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https://en.m.wikipedia.org/wiki/Dicke_model
The Dicke model is a quantum mechanical model that describes the coupling between a single-mode cavity and two-level systems, or equivalently spin-½ degrees of freedom. The model was first introduced in 1973 by K. Hepp and E. H. Lieb. Their study was inspired by the pioneering work of R. H. Dicke on the superradiant emission of light in free space and named after him.

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The Dicke model is a quantum mechanical model that describes the coupling between a single-mode cavity and two-level systems, or equivalently spin-½ degrees of freedom. The model was first introduced in 1973 by K. Hepp and E. H. Lieb. Their study was inspired by the pioneering work of R. H. Dicke on the superradiant emission of light in free space and named after him.

Like any other model in quantum mechanics, the Dicke model includes a set of quantum states (the Hilbert space) and a total-energy operator (the Hamiltonian). 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 Hilbert space of the cavity can be spanned by Fock states with photons, denoted by . These states can be constructed from the vacuum state using the canonical ladder operators, and , which add and subtract a photon from the cavity, respectively. The states of each two-level system are referred to as up and down and are defined through the spin operators , satisfying the spin algebra . Here is the Planck constant and indicates a specific two-level system.

The Hamiltonian of the Dicke model is

H = ℏ ω c a † a + ω z ∑ j = 1 N σ j z + 2 λ N ( a + a † ) ∑ j σ j x . {\displaystyle H=\hbar \omega _{c}a^{\dagger }a+\omega _{z}\sum _{j=1}^{N}\sigma _{j}^{z}+{\frac {2\lambda }{\sqrt {N}}}(a+a^{\dagger })\sum _{j}\sigma _{j}^{x}\;.} (1)

Here, the first term describes the energy of the cavity and equals to the product of the energy of a single cavity photon (where is the cavity frequency), times the number of photons in the cavity, . The second term describes the energy of the two-level systems, where is the energy difference between the states of each two-level system. The last term describes the coupling between the two-level systems and the cavity and is assumed to be proportional to a constant, , times the inverse of the square root of the number of two-level systems. This assumption allows one to obtain a phase transition in the limit of (see below). The coupling can be written as the sum of two terms: a co-rotating term that conserves the number of excitations and is proportional to and a counter-rotating term proportional to , where are the spin ladder operators.

The Hamiltonian in Eq. 1 assumes that all the spins are identical (i.e. have the same energy difference and are equally coupled to the cavity). Under this assumption, one can define the macroscopic spin operators , with , which satisfy the spin algebra, . Using these operators, one can rewrite the Hamiltonian in Eq. 1 as

H = ℏ ω c a † a + ω z S z + 2 λ N ( a + a † ) S x . {\displaystyle H=\hbar \omega _{c}a^{\dagger }a+\omega _{z}S^{z}+{\frac {2\lambda }{\sqrt {N}}}(a+a^{\dagger })S^{x}.} (2)

This notation simplifies the numerical study of the model because it involves a single spin-S with , whose Hilbert space has size , rather than spin-1/2, whose Hilbert space has size .

The Dicke model has one global symmetry,

P : ( a , σ ± ) → ( − a , − σ ± ) . {\displaystyle {\mathcal {P}}:(a,\sigma ^{\pm })\to (-a,-\sigma ^{\pm })\;.} (3)

Because squares to unity (i.e. if applied twice, it brings each state back to its original state), it has two eigenvalues, and . This symmetry is associated with a conserved quantity: the parity of the total number of excitations, , where

N e x = a † a + ∑ j = 1 N σ j z . {\displaystyle N_{ex}=a^{\dagger }a+\sum _{j=1}^{N}\sigma _{j}^{z}\;.} (4)

This parity conservation can be seen from the fact that each term in the Hamiltonian preserves the excitation number, except for the counter-rotating terms, which can only change the excitation number by . A state of the Dicke model is said to be normal when this symmetry is preserved, and superradiant when this symmetry is spontaneously broken.

Related models
The Dicke model is closely related to other models of quantum optics. Specifically, the Dicke model with a single two-level system, , is called the Rabi model. In the absence of counter-rotating terms, the model is called Jaynes-Cummings for and Tavis-Cummings for . These two models conserve the number of excitations and are characterized by a symmetry. The spontaneous breaking of this symmetry gives rise to a lasing state (see below).

The relation between the Dicke model and other models is summarized in the table below
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