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Optics and Spectroscopy

Operator method for calculating the spectrum of states in the framework of the extended dicke model

O. A. Boyarshinova, И. Д. Феранчук

Published Aug 1, 2007• 0 citations
Publisher

Abstract

A system of N two-level atoms interacting with a resonant single-mode quantum field (the Dicke model) is described using the operator method for solving the Schrödinger equation. The spectrum of states is calculated without recourse to the rotating-wave approximation and the assumption regarding smallness of the linear sizes of the system as compared to the radiation wavelength. Analytical approximate expressions are obtained for the energy spectrum. These expressions approximate the energy spectrum over the entire range of Hamiltonian parameters in the normal and collective states of the system and make it possible to calculate the thermodynamic characteristics.

DOI
10.1134/s0030400x0708019x

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The Dicke Hamiltonian is linearized by expanding the field and atomic shift operators about disposable $c$- number parameters. When these parameters are chosen to be the expectation values of the corresponding operators with respect to the linearized Hamiltonian, the difference between the free energy of the original Hamiltonian and the free energy of the linearized Hamiltonian is minimized. This difference (per particle) vanishes in the thermodynamic limit; so the equilibrium statistical mechanics of the system is described by the linearized Hamiltonian. The disposable parameters are order parameters of the linearized Hamiltonian. The field and atom order parameters obey a system of coupled nonlinear equations characteristic of meanfield theories. These equations determine the critical temperature and the ordered-state behavior. The thermodynamic state of each subsystem is a statistical superposition of thermal noise and a coherent state produced by the complementary subsystem acting as a classical source. The classical driving terms are the order parameters. The order parameter and coherent-state parameter are equal for the field subsystem but not for the atomic subsystem. The ordered state is characterized by an enhanced condensation of the atomic subsystem into the state of maximum cooperation number $r$. This enhancement is attributed entirely to the Starksplitting of the atomic energy levels due to the classical driving field. A different but equivalent system of coupled nonlinear order-parameter equations is derived from the order-parameter equations of motion in thermodynamic equilibrium in the thermodynamic limit. This alternative form of the coupled self-consistent equations leads to a very simple method for locating the critical temperature and determining the ordered-state behavior of interacting systems. This method is illustrated by determining the gap equations for three model Hamiltonians. These describe (a) interaction of spin- $j$ systems with a single mode of the radiation field, (b) interaction of two-level atoms with a finite number of modes of the radiation field, and (c) interaction of two-level atoms with one or two modes of the radiation field through double photon absorption and emission processes.

Physical Review A•May 1, 1976•41 citations
A system of N two-level atoms interacting with a resonant single-mode quantum field (the Dicke model) is described using the operator method for solving the Schrödinger equation. The spectrum of states is calculated without recourse to the rotating-wave approximation and the assumption regarding smallness of the linear sizes of the system as compared to the radiation wavelength. Analytical approximate expressions are obtained for the energy spectrum. These expressions approximate the energy spectrum over the entire range of Hamiltonian parameters in the normal and collective states of the system and make it possible to calculate the thermodynamic characteristics.
In this paper we analyze the energy eigenvalues of the Dicke Hamiltonian which describes the interaction of NNN two-level atoms with an electromagnetic field. Using as a starting point the classification of the collective states proposed by Tavis and Cummings, we first derive approximate analytic expressions for the energy eigenvalues which are appropriate for special values of the collective parameters. We next evaluate a first-order correction which allows the application of our results to a wider class of collective states. An accurate description of the energy spectrum for arbitrary values of the collective parameters is provided using a method first introduced by Scharf for the case of resonant interaction, suitably generalized to account for possible frequency detuning between the radiation and the atomic resonant frequency. The limit of validity of the approximation is analyzed and the results compared with the numerical diagonalization of the Dicke Hamiltonian. With the exception of asymptotically small ranges of values of the parameters, the analytic expression for the density of energy eigenvalues derived in this paper compares very well with the results of the numerical diagonalization.
The Dicke model describes an ensemble of N identical two-level atoms (qubits) coupled to a single mode of a bosonic field. The fermion Dicke model should be obtained by changing the atomic pseudo-spin operators by a linear combination of Fermi operators. The generalized fermion Dicke model is defined introducing different coupling constants between the single mode of the bosonic field and the reservoir. In the thermodynamic limit, the fermion Dicke model can be analized using the path integral approach with functional method. The system exhibits a second order phase transition from normal to superrandiance at some critical temperature with the presence of a condensate. We evaluate the critical transition temperature and present the spectrum of the collective bosonic excitations. There is quantum phantum critical behavior when the coupling constants satisfy an especific condition. Two particular situations are analyzed. First, we present the spectrum of the collective bosonic excitations in the case using the rotating-wave approximation, recovering the well know results. Second, the case only considering virtual processes. In the last case, it is possible to have a superradiance phase when only virtual processes are introduced in the interaction Hamiltonian. Here also appears a quantum phase transition at some critical coupling, and for larger values for the critical coupling, the system enter in this superradiant phase with a Goldstone mode.
Abstract In the present paper we show that the Hamiltonian describing the resonant interaction of N two-level systems with a single-mode electromagnetic quantum field in the Coulomb gauge can be diagonalized with a high degree of accuracy using a simple basis set of states. This allows one to find an analytical approximation for the eigenvectors and eigenvalues of the system, which interpolates the numerical solution in a broad range of the coupling constant values. For the first time, we show that the energy spectrum of Dicke model (DM) as a function of N (with N ≫ 1 ) satisfies the simple scaling relation. In addition, the introduced basis states provide a regular way of calculating the corrections and estimating the convergence to the exact numerical solution. Analytical approximations and numerical results describing the energy spectrum are valid for both quantum Rabi model ( N = 1) and DM for N ⩾ 2 atoms.
The existence of sub-radiance is demonstrated for a generalized rotating-wave approximation Dicke Hamiltonian model characterized by an arbitrary space dependence of the coupling constants of N two-level atoms to a single mode of a radiation field. To deduce this result we first establish the form of a characteristic equation for sub-radiance and then solve it by introducing a suitable non-unitary operator. In this way the explicit form of all the sub-radiant states of the spatially extended model are easily obtained from those that are well known and relative to the point-like Dicke model. Our investigation clearly shows that inhomogeneous coupling of the N atoms with the field is compatible with sub-radiant behaviour.
The Dicke model describes an ensemble of N identical two-level atoms (qubits) coupled to a single quantized mode of a bosonic field. The fermion Dicke model should be obtained by changing the atomic pseudo-spin operators by a linear combination of Fermi operators. The generalized fermion Dicke model is defined introducing different coupling constants between the single mode of the bosonic field and the reservoir, g1 and g2 for rotating and counter-rotating terms respectively. In the limit N → ∞, the thermodynamic of the fermion Dicke model can be analyzed using the path integral approach with functional method. The system exhibits a second order phase transition from normal to superradiance at some critical temperature with the presence of a condensate. We evaluate the critical transition temperature and present the spectrum of the collective bosonic excitations for the general case (g1 ̸ = 0 and g2 ̸ = 0). There is quantum critical behavior when the coupling constants g1 and g2 satisfy g1 + g2 = (ω0 Ω) 1 2, where ω0 is the frequency of the mode of the field and Ω is the energy gap between energy eigenstates of the qubits. Two particular situations are analyzed. First, we present the spectrum of the collective bosonic excitations, in the case g1 ̸ = 0 and g2 = 0, recovering the well known results. Second, the case g1 = 0 and g2 ̸ = 0 is studied. In this last case, it is possible to have a superradiant phase when only virtual processes are introduced in the interaction Hamiltonian. Here also appears a quantum phase transition at the critical coupling g2 = (ω0 Ω) 1 2, and for larger values for the critical coupling, the system enter in this superradiant phase with a Goldstone mode.
The Dicke Hamiltonian is linearized by expanding the field and atomic shift operators about disposable ccc- number parameters. When these parameters are chosen to be the expectation values of the corresponding operators with respect to the linearized Hamiltonian, the difference between the free energy of the original Hamiltonian and the free energy of the linearized Hamiltonian is minimized. This difference (per particle) vanishes in the thermodynamic limit; so the equilibrium statistical mechanics of the system is described by the linearized Hamiltonian. The disposable parameters are order parameters of the linearized Hamiltonian. The field and atom order parameters obey a system of coupled nonlinear equations characteristic of meanfield theories. These equations determine the critical temperature and the ordered-state behavior. The thermodynamic state of each subsystem is a statistical superposition of thermal noise and a coherent state produced by the complementary subsystem acting as a classical source. The classical driving terms are the order parameters. The order parameter and coherent-state parameter are equal for the field subsystem but not for the atomic subsystem. The ordered state is characterized by an enhanced condensation of the atomic subsystem into the state of maximum cooperation number rrr. This enhancement is attributed entirely to the Starksplitting of the atomic energy levels due to the classical driving field. A different but equivalent system of coupled nonlinear order-parameter equations is derived from the order-parameter equations of motion in thermodynamic equilibrium in the thermodynamic limit. This alternative form of the coupled self-consistent equations leads to a very simple method for locating the critical temperature and determining the ordered-state behavior of interacting systems. This method is illustrated by determining the gap equations for three model Hamiltonians. These describe (a) interaction of spin- jjj systems with a single mode of the radiation field, (b) interaction of two-level atoms with a finite number of modes of the radiation field, and (c) interaction of two-level atoms with one or two modes of the radiation field through double photon absorption and emission processes.