Full characterization of polarization states of light via direct measurement

Journal name:
Nature Photonics
Year published:
Published online


Ascertaining the physical state of a system is vital in order to understand and predict its behaviour. However, due to their fragile nature, the direct observation of quantum states has, until recently, been elusive. Historically, determination of the quantum state has been performed indirectly through the use of tomography. We report on two experiments showing that an alternative approach can be used to determine the polarization quantum state in a simple, fast and general manner. The first experiment entails the direct measurement of the probability amplitudes describing pure polarization states of light, the first such measurement on a two-level system. The second experiment entails the direct measurement of the Dirac distribution (a phase-space quasi-probability distribution informationally equivalent to the density matrix), demonstrating that the direct measurement procedure is applicable to general (that is, potentially mixed) quantum states. Our work has applications to measurements in foundational quantum mechanics, quantum information and quantum metrology.

At a glance


  1. Schematic representation of the experiment.
    Figure 1: Schematic representation of the experiment.

    a, The output of the single-mode fibre (SMF) is a near-Gaussian transverse mode of light. A polarizing beamsplitter (PBS) and waveplate(s) create a known pure polarization state. b, A quartz crystal at an oblique angle performs the weak measurement by introducing a small (compared to the beam waist) lateral displacement between horizontal and vertical polarization components. c, A strong measurement in a basis (diagonal/antidiagonal) mutually unbiased from the weak measurement is used to complete the direct measurement. Inset: To measure the wavefunction, a linear polarizer oriented to transmit diagonally polarized light performs the strong measurement and post-selection. To measure the Dirac distribution, a λ/2 wave-plate and calcite beam displacer carry out the strong measurement. d, A 50:50 non-polarizing beamsplitter (NPBS) splits the light into two sub-ensembles. These are imaged in the near-field (dotted line, NF) and far-field (dashed line, FF) of the quartz crystal onto non-overlapping regions of the CCD camera.

  2. Results of experiment 1 with linearly polarized probe states.
    Figure 2: Results of experiment 1 with linearly polarized probe states.

    a, Measured average weak values plotted as a function of input polarization angle; 0° is defined as parallel to the optical table. For clarity, error bars are shown only for the red points, and indicate the standard deviation of 100 independently measured weak values. b, Real and imaginary components of the probability amplitudes determined by normalizing the weak values of each test state, where |ψright fence = α|Hright fence + β|Vright fence. For both panels, the solid lines are the theoretical predictions of the real components, and the dotted lines are the theoretical predictions of the imaginary components. Inset: a Poincaré sphere with the path taken indicated by the blue line.

  3. Measured states on the Poincare sphere.
    Figure 3: Measured states on the Poincaré sphere.

    A Poincaré sphere with the set of directly measured states, indicated by their calculated Stokes parameters left fenceσxright fence = left fenceψ | (πD − πA)|ψright fence, left fenceσyright fence = left fenceψ| (πR − πL)|ψright fence, left fenceσzright fence = left fenceψ| (πH − πV)|ψright fence. Blue points indicate states created by rotating the half-wave plate. Red (green) points indicate calculated Stokes parameters for states created by rotating the half-wave plate, followed by a quarter-wave plate at fixed angle of 0° (45°). Solid lines indicate paths taken for each data set.

  4. Results of experiment 2.
    Figure 4: Results of experiment 2.

    a,b, The directly measured Dirac distributions (left) and corresponding density matrices (right) for the horizontal linear polarization state |Hright fence (a) and left-hand circular polarization state |Lright fence = 1/2((1 + i)|Hright fence + (1 — i)|Vright fence) (b). The axes of the Dirac distribution are the mutually unbiased bases H/V and D/A; the density matrix describes the state in terms of only one basis, H/V. Red towers indicate negative values.


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  1. Department of Physics, University of Ottawa, Ottawa, Canada K1N 6N5

    • Jeff Z. Salvail,
    • Megan Agnew,
    • Allan S. Johnson,
    • Eliot Bolduc,
    • Jonathan Leach &
    • Robert W. Boyd
  2. Institute of Optics, University of Rochester, Rochester, New York 14627, USA

    • Robert W. Boyd


J.Z.S. initiated the study. The experiment was designed by J.Z.S., A.S.J., E.B. and J.L. The experiment was performed by J.Z.S., M.A. and A.S.J., and data analysis was performed by J.Z.S. R.W.B. supervised all aspects of the project. All authors contributed to the text of the manuscript.

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