In this thesis, we explore the Dicke model, well-known for its explanation of the superradiant
phase transition and signatures of Classical and Quantum Chaos. We
then propose a variational approach to determine the semi-classical (SC) equations
of motion, using coherent states. The SC dynamics exhibits signs of classical
chaos both with increasing coupling and as the energy of the system is gradually
increased, eventually leading to it completely engulfing the 2 dimensional phase
space of the pseudospin, which we examine through Poincare sections. We then
use numerical diagonalization to study the Quantum Dicke model - it is observed
that at and after exactly the critical value of the coupling, the system shows
a net macroscopic photon expectation value, and the pseudospin component in
the z direction starts to steadily decline - implying a net fraction of the atoms
are shifting to the excited state. We then study the dynamics of the Quantum
Dicke Hamiltonian using coherent states and compare the expectation values of
the dimensional with the classical trajectories from the semi-classical dynamics
for stable and chaotic orbits. We then introduce a linear quench which drives
the Dicke model across the Superradiant phase transition by changing the coupling
strength. The quenched dynamics is simulated using both semi-classical and
quantum dynamics - it is seen that dynamically varying quantities are periodically
oscillating around their instantaneous steady-state coupling values.
In this thesis, we explore the Dicke model, well-known for its explanation of the superradiant
phase transition and signatures of Classical and Quantum Chaos. We
then propose a variational approach to determine the semi-classical (SC) equations
of motion, using coherent states. The SC dynamics exhibits signs of classical
chaos both with increasing coupling and as the energy of the system is gradually
increased, eventually leading to it completely engulfing the 2 dimensional phase
space of the pseudospin, which we examine through Poincare sections. We then
use numerical diagonalization to study the Quantum Dicke model - it is observed
that at and after exactly the critical value of the coupling, the system shows
a net macroscopic photon expectation value, and the pseudospin component in
the z direction starts to steadily decline - implying a net fraction of the atoms
are shifting to the excited state. We then study the dynamics of the Quantum
Dicke Hamiltonian using coherent states and compare the expectation values of
the dimensional with the classical trajectories from the semi-classical dynamics
for stable and chaotic orbits. We then introduce a linear quench which drives
the Dicke model across the Superradiant phase transition by changing the coupling
strength. The quenched dynamics is simulated using both semi-classical and
quantum dynamics - it is seen that dynamically varying quantities are periodically
oscillating around their instantaneous steady-state coupling values.
Now, many different approaches have been presented to study the different semi-classical models derived from the Dicke Hamiltonian, which reflect a fact that the quantum-mechanical spin possesses no direct classical analog. The Hartree-Fock-type approximation is one of the widely used approaches, with which we derive the Heisenberg equations of motion for the system and replace the operators in these equations with the corresponding expectation values. In the present paper, we investigate the role of quantum phase transition in determining the chaotic property of the time-dependent driven Dicke model. The semi-classical Hamiltonian is derived by evaluating the expectation value of the Dicke Hamiltonian in a state, which is a product state of photonic and atomic coherent states. Based on the inverse of the relations between the position-momentum representation and the Bosonic creation-annihilation operators, the Hamiltonian is rewritten in the position-momentum representation and it undergoes a spontaneous symmetry-breaking phase transition, which is directly analogous to the quantum phase transition of the quantum system. In order to depict the Poincaré sections, which are used to analyze the trajectories through the four-dimensional phase space, we give the equations of motion of system from the derivatives of the semi-classical Hamiltonian for a variety of different parameters and initial conditions. According to the Dicke quantum phase transition observed from the experimental setup , we study the effect of a monochromatic non-adiabatic modulation of the atom-field coupling in Dicke model (i.e., the driven Dicke model) on the system chaos by adjusting the pump laser intensity. The change from periodic track to chaotic figure reflects the quantum properties of the system, especially the quantum phase transition point, which is a key position for people to analyse the shift from periodic orbit to chaos. In an undriven case, the system reduces to the standard Dicke model. We discover from the Poincaré sections that the system undergoes a change from the classical periodic orbit to a number of chaotic trajectories and in the superradiant phase area, the whole phase space is completely chaotic. When the static and driving coupling both exist, the system shows rich chaotic motion. The ground state properties are mainly determined by the static coupling, while the orbit of the system is adjusted by the driving coupling. If the static coupling is greater than the critical coupling, the system displays completely chaotic images in the Poincaré sections, and the periodic orbits in the chaos can also be adjusted by the strong driving coupling. While the static coupling is less than the critical coupling, the system can also show the chaotic images by adjusting the driving coupling strength and oscillation frequency. Moreover, by tuning the oscillation frequency, the Poincaré sections may change from the classical orbits to the chaos, and back to the classical orbits in the normal phase of the system.
The emergence of chaos in an atom-field system is studied employing both semiclassical and numerical quantum techniques, taking advantage of the algebraic character of the Hamiltonian. A semiclassical Hamiltonian is obtained by considering the expectation value of the quantum Hamiltonian in Glauber (for the field) and Bloch (for the atoms) coherent states. Regular and chaotic regions are identified by looking at the Poincaré sections for different energies and parameter values. An analytical expression for the semiclassical energy density of states is obtained by integrating the available phase space, which provides an exact unfolding to extract the fluctuations in the level statistics. Quantum chaos is recognized in these fluctuations, as a function of the coupling strength, for different regions in the energy spectrum, evaluating the Anderson–Darling (A–D) parameter, which distinguishes the Wigner- or Poisson-like distributions. Peres lattices play a role similar to the Poincaré section for quantum states. They are calculated employing efficient numerical solutions and are a powerful visual tool to identify individual states belonging to a regular or chaotic region, classified by utilizing the Poincaré sections and the A–D parameter. Finally, the quantum Husimi function for selected excited states is shown to have a noticeable similitude with the Poincaré sections at the same energy.
We study the dynamical properties of the Dicke model for increasing spin length, as the system approaches the limit of a classical spin. First, we describe the emergence of collective excitations above the ground state that converge to the coupled spin-oscillator oscillations found in the classical limit. The corresponding Green functions reveal quantum dynamical signatures close to the superradiant quantum phase transition. Second, we identify signatures of classical quasiperiodic orbits in the quantum time evolution using numerical time propagation of the wave function. The resulting phase-space plots are compared to the classical trajectories. We complete our study with the analysis of individual eigenstates close to the quasiperiodic orbits.
We study the quantum critical behavior of the Dicke Hamiltonian with finite number of atoms and explore the signature of quantum chaos using measures like the ground-state fidelity and the Loschmidt echo and the time-averaged Loschmidt echo. We show that these quantities clearly point to the classically chaotic nature of the system in the superradiant (SR) phase. While the ground-state fidelity shows aperiodic oscillations as a function of the coupling strength, the echo shows aperiodic oscillations in time and decays rapidly when the system is in the SR phase. We clearly demonstrate how the time-averaged value of the echo already incorporates the information about the ground-state fidelity and stays much less than unity, indicating the classically chaotic nature of the model in the SR phase.
In recent years there have been significant advances in the study of many-body interactions between atoms and light confined to optical cavities.One model that has received widespread attention of late is the Dicke model, which under certain conditions exhibits a quantum phase transition to a state in which the atoms collectively emit light into the cavity mode, known as superradiance.We consider a generalization of this model that features independently controllable strengths of the corotating and counter-rotating terms of the interaction Hamiltonian.We study this system in the semiclassical (mean field) limit, i.e., neglecting the role of quantum fluctuations.Under this approximation, the model is described by a set of nonlinear differential equations, which determine the system's semiclassical evolution.By taking a dynamical systems approach, we perform a comprehensive analysis of these equations to reveal an abundance of complex dynamics.Examples of the phenomena that we observe are the emergence of superradiant oscillations arising due to Hopf bifurcations, and the appearance of a pair of chaotic attractors arising from period-doubling cascades, followed by their collision to form a single, larger chaotic attractor via a sequence of infinitely many homoclinic bifurcations.Moreover, we find that a flip of the collective spin can result in the sudden emergence of chaotic dynamics.Overall, we provide a comprehensive road map of the possible dynamics that arise in the unbalanced, open Dicke model in the form of a phase diagram in the plane of the two interaction strengths.Hence, we lay out the foundations to make further advances in the study of the fingerprint of semiclassical chaos when considering the master equation of the unbalanced Dicke model, thus, providing the possibility of studying a manifestation of quantum chaos in a specific, experimentally realizable system.
The Dicke model is a staple of theoretical cavity quantum electrodynamics (cavity QED), describing the interaction between an ensemble of atoms and a single radiation mode of an optical cavity. It is studied both quantum mechanically and semiclassically for two-level atoms, and demonstrates a rich variety of dynamics such as phase transitions, phase multistability, and chaos. In this work we explore an open, spin-1 Dicke model with independent co and counterrotating coupling terms as well as a quadratic Zeeman shift enabling control over the atomic energy-level structure. We investigate the stability of operator and moment equations under two approximations and show the system undergoes phase transitions. To compliment these results, we relax the aforementioned approximations and investigate the system semiclassically. We show evidence of phase transitions to steady-state and oscillatory superradiance in this semiclassical model, as well as the emergence of chaotic dynamics. The varied and complex behaviors admitted by the model highlights the need to more rigorously map its dynamics.