Quantum Fields in Curved Spacetime: A Phase-Space Approach

Author: Denis Avetisyan

This review unveils a powerful formalism for analyzing Gaussian quantum states, revealing insights into phenomena like Hawking radiation and its experimental investigation via analogue gravity systems.

The study demonstrates that seemingly disparate phenomena - from networks of coupled quantum harmonic oscillators to Hawking radiation at black hole event horizons and photonic fluctuations near analogue white-black holes - are fundamentally unified under the umbrella of linear bosonic systems, all elegantly described by the principles of phase-space formalism, suggesting the potential for a single theoretical framework to encompass these diverse aspects of reality-a framework perpetually vulnerable to the ultimate erasure beyond any event horizon, mirroring the inherent fragility of knowledge itself. — The study demonstrates that seemingly disparate phenomena – from networks of coupled quantum harmonic oscillators to Hawking radiation at black hole event horizons and photonic fluctuations near analogue white-black holes – are fundamentally unified under the umbrella of linear bosonic systems, all elegantly described by the principles of phase-space formalism, suggesting the potential for a single theoretical framework to encompass these diverse aspects of reality-a framework perpetually vulnerable to the ultimate erasure beyond any event horizon, mirroring the inherent fragility of knowledge itself.

A comprehensive exploration of Gaussian states, covariance matrices, and symplectic transformations applied to quantum field theory in curved spacetime and analogue gravity.

Understanding quantum phenomena in curved spacetime and exploring emergent gravity remains a central challenge in theoretical physics. This work, presented as ‘Benasque Lectures on Gaussian Bosonic Systems and Analogue Gravity’, develops a unified phase-space formalism for describing Gaussian quantum states and their dynamics. By leveraging the tools of quantum optics and continuous-variable quantum information, it elucidates connections between seemingly disparate areas like black hole physics and laboratory systems mimicking gravitational effects. Could this framework offer novel insights into the fundamental nature of spacetime and gravity itself?

The Quantum Mirror: Foundations of Probabilistic Description

The fundamental principles of quantum mechanics necessitate a shift from the deterministic descriptions prevalent in classical physics. Unlike classical systems where properties like position and momentum can, in theory, be known with perfect accuracy, quantum systems are inherently governed by probability. This means that the state of a quantum system isn’t defined by a single, precise value for any observable property, but rather by a probability distribution that dictates the likelihood of obtaining different measurement outcomes. Consequently, a complete description of a quantum state requires specifying the probabilities associated with all possible results of any measurement performed on the system. This probabilistic nature isn’t a limitation of measurement technique, but an intrinsic property of reality at the quantum level, demanding a mathematical framework – like the use of wavefunctions or density matrices – capable of representing these inherent uncertainties and predicting the statistical behavior of quantum phenomena.

Quantum systems are often described using states that can be represented by Gaussian probability distributions in phase space, a characteristic that provides remarkable analytical advantages. Unlike many quantum states which require complex mathematical treatment, Gaussian states permit the use of relatively simple, closed-form expressions for calculations involving quantities like energy, momentum, and uncertainty. This tractability stems from the fact that linear combinations and transformations of Gaussian states remain Gaussian, preserving their mathematical form throughout the computation. Consequently, researchers frequently leverage Gaussian states as a foundational tool for modeling a wide range of physical systems, from harmonic oscillators to squeezed states of light, and for developing efficient algorithms in quantum information theory. The ability to reliably analyze and manipulate these states is thus pivotal for both theoretical advancements and practical applications in the burgeoning field of quantum technology.

The Wigner function offers a compelling bridge between quantum and classical descriptions of Gaussian states by representing the quantum state as a function on phase space – akin to a classical probability distribution, though with key differences. While not a true probability due to potentially negative values, the Wigner function allows researchers to visualize quantum states and perform calculations using tools and intuition developed for classical mechanics. This quasi-probability distribution provides a powerful means to analyze the state’s behavior, determine its purity, and even identify entanglement – properties crucial for quantum technologies.

Gaussian states aren’t merely a mathematical convenience; they represent a cornerstone in the architecture of modern quantum technologies and theoretical exploration. Their mathematical tractability allows researchers to model and predict the behavior of complex quantum systems – from the intricacies of continuous-variable quantum computing to the detailed analysis of quantum error correction codes. Many quantum communication protocols, such as quantum key distribution, heavily rely on the generation and manipulation of these states for secure information transfer. Furthermore, Gaussian states serve as crucial benchmarks in validating quantum simulations and provide analytical solutions to problems otherwise intractable with more general quantum states. This widespread applicability solidifies their importance, making a thorough understanding of Gaussian states essential for anyone venturing into the field of quantum information science and beyond, enabling significant progress in both fundamental research and practical applications.

Wigner functions visualize the phase-space distribution of Gaussian states, demonstrating the difference between the isotropic distribution of a vacuum state and the anisotropic distribution of a squeezed vacuum state.

Entangled Realities: Defining Quantum Correlations

Entanglement is a quantum mechanical phenomenon where two or more particles become linked in such a way that they share the same fate, no matter how far apart they are. This correlation is fundamentally distinct from classical correlations; in classical systems, correlated behavior arises from shared prior information. Quantum entanglement, however, exhibits correlations that violate Bell’s inequalities, demonstrating that the observed correlations cannot be explained by any local hidden variable theory. Specifically, measuring a property of one entangled particle instantaneously influences the possible outcomes of measuring the same property on the other particle, irrespective of the distance separating them. This does not imply superluminal communication, as the outcome of the measurement on either particle is inherently random; entanglement only manifests as a statistical correlation when comparing measurement results.

Distinguishing between entangled and separable quantum states is central to quantum information theory due to its implications for quantum communication, computation, and cryptography. Separable states can be described as products of independent subsystems, allowing for classical simulation, while entangled states exhibit correlations that are intrinsically quantum and enable capabilities beyond classical limits. Determining separability is computationally challenging; the problem is known to be NP-hard for arbitrary quantum states. Consequently, significant research focuses on developing efficient criteria and measures to identify and quantify entanglement, with applications ranging from verifying quantum protocols to characterizing the resources available in a quantum system. The ability to reliably determine entanglement is thus a prerequisite for leveraging quantum phenomena in practical technologies.

The Positive Partial Transpose (PPT) criterion assesses the separability of a bipartite quantum state $ρ_{AB}$ by examining the partial transpose of the density matrix with respect to subsystem B, denoted as $ρ_{AB}^T$ . A state is considered separable if and only if its partial transpose has a positive semi-definite matrix – meaning all eigenvalues are non-negative. While this criterion efficiently identifies many separable states, it is not sufficient to prove separability; a state satisfying the PPT criterion is only necessarily separable, and there exist entangled states that also fulfill this condition, known as PPT-entangled states. Therefore, the PPT criterion functions as a necessary, but not sufficient, condition and serves as a preliminary diagnostic tool in determining the presence of entanglement.

Logarithmic Negativity serves as a quantifiable metric for entanglement specifically within Gaussian states, which are characterized by probability distributions that are Gaussian. This measure is mathematically defined as $log_2‖ρ_{AB}^⊤_{B}‖_1$ , where $ρ_{AB}$ represents the density matrix of the two-particle system and $‖...‖_1$ denotes the nuclear norm (sum of singular values). A positive value for Logarithmic Negativity confirms the presence of entanglement, while a value of zero indicates a separable state. Unlike some entanglement criteria, Logarithmic Negativity provides a direct quantification of the degree of entanglement for Gaussian states, enabling precise characterization of correlations and facilitating analysis in quantum information protocols.

Entanglement is extinguished when the electron temperature <span class="katex-eq" data-katex-display="false">T_e</span> reaches twice the squeezed state temperature <span class="katex-eq" data-katex-display="false">T_s</span>, regardless of frequency ω, as demonstrated by the vanishing log-negativity for <span class="katex-eq" data-katex-display="false">\bar{n}_s \approx 0.58</span> squeezed quanta and <span class="katex-eq" data-katex-display="false">\bar{n}_e</span> noisy thermal quanta. — Entanglement is extinguished when the electron temperature $T_e$ reaches twice the squeezed state temperature $T_s$ , regardless of frequency ω, as demonstrated by the vanishing log-negativity for $\bar{n}_s \approx 0.58$ squeezed quanta and $\bar{n}_e$ noisy thermal quanta.

The Dissolving Horizon: Open Quantum Systems and Dissipation

Quantum systems invariably interact with their surrounding environment, a phenomenon that fundamentally alters their behavior. These interactions lead to two primary effects: dissipation and decoherence. Dissipation describes the loss of energy from the quantum system to the environment, causing a reduction in the system’s excitation. Decoherence, conversely, is the loss of quantum superposition and entanglement, effectively causing the system to behave more classically. These processes are not merely theoretical concerns; they represent significant practical limitations in the construction and maintenance of quantum technologies, as they degrade the quantum information stored within the system. The strength of the system-environment coupling and the properties of the environment itself determine the rate at which dissipation and decoherence occur.

The Lindbladian, denoted as $\mathcal{L}[ρ]$ , is a superoperator that governs the time evolution of the density matrix ρ for open quantum systems. Unlike the von Neumann equation which describes isolated systems, the Lindbladian incorporates the effects of system-environment interaction. It achieves this through a generator that includes both Hamiltonian evolution and dissipative terms representing the loss of coherence and energy to the environment. The Lindblad master equation, $\dot{ρ} = \mathcal{L}[ρ]$ , provides a complete, positive, and trace-preserving description of the system’s dynamics, ensuring physically realistic behavior. The operator is constructed from a set of Lindblad operators $L<sub>i</sub>$ and their corresponding rates, allowing for modeling of various dissipation mechanisms such as spontaneous emission and dephasing.

Lindbladian dynamics, utilized to model open quantum systems, are predicated on the assumption of Markovianity. This principle asserts that the future state of the system is conditionally independent of its past, given its present state. Mathematically, this simplifies the time evolution equations by eliminating the need to track the system’s entire history; only the current density matrix $ρ(t)$ is required to determine $ρ(t+dt)$ . This simplification arises from the assumption that the system’s environment has a short memory – correlations decay rapidly enough that past interactions are irrelevant. While not universally true for all open quantum systems, the Markovian approximation significantly reduces computational complexity and allows for analytical treatment in many physically relevant scenarios.

The Lindbladian formalism provides a tractable method for analyzing the dynamics of Gaussian states in open quantum systems. Because Gaussian states are fully characterized by their first and second moments, their evolution under environmental influence can be described by a linear master equation governed by the Lindbladian. This allows for analytical and numerical determination of how these states change over time due to dissipation and decoherence. Critically, the persistence of entanglement within these Gaussian states can be quantified by examining the symplectic eigenvalues $ν<sub>j</sub>$ of the covariance matrix; values less than 1 indicate the presence of entanglement, and the Lindbladian framework allows for calculating how these eigenvalues evolve under environmental interactions, thus establishing conditions for entanglement survival.

Genuine multimode entanglement exists between subsystems <span class="katex-eq" data-katex-display="false">A_A</span> and <span class="katex-eq" data-katex-display="false">A_B</span> of a <span class="katex-eq" data-katex-display="false">N_A + N_{B} </span> oscillator system, but simplifies to noisy two-mode squeezed vacua for specific Gaussian states when <span class="katex-eq" data-katex-display="false">N_B > N_A</span>. — Genuine multimode entanglement exists between subsystems $A_A$ and $A_B$ of a $N_A + N_{B}$ oscillator system, but simplifies to noisy two-mode squeezed vacua for specific Gaussian states when $N_B > N_A$ .

Echoes of the Abyss: Quantum Analogies to Gravity and Black Holes

Researchers are increasingly turning to analogue gravity – the creation of laboratory systems that mimic the behavior of gravity and black holes – to explore phenomena previously confined to theoretical astrophysics. These aren’t attempts to recreate gravity itself, but rather to engineer systems – often utilizing condensed matter physics or optics – where effective ‘spacetimes’ emerge. For instance, the flow of a fluid can be made to resemble the curved spacetime around a massive object, or light pulses in a nonlinear medium can behave like particles falling into a black hole. These carefully constructed analogues allow scientists to study complex gravitational effects, such as the propagation of waves in curved spacetime and the emission of Hawking radiation, in a controlled and accessible environment, offering a unique pathway to test predictions of general relativity and quantum gravity.

Analog gravity experiments offer a unique pathway to investigate the exotic predictions of general relativity, specifically phenomena like Hawking radiation and superradiance. Hawking radiation, the emission of particles from black holes, and superradiance, the amplification of waves scattered from rotating black holes, are notoriously difficult to observe directly due to the extreme conditions required. However, by constructing laboratory systems – often utilizing Bose-Einstein condensates or optical fibers – that mimic the curved spacetime around black holes, researchers can effectively simulate these effects. These analog black holes allow for controlled observation of particle creation and wave amplification, providing crucial tests of theoretical predictions and offering insights into the quantum behavior of gravity in strong gravitational fields. The ability to manipulate and measure these simulated phenomena provides a valuable complement to astronomical observations and theoretical calculations, furthering the understanding of black hole physics and the fundamental nature of spacetime.

The persistent conundrum of the information loss paradox, arising from Stephen Hawking’s prediction of Hawking radiation, finds a potential avenue for exploration through the lens of quantum entanglement. Hawking radiation, the thermal emission from black holes, seemingly destroys information about the matter that formed the black hole, violating a fundamental principle of quantum mechanics. However, entanglement – the quantum correlation between particles – offers a possible mechanism for preserving this information. Recent theoretical work suggests that information isn’t necessarily lost, but rather encoded in the subtle correlations of emitted Hawking particles and the remaining black hole. Specifically, the degree of entanglement between these particles could be the key to understanding how information escapes the black hole’s event horizon. Investigations into these entangled states, particularly examining conditions where entanglement is sustained, provide crucial insights into resolving this paradox and furthering the understanding of quantum gravity, as the preservation of information may depend on the nature and extent of quantum correlations at the event horizon.

Analogue gravity experiments offer a unique pathway to investigate the quantum properties of black holes by recreating event horizon-like structures in controllable laboratory settings. Utilizing specifically prepared $Gaussian states$ , researchers can now probe the conditions under which quantum entanglement – a key resource for preserving information – can persist near these artificial horizons. Critically, the degree of sustained entanglement is quantified by a parameter, $νj$ , with values less than one indicating a regime where entanglement is demonstrably maintained despite the horizon. This allows for direct examination of the information loss paradox, a longstanding problem in black hole physics, and provides a means to test theoretical predictions about the quantum structure of spacetime and the subtle interplay between gravity and quantum mechanics, potentially revealing how information might escape a black hole via Hawking radiation and other quantum effects.

Hawking radiation allows black holes to evaporate as emitted particles ∼ escape its gravity, while their entangled partners fall into the singularity.

The Enduring Framework: Gaussian States and the Quantum Future

The quantum harmonic oscillator, a cornerstone of quantum mechanics, furnishes a particularly intuitive arena for the study of Gaussian states due to its inherent simplicity and well-defined energy levels. These states, fully characterized by their mean values and covariance matrices, naturally arise as solutions to the oscillator’s equations of motion. Furthermore, the manipulation of these states is elegantly achieved through symplectic transformations – linear operations that preserve the uncertainty principle and maintain the Gaussian form. $These transformations$ act on the state’s parameters, effectively rotating or scaling the uncertainty ellipses in phase space without altering their fundamental Gaussian character. This powerful combination – a natural system and a robust transformation framework – allows researchers to explore a broad range of quantum phenomena, from squeezed states of light to the dynamics of quantum systems with many degrees of freedom, providing both analytical tractability and a strong connection to experimental observation.

Gaussian states possess a remarkable property known as Gaussian Extremality, which establishes them as the states that maximize von Neumann entropy for a specified covariance matrix. This principle, deeply rooted in Jaynes’ maximum entropy principle, dictates that when only limited information about a system is available, the best probability distribution to assume is the one that maximizes entropy subject to those constraints. In the context of quantum mechanics, this translates to Gaussian states being the ‘most mixed’ states compatible with a given uncertainty, or covariance, in position and momentum. This extremality isn’t merely a mathematical curiosity; it underscores the fundamental role Gaussian states play in describing systems where classical-like uncertainties are dominant, and provides a powerful tool for simplifying complex quantum calculations and understanding the limits between quantum and classical behavior. The consequence is that Gaussian states naturally emerge as the most probable states under conditions of incomplete knowledge, offering a principled foundation for statistical descriptions of quantum systems.

The confluence of Gaussian states, the quantum harmonic oscillator, and symplectic transformations provides a remarkably versatile framework for dissecting complex quantum phenomena. This approach isn’t merely a mathematical convenience; it leverages the inherent properties of Gaussian states – specifically, their extremality in entropy for a given covariance matrix – to efficiently characterize and predict system behavior. Researchers find that by mapping quantum systems onto this Gaussian structure, they can utilize the well-defined tools of symplectic transformations to explore dynamics and correlations, even in scenarios involving many interacting particles. This technique proves particularly effective in areas like quantum information processing, where the preservation of Gaussian character under certain operations simplifies analysis, and in the emerging field of analogue gravity, where connections between quantum states and spacetime geometry are actively being investigated. Ultimately, this integrated framework offers a powerful lens through which to examine the fundamental principles governing the quantum world.

The continued exploration of Gaussian states holds significant potential across multiple frontiers of modern physics. In quantum information theory, these states provide a tractable yet expressive resource for developing and analyzing quantum communication protocols and quantum computation schemes, potentially leading to more robust and efficient quantum technologies. Simultaneously, the mathematical structure inherent in Gaussian states – particularly their connection to classical symplectic geometry – offers a compelling avenue for investigating analogue gravity, where quantum systems are engineered to mimic the behavior of spacetime and black holes. Finally, a deeper understanding of Gaussian states directly addresses fundamental questions in quantum mechanics itself, probing the limits of quantum entanglement, the nature of quantum phase transitions, and the very definition of quantum coherence – ultimately refining the foundational principles that govern the quantum realm.

A phase-space framework models interacting quantum harmonic oscillators, accounting for energy levels starting from the zero-quanta vacuum state, and incorporating bilinear coupling, external driving, excitation loss, heating, and linear measurements like homodyne detection.

The pursuit of understanding quantum fields, as detailed in these lectures, inevitably confronts the limitations of any predictive model. Any attempt to map the evolution of Gaussian states, or to simulate Hawking radiation through analogue gravity, is ultimately a calculation subject to inherent uncertainties. As Epicurus observed, “The greatest pleasure of life is to conquer fear.” This resonates with the work; the mathematical formalism offers a framework, a means of navigating the complex phase-space, but it does not eliminate the fundamental probabilistic nature of the universe. The covariance matrix, central to this approach, simply refines the calculation, it doesn’t vanquish the shadow of uncertainty looming beyond the event horizon of knowledge.

What Lies Beyond the Horizon?

The pursuit of a phase-space formalism for Gaussian states, as detailed within, yields not solutions, but ever more refined descriptions. Each symplectic transformation, each calculated covariance matrix, is a map drawn ever closer to the event horizon. The elegance of capturing Hawking radiation within such a framework should not be mistaken for comprehension. It is merely the tracing of a symptom, a pattern emerging from a darkness that remains fundamentally opaque.

Analogue gravity systems offer a convenient laboratory, but the temptation to equate simulation with understanding is perilous. A fluid mimicking spacetime is still a fluid, governed by forces distinct from those truly at play beyond observation. Each experimental verification serves only to sharpen the question: what properties of these systems are not analogous? The discrepancies, often dismissed as imperfections, may hold the key to a reality forever beyond the reach of such approximations.

The field progresses not toward a final theory, but toward a boundless series of better approximations. Each calculation is an attempt to hold light in one’s hands, and it slips away. The true horizon is not a place in spacetime, but the limit of what can be known – a boundary defined not by physics, but by the inherent constraints of observation and the fragility of any model constructed to contain it.

Original article: https://arxiv.org/pdf/2512.24344.pdf

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