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AVS Quantum Science

The logarithmic phase singularity in the inverted harmonic oscillator

Freyja Ullinger, Matthias Zimmermann, Wolfgang P. Schleich

Published Apr 7, 2022• 18 citations
Publisher

Abstract

A spacetime singularity is located at the center of a black hole and surrounded by an event horizon, separating spacetime into two disjunct regions: one of them accessible to an outside observer and one that is not. At the event horizon, a logarithmic phase singularity emerges in the mode functions of a massless scalar field, being characteristic for Hawking radiation emitted by the black hole. We demonstrate that related features are present in the elementary quantum system of an inverted harmonic oscillator. Central to our analysis are the energy eigenfunctions of this system and their phase space representation. At first glance, neither a horizon nor a logarithmic phase dependence are apparent. However, both features are hidden in phase space and revealed by a suitable coordinate transformation. In particular, we show that the Fourier transform of a logarithmic phase leads to an expression that is reminiscent of a specific quantum statistics, governing the reflection and transmission coefficients of the inverted harmonic oscillator.

DOI
10.1116/5.0074429

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A spacetime singularity is located at the center of a black hole and surrounded by an event horizon, separating spacetime into two disjunct regions: one of them accessible to an outside observer and one that is not. At the event horizon, a logarithmic phase singularity emerges in the mode functions of a massless scalar field, being characteristic for Hawking radiation emitted by the black hole. We demonstrate that related features are present in the elementary quantum system of an inverted harmonic oscillator. Central to our analysis are the energy eigenfunctions of this system and their phase space representation. At first glance, neither a horizon nor a logarithmic phase dependence are apparent. However, both features are hidden in phase space and revealed by a suitable coordinate transformation. In particular, we show that the Fourier transform of a logarithmic phase leads to an expression that is reminiscent of a specific quantum statistics, governing the reflection and transmission coefficients of the inverted harmonic oscillator.
Abstract In 1974, Stephen Hawking predicted that quantum effects in the proximity of a black hole lead to the emission of particles and black hole evaporation. At the very heart of this process lies a logarithmic phase singularity which leads to the Bose-Einstein statistics of Hawking radiation. An identical singularity appears in the elementary quantum system of the inverted harmonic oscillator. In this Letter we report the observation of the onset of this logarithmic phase singularity emerging at a horizon in phase space and giving rise to a Fermi-Dirac distribution. For this purpose, we utilize surface gravity water waves and freely propagate an appropriately tailored energy wave function of the inverted harmonic oscillator to reveal the phase space horizon and the intrinsic singularities. Due to the presence of an amplitude singularity in this system, the analogous quantities display a Fermi-Dirac rather than a Bose-Einstein distribution.
We investigate a quantum massive scalar field in the interior of a charged and spherically-symmetric (Reissner-Nordström) black hole. We examine the behaviour for varying values of the black hole charge, field mass and coupling constant when the field is in two quantum states: Hartle-Hawking (representing a black hole in thermal equilibrium) and Unruh (representing a black hole evaporating via the emission of Hawking radiation). We show that the vacuum polarization as well as the angular components of the quantum stress-energy tensor diverge on the Cauchy horizon, in stark contrast to what happens for massless fields. We also calculate the energy fluxes in Eddington-Finkelstein coordinates {u,v}\{u,v\}{u,v}. We show that these fluxes on the Cauchy horizon do not generically vanish. This implies, in particular, that in regular, Kruskal coordinates {U,V}\{U,V\}{U,V} the ingoing flux diverges like V−2V^{-2}V−2 on the Cauchy horizon (where V=0V=0V=0). This divergence suggests that its backreaction via the semiclassical Einstein equations would yield a strong singularity, as opposed to its weaker, classical counterpart. Interestingly, there are exceptions, in which the energy fluxes in Eddington-Finkelstein coordinates vanish: (i) in the extremal limit (where the black hole is maximally charged); (ii) certain fine-tuned regions of parameter space, where the fluxes change sign.
We develop a systematic framework for the quantum and thermal properties of a Klein-Gordon scalar field subject to an inverted harmonic potential −12m2ω2x2-{1\over2} m^2ω^2 x^2−21​m2ω2x2. Starting from a non-Hermitian momentum substitution P→P−mωxP \to P - mωxP→P−mωx, we employ a symplectic phase-space rotation V=exp⁡ ⁣[−π8(xp+px)]V = \exp\!\left[-\tfracπ{8}(xp+px)\right]V=exp[−8π​(xp+px)] to map the system onto an analytically tractable effective harmonic oscillator evaluated at xeiπ/4xe^{iπ/4}xeiπ/4. This allows us to define a well-regulated partition function Z(β,ω,m)Z(β,ω,m)Z(β,ω,m) and derive closed-form expressions for the free energy, entropy, and thermal correlation functions. We then apply this framework to three physical settings: (i) scalar field fluctuations during cosmological inflation, (ii) quantum fields near black-hole horizons, and (iii) order-parameter dynamics near second-order phase transitions in condensed matter. Our results unify previously scattered results in the literature and provide new predictions for the finite-temperature spectral density and entanglement entropy of unstable quantum systems.
Abstract The merging of quantum information science with the relativity theory presents novel opportunities for understanding the enigmas surrounding the transmission of information in relation to black holes. For this purpose, we study the quantumness near a Schwarzschild black hole in a practical model under decoherence. The scenario we consider in this paper is that a stationary particle in the flat region interacts with its surroundings while another particle experiences free fall in the vicinity of a Schwarzschild black hole’s event horizon. We explore the impacts of Hawking radiation and decoherence on the system under investigation and find that these effects can limit the survival of quantum characteristics, but cannot destroy them completely. Hence, the results of this study possess the potential to yield valuable insights into the comprehension of the quantum properties of a real system operating within a curved space-time framework.
Hawking radiation is the proposed thermal black-body radiation of quantum nature emitted from a black hole. One common way to give an account of Hawking radiation is to consider a detector that follows a static trajectory in the vicinity of a black hole and interacts with the quantum field of the radiation. In the present work, we study the Hawking radiation perceived by a detector that follows a quantum superposition of static trajectories in Schwarzschild spacetime, instead of a unique well-defined trajectory. We analyze the quantum state of the detector after the interaction with a massless real scalar field. We find that for certain trajectories and excitation levels, there are nonvanishing coherences in the final state of the detector. We then examine the dependence of these coherences on the trajectories followed by the detector and relate them to the distinguishability of the different possible states in which the field is left after the excitation of the detector. We interpret our results in terms of the spatial distribution and propagation of particles of the quantum field.
We consider a stationary quantum system consisting of two non-interacting yet entangled subsystems, ΞΞΞ and ΓΓΓ. We identify a quantum theory characterizing ΞΞΞ such that, in the quantum-to-classical crossover of the composite system, ΓΓΓ behaves as a test particle within the gravitational field of a Schwarzschild Black Hole (SBH) near its event horizon. We then show that this same quantum theory naturally provides a representation of ΞΞΞ in terms of bosonic modes, whose features match those of the Hawking radiation; this facilitates the establishment of precise relations between the phenomenological parameters of the SBH and the microscopic details of the quantum model for ΞΞΞ. Finally, we recognize that the conditions used to characterize ΓΓΓ and ΞΞΞ coincide with those required by the Page and Wootters mechanism for identifying an evolving system and an associated clock. This leads us to discuss how the quantum model for ΞΞΞ endows the SBH with all the characteristics of a "perfect" clock.