This thesis is a collection of six papers. The first four constitute the heart of the thesis; they are concerned with quantum mechanical properties of certain harmonic-oscillator states. The first paper is a discourse on single-mode and two-mode Gaussian pure states (GPS), states produced when harmonic oscillators in their ground states are exposed to potentials that are linear or quadratic in oscillator position and moment um variables (creation and annihilation operators). The second and third papers develop a formalism for analyzing two-photon devices (e.g., parametric amplifiers and phase-conjugate mirrors), in which photons in the ouput modes arise from two-photon transitions, i.e., are created or destroyed two at a time. The states produced by such devices are single-mode and two-mode "squeezed states", special kinds of GPS whose low-noise properties make them attractive for applications in such fields as optical communications and gravitational wave detection. The fourth paper is an analysis of the noise in homodyne detection, a phase-sensitive detection scheme in which the special properties of (single-mode) squeezed states are revealed as an improved signal-to-noise ratio relative to that obtained with coherent states (the states produced, e.g., by a laser). The fifth and sixth papers deal with problems of a different nature from that of the previous papers. The fifth paper considers the validity of the "standard quantum limit" (SQL) for measurements which monitor the position of a free mass. It shows specifically that when the pre-measurement wave functions of the free mass and the measuring apparatus(es) are Gaussian (in the general sense, which includes so-called "contractive states"), measurements described by linear couplings to the position or to both the position and momentum are limited by the SQL. The sixth paper develops the mathematical theory of torsional (toroidal) oscillations in fully general relativistic, nonrotating, spherical stellar models, and of the gravitational waves they emit.
This thesis is a collection of six papers. The first four constitute the heart of the thesis; they are concerned with quantum mechanical properties of certain harmonic-oscillator states. The first paper is a discourse on single-mode and two-mode Gaussian pure states (GPS), states produced when harmonic oscillators in their ground states are exposed to potentials that are linear or quadratic in oscillator position and moment um variables (creation and annihilation operators). The second and third papers develop a formalism for analyzing two-photon devices (e.g., parametric amplifiers and phase-conjugate mirrors), in which photons in the ouput modes arise from two-photon transitions, i.e., are created or destroyed two at a time. The states produced by such devices are single-mode and two-mode "squeezed states", special kinds of GPS whose low-noise properties make them attractive for applications in such fields as optical communications and gravitational wave detection. The fourth paper is an analysis of the noise in homodyne detection, a phase-sensitive detection scheme in which the special properties of (single-mode) squeezed states are revealed as an improved signal-to-noise ratio relative to that obtained with coherent states (the states produced, e.g., by a laser). The fifth and sixth papers deal with problems of a different nature from that of the previous papers. The fifth paper considers the validity of the "standard quantum limit" (SQL) for measurements which monitor the position of a free mass. It shows specifically that when the pre-measurement wave functions of the free mass and the measuring apparatus(es) are Gaussian (in the general sense, which includes so-called "contractive states"), measurements described by linear couplings to the position or to both the position and momentum are limited by the SQL. The sixth paper develops the mathematical theory of torsional (toroidal) oscillations in fully general relativistic, nonrotating, spherical stellar models, and of the gravitational waves they emit.
In recent years advances in technology, especially in the field of quantum optics, have made possible repetitive quantum measurements on a single quantum object, an issue that remained an academic curiosity until 25 years ago. This book, first published in 1992, provides a tutorial introduction to this field. The book is coauthored by an experimentalist and a theorist: the former - Braginsky - was the first experimentalist after Weber to develop techniques for gravitational-wave detection; the latter - Khalili - contributed with Braginsky to develop the formal apparatus to describe the experimental devices for measurement schemes. After the first part, intended as a pedagogical appendix of a standard textbook, and devoted to the modern approach to quantum measurement, the second part is a review of some new methods of quantum measurements for macroscopic objects. The central problem is the detection of a small time-dependent classical force that acts on a quantum probe. This is the typical situation that occurs in gravitational-wave detection, where the probe is a macroscopic moving mass weakly coupled to a classical wave. The problem of gravitational-wave detection and the limitation to accuracy of measurement of position of a moving mass - the so-called standard quantum limit (SQL), which is a consequence of the random back-action of the measuring device on the mass - triggered the development of new quantum nondemolition (QND) measurements, i.e. measurements where the measured quantity is left unperturbed after the measurement. SQL and QND measurements (both Braginsky inventions) permeate the whole book as a transversal chapter. The aim is to give simple criteria and recipes to design a measuring device in order to optimize the accuracy of the measurement, and, at the same time, to minimize the perturbation that the device exerts on the measured object, so that it will not interfere with successive measurements on the same object. The continuous measurements are regarded as the limiting case of a sequence of repeated discrete measurements, when the interval between the measurements is much smaller than the characteristic time-scale on which the measured quantity changes. The analysis of continuous measurements give to the authors the opportunity to illustrate the watchdog effect and the quantum Zeno paradox (real effect), while deriving the master equation for the evolution of the object state. Then, an entire chapter is devoted to the analysis of devices for continuous measurement of small mechanical displacements. The book is easy to understand, and can serve as a complementary reading for graduate level courses in quantum mechanics. The material of the book can be useful for all scientists and engineers involved in researches related to high-sensitive quantum measurements, quantum information theory and quantum computation. The reader, however, should be aware that this book presents a very particular point of view, with no room left to very relevant concepts in quantum measurement theory, as, for example, squeezing, quantum amplification, joint-measurements of non-commuting observables and phase-measurements. More regrettably, the list of references is unsatisfactorily incomplete, with no mention of alternative developments, for example, regarding the SQL for measuring the position of a free mass. Here, schemes to breach the SQL have been proposed in the literature (Yuen H P 1983 Phys. Rev. Lett. 51 719, Ozawa M 1988 Phys. Rev. Lett. 60 385), whereas from the book one could conclude that such SQL would be unsurpassable. Also, a criticism is in order of the way of deriving some SQL's, based solely on the Heisenberg relation, even in consideration that the aim of such derivations is mostly pedagogical.
This thesis concerns quantum states that are superpositions of discrete or of continuum energy levels. It develops mathematical methods to characterize the physical properties of these states and finds important consequences. Regarding rational extensions of the harmonic oscillator, no other authors had constructed the coherent states associated with the ladder operators of Marquette and Quesne. It was our aim to proceed with this construction, then to calculate physical properties of these coherent states, looking for classical or quantum behaviour. The method of partial wave analysis was known to be not applicable to the Coulomb potential for the scattering of plane waves. It was my aim to repeat the calculation for wavepacket scattering, to investigate whether the method became applicable. Other authors have constructed relativistic probability amplitudes and compared them to the scalar Klein-Gordon amplitude. In this context, it was my aim to question the conclusion of a minimum width for a massive particle. In the same context, I investigated the possibility of a four-vector probability current with the Newton-Wigner position probability density as the zero component. In Chapter 12, my aim was to construct a measure of localization for the photon, knowing from the work of other authors that a point-localized photon state is not possible.In Chapters 3, 4 and 5, our measures of classicality for the Barut-Girardello coherent states included plotting energy expectations as functions of the magnitude of the coherent state para- meter, z, position probability densities for the coherent states and the even and odd cat states, the Wigner functions for particular choices of z, the two-photon number probability distribu- tion and the linear entropy for a coherent state on a beamsplitter, position spreads to look for squeezing and the Mandel Q parameter. In Chapter 6, it was necessary to approximate an integral containing a particular Wigner rotation matrix. An approximation of general Wigner rotation matrices, at low angle, was derived and its precision evaluated numerically in Chapter 7. The exact values for the Coulomb phase shifts were used to construct the incoming wavefunction. Time reversal and a rotation were used to construct the outgoing wavefunction. The sums over l were evaluated numerically. In Chapter 10, four arguments were used against the minimum width. Comparison of the scalar amplitude and the relativistic probability amplitude for a Newton-Wigner localized state showed the root of the problem. A Lorentz transformation was applied to the scalar function to see if the “minimum” width became even smaller. A simple construction produced scalar functions with arbitrarily small widths. A relevant argument from Almeida and Jabs was included here. In Chapter 11, the algebra involving the four components of a four-vector current (at the origin) and the boost generators was derived. All of these commutators must be satisfied by a four-vector, providing a test for the claimed four-vector probability currents of other authors. In Chapter 12, field expectations were taken in a coherent state with a mean photon number of unity, built from identical single-photon states.In Chapters 3, 4 and 5, we found a mix of classical and quantum behaviour for the coherent states. In none of the systems studied did we see the high degree of classicality of the unmodified harmonic oscillator. The investigation of cat states was relevant to the fundamental problem of measurement in quantum mechanics. In Chapter 6, the use of wavepackets introduced a convergence factor into the sum over l, which diverges in the plane wave treatment. The elimination of a divergence is the most significant result of this thesis. Scattering probabilities remained finite at all angles, in disagreement with the diverging Rutherford formula, but only on small angular regions around forward scattering. Future work will investigate the measurability of the phenomena predicted here. In Chapter 10, all four arguments pointed to no minimum width for a massive particle. In Chapter 11, it was found that there can be no four-vector current density with the Newton-Wigner position probability density as the zero component, for any spin. Other authors were found to be in error when they claimed to have produced such a four-vector current density. This result strengthens the position of the Dirac current as the appropriate current for an electron. In Chapter 12, the coherent state field expectations were found to provide a meaningful measure of localization. In summary, wavepackets give a realism to the description of a scattering experiment that is absent with the use of plane waves. Phenomena not captured by the plane wave scattering amplitude, such as time delays, are calculable in the wavepacket context. Relativistic probability amplitudes provide a complete description of free particles, consistent with quantum mechanics and with special relativity. They predict agreement with the Heisenberg uncertainty principle and relativistic effects such as Lorentz contraction. The absence of relativistic probability amplitudes from standard textbooks on relativistic quantum mechanics is a serious omission.
This dissertation deals with theoretical and experimental research on nonclassical, or entangled, states of atoms and photons. In the first part, I describe two approaches to the preparation of entangled states of a large number of atoms. The first approach is based on the transfer of quantum correlations from non-classical light to the atomic spins. I consider three different situations: (a) the atoms are placed in a loss-free cavity and no relaxation of any kind is present; (b) the atoms are placed in a cavity with input and output for the electromagnetic field and the atoms spontaneously decay from the upper state; (c) there is no cavity around the atoms and spontaneous emission is present. I show that in all three situations with judicious choice of parameters, entangled samples of atoms can be produced. The second approach is based on quantum-nondemolition (QND) measurements of collective atomic operators. I describe experiments with pulsed and cw light as the probe, that result in squeezed spin states of atoms. When a rf magnetic field is applied to the atomic sample, sub- shot noise performance of an atomic spin interferometer is demonstrated. In the second part of the thesis I describe investigations of phase properties of two-photon and single-photon states. These experiments make use of spontaneous type-II down-conversion in a nonlinear crystal. First, interferometric measurement of a phase shift at the Heisenberg limit for a two-photon state is described. Next, a violation of Bell-type inequalities in phase space is demonstrated for the quantum optical version of the Einstein-Podolsky-Rosen state. Finally, some recent work directed toward homodyning a single photon against a strong local oscillator field is described.
In this thesis we examine a variety of question regarding several quantum information-theoretic concepts as they manifest in the context of relativistic quantum fields. In particular, we study the qualitative and quantitative structure of entanglement (as well as more generalized quantum correlations) in field theory. Primarily we study the nature of vacuum entanglement, but consider spatial correlations in more general field states as well. We also discuss several novel aspects of particle creation phenomena, which generally go hand in hand with the presence of field entanglement. Such investigations are important from two perspectives. First, we gain enlightening new insight into the nature of quantum fields and, therefore, into the fundamental constituents of the matter contained in our universe. Second, we are able to consider the possibility of utilizing the relativistic nature of quantum fields as a resource for quantum technological tasks such as entanglement distribution, information processing, and metrology. \n \nAs a general guiding principle, here we approach such questions from an operational perspective. That is, we focus less on the mathematical facts derivable from a given theory and instead attempt to answer in what concretely physical ways such facts manifest themselves. For example, in what ways can vacuum entanglement actually be experimentally detected and measured, at least in principle? Such an approach allows us to circumvent otherwise problematic interpretational issues. It also lends itself more naturally to the proposal of real world experiments regarding, and utilizations of, the physics being investigated. In this thesis we utilize two models of detection as applied to field theory. \n \nThe first is one that we will develop from scratch and use to great effect in a number of studies; this is the non-perturbative oscillator detector. In this context, a ``detector" is simply some quantum system that is allowed to interact in some way with a quantum field. The response of this system then gives us information about the field properties. Typically such studies are done perturbatively, which presents several severe limitations on what may be investigated. We develop an alternative model, in which the detector is identified as a quantum harmonic oscillator, that allows non-perturbative solutions. We go on to apply this model to a variety of studies. We start by the examining the Unruh effect in a cavity setting. Not only are we able to examine the thermalization behavior of such a detector (which is stronger than what can be achieved perturbatively), we also demonstrate that the Unruh response is largely independent of the cavity boundary conditions, indicating a surprising amount of generality and universality to the phenomenon. We also consider the spacelike extraction of spatial entanglement by a set of such detectors. This includes the harvesting of genuine tripartite entanglement out of the vacuum, as well as the observation that the extractable quantum discord (a generalized measure of quantum correlations) can be greatly amplified by thermal fluctuations in the field, in opposite behavior to entanglement. This calls into question the nature of quantum discord, but also suggests at the possibility of actually using environmental noise as a quantum computational resource. Next, we use the oscillator detector model to great effect by proposing a method of reliably and sustainably generating and distributing entanglement via a cavity field. This protocol is inherently non-perturbative, and we refer to it as entanglement farming. Lastly, we modify the farming scenario to discover a very sensitive resonance effect that generates a large difference in the protocol output due to a minute change in system parameters, including the length of cavity being used. We put this forth as a potential method of quantum seismic metrology. \n \nThe second model that we will consider simply involves performing projective measurements onto a given set of field modes. We will find, however, that such an approach allows us to solve previously intractable problems. In particular, we apply this procedure to the physically sensible choice of localized, positive frequency field modes. Using this model we perform two different studies. In the first, we consider the response of an observer accelerating not through the vacuum state but rather a thermal state. Such a setting has not been (properly) studied previously, due to the computational challenges involved. By applying the localized projective measurement model we are able determine that such an observer sees both an Unruh response from acceleration and a response from the field thermality and that, critically, these responses are distinguishable. We then go on to examine the degradation of quantum correlations between two observers due to the acceleration of one of the observers. This is a prototypical scenario of interest and, by applying realistic measurement assumptions, we obtain results qualitatively different from many previous works on the issue. \n \nIn addition to these models, we question the nature of vacuum entanglement and particle creation by considering an operationally straightforward procedure that greatly demystifies these phenomena and puts them on to solid, physically concrete grounds. This operation is that of very quickly introducing a boundary condition onto one's field, for example by creating a mirror. The real particles generated in this process will then be quantum entangled with each other, even if they are created at spatially separate locations, and this entanglement derives exactly from the previously present vacuum entanglement. Not only does this realization provide an operationally satisfying interpretation of vacuum entanglement, it also suggests a straightforward procedure for its experimental verification. \n \nIn all of this, the computational framework that we will rely upon is that of Gaussian quantum mechanics. This is a restriction of continuous variable quantum mechanics that, while limited in its applicability, provides extremely powerful computational tools in the cases that it may be used. The scenarios of interest in this thesis fall within this category and lend themselves naturally to a Gaussian approach. With this we are able to perform, near trivially, many otherwise intractable studies.
In the present PhD work, we focus on a specific class of quantum states of light: the non-Gaussian states. These states have the particularity of exhibiting Wigner functions with some negative values. This quantum feature is a necessary condition to perform some quantum computation task; furthermore it is also useful for various other applications, including quantum communication and metrology. Different strategies can be used to generate these states. Here, we start from Gaussian states produced by optical parametric oscillators in the continuous wave regime, (i.e. single-mode and two-mode squeezed vacuum states). The non-Gaussian feature can only be obtained by non-linear phenomena (over-quadratic Hamiltonian). In our case, the non-linearity is induced by photon-counting-based measurements (also called non-Gaussian measurements). This study is mainly divided into two parts. First, the generation of non-classical states associated with two types of qubit encoding: the single-photon state, used for quantum information with discrete variables, and the coherent state superposition (the so-called optical Schrodinger cat state), used for quantum information with continuous variables. These two states have then been used to perform some quantum information protocols. The first one addresses the problem of single-photon entanglement witness, and the other the generation of entanglement between the two encodings (also called hybrid entanglement).
The quantum statistics of a laser result in noise when measurements of the beam are made. This noise sets a classical limit beyond which a laser cannot be used with increasing sensitivity. This quantum noise limit is imposed on many of the uses of lasers currently, especially in power limited devices such as optical communications. The statistics of the laser photon field can be modified to produce a non-classical state resulting in lower noise than the quantum noise limit when detected appropriately. This state, called a squeezed state, has been measured previously from a cavity enhanced optical parametric oscillator (OPO) only at frequency sidebands within the linewidth of the cavity. ¶ This thesis reports measurements of squeezing at microwave frequency sidebands on an optical beam produced by an optical parametric oscillator. This is the first reported measurement of squeezing at frequency sidebands at higher longitudinal modes of the cavity from an OPO. Noise reduction below the quantum noise limit is measured at sideband frequencies of 5 MHz, 1.7 GHz, 3.4 GHz and 5.1 GHz, corresponding to the zeroth, first, second and third longitudinal modes from the squeezed beam. These results are the highest frequency sideband measurements of squeezing to date. In addition to measuring squeezing at different longitudinal modes for the fundamental Gaussian spatial mode, non-classical noise reduction is measured at the same frequencies for a squeezed higher order spatial mode, TEM10. ¶ A single mode theoretical model of the OPO is presented, based on the work of ref. [1]. Computer simulations of the squeezing predicted by this model are developed and compared to the experimental results, showing excellent agreement between the different longitudinal modes for each of the two spatial modes measured.