3D imaging of the biphoton spatiotemporal wave packet

Photons are among the most important carriers of quantum information owing to their rich degrees of freedom (DoFs), including various spatiotemporal structures. The ability to characterize these DoFs, as well as the hidden correlations among them, directly determines whether they can be exploited for quantum tasks. While various methods have been developed for measuring the spatiotemporal structure of classical light fields, owing to the technical challenges posed by weak photon flux, there have so far been no reports of observing such structures in their quantum counterparts, except for a few studies limited to correlations within individual DoFs. Here, we propose and experimentally demonstrate a self-referenced, high-efficiency, and all-optical method, termed 3D imaging of photonic wave packets, for comprehensive characterization of the spatiotemporal structure of a quantum light field, i.e., the biphoton spatiotemporal wave packet. Benefiting from this developed method, we successfully observe the spatial-spatial, spectral-spectral, and spatiotemporal correlations of biphotons generated via spontaneous parametric down-conversion, revealing rich local and nonlocal spatiotemporal structure in quantum light fields. This method will further advance the understanding of the dynamics in nonlinear quantum optics and expand the potential of photons for applications in quantum communication and quantum computing.

💡 Research Summary

The paper introduces a novel, self‑referenced, high‑efficiency all‑optical technique—termed three‑dimensional (3D) imaging of photonic wave packets—to fully characterize the spatiotemporal wave packet of biphoton states generated by spontaneous parametric down‑conversion (SPDC). While classical light fields can be measured in space, time, frequency, and polarization using a variety of well‑established methods, extending these techniques to the quantum regime has been hampered by the extremely low photon flux and the lack of a well‑defined phase reference. Existing quantum‑optical methods have either measured only intensity correlations in a single degree of freedom (DoF) or required an external, well‑characterized reference pulse, which introduces inefficiencies and circular dependencies.

The authors overcome these limitations by developing a joint spatiotemporal amplitude (JSTA) measurement based on an extension of spectral shearing interferometry (SSI) into the spatial domain. The core of the method is a photonic crystal fiber (PCF) that provides a constant frequency shift to the signal photon through cross‑phase modulation with a strong classical control pulse. This shift creates a sheared replica of the unknown quantum state that serves as its own reference, eliminating the need for any external reference beam. By recombining the original and sheared replicas in a Mach–Zehnder interferometer, the resulting interference pattern encodes the gradient of the joint spectral phase. Integration of this gradient yields the full complex amplitude ψ(q_s, q_i, ω_s, ω_i), where q denotes transverse momentum (or spatial coordinate) and ω denotes angular frequency for signal (s) and idler (i) photons.

Mathematically, the biphoton state is expressed as |ψ⟩ = ∫∫ ψ(k_s, k_i) |1:k_s⟩|1:k_i⟩ dk_s dk_i, with ψ(k_s, k_i) generally non‑separable, reflecting non‑local spatiotemporal correlations. The authors decompose ψ into a phase‑matching function Φ(Δq, Δν) determined by the crystal length and internal structure, and a pump envelope ˜E(q_p, ν_p) that reflects the spatial‑spectral profile of the pump pulse. In the regime where the crystal transverse dimensions far exceed the pump waist and the interaction time exceeds the pump pulse duration, transverse momentum conservation (Δq = q_s – q_i = 0) and energy conservation (Δν = ν_s – ν_i = 0) are simultaneously satisfied, leading to anti‑correlated transverse momenta and positively correlated frequencies. The pump’s Gaussian spatial and spectral profiles further shape the joint wave packet, imprinting additional correlations.

Experimentally, the authors generate photon pairs via type‑II SPDC in a nonlinear crystal. The signal photon passes through the PCF together with a synchronized classical control pulse, acquiring a fixed frequency shift. The idler photon is post‑selected in various spatial positions and spectral modes using slits and tunable filters. Coincidence detection between the idler and the interferometrically processed signal yields a set of interference fringes for each post‑selection condition. By scanning the spatial position of the idler and the spectral shear applied to the signal, the full four‑dimensional joint amplitude is reconstructed. The method simultaneously provides (i) spatial‑spatial intensity correlations, confirming anti‑correlation of transverse momenta; (ii) spectral‑spectral intensity correlations, confirming energy conservation and the shape of the phase‑matching bandwidth; and (iii) genuine spatiotemporal correlations, revealing how spatial and spectral variables intertwine—a direct observation of non‑local spatiotemporal structure in a biphoton state.

Key advantages of this technique are: (1) self‑referencing eliminates the need for an externally calibrated reference pulse, avoiding circular calibration problems; (2) the shearing interferometer operates efficiently at the single‑photon level, achieving high sampling efficiency compared with multi‑plane intensity‑only methods; (3) the approach naturally extends to joint measurements across multiple DoFs, paving the way for full characterization of hyper‑entangled states (simultaneous entanglement in polarization, spatial mode, frequency, etc.). Consequently, the method opens new avenues for quantum information protocols that exploit high‑dimensional entanglement, such as high‑capacity quantum key distribution, quantum computing with multi‑mode photonic qudits, and ultrafast quantum optics where sub‑cycle temporal features become relevant.

In summary, the paper demonstrates that by integrating cross‑phase modulation‑based spectral shearing with spatial interferometry, one can directly image the full 3D spatiotemporal wave packet of biphotons. This breakthrough provides the experimental community with a powerful tool to probe and harness the rich, multidimensional structure of quantum light, moving beyond intensity‑only measurements toward complete complex‑amplitude reconstruction in the quantum regime.