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The coherence of light is fundamentally tied to the quantum coherence of the emitting particle

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Science Advances  30 Apr 2021:
Vol. 7, no. 18, eabf8096
DOI: 10.1126/sciadv.abf8096
  • Fig. 1 Excitation of waves by free particles: classical versus quantum theory.

    (A) Classical wave dynamics. A point particle with velocity v passes through an optical medium and emits waves that may interfere coherently. The classical emitter current density J(r, t) = evδ(rvt) emits a temporally coherent shock wave. (B) Quantum description. A quantum particle is described by a delocalized wave function ψ(r, t). A current operator Ĵ(r,t) is then associated with the particle. Even when the initial particle is only described by a single momentum ki, it may spontaneously emit many wave quanta (momenta q, q′, …). The waves are then entangled with the particle because of momentum conservation (leaving the final particle having momenta kf, kf′, … respectively). When only the emitted waves are observed, this entanglement can lead to quantum decoherence and lack of interference visibility, resulting in the emission of incoherent radiation.

  • Fig. 2 How can particle momentum uncertainty determine the interference of waves emitted by that particle?

    (A) A quantum particle with a coherent momentum uncertainty δpe that equals its total momentum uncertainty Δpe displays a broad quantum coherence between its initial momenta pi (yellow glow). When the particle transitions to any final momentum pf, the emitted wave inherits this initial coherence because of the “which path” interference between the initial particle states. Hence, different wave vector components of the wave are coherent (red glow). (B) A quantum particle in a mixture of momenta (total uncertainty Δpe) with low coherent uncertainty δpe ≪ Δpe emits temporally incoherent waves. The limited interference inhibits the pulse formation, and its length exceeds the classical prediction. (C and D) The temporal field autocorrelations, 2r2ϵ0ncE(−)(t)E(+)(t)⟩ (in μW), for 1-MeV electrons in silica in the visible range. The electrons are modeled as spherical Gaussian wave packets with coherent energy uncertainty (A) Δεe = 3.72 eV (wave packet radius ~50 nm) and (B) Δεe = 0.19 eV (wave packet radius ~1 μm). The diagonal (t = t) indicates the temporal power envelope, P(t), being transform-limited in (A) and incoherent in (B). Insets show a scaled comparison between P(t) and the degree of first-order coherence of the light, g(1)(τ). For both (A) and (B), the classically expected shock wave full width at half maximum is 1.4 fs.

  • Fig. 3 Quantum optical analysis of CR—for measuring the emitter’s wave function.

    (A) A charged particle wave packet ψ(r, t) of finite size Δ and carrier velocity v0 impinges on a Cherenkov detector with material dispersion n(ω). The particle spontaneously emits quantum shock waves of light into a cone with opening half-angle θc(ω) = acos[1/βn(ω)]. Collection optics is situated along the cone in the direction r̂c in the far field. (B) Detection scheme for measuring the spectral field autocorrelations ⟨E(ω)E(ω′)⟩ using an interference between spectrally/temporally sheared fields (57). (C) The reconstructed photon density matrix determines the spatial probability distribution ∣ψ(r)∣2. (D and E) Simulation of particle wave function size reconstruction from the photon density matrix. A single 1-MeV electron (β = 0.94) in a silica Cherenkov detector [dispersion taken from (77)] emits CR that is collected within the visible range (λ = 400 to 700 nm, centered at λ0 = 550 nm). The electron wave function envelope is Gaussian and spherically symmetric, with position uncertainty of (d) Δxe = 254 nm and (E) Δxe = 1016 nm (bottom insets). In both (D) and (E), the measured photon density matrix, ρph(ω, ω′), is plotted. The wave function–independent diagonal ρph(ω, ω) that denotes the photodetection probability is the same for both cases. However, the off-diagonal spectral coherence ρph(ω − ω′) is strongly dependent on the wave function. Measuring its width Δωcoh (top insets) and using the approximate Eq. 7 provide the estimates (D) Δxe=290 nm and (E) Δxe=1006 nm.

  • Fig. 4 Quantum optical analysis of CR—emitted from a laser-driven electron wave function.

    (A) A free electron wave function is shaped by the interaction with a strong laser field of frequency Ω (here, Ω = 2π × 200 THz), as done in photon-induced near-field electron microscopy (61). The result is a coherent electron energy ladder, manifested as a temporal pulse train. (B) Cherenkov photon autocorrelations reveal the electron wave function spectral interference pattern, matching the laser frequency. The measurement scheme is the same as in Fig. 3 (A to C).

Supplementary Materials

  • Supplementary Materials

    The coherence of light is fundamentally tied to the quantum coherence of the emitting particle

    Aviv Karnieli, Nicholas Rivera, Ady Arie, Ido Kaminer

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