Hologram of a single photon

Journal name:
Nature Photonics
Year published:
Published online

The spatial structure of single photons1, 2, 3 is becoming an extensively explored resource to facilitate free-space quantum communication4, 5, 6, 7 and quantum computation8 as well as for benchmarking the limits of quantum entanglement generation3 with orbital angular momentum modes1, 9 or reduction of the photon free-space propagation speed10. Although accurate tailoring of the spatial structure of photons is now routinely performed using methods employed for shaping classical optical beams3, 10, 11, the reciprocal problem of retrieving the spatial phase-amplitude structure of an unknown single photon cannot be solved using complementary classical holography techniques12, 13 that are known for excellent interferometric precision. Here, we introduce a method to record a hologram of a single photon that is probed by another reference photon, on the basis of a different concept of the quantum interference between two-photon probability amplitudes. As for classical holograms, the hologram of a single photon encodes the full information about the photon's ‘shape’ (that is, its quantum wavefunction) whose local amplitude and phase are retrieved in the demonstrated experiment.

At a glance


  1. Quantum interference of two spatially structured photons.
    Figure 1: Quantum interference of two spatially structured photons.

    a, In analogy to classical holography we repeatedly overlap an unknown photon |ψu〉 with a reference (known) photon |ψr〉 with a constant local phase profile on a 50/50 beam splitter. Coincidence events localized in x and x′ provide the joint probability distribution |Ψ(x, x′)|2, which is sensitive to any differences between the quantum wavefunctions of the photons ψu(x) and ψr(x), including the local variations of their phases. b, The spatially localized coincidence events (x, x′) originate from the non-destructive interference of the probability amplitudes of two classically exclusive, but quantum mechanically coexisting scenarios. Left: The unknown photon in x and the reference photon in x′ have passed through the beam splitter. Right: Both photons localized conversely in x′ and x have been reflected from the beam splitter.

  2. Encoding of the local phase of the quantum wavefunction in the HSP.
    Figure 2: Encoding of the local phase of the quantum wavefunction in the HSP.

    The HSP that emerges from a joint probability distribution of the coincidence events |Ψ(x, x′)|2 encodes (see equation (2)) the local phase profile of the unknown photon φ(x). ac, To illuminate this feature, which originates from the local phase sensitivity of the quantum interference, we depict the expected HSP structure (the false colours denote the computed probability gradations) for photons in two Gaussian modes with identical amplitudes |ψu(x)| = |ψr(x)|, differing by the local phase profile of the unknown photon presented below each plot. a, For the experimental demonstration, the purely quadratic local phase profile has been chosen. b, HSP for the fourth-order polynomial local phase profile. c, HSP for a non-polynomial local phase profile that resembles a fragment of the Warsaw skyline. The influence of dark counts and non-unit spectral visibility on the HSP structure is discussed in the Supplementary Information.

  3. Experimental set-up for measuring the HSP.
    Figure 3: Experimental set-up for measuring the HSP.

    The orthogonally polarized unknown and reference photons, generated in an SPDC process, are prepared in the same spectral mode. The photons are transmitted through the single-mode fibre (SMF), separated by the polarization beam splitter (PBS) and then, at the output beam waist, the local phase profile φ(x) is imprinted on the unknown photon during its double-pass propagation through a phase mask (a cylindrical lens (CL1) for the quadratic phase as in Fig. 2a). We localized the photons outgoing from two distinct ports of a beam splitter, here implemented collinearly as a half-wave plate (λ/2) and calcite crystal, by means of a state-of-the-art I-sCMOS camera18. Both the beam-waist surface of the reference photon and the phase-mask surface were mapped onto the camera with a phase-preserving imaging system consisting of two spherical lenses (SL).

  4. Measured and reconstructed HSP and the full retrieval of the encoded quantum wavefunction.
    Figure 4: Measured and reconstructed HSP and the full retrieval of the encoded quantum wavefunction.

    a, Directly measured joint probability distribution |Ψ(x, x′)|2 forms an empirical HSP. b, Numerical reconstruction of the HSP that best matches the raw experimental data using independently measured, nearly identical amplitudes of the quantum wavefunctions |ψu(x)|, |ψr(x)|. This closely resembles the theoretical pattern presented in Fig. 2a. c, The measurements, followed by the numerical reconstruction, yield the complex quantum wavefunction of the unknown photon ψu(x), in particular its phase (right-hand y axis). The error bars and uncertainty ranges show one standard deviation (see Methods for details).


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  1. Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland

    • Radosław Chrapkiewicz,
    • Michał Jachura,
    • Konrad Banaszek &
    • Wojciech Wasilewski


W.W. proposed the idea of wavefunction phase retrieval. R.C. designed and programmed the experiment, developed HSP methods, analysed the data and prepared figures. M.J. built the set-up and performed the measurements. R.C. and M.J. wrote the manuscript assisted by W.W. and K.B, who supervised the work and contributed to data analysis.

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