Skip to main content
SpringerLink
Log in

Quantum filtering for a two-level atom driven by two counter-propagating photons

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

The purpose of this paper is to propose quantum filters for a two-level atom driven by two continuous-mode counter-propagating photons and under continuous measurements. Two scenarios of multiple measurements are discussed: (1) homodyne detection plus photodetection and (2) two homodyne detections. Filtering equations for both cases are derived explicitly. As demonstration, the two input photons with rising exponential and Gaussian pulse shapes are used to excite a two-level atom under two homodyne detection measurements. Simulations reveal scaling relations between atom-photon coupling and photonic pulse shape for maximum atomic excitation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Amini, H., Somaraju, R.A., Dotsenko, I., Sayrin, C., Mirrahimi, M., Rouchon, P.: Nondemolition measurements, feedback stabilization of discrete-time quantum systems subject to non-demolition measurements with imperfections and delays. Automatica 49(9), 2683–2692 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bachor, H.A., Ralph, T.C.: A Guide to Experiments in Quantum Optics. Wiley, Hoboken (2004)

    Book  Google Scholar 

  3. Baragiola, B.Q., Combes, J.: Quantum trajectories for propagating fock states. Phys. Rev. A 96, 023819 (2017)

    Article  ADS  Google Scholar 

  4. Baragiola, B.Q., Cook, R.L., Brańczyk, A.M., Combes, J.: N-photon wave packets interacting with an arbitrary quantum system. Phys. Rev. A 86(1), 013811 (2012)

    Article  ADS  Google Scholar 

  5. Barchielli, A., Gregoratti, M.: Quantum Trajectories and Measurements in Continuous Time: The Diffusive Case. Springer, Berlin (2009)

    Book  MATH  Google Scholar 

  6. Belavkin, V.P.: Nondemolition measurements, nonlinear filtering and dynamic programming of quantum stochastic processes. In: Blaquiére, A. (ed.) Modeling and Control of Systems. Lecture Notes in Control and Information Sciences, vol. 121, pp. 245–265. Springer, Berlin, Heidelberg (1989)

    Chapter  Google Scholar 

  7. Belavkin, V.P.: Quantum stochastic calculus and quantum nonlinear filtering. J. Multivar. Anal. 42(2), 171–201 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Belavkin, V.P.: Quantum filtering of markov signals with white quantum noise. In: Belavkin, V.P., Hirota, O., Hudson, R.L. (eds.) Quantum Communications and Measurement, pp. 381–391. Springer, Boston, MA (1995)

    Chapter  MATH  Google Scholar 

  9. Belavkin, V.P.: Quantum quasi-markov processes in eventum mechanics dynamics, observation, filtering and control. Quantum Inf. Process. 12, 1539–1626 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Bouten, L., Handel, R.V., James, M.R.: An introduction to quantum filtering. SIAM J. Control Optim. 46(6), 2199–2241 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Carvalho, A.R.R., Hush, M.R., James, M.R.: Cavity driven by a single photon: conditional dynamics and nonlinear phase shift. Phys. Rev. A 86(2), 023806 (2012)

    Article  ADS  Google Scholar 

  12. Chia, A., Wiseman, H.M.: Quantum theory of multiple-input-multiple-output Markovian feedback with diffusive measurements. Phys. Rev. A 84(1), 012120 (2011)

    Article  ADS  Google Scholar 

  13. Combes, J., Kerckhoff, J., Sarovar, M.: The SLH framework for modeling quantum input–output networks. Adv. Phys. X 2(3), 784–888 (2017)

    Google Scholar 

  14. Dong, Z., Zhang, G., Amini, H.: Quantum filtering for multiple measurements driven by fields in single-photon states. In: American Control Conference (ACC), pp. 4754–4759 (2016)

  15. Dong, Z., Zhang, G., Amini, H.: Quantum filtering for multiple measurements driven by two single-photon states. In: 12th World Congress on Intelligent Control and Automation (WCICA), pp. 3011–3015 (2016)

  16. Dong, Z., Zhang, G., Amini, H.: Single-photon quantum filtering with multiple measurements. Int. J. Adapt. Control Signal Process. 32(3), 528–546 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  17. Dum, R., Parkins, A.S., Zoller, P., Gardiner, C.W.: Monte Carlo simulation of master equations in quantum optics for vacuum, thermal, and squeezed reservoirs. Phys. Rev. A 46(7), 4382 (1992)

    Article  ADS  Google Scholar 

  18. Emzir, M.F., Woolley, M.J., Petersen, I.R.: Quantum filtering for multiple diffusive and Poissonian measurements. J. Phys. A: Math. Theor. 48(38), 385302 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Gao, Q., Dong, D., Petersen, I.R.: Fault tolerant quantum filtering and fault detection for quantum systems. Automatica 71, 125–134 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gao, Q., Dong, D., Petersen, I.R., Rabitz, H.: Fault tolerant filtering and fault detection for quantum systems driven by fields in single photon states. J. Math. Phys. 57(6), 062201 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Gao, Q., Zhang, G., Petersen, I.R.: An exponential quantum projection filter for open quantum systems. Automatica 99, 59–68 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  22. Gardiner, C.W., Zoller, P.: Quantum Noise. Springer, Berlin (2004)

    MATH  Google Scholar 

  23. Gough, J.E., Belavkin, V.P.: Quantum control and information processing. Quantum Inf. Process. 12, 1397–1415 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Gough, J.E., James, M.R.: The series product and its application to quantum feedforward and feedback networks. IEEE Trans. Autom. Contr. 54(11), 2530–2544 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gough, J.E., James, M.R., Nurdin, H.I.: Single photon quantum filtering using non-Markovian embeddings. Philos. Trans. R. Soc. A 370(1979), 5408–5421 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Gough, J.E., James, M.R., Nurdin, H.I.: Quantum filtering for systems driven by fields in single photon states and superposition of coherent states using non-Markovian embeddings. Quantum Inf. Process. 12(3), 1469 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Gough, J.E., James, M.R., Nurdin, H.I.: Quantum trajectories for a class of continuous matrix product input states. New J. Phys. 16(7), 075008 (2014)

    Article  ADS  Google Scholar 

  28. Gough, J.E., James, M.R., Nurdin, H.I., Combes, J.: Quantum filtering for systems driven by fields in single-photon states or superposition of coherent states. Phys. Rev. A 86(4), 043819 (2012)

    Article  ADS  Google Scholar 

  29. Gough, J.E., Zhang, G.: Generating nonclassical quantum input field states with modulating filters. EPJ Quantum Technol. 2(1), 15 (2015)

    Article  Google Scholar 

  30. Lodahl, P., Mahmoodian, S., Stobbe, S.: Interfacing single photons and single quantum dots with photonic nanostructures. Rev. Mod. Phys. 87(2), 347 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  31. Lodahl, P., Mahmoodian, S., Stobbe, S., Rauschenbeutel, A., Schneeweiss, P., Volz, J., Pichler, H., Zoller, P.: Chiral quantum optics. Nature 541(7638), 473 (2017)

    Article  ADS  Google Scholar 

  32. Loudon, R.: The Quantum Theory of Light. Oxford University Press, Oxford (2000)

    MATH  Google Scholar 

  33. Nurdin, H.I.: Quantum filtering for multiple input multiple output systems driven by arbitrary zero-mean jointly Gaussian input fields. Russ. J. Math. Phys. 21(3), 386–398 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  34. Nysteen, A., Kristensen, P.T., McCutcheon, D.P., Kaer, P., Mørk, J.: Scattering of two photons on a quantum emitter in a one-dimensional waveguide: exact dynamics and induced correlations. New J. Phys. 17(2), 023030 (2015)

    Article  ADS  Google Scholar 

  35. Ogawa, H., Ohdan, H., Miyata, K., Taguchi, M., Makino, K., Yonezawa, H., Yoshikawa, Ji, Furusawa, A.: Real-time quadrature measurement of a single-photon wave packet with continuous temporal-mode matching. Phys. Rev. Lett. 116, 233602 (2016)

    Article  ADS  Google Scholar 

  36. Pan, Y., Dong, D., Zhang, G.: Exact analysis of the response of quantum systems to two-photons using a QSDE approach. New J. Phys. 18(3), 033004 (2016)

    Article  ADS  Google Scholar 

  37. Pan, Y., Zhang, G., James, M.R.: Analysis and control of quantum finite-level systems driven by single-photon input states. Automatica 69, 18–23 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  38. Raymer, M.G., Noh, J., Banaszek, K., Walmsley, I.A.: Pure-state single-photon wave-packet generation by parametric down-conversion in a distributed microcavity. Phys. Rev. A 72(2), 023825 (2005)

    Article  ADS  Google Scholar 

  39. Rag, H.S., Gea-Banacloche, J.: Two-level-atom excitation probability for single- and \(N\)-photon wave packets. Phys. Rev. A 96(3), 033817 (2017)

    ADS  Google Scholar 

  40. Rephaeli, E., Fan, S.: Stimulated emission from a single excited atom in a waveguide. Phys. Rev. Lett. 108, 143602 (2012)

    Article  ADS  Google Scholar 

  41. Rephaeli, E., Shen, J.T., Fan, S.: Full inversion of a two-level atom with a single-photon pulse in one-dimensional geometries. Phys. Rev. A 82, 033804 (2010)

    Article  ADS  Google Scholar 

  42. Roulet, A., Scarani, V.: Solving the scattering of N photons on a two-level atom without computation. New J. Phys. 18(9), 093035 (2016)

    Article  ADS  Google Scholar 

  43. Sarma, G., Hamerly, R., Tezak, N., Pavlichin, D.S., Mabuchi, H.: Transformation of quantum photonic circuit models by term rewriting. IEEE Photonics J. 5(1), 7500111 (2013)

    Article  ADS  Google Scholar 

  44. Song, H., Zhang, G., Xi, Z.: Continuous-mode multi-photon filtering. SIAM J. Control Optim. 54(3), 1602–1632 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  45. Stobinska, M., Alber, G., Leuchs, G.: Perfect excitation of a matter qubit by a single photon in free space. EPL (Europhysics Letters) 86(1), 14007 (2009)

    Article  ADS  Google Scholar 

  46. Sun, S., Kim, H., Luo, Z., Solomon, G.S., Waks, E.: A single-photon switch and transistor enabled by a solid-state quantum memory. Science 361(6397), 57–60 (2018)

    Article  ADS  Google Scholar 

  47. Tezak, N., Niederberger, A., Pavlichin, D.S., Sarma, G., Mabuchi, H.: Specification of photonic circuits using quantum hardware description language. Philos. Trans. R. Soc. A 370(1979), 5270–5290 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  48. Wang, Y., Minář, J., Sheridan, L., Scarani, V.: Efficient excitation of a two-level atom by a single photon in a propagating mode. Phys. Rev. A 83(6), 063842 (2011)

    Article  ADS  Google Scholar 

  49. Wiseman, H.M., Milburn, G.J.: Quantum Measurement and Control. Cambridge University Press, Cambridge (2009)

    Book  MATH  Google Scholar 

  50. Yukawa, M., Miyata, K., Mizuta, T., Yonezawa, H., Marek, P., Filip, R., Furusawa, A.: Generating superposition of up-to three photons for continuous variable quantum information processing. Opt. Express 21(5), 5529–5535 (2013)

    Article  ADS  Google Scholar 

  51. Zhang, G., James, M.R.: Quantum feedback networks and control: a brief survey. Chin. Sci. Bull. 57(18), 2200–2214 (2012)

    Article  Google Scholar 

  52. Zhang, J., Liu, Y., Wu, R., Jacobs, K., Nori, F.: Quantum feedback: theory, experiments, and applications. Phys. Rep. 679, 1–60 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We wish to thank financial supports from the Hong Kong Research Grant Council under Grants 15206915 and 15208418, JCJC INS2I 2016 “QIGR3CF” Project and JCJC INS2I 2017 “QFCCQI” project.

Author information

Authors and Affiliations

  1. Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen, 518055, China

    Zhiyuan Dong

  2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China

    Guofeng Zhang

  3. Laboratoire des signaux et systèmes (L2S), CNRS-CentraleSupélec-Université Paris-Sud, Université Paris-Saclay, 3, rue Joliot Curie, 91190, Gif-sur-Yvette, France

    Nina H. Amini

Authors
  1. Zhiyuan Dong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guofeng Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Nina H. Amini
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guofeng Zhang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Appendix A

In this appendix, we give the stochastic differential equations for the quantum filter mentioned in Theorem 1. (The dynamics of \(\rho ^{11;11}(t)\) has already been given in (32))

$$\begin{aligned} \mathrm{d}\rho ^{10;11}(t)= & {} \Big \{(\kappa _1+\kappa _2){\mathcal {D}}_{\sigma _-}^\star \rho ^{10;11}(t)+\sqrt{\kappa _1}\xi _1(t)[\rho ^{00;11}(t),\sigma _+]+\sqrt{\kappa _2}\xi _2(t)[\rho ^{10;01}(t),\sigma _+]\\&+\sqrt{\kappa _2}\xi _2^*(t)[\sigma _-,\rho ^{10;10}(t)]\Big \}\mathrm{d}t\\&+\Big \{\sqrt{1-r^2}\left[ \xi _1(t)\rho ^{00;11}(t)+\sqrt{\kappa _1}\rho ^{10;11}(t)\sigma _++\sqrt{\kappa _1}\sigma _-\rho ^{10;11}(t)\right] \\&+r\left[ \xi _2^*(t)\rho ^{10;10}(t)+\xi _2(t)\rho ^{10;01}(t)+\sqrt{\kappa _2}\rho ^{10;11}(t)\sigma _++\sqrt{\kappa _2}\sigma _-\rho ^{10;11}(t)\right] \\&-\rho ^{10;11}(t)\left[ \sqrt{1-r^2}z_{11}(t)+rz_{12}(t)\right] \Big \}\mathrm{d}W_1(t)\\&+\bigg \{K_p^{-1}(t)\Big \{r^2\Big [2\sqrt{\kappa _1}\xi _1(t)\rho ^{00;11}(t)\sigma _++\kappa _1\sigma _-\rho ^{10;11}(t)\sigma _+\Big ]\\&-r\sqrt{1-r^2}\Big [2\sqrt{\kappa _1}\xi _2(t)\rho ^{10;01}(t)\sigma _+\\&+2\xi _1(t)\xi _2^*(t)\rho ^{00;10}(t)+\sqrt{\kappa _1}\xi _2^*(t)\sigma _+\rho ^{10;10}(t)+2\sqrt{\kappa _2}\xi _1(t)\rho ^{00;11}(t)\sigma _+\Big ]\\&+(1-r^2)\Big [2|\xi _2(t)|^2\rho ^{10;00}(t)+\sqrt{\kappa _2}\xi _2^*(t)\sigma _-\rho ^{10;10}(t)+2\sqrt{\kappa _2}\xi _2(t)\rho ^{10;01}(t)\sigma _+\\&+\kappa _2\sigma _-\rho ^{10;11}(t)\sigma _+\Big ]\Big \}-\rho ^{10;11}(t)\bigg \}\mathrm{d}N(t),\\ \mathrm{d}\rho ^{00;11}(t)= & {} \Big \{(\kappa _1+\kappa _2){\mathcal {D}}_{\sigma _-}^\star \rho ^{00;11}(t)+\sqrt{\kappa _2}\xi _2(t)[\rho ^{00;01}(t),\sigma _+]+\sqrt{\kappa _2}\xi _2^*(t)[\sigma _-,\rho ^{00;10}(t)]\Big \}\mathrm{d}t\\&+\Big \{\sqrt{1-r^2}\left[ \sqrt{\kappa _1}\rho ^{00;11}(t)\sigma _++\sqrt{\kappa _1}\sigma _-\rho ^{00;11}(t)\right] \\&+r\left[ \xi _2^*(t)\rho ^{00;10}(t)+\xi _2(t)\rho ^{00;01}(t)+\sqrt{\kappa _2}\rho ^{00;11}(t)\sigma _++\sqrt{\kappa _2}\sigma _-\rho ^{00;11}(t)\right] \\&-\rho ^{00;11}(t)\left[ \sqrt{1-r^2}z_{11}(t)+rz_{12}(t)\right] \Big \}\mathrm{d}W_1(t)\\&+\bigg \{K_p^{-1}(t)\Big \{r^2\Big [\kappa _1\sigma _-\rho ^{00;11}(t)\sigma _+\Big ]-r\sqrt{1-r^2}\Big [2\sqrt{\kappa _1}\xi _2(t)\rho ^{00;01}(t)\sigma _+\\&+\sqrt{\kappa _1}\xi _2^*(t)\sigma _-\rho ^{00;10}(t)\Big ]+(1-r^2)\Big [2|\xi _2(t)|^2\rho ^{00;00}(t)+\sqrt{\kappa _2}\xi _2^*(t)\sigma _-\rho ^{00;10}(t)\\&+2\sqrt{\kappa _2}\xi _2(t)\rho ^{00;01}(t)\sigma _++\kappa _2\sigma _-\rho ^{00;11}(t)\sigma _+\Big ]\Big \}-\rho ^{00;11}(t)\bigg \}\mathrm{d}N(t), \end{aligned}$$
$$\begin{aligned} \mathrm{d}\rho ^{11;10}(t)= & {} \Big \{(\kappa _1+\kappa _2){\mathcal {D}}_{\sigma _-}^\star \rho ^{11;10}(t)+\sqrt{\kappa _1}\xi _1(t)[\rho ^{01;10}(t),\sigma _+]+\sqrt{\kappa _1}\xi _1^*(t)[\sigma _-,\rho ^{10;10}(t)]\\&+\sqrt{\kappa _2}\xi _2(t)[\rho ^{11;00}(t),\sigma _+]\Big \}\mathrm{d}t\\&+\Big \{\sqrt{1-r^2}\left[ \xi _1^*(t)\rho ^{10;10}(t)+\xi _1(t)\rho ^{01;10}(t)+\sqrt{\kappa _1}\rho ^{11;10}(t)\sigma _++\sqrt{\kappa _1}\sigma _-\rho ^{11;10}(t)\right] \\&+r\left[ \xi _2(t)\rho ^{11;00}(t)+\sqrt{\kappa _2}\rho ^{11;10}(t)\sigma _++\sqrt{\kappa _2}\sigma _-\rho ^{11;10}(t)\right] \\&-\rho ^{11;10}(t)\left[ \sqrt{1-r^2}z_{11}(t)+rz_{12}(t)\right] \Big \}\mathrm{d}W_1(t)\\&+\bigg \{K_p^{-1}(t)\Big \{r^2\Big [2|\xi _1(t)|^2\rho ^{00;10}(t)+\sqrt{\kappa _1}\xi _1^*(t)\sigma _-\rho ^{10;10}(t)+2\sqrt{\kappa _1}\xi _1(t)\rho ^{01;10}(t)\sigma _+\\&+\kappa _1\sigma _-\rho ^{11;10}(t)\sigma _+\Big ]\\&-r\sqrt{1-r^2}\Big [2\xi _1^*(t)\xi _2(t)\rho ^{10;00}(t)+\sqrt{\kappa _2}\xi _1^*(t)\sigma _-\rho ^{10;10}(t)+2\sqrt{\kappa _1}\xi _2(t)\rho ^{11;00}(t)\sigma _+\\&+2\sqrt{\kappa _2}\xi _1(t)\rho ^{01;10}(t)\sigma _+\Big ]+(1-r^2)\Big [2\sqrt{\kappa _2}\xi _2(t)\rho ^{11;00}(t)\sigma _++\kappa _2\sigma _-\rho ^{11;10}(t)\sigma _+\Big ]\Big \}\\&-\rho ^{11;10}(t)\bigg \}\mathrm{d}N(t),\\ \end{aligned}$$
$$\begin{aligned} \mathrm{d}\rho ^{10;10}(t)= & {} \Big \{(\kappa _1+\kappa _2){\mathcal {D}}_{\sigma _-}^\star \rho ^{10;10}(t)+\sqrt{\kappa _1}\xi _1(t)[\rho ^{00;10}(t),\sigma _+]+\sqrt{\kappa _2}\xi _2(t)[\rho ^{10;00}(t),\sigma _+]\Big \}\mathrm{d}t\\&+\Big \{\sqrt{1-r^2}\left[ \xi _1(t)\rho ^{00;10}(t)+\sqrt{\kappa _1}\rho ^{10;10}(t)\sigma _++\sqrt{\kappa _1}\sigma _-\rho ^{10;10}(t)\right] \\&+r\left[ \xi _2(t)\rho ^{10;00}(t)+\sqrt{\kappa _2}\rho ^{10;10}(t)\sigma _++\sqrt{\kappa _2}\sigma _-\rho ^{10;10}(t)\right] \\&-\rho ^{10;10}(t)\left[ \sqrt{1-r^2}z_{11}(t)+rz_{12}(t)\right] \Big \}\mathrm{d}W_1(t)\\&+\bigg \{K_p^{-1}(t)\Big \{r^2\Big [2\sqrt{\kappa _1}\xi _1(t)\rho ^{00;10}(t)\sigma _++\kappa _1\sigma _-\rho ^{10;10}(t)\sigma _+\Big ]\\&-r\sqrt{1-r^2}\Big [2\sqrt{\kappa _1}\xi _2(t)\rho ^{10;00}(t)\sigma _++2\sqrt{\kappa _2}\xi _1(t)\rho ^{00;10}(t)\sigma _+\Big ]\\&+(1-r^2)\Big [2\sqrt{\kappa _2}\xi _2(t)\rho ^{10;00}(t)\sigma _++\kappa _2\sigma _-\rho ^{10;10}(t)\sigma _+\Big ]\Big \}-\rho ^{10;10}(t)\bigg \}\mathrm{d}N(t),\\ \mathrm{d}\rho ^{01;10}(t)= & {} \Big \{(\kappa _1+\kappa _2){\mathcal {D}}_{\sigma _-}^\star \rho ^{01;10}(t)+\sqrt{\kappa _1}\xi _1^*(t)[\sigma _-,\rho ^{00;10}(t)]+\sqrt{\kappa _2}\xi _2(t)[\rho ^{01;00}(t),\sigma _+]\Big \}\mathrm{d}t\\&+\Big \{\sqrt{1-r^2}\left[ \xi _1^*(t)\rho ^{00;10}(t)+\sqrt{\kappa _1}\rho ^{01;10}(t)\sigma _++\sqrt{\kappa _1}\sigma _-\rho ^{01;10}(t)\right] \\&+r\left[ \xi _2(t)\rho ^{01;00}(t)+\sqrt{\kappa _2}\rho ^{01;10}(t)\sigma _++\sqrt{\kappa _2}\sigma _-\rho ^{01;10}(t)\right] \\&-\rho ^{01;10}(t)\left[ \sqrt{1-r^2}z_{11}(t)+rz_{12}(t)\right] \Big \}\mathrm{d}W_1(t)\\&+\bigg \{K_p^{-1}(t)\Big \{r^2\Big [\sqrt{\kappa _1}\xi _1^*(t)\sigma _-\rho ^{00;10}(t)+\kappa _1\sigma _-\rho ^{01;10}(t)\sigma _+\Big ]\\&-r\sqrt{1-r^2}\Big [2\xi _1^*(t)\xi _2(t)\rho ^{00;00}(t)+\sqrt{\kappa _2}\xi _1^*(t)\sigma _-\rho ^{00;10}(t)+2\sqrt{\kappa _1}\xi _2(t)\rho ^{01;00}(t)\sigma _+\Big ]\\&+(1-r^2)\Big [2\sqrt{\kappa _2}\xi _2(t)\rho ^{01;00}(t)\sigma _++\kappa _2\sigma _-\rho ^{01;10}(t)\sigma _+\Big ]\Big \}-\rho ^{01;10}(t)\bigg \}\mathrm{d}N(t),\\ \mathrm{d}\rho ^{00;10}(t)= & {} \Big \{(\kappa _1+\kappa _2){\mathcal {D}}_{\sigma _-}^\star \rho ^{00;10}(t)+\sqrt{\kappa _2}\xi _2(t)[\rho ^{00;00}(t),\sigma _+]\Big \}\mathrm{d}t\\&+\Big \{\sqrt{1-r^2}\left[ \sqrt{\kappa _1}\rho ^{00;10}(t)\sigma _++\sqrt{\kappa _1}\sigma _-\rho ^{00;10}(t)\right] +r\big [\xi _2(t)\rho ^{00;00}(t)\\&+\sqrt{\kappa _2}\rho ^{00;10}(t)\sigma _++\sqrt{\kappa _2}\sigma _-\rho ^{00;10}(t)\big ]-\rho ^{00;10}(t)\left[ \sqrt{1-r^2}z_{11}(t)+rz_{12}(t)\right] \Big \}\mathrm{d}W_1(t)\\&+\bigg \{K_p^{-1}(t)\Big \{r^2\Big [\kappa _1\sigma _-\rho ^{00;10}(t)\sigma _+\Big ]-r\sqrt{1-r^2}\Big [2\sqrt{\kappa _1}\xi _2(t)\rho ^{00;00}(t)\sigma _+\Big ]\\&+(1-r^2)\Big [2\sqrt{\kappa _2}\xi _2(t)\rho ^{00;00}(t)\sigma _++\kappa _2\sigma _-\rho ^{00;10}(t)\sigma _+\Big ]\Big \}-\rho ^{00;10}(t)\bigg \}\mathrm{d}N(t), \end{aligned}$$
$$\begin{aligned} \mathrm{d}\rho ^{11;00}(t)= & {} \Big \{(\kappa _1+\kappa _2){\mathcal {D}}_{\sigma _-}^\star \rho ^{11;00}(t)+\sqrt{\kappa _1}\xi _1(t)[\rho ^{01;00}(t),\sigma _+]+\sqrt{\kappa _1}\xi _1^*(t)[\sigma _-,\rho ^{10;00}(t)]\Big \}\mathrm{d}t\\&+\Big \{\sqrt{1-r^2}\left[ \xi _1^*(t)\rho ^{10;00}(t)+\xi _1(t)\rho ^{01;00}(t)+\sqrt{\kappa _1}\rho ^{11;00}(t)\sigma _++\sqrt{\kappa _1}\sigma _-\rho ^{11;00}(t)\right] \\&+r\left[ \sqrt{\kappa _2}\rho ^{11;00}(t)\sigma _++\sqrt{\kappa _2}\sigma _-\rho ^{11;00}(t)\right] -\rho ^{11;00}(t)\left[ \sqrt{1-r^2}z_{11}(t)+rz_{12}(t)\right] \Big \}\mathrm{d}W_1(t)\\&+\bigg \{K_p^{-1}(t)\Big \{r^2\Big [2|\xi _1(t)|^2\rho ^{00;00}(t)+\sqrt{\kappa _1}\xi _1^*(t)\sigma _-\rho ^{10;00}(t)\\&+2\sqrt{\kappa _1}\xi _1(t)\rho ^{01;00}(t)\sigma _++\kappa _1\sigma _-\rho ^{11;00}(t)\sigma _+\Big ]\\&-r\sqrt{1-r^2}\Big [\sqrt{\kappa _2}\xi _1^*(t)\sigma _-\rho ^{10;00}(t)+2\sqrt{\kappa _2}\xi _1(t)\rho ^{01;00}(t)\sigma _+\Big ]\\&+(1-r^2)\Big [\kappa _2\sigma _-\rho ^{11;00}(t)\sigma _+\Big ]\Big \}-\rho ^{11;00}(t)\bigg \}\mathrm{d}N(t),\\ \mathrm{d}\rho ^{10;00}(t)= & {} \Big \{(\kappa _1+\kappa _2){\mathcal {D}}_{\sigma _-}^\star \rho ^{10;00}(t)+\sqrt{\kappa _1}\xi _1(t)[\rho ^{00;00}(t),\sigma _+]\Big \}\mathrm{d}t\\&+\Big \{\sqrt{1-r^2}\left[ \xi _1(t)\rho ^{00;00}(t)+\sqrt{\kappa _1}\rho ^{10;00}(t)\sigma _++\sqrt{\kappa _1}\sigma _-\rho ^{10;00}(t)\right] \\&+r\left[ \sqrt{\kappa _2}\rho ^{10;00}(t)\sigma _++\sqrt{\kappa _2}\sigma _-\rho ^{10;00}(t)\right] \\&-\rho ^{10;00}(t)\left[ \sqrt{1-r^2}z_{11}(t)+rz_{12}(t)\right] \Big \}\mathrm{d}W_1(t)\\&+\bigg \{K_p^{-1}(t)\Big \{r^2\Big [2\sqrt{\kappa _1}\xi _1(t)\rho ^{00;00}(t)\sigma _++\kappa _1\sigma _-\rho ^{10;00}(t)\sigma _+\Big ]\\&-r\sqrt{1-r^2}\Big [2\sqrt{\kappa _2}\xi _1(t)\rho ^{00;00}(t)\sigma _+\Big ]\\&+(1-r^2)\Big [\kappa _2\sigma _-\rho ^{10;00}(t)\sigma _+\Big ]\Big \}-\rho ^{10;00}(t)\bigg \}\mathrm{d}N(t),\\ \mathrm{d}\rho ^{00;00}(t)= & {} \Big \{(\kappa _1+\kappa _2){\mathcal {D}}_{\sigma _-}^\star \rho ^{00;00}(t)\Big \}\mathrm{d}t\\&+\Big \{\sqrt{1-r^2}\left[ \sqrt{\kappa _1}\rho ^{00;00}(t)\sigma _++\sqrt{\kappa _1}\sigma _-\rho ^{00;00}(t)\right] +r\big [\sqrt{\kappa _2}\rho ^{00;00}(t)\sigma _+\\&+\sqrt{\kappa _2}\sigma _-\rho ^{00;00}(t)\big ]-\rho ^{00;00}(t)\left[ \sqrt{1-r^2}z_{11}(t)+rz_{12}(t)\right] \Big \}\mathrm{d}W_1(t)\\&+\bigg \{K_p^{-1}(t)\Big \{r^2\Big [\kappa _1\sigma _-\rho ^{00;00}(t)\sigma _+\Big ]+(1-r^2)\Big [\kappa _2\sigma _-\rho ^{00;00}(t)\sigma _+\Big ]\Big \}-\rho ^{00;00}(t)\bigg \}\mathrm{d}N(t). \end{aligned}$$

Appendix B

In this appendix, we give the stochastic differential equations for the quantum filter mentioned in Theorem 2. (The dynamics of \(\rho ^{11;11}(t)\) has already been given in (41))

$$\begin{aligned} \mathrm{d}\rho ^{10;11}(t)= & {} \Big \{(\kappa _1+\kappa _2){\mathcal {D}}_{\sigma _-}^\star \rho ^{10;11}(t)+\sqrt{\kappa _1}\xi _1(t)[\rho ^{00;11}(t),\sigma _+]+\sqrt{\kappa _2}\xi _2(t)[\rho ^{10;01}(t),\sigma _+]\\&+\sqrt{\kappa _2}\xi _2^*(t)[\sigma _-,\rho ^{10;10}(t)]\Big \}\mathrm{d}t\\&+\Big \{\sqrt{1-r^2}\left[ \xi _1(t)\rho ^{00;11}(t)+\sqrt{\kappa _1}\rho ^{10;11}(t)\sigma _++\sqrt{\kappa _1}\sigma _-\rho ^{10;11}(t)\right] \\&+r\left[ \xi _2^*(t)\rho ^{10;10}(t)+\xi _2(t)\rho ^{10;01}(t)+\sqrt{\kappa _2}\rho ^{10;11}(t)\sigma _++\sqrt{\kappa _2}\sigma _-\rho ^{10;11}(t)\right] \\&-\rho ^{10;11}(t)\left[ \sqrt{1-r^2}z_{11}(t)+rz_{12}(t)\right] \Big \}\mathrm{d}W_1(t)\\&+\Big \{-r\left[ \xi _1(t)\rho ^{00;11}(t)+\sqrt{\kappa _1}\rho ^{10;11}(t)\sigma _++\sqrt{\kappa _1}\sigma _-\rho ^{10;11}(t)\right] \\&+\sqrt{1-r^2}\left[ \xi _2^*(t)\rho ^{10;10}(t)+\xi _2(t)\rho ^{10;01}(t)+\sqrt{\kappa _2}\rho ^{10;11}(t)\sigma _++\sqrt{\kappa _2}\sigma _-\rho ^{10;11}(t)\right] \\&-\rho ^{10;11}(t)\left[ -rz_{11}(t)+\sqrt{1-r^2}z_{12}(t)\right] \Big \}\mathrm{d}W_2(t),\\ \mathrm{d}\rho ^{00;11}(t)= & {} \Big \{(\kappa _1+\kappa _2){\mathcal {D}}_{\sigma _-}^\star \rho ^{00;11}(t)+\sqrt{\kappa _2}\xi _2(t)[\rho ^{00;01}(t),\sigma _+]+\sqrt{\kappa _2}\xi _2^*(t)[\sigma _-,\rho ^{00;10}(t)]\Big \}\mathrm{d}t\\&+\Big \{\sqrt{1-r^2}\left[ \sqrt{\kappa _1}\rho ^{00;11}(t)\sigma _++\sqrt{\kappa _1}\sigma _-\rho ^{00;11}(t)\right] \\&+r\left[ \xi _2^*(t)\rho ^{00;10}(t)+\xi _2(t)\rho ^{00;01}(t)+\sqrt{\kappa _2}\rho ^{00;11}(t)\sigma _++\sqrt{\kappa _2}\sigma _-\rho ^{00;11}(t)\right] \\&-\rho ^{00;11}(t)\left[ \sqrt{1-r^2}z_{11}(t)+rz_{12}(t)\right] \Big \}\mathrm{d}W_1(t)\\&+\Big \{-r\left[ \sqrt{\kappa _1}\rho ^{00;11}(t)\sigma _++\sqrt{\kappa _1}\sigma _-\rho ^{00;11}(t)\right] \\&+\sqrt{1-r^2}\left[ \xi _2^*(t)\rho ^{00;10}(t)+\xi _2(t)\rho ^{00;01}(t)+\sqrt{\kappa _2}\rho ^{00;11}(t)\sigma _++\sqrt{\kappa _2}\sigma _-\rho ^{00;11}(t)\right] \\&-\rho ^{00;11}(t)\left[ -rz_{11}(t)+\sqrt{1-r^2}z_{12}(t)\right] \Big \}\mathrm{d}W_2(t),\\ \mathrm{d}\rho ^{11;10}(t)= & {} \Big \{(\kappa _1+\kappa _2){\mathcal {D}}_{\sigma _-}^\star \rho ^{11;10}(t)+\sqrt{\kappa _1}\xi _1(t)[\rho ^{01;10}(t),\sigma _+]+\sqrt{\kappa _1}\xi _1^*(t)[\sigma _-,\rho ^{10;10}(t)]\\&+\sqrt{\kappa _2}\xi _2(t)[\rho ^{11;00}(t),\sigma _+]\Big \}\mathrm{d}t\\&+\Big \{\sqrt{1-r^2}\left[ \xi _1^*(t)\rho ^{10;10}(t)+\xi _1(t)\rho ^{01;10}(t)+\sqrt{\kappa _1}\rho ^{11;10}(t)\sigma _++\sqrt{\kappa _1}\sigma _-\rho ^{11;10}(t)\right] \\&+r\left[ \xi _2(t)\rho ^{11;00}(t)+\sqrt{\kappa _2}\rho ^{11;10}(t)\sigma _++\sqrt{\kappa _2}\sigma _-\rho ^{11;10}(t)\right] \\&-\rho ^{11;10}(t)\left[ \sqrt{1-r^2}z_{11}(t)+rz_{12}(t)\right] \Big \}\mathrm{d}W_1(t)\\&+\Big \{-r\left[ \xi _1^*(t)\rho ^{10;10}(t)+\xi _1(t)\rho ^{01;10}(t)+\sqrt{\kappa _1}\rho ^{11;10}(t)\sigma _++\sqrt{\kappa _1}\sigma _-\rho ^{11;10}(t)\right] \\&+\sqrt{1-r^2}\left[ \xi _2(t)\rho ^{11;00}(t)+\sqrt{\kappa _2}\rho ^{11;10}(t)\sigma _++\sqrt{\kappa _2}\sigma _-\rho ^{11;10}(t)\right] \\&-\rho ^{11;10}(t)\left[ -rz_{11}(t)+\sqrt{1-r^2}z_{12}(t)\right] \Big \}\mathrm{d}W_2(t),\\ \mathrm{d}\rho ^{10;10}(t)= & {} \Big \{(\kappa _1+\kappa _2){\mathcal {D}}_{\sigma _-}^\star \rho ^{10;10}(t)+\sqrt{\kappa _1}\xi _1(t)[\rho ^{00;10}(t),\sigma _+]+\sqrt{\kappa _2}\xi _2(t)[\rho ^{10;00}(t),\sigma _+]\Big \}\mathrm{d}t\\&+\Big \{\sqrt{1-r^2}\left[ \xi _1(t)\rho ^{00;10}(t)+\sqrt{\kappa _1}\rho ^{10;10}(t)\sigma _++\sqrt{\kappa _1}\sigma _-\rho ^{10;10}(t)\right] \\&+r\left[ \xi _2(t)\rho ^{10;00}(t)+\sqrt{\kappa _2}\rho ^{10;10}(t)\sigma _++\sqrt{\kappa _2}\sigma _-\rho ^{10;10}(t)\right] \\&-\rho ^{10;10}(t)\left[ \sqrt{1-r^2}z_{11}(t)+rz_{12}(t)\right] \Big \}\mathrm{d}W_1(t)\\&+\Big \{-r\left[ \xi _1(t)\rho ^{00;10}(t)+\sqrt{\kappa _1}\rho ^{10;10}(t)\sigma _++\sqrt{\kappa _1}\sigma _-\rho ^{10;10}(t)\right] \\&+\sqrt{1-r^2}\left[ \xi _2(t)\rho ^{10;00}(t)+\sqrt{\kappa _2}\rho ^{10;10}(t)\sigma _++\sqrt{\kappa _2}\sigma _-\rho ^{10;10}(t)\right] \\&-\rho ^{10;10}(t)\left[ -rz_{11}(t)+\sqrt{1-r^2}z_{12}(t)\right] \Big \}\mathrm{d}W_2(t),\\ \mathrm{d}\rho ^{01;10}(t)= & {} \Big \{(\kappa _1+\kappa _2){\mathcal {D}}_{\sigma _-}^\star \rho ^{01;10}(t)+\sqrt{\kappa _1}\xi _1^*(t)[\sigma _-,\rho ^{00;10}(t)]+\sqrt{\kappa _2}\xi _2(t)[\rho ^{01;00}(t),\sigma _+]\Big \}\mathrm{d}t\\&+\Big \{\sqrt{1-r^2}\left[ \xi _1^*(t)\rho ^{00;10}(t)+\sqrt{\kappa _1}\rho ^{01;10}(t)\sigma _++\sqrt{\kappa _1}\sigma _-\rho ^{01;10}(t)\right] \\&+r\left[ \xi _2(t)\rho ^{01;00}(t)+\sqrt{\kappa _2}\rho ^{01;10}(t)\sigma _++\sqrt{\kappa _2}\sigma _-\rho ^{01;10}(t)\right] \\&-\rho ^{01;10}(t)\left[ \sqrt{1-r^2}z_{11}(t)+rz_{12}(t)\right] \Big \}\mathrm{d}W_1(t)\\&+\Big \{-r\left[ \xi _1^*(t)\rho ^{00;10}(t)+\sqrt{\kappa _1}\rho ^{01;10}(t)\sigma _++\sqrt{\kappa _1}\sigma _-\rho ^{01;10}(t)\right] \\&+\sqrt{1-r^2}\left[ \xi _2(t)\rho ^{01;00}(t)+\sqrt{\kappa _2}\rho ^{01;10}(t)\sigma _++\sqrt{\kappa _2}\sigma _-\rho ^{01;10}(t)\right] \\&-\rho ^{01;10}(t)\left[ -rz_{11}(t)+\sqrt{1-r^2}z_{12}(t)\right] \Big \}\mathrm{d}W_2(t),\\ \end{aligned}$$
$$\begin{aligned} \mathrm{d}\rho ^{00;10}(t)= & {} \Big \{(\kappa _1+\kappa _2){\mathcal {D}}_{\sigma _-}^\star \rho ^{00;10}(t)+\sqrt{\kappa _2}\xi _2(t)[\rho ^{00;00}(t),\sigma _+]\Big \}\mathrm{d}t\\&+\Big \{\sqrt{1-r^2}\left[ \sqrt{\kappa _1}\rho ^{00;10}(t)\sigma _++\sqrt{\kappa _1}\sigma _-\rho ^{00;10}(t)\right] +r\big [\xi _2(t)\rho ^{00;00}(t)\\&+\sqrt{\kappa _2}\rho ^{00;10}(t)\sigma _++\sqrt{\kappa _2}\sigma _-\rho ^{00;10}(t)\big ]-\rho ^{00;10}(t)\left[ \sqrt{1-r^2}z_{11}(t)+rz_{12}(t)\right] \Big \}\mathrm{d}W_1(t)\\&+\Big \{-r\left[ \sqrt{\kappa _1}\rho ^{00;10}(t)\sigma _++\sqrt{\kappa _1}\sigma _-\rho ^{00;10}(t)\right] +\sqrt{1-r^2}\big [\xi _2(t)\rho ^{00;00}(t)\\&+\sqrt{\kappa _2}\rho ^{00;10}(t)\sigma _++\sqrt{\kappa _2}\sigma _-\rho ^{00;10}(t)\big ]-\rho ^{00;10}(t)\left[ -rz_{11}(t)+\sqrt{1-r^2}z_{12}(t)\right] \Big \}\mathrm{d}W_2(t),\\ \mathrm{d}\rho ^{11;00}(t)= & {} \Big \{(\kappa _1+\kappa _2){\mathcal {D}}_{\sigma _-}^\star \rho ^{11;00}(t)+\sqrt{\kappa _1}\xi _1(t)[\rho ^{01;00}(t),\sigma _+]+\sqrt{\kappa _1}\xi _1^*(t)[\sigma _-,\rho ^{10;00}(t)]\Big \}\mathrm{d}t\\&+\Big \{\sqrt{1-r^2}\left[ \xi _1^*(t)\rho ^{10;00}(t)+\xi _1(t)\rho ^{01;00}(t)+\sqrt{\kappa _1}\rho ^{11;00}(t)\sigma _++\sqrt{\kappa _1}\sigma _-\rho ^{11;00}(t)\right] \\&+r\left[ \sqrt{\kappa _2}\rho ^{11;00}(t)\sigma _++\sqrt{\kappa _2}\sigma _-\rho ^{11;00}(t)\right] -\rho ^{11;00}(t)\left[ \sqrt{1-r^2}z_{11}(t)+rz_{12}(t)\right] \Big \}\mathrm{d}W_1(t)\\&+\Big \{-r\left[ \xi _1^*(t)\rho ^{10;00}(t)+\xi _1(t)\rho ^{01;00}(t)+\sqrt{\kappa _1}\rho ^{11;00}(t)\sigma _++\sqrt{\kappa _1}\sigma _-\rho ^{11;00}(t)\right] \\&+\sqrt{1-r^2}\left[ \sqrt{\kappa _2}\rho ^{11;00}(t)\sigma _++\sqrt{\kappa _2}\sigma _-\rho ^{11;00}(t)\right] \\&-\rho ^{11;00}(t)\left[ -rz_{11}(t)+\sqrt{1-r^2}z_{12}(t)\right] \Big \}\mathrm{d}W_2(t),\\ \mathrm{d}\rho ^{10;00}(t)= & {} \Big \{(\kappa _1+\kappa _2){\mathcal {D}}_{\sigma _-}^\star \rho ^{10;00}(t)+\sqrt{\kappa _1}\xi _1(t)[\rho ^{00;00}(t),\sigma _+]\Big \}\mathrm{d}t\\&+\Big \{\sqrt{1-r^2}\left[ \xi _1(t)\rho ^{00;00}(t)+\sqrt{\kappa _1}\rho ^{10;00}(t)\sigma _++\sqrt{\kappa _1}\sigma _-\rho ^{10;00}(t)\right] \\&+r\left[ \sqrt{\kappa _2}\rho ^{10;00}(t)\sigma _++\sqrt{\kappa _2}\sigma _-\rho ^{10;00}(t)\right] \\&-\rho ^{10;00}(t)\left[ \sqrt{1-r^2}z_{11}(t)+rz_{12}(t)\right] \Big \}\mathrm{d}W_1(t)\\&+\Big \{-r\left[ \xi _1(t)\rho ^{00;00}(t)+\sqrt{\kappa _1}\rho ^{10;00}(t)\sigma _++\sqrt{\kappa _1}\sigma _-\rho ^{10;00}(t)\right] \\&+\sqrt{1-r^2}\left[ \sqrt{\kappa _2}\rho ^{10;00}(t)\sigma _++\sqrt{\kappa _2}\sigma _-\rho ^{10;00}(t)\right] \\&-\rho ^{10;00}(t)\left[ -rz_{11}(t)+\sqrt{1-r^2}z_{12}(t)\right] \Big \}\mathrm{d}W_2(t),\\ \mathrm{d}\rho ^{00;00}(t)= & {} \Big \{(\kappa _1+\kappa _2){\mathcal {D}}_{\sigma _-}^\star \rho ^{00;00}(t)\Big \}\mathrm{d}t\\&+\Big \{\sqrt{1-r^2}\left[ \sqrt{\kappa _1}\rho ^{00;00}(t)\sigma _++\sqrt{\kappa _1}\sigma _-\rho ^{00;00}(t)\right] +r\big [\sqrt{\kappa _2}\rho ^{00;00}(t)\sigma _+\\&+\sqrt{\kappa _2}\sigma _-\rho ^{00;00}(t)\big ]-\rho ^{00;00}(t)\left[ \sqrt{1-r^2}z_{11}(t)+rz_{12}(t)\right] \Big \}\mathrm{d}W_1(t)\\&+\Big \{-r\left[ \sqrt{\kappa _1}\rho ^{00;00}(t)\sigma _++\sqrt{\kappa _1}\sigma _-\rho ^{00;00}(t)\right] +\sqrt{1-r^2}\big [\sqrt{\kappa _2}\rho ^{00;00}(t)\sigma _+\\&+\sqrt{\kappa _2}\sigma _-\rho ^{00;00}(t)\big ]-\rho ^{00;00}(t)\left[ -rz_{11}(t)+\sqrt{1-r^2}z_{12}(t)\right] \Big \}\mathrm{d}W_2(t). \end{aligned}$$

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dong, Z., Zhang, G. & Amini, N.H. Quantum filtering for a two-level atom driven by two counter-propagating photons. Quantum Inf Process 18, 136 (2019). https://doi.org/10.1007/s11128-019-2258-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-019-2258-x

Keywords

Search

Navigation