Gaussian Zoos

vacuumstate([Tm=Vector{Float64}, Tc=Matrix{Float64}], basis::SymplecticBasis)

Gaussian state with zero photons, known as the vacuum state. The symplectic representation is defined by basis.

Mathematical description of a vacuum state

A vacuum state |0⟩ is characterized by the zero mean vector and covariance matrix (ħ/2)I.

Example

julia> vacuumstate(QuadPairBasis(1))
GaussianState for 1 mode.
  symplectic basis: QuadPairBasis
mean: 2-element Vector{Float64}:
 0.0
 0.0
covariance: 2×2 Matrix{Float64}:
 1.0  0.0
 0.0  1.0

thermalstate([Tm=Vector{Float64}, Tc=Matrix{Float64},] basis::SymplecticBasis, photons<:Int)

Gaussian state at thermal equilibrium, known as the thermal state. The symplectic representation is defined by basis. The mean photon number of the state is given by photons.

Mathematical description of a thermal state

A thermal state |n̄⟩, where n̄ is the mean number of photons, is characterized by the zero mean vector and covariance matrix ħ(n̄+1/2)I.

Example

julia> thermalstate(QuadPairBasis(1), 4)
GaussianState for 1 mode.
  symplectic basis: QuadPairBasis
mean: 2-element Vector{Float64}:
 0.0
 0.0
covariance: 2×2 Matrix{Float64}:
 9.0  0.0
 0.0  9.0

coherentstate([Tm=Vector{Float64}, Tc=Matrix{Float64},] basis::SymplecticBasis, alpha<:Number)

Gaussian state that is the quantum analogue of a monochromatic electromagnetic field, known as the coherent state. The symplectic representation is defined by basis. The complex amplitude of the state is given by alpha.

Mathematical description of a coherent state

A coherent state |α⟩, where α is the complex amplitude, is characterized by the mean vector √2ħ [real(α), imag(α)] and covariance matrix (ħ/2)I.

Example

julia> coherentstate(QuadPairBasis(1), 1.0+im)
GaussianState for 1 mode.
  symplectic basis: QuadPairBasis
mean: 2-element Vector{Float64}:
 2.0
 2.0
covariance: 2×2 Matrix{Float64}:
 1.0  0.0
 0.0  1.0

squeezedstate([Tm=Vector{Float64}, Tc=Matrix{Float64},] basis::SymplecticBasis, r<:Real, theta<:Real)

Gaussian state with quantum uncertainty in its phase and amplitude, known as the squeezed state. The symplectic representation is defined by basis. The amplitude and phase squeezing parameters are given by r and theta, respectively.

Mathematical description of a squeezed state

A squeezed state |r, θ⟩, where r is the amplitude squeezing parameter and θ is the phase squeezing parameter, is characterized by the zero mean vector and covariance matrix (ħ/2) (cosh(2r)I - sinh(2r)R(θ)), where R(θ) is the rotation matrix.

Example

julia> squeezedstate(QuadPairBasis(1), 0.5, pi/4)
GaussianState for 1 mode.
  symplectic basis: QuadPairBasis
mean: 2-element Vector{Float64}:
 0.0
 0.0
covariance: 2×2 Matrix{Float64}:
  0.712088  -0.830993
 -0.830993   2.37407

eprstate([Tm=Vector{Float64}, Tc=Matrix{Float64},] basis::SymplecticBasis, r<:Real, theta<:Real)

Gaussian state that is a two-mode squeezed state, known as the Einstein-Podolski-Rosen (EPR) state. The symplectic representation is defined by basis. The amplitude and phase squeezing parameters are given by r and theta, respectively.

Mathematical description of an EPR state

An EPR state |r, θ⟩ₑₚᵣ, where r is the amplitude squeezing parameter and θ is the phase squeezing parameter, is characterized by the zero mean vector and covariance matrix (ħ/2)[cosh(2r)I -sinh(2r)R(θ); -sinh(2r)R(θ) cosh(2r)I], where R(θ) is the rotation matrix.

Example

julia> eprstate(QuadPairBasis(2), 0.5, pi/4)
GaussianState for 2 modes.
  symplectic basis: QuadPairBasis
mean: 4-element Vector{Float64}:
 0.0
 0.0
 0.0
 0.0
covariance: 4×4 Matrix{Float64}:
  1.54308    0.0       -0.830993  -0.830993
  0.0        1.54308   -0.830993   0.830993
 -0.830993  -0.830993   1.54308    0.0
 -0.830993   0.830993   0.0        1.54308

displace([Tm=Vector{Float64}, Ts=Matrix{Float64}], basis::SymplecticBasis, alpha<:Number)
displace([Tm=Vector{Float64}, Ts=Matrix{Float64}], basis::SymplecticBasis, alpha<:Number, noise::Ts)

Gaussian operator that displaces the vacuum state into a coherent state, known as the displacement operator. The symplectic representation is given by basis. The complex amplitude is given by alpha. Noise can be added to the operation with noise.

Mathematical description of a displacement operator

A displacement operator D(α) is defined by the operation D(α)|0⟩ = |α⟩, where α is a complex amplitude. The operator D(α) is characterized by the displacement vector √2ħ [real(α), imag(α)] and symplectic matrix I.

Example

julia> displace(QuadPairBasis(1), 1.0+im)
GaussianUnitary for 1 mode.
  symplectic basis: QuadPairBasis
displacement: 2-element Vector{Float64}:
 2.0
 2.0
symplectic: 2×2 Matrix{Float64}:
 1.0  0.0
 0.0  1.0

squeeze([Tm=Vector{Float64}, Ts=Matrix{Float64}], basis::SymplecticBasis, r<:Real, theta<:Real)
squeeze([Tm=Vector{Float64}, Ts=Matrix{Float64}], basis::SymplecticBasis, r<:Real, theta<:Real, noise::Ts)

Gaussian operator that squeezes the vacuum state into a squeezed state, known as the squeezing operator. The symplectic representation is given by basis. The amplitude and phase squeezing parameters are given by r and theta, respectively. Noise can be added to the operation with noise.

Mathematical description of a squeezing operator

A squeeze operator S(r, θ) is defined by the operation S(r, θ)|0⟩ = |r, θ⟩, where r and θ are the real amplitude and phase parameters, respectively. The operator S(r, θ) is characterized by the zero displacement vector and symplectic matrix cosh(r)I - sinh(r)R(θ), where R(θ) is the rotation matrix.

Example

julia> squeeze(QuadPairBasis(1), 0.25, pi/4)
GaussianUnitary for 1 mode.
  symplectic basis: QuadPairBasis
displacement: 2-element Vector{Float64}:
 0.0
 0.0
symplectic: 2×2 Matrix{Float64}:
  0.852789  -0.178624
 -0.178624   1.21004

twosqueeze([Tm=Vector{Float64}, Ts=Matrix{Float64}], basis::SymplecticBasis, r<:Real, theta<:Real)
twosqueeze([Tm=Vector{Float64}, Ts=Matrix{Float64}], basis::SymplecticBasis, r<:Real, theta<:Real, noise::Ts)

Gaussian operator that squeezes a two-mode vacuum state into a two-mode squeezed state, known as the two-mode squeezing operator. The symplectic representation is given by basis. The amplitude and phase squeezing parameters are given by r and theta, respectively. Noise can be added to the operation with noise.

Mathematical description of a two-mode squeezing operator

A two-mode squeeze operator S₂(r, θ) is defined by the operation S₂(r, θ)|0⟩ = |r, θ⟩, where r and θ are the real amplitude and phase parameters, respectively. The operator S₂(r, θ) is characterized by the zero displacement vector and symplectic matrix [cosh(r)I -sinh(r)R(θ); -sinh(r)R(θ) cosh(r)I], where R(θ) is the rotation matrix.

Example

julia> twosqueeze(QuadPairBasis(2), 0.25, pi/4)
GaussianUnitary for 2 modes.
  symplectic basis: QuadPairBasis
displacement: 4-element Vector{Float64}:
 0.0
 0.0
 0.0
 0.0
symplectic: 4×4 Matrix{Float64}:
  1.03141    0.0       -0.178624  -0.178624
  0.0        1.03141   -0.178624   0.178624
 -0.178624  -0.178624   1.03141    0.0
 -0.178624   0.178624   0.0        1.03141

phaseshift([Tm=Vector{Float64}, Ts=Matrix{Float64}], basis::SymplecticBasis, theta<:Real)
phaseshift([Tm=Vector{Float64}, Ts=Matrix{Float64}], basis::SymplecticBasis, theta<:Real, noise::Ts)

Gaussian operator that rotates the phase of a given Gaussian mode by theta, as the phase shift operator. The symplectic representation is given by basis. Noise can be added to the operation with noise.

Mathematical description of a phase shift operator

A phase shift operator is defined by the operation U(θ) = exp(-iθâᵗâ), where θ is the phase parameter, and âᵗ and â are the raising and lowering operators, respectively. The operator U(θ) is characterized by the zero displacement vector and symplectic matrix [cos(θ) sin(θ); -sin(θ) cos(θ)].

Example

julia> phaseshift(QuadPairBasis(1), 3pi/4)
GaussianUnitary for 1 mode.
  symplectic basis: QuadPairBasis
displacement: 2-element Vector{Float64}:
 0.0
 0.0
symplectic: 2×2 Matrix{Float64}:
 -0.707107   0.707107
 -0.707107  -0.707107

beamsplitter([Tm=Vector{Float64}, Ts=Matrix{Float64}], basis::SymplecticBasis, transmit<:Real)
beamsplitter([Tm=Vector{Float64}, Ts=Matrix{Float64}], basis::SymplecticBasis, transmit<:Real, noise::Ts)

Gaussian operator that serves as the beam splitter transformation of a two-mode Gaussian state, known as the beam splitter operator. The symplectic representation is given by basis. The transmittivity of the operator is given by transmit. Noise can be added to the operation with noise.

Mathematical description of a beam splitter operator

A beam splitter operator B(τ) is defined by the operation B(τ) = exp(θ(âᵗb̂ - âb̂ᵗ)), where θ is defined by τ = cos²θ, and â and b̂ are the annihilation operators of the two modes, respectively. The operator B(τ) is characterized by the zero displacement vector and symplectic matrix [√τI √(1-τ)I; -√(1-τ)I √τI].

Example

julia> beamsplitter(QuadPairBasis(2), 0.75)
GaussianUnitary for 2 modes.
  symplectic basis: QuadPairBasis
displacement: 4-element Vector{Float64}:
 0.0
 0.0
 0.0
 0.0
symplectic: 4×4 Matrix{Float64}:
  0.5        0.0       0.866025  0.0
  0.0        0.5       0.0       0.866025
 -0.866025   0.0       0.5       0.0
  0.0       -0.866025  0.0       0.5

attenuator([Td=Vector{Float64}, Tt=Matrix{Float64},] basis::SymplecticBasis, theta<:Real, n<:Int)

Gaussian channel describing the coupling of an input single mode Gaussian state and its environment via a beam splitter operation. The channel is paramatrized by beam splitter rotation angle theta and thermal noise n.

Mathematical description of an attenuator channel

An attenuator channel, E(θ, nₜₕ), where θ is the beam splitter rotation parameter and nₜₕ ≥ 1 is the thermal noise parameter, is characterized by the zero displacement vector, transformation matrix cos(θ)I, and noise matrix nₜₕsin²(θ)I.

Example

julia> attenuator(QuadPairBasis(1), pi/6, 3)
GaussianChannel for 1 mode.
  symplectic basis: QuadPairBasis
displacement: 2-element Vector{Float64}:
 0.0
 0.0
transform: 2×2 Matrix{Float64}:
 0.866025  0.0
 0.0       0.866025
noise: 2×2 Matrix{Float64}:
 0.75  0.0
 0.0   0.75

amplifier([Td=Vector{Float64}, Tt=Matrix{Float64},] basis::SymplecticBasis, r<:Real, n<:Int)

Gaussian channel describing the interaction of an input single mode Gaussian state and its environment via a two-mode squeezing operation. The channel is paramatrized by squeezing amplitude parameter r and thermal noise n.

Mathematical description of an amplifier channel

An amplifier channel, A(r, nₜₕ), where r is the squeezing amplitude parameter and nₜₕ ≥ 1 is the thermal noise parameter, is characterized by the zero displacement vector, transformation matrix cosh(r)I, and noise matrix nₜₕsinh²(r)I.

Example

julia> amplifier(QuadPairBasis(1), 2.0, 3)
GaussianChannel for 1 mode.
  symplectic basis: QuadPairBasis
displacement: 2-element Vector{Float64}:
 0.0
 0.0
transform: 2×2 Matrix{Float64}:
 3.7622  0.0
 0.0     3.7622
noise: 2×2 Matrix{Float64}:
 39.4623   0.0
  0.0     39.4623