Analytical Fock Representation of Two-Mode Squeezing for Quantum Interference

Two-mode squeezing is central to entangled-photon generation and nonlinear interferometry, yet standard perturbative low-gain treatments and Gaussian formalisms can obscure the interference of photon-number amplitudes, especially in nonlinear interferometers and at high gain. Here we derive a closed-form Fock-basis expression for the action of the two-mode squeezing operator on arbitrary number states, enabling the direct analysis of nonlinear interferometers in the photon-number basis at arbitrary squeezing strength. Within this framework, we provide intuitive physical interpretations of several known quantum-interference effects and identify a new multi-photon interference phenomenon in a four-crystal geometry that could readily be observed in laboratories. Our work provides a compact analytic toolkit and explicit design rules for engineering multi-photon interference, with applications in quantum sensing, precision metrology, and quantum state generation.

💡 Research Summary

The paper presents a closed‑form expression for the action of the two‑mode squeezing operator on arbitrary Fock states and demonstrates how this analytic tool can be used to understand and design multi‑photon interference effects in nonlinear interferometers across a wide range of gain. Starting from the SU(1,1) disentangling formula, the authors rewrite the squeezing operator (S_{2}(\zeta)=\exp(\zeta^{*}ab-\zeta a^{\dagger}b^{\dagger})) in normal‑ordered form, introducing the hyperbolic functions (\tanh r) and (\operatorname{sech} r) (with (\zeta = r e^{i\theta})). Applying this normal‑ordered operator to a general two‑mode Fock state (|p,q\rangle) yields a double‑sum expression (Eq. 3) that explicitly gives the amplitude for every possible output photon‑number pair. Although mathematically known, the authors emphasize that this formula is rarely used as a practical design tool in quantum optics.

Using the formula, they analyze four configurations:

1. Single crystal with a (|1,1\rangle) seed – Two indistinguishable quantum pathways (direct transmission vs. annihilation of the seed followed by pair creation) interfere. The output amplitude for (|1,1\rangle) becomes (A(r)=1-\sinh^{2}r,\cosh^{3}r), which vanishes at (r=\operatorname{arcsinh}(1)\approx0.88). This reproduces the Cerf‑Jabbour “time‑reversed Hong‑Ou‑Mandel” effect and provides a simple phase‑amplitude picture.

2. Two‑crystal SU(1,1) interferometer – The first crystal creates a pair, the second may also create a pair; a relative phase (\phi) is inserted between them. The exact amplitude for detecting one photon in each output mode is given by Eq. (8). In the low‑gain limit the probability scales as (4r^{2}\cos^{2}(\phi/2)), i.e., a sinusoidal fringe. In the high‑gain regime the curvature of the fringe at (\phi=\pi) grows as (\sinh^{2}(2r)\sim e^{4r}), indicating exponentially enhanced phase sensitivity, a key resource for quantum metrology.

3. Three‑crystal cascade – With two phase shifters (\phi_{1},\phi_{2}) the (|1,1\rangle) amplitude (Eq. 13) contains four interfering terms. At low gain the condition for perfect destructive interference reduces to (\phi_{1}=\phi_{2}=2\pi/3) (or (4\pi/3)), reflecting the geometric fact that three unit vectors at 120° sum to zero. At higher gain the phases must be adjusted according to Eq. (17‑18); the authors provide a compact analytic condition that works for any gain. This demonstrates that the “frustrated pair creation” effect persists but its exact cancellation point drifts with the squeezing strength.

4. Four‑crystal square geometry (four‑photon interference) – Two rows of two crystals each are pumped; a single phase shifter (\phi) is placed between the rows. The only way to obtain exactly one photon in each of the four detectors is either both upper crystals generate a pair or both lower crystals do so. The amplitude for the four‑photon state (|1,1,1,1\rangle) (Eq. 19) is proportional to (\operatorname{sech}^{4}r) times a cubic polynomial in (\tanh r) and (e^{i\phi}). For equal squeezing, (\phi=\pi) yields complete cancellation, while (\phi=0) gives constructive interference. Importantly, even with asymmetric pump strengths (e.g., (r_{4}=2r_{3})) the same cancellation can be achieved by appropriate choice of (\phi). This constitutes a novel multi‑photon, non‑local interference effect that extends the two‑photon frustrated creation to the four‑photon sector.

Across all cases the authors emphasize that the Fock‑basis description makes the origin of each term transparent: each term corresponds to a specific sequence of annihilation‑creation events across the crystals. By keeping the full photon‑number information, the method reveals interference that is invisible in Gaussian (quadrature) treatments, especially at high gain where many photon pairs are present.

The paper also discusses practical considerations: the need for photon‑number‑resolving detectors, the impact of loss and imperfect phase control, and the scalability of the analytic formulas to larger networks. The derived design rules (e.g., required phase settings for perfect cancellation given any set of squeezing parameters) provide a ready‑to‑use toolkit for experimentalists aiming to exploit multi‑photon interference for quantum sensing, entangled‑state generation, or quantum information processing.

In summary, the work delivers a mathematically exact, yet experimentally friendly, Fock‑state representation of two‑mode squeezing, applies it to elucidate known low‑gain phenomena, extends the analysis to the high‑gain regime, and predicts a new four‑photon interference effect. This bridges the gap between abstract algebraic treatments and concrete interferometer design, offering a powerful new approach for advancing quantum optics and metrology.