Ultimate bounds for quantum and Sub-Rayleigh imaging

Transcription

1 Advances in Optical Metrology June 2016 Ultimate boun for quantum and Sub-Rayleigh imaging Cosmo Lupo & Stefano Pirandola University of York arxiv:

2 Introduction Quantum imaging for metrology - Linear - Diffraction-limited - Paraxial approximation - Far field - Thin lens Rayleigh length x R D R

3 Introduction Quantum imaging for metrology Communication theory: how much information can be transmitted through the optical system? 1 Hypothesis testing: 2 is the light emitted by one source or two? Parameter estimation: how much is the distance between the two sources? 3 Cramer-Rao bound s 1 QFI s Quantum Fisher information QFI s Tsang, Nair, Lu arxiv:

4 Introduction Quantum imaging for metrology What are the optimal sources? what happen for non-classical or entangled light? How much can we resolve sub-rayleigh features? what is the ultimate bound for sub-wavelength imaging? How is the scaling of s with the number of photons? 1 Shot noise s vs Heisenberg N s 1 N

5 An intuition Any linear optical imaging system is formally equivalent to a circuit made of beam splitter and phase shifter Reck, Zeilinger, Bernstein, Bertani, PRL 1994

6 Going abstract c 1 a 1 c 2 a 2

7 Going abstract c 1 a 1 c 2 a 2 c a c a c c c a a a 1 2 2(1 )

8 Dynamical equations c 1 a 1 c 2 a 2 da

9 Dynamical equations c 1 a 1 c 2 a 2 da i H, a

10 Dynamical equations c 1 a 1 c 2 a 2 da i H, a a s

11 Dynamical equations c 1 a 1 c 2 a 2 da i H, a a s da i H eff, a

12 Dynamical equations c 1 a 1 c 2 a 2 da i H, a a s da i H eff, a H i a b a b eff

13 Dynamical equations c 1 a 1 c 2 a 2 da i H, a a s da i H eff, a eff H i a b a b Beam-splitter Hamiltonian

14 Beam-splitter (review) Adesso et al., PRA s Photon-counting N QFI N (1 ) QFI s

15 Ultimate precision bound max QFI 2 N s 2 (1 ) (1 ) x R d s 1 QFI s

16 Optimal and almost-optimal sources Entangled sources (high efficiency) Thermal / incoherent sources (high attenuation)

17 Sub-optimal measurement 0 s QFI Even/odd photon counting

18 Conclusion Problem solved at a formal level Optimal measurements? Standard quantum limit (shot-noise scaling) Entangled sources are optimal (but only by a small factor) Thermal sources are practically optimal Non-linear imaging, non-paraxial

19 Quantum metrology for imaging Stefano and me have been looking at quantum imaging for a while, but we were looking at it through the glass of communication theory. We have been asking questions like how much information you can transmit or read using an optical, diffraction limited, imaging system. Or we were considering problems as decision making, hypothesis testing, typically deciding if the image you capture has been produced by one source or by to sources that are very close to each other. Clearly, these problems are all related to quantum metrology. But the approach of Mankei makes the connection more direct. When we saw their paper it was illuminating: that was an interesting way of looking at it! And their results were brilliant, unexpected, surprinsing. We did not understand it at the beginning, we asked them a lot of questions, and they explained to us how it worked. At that point we wanted to know more. Is their result valid in general? Or only for attenuated/incoherent/thermal sources? How we can solve for general sources? What is the optimal source? What is the role of Rayleigh length? Sub-Rayleigh One problem is how you can image Sub-Rayleigh details. Heisenber vs Standard Quantum Limit Another problem is the scaling with the number of photon, SQL vs H (add more details) Ultimate optical imaging technology What we have done is that we have found the ultimate precision bound. This tells us what are the optimal sources. What is the optimal resolution of sub-rayleigh features, and what is the scaling. We have solve this problem using one of the standard problem-solving techniques: reducing the problem to a problem for which you know the solution!