Orbital angular momentum of entangled photons as a probe for relativistic effects

TL;DR

The paper develops a relativistic metrology scheme using orbital angular momentum (OAM) entanglement to probe Lorentz boosts. By modeling length contraction along the transverse plane as φ' = arctan(γ tan φ), it derives a γ-dependent joint OAM distribution P_{ℓ_Aℓ_B} that broadens with increasing γ and breaks the orthogonality of projection measurements. Experimentally, nondegenerate SPDC photon pairs are projected with contracted OAM modes via spatial light modulators to simulate relativistic effects, and γ is extracted from observed spectra, achieving good agreement with theory and simulating velocities up to 0.99c. The work demonstrates a concrete pathway for OAM-based metrology in relativistic conditions and motivates extensions to accelerated frames and gravitational contexts. Overall, it provides a quantitative link between relativistic motion and observable changes in the OAM correlations of entangled photons, enabling Lorentz-factor estimation from structured light experiments.

Abstract

Orbital angular momentum (OAM) as both classical and quantum states of light has proven essential in numerous applications, from high-capacity information transfer to enhanced precision and accuracy in metrology. Here, we extend OAM metrology to relativistic scenarios to determine the Lorentz factor of a moving reference frame, exploiting the fact that OAM is not Lorentz invariant. We show that the joint OAM spectrum from entangled states is modified by length contraction when measured by two observers moving relative to the entanglement source. This relative motion rescales the spatial dimensions, thus breaking the orthogonality of the OAM measurement process and resulting in a broadening of the joint OAM spectrum that can precisely determine the Lorentz factor. We experimentally simulate velocities up to $0.99c$, confirm the predicted broadening, and use the measurement outcomes to extract the Lorentz factor. Our work provides a pathway for novel measurement techniques suitable for relativistic conditions that leverage OAM structured light as a resource.

Figures (3)

• Figure 1: (a) Two photons entangled in their OAM degree of freedom are sent to two independent detectors, Alice (A) and Bob (B), that are moving at a speed $v \sim c$. In Charlie's reference frame, this is seen as a length contraction in the x-direction of the detectors, mapping the coordinates $(x, y) \rightarrow (x/ \gamma, y)$. The detectors project onto the OAM eigenstates $\ket{\ell_A}$ and $\ket{\ell_B}$, where Charlie observes their OAM phase patterns as shown in (b) for Alice and Bob at rest ($v = 0, \gamma=1$) or moving at relativistic speeds ($v = 0.99c, \gamma=10$). The resulting OAM correlations are shown in (c) for $\gamma = 1$ and 10 with insets indicating the conditional probability, $P_{\ell_B| \ell_A = 0 }$, in the range $\ell_B = \{-20, 20\}$. For $\gamma =1$ the plot shows that the joint detection probability has OAM anti-correlations, i.e. non-zero detection probability for $\ell_A=-\ell_B$. (d) However, as $\gamma$ increases, the orthogonality is compromised and the number of contributing modes, $\Omega$ increases with the increment of the Lorentz factor. (e) Consequently, the Lorentz factor can be extracted from the observed OAM spectrum using the measurement probabilities $\mathcal{M}$ computed from Eq. (\ref{['eqn:gammaextract']}) which is monotonic and near linear with respect to $\gamma$.
• Figure 2: Schematic diagram of the non-degenerate quantum experiment used to create OAM entangled photons. A non-linear crystal (NLC) is used to produce dual-wavelength SPDC at $\lambda_1$ = 810 nm and $\lambda_2$ = 1550 nm. The plane of the crystal is imaged onto two spatial light modulators (SLM). OAM projection holograms are displayed on each SLM with the $x$-coordinate contracted as $x\rightarrow x/\gamma$ to perform joint projections. The photons are thereafter coupled into SMFs connected to single-photon detectors (Det. A(B)).
• Figure 3: (a) Measured coincidences for encoded $\gamma=$ 1, 2, 5, 10 and 20. Insets show the conditional probability, $P_{\ell_B| \ell_A = 0 }$, for a fixed $\ell_A =0$ in the range $\ell_B = \{-20, 20\}$. (b) Measured $\Omega$ (number of contributing modes) vs the encoded Lorentz factors $\gamma$. (c) The encoded Lorentz factor $\gamma$ compared with the measured factor, $\gamma_\text{meas}$. The fitted $\gamma$ values using Eq. \ref{['eq:sumprob']} are represented by pink crosses. (d) A plot for the related rapidity $\eta$, computed using the experimentally measured Lorentz factor in comparison with theoretical $\eta$. The inset represents the velocities determined from $\gamma_\text{meas}$, showing simulated velocities up to $0.99c$.