Heralded generation of a three-mode NOON state

TL;DR

We report heralded generation of a three-mode two-photon NOON state $|psi_3^2> = (|200> + e^{i alpha1}|020> + e^{i alpha2}|002>)/\sqrt{3}$ using a four-mode linear-optical unitary realized with a displaced Sagnac interferometer acting on three single photons; heralding is achieved by detecting a photon in an auxiliary mode. The experiment yields a nominal success probability of $|\gamma|^2 = 0.25$ with a measured value of $0.237 \pm 0.009$ and a fidelity to the target state of $F = 0.823 \pm 0.018$, with fidelity bounds $F \in [0.818, 0.836]$ derived from coherence measurements and populations. The state certifies genuine multipartite entanglement, exceeding the biseparable threshold $F_{bs} = 2/3$ by more than eight standard deviations. This heralded, multi-mode entangled-state generation provides a practical stepping stone toward scalable linear-optical quantum information processing and could be extended to integrated photonics for multi-phase sensing and distributed quantum networks.

Abstract

Entangled states of photons form the foundation of quantum communication, computation, and metrology. Yet their generation remains fundamentally constrained: in the absence of intrinsic photon-photon interactions, the generation of such states is inherently probabilistic rather than deterministic. The prevalent technique of post-selection verifies the creation of an entangled state by detecting and thus destroying it. Heralding offers a solution in which measuring ancillary photons in auxiliary modes signals the state generation without the need to measure it. Here, we report an experiment to generate a three-mode two-photon NOON state, where the detection of a single photon in one heralding mode signifies the presence of the state in three target modes. We validate the generated state by estimating a fidelity of 0.823 +/- 0.018 with respect to an ideal three-mode NOON state and certifying genuine multipartite entanglement. By virtue of the high success probability and small resource overhead of our scheme, our work provides a theoretical and experimental stepping stone for entangled multi-mode state generation, which is realizable with current technology. These multi-mode entangled states represent a key direction for linear optical quantum information that is complementary to multi-qubit state encoding.

Figures (7)

• Figure 1: Heralded generation of a three-mode NOON state via linear optics: (a) From an input of three single photons, a four-mode unitary transformation generates the desired coherent superposition of two photons in three modes (A,B,C) upon detecting a photon in the fourth mode. (b) The experimental setup for state generation consists of a displaced Sagnac interferometer constructed using a polarizing beam splitter and wave plates. The heralding process is schematically indicated, highlighting the projection onto the three-mode NOON state in the target modes upon detection of a photon in the heralding mode. The interferometer provides a stable and flexible platform for realizing four-mode unitary transformations EnglertgateGuanggate2020. QWP: quarter-wave plate; HWP: half-wave plate; M: mirror; PBS: polarizing beam splitter. The red and blue lines with arrows indicate the two spatial paths of the interferometer.
• Figure 2: Interferometric scheme for heralded NOON state generation and characterization. State preparation relies upon the HWP in input 1 and the first interferometer (unitary operation panel), followed by a QWP in the blue path in the subsequent interferometer. The measurement stage is realized using the second interferometer (measurement panel) and polarization projection in both outputs. The wave plates inside the interferometers have a hole in the center such that the polarization of only one path is transformed. Detection is performed using SNSPDs, with fiber beam splitters enabling pseudo-photon-number-resolving measurements on all three target modes. QWP: quarter-wave plate; HWP: half-wave plate; PBS: polarizing beam splitter; FBS: fiber beam splitter.
• Figure 3: The heralded three-mode state is projected onto a two-photon two-mode subspace by conditioning on vacuum in the remaining mode. To probe the coherence of the state in this subspace, we interfere the two modes after mapping them into the polarization basis of a single spatial mode. The resulting coincidence probabilities are shown as a function of the measurement half-wave plate angle $\theta$ for pairwise combinations of modes A and B (a), B and C (b), and A and C (c). Each probability data point (blue points) is obtained from an average of 1039 four-fold coincidence counts accumulated over 1800 s. The resulting data are fitted with the function given in Eq. \ref{['eq:coincidence']} (red curve), and the uncertainties in the measured counts are estimated assuming Poissonian statistics.
• Figure S1: Simplified schematic for the characterization of the setup with classical light. QWP: quarter-wave plate; HWP: half-wave plate; M: mirror; PBS: polarizing beam splitter; SQT: single-qubit tomography setup. The orange and blue lines with arrows indicate the two spatial paths of the interferometer.
• Figure S2: Schematic summary of the measurement sequence used to extract the interference fringes presented in Fig 3 of the main text.