Quantum estimation to enhance classical wavefront sensing

Transcription

1 Quantum estimation to enhance classical wavefront sensing L. L. Sánchez-Soto B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil

2 Quantum Motivation tomography Importance of wavefronts (3D imaging) Holography Interferometry Shack-Hartmann

3 Quantum Motivation tomography Applications Noncontact metrology Adaptive optics Vision correction

4 Shack-Hartmann Quantum tomography detection LPHYS 14. Mesoscopic Sofia systems. (Bulgaria), Brasov July (Romania) September 2015

5 Shack-Hartmann Quantum tomography detection

6 Shack-Hartmann Quantum tomography detection local wavefront tilts are measured wavefront is reconstructed what if there is no well-defined wavefront? x p object scanning aperture near field focal plane far field

7 Coherence Quantum theory tomography in a nutshell (I) Propagation Optical transfer function Detection Pupil function Focusing Z.Hradil, J. Rehacek, L.L.Sanchez-Soto, Phys. Rev. Lett. 105, (2010)

8 Coherence Quantum theory in tomography a nutshell (II) Y.S.Teo, H.Zhu, B.-G.Englert, J. Rehacek, Z.Hradil, L. L. Sanchez-Soto: Phys. Rev. Lett. 107, (2011)

9 Coherence Quantum theory in tomography a nutshell (III) Signal Y.S.Teo, H.Zhu, B.-G.Englert, J. Rehacek, Z.Hradil, L. L. Sanchez-Soto: Phys. Rev. Lett. 107, (2011)

10 Coherence Quantum theory in tomography a nutshell (IV) Limiting cases Pointlike microlenses information about the transversal momentum is lost Large microlenses complete loss of position sensitivity Gaussian approximation simultaneous detection of position and angular spectrum: determine the Q function

11 Quantum estimation tomography Q function The knowledge of the Q function is enough to reconstruct the mutual cohere function (in principle, infinite points) Maximum likelihood method Any realistic detection can be seen as an estimation problem, and one should look for the most likely states from the point of view of registered data J. Rehacek, M. Paris: Quantum state estimation (Springer, Berlin, 2008)

12 Tomography in a nutshell (I)

13 Tomography in a nutshell (II)

14 Radon transform R(, x 0 )= Z f(x, y) (x cos + y sin x 0 ) dxdy

15 Radon transform

16 Quantum tomography Quantum tomography is the process by which a quantum state is reconstructed using measurements on an ensemble of identical quantum states. Because measurement of a quantum state (in general) changes the state being measured, getting a complete picture of that state requires measurements on many state copies. Pauli problem (1933): Can the wave function be determined uniquely from the distribution of position and momentum: No Such data are not complete Generalized Pauli problem: Can the wave function be inferred from a series of measurements on a large collection of identically prepared particles: Yes Bertrand (1979)

17 Quantum tomography Quantum state Measurement {A j } A j 0 Σ j A j = 1 Probabilities p j =Tr( A j ) Problem: Inversion of the measured frequencies 0 Tr( ) =1 f j =Tr( A j ) f j 6= p j

18 Quantum Tomography Measure diffferent projections of several copies of a state reconstruct the state Q = Q cos + P sin Importance: Ability to rotate the quadrature projections on phase space Robust inversion formula to reconstruct state from tomograms

19 Estimation General estimation scheme operation measurement estimation The variable of interest is a c-number θ true which cannot be addressed directly: only some variable-dependent data D can be detected. The presence of the variable θ true is manifested by the conditional probability p(d θ true ) The estimator θ = θ(d) relates the data to the variable of interest, but due to the stochastic nature of data there is no unique mapping between D and θ.

20 Estimation The inversion can be formulated just in statistical sense by Bayes theorem likelihood prior information The quality of estimation should be assessed by the cost function C(θ, θ true ) -least square fit C(θ, θ true )= (θ- θ true ) 2 -maximum likelihood fit C(θ, θ true ) =- δ(θ- θ true ) The risk function R(θ D) = dθ true C(θ, θ true ) p(θ true D)

21 Maximum likelihood Maximum Likelihood (MaxLik) principle is not a rule that requires justification: Bet always on the highest chance Likelihood L quantifies the degree of belief in certain hypothesis under the condition of the given data log L = X f j log P j j MaxLik principle selects the most likely configuration Information is updated according to the Bayes rule prior probability è posterior probability

22 Maximum likelihood tomography Generic measurement p i = Tr(ρ Ai ) Maximize the likelihood L(ρ) = Π i p j Ni Tr(ρ) = 1 ρ 0 Jensen inequality (inequality between geometric and arithmetic means) Π i (x i /a i ) fi i f i x i /a i L(ρ) 1/N = Π i p j fi (Π i a i fi ) Tr(R ρ) We choose a i = Tr(ρA i ) R = i (f i /a i ) A i

23 Maximum likelihood tomography Maximization Solution log L(ρ) = i N i log p j (ρ) λ Tr(ρ) R ρ = ρ Likelihood is a convex functional defined on the convex manifold of the density matrices

24 Characterization: Quantum tomography vortex beams p = 0 p = 1 p = 2 l = 0 l = 1 l = 2

25 Angular momentum of light Orbital angular momentum l = 0 l= 1 l = 2 l = 3 INAOE (Puebla). November 2013

26 Generation of vortex beams Holographic mask Topological charge

27 Vortex beams: Quantum experimental tomographysetup B. Stoklasa, L. Motka, J. Rehacek, Z.Hradil, L.L.Sanchez-Soto, Nat. Commun. 5, 3275 (2014)

28 Vortex beams: Quantum experimental tomography reconstruction a b

29 Quantum Propagation tomography a 0 b 0 c 0

30 Quantum tomography Propagation Mesoscopic systems. Brasov (Romania). 1 September 2015

31 LPHYS 14. Sofia (Bulgaria), July 2014 Some Quantum conjectures tomography

32 Some Quantum conjectures tomography Planck Mission LPHYS 14. Sofia (Bulgaria), July 2014

33 Quantum Conclusions tomography realistic theory of S-H detection presented applications discussed: characterization of vortex fields digital propagation and 3D imaging possible extensions optimization alternative inversion methods