Analytical Fock Representation of Two-Mode Squeezing for Quantum Interference

Abstract

Two-mode squeezing is central to entangled-photon generation and nonlinear interferometry, yet standard perturbative low-gain and Gaussian treatments obscure how photon-number amplitudes interfere, especially in multi-crystal geometries and at high gain. Here, we derive the exact analytic action of the two-mode squeezing operator on arbitrary Fock states to analyze nonlinear interferometers directly in the number basis at arbitrary squeezing strength. Within this framework, we find new physical interpretations of previously known quantum interference effects, and theoretically discover a new and unusual multi-photon interference effect in an experimental four-crystal geometry that could readily be observed in laboratories. Our work provides a compact analytic toolkit and concrete design rules for engineering multi-photon interference, with applications in quantum sensing, precision metrology, and advanced quantum state generation.

Figures (4)

• Figure 1: A single nonlinear crystal seeded with two single photons and the two-photon coincidence probability. (a) Two indistinguishable single photons (blue dots) enter a nonlinear crystal and the output detection paths are $a$ and $b$. (b) The coincidence probability $P_{|1,1\rangle}$ is plotted versus the squeezing parameter $r$. At $r=\operatorname{arcsinh}(1)\approx 0.88$, the $|1,1\rangle$ amplitude vanishes, in agreement with Ref. cerf2020two.
• Figure 2: Two-crystal setup and two-photon coincidence probability. (a) The experimental scheme shows a phase shifter placed between nonlinear crystals I and II, with detections in the output paths $a$ and $b$. (b) The two-photon coincidence probability $P_{|1,1\rangle}$ (one photon in each output) is plotted as a function of the relative phase for different squeezing strengths $r$, assuming $r_{1}=r_{2}=r$.
• Figure 3: Three-crystal setup and two-photon coincidence probability. (a) The experimental scheme includes two phase shifters placed between the three sequential nonlinear crystals. (b) The coincidence probability $P_{|1,1\rangle}$ for equal squeezing strengths $r_{1}=r_{2}=r_{3}=1.0$ shows a coincidence zero due to perfect destructive interference (black dots). (c) The coincidence probability for asymmetric pump powers ($r_{1}=1.0$, $r_{2}=0.5$, $r_{3}=0.6$) also exhibits destructive interference of photon pairs (black dots).
• Figure 4: Four-crystal setup and four-photon coincidence probability. (a) The experimental scheme includes a phase shifter placed between crystals I and III in path $a$. (b) The four-photon coincidence probability $P_{|1,1,1,1\rangle}$ is shown for phase $\phi=\pi$, with squeezing strengths set to $r_{1}=r_{2}$ and $r_{3}=r_{4}$. (c) The probability for phase $\phi=0$ is shown for high squeezing, with $r_{1}=r_{2}$ and $r_{4}=2r_{3}$.