.B.: Quantum theory of open systems. IMA (1976)Google Scholar
  14. 14.
    Dressel, J., Agarwal, S., Jordan, A.N.: Contextual values of observables in quantum measurements. Phys. Rev. Lett. 104, 240401 (2010)CrossRefGoogle Scholar
  15. 15.
    Dressel, J., Jordan, A.N.: Contextual-value approach to the generalized measurement of observables. Phys. Rev. A 85, 022123 (2012)CrossRefGoogle Scholar
  16. 16.
    Dum, R., Zoller, P., Ritsch, H.: Monte Carlo simulation of the atomic master equation for spontaneous emission. Phys. Rev. A 45, 4879 (1992)CrossRefGoogle Scholar
  17. 17.
    Elouard, C., Auffeves, A., Clusel, M.: Stochastic thermodynamics in the quantum regime. arXiv preprint arXiv:1507.00312 (2015)
  18. 18.
    Gardiner, C.W.: Handbook of stochastic methods. Springer, Berlin (1985)Google Scholar
  19. 19.
    Hatridge, M., Shankar, S., Mirrahimi, M., Schackert, F., Geerlings, K., Brecht, T., Sliwa, K., Abdo, B., Frunzio, L., Girvin, S.M., et al.: Quantum back-action of an individual variable-strength measurement. Science 339, 178–181 (2013)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Holevo, A.S.: Statistical structure of quantum theory. Springer Science & Business Media (2001)Google Scholar
  21. 21.
    Jordan, A.N., Büttiker, M.: Continuous quantum measurement with independent detector cross correlations. Phys. Rev. Lett. 95, 220401 (2005)CrossRefGoogle Scholar
  22. 22.
    Jordan, A.N., Korotkov, A.N.: Qubit feedback and control with kicked quantum nondemolition measurements: a quantum bayesian analysis. Phys. Rev. B 74, 085307 (2006)CrossRefGoogle Scholar
  23. 23.
    Katz, N., Ansmann, M., Bialczak, R.C., Lucero, E., McDermott, R., Neeley, M., Steffen, M., Weig, E.M., Cleland, A.N., Martinis, J.M., et al.: Coherent state evolution in a superconducting qubit from partial-collapse measurement. Science 312, 1498–1500 (2006)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Katz, N., Neeley, M., Ansmann, M., Bialczak, R.C., Hofheinz, M., Lucero, E., O’Connell, A., Wang, H., Cleland, A., Martinis, J.M., et al.: Reversal of the weak measurement of a quantum state in a superconducting phase qubit. Phys. Rev. Lett. 101, 200401 (2008)CrossRefGoogle Scholar
  25. 25.
    Korotkov, A.N.: Continuous quantum measurement of a double dot. Phys. Rev. B 60, 5737 (1999)CrossRefGoogle Scholar
  26. 26.
    Korotkov, A.N.: Selective quantum evolution of a qubit state due to continuous measurement. Phys. Rev. B 63, 115403 (2001)CrossRefGoogle Scholar
  27. 27.
    Korotkov, A.N., Jordan, A.N.: Undoing a weak quantum measurement of a solid-state qubit. Phys. Rev. Lett. 97, 166805 (2006)CrossRefGoogle Scholar
  28. 28.
    Kraus, K., Böhm, A., Dollard, J.D., Wootters, W.: States, effects, and operations fundamental notions of quantum theory. Springer, Berlin Heidelberg (1983)CrossRefGoogle Scholar
  29. 29.
    Murch, K.W., Weber, S.J., Macklin, C., Siddiqi, I.: Observing single quantum trajectories of a superconducting quantum bit. Nature 502, 211–214 (2013)CrossRefGoogle Scholar
  30. 30.
    Nielsen, M.A., Chuang, I.L.: Quantum computation and quantum information. Cambridge University Press (2010)Google Scholar
  31. 31.
    Rouchon, P.: Models and feedback stabilization of open quantum systems. arXiv preprint arXiv:1407.7810 (2014)
  32. 32.
    Rouchon, P., Ralph, J.F.: Efficient quantum filtering for quantum feedback control. Phys. Rev. A 91, 012118 (2015)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Six, P., Campagne-Ibarcq, P., Bretheau, L., Huard, B., Rouchon, P.: Parameter estimation from measurements along quantum trajectories. arXiv preprint arXiv:1503.06149 (2015)
  34. 34.
    Weber, S.J., Chantasri, A., Dressel, J., Jordan, A.N., Murch, K.W., Siddiqi, I.: Mapping the optimal route between two quantum states. Nature 511, 570–573 (2014)CrossRefGoogle Scholar
  35. 35.
    Wiseman, H.M., Milburn, G.J.: Interpretation of quantum jump and diffusion processes illustrated on the Bloch sphere. Phys. Rev. A 47, 1652 (1993)CrossRefGoogle Scholar
  36. 36.
    Wiseman, H.M., Milburn, G.J.: Quantum measurement and control. Cambridge University Press (2009)Google Scholar
  37. 37.
    Wong, E., Zakai, M.: On the relation between ordinary and stochastic differential equations. Int. J. Eng. Sci. 3, 213–229 (1965)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Xu, Q., Greplova, E., Julsgaard, B., Mølmer, K.: Correlation functions and conditioned quantum dynamics in photodetection theory. Physica Scripta 90, 128004 (2015)CrossRefGoogle Scholar

Copyright information

© Chapman University 2016

Authors and Affiliations

  • Andrew N. Jordan
    • 1
    • 2
    • 3
  • Areeya Chantasri
    • 1
    • 2
  • Pierre Rouchon
    • 4
  • Benjamin Huard
    • 5
  1. 1.Department of Physics and AstronomyUniversity of RochesterRochesterUSA
  2. 2.Center for Coherence and Quantum Optics, University of RochesterRochesterUSA
  3. 3.Institute for Quantum Studies, Chapman UniversityOrangeUSA
  4. 4.Centre Automatique et Systèmes, Mines-ParisTechPSL Reseach UniversityParisFrance
  5. 5.Laboratoire Pierre Aigrain, Ecole Normale Supérieure-PSL Research UniversityCNRS, Université Pierre et Marie Curie-Sorbonne Universités, Université Paris Diderot-Sorbonne Paris CitéParisFrance

Personalised recommendations