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This report presents a selfcontained derivation of Hawking radiation, and discusses the consequent information loss paradox.

The black hole information paradox arises when quantum mechanical effects are considered in the vicinity of the event horizon of a black hole. In this report we describe the fundamental properties of quantum mechanical systems and black holes that lead to the information paradox, with focus on quantum entanglement. While first presented in 1976, the information paradox is as of yet an unsolved problem. Two of the proposed solutions, black hole complementarity and firewalls, are discussed. / Svarta hålets informationsparadox uppkommer när man tar hänsyn till kvantmekaniska effekter i närheten av händelsehorisonten av ett svart hål. I denna rapport beskrivs de grundläggande egenskaper hos kvantmekaniska system och svarta hål som leder till informationsparadoxen, med fokus på kvantintrassling. Paradoxen, som presenterades 1976 men än idag är ett olöst problem, förklaras sedan. Två av de förslagna lösningarna till paradoxen, svarta hål-komplementaritet och firewalls, diskuteras.

In this thesis we investigate two different aspects of the AdS/CFT correspondence. We first investigate the holographic AdS/CMT correspondence. Gravitational backgrounds in d+2 dimensions have been proposed as holographic duals to Lifshitz-like theories describing critical phenomena in d+1 dimensions with critical exponent z>1. We numerically explore a dilaton-Einstein-Maxwell model admitting such backgrounds as solutions. We show how to embed these solutions into AdS space for a range of values of z and d. We next investigate the AdS3/CFT2 correspondence and focus on the microscopic CFT description of the D1-D5 system on T^4*S_1. In the context of the fuzzball programme, we investigate deforming the CFT away from the orbifold point and study lifting of the low-lying string states. We start by considering general 2D orbifold CFTs of the form M^N/S_N, with M a target space manifold and S_N the symmetric group. The Lunin-Mathur covering space technique provides a way to compute correlators in these orbifold theories, and we generalize this technique in two ways. First, we consider excitations of twist operators by modes of fields that are not twisted by that operator, and show how to account for these excitations when computing correlation functions in the covering space. Second, we consider non-twist sector operators and show how to include the effects of these insertions in the covering space. Using the generalization of the Lunin-Mathur symmetric orbifold technology and conformal perturbation theory, we initiate a program to compute the anomalous dimensions of low-lying string states in the D1-D5 superconformal field theory. Our method entails finding four-point functions involving a string operator O of interest and the deformation operator, taking coincidence limits to identify which other operators mix with O, subtracting conformal families of these operators, and computing their mixing coefficients. We find evidence of operator mixing at first order in the deformation parameter, which means that the string state acquires an anomalous dimension. After diagonalization this will mean that anomalous dimensions of some string states in the D1-D5 SCFT must decrease away from the orbifold point while others increase. Finally, we summarize our results and discuss some future directions of research.

In this thesis we investigate two different aspects of the AdS/CFT correspondence. We first investigate the holographic AdS/CMT correspondence. Gravitational backgrounds in d+2 dimensions have been proposed as holographic duals to Lifshitz-like theories describing critical phenomena in d+1 dimensions with critical exponent z>1. We numerically explore a dilaton-Einstein-Maxwell model admitting such backgrounds as solutions. We show how to embed these solutions into AdS space for a range of values of z and d. We next investigate the AdS3/CFT2 correspondence and focus on the microscopic CFT description of the D1-D5 system on T^4*S_1. In the context of the fuzzball programme, we investigate deforming the CFT away from the orbifold point and study lifting of the low-lying string states. We start by considering general 2D orbifold CFTs of the form M^N/S_N, with M a target space manifold and S_N the symmetric group. The Lunin-Mathur covering space technique provides a way to compute correlators in these orbifold theories, and we generalize this technique in two ways. First, we consider excitations of twist operators by modes of fields that are not twisted by that operator, and show how to account for these excitations when computing correlation functions in the covering space. Second, we consider non-twist sector operators and show how to include the effects of these insertions in the covering space. Using the generalization of the Lunin-Mathur symmetric orbifold technology and conformal perturbation theory, we initiate a program to compute the anomalous dimensions of low-lying string states in the D1-D5 superconformal field theory. Our method entails finding four-point functions involving a string operator O of interest and the deformation operator, taking coincidence limits to identify which other operators mix with O, subtracting conformal families of these operators, and computing their mixing coefficients. We find evidence of operator mixing at first order in the deformation parameter, which means that the string state acquires an anomalous dimension. After diagonalization this will mean that anomalous dimensions of some string states in the D1-D5 SCFT must decrease away from the orbifold point while others increase. Finally, we summarize our results and discuss some future directions of research.

We explore the optical parametric amplifier, an optical device where a pump field creates a pair of lower-frequency fields: signal and idler. The pump field is usually treated classically, but this thesis focuses on scenarios where the pump must be treated quantum mechanically. One of these scenarios is the growing field of nonlinear interferometry, where the fundamental sensitivity of a probed relative phase can beat the classical bounds and reach the maximum limit allowed by quantum mechanics, the Heisenberg limit. Indeed, we show that a fully quantum nonlinear interferometer displays a Heisenberg scaling in terms of the mean number of input pump photons. This result goes beyond the well-accepted Heisenberg scaling with respect to the down-converted photons inside the interferometer, which predicts unphysical phase sensitivities starting at a particular input pump energy. Our theoretical findings are particularly useful when designing a nonlinear interferometer with bright pump fields or optimized optical parametric amplifiers for quantum metrology and quantum imaging applications. The quantum nature of the pump field may also play a central role concerning other physical phenomena, like Hawking radiation in the context of black holes. As suggested by several authors, both the optical parametric amplifier and Hawking radiation comprise the creation of fundamental particle pairs. Thus, if the optical parametric amplifier is fully treated quantum mechanically, we may get insight into an open problem in modern physics, namely the black hole information paradox. According to this paradox, the information stored in a black hole can be destroyed once the black hole has evaporated by emitting Hawking radiation, contradicting quantum mechanics. Despite the experimental efforts to build systems that reproduce event horizons and gravitational effects in the laboratory, the evaporation of black holes due to the emission of Hawking radiation remains a challenging task. In this thesis, we experimentally investigate the impact of an evolving pump field in an optical parametric amplifier by optimizing a parametric down-conversion process. We measure the pump and signal photon number properties, finding that the pump field gets chaotic and the signal coherent when the pump displays some sizeable depletion. We arrive at similar conclusions about the pump field from its measured Wigner function. Our experiment is the first step towards a successful experiment that could suggest that information in the black hole is not destroyed but encoded in the emitted Hawking radiation starting at some point in the black hole evolution. We finally discuss further experimental improvements to investigate the parallel between the optical parametric amplifier and Hawking radiation.

Hawking theorized in 1974 that black holes emit particles as a quantum effect. It follows from this fact that a black hole that emits particles while absorbing none ends up evaporating. The process of black hole evaporation studied from semiclassical gravity violates quantum mechanics leading to serious problems. This is the black hole information paradox, one of the most famous paradoxes in theoretical physics first pointed out by Hawking in 1975 and still unsolved today. Nowadays the widespread interpretation is that quantum mechanics cannot be violated and that the semiclassical gravity approach is not good enough. We need to go beyond semiclassical physics to understand this process. The paradox as originally stated by Hawking is that a pure state evolves into a mixed state, violating unitarity and losing information in the process. There is also an alternative way to state the paradox using the so-called Page curve, which involves working with entropies rather than states. In a unitary process, the entanglement entropy of the radiation will follow the Page curve. In 2019, it was shown explicitly using holographic tools that an evaporating black hole in an Anti-de Sitter spacetime follows the Page curve. Holography is a property of quantum gravity stating that a spatial region can be described by its area rather than its volume. These recent developments also involve the famous island rule as the formula that reproduces the Page curve. This master thesis reviews the current understanding of the paradox, exploring the original paradox as well as the recent developments in the field.

Classical theory cannot provide a satisfying scenario for a unitary thermodynamic evolution of black holes. To preserve information one requires quantum mechanical effects on scales reaching beyond the traditional horizon radius. Therefore, common to many of the theories attempting to resolve the paradox is the existence of exotic horizon structures. The recent advent of gravitational wave astronomy provides a possible means for detecting the existence of such structures through gravitational wave emission in the ringdown phase of binary black hole mergers. Such emission is described by quasinormal modes (QNMs) in which the gravitational waves originates outside the black hole, in the vicinity of the photon spheres. Requiring reflective properties of the horizon structure results in the existence of gravitational echoes that may be detected by facilities such as LIGO etc.. This thesis studies geodesic motion of such echoes in the equatorial plane of a rotating black hole. Depending on the extent of the horizon structure, and the particular mode of emission, one can expect different timescales for the echoes. For a horizon structure extending <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5CLambda%20r%20=" /> <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?10%5E%7B-12%7D" /> <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?M" /> outside the traditional horizon of a  <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?M%20=%2022.6%20M_%7B%5Codot%7D" />, <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?a%20=%200.74%20M" /> black hole one would ideally find echoes appearing as integer multiples of <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5CDelta%20t_%7Becho%7D" /><img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?=%200.0204%20s" /> after the primary signal. The time delay is expected to increase by at least an order of magnitude if one lets <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5CDelta%20r%20%5Csim%2010%5E%7B-80%7D%20M" />. The expected echo timescales for gravitational waves emitted from any point around the black hole, in arbitrary modes, is an interesting further study.