We theoretically study an extension of the Dicke model, where the single-particle Hamiltonian has three energy levels in Lambda configuration (i.e., the excited state is coupled to two nondegenerate ground states via two independent quantized light fields). The corresponding many-body Hamiltonian can be diagonalized in the thermodynamic limit with the help of a generalized Holstein-Primakoff transformation. Analyzing the ground-state energy and the excitation energies, we identify one normal and two superradiant phases, separated by phase transitions of both first and second oder. A phase with both superradiant states coexisting is not stable. In addition, in the limit of two degenerate ground states a dark state emerges, which seems to be analogous to the dark state appearing in the well-known stimulated Raman adiabatic passage scheme.
We theoretically study an extension of the Dicke model, where the single-particle Hamiltonian has three energy levels in Lambda configuration (i.e., the excited state is coupled to two nondegenerate ground states via two independent quantized light fields). The corresponding many-body Hamiltonian can be diagonalized in the thermodynamic limit with the help of a generalized Holstein-Primakoff transformation. Analyzing the ground-state energy and the excitation energies, we identify one normal and two superradiant phases, separated by phase transitions of both first and second oder. A phase with both superradiant states coexisting is not stable. In addition, in the limit of two degenerate ground states a dark state emerges, which seems to be analogous to the dark state appearing in the well-known stimulated Raman adiabatic passage scheme.
In this doctoral thesis, the thermodynamic phases und phase transitions of a generalised Dicke model are studied and characterised. Both, finite and vanishing temperatures are considered. The Dicke model of quantum optics describes collective phenomena which occur when light interacts with a many-atom system. In the thermodynamic limit, a so-called Hepp–Lieb superradiant phase transition sets in. A superradiant phase develops for low temperatures or strong coupling between light and atoms, which is characterised by a macroscopic excitation of the light field and a spontaneous, collective polarisation of the atoms. This thesis discusses a generalised version of the Dicke model, which is described by a quantum mechanical system consisting of three-level atoms in Lambda-configuration and two modes of a resonator. By means of the Holstein–Primakoff transformation, the Hamiltonian of this system is written in terms of four interacting, non-linear oscillators, which can be linearised in the thermodynamic limit, yielding the ground-state energy as well as the low-energy excitations. The phase diagram consisting of two superradiant phases separated by continuous and first-order phase transitions is derived. In order to clarify the question whether or not the superradiant phase transition of this generalised Dicke model can be observed experimentally for real atoms, the significance of the diamagnetic term is discussed. In contrast to the original Dicke model, superradiant phase transitions are possible in principle. This is due to the first-order phase transitions. In addition, a no-go theorem for continuous superradiant phase transitions is presented. The argument is based on the Thomas–Reiche–Kuhn sum rule. Last, we study the superradiant phase transition of the generalised Dicke model at finite temperatures. Therefore, the partition sum is computed and analysed in the thermodynamic limit using Laplace’s method. At finite temperatures, the properties of the phase diagram and phases remain. However, here all phase transitions are of first order.
We consider an ensemble of three-level particles in Lambda configuration interacting with two bosonic modes. The Hamiltonian has the form of a generalized Dicke model. We show that in the thermodynamic limit this model supports a superradiant quantum phase transition. Remarkably, this can be both a first- and a second-order phase transition. A connection of the phase diagram to the symmetries of the Hamiltonian is also given. In addition, we show that this model can describe atoms interacting with an electromagnetic field in which the microscopic Hamiltonian includes a diamagnetic contribution. Even though the parameters of the atomic system respect the Thomas-Reiche-Kuhn sum rule, the system still shows a superradiant phase transition.
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 employ a generalized Dicke model to study theoretically the quantum criticality of an extended two-level atomic ensemble interacting with a single-mode quantized light field. Effective Hamiltonians are derived and diagonalized to investigate numerically their eigenfrequencies for different quantum phases in the system. Based on the analysis of the eigenfrequencies, an intriguing quantum-phase transition from a normal phase to a superradiant phase is revealed clearly, which is quite different from that observed with a standard Dicke model.
Dicke model describes a collective interaction between the two-level atoms and the light cavity and has been predicted to show a peculiar quantum phase transition, which is a second-order phase transition from a normal phase (in a weak-coupling strength) to a superradiant phase (in a strong-coupling strength). The model plays an important role in illustrating the quantum ground-state properties of many-body macroscopic quantum states. In the experiment, Dicke quantum phase transition in an open system could be formed by a Bose-Einstein condensate coupled to a high-finesse optical cavity. This experiment on the Bose-Einstein condensate trapped in the optical cavity have opened new frontiers, which could combine the cold atoms with quantum optics and makes it possible to enter into the strongly coupled regime of cavity quantum electrodynamics. In strong coupled regime, the atoms exchange the photons many times before spontaneous emission and cavity losses set in. It has become a hot research topic in recent years and plays an important role in many fields of modern physics, such as condensed matter physics, nuclear physics, etc. It can be applied to the manipulation of the geometric phase and entanglement in quantum information and computing. Quantum phase transition has been widely studied for the Dicke model as a typical example. Many different research methods about the mean-field approximation have been used to analyze the ground state properties of the Dicke model. In this paper, we study the ground state properties of two-component Bose-Einstein condensate in a single-mode cavity. Meanwhile, the associated quantum phase transition is described by the spin-coherent-state variational method, whose advantage is that the ground state energy and wave function can be obtained without the thermodynamic limit. By taking the average in the boson coherent state, we obtain an equivalent effective pesudospin Hamiltonian, which will be diagonalized by using the spin coherent state. Finally, we can obtain the energy functional, which is the basics of the variation to obtain the numerical solution of photon number and the expression of the atomic number and the ground state energy. This paper presents a rich phase diagram, which can be manipulated by changing the atom-field coupling imbalance between two components and the atom-field frequency detuning. While in the single-mode Dicke model there exist only the normal phase and the superradiation phase. When the frequency of one component atom is zero or the frequency of the two component atoms are equal in optical cavity, the system returns to the standard Dicke model, in which there occurs the second-order phase transition from the normal phase to the superradiant phase by adjusting the atom-field coupling. In conclusion, we discover that the stimulated radiation comes from the collective state of atomic population inversion, which does not exist in the single-mode Dicke model. Meanwhile, the new stimulated-radiation state S and S, which can only be produced by one component of the atom, are observed in the two-component Bose-Einstein condensates in the single-mode optical cavity. By adjusting the atom-field coupling imbalance and the atom-field frequency detuning (the blue or red detuning), the order of the superradiation state and the stimulated-radiation states can be exchanged between the two components of the atom.
In the present article, we investigate the behavior of a large collection of three-level $\mathrm{\ensuremath{\Lambda}}$ atoms interacting with two quantized electromagnetic fields inside a second-order nonlinear quantum cavity. The nonlinearity of the cavity is supposed to be activated by the application of two classical pump fields. The total Hamiltonian of the combination, with due attention to the nonlinearity effects, is diagonalized. The corresponding ground-state energy then follows from minimizing the total Hamiltonian. The structure of the ground state indicates that four distinct optical phases can occur in such a system. These possible phases turn out to be trivial, dark, left-arm and right-arm superradiant ones. Conditions under which any of the four optical phases can actually occur are also analyzed and discussed. The analysis of the conditions, accompanied by several figures, then reveals that with a suitable choice of the pump field amplitudes and/or geometrical phases, one can intensify the two superradiant phases drastically. Moreover, we demonstrate that the dark optical phase cannot occur at all, while the trivial and superradiant ones can, in fact, coexist. Our calculations also show that transition from the trivial phase to left-arm (right-arm) superradiant one is continuous (discrete) and second (first) order in nature. Another important result of our investigation is that by adjusting the pump field strength, one can switch from the left-arm superradiant phase to the right-arm one and vice versa. This point, in turn, provides an alternative mechanism for the development of quantum optical switching devices.