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 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.
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.
We study the quantum phase transition of the Dicke model in the classical oscillator limit, where it occurs already for finite spin length. In contrast to the classical spin limit, for which spin-oscillator entanglement diverges at the transition, entanglement in the classical oscillator limit remains small. We derive the quantum phase transition with identical critical behavior in the two classical limits and explain the differences with respect to quantum fluctuations around the mean-field ground state through an effective model for the oscillator degrees of freedom. With numerical data for the full quantum model we study convergence to the classical limits. We contrast the classical oscillator limit with the dual limit of a high-frequency oscillator, where the spin degrees of freedom are described by the Lipkin-Meshkov-Glick model. An alternative limit can be defined for the Rabi case of spin length one-half, in which spin frequency renormalization replaces the quantum phase transition.
We study the numerical solutions of the Dicke Hamiltonian, which describes a system of many two level atoms interacting with a monochromatic radiation field into a cavity. The Dicke model is an example of a quantum collective behavior which shows superradiant quantum phase transitions in the thermodynamic limit. Results obtained employing two different bases are compared. Both of them use the pseudospin basis to describe the atomic states. For the photon states we use in one case Fock states, while in the other case we use a basis built over a particular coherent state, associated to each atomic state. It is shown that, when the number of atoms increases, the description of the ground state of the system in the superradiant phase requires an equivalent number of photons to be included. This imposes a strong limit to the states that can be calculated using Fock states, while the dimensionality needed to obtain convergent results in the other basis decreases when the atomic number increases, allowing calculations that are very difficult in the Fock basis. Naturally, it reduces also the computing time, economizing computing resources. We show results for the energy, the photon number and the number of excited atoms, for the ground and the first excited state.
We investigate a generalized Dicke model by introducing two interacting spin ensembles coupled with a single-mode bosonic field. Apart from the normal to superradiant phase transition induced by the strong spin-boson coupling, interactions between the two spin ensembles enrich the phase diagram by introducing ferromagnetic, antiferromagnetic and paramagnetic phases. The mean-field approach reveals a phase diagram comprising three phases: paramagnetic-normal phase, ferromagnetic-superradiant phase, and antiferromagnetic-normal phase. Ferromagnetic spin-spin interaction can significantly reduce the required spin-boson coupling strength to observe the superradiant phase, where the macroscopic excitation of the bosonic field occurs. Conversely, antiferromagnetic spin-spin interaction can strongly suppress the superradiant phase. To examine higher-order quantum effects beyond the mean-field contribution, we utilize the Holstein-Primakoff transformation, which converts the generalized Dicke model into three coupled harmonic oscillators in the thermodynamic limit. Near the critical point, we observe the close of the energy gap between the ground and the first excited states, the divergence of entanglement entropy and quantum fluctuation in certain quadrature. These observations further confirm the quantum phase transition and offer additional insights into critical behaviors.
We demonstrate quantum signatures of deterministic nonlinear dynamics in the transition to superradiance of a generalized open Dicke model with different coupling strengths for the co- and counterrotating light-matter interaction terms. A first-order phase transition to coexisting normal and superradiant phases is observed, corresponding with the emergence of switching dynamics between these two phases, driven by quantum fluctuations. We show that this phase coexistence gives rise to a hysteresis loop also for the quantum mechanical system. Additionally, a transition to a superradiant oscillatory phase is observed clearly in quantum simulations.
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.