Quantum limits in optical interferometry

1. Quantum limits in optical interferometry R. Demkowicz-Dobrzański1, K. Banaszek1,J. Kołodyński1, M. Jarzyna1, M. Guta2, K. Macieszczak1,2, R. Schnabel3, M. Fraas4 1Faculty of Physics, University of Warsaw, Poland 2 School of Mathematical Sciences, University of Nottingham, United Kingdom 3Max-Planck-Institut fur Gravitationsphysik, Hannover, Germany4Theoretische Physik, ETH Zurich, 8093 Zurich, Switzerland

2. Quantum enhncementin an imperfectMach-Zehnderinterferometer imperfectvisibility loss for classicallight: shotnoise Whatisthemaximal quantum enhanced precision we cangetusingnon-classicalstates of lightwithfixedtotal energy attheinput? Quantum Cramer-Raobound Quantum Fisher Information Symmetriclogarithmicderrivative MaximizeFQoverinputstates

3. Modevsparticledescription of light A general Nphotontwomode state: a b Writteninthelanguage of Nformallydistinguishableparticles: symetrization Modevsparticleentanglement enhancedsensitivity whenprojected on a fixedphotonnumbersectoryields a particleentangledstates Hong-Ou-Mandelinterference Itistheparticleentanglementthatisthefundamentalsource for quantum precision enhancement

4. Quantum enhanced interferometry usingtheparticledescription phaseencoding decoherence imperfectviisbility– loss of coherencebetweenthemodes (localqubitdephasing) loss– we usethreedimensionaloutputspace uncorrelatednoisemodels commutewiththephaseencoding Findthebounds on the quantum Fisher information as a function of N photonsurvives lostinmode b lostinmode a

5. Classicalsimulation of a quantum channel Convex set of quantum channels

6. Classicalsimulation of a quantum channel Convex set of quantum channels Parameterdependencemoved to mixingprobabilities Before: After: By Markov property…. • K. Matsumoto, arXiv:1006.0300 (2010)

7. Classicalsimulation of Nchannelsusedinparallel

8. Classicalsimulation of Nchannelsusedinparallel =

9. Classicalsimulation of Nchannelsusedinparallel =

10. Precision boundsthanks to classicalsimulation • For unitarychannels Heisenberg scalingpossible • Genericdecoherence model will manifest shotnoisescaling • To getthetighestbound we need to findtheclassicalsimulationwithlowestFcl

11. Precision boundsthanks to classicalsimulation • For unitarychannels Heisenberg scalingpossible • Genericdecoherence model will manifest shotnoisescaling • To getthetighestbound we need to findtheclassicalsimulationwithlowestFcl RDD,J. Kolodynski, M. Guta, Nature Communications 3, 1063 (2012)

12. Example: dephasing dephasing For „classicalstrategies” Maximal quantum enhancment RDD,J. Kolodynski, M. Guta, Nature Communications 3, 1063 (2012)

13. Example: loss Lossyinterferometer photontransmitted photonlostfrom theupperarm Bounduseless photonlostfrom thelowerarm Need to generalizethe idea of classicalsimmulation

14. Quantum simulation Classicalsimulation = =

15. Quantum simulation Quantum simulation = arbitrary state arbitrary map

16. Quantum simulation Fisher informationcannotincrease under parameter independent CP map We shouldlook for the ,,worst’’ quantum simulation to getthetightestbounds

17. Search for the,,worst’’ Quantum simulation A semi-definiteprogramm Lossyinterferometer dephasing the same as fromclassicalsimulation lossyinterferometer -> dephasing Heisenberg 1/Nscalinglost! RDD,J. Kolodynski, M. Guta, Nature Communications 3, 1063 (2012) J. Kolodynski, RDD, New J. Phys. 15, 073043 (2013)

18. Search for the,,worst’’ Quantum simulation A semi-definiteprogramm dephasing Lossyinterferometer dephasing= losses+ sending back decoheredphotons RDD,J. Kolodynski, M. Guta, Nature Communications 3, 1063 (2012) J. Kolodynski, RDD, New J. Phys. 15, 073043 (2013)

19. Explicitexample of a quantum simulation a lossy interferometr: we will provethisbound for b photonlostwithprobability 1/2 quantum simulation:

20. Saturatingthefundamentalboundsissimple! Fundamentalbound For strongbeams: Simple estimatorbasedon n1- n2measurement C. Caves, Phys. Rev D 23, 1693 (1981) Weaksquezing + simplemeasurement + simpleestimator = optimalstrategy! The same istrue for dephasing (alsoatomicdephasing – spin squeezedstatesareoptimal) S. Huelga, et al. Phys. Rev. Lett79,3865 (1997), B. M. Escher, R. L. de Matos Filho, L. Davidovich Nature Phys.7, 406–411 (2011), D. Ulam-Orgikh and M. Kitagawa, Phys. Rev. A 64, 052106(2001).

21. GEO600 interferometeratthefundamental quantum bound coherentlight +10dB squeezed fundamentalbound The most general quantum strategiescouldimprovethe precision by at most 8% RDD, K. Banaszek, R. Schnabel, Phys. Rev. A, 041802(R) (2013)

22. Definitevs. indefinitephotonnumber boundderrived for Nphotonstates Typically we usestateswithindefinitephotonnumber (coherent, squeezed)

23. Definitevs. indefinitephotonnumber boundderrived for Nphotonstates Typically we usestateswithindefinitephotonnumber (coherent, squeezed) If no otherphasereferencebeamisused: no coherencebetweendifferenttotalphotonnumbersectors Thanks to convexity of Fisher information

24. Take home… • Precision boundsin quantum metrologywithuncorrelatednoisecan be derrivedusingclassical/quantum simulationsideas • RDD, J. Kolodynski, M. Guta, , Nature Communications 3, 1063 (2012) • Boundsarealsovalid for indefinitephotonnumberstates, and can be applied to realsetups (GEO600): • RDD, K. Banaszek, R. Schnabel, Phys. Rev. A, 041802(R) (2013) • Errorcorrection: addingancillas and peformingadaptivemeasurementsdoes not affectthebounds. • paperswitherrorcorrectioninmetrology, usetransversalnoise: arxiv:1310.3750, arXiv:1310.3260 • Boundsare not guaranteed to be tight, but areincase of loss and dephasing • seee.g. S. Knysh, E. Chen, G. Durkin, arXiv:1402.0495 • Review paper iscomming:RDD, M. Jarzyna, J. Kolodynski, Quantum limitsinoptical interferometry, Progress inOptics, ??? • Frequency estimation, Bayesian approach • K. Macieszczak, RDD, M. Fraas, arXiv:1311.5576