Performing sequential WMs to reconstruct the density matrix of a twolevel. Seminario de Óptica Cuántica Mar

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1 Performing sequential WMs to reconstruct the density matrix of a twolevel system OMAR CALDERÓN LOSADA Seminario de Óptica Cuántica Mar

2 Outline Initial considerations Elements of reality Quantum measurements: Strong vs weak Standard quantum state tomography vs Direct Measurement In brief: More questions than conclusions 23/03/2017 O.CALDERON-LOSADA/PERFORMING SEQUENTIAL... 2

3 Elements of reality Vaidman s elements of reality For any definite result of a measurement there is corresponding element of reality. As definite shift of the probability distribution of the pointer variable. [L. Vaidman. Foundations of Physics. Vol. 26. No ] 23/03/2017 O.CALDERON-LOSADA/PERFORMING SEQUENTIAL... 3

4 Elements of reality Vaidman s elements of reality For any definite result of a measurement there is corresponding element of reality. As definite shift of the probability distribution of the pointer variable. [L. Vaidman. Foundations of Physics. Vol. 26. No ] 23/03/2017 O.CALDERON-LOSADA/PERFORMING SEQUENTIAL... 4

5 QUANTUM MEASUREMENT Consists on [Von Neumann approach ] Measuring device (Meter) e.g. Spatial or temporal profile of a beam [Naimark extension] Coupling two systems Measured system e.g. Polarization of light The reading of the pointer variable in the end of the measurement almost always yields the value of the shift (the eigenvalue of the observable) 23/03/2017 O.CALDERON-LOSADA/PERFORMING SEQUENTIAL... 5

6 QUANTUM MEASUREMENT Consists on Measuring device (Meter) e.g. Spatial or temporal profile of a beam Or increasing the spread in the pointer distribution [Naimark extension] Coupling two systems Decreasing strength coupling WEAK MEASUREMENT Measured system e.g. Polarization of light Introduces a new concept called Weak Value, a complex number with different interpretations and applications. weak-measurement element of reality 23/03/2017 O.CALDERON-LOSADA/PERFORMING SEQUENTIAL... 6

7 Strong vs weak Measurements [L. J. Salazar-Serrano. Opt. Express 23, (2015)] 23/03/2017 O.CALDERON-LOSADA/PERFORMING SEQUENTIAL... 7

8 Standard Quantum State Tomography Single Qubit SQST Projection states QWP HWP H 0 0 R L A [D. F. V. James et al. Phys. Rev. A, 64, (2011)] 23/03/2017 O.CALDERON-LOSADA/PERFORMING SEQUENTIAL... 8

9 Standard Quantum State Tomography Two Qubit SQST Projections QWPA HWPA QWPB HWPB HH HV HD HR VH VV VD VR DH DV DD DR RH RV RD RR Maximum Likelihood! [D. F. V. James et al. Phys. Rev. A, 64, (2011)] 23/03/2017 O.CALDERON-LOSADA/PERFORMING SEQUENTIAL... 9

10 Direct measurement of the density matrix One drawback of conventional quantum state tomography is that it does not readily provide access to single density matrix elements since it requires a global reconstruction. 23/03/2017 O.CALDERON-LOSADA/PERFORMING SEQUENTIAL... 10

11 Direct measurement of the wave function A previous work: The complex nature of the weak value is what enables the authors to directly measure the real and imaginary parts of the wave function [Lundeen J, et al. Nature 474, 188 (2011)] 23/03/2017 O.CALDERON-LOSADA/PERFORMING SEQUENTIAL... 11

12 Direct measurement of the wave function Single photon reconstructed quantum wave function: [Lundeen J, et al. Nature 474, 188 (2011)] 23/03/2017 O.CALDERON-LOSADA/PERFORMING SEQUENTIAL... 12

13 Direct measurement of the density matrix Key ideas: Weak average It differs from the weak value, there is no postselection Weak average and density matrix elements connection Let A and B two observables maximally incompatible or complementaty Can be replaced by a strong measurement without affecting the weak average 23/03/2017 O.CALDERON-LOSADA/PERFORMING SEQUENTIAL... 13

14 Direct measurement of the density matrix Experimental setup: σ = 250 μm δ x = δ y = 176 μm I, J = {H, V} [G. S. Thekkadath et. al. PRL 117, (2016)] 23/03/2017 O.CALDERON-LOSADA/PERFORMING SEQUENTIAL... 14

15 Direct measurement of the density matrix Key ideas: The measurements The final connection: Join position and momentum expectation values with density matrix elements relations 23/03/2017 O.CALDERON-LOSADA/PERFORMING SEQUENTIAL... 15

16 Direct measurement of the density matrix Experimental results: Pure states 23/03/2017 O.CALDERON-LOSADA/PERFORMING SEQUENTIAL... 16

17 Direct measurement of the density matrix Experimental results: Mixed states 23/03/2017 O.CALDERON-LOSADA/PERFORMING SEQUENTIAL... 17

18 In brief 23/03/2017 O.CALDERON-LOSADA/PERFORMING SEQUENTIAL... 18

19 THANKS FOR YOUR ATTENTION AND YOUR QUESTIONS! 23/03/2017 O.CALDERON-LOSADA/PERFORMING SEQUENTIAL... 19

20 En resumen Para medir directamente el estado de un sistema cuántico arbitrario: Elegir una base { a } cualquiera asociada con un observable A. Medir débilmente un proyector de esta base, π a a a. Realizar una post-selección en un valor b 0 asociado al observable B complementario a A, es decir A, B 0. El estado se obtiene π a w = b 0 a a Ψ Ψ b 0 = a Ψ /ν Ψ = ν a π a w a Qué y cómo medimos esto?