Macroscopic Quantum State Analyzed Particle by Particle

Macroscopic quantum phenomena, e.g., superconductivity and squeezing, are believed to result from entanglement of macroscopic numbers of particles. We report the first direct study of this kind of entanglement: we use discrete quantum tomography to reconstruct the joint quantum state of photon pairs extracted from polarization-squeezed light. Our observations confirm several predictions from spin-squeezing theory [Beduini et al., Phys. Rev. Lett. 111, 143601 (2013)], including strong entanglement and entanglement of all photon pairs within the squeezing coherence time. This photon-by-photon analysis may give insight into other macroscopic many-body systems, e.g., photon Bose-Einstein condensates.

Figure 1

Photon-level analysis of a polarization-squeezed state. (a) Simplified experimental schematic (full schematic in the Supplemental Material [28]). (red box) A 795 nm continuous-wave laser is locked to the Rb $D_1$ line, frequency doubled, and used to pump a subthreshold optical parametric oscillator (OPO) producing 795 nm squeezed vacuum (SV) with $V$ polarization. A portion of the laser is also taken to produce an $H$ polarization coherent state (CS), and the two components are combined on a polarizing beam splitter (PBS) to produce PS light. Two piezoelectric actuators (PZTs) act on the relative phase of the SV and CS components of the state. (blue box) A galvanometer mirror (GM) directs the PS beam alternately to a continuous-variable (CV) detector, sensitive to the macroscopic polarization of the state or to a discrete-variable (DV) photon-counting system. The CV detector consists of a half wave plate (HWP), PBS, and linear differential detector. Its electronic signals are fed back to ${\mathrm{PZT}}_1$ and ${\mathrm{PZT}}_2$ to stabilize the SV and CS relative phase. (green box) The DV detector consists of a quarter wave plate (QWP), HWP, polarization-maintaining fiber (PMF), and beam displacer, together acting as a polarization analyzer splitting the two polarizations (in a basis determined by the QWP and HWP) to two parallel output channels. A Faraday anomalous-dispersion optical filter (FADOF) is used to eliminate photons from nondegenerate modes of the OPO. The two output paths are each split to two single-photon avalanche photodiodes (SPADs) to allow detection of both same- and opposite-polarization photon pairs. (b) Reconstructed density matrices (magnitudes only) of photons extracted from a PS state with ${\mathrm{\Phi }}_C=9.6\times 10^5\mathrm{ph}/\mathrm{s}$ and ${\mathrm{\Phi }}_S=1.9\times 10^5\mathrm{ph}/\mathrm{s}$ [orange square in (d)] for mean arrival time differences $⟨|\tau |⟩=6$, 30, 48, and 66 ns. We include all events within the coincidence window $⟨|\tau |⟩-6\mathrm{ns}\le |\tau |\le ⟨|\tau |⟩+6\mathrm{ns}$. Blue bars indicate the “NooN” portion of the state, a superposition of $|HH⟩$ and $|VV⟩$, green bars indicate the “$W$” portion of the state, $\propto |HV⟩+|VH⟩$, and pink bars indicate anomalous coherences (see text). (c) (upper) Concurrence of the two-photon density matrices versus $⟨|\tau |⟩$ under the same conditions. Error bars indicate $\pm 1\sigma$ statistical error estimated by a bootstrapping procedure (see the Supplemental Material). Results show entanglement for photons separated by up to 60 ns, as predicted by the theory (solid line). (lower) SV (red) and CS (green-flat line) contributions to the PS state. Shown are photon pair detection rates as a function of delay time for one or the other contribution indicating a crossover in source brightness at $\tau =15\mathrm{ns}$, which corresponds to balanced $|HH⟩$, $|VV⟩$ amplitudes and maximum concurrence. (d) Comparison between the theoretical concurrence (surface) and the experimental observations (filled squares) for a coincidence time window of 26 ns centered on $\tau =0$. The contour plot and the empty squares on the bottom plane are the projection of the theoretical surface and of the experimental data on the space of SV and CS photon fluxes, ${\mathrm{\Phi }}_S$ and ${\mathrm{\Phi }}_C$, respectively. The upper empty squares lie on the surface and represent the expected concurrence for the measured density matrices. Gray bars indicate $\pm 1\sigma$ statistical errors calculated by bootstrapping. For all cases, we obtain theoretical predictions by integrating the elements of the expected density matrix $\rho$ over the time window considered.