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.
In this dissertation we proposed and modeled five ultra cold atoms experiments that make use of quantum effects to produce novel macroscopic phenomena. The first of these experiments is a proposal to produce a p-wave superfluid using two different atomic species.The pairing species is trapped in two-dimensions and the other species is allowed to move in three-dimensions and mediates the p-wave pairing interaction. We confirm our predictions using the functional Renormalization Group method.The next experiment studies how macroscopic cat states can be produced and detected in bosonic interferometers. A bosonic interferometer is composed of ultra-cold bosons that can condense in one of two modes. Tunneling and interactions are the primary contributions to the dynamics and in cooperation can produce highly entangled states such as Schrodinger cat states. We show how the quantum Fisher information, a tool in quantum metrology, can be used to perform a partial state tomography to determine the nature of the superposition in experimental cat states.In the last three proposals, we study the long time equilibration dynamics of three different proposed experiments. The first proposed experiment takes place in the same bosonic interferometer as discussed in the Schr{\\"o}dinger cat work, and we identify a novel mechanism for the breakdown of thermalization with the slow dynamics of an unstable fixed point in a semi-classical approximation. The second proposed experiment also takes place in the same bosonic model, but involves a periodic kick to the interaction strength. We find that these dynamics can produce a time-crystal-like state where discrete time-translation symmetry is broken in the long time dynamics. Finally, we study the long time dynamics of a clean fermionic chain and a disordered fermionic chain that are coupled by density density interactions. In this proposal, we find that the effects of disorder can be transferred to the clean chain and prevent thermalization.
In my dissertation, I will discuss unconventional superfluid states of various bosonic systems and present benchmark calculations for phase diagrams. The method of study is based on quantum Monte Carlo simulations by the Worm algorithm. The two qualitatively different systems to be discussed in the thesis are: (i) superfluidity of lower dimensional objects, i.e dislocations and defects, embedded in solid structures both in optical lattices and in realistic systems such as 4He . (ii) the two-component Bose-Hubbard model. Besides the interesting phases and phase diagrams that they exhibit, bosonic optical lattice systems are also interesting for studies of quantum magnetism where under certain conditions they can be mapped onto various spin Hamiltonians. On the other hand, superfluid dislocations in 4He is an important and interesting subject for it's relevance to supersolid behavior observed in solid 4 He . I will also present technical details of the path integral Monte Carlo and the Worm algorithm and generalization of the algorithm to the two-component bosonic systems.
1. Define Bose-Einstein condensation in an interacting boson system in term of macroscopic occupation. 2. Introduce the Gross-Pitaevskii equation and its applicable conditions. 3. Derive the hydrodynamic equations and various physical consequences of the hydrodynamics with or without harmonic trap. 4. Introduce sound wave and emphasize the difference between the wave behavior and the diffusive behavior. 5. Introduce the criterion for superfluidity and determine the superfluid critical velocity. 6. Introduce two-fluid hydrodynamics. 7. Introduce the Bogoliubov theory. 8. Introduce various concepts related to quasi-particle, including ground state as vacuum of quasi-particles, the zero-point energy of quasi-particle, quasi-particle lifetime and vacuum fluctuation. 9. Introduce healing length as the characteristic length scale for low-energy physics of an interacting Bose condensate. 10. Summarize the difference between an interacting and non-interacting Bose condensate. 11. Introduce the Bragg spectroscopy. 12. Show that the Bogoliubov theory fails in one-dimension due to the infrared divergence. 13. Discuss the basic idea of the Bethe-ansatz solution and the Tonks-Girardeau limit. 14. Introduce scale invariant quantum gases. 15. Introduce the Josephson effect and the self-trapping as two different transport phenomena driven by phase coherence. 16. Discuss the conjugate relation between the relative phase fluctuation and the relative particle number fluctuation, and show strong enough phase fluctuation renders the system into a Fock state. 17. Discuss an example of quantum measurement that can project two independent condensates to a coherent superposition state. 18. Discuss interference effects in density and density-density correlations. 19. Discuss the relation between the instability of the Sch\"odinger Cat state and the stability of symmetry breaking.
Macro-orbital representation of a particle (detailed account given in cond-mat/0603784) has been used to develop the microscopic theory of a system of interacting bosons. It concludes that: (i) below certain temperature (say,
Tλ), particles assume a state of (q, -q) bound pairs, (ii) the
λ−transition is a consequence of inter-particle quantum correlations clubbed with zero-point repulsion and inter-particle attraction and represents an onset of the order-disorder of particles in their
ϕ−space followed simultaneously by their BEC as (q, -q) bound pairs in a state of q =
qo=π/d and K = 0, (iii) particles at
T≤Tλ acquire collective binding which locks them at = 0, =
λ/2 and
Δϕ=2nπ (with n = 1, 2, 3, ...), (iv) the entire system assumes mechanical strain in inter-particle bonds and behaves like a single macroscopic molecule, (v) there exists an energy gap between the superfluid and normal fluid phases of the system, (vi) the
λ−transition represents the twin phenomena of broken gauge symmetry and phase coherence, (vii) the system does not have p = 0 condensate, (viii) a new kind of quantum quasi-particle "omon" (a phononlike wave of the oscillations of the momentum coordinates of particles) exists in superfluid phase, etc. It explains the properties of He-II, including the origin of quantized vortices, critical velocities, logarithmic singularity of specific heat, etc. at quantitative level and provides microscopic foundation to two fluid theory,
Ψ−theory, idea of macroscopic wave function, etc. The framework of the theory can unify the physics of interacting bosons and fermions.
Recent experiments claiming the formation of quantum superposition states in near-macroscopic systems raise the question of how the sizes of general quantum superposition states in an interacting system are to be quantified. We propose here a measure of size for such superposition states that is based on what measurements can be performed to probe and distinguish the different branches of the state. The measure allows the comparison of the effective size for superposition states in very different physical systems. It can be applied to a very general class of superposition states and reproduces known results for near-ideal cases. Comparison with a prior measure based on analysis of coherence between branches indicates that significantly smaller effective superposition sizes result from our measurement-based measure. Application to a system of interacting bosons in a double-well trapping potential shows that the effective superposition size is strongly dependent on the relative magnitude of the barrier height and interparticle interaction.
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.