In the framework of the semi-classical approximation for the Dicke model we show that there is an analogy between the dynamics in a system of two-level atoms coherently interacting with a monochromatic field at resonance or just showing superradiance and the one-dimensional classical motion of a point particle in a parameter-dependent potential, whose position is the time integral of the population inversion.
In the framework of the semi-classical approximation for the Dicke model we show that there is an analogy between the dynamics in a system of two-level atoms coherently interacting with a monochromatic field at resonance or just showing superradiance and the one-dimensional classical motion of a point particle in a parameter-dependent potential, whose position is the time integral of the population inversion.
A very approximate second integral of motion of the Dicke model is identified within a broad energy region above the ground state, and for a wide range of values of the external parameters. This second integral, obtained from a Born-Oppenheimer approximation, classifies the whole regular part of the spectrum in bands, coming from different semi-classical energy surfaces, and labelled by its corresponding eigenvalues. Results obtained from this approximation are compared with exact numerical diagonalization for finite systems in the superradiant phase, obtaining a remarkable accord. The region of validity of our approach in the parameter space, which includes the resonant case, is unveiled. The energy range of validity goes from the ground state up to a certain upper energy where chaos sets in, and extends far beyond the range of applicability of a simple harmonic approximation around the minimal energy configuration. The upper energy validity limit increases for larger values of the coupling constant and the ratio between the level splitting and the frequency of the field. These results show that the Dicke model behaves like a two-degree-of-freedom integrable model for a wide range of energies and values of the external parameters.
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 Rabi model describes a single mode electromagnetic field interacting with a two-level atom. As is well-known, the quantum treatment and the semi-classical treatment (where the field is treated classically rather than quantum mechanically) give qualitatively different results. In this paper, we analyse the model using a different semi-classical approximation based on Bohmian mechanics. In this approximation, the back-reaction from the two-level atom onto the classical field is mediated by the Bohmian configuration of the two-level atom. We find that the Bohmian semi-classical approximation gives results comparable to the usual mean field one for the transition between ground and first excited state. However it does not always perform well when considering a higher excited state. Both semi-classical approximations tend to reproduce the collapse of the population inversion, but fail to reproduce the revival, which is characteristic of the full quantum description.
Abstract Analytic solution of semiclassical dynamics equations of the Dicke model in a superradiant state is presented. The time dependences of the amplitudes of superradiant photonic condensate and coherent population of two-level atomic array in the microwave cavity prove to be expressed via Jacobi elliptic functions of real time and manifest existence of an adiabatic invariant of motion in the strongly coupled system. The periodic beatings of the photonic and atomic coherent state amplitudes are shifted in time revealing an effect of “bound luminosity,” when energy stored in the two-level system during “darkness” in the cavity is suddenly converted into photonic condensate that “illuminates” the cavity for half a period, before it plunges into “darkness” again.
Bohmian mechanics is an alternative to standard quantum mechanics that does not suffer from the measurement problem. While it agrees with standard quantum mechanics concerning its experimental predictions, it offers novel types of approximations not suggested by the latter. Of particular interest are semi-classical approximations, where part of the system is treated classically. Bohmian semi-classical approximations have been explored before for systems without electromagnetic interactions. Here, the Rabi model is considered as a simple model involving light-matter interaction. This model describes a single mode electromagnetic field interacting with a two-level atom. As is well-known, the quantum treatment and the semi-classical treatment (where the field is treated classically rather than quantum mechanically) give qualitatively different results. We analyze the Rabi model using a different semi-classical approximation based on Bohmian mechanics. In this approximation, the back-reaction from the two-level atom onto the classical field is mediated by the Bohmian configuration of the two-level atom. We find that the Bohmian semi-classical approximation gives results comparable to the usual mean field one for the transition between ground and first excited state. Both semi-classical approximations tend to reproduce the collapse of the population inversion, but fail to reproduce the revival, which is characteristic of the full quantum description. Also an example of a higher excited state is presented where the Bohmian approximation does not perform so well.