The objective of this thesis is to investigate two fundamental problems of quantum optics: macroscopically distinguishable quantum superposition states and quantum chaos. After a brief introduction, Chapter II and Chapter III investigate the problem of superposition states. A method for generating the macroscopically distinguishable superposition states via single two-level atom dispersion was proposed by Savage et a/.[Opt.Lett. 15, 628(1990)]. In chapter II, an extension of this model to a three-level atom is investigated and the result shows that the superposition state can be obtained from the three-level model sytem in the dispersive limit. In chapter III, S ’ the effect of the atomic linewidth is considered. If the atomic linewidth is included, there will be two contradictory conditions for the generation of distinguishable superpositon states. This shows that the atomic linewidth prevents the formation of macroscopically distinguishable superpositon states by the proposed method. From chapter IV to chapter VI, we investigate the transition from a dissipative quantum system to a classically chaotic system. Specifically, wre apply the method of quantum trajectories to the case of optical second harmonic generation. In chapter V, the Q function is obtained numerically by this method and found to agree with the results obtained by solving the master equation. In chapter VI, the field amplitudes and mean photon number associated with a single quantum trajectory are investigated. The results show that the trajectory becomes more classical and the configuration of the classical strange attractor is approached as the field strength is increased. This is because the mean values of operator products factorize into products of mean values in the classical limit.
The objective of this thesis is to investigate two fundamental problems of quantum optics: macroscopically distinguishable quantum superposition states and quantum chaos. After a brief introduction, Chapter II and Chapter III investigate the problem of superposition states. A method for generating the macroscopically distinguishable superposition states via single two-level atom dispersion was proposed by Savage et a/.[Opt.Lett. 15, 628(1990)]. In chapter II, an extension of this model to a three-level atom is investigated and the result shows that the superposition state can be obtained from the three-level model sytem in the dispersive limit. In chapter III, S ’ the effect of the atomic linewidth is considered. If the atomic linewidth is included, there will be two contradictory conditions for the generation of distinguishable superpositon states. This shows that the atomic linewidth prevents the formation of macroscopically distinguishable superpositon states by the proposed method. From chapter IV to chapter VI, we investigate the transition from a dissipative quantum system to a classically chaotic system. Specifically, wre apply the method of quantum trajectories to the case of optical second harmonic generation. In chapter V, the Q function is obtained numerically by this method and found to agree with the results obtained by solving the master equation. In chapter VI, the field amplitudes and mean photon number associated with a single quantum trajectory are investigated. The results show that the trajectory becomes more classical and the configuration of the classical strange attractor is approached as the field strength is increased. This is because the mean values of operator products factorize into products of mean values in the classical limit.
The topic of this thesis is a theoretical study of nonequilibrium transitions and quantum statistical properties of nonlinear quantum optical systems driven by an external radiation source. In certain limiting cases, a comparison is made between these transitions and the phase transitions found in equilibrium physical systems. In chapters one and two, the mathematical tools are introduced. In operator terms, the time development is described by a Markovian master equation in the interaction picture. This is equivalent to a corresponding time development equation or Fokker-Planck equation in a vector space of c-numbers. In order to deal with the type of Fokker-Planck equation that results, we introduce a quantum classical correspondence resulting in a distribution function over a complex phase-space, which is a generalisation of the Glauber-Sudarshan P-function. In chapter three this is applied to a model of a coherently driven mode with nonlinear dispersion and absorption. We find in the limit of zero temperature, that the spectrum is symmetric relative to the input frequency, and an exact solution is obtained for the distribution function. For a detuned driving field and nonlinear dispersion, optical bistability can occur. In chapter four a model of sub/second harmonic generation is introduced. This has several non-equilibrium transitions, including dispersive optical bistability, and bistable behaviour with coherent phase-locked input to both modes. Exact solutions occur in the limit of zero temperature and adiabatic elimination of one mode. In both chapters three and four, steady-state photon anti-bunching occurs with an absorptive nonlinearity. In chapter five we include interactions between the radiation mode and a fluorescent atomic system. In this case different behaviour occurs depending on the relative decay rates of the individual atoms and of the radiation mode. In the case of a high-Q interferometer, the atomic variables can be adiabatically eliminated. Both dispersive and absorptive bistability can occur. We show by analytic and numerical calculations (in the case of inhomogeneous broadening) that dispersive operation has advantages in requiring a lower atomic density and input field to observe bistability. When the input field has Gaussian (rather than coherent) photon statistics, there is no bistability, but enhanced photon bunching occurs. Finally, there is a different type of behaviour when the field mode decays rapidly and can be adiabatically eliminated. Within the cooperation lifetime the system can be described by a J²-invariant Hamiltonian, giving the special case of only collective damping. The result is a new type of critical point transition in the thermodynamic limit, with the appearance of a family of solutions like Lotka-Volterra cycles for a coherent driving field above threshold.
A method for generating macroscopically distinguishable superposition states via single atom dispersion was proposed by Savage et al. see Optics Letters, vol.15, p.628, 1990. In that model, the atomic linewidth was not considered. However, including the atomic linewidth, the authors get two contradictory conditions for the generation of distinguishable superposition states. This shows that the atomic linewidth prevents the formation of macroscopically distinguishable superposition states by the proposed method.
This thesis consists of two parts. Although both are concerned with macroscopic quantum phenomena, I choose to present the two parts quite separately. Macroscopic Quantum Phenomena: Tests of Quantum Mechanics The correctness of quantum mechanics has been verified in numerous situations. A particularly strong test of quantum mechanics was suggested by Bell, who showed that the predictions of all classical theories, based as they are on the assumptions of local realism, contradict those of quantum mechanics. Later work by Greenberger, Horne, and Zeilinger showed that the GHZ phenomena also contradicts all classical theories. Experimental tests of Bells inequality performed to date support quantum mechanics (No experimental test of the GHZ phenomenon has yet been attempted). However, such tests have been so far restricted to microscopic systems in the sense that one particle is at a detector at a time. Our interest here in the first part of this thesis is in tests of quantum mechanics against local realistic theories in macroscopic or mesoscopic systems where there is a significant number of particles incident on each analyser. The states considered in the first two chapters are not simple quantum superpositions of macroscopically distinct states, but we nevertheless show how one can obtain tests of quantum mechanics in situations involving large numbers of particles. A more elementary macroscopic quantum state, and one in which the deviation from classical interpretations is striking, is the Schrodinger Cat state considered by Schrodinger in his famous "Schrodinger Cat" paradox. In Chapter Three we consider a scheme for generating such a state. The Quantum Brownian Motion Master Equation The second section of this thesis involves an examination of various aspects of the Quantum Brownian Motion Master Equation. We begin with a discussion of the recent problem occurring from the study of the quantum Brownian motion (QBM) master equation, which can occur because the QBM master equation is not of the Lindblad form. We examine two specific examples, namely the damped free particle and the two level atom. More specifically, in the case of the evolution of a free particle, we show that the equation of motion for the Wigner function (which is exactly the same as the Fokker-Planck equation for classical Brownian motion) gives unphysical results if the initial position distribution is well localised. In the case of the two level atom we show that unphysical results can also occur even in the steady state. The last chapter of the thesis involves using the quantum Brownian motion master equation to estimate the rate of diagonalisation of the off diagonal elements of a Fermion system. These elements contain the "quantum information" and hence knowledge of how long they exist can determine the type of model required to describe the semiconductor medium. Using a very simplistic approach we can show that there are two main time scales present and for normal parameters one of these is very small. If we neglect electron-electron scattering we can show that the the many electron system solution is simply a product of the one electron solutions.
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.
In this dissertation, I analyze certain problems in the following areas: 1) quantum dynamical phenomena in macroscopic systems of interacting, degenerate bosons (Parts II, III, and V), and 2) measures of macroscopicity for a large class of two-branch superposition states in separable Hilbert space (Part IV). Part I serves as an introduction to important concepts recurring in the later Parts. In Part II, a microscopic derivation of the effective action for the relative phase of driven, aperture-coupled reservoirs of weakly-interacting condensed bosons from a (3 + 1)D microscopic model with local U(1) gauge symmetry is presented. The effective theory is applied to the transition from linear to sinusoidal current vs. phase behavior observed in recent experiments on liquid 4He driven through nanoaperture arrays. Part III discusses path-integral Monte Carlo (PIMC) numerical simulations of quantum hydrodynamic properties of reservoirs of He II communicating through simple nanoaperture arrays. In addition to calculating the local superfluid density in these systems, new estimators for hydrodynamic observables and novel methods for extracting the length scale characterizing the decay of superfluidity at the system boundary from PIMC data are introduced with the aim of exploring the mechanism of superfluid weak-link formation in nanoscale containers.Part IV consists of an analysis of macroscopicity measures for a large class of Schrodinger cat states of N-mode photonic systems. For cat states of this class, it is shown that a well-known measure of superposition size based on the optimal distinguishability of the branches and another based on metrological usefulness of the superposition relative to its branches agree (i.e., designate the same superpositions as macroscopic) when the inner product of the branches of the superposition is sufficiently small. For certain superpositions in this class, a technique is presented for deriving a state-specific textit{metrological macroscopicity algebra} of observables. The dynamics of superposition size measures is also considered, leading to the notion of "distinguishability time,'' a generalization of the orthogonalization times of Mandelstam-Tamm and Margolus-Levitin. In Part V, an investigation of the nonequilibrium dynamics of a quantum dot-Josephson junction-quantum dot nanodevice is presented. The nonequilibrium action of the system is calculated by making use of the Keldysh formalism and a potential use of the device for transferring macroscopic superposition states between a spin-system and an s-wave superconductor is discussed.
Abstract This book is an introduction to quantum optics for students who have studied electromagnetism and quantum mechanics at an advanced undergraduate or graduate level. It provides detailed expositions of theory with emphasis on general physical principles. Foundational topics in classical and quantum electrodynamics, including the semiclassical theory of atom-field interactions, the quantization of the electromagnetic field in dispersive and dissipative media, uncertainty relations, and spontaneous emission, are addressed in the first half of the book. The second half begins with a chapter on the Jaynes-Cummings model, dressed states, and some distinctly quantum-mechanical features of atom-field interactions, and includes discussion of entanglement, the no-cloning theorem, von Neumann’s proof concerning hidden variable theories, Bell’s theorem, and tests of Bell inequalities. The last two chapters focus on quantum fluctuations and fluctuation-dissipation relations, beginning with Brownian motion, the Fokker-Planck equation, and classical and quantum Langevin equations. Detailed calculations are presented for the laser linewidth, spontaneous emission noise, photon statistics of linear amplifiers and attenuators, and other phenomena. Van der Waals interactions, Casimir forces, the Lifshitz theory of molecular forces between macroscopic media, and the many-body theory of such forces based on dyadic Green functions are analyzed from the perspective of Langevin noise, vacuum field fluctuations, and zero-point energy. There are numerous historical sidelights throughout the book, and approximately seventy exercises.