1. Polarization descriptions of quantized fields Anita Sehat, Jonas Söderholm, Gunnar Björk Royal Institute of Technology Stockholm, Sweden Pedro Espinoza, Andrei B. Klimov Universidad de Guadalajara, Jalisco, Mexico Luis L. Sánchez-Soto Universidad Complutense, Madrid, Spain

2. Outline Motivation Stokes parameters and Stokes operators Unpolarized light – hidden polarization Quantification of polarization for quantized fields Generalized visibility Polarization of pure N-photon states Orbits and generating states Arbitrary pure states Summary

3. Motivation The polarization state of a propagating electromagnetic field is relatively robust The polarization state is relatively simple to transform Transformation of the polarization state introduces only marginal losses The polarization state can easily and relatively efficiently be measured  The polarization is an often used property to encode quantum information Typically, photon counting detectors are used to measure the polarization => The post-selected polarization states are number states A (semi)classical description of polarization is insufficient.

4. The Stokes parameters In 1852, G. G. Stokes introduced operational parameters to classify the polarization state of light If P=0, then the light is (classically) unpolarized tests + linear polarization tests x linear polarization tests circular polarization

5. The Stokes operators E. Collett, 1970: E. Collett, Am. J. Phys. 38, 563 (1970). Two-mode thermal state Any two-mode coherent state

6. A problem with P SC A two-mode coherent state, arbitrarily close to the vacuum state is fully polarized according to the semiclassical definition!

7. SU(2) transformations – realized by geometrical rotations and differential-phase shifts Only waveplates, rotating optics holders, and polarizers needed for all SU(2) transformations and measurements.

8. Another problem: Unpolarized light – hidden polarization P SC = 0 => Is the corresponding state is unpolarized? Frequency doubled pulsed Ti:Sapphire laser =780 nm  =390 nm BBO Type II PBSDetector Counter HWP

9. Experimental results HWP@ QWP@ HWP@22.5+QWP@ light off Single counts per 10 sec Phase plate rotation, deg HWP QWP The state is unpolarized according to the classical definition P. Usachev, J. Söderholm, G. Björk, and A. Trifonov, Opt. Commun. 193, 161 (2001).

10. Unpolarized light in the quantum world A quantum state which is invariant under any combination of geometrical rotations (around its axis of propagation) and differential phase-shifts is unpolarized. H. Prakash and N. Chandra, Phys. Rev. A 4, 796 (1971). G. S. Agarwal, Lett. Nuovo Cimento 1, 53 (1971). J. Lehner, U. Leonhardt, and H. Paul, Phys. Rev. A 53, 2727 (1996).

11. A coincidence count experiment Since the state is not invariant under geometrical rotation, it is not unpolarized. The raw data coincidence count visibility is ~ 76%, so the state has a rather high degree of (quantum) polarization although by the classical definition the state is unpolarized. This is referred to as “hidden” polarization. HWP@ curve fit Coincidence counts per 10 sec Half-wave plate rotation, deg BBO Type II PBS Detector Coincidence counter HWP D. M. Klyshko, Phys. Lett. A 163, 349 (1992).

12. States invariant to differential phase shifts - “Linearly” polarized quantum states Vertical Classical polarization Horizontal Unpolarized! Vertical Quantum polarization Horizontal Neutral, but fully polarized? The “linear” neutrally polarized state lacks polarization direction (it is symmetric with respect to permutation of the vertical and horizontal directions). It has no classical counterpart. For all even total photon numbers such states exist.

13. Rotationally invariant states - Circularly polarized quantum states Left handed Classical polarization Right handed Unpolarized Left handed Quantum polarization Right handed Neutral, but fully polarized? The circular neutrally polarized state is rotationally invariant but lacks chirality. It has no classical counterpart. For all even total photon numbers such states exist.

14. States with quantum resolution of geometric rotations A geometrical rotation of this state by  /3 (60 degrees) will yield the state: A rotation of by 2  /3 (120 degrees) or by -  /3 will yield the state:  Complete set of orthogonal two-mode two photon states. There states are not the “linearly” polarized quantum states P SC = 0 for these states => Semiclassically unpolarized, “hidden” polarization Consider:

15. Experimental demonstration Polarization rotation angle (deg) Coincidence counts per 500 s 0180-180 0 2500 Back- ground level 2000 1500 1000 500 -120-6060120 T. Tsegaye, J. Söderholm, M. Atatüre, A. Trifonov, G. Björk, A.V. Sergienko, B. E. A. Saleh, and M. C. Teich, Phys. Rev. Lett., vol. 85, pp. 5013-5016, 2000. Measured data (dots) and curve fit for the overlap

16. Existing proposals for quantum polarization quantification The measures quantify to what extent the state’s SU(2) Q-function is spread out over the spherical coordinates. That is, how far is it from being a Stokes operator minimum uncertainty state? A. Luis, Phys. Rev. A 66, 013806 (2002).

17. Examples That is, the vacuum state is unpolarized and highly excited states are polarized Note that: NMax{P SU(2) } 11/4 24/9 416/25

18. Degree of polarization based on distance to unpolarized state Another proposal is to define the degree of polarization as the distance (the distinguishability) to a proximal unpolarized state. Will be covered in L. Sánchez-Soto’s talk.

19. Proposal for quantification of polarization – Generalized visibility Original state Transformed state How orthogonal (distinguishable) can the original and a transformed state become under any polarization transformation?

20. All pure, two-mode N-photon states are polarized One can show that all pure, two-mode N-photon states with N ≥ 1 have unit degree of polarization using this definition, even those states that are semiclassically unpolarized => No ”hidden” polarization.

21. Orbits The set of all such states define an orbit. If one state in an orbit can be generated, then we can experimentally generate all states in the orbit.

22. Orbit generating states

23. Orbit generating states where the orbit spans the whole Hilbert space Moreover, to generate the basis set we need only make geometrical rotations or differential phase shifts. In higher excitation manifolds it is not known if it is possible to find complete- basis generating orbits, but it seems unlikely. Such orbits are of particular interest for experimentalists to implement 3- dimensional quantum information protocols, and to demonstrate effects of two- photon interference.

24. Summary Polarization is a useful and often used characteristic for coding of quantum info. The classical, and semiclassical description of polarization is unsatisfactory for quantum states. Other proposed measures have been discussed and compared. We have proposed to use the generalized visibility under (linear) polarization transformations as a quantitative polarization measure. Polarization orbits naturally appears under this quantitative measure. Orbits spanning the complete N-photon space have special significance and interest for experiments and applications.

25. Schematic experimental setup Phase shiftHWPPBSBBO Type II Phase shift Generated state: Projection onto the state. (This state causes coincidence counts.) Detector Coincidence