DickeModel.ClassicalDicke

abstract type DickeSystem <: ClassicalSystems.ClassicalSystem

Abstract subtype of ClassicalSystems.ClassicalSystem which represents the classical Dicke model. To generate it, use the concrete subtypes ClassicalDickeSystem or DickeBCE.QuantumDickeSystem.

struct ClassicalDickeSystem <: DickeSystem

Subtype of ClassicalSystems.ClassicalSystem which represents the classical Dicke model with the given parameters $ω_0$, $ω$, and $γ$. See Eq. (5) of Ref. [13]. To generate this struct, use the constructor

ClassicalDickeSystem(;ω₀::Real,ω::Real,γ::Real)

For example, system = ClassicalDickeSystem(ω₀=1, ω=1, γ=1).

This struct may be passed to all functions in this module that require an instance of DickeSystem, as well as functions in other modules that require ClassicalSystems.ClassicalSystem, such as ClassicalSystems.integrate.

function hamiltonian(system::DickeSystem)

Returns a classical Hamiltonian function h(x) where x=[Q,q,P,p], which is given by Eq. (5) of Ref. [13].

Arguments

• system should be a subtype of DickeSystem.

function Point(;Q::Real,q::Real,P::Real,p::Real)

Returns the array [Q,q,P,p]. This ordering is used throughout this package.

function Point(system::DickeSystem;
    Q::Real,
    P::Real,
    p::Real,
    ϵ::Real,
    sgn::Union{typeof(-),typeof(+)} = +)

Returns a list [Q,q,P,p], where q is calculated with ClassicalDicke.q_of_ϵ. If there are no solutions for $q$, an error is raised.

function Pointθϕ(;θ::Real,ϕ::Real,q::Real,p::Real)

Returns a list [Q,q,P,p], where Q and P are calculated from θ and ϕ using PhaseSpaces.Q_of_θϕ and PhaseSpaces.P_of_θϕ.

function energy_shell_volume(system::DickeSystem,ϵ::Real)

Returns the volume of the classical energy shell in the phase space, that is,

\[\mathcal{V}(\mathcal{M}_ϵ) = \int_\mathcal{M} \text{d}\mathbf{x} \, \delta(h_\text{cl}(\mathbf{x})- \epsilon).\]

This is computed using a modified version of Eq. (A2) in [15], (the integral is performed in the variable $Q$ instead of $y$). That equation was originally derived in Ref. [3], with different units for $\epsilon$.

The integration is performed using QuadGK.

Arguments

• system should be a subtype of DickeSystem.

• ϵ is the energy.

function phase_space_dist_squared(x::AbstractVector{<:Real},y::AbstractVector{<:Real})

Returns the phase-space distance $d_{\mathcal{M}}(\mathbf{x},\mathbf{y})$ (See App. C of Ref [13]), where x and y are vectors in the form [Q,q,P,p].

function classical_path_random_sampler(system::DickeSystem; ϵ::Real, dt::Real=3)

This function returns a function sample(), which produces random points within the classical energy shell at energy ϵ. The function sample() returns points from a fixed chaotic trajectory picked at random separated by a fixed time interval dt. If the classical dynamics are ergodic, which only happens in chaotic regions, this produces random points in the energy shell.

Arguments

• system should be a subtype of DickeSystem.

Keyword arguments

• ϵ is the energy of the energy shell from where to sample.
• dt is the fixed time interval that separates the points that are returned by sample().

function minimum_energy(system::DickeSystem)

Returns the ground-state energy, as given by multiplying Eq. (15) of Ref. [3] by $\omega_0$ (because in that work they use $\epsilon =E/(\omega_0 j)$, and in this module we use $\epsilon = E/j$).

Arguments

• system should be a subtype of DickeSystem.

function minimum_ϵ_for(system::DickeSystem;
    Q::Union{Real,Nothing}=nothing,
    q::Union{Real,Nothing}=nothing,
    P::Union{Real,Nothing}=nothing,
    p::Union{Real,Nothing}=nothing)

Returns the minimum energy $\epsilon$ when constraining the system to three fixed values of the coordinates $Q$, $q$, $P$, $p$.

Arguments

• system should be a subtype of DickeSystem.

Keyword arguments

• You may pass either $(Q,q,P)$ or $(q,P,p)$. The other combinanations are not implemented.

This function can be especially useful to draw contours of the available phase space (see Drawing contours of the available phase space)

function minimum_energy_point(system::DickeSystem,
    Qsign::Union{typeof(-),typeof(+)} = +)

Returns the ground-state coordinate, that is, (0, 0, 0, 0) for the normal phase and $\mathbf{x}_\text{GS}$ below Eq. (7) of Ref. [13] for the superradiant phase.

Arguments

• system should be a subtype of DickeSystem.

• Qsign toggles the sign of the $Q$ coordinate in the superradiant phase, that is, + for $\mathbf{x}_\text{GS}$ and - for $\widetilde{\mathbf{x}}_\text{GS}$ . Defaults to +.

function normal_frequency(system::DickeSystem,
    sgn::Union{typeof(-),typeof(+)} = +)

Returns the ground-state normal frequency, that is, $\Omega_{\epsilon_{\text{GS}}}^{A,B}$ at the bottom of page 3 of Ref. [13].

Arguments

• system should be a subtype of DickeSystem.

• sgn is - for $\Omega^A$ and + for $\Omega^B$. Defaults to +.

Note: This function currently only works for the supperadiant phase.

function discriminant_of_q_solution(system::DickeSystem; 
    Q::Real,
    P::Real,
    p::Real,
    ϵ::Real)

Returns the discriminant of the second degree equation in $q$ given by

\[ h_\text{cl}(Q,q,P,p)=\epsilon,\]

where $h_\text{cl}$ is given by Eq. (5) of Ref. [13].

Arguments

• system should be a subtype of DickeSystem.

Keyword arguments

• Q, P, p, and ϵ are the values of $Q$, $P$, $p$, and $\epsilon$, respectively.

function q_of_ϵ(system::DickeSystem;
    Q::Real,
    P::Real,
    p::Real,
    ϵ::Real,
    sgn::Union{typeof(-),typeof(+)}=+,
    returnNaNonError::Bool=true)

Returns the solutions $q_\pm$ of the second degree equation in $q$ given by

\[ h_\text{cl}(Q,q,P,p)=\epsilon,\]

where $h_\text{cl}$ is given by Eq. (5) of Ref. [13].

Arguments

• system should be a subtype of DickeSystem.

Keyword arguments

• Q, P, p, and ϵ are values of $Q$, $P$, $p$, and $\epsilon$, respectively.
• sgn is + for $q_+$ and - for $q_-$
• If returnNaNonError is true, then NaN is returned if there are no solutions. If it is false, and error is raised.

function q_sign(system::DickeSystem,
    x::AbstractVector{<:Real},
    ϵ::Real=hamiltonian(system)(x))

Returns the sign of the root of the second degree equation in $q$ given by

\[ h_\text{cl}(Q,q,P,p)=\epsilon.\]

That is, this function returns + if q=x[2] ≈ q_of_ϵ(system;Q=x[1],P=x[3],p=x[4],ϵ=ϵ,sgn=+) and returns - if q=x[2] ≈ q_of_ϵ(system;Q=x[1],P=x[3],p=x[4],ϵ=ϵ,sgn=-).

Arguments

• system should be a subtype of DickeSystem.

• x is a vector in the form [Q,q,P,p].

• ϵ should be the energy of x. If this is not passed, it is computed using hamiltonian(system)(x).

function Point(system::DickeSystem;
    Q::Real,
    P::Real,
    p::Real,
    ϵ::Real,
    sgn::Union{typeof(-),typeof(+)} = +)

Returns a list [Q,q,P,p], where q is calculated with ClassicalDicke.q_of_ϵ. If there are no solutions for $q$, an error is raised.

function minimum_nonnegative_Q_for_ϵ(system::DickeSystem,ϵ::Real)

Computes the minimum nonnegative value of the parameter $Q$ accessible to the system at energy $\epsilon$.

Arguments

• system should be a subtype of DickeSystem.

• ϵ is the energy.

function maximum_Q_for_ϵ(system::DickeSystem,ϵ::Real)

Computes the maximum value of the parameter $Q$ accessible to the system at energy $\epsilon$.

Arguments

• system should be a subtype of DickeSystem.

• ϵ is the energy.

function maximum_P_for_ϵ(system::DickeSystem,ϵ::Real)

Computes the maximum value of the parameter $P$ accessible to the system at energy $\epsilon$.

Arguments

• system should be a subtype of DickeSystem.

• ϵ is the energy.