Dicke model

This transition belongs to the Ising universality class and was realized in cavity quantum electrodynamics experiments.

[1] Their study was inspired by the pioneering work of R. H. Dicke on the superradiant emission of light in free space [2] and named after him.

The Hilbert space of the Dicke model is given by (the tensor product of) the states of the cavity and of the two-level systems.

The last term describes the coupling between the two-level systems and the cavity and is assumed to be proportional to a constant,

In their original derivation, Hepp and Lieb[1] neglected the effects of counter-rotating terms and, thus, actually considered the Tavis-Cummings model (see above).

Further studies of the full Dicke model found that the phase transition still occurs in the presence of counter-rotating terms, albeit at a different critical coupling.

These two values correspond to physical states of the cavity field with opposite phases (see Eq.

The simplest way to describe the superradiant transition is to use a mean-field approximation, in which the cavity field operators are substituted by their expectation values.

At thermal equilibrium (see above), one finds that the free energy per two-level system is[6] The critical coupling of the transition can be found by the condition

For a finite system size, there is a classical and quantum correspondence that breaks down at the Ehrenfest time, which is inversely proportional to

The Dicke model provides an ideal system to study the quantum-classical correspondence and quantum chaos.

1 assumes that the cavity mode and the two-level systems are perfectly isolated from the external environment.

In actual experiments, this assumption is not valid: the coupling to free modes of light can cause the loss of cavity photons and the decay of the two-level systems (i.e. dissipation channels).

The various dissipation channels can be described by adding a coupling to additional environmental degrees of freedom.

By averaging over the dynamics of these external degrees of freedom one obtains equations of motion describing an open quantum system.

According to the common Born-Markov approximation, one can describe the dynamics of the system with the quantum master equation in Lindblad form [17] Here,

Some common decay processes that are relevant to experiments are given in the following table: In the theoretical description of the model, one often considers the steady state where

Dicke superradiance is a collective phenomenon in which many two-level systems emit photons coherently in free space.

[2][18] It occurs if the two-level systems are initially prepared in their excited state and placed at a distance much smaller than the relevant photon's wavelength.

Under ideal conditions, the pulse duration is inversely proportional to the number of two-level systems,

In this system, the observation of the superradiant transition is hindered by two possible problems: (1) The bare coupling between atoms and cavities is usually weak and insufficient to reach the critical value

Both limitations can be circumvented by applying external pumps on the atoms and creating an effective Dicke model in an appropriately rotating frame.

[20][21] In 2010, the superradiant transition of the open Dicke model was observed experimentally using neutral Rubidium atoms trapped in an optical cavity.

This two-photon process causes the two-level system to change its state from down to up, or vice versa, and emit or absorb a photon into the cavity.

[25] [26] In contrast, later experiments[27][28] used two different hyperfine levels of the Rubidium atoms in a magnetic field.

In both experiments, the system is time-dependent and the (generalized) Dicke Hamiltonian is realized in a frame that rotates at the pump's frequency.

The Dicke model can be generalized by considering the effects of additional terms in the Hamiltonian of Eq.

[6] For example, a recent experiment[28] realized an open Dicke model with independently tunable rotating and counter-rotating terms.

Both transitions are of a mean-field type and can be understood in terms of the dynamics of a single degree of freedom.

[6] This article was adapted from the following source under a CC BY 4.0 license (2020) (reviewer reports): Mor M Roses; Emanuele Dalla Torre (4 September 2020).

Schematic plot of the order parameter of the Dicke transition, which is zero in the normal phase and finite in the superradiant phase. The inset shows the free energy in the normal and superradiant phases, see Eq. 5 .
Percentage of classical trajectories with positive Lyapunov exponent as a function of the energy per particle and the coupling parameter (divided by the critical coupling ). The parameters are .
Schematic representation of the difference between Dicke superradiance and the superradiant transition of the open Dicke model.
Schematic representation of two schemes to experimentally realize the Dicke model: on the left, the equilibrium approach based on the dipole coupling between the two levels and, on the right, the nonequilibrium approach based on two-photon processes, namely stimulated Raman scattering. Only the latter scheme is used to realize the Dicke model.