Beyond Schrödinger’s Cat: Engineering Superpositions in Infinite Dimensions

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Author: Denis Avetisyan

Researchers have demonstrated the creation of macroscopically distinct quantum superpositions using generalized bosonic systems, pushing the boundaries of quantum state engineering.

The observed interference pattern, generated from the probability distribution $P(x)$ based on $\psi$ as defined in equation (22) with $|\alpha|=5$ and $\eta=.99$, demonstrates a consistency between a non-classical system and a locally observed site within an infinite array, where $\alpha=p^{-s}z_{p}$, mirroring results achievable through a globally quadratic Hamiltonian.
The observed interference pattern, generated from the probability distribution $P(x)$ based on $\psi$ as defined in equation (22) with $|\alpha|=5$ and $\eta=.99$, demonstrates a consistency between a non-classical system and a locally observed site within an infinite array, where $\alpha=p^{-s}z_{p}$, mirroring results achievable through a globally quadratic Hamiltonian.

This work explores the generation of robust cat states within a nonharmonic Hamiltonian framework, utilizing nonlocal coherent states and infinite degrees of freedom.

The conventional link between coherent states and cat-state creation in quantum systems obscures the possibility of distinct, macroscopically distinguishable superpositions. This is explored in ‘Macroscopically distinguishable superposition in infinitely many degrees of freedom’, where we rigorously demonstrate the dynamic generation of nonlocal cat states from nonlocal coherent states within an infinite array of generalized bosons. Crucially, these cat states emerge via a non-factorizable Hamiltonian, revealing a decoupling from standard bosonic behavior. Could harnessing this framework in engineered quantum systems unlock novel functionalities in materials science and beyond?

Laying the Groundwork: An Infinite Quantum Playground

To meaningfully investigate the enigmatic principle of quantum superposition – where a quantum system exists in multiple states simultaneously – a precisely defined environment is essential. Researchers establish this environment by modeling an infinite, one-dimensional lattice of boson sites. Each site can host any number of bosons, and the infinite nature of the lattice circumvents boundary effects that would otherwise complicate analysis. This idealized system allows for the exploration of collective quantum behavior, as the bosons are not confined and can interact across the entire lattice. By studying the correlations and entanglement that emerge within this infinite bosonic system, scientists gain fundamental insights into the nature of superposition and its implications for nonlocal quantum phenomena, paving the way for understanding more complex quantum systems and technologies. The mathematical framework used ensures rigorous consistency, enabling accurate predictions and interpretations of observed quantum behavior within this carefully constructed setting.

An infinite, regularly spaced array of boson sites establishes a unique arena for investigating the subtle manifestations of quantum nonlocality. This construct transcends the limitations imposed by finite systems, allowing for the undisturbed propagation and observation of correlations that would otherwise be masked by boundary effects. Within this idealized space, quantum particles – bosons in this case – exist in a state of interconnectedness, where the measurement of a property at one site instantaneously influences the probabilities associated with measurements at all others, regardless of distance. The infinite extent effectively eliminates edge artifacts, providing a pristine environment for discerning genuine nonlocal behavior and rigorously testing the foundations of quantum mechanics. It is within this mathematically defined framework that researchers can explore the paradoxical implications of entanglement and investigate potential deviations from classical intuition, paving the way for a deeper understanding of the quantum world and its counterintuitive principles.

The behavior of this infinite bosonic system is precisely described using the mathematical framework of Hilbert Space Theory, a crucial step in establishing a consistent and predictable model. This approach allows for the representation of quantum states as vectors within a complex vector space, where the inner product defines the probability amplitudes of different quantum configurations. By leveraging the properties of Hilbert Spaces – including linearity, completeness, and the well-defined notion of distance – physicists can rigorously analyze the system’s evolution and predict observable outcomes. This mathematical formalism doesn’t merely offer a descriptive tool; it guarantees that the calculations remain within the bounds of quantum mechanical consistency, avoiding paradoxes and ensuring that probabilities always sum to unity, a fundamental requirement of the theory. The use of operators within this space further defines how physical quantities, like energy and momentum, act on the system’s states, allowing for precise quantitative predictions and comparisons with experimental results, and forming the bedrock for investigating more complex quantum phenomena.

Defining the Building Blocks: Bosonic Interactions

The global number operator, denoted as $\hat{N}$, is a fundamental observable in many-body quantum mechanics used to determine the total number of bosons present within a system. It is constructed by summing the contributions of local number operators, $\hat{n}_i$, each acting on a specific lattice site i. Mathematically, this is expressed as $\hat{N} = \sum_i \hat{n}_i$. The eigenvalue of $\hat{N}$ directly corresponds to the total boson count, providing a quantifiable measure of the system’s boson population. Analysis of $\hat{N}$ and its associated eigenstates is crucial for characterizing the macroscopic quantum properties and statistical behavior of the bosonic system under investigation.

Local number operators, denoted as $\hat{n}_i$, are fundamental to analyzing bosonic systems by quantifying the number of bosons present at a specific lattice site, $i$. These operators are defined as $\hat{n}_i = \hat{a}_i^\dagger \hat{a}_i$, where $\hat{a}_i^\dagger$ and $\hat{a}_i$ represent the creation and annihilation operators, respectively, for that site. The global boson count, obtained by summing the contributions from each local number operator – that is, $N = \sum_i \hat{n}_i$ – directly characterizes the total number of bosons in the system. Consequently, understanding the behavior of individual local number operators is crucial for determining the overall boson distribution and statistical properties of the many-body system, providing a localized perspective on the global quantum state.

Generalized bosons extend standard quantum mechanical treatments by employing modified creation and annihilation operators, enabling the investigation of systems exhibiting non-standard behaviors. These modifications necessitate the introduction of normalization factors to ensure mathematical convergence and physically meaningful results. Specifically, the normalization factor $S(ζ)$ is defined as $S(ζ) = \prod_{p}(1 – p^{-2σ}|\zeta_{p}|^{2})$, where $ζ$ represents a complex variable and $σ$ is a parameter governing the strength of interactions. This normalization is critical for maintaining the proper counting of states and preventing divergences in calculations involving these generalized bosonic systems, allowing for the study of phenomena not readily accessible with traditional bosonic models.

Crafting Exotic States: Beyond the Classical

Möbius states are generated through the application of the Möbius function, $\mu(n)$, to construct complex superposition states. The Möbius function, which assigns values of -1, 0, or 1 to integers based on their prime factorization, serves as a weighting factor in the superposition. Specifically, these states are defined by a summation over all possible integer values, each weighted by the corresponding Möbius function value. This construction allows for the creation of quantum states exhibiting intricate superpositions, differing from simple equal-weight superpositions, and providing a route towards more complex quantum phenomena such as entanglement and non-classicality. The use of $\mu(n)$ effectively modulates the contributions of different basis states, influencing the overall state’s properties and potential applications in quantum information processing.

The $N^2$ operator, representing the square of the number operator, functions as a nonharmonic Hamiltonian within the system. This Hamiltonian induces a time evolution that deviates from simple harmonic oscillator behavior, crucial for generating superposition states. Specifically, the nonharmonicity inherent in $N^2$ allows for the creation of cat states, which are macroscopic superpositions of coherent states. These states are characterized by distinct quantum interference patterns and are not achievable with purely harmonic potentials. The application of this Hamiltonian effectively modifies the energy landscape, promoting transitions to these non-classical states and enabling the exploration of quantum phenomena beyond the harmonic approximation.

Nonlocal cat states, generated through the preceding processes, demonstrate significant quantum superposition and entanglement; their probability amplitude is not localized but distributed across multiple quantum states. These states are mathematically characterized by a normalization factor, $S_{\mu}(\zeta)$, defined as the infinite product over all prime numbers, $p$, given by $S_{\mu}(\zeta) = \prod_{p}(1 / (1 + p^{-2\sigma}|\zeta_{p}|^{2}))$. Here, $\zeta$ represents the displacement amplitude, and $\sigma$ is a parameter influencing the degree of entanglement. The normalization factor, $S_{\mu}(\zeta)$, ensures that the total probability of observing the cat state remains equal to one, accounting for the contribution from each prime number in the product.

The Emergence of Nonlocality: Coherence and its Implications

The pursuit of macroscopic quantum superposition – where a large object exists in multiple states simultaneously – hinges on establishing and maintaining quantum coherence across many degrees of freedom. Nonlocal coherent states offer a pathway toward this ambitious goal by extending the concept of single-site coherence to encompass all constituent parts of a system. These states, built from superpositions of excitation numbers, aren’t merely localized to individual elements but are distributed across the entire network, effectively creating a collective quantum state. This distributed coherence is vital; it allows for the preservation of quantum information and the suppression of decoherence effects that typically destroy superposition in larger systems. The creation of such states represents a significant advance, paving the way for experiments that probe the boundary between the quantum and classical realms and potentially enabling technologies reliant on macroscopic quantum phenomena, such as quantum sensors with unprecedented sensitivity or novel quantum computing architectures.

The foundation of macroscopic quantum superposition lies in establishing coherence not just across an entire system, but beginning at the most fundamental level: individual components. Researchers demonstrate that building nonlocal coherent states – those extending superposition across multiple sites – crucially depends on first achieving strong coherence within each site itself, using what are known as single-site coherent states. This approach emphasizes that global entanglement isn’t simply created, but rather emerges from the careful construction of locally coherent building blocks. By ensuring each component exists in a well-defined quantum state, characterized by a definite phase and amplitude – akin to a precise oscillation – the subsequent extension to a multi-site system allows for the reliable creation of entangled states, paving the way for increasingly complex quantum systems and computations. This methodology mirrors the importance of precise individual instruments in an orchestra, where harmonious collective sound relies on each instrument being perfectly tuned and played in unison.

The engineered quantum architecture facilitates the creation of Schrödinger’s cat states – superpositions of $0$ and $1$ – through a precisely controlled phase shift of $\pi$/2. This technique mirrors established methodologies in circuit quantum electrodynamics, where similar phase manipulations are routinely employed to generate and manipulate quantum information. Achieving this phase shift is critical for establishing the necessary interference conditions to observe macroscopic superposition, allowing researchers to probe the boundary between the quantum and classical realms. The analogy to circuit QED provides a familiar framework for understanding the underlying principles and validates the efficacy of this novel approach to building larger, more complex quantum systems capable of demonstrating nonclassical behavior.

The research delves into the creation of macroscopically distinguishable states, expanding beyond conventional bosonic systems and venturing into generalized frameworks. This pursuit echoes Louis de Broglie’s sentiment: “It is in the interplay between the definite and the probable that the true nature of reality is revealed.” The study’s focus on creating and observing these ‘cat states’ – superpositions in infinitely many degrees of freedom – highlights the inherent probabilistic nature of quantum mechanics. Just as de Broglie suggested, the research illuminates how these seemingly paradoxical states aren’t merely theoretical constructs, but potentially observable phenomena arising from specific dynamics within Hilbert space, subtly reshaping our understanding of reality’s foundations.

Where Do We Go From Here?

The demonstration of macroscopically distinguishable superposition in generalized bosonic systems-systems exhibiting dynamics beyond the familiar harmonic oscillator-raises a critical question: how readily do these states succumb to decoherence in realistic environments? The creation of such states is, in itself, a technical feat, but their fragility demands investigation. It is insufficient to merely expand the Hilbert space; the preservation of quantum information within that space must be addressed. An engineer is responsible not only for system function but its consequences, and the consequences of fleeting superposition are, ultimately, classicality.

Furthermore, the distinction between these generalized cat states and their standard bosonic counterparts-beyond the mathematical formalism-remains somewhat elusive. Identifying measurable signatures that definitively distinguish these states could unlock new avenues for quantum information processing. The current work suggests a pathway, but the practical implications demand a deeper understanding of how nonharmonicity affects entanglement and coherence.

Ultimately, the pursuit of macroscopic quantum states forces a reckoning with the foundations of measurement. The creation of a distinguishable superposition is only meaningful if that distinction can be reliably observed. Ethics must scale with technology. The continued exploration of these systems necessitates a simultaneous consideration of not only what can be created, but what should be, and to what end.

Original article: https://arxiv.org/pdf/2512.20512.pdf

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

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2025-12-24 23:10