1. Displaced-photon counting for coherent optical communication Shuro Izumi

2. 1.Discrimination of phase-shift keyed coherent states 2.Super resolution with displaced-photon counting 3.Phase estimation for coherent state

3. 1.Discrimination of phase-shift keyed coherent states 2.Super resolution with displaced-photon counting 3.Phase estimation for coherent state

4. Optical communication Encode the information on the optical states Squeezed states Entangled states Photon number states Super position states Receiver Sender Laser Detector Optical state Excellent properties Decision However.. Changed to mixed states by losses Non-classical states are not optimal for signal carriers Non-classical states

5. Motivation Optical communication with coherent states ✓ Remain pure state under loss condition ✓ Easily generated compared with non-classical states Coherent state is the best signal carrier under the losses because However ✓ It is impossible to discriminate coherent states without error because of their non-orthogonality

6. Achievable minimum Error Probability Standard Quantum Limit…. Achievable Error probability by measurement of the observable which characterizes the states Helstrom bound…. Achievable Error probability for given states Overcome the SQL and approach the Helstrom bound!! C. W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976). 0101 Helstrom bound SQL Error Phase-shift keyed→ Homodyne measurement Binary phase-shift keyed (BPSK)

7. on off Photon counter Displacement operation Near-optimal receiver for BPSK signals Displaced-photon counting R. S. Kennedy, Research Laboratory of Electronics, MIT, Technical Report No. 110, 1972 Local Oscillator Beam splitter Error Helstrom bound SQL (Homodyne measurement) Displaced-photon counting

8. k. Tsujino et al., Phys. Rev. Lett. 106, 250503(2011) Experimental demonstration of near-optimal receiver for BPSK signals ✓ Detector with high detection efficiency Transition edge sensor (TES) →Detection efficiency : 95 % for 853nm Photon counter Classical (electrical) feedback or Optimal receiver for BPSK signals Displaced-photon counting with feedback operation (Dolinar receiver) S. J. Dolinar, Research Laboratory of Electronics, MIT, Quarterly Progress Report No. 111, 1973 R. L. Cook, et al., Nature 446, 774, (2007) ✓ Displacement optimization→ Optimize the amount of

9. p x QPSK signals p x ✓ Displaced-photon counting → Near-optimal ✓ Displaced-photon counting with feedback → Optimal

10. R. S. Bondurant,5 Opt. Lett. 18, 1896 (1993) Near-optimal receiver for QPSK signals Photon counter Classical (electrical) feedback p x Displaced-photon counting with feedback receiver Helstrom Heterodyne measurement (SQL) Bondurant receiver

11. x p on off Evaluation for finite feedforward steps Change the displacement operation depending on previous results S. Izumi et al., PRA. 86, 042328(2012) S. Izumi et al., PRA. 86, 042328 (2012) M. Takeoka et al., PRA. 71, 022318(2005) M. Takeoka et al., PRA. 71, 022318 (2005) N→∞ ⁼Bondurant receiver

12. Displaced-photon counting without feedforward Helstrom Heterodyne measurement (SQL) Bondurant N=∞ N=20 N=10 N=5 N=4 N=3 Numerical evaluation S. Izumi et al., PRA. 86, 042328(2012) S. Izumi et al., PRA. 86, 042328 (2012)

13. x p Displaced-photon counting with Feedforward operation(Dolinar receiver ) Change the displacement operation depending on previous results Photon-number resolving detector *Symbol selection Bayesian estimation →The signal which maximizes the posteriori probability S. Izumi et al., PRA. 87, 042328(2013) S. Izumi et al., PRA. 87, 042328 (2013)

14. Heterodyne measurement (SQL) Helstrom bound N=10 N=4 N=3 On-off detector Photon-number-resolving detector N=5 Numerical evaluation S. Izumi et al., PRA. 87, 042328(2013) S. Izumi et al., PRA. 87, 042328 (2013)

15. Numerical evaluation with detector’s imperfection On-off detector PNRD S. Izumi et al., PRA. 87, 042328(2013) S. Izumi et al., PRA. 87, 042328 (2013)

16. Experimental realization of feedforward receiver for QPSK NIST demonstrated the feedforward (feedback) receiver F. E. Becerra et al., Nature Photon. 7, 147 (2013) C. R. Muller et al., New J. Phys. 14, 083009 (2012) F. E. Becerra et al., Nature Photon. 9, 48 (2015) With on-off detector With PNRD Homodyne + Displaced-photon counting Hybrid scheme from Max-Plank institute Feedforward operation dependent on the result of homodyne measurement

17. Summary ✓ We propose and numerically evaluated the receiver for QPSK signals ✓ Displaced-photon counting with PNRD based feedforward operation improve the performance for QPSK discrimination

18. 1.Discrimination of phase-shift keyed coherent states 2.Super resolution with displaced-photon counting 3.Phase estimation for coherent state

19. Phase sensing with displaced-photon counting ✓ Better performance than homodyne measurement Displaced-photon counting is near-optimal receiver for signal discrimination Can displaced-photon counting make improvements in phase sensing? ✓ Super resolution ✓ Approach the Helstrom bound ✓ Phase estimation

20. Super resolution and Sensitivity Input state Quantum measurement Phase shift N00N state Coherent state Nagata et al., Science 316, 726 (2007) Xiang et al., Nature Photonics 5, 268 (2010) Sensitivity Resolution →Interference pattern Coherent state with particular quantum measurements Super resolution Narrower width Y. Gao et al., J. Opt. Soc. Am. B. 27, No.6 (2010) E. Distant et al., Phys. Rev. Lett. 111, 033603(2013) K. Jiang et al., J. Appl. Phys. 114, 193102(2013) K. J. Resch et al., Phys. Rev. Lett. 98, 223601 (2007)

21. Standard two-port intensity difference monitoring Input state Intensity difference monitoring －

22. Super resolution with parity detection Input state Parity detection Even Odd PNRD Y. Gao et al., J. Opt. Soc. Am. B. 27, No.6 (2010) Super resolution

23. Super resolution with homodyne measurement E. Distant et al., Phys. Rev. Lett. 111, 033603(2013) － Homodyne measurement Super resolution

24. Super resolution with homodyne measurement Threshold homodyne measurement POVM Normalized E. Distant et al., Phys. Rev. Lett. 111, 033603(2013) →Count probability with the phase shift a=0.1 a=1.0 a=2.0 a=0.1 a=1.0 a=2.0

25. Evaluation of sensitivity E. Distant et al., Phys. Rev. Lett. 111, 033603(2013)

26. Super resolution with displaced-photon counting Photon counter Displacement operation General phase detection scheme Mach-Zehnder phase detection scheme →Count probability with the phase shift

27. Super resolution with displaced-photon counting Homodyne measurement (with normalization) Parity detection (same input power to the phase shifter) Displaced-photon counting Super resolution Width Width :

28. Evaluation of resolution and sensitivity a=0.1 Displaced- photon counting a=1.0 Parity detection Resolution Sensitivity a=0.1 Displaced- photon counting a=1.0 Parity detection Shot noise limit

29. Summary ✓ Displaced-photon counting shows both super resolution and good sensitivity Super resolution can be observed with coherent state and quantum measurement →parity detection, homodyne measurement a=0.1 Displaced-photon counting a=1.0 Parity detection Shot noise limit

30. 1.Discrimination of phase-shift keyed coherent states 2.Super resolution with displaced-photon counting 3.Phase estimation for coherent state

31. Phase estimation Quantum measurement Estimator Input state Phase shift Optimal input state Optimal measurement Optimize for good estimation Figure of merit →Variance of the estimator

32. Cramer-Rao bound The variance of estimator must be larger than inverse of Fisher information. For M states, B.R.Frieden, “Science from Fisher Information”,CAMBRIDGE UNI.PRESS(2004)

33. Fisher information (FI) Quantum FI Classical FI depends only on input state. depends on input state and measurement. Possible to derive the minimum variance for given state S.L.Braunstein and C.M.Caves, PRL, 72, 3439 (1994) Possible to derive the minimum variance for given state and measurement How much information state has How much information we can extract from the state by measurement

34. Fisher information for coherent state Phase shift Quantum FI Quantum measurement Homodyne measurement Heterodyne measurement Classical FI S.Olivares et, al., J.Phys.B, Mol. Opt. Phys, 42(2009) Homodyne Heterodyne

35. Fisher information for coherent state with displaced-photon counting (PNRD) PNRD Displaced-PNRD Fisher information for discrete variable

36. Fisher information

37. Experimental setup ~Preliminary experiment 99:1 BS LO probe PNRD PZT Transition edge sensor (TES) ✓ Photon-number resolving up to 8-photon ✓ Detection efficiency 92% Fukuda et al., (AIST) Metrologia, 46, S288 (2009) Laser 1550 nm

38. Experimental condition Probe amplitude Displacement amplitude Detection efficiency Visibility

39. Experimental results ~Preliminary experiment Heterodyne Displaced-PNRD Experiment Displaced-PNRD Theory Homodyne Displaced-PNRD Theory with imperfections # of measurement Expectation value # of measurement Variance Expectation value Variance

40. Summary ✓ Displaced-photon counting gives higher fisher information than homodyne measurement around Θ=0 →Is it possible to use this result for phase sensing? ✓ We demonstrated preliminary experiment →We experimentally show that displaced-photon counting gives better performance in particular condition →Adjustment of the experimental setup more carefully is required