Abstract We investigate the ground state of two physically motivated modifications of the Dicke model. The first modification corresponds to particles whose phase space contains only two states, for example, artificially created qubits. The second modification describes two-level systems that arise as a result of truncating the full Hilbert space of atoms to two levels that are in resonance with the electromagnetic field and are described by the gauge-invariant Dicke model. We demonstrate that the behavior of these systems is qualitatively distinct in both cases. In particular, in the first scenario, a superradiant phase transition (SPT) into a state with a non-zero amplitude of the classical field is possible, while in the second case, the so-called order parameter η = 〈 a ˆ 〉 which would indicate condensation into a coherent state remains zero, meaning that the SPT is forbidden. It resolves the long-standing discussion about the SPT in the Dicke model. At the same time, the average number of photons n ¯ = 〈 a ˆ † a ˆ 〉 ≠ 0 , and the collective excitation in the system manifests as a non-classical ‘squeezed’ state of the field. We analyze the observable characteristics of both systems in a wide range of variation of their parameters.
Abstract We investigate the ground state of two physically motivated modifications of the Dicke model. The first modification corresponds to particles whose phase space contains only two states, for example, artificially created qubits. The second modification describes two-level systems that arise as a result of truncating the full Hilbert space of atoms to two levels that are in resonance with the electromagnetic field and are described by the gauge-invariant Dicke model. We demonstrate that the behavior of these systems is qualitatively distinct in both cases. In particular, in the first scenario, a superradiant phase transition (SPT) into a state with a non-zero amplitude of the classical field is possible, while in the second case, the so-called order parameter η = 〈 a ˆ 〉 which would indicate condensation into a coherent state remains zero, meaning that the SPT is forbidden. It resolves the long-standing discussion about the SPT in the Dicke model. At the same time, the average number of photons n ¯ = 〈 a ˆ † a ˆ 〉 ≠ 0 , and the collective excitation in the system manifests as a non-classical ‘squeezed’ state of the field. We analyze the observable characteristics of both systems in a wide range of variation of their parameters.
Abstract In this article, we use the spin coherent state transformation and the ground state variational method to theoretically calculate the ground function. In order to consider the influence of the atom-atom interaction on the extended Dicke model’s ground state properties, the mean photon number, the scaled atomic population and the average ground energy are displayed. Using the self-consistent field theory to solve the atom-atom interaction, we discover the system undergoes a first-order quantum phase transition from the normal phase to the superradiant phase, but a famous Dicke-type second-order quantum phase transition without the atom-atom interaction. Meanwhile, the atom-atom interaction makes the phase transition point shift to the lower atom-photon collective coupling strength.
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.
In view of recent proposals for the realization of anisotropic light-matter interaction in such platforms as (i) nonstationary or inductively and capacitively coupled superconducting qubits, (ii) atoms in crossed fields, and (iii) semiconductor heterostructures with spin-orbital interaction, the concept of a generalized Dicke model, where coupling strengths of rotating wave and counter-rotating wave terms are unequal, has attracted great interest. For this model we study photon fluctuations in the critical region of normal-to-superradiant phase transition when both the temperatures and numbers of two-level systems are finite. In this case, the superradiant quantum phase transition is changed to a fluctuational region in the phase diagram that reveals two types of critical behaviors. These are regimes of Dicke model (with discrete Z2 symmetry), and that of anti-Tavis-Cummings and Tavis-Cummings U(1) models. We show that squeezing parameters of photon condensate in these regimes show distinct temperature scalings. Besides, relative fluctuations of a photon number take universal values. We also find a temperature scale below which one approaches a zero-temperature quantum phase transition where quantum fluctuations dominate. Our effective theory is provided by a non-Goldstone functional for condensate mode and by Majorana representation of Pauli operators. We also discuss the Bethe ansatz solution for integrable U(1) limits.
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 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.
The Dicke Hamiltonian describes the simplest quantum system with atoms interacting with photons: N two-level atoms inside a perfectly reflecting cavity, which allows only one electromagnetic mode. It has also been successfully employed to describe superconducting circuits that behave as artificial atoms coupled to a resonator. The system exhibits a transition to a superradiant phase at zero temperature. When the interaction strength reaches its critical value, both the number of photons and atoms in excited states in the cavity, together with their fluctuations, exhibit a sudden increase from zero. By employing symmetry-adapted coherent states, it is shown that these properties scale with the number of atoms, their reported divergences at the critical point represent the limit when this number goes to infinity, and, in this limit, they remain divergent in the superradiant phase. Analytical expressions are presented for all observables of interest, for any number of atoms. Comparisons with exact numerical solutions strongly support the results.