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Thermodynamics of the extended Dicke model

Philipp Pilar

Published Feb 1, 2019• 0 citations
Publisher

Abstract

The derivation of effective light-matter Hamiltonians in the context of cavity QED is revisited and the importance of making the correct choice of gauge is demonstrated. Controversies surrounding the quantum Rabi model as to whether the ultrastrong coupling regime can be reached are resolved and the adequacy of the effective models when applied to a superconducting circuit setup is shown. The effects of finite temperatures on the superradiant phase transition and the crossover to the subradiant phase in the extended Dicke model are exhibited. Applicability and limitations of the Holstein-Primakoff approximation as well as a mean field approach to the extended Dicke model are investigated. The heat capacity of the extended Dicke model is examined for a broad range of parameters.

DOI
10.34726/hss.2019.50580

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The derivation of effective light-matter Hamiltonians in the context of cavity QED is revisited and the importance of making the correct choice of gauge is demonstrated. Controversies surrounding the quantum Rabi model as to whether the ultrastrong coupling regime can be reached are resolved and the adequacy of the effective models when applied to a superconducting circuit setup is shown. The effects of finite temperatures on the superradiant phase transition and the crossover to the subradiant phase in the extended Dicke model are exhibited. Applicability and limitations of the Holstein-Primakoff approximation as well as a mean field approach to the extended Dicke model are investigated. The heat capacity of the extended Dicke model is examined for a broad range of parameters.
We revisit the derivation of Rabi- and Dicke-type models, which are commonly used for the study of quantum light-matter interactions in cavity and circuit QED. We demonstrate that the validity of the two-level approximation, which is an essential step in this derivation, depends explicitly on the choice of gauge once the system enters the ultrastrong-coupling regime. In particular, while in the electric dipole gauge the two-level approximation can be performed as long as the Rabi frequency remains much smaller than the energies of all higher-lying levels, it can dramatically fail in the Coulomb gauge, even for systems with an extremely anharmonic spectrum. We extensively investigate this phenomenon both in the single-dipole (Rabi) and multidipole (Dicke) case, and consider the specific examples of dipoles confined by double-well and square-well potentials, and of circuit QED systems with flux qubits coupled to an LC resonator.
We study effective light-matter interactions in a circuit QED system consisting of a single LCLCLC resonator, which is coupled symmetrically to multiple superconducting qubits. Starting from a minimal circuit model, we demonstrate that, in addition to the usual collective qubit-photon coupling, the resulting Hamiltonian contains direct qubit-qubit interactions, which have a drastic effect on the ground- and excited-state properties of such circuits in the ultrastrong-coupling regime. In contrast to the superradiant phase transition expected from the standard Dicke model, we find an opposite mechanism, which at very strong interactions completely decouples the photon mode and projects the qubits into a highly entangled ground state. These findings resolve previous controversies over the existence of superradiant phases in circuit QED, but they more generally show that the physics of two- or multiatom cavity QED settings can differ significantly from what is commonly assumed.
In order to examine whether or not the quantum phase transition of Dicke type exists in realistic systems, we revisit the model setup of the superconducting circuit QED from a microscopic many-body perspective based on the BCS theory with pseudo-spin presentation. By deriving the Dicke model with the correct charging terms from the minimum coupling principle, it is shown that the circuit QED system can exhibit superradiant quantum phase transition in the limit N\rightarrow\infty. The critical point could be reached at easiness by adjusting the extra parameters, the ratio of Josephson capacitance C_{J} to gate capacitance C_{g}, as well as the conventional one, the ratio of Josephson energy E_{J} to charging energy E_{C}.
The long-standing debate whether the phase transition in the Dicke model can be realized with dipoles in electromagnetic fields is still an unsettled one. The well-known statement often referred to as the ``no-go theorem'' asserts that the so-called A2{A}^{2}A2 term, missing from the Dicke model, is relevant and prevents the system from undergoing a phase transition. At variance with this common belief, in this paper we prove that the Dicke model does give a proper description of the interaction of light field with the internal excitation of atoms. The phase transition and the spontaneous formation of a transverse-electric mean field cannot be excluded by principle. We point out that the proper treatment of the single-mode approximation is crucial, wherefore we present a systematic derivation of the dipole gauge inside a perfect Fabry-P\'erot cavity from first principles. Besides the impact on the debate around the Dicke phase transition, such a cleanup of the theoretical ground of cavity QED is important because currently there are many emerging experimental approaches to reach strong or even ultrastrong coupling between dipoles and photons, which demand a correct treatment of the Dicke model parameters.
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.
Abstract One of the long-standing problems in quantum optics and laser physics is the observation of both equilibrium and non-equilibrium (Dicke) superradiant states in two-level (qubit) systems interacting with the quantized electromagnetic (EM) field. We demonstrate that the superradiant phase transition (PT) at thermal equilibrium in such a system is close to ultrastrong light–matter coupling (USC) condition, which is currently available for superconducting devices. We examine the USC regime for EM fields interacting with two-level systems located in highly connected graph nodes of photonic networks beyond the rotating wave approximation. We find that a critical matter–field coupling parameter reduces ⟨ k ⟩ times due to network peculiarities providing suitable conditions for equilibrium PT observation; ⟨ k ⟩ is a network average node degree. We study the non-equilibrium (Dicke) superradiance for the same network system and show that superradiance appears as a non-equilibrium PT to the superradiant laser operating within a bad cavity limit. We show that population inversion in this case plays the same role as the reciprocal temperature inherent to the equilibrium superradiant PT. Our results open new perspectives for experimental observation of superradiance in cavity-free systems in the gigahertz photonic regime and beyond.