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Observation of three-photon bound states in a quantum nonlinear medium

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Science  16 Feb 2018:
Vol. 359, Issue 6377, pp. 783-786
DOI: 10.1126/science.aao7293
  • Fig. 1 Qualitative descriptions of the experiment.

    (A and B) Setup and atomic-level scheme. The atoms are optically pumped into the hyperfine (F) and magnetic (mF) sublevel Embedded Image. The weak coherent probe light is coupled to the Rydberg state Embedded Image (mJ, projection of the total electron angular momentum along the quantization axis) via an intermediate state Embedded Image, with linewidth Γ/2π = 6.1 MHz, by means of a counterpropagating control field that is detuned by ∆ below the resonance frequency of the upper transition, Embedded Image. Strong interactions between probe photons are detected via photon correlations of the transmitted light, which is split onto three single-photon detectors with equal intensities. To perform phase measurements, a local oscillator is mixed into detector D3. Embedded Image, photon annihilation operator of the time bin mode t; Ωc, control laser Rabi frequency; g, correlation function; Embedded Image, phase; LO, local oscillator. (C) Transmission (top) and phase Embedded Image (bottom) as a function of probe frequency measured at a low input photon rate (0.5 μs−1). Embedded Image is measured without conditioning on the detection of other photons. The control laser is set at Δ/2π = 30 MHz below the Embedded Imagetransition, with Rabi frequency Ωc/2π = 10 MHz. The blue and red traces are from measurements with and without a control beam, respectively. The blue and red dashed lines in the bottom graph are theoretical expectations. The vertical yellow dashed line marks electromagnetically induced transparency (EIT) resonance. (D) Rate dependence of transmission (top) and unconditional phase (bottom) on the two-photon resonance Embedded Image, with a one-photon detuning of Δ/2π = 30 MHz and a control Rabi frequency Ωc/2π = 10 MHz. Whereas the transmission is rate-independent, the phase is strongly rate-dependent (slope is 0.4 rad·μs). (E) Schematic correlation functions for two (top) and three (bottom) photons as a function of their time separation τ. The attractive interaction leads to photon bunching, with three photons being more tightly bound together than two photons.

  • Fig. 2 Photon correlation functions with tighter bunching due to the three-photon bound state.

    Photon correlation functions were measured on EIT resonance and at one-photon detuning Δ/2π = 30 MHz, control Rabi frequency Ωc/2π = 10 MHz, an input photon rate of 1 μs−1. (A) 2D representation of the three-photon correlation function g(3)(t1, t2, t3), with ti being the photon detection time at detector Di. Three-photon bunching corresponds to the central region, two-photon bunching to the stripes. (B) g(3)(t, t, t + |τ|) (blue data points) and g(2)(t, t + |τ|) (brown data points), with the decay constants calculated from the exact solution for the bound states Embedded Image and Embedded Image, respectively (dashed lines). The calculated exponential decay is scaled to match the initial point of the measured intensity correlation functions. The approximately twofold smaller decay length of the three-photon correlation function shows that a photon is more strongly bound to two photons than to one. The fitted exponential decay constants with zero offset for g(3) and g(2) are τ3 = 0.14(2) μs and τ2 = 0.31(6) μs, respectively, in agreement with the calculated values. (C) Three representative plots of g(3)(t1, t2, t3)/g(2)(t1, t2) for fixed time separation T ≡ |t1t2| = 0 μs (i), T = 0.2 μs (ii), and T = 1.8 μs (iii), within a 50-ns window. As the two photons get farther and farther away from each other, the sharply decaying g(3) function transitions to a slower decaying g(2) function. For intermediate time separations (ii), interference occurs between all states, including the dimer and trimer. All permutations of the detectors are used to generate the data in (B) and (C). Error bars indicate 1 SD.

  • Fig. 3 Larger nonlinear phase for three photons.

    Nonlinear phase was measured under identical conditions as the data in Fig. 2. (A) Conditional phase Embedded Image, where t1 and t2 correspond to photon detection events at detectors D1 and D2, and a heterodyne measurement is performed on detector D3 at time t3. (B) Diagonal cut Embedded Image (blue), with the two conditioning probe photons within 40 ns of each other, and Embedded Image (brown), showing a larger phase when conditioning on two other near-simultaneous photons Embedded Image than on one near-simultaneous photon Embedded Image. Embedded Image is referenced to its own average value when all N photons are too far apart to be correlated. Specifically, Embedded Image and Embedded Image. At large |τ|, Embedded Image asymptotically goes to Embedded Image because Embedded Image. Error bars indicate 1 SD.

  • Fig. 4 Comparison of the phase ratio with the effective field theory (EFT) predictions.

    (A) Potential (solid black and gray lines) that the third photon, at position r′, experiences because of the other two photons, at positions ±r/2. (i) When the two photons are separated by more than twice the blockade radius (r > 2rB), each of them creates its own square potential with a width of 2rB. (ii) When the two photons overlap (rB < r < 2rB), the potential is partially saturated. Dashed lines denote the overlap of the two interaction potentials. (iii) When the two photons are within one blockade radius (r < rB), because there can be no more than one Rydberg excitation within rB, the potential is not deeper than that created by one photon. Therefore, we overestimate the attractive potential by considering pairwise interaction only, and a repulsive effective three-photon force is required to correctly account for the saturation of the Rydberg blockade. U, interaction potential between two photons. (B) Measured phase ratio Embedded Image (blue) and EFT predictions (with the effective three-photon force in red; without in green) as a function of Embedded Image, where Embedded Image refers to the average over the Gaussian profile of the atomic density, and ODB is optical depth per blockade radius. The quantity Embedded Image is a quantitative measure of the interaction strength in this system. The control Rabi frequency Ωc/2π = {22, 18, 10, 10, 8} MHz for Δ/2π = {54, 42, 30, 24, 18} MHz is chosen such that the transmission is insensitive to the input photon rate (Fig. 1C). We also change the input photon rate to {0.7, 1, 1, 1.3, 2.5} photons/μs to achieve similar data-acquisition rates, because the losses are larger at smaller detunings. For a fully saturated medium, one expects Embedded Image, as indicated by the pink dashed line. For bound states in a long medium and no effective three-photon force, one expects Embedded Image, as indicated by the light blue dashed line (see text). For a dispersionless Kerr medium, one expects ϕ(3)(2) = 3, as indicated by the gray dashed line. EFT results are calculated with parameters from independent measurements, and the two-photon detuning from the EIT resonance is the only parameter varied within the experimental uncertainty to fit the two-photon phase. Error bars in the EFT with the effective three-photon force arise from the variations with the choice of matching conditions for the three-body scattering amplitudes (27). Error bars in the experimental data indicate 1 SD.

Supplementary Materials

  • Observation of three-photon bound states in a quantum nonlinear medium

    Qi-Yu Liang, Aditya V. Venkatramani, Sergio H. Cantu, Travis L. Nicholson, Michael J. Gullans, Alexey V. Gorshkov, Jeff D. Thompson, Cheng Chin, Mikhail D. Lukin, Vladan Vuletić

    Materials/Methods, Supplementary Text, Tables, Figures, and/or References

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    • Materials and Methods 
    • Supplementary Text
    • Figs. S1 to S4
    • Tables S1 and S2
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