Squeezed Light, Quantized Interference

Author: Denis Avetisyan

A new analytical framework unlocks deeper understanding of multi-photon interactions using squeezed light states.

The experimental setup utilizes a phase shifter between two nonlinear crystals to demonstrate that the two-photon coincidence probability, $P|1,1\rangle$, varies predictably with relative phase and is directly influenced by the squeezing strength, $r$, where equal squeezing is applied to both crystals ($r_1 = r_2 = r$).

Researchers derive an exact Fock-basis representation of the two-mode squeezing operator for improved analysis of quantum interference phenomena and potential advancements in quantum metrology.

While two-mode squeezing is crucial for generating entangled photons and performing nonlinear interferometry, existing perturbative treatments obscure the underlying mechanisms of photon-number interference, particularly at high gain. This work, ‘Analytical Fock Representation of Two-Mode Squeezing for Quantum Interference’, derives an exact analytic representation of the two-mode squeezing operator directly in the Fock basis, revealing new physical interpretations of quantum interference effects. We predict a novel multi-photon interference phenomenon observable in a four-crystal geometry, offering a compact toolkit for engineering complex quantum states. Could this framework unlock enhanced precision in quantum sensing and metrology through optimized multi-photon interference designs?

Entangled Photons: The Building Blocks of Quantum Reality

The creation of correlated photon pairs, known as entangled photons, represents a cornerstone of modern quantum technology. These aren’t simply two photons emitted simultaneously; rather, they share a quantum link, possessing correlated properties like polarization or momentum, regardless of the physical distance separating them. This peculiar connection allows for applications ranging from quantum cryptography – where entanglement ensures secure communication – to quantum computing, where entangled photons can represent and manipulate qubits. Furthermore, the ability to reliably generate and control entangled photons is vital for quantum sensors, promising unparalleled precision in measurements of time, gravity, and magnetic fields. The development of efficient sources of entangled photons, therefore, remains a primary focus for researchers striving to build practical and scalable quantum devices, pushing the boundaries of what’s possible with information processing and sensing.

Accurately characterizing the quantum state of light, particularly when dealing with entangled photons, necessitates a sophisticated mathematical framework beyond simple intensity measurements. The Fock basis representation provides this crucial tool, allowing physicists to describe the number of photons in a given quantum state – a statistic vital for understanding and manipulating quantum light sources. Unlike classical descriptions, this approach doesn’t assume a continuous range of light intensities; instead, it quantizes the electromagnetic field, treating it as a collection of discrete energy packets, or photons. Each possible photon number is represented as a basis state, denoted $|n\rangle$, where $n$ is a non-negative integer. The complete quantum state is then expressed as a superposition of these Fock states, for example, $c_0|0\rangle + c_1|1\rangle + c_2|2\rangle$, where the coefficients $c_i$ determine the probability of finding $i$ photons. This detailed statistical description is essential not only for verifying entanglement but also for optimizing the performance of quantum technologies reliant on precisely controlled photon numbers, such as quantum cryptography and quantum computation.

Entangled photons are not simply a fascinating prediction of quantum mechanics, but a crucial resource for creating squeezed states of light. These unique states exhibit a reduction in quantum noise – specifically, noise in one quadrature of the electromagnetic field – below the standard quantum limit. This noise reduction isn’t achieved at the expense of all quantum properties; rather, the uncertainty is redistributed, minimizing it in the chosen quadrature while increasing it in its complementary partner, adhering to the fundamental Heisenberg uncertainty principle. Squeezed states, generated through techniques leveraging entangled photons, have become indispensable tools in precision measurements, enabling enhanced sensitivity in applications like gravitational wave detection, quantum communication, and advanced spectroscopy, where minimizing noise is paramount to discerning faint signals.

Seeding a nonlinear crystal with two photons demonstrates a vanishing two-photon coincidence probability at a squeezing parameter of approximately 0.88, consistent with previous findings.

Modeling Quantum Dynamics: The Two-Mode Squeezing Operator

The two-mode squeezing operator is a mathematical tool utilized to model parametric down-conversion (PDC), a nonlinear optical process where a pump photon spontaneously splits into two lower-energy photons, known as the signal and idler. This operator facilitates the description of correlations between the generated photon pairs, which are fundamental to quantum entanglement. Specifically, the operator acts on the initial vacuum state of the signal and idler modes, creating a superposition of Fock states that represent the entangled output. By transforming the creation and annihilation operators for these modes, the squeezing operator effectively mixes the quantum fluctuations, resulting in correlations that violate classical expectations and are characteristic of entangled photons produced via PDC. This framework allows for precise calculations of the quantum state, enabling the prediction and control of entanglement properties.

The two-mode squeezing operator facilitates a detailed mathematical treatment of quantum state evolution during parametric down-conversion by employing an exact Fock-basis representation. This representation allows for the complete expansion of the quantum state in terms of Fock states – eigenstates of the number operator – without approximation. Consequently, any quantum state can be fully described as a superposition of these states, each characterized by a specific number of photons in each mode. This exactness is critical for accurately predicting the statistical properties of the generated squeezed states and analyzing the correlations between photons, as it avoids the limitations inherent in perturbative or mean-field approaches. The operator’s action on these Fock states is defined by specific transformation rules that preserve the overall bosonic commutation relations, ensuring the physical validity of the resulting quantum state.

The two-mode squeezing operator’s behavior directly dictates the characteristics of generated squeezed states, allowing for the prediction of quantum state evolution and the establishment of conditions for specific outcomes. Specifically, analytical expressions like $cosh(2λ) = 1 – 2βδε^{-γ}$ are derived through manipulation of this operator, describing parameters necessary for destructive interference. In this equation, $λ$ represents the squeezing parameter, while $β$, $δ$, and $γ$ define the interaction strength, detuning, and cavity decay rate, respectively. Precise control over these parameters, facilitated by understanding the operator’s action, enables the tailoring of squeezed state properties for applications in quantum communication and computation.

Four-photon coincidence probabilities, measured with a four-crystal setup incorporating a phase shifter, demonstrate squeezing effects dependent on phase and relative crystal settings.

Observing Quantum Interference: Nonlinear Interferometers as Probes

Nonlinear interferometers are critical for the observation of quantum interference due to their ability to manipulate and amplify quantum effects related to squeezed light. Squeezed light, a non-classical state of light exhibiting noise properties below the standard quantum limit, is particularly sensitive to interference. These interferometers utilize nonlinear optical crystals to generate and interact with squeezed states, allowing for the precise control and measurement of quantum interference phenomena. The nonlinear interaction within the crystal modifies the photon statistics and correlations, making it possible to observe interference effects that are otherwise obscured by classical noise. This capability is essential for exploring fundamental aspects of quantum mechanics and developing advanced quantum technologies, including quantum communication and computation.

Nonlinear interferometers utilize nonlinear optical crystals to generate and manipulate quantum states of light. Initial configurations employed a single crystal, typically $\chi^{(2)}$ or $\chi^{(3)}$ nonlinear media, to achieve frequency doubling or parametric down-conversion for interference experiments. However, to improve signal strength, enhance specific nonlinear processes, or achieve more precise control over quantum states, researchers have moved towards multi-crystal arrangements. These setups involve two, three, or four crystals strategically aligned to cascade nonlinear interactions, compensate for dispersion, or tailor the spectral properties of the interfering photons. Increasing the number of crystals allows for greater flexibility in engineering the quantum state and optimizing the observation of interference effects, particularly in complex scenarios involving squeezed light and multi-photon correlations.

Nonlinear interferometers exhibit two distinct types of interference: dispersive and nonclassical. Dispersive interference arises from the phase accumulation of photons traversing different paths, a phenomenon explainable by classical wave optics. However, nonclassical interference, observed through the correlation functions of the output fields, violates classical predictions and is a direct consequence of the quantum nature of light. Specifically, these interferometers demonstrate four-photon interference patterns that are sensitive to the degree of squeezing – a nonclassical state where the quantum fluctuations in one quadrature are reduced at the expense of increased fluctuations in the other. Precise control over the squeezing parameters, achieved through manipulation of the nonlinear crystal properties and interferometer configuration, allows for observation and characterization of these nonclassical effects and the resulting novel interference patterns, which cannot be replicated with coherent states.

A three-crystal setup with tunable phase shifters demonstrates destructive interference of photon pairs, resulting in a coincidence minimum even with asymmetric pump powers.

Operational Regimes and the Limits of Approximation

Nonlinear interferometers exhibit markedly different behaviors contingent upon their operational regime, specifically whether they function in the low-gain or high-gain domain. Low-gain operation allows for treatment as a nearly linear system, simplifying analysis and enabling perturbative approaches to characterize the quantum state evolution. Conversely, high-gain regimes introduce significant nonlinearities, dramatically altering the interference patterns and demanding more complex theoretical frameworks for accurate modeling. This shift impacts the very nature of the observed interference; while low-gain scenarios often produce readily interpretable fringes, high-gain operation can generate intricate, multi-peak structures or even complete signal distortion. Consequently, understanding which regime governs a particular interferometer is crucial for both experimental design and the correct interpretation of results, as the underlying physics and applicable analytical tools diverge substantially between the two.

When dealing with nonlinear interferometers operating in the low-gain regime, a perturbative approach offers a powerful means of characterizing the resulting quantum state. This method relies on approximating the complex interactions within the interferometer by treating them as small deviations from a simpler, well-understood system. By expanding the relevant equations to first or second order, researchers can gain valuable insights into the state’s properties-such as its photon number distribution and coherence-without needing to solve the full, often intractable, nonlinear equations. The efficacy of perturbative analysis hinges on the assumption that these small deviations are sufficient to accurately describe the system’s behavior, making it a computationally efficient and analytically tractable alternative to more complex simulations. This technique is particularly useful for understanding subtle quantum effects and validating theoretical models in regimes where the signal is weak but still measurable.

Interferometers, particularly those operating in nonlinear regimes, are susceptible to phase cancellation – a phenomenon where constructive interference is diminished or even eliminated. This isn’t a simple absence of signal, but a complex interplay of quantum states that dramatically alters observed interference patterns. The degree of cancellation is not random; it’s fundamentally linked to the interferometer’s parameters through relationships like $γ = -2ln(cosh λ)$, where γ represents the cancellation factor and λ is related to the intensity of the light. Understanding this connection is crucial for accurate data interpretation, as even small variations in input conditions can significantly affect the observed signal due to these phase-dependent effects. Consequently, careful consideration of this cancellation, and its governing equations, is essential when designing experiments and analyzing results from nonlinear interferometers.

The derivation presented meticulously constructs a Fock representation of two-mode squeezing, a process inherently reliant on probabilistic interpretations of quantum states. This echoes a fundamental principle: data isn’t truth – it’s the tension between noise and model. The ability to precisely represent these squeezed states within the Fock basis isn’t merely a mathematical convenience; it’s a necessary condition for accurately predicting multi-photon interference. As Richard Feynman observed, “The first principle is that you must not fool yourself – and you are the easiest person to fool.” A beautiful mathematical formulation, like any model, demands rigorous validation against experimental results, lest it become a self-deception masked as insight. The presented work, therefore, represents a step toward minimizing that deception by providing a robust framework for analyzing complex quantum phenomena.

What’s Next?

The derivation of an exact Fock-basis representation, while mathematically satisfying, merely shifts the burden of approximation. One suspects the true limitation isn’t the operator itself, but the implicit assumption that ‘squeezed’ adequately describes a state inevitably born from probabilistic parametric down-conversion. A perfect squeeze, after all, implies a degree of control nature rarely grants. Future work will likely focus on quantifying the deviations from ideal squeezing – the ‘noise’ that stubbornly resists purification – and understanding how these imperfections propagate through complex interferometers.

The promise of enhanced quantum metrology hinges, as always, on the practical realization of these states. The current formalism provides a powerful tool for predicting interference patterns, but prediction is cheap. The challenge remains to demonstrate a measurable advantage in parameter estimation – a signal exceeding the standard quantum limit – when confronted with the realities of loss, detector inefficiency, and, inevitably, imperfect squeezing. One anticipates a proliferation of ‘optimized’ schemes, each claiming superiority, until a truly robust and demonstrably advantageous protocol emerges.

Ultimately, this work is a reminder that a model is a compromise between knowledge and convenience. The Fock basis, while elegant, is not the only path. Perhaps the most fruitful avenue for future research lies in exploring alternative representations – those that more naturally account for the inherent stochasticity of the process, or that are better suited for describing interactions beyond the confines of simple linear optics. The pursuit of ‘optimal’ control, it seems, is a perpetual refinement of this compromise.

Original article: https://arxiv.org/pdf/2511.16529.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

Entangled Photons: The Building Blocks of Quantum Reality

Modeling Quantum Dynamics: The Two-Mode Squeezing Operator

Observing Quantum Interference: Nonlinear Interferometers as Probes

Operational Regimes and the Limits of Approximation

What’s Next?

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