Analytical Fock Representation Reveals Multi-Photon Interference in Two-Mode Squeezing for Quantum Interferometry

Two-mode squeezing lies at the heart of generating entangled photons and performing nonlinear interferometry, but conventional approaches struggle to accurately describe how photon numbers interact, particularly in complex systems and at high intensities. Xuemei Gu, alongside Carlos Ruiz-Gonzalez and Mario Krenn from the Max Planck Institute for the Science of Light and the University of Tuebingen, now present a new analytical method that precisely calculates the effect of two-mode squeezing on individual photons. This breakthrough allows scientists to directly analyse nonlinear interferometers using a simple number-based approach, revealing fresh insights into known interference phenomena and, crucially, predicting a previously unseen multi-photon interference effect achievable with current laboratory technology. The team’s work delivers a powerful, compact toolkit and clear design principles for controlling multi-photon interference, promising advances in precision measurement and the creation of complex quantum states.

Fock Representation Simplifies Squeezed State Analysis

Two-mode squeezing is fundamental to generating entangled photons and exploring nonlinear optical phenomena, offering significant advantages for quantum technologies. Researchers have developed an analytical Fock representation to describe two-mode squeezed states, providing a powerful tool for analysing quantum interference effects. This approach avoids the complexities of traditional methods, enabling a clearer and more efficient treatment of quantum states, and allows for the direct calculation of interference patterns without approximation. This analytical representation facilitates the study of complex quantum interference phenomena, such as those arising in multi-photon experiments and quantum imaging, and provides insights into the fundamental properties of entangled photons. The method accurately predicts the visibility of interference fringes, even for highly squeezed states, and offers a valuable resource for designing and optimising quantum experiments, extending its versatility and practical relevance.

Conventional linear interferometry and perturbative treatments obscure how photon-number amplitudes interfere, particularly in multi-crystal geometries and at high gain. Researchers derive the exact analytical action of the two-mode squeezing operator on arbitrary Fock states to analyse nonlinear interferometers directly in the number basis at any squeezing strength. Within this framework, they find new physical interpretations of previously known quantum interference effects, and theoretically discover a novel multi-photon interference effect in a four-crystal geometry that could readily be observed in laboratories.

Multi-Crystal Squeezed Light State Derivation

This work details a derivation of the quantum state for multi-crystal squeezed light generation. The core idea is to iteratively build up the quantum state as squeezed light passes through successive crystals, each introducing squeezing and potential phase shifts. The derivation uses the squeezing operator to represent the transformation applied by each crystal, and the phase shift operator to account for any phase changes, beginning with the vacuum state and sequentially applying these operators. Expanding the squeezed states in terms of Fock states provides a concrete representation of the quantum state, extending to multiple crystals and building on simpler cases.

The derivation involves expressing the squeezed states as a superposition of Fock states, where the coefficients represent the probability amplitudes for finding the system in a particular state. The complexity increases with the number of crystals, requiring careful consideration of the mathematical relationships between the operators and the resulting state. Careful attention must be paid to the explicit forms of the squeezing and phase shift operators, and the limits of summation used in the calculations, while considering the commutation relations between the creation and annihilation operators. The final quantum state is a complex superposition of Fock states, representing the probability amplitudes for finding the system in each possible configuration. The distribution of these amplitudes reveals the squeezing and anti-squeezing effects, and the multi-mode nature of the squeezed state, likely exhibiting entanglement. To strengthen the derivation, it is important to state the explicit forms of the operators, justify the summation limits, and verify that the final state is normalized, with numerical calculations helping to validate the results.

Four-Crystal Geometry Reveals Novel Interference Effect

This work presents a new analytical framework for understanding multi-mode squeezing, a process central to generating entangled photons and performing precise measurements. By deriving an exact mathematical description of how squeezing affects photon number states, researchers have gained new insights into interference effects, particularly in complex multi-crystal setups. The team demonstrates that existing interpretations of interference can be refined with this new approach, and importantly, they have theoretically discovered a novel multi-photon interference effect observable in a four-crystal geometry. This discovery centres on a specific configuration where detecting a photon in each of four detectors requires the simultaneous creation of photon pairs in either the upper or lower rows of crystals, but not both.

The researchers show that by carefully controlling the phase between crystals, it is possible to eliminate the contribution from certain interference pathways, revealing this unique quantum effect. This analytical toolkit and the resulting design rules offer a means to engineer multi-photon interference with greater precision, potentially advancing applications in areas such as metrology and the generation of complex quantum states. The authors acknowledge that their analysis relies on certain approximations when considering higher-order effects, and that experimental verification is crucial to fully validate the theoretical predictions. Future research will likely focus on exploring the practical implications of this new interference effect, and investigating its potential for enhancing quantum technologies, with the framework potentially extended to analyse even more complex multi-crystal configurations.