Spin shuttling model defined by a stochastic field, the realization of the stochastic field is specified by the paths of the shuttled spins.

Arguments

• n::Int: Number of spins
• Ψ::Vector{<:Complex}: Initial state of the spin system, the length of the vector must be `2^n
• T::Real: Maximum time
• N::Int: Time discretization
• B::GaussianRandomField: Noise field
• X::Vector{<:Function}: Shuttling paths, the length of the vector must be `n
• initialize::Bool: Initialize the random function

Fields

• R::RandomFunction: Random function defined by the noise field and the paths

Example

model = ShuttlingModel(1, [1+0im, 0+0im], 1.0, 100, OrnsteinUhlenbeckField([1.0, 1.0, 1.0]), [t->t])

General one spin shuttling model initialized at initial state |Ψ₀⟩, with arbitrary shuttling path x(t).

Arguments

• Ψ::Vector{<:Complex}: Initial state of the spin system, the length of the vector must be `2^n
• T::Real: Maximum time
• N::Int: Time discretization
• B::GaussianRandomField: Noise field
• x::Function: Shuttling path
• initialize::Bool: Initialize the random function

Example

model = OneSpinModel(1 / √2 * [1+0im, 1+0im], 1.0, 100, OrnsteinUhlenbeckField([1.0, 1.0, 1.0]), t->t)

One spin shuttling model initialzied at |Ψ₀⟩=|+⟩. The qubit is shuttled at constant velocity along the path x(t)=L/T*t, with total time T in μs and length L in μm.

Arguments

• T::Real: Maximum time
• L::Real: Length of the path
• N::Int: Time discretization
• B::GaussianRandomField: Noise field
• v::Real: Velocity of the shuttling

Example

model = OneSpinModel(1.0, 1.0, 100, OrnsteinUhlenbeckField([1.0, 1.0, 1.0]), 1.0)

One spin shuttling model initialzied at |Ψ₀⟩=|+⟩. The qubit is shuttled at constant velocity along a forth-back path x(t, T, L) = t<T/2 ? 2L/T*t : 2L/T*(T-t), with total time T in μs and length L in μm.

Arguments

• T::Real: Maximum time
• L::Real: Length of the path
• N::Int: Time discretization
• B::GaussianRandomField: Noise field
• v::Real: Velocity of the shuttling

Example

model = OneSpinForthBackModel(1.0, 1.0, 100, OrnsteinUhlenbeckField([1.0, 1.0, 1.0]))

General two spin shuttling model initialized at initial state |Ψ₀⟩, with arbitrary shuttling paths x₁(t), x₂(t).

Arguments

• Ψ::Vector{<:Complex}: Initial state of the spin system, the length of the vector must be `2^n
• T::Real: Maximum time
• N::Int: Time discretization
• B::GaussianRandomField: Noise field
• x₁::Function: Shuttling path for the first spin
• x₂::Function: Shuttling path for the second spin

Example

model = TwoSpinModel(1 / √2 * [1+0im, 1+0im, 1+0im, 1+0im], 1.0, 100, OrnsteinUhlenbeckField([1.0, 1.0, 1.0]), t->t, t->t+1)

Two spin shuttling model initialized at the singlet state |Ψ₀⟩=1/√2(|↑↓⟩-|↓↑⟩). The qubits are shuttled at constant velocity along the path x₁(t)=L/T₁*t and x₂(t)=L/T₁*(t-T₀). The delay between the them is T₀ and the total shuttling time is T₁+T₀. It should be noticed that due to the exclusion of fermions, x₁(t) and x₂(t) cannot overlap.

Arguments

• T₀::Real: Delay time
• T₁::Real: Shuttling time
• L::Real: Length of the path
• N::Int: Time discretization
• B::GaussianRandomField: Noise field

Example

model = TwoSpinSequentialModel(1.0, 1.0, 1.0, 100, OrnsteinUhlenbeckField([1.0, 1.0, 1.0]))

Two spin shuttling model initialized at the singlet state |Ψ₀⟩=1/√2(|↑↓⟩-|↓↑⟩). The qubits are shuttled at constant velocity along the 2D path x₁(t)=L/T*t, y₁(t)=0 and x₂(t)=L/T*t, y₂(t)=D. The total shuttling time is T and the length of the path is L in μm.

Arguments

• T::Real: Total time
• D::Real: Distance between the two qubits
• L::Real: Length of the path
• N::Int: Time discretization
• B::GaussianRandomField: Noise field

Example

model = TwoSpinParallelModel(1.0, 1.0, 1.0, 100, OrnsteinUhlenbeckField([1.0, 1.0, 1.0]))

Calculate the dephasing matrix of a given spin shuttling model.

Arguments

• model::ShuttlingModel: The spin shuttling model
• method::Symbol: The method to calculate the dephasing factor, :trapezoid, :simpson, :adaptive

Returns

• W::Matrix{<:Real}: The dephasing matrix

Example

model = OneSpinModel(1.0, 1.0, 100, OrnsteinUhlenbeckField([1.0, 1.0, 1.0]), t->t)
W = dephasingmatrix(model)

Sample the dephasing matrix array for a given normal random vector.

Arguments

• model::ShuttlingModel: The spin shuttling model
• randseq::Vector{<:Real}: The random sequence
• isarray::Bool: Return the dephasing matrix array for each time step

Sampling an observable that defines on a specific spin shuttling model

Arguments

• model::ShuttlingModel: The spin shuttling model
• objective::Function: The objective function objective(mode::ShuttlingModel; randseq)`
• M::Int: Monte-Carlo sampling size

Calculate the state fidelity of a spin shuttling model using numerical integration of the covariance matrix.

Arguments

• model::ShuttlingModel: The spin shuttling model
• method::Symbol: The method to calculate the dephasing factor, :trapezoid, :simpson, :adaptive

Returns

• Real: The state fidelity

Sample the state fidelity of a spin shuttling model using Monte-Carlo sampling.

Arguments

• model::ShuttlingModel: The spin shuttling model
• randseq::Vector{<:Real}: The random sequence
• isarray::Bool: Return the dephasing matrix array for each time step

Analytical dephasing factor of a one-spin shuttling model.

Arguments

• T::Real: Total time
• L::Real: Length of the path
• B<:GaussianRandomField: Noise field, Ornstein-Uhlenbeck or Pink-Lorentzian
• path::Symbol: Path of the shuttling model, :straight or :forthback

Analytical dephasing factor of a sequenced two-spin EPR pair shuttling model.

Ornstein-Uhlenbeck field, the correlation function of which is σ^2 * exp(-|t₁ - t₂|/θₜ) * exp(-|x₁-x₂|/θₓ) where t is time and x is position.

Pink-Lorentzian Field, the correlation function of which is σ^2 * (expinti(-γ[2]abs(t₁ - t₂)) - expinti(-γ[1]abs(t₁ - t₂)))/log(γ[2]/γ[1]) * exp(-κ|x₁-x₂|) where expinti is the exponential integral function.

Pink-HeavisidePi Field, the correlation function of which is σ^2 * (expinti(-γ[2]abs(t₁ - t₂)) - expinti(-γ[1]abs(t₁ - t₂)))/log(γ[2]/γ[1]) * sinc(-κ|x₁-x₂|) where expinti is the exponential integral function.

Generate a random time series from a Gaussian random field.

R() generates a random time series from a Gaussian random field R R(randseq) generates a random time series from a Gaussian random field R with a given random sequence randseq.

Create a new random function composed by a linear combination of random processes. The input random function represents the direct sum of these processes. The output random function is a tensor contraction from the input.

Arguments

• R::RandomFunction: a direct sum of random processes R₁⊕ R₂⊕ ... ⊕ Rₙ
• c::Vector{Int}: a vector of coefficients
• initialize::Bool=true: whether to initialize the Cholesky decomposition of the covariance matrix

Returns

• RandomFunction: a new random function composed by a linear combination of random processes

Restrict the random field along a parameteized curve.

Arguments

• X::Vector{<:Function}: a vector of functions [x₁(t), x₂(t), ...]
• t::AbstractArray: time array
• B::GaussianRandomField: a Gaussian random field
• initialize::Bool: initialize the random function

Returns

• RandomFunction: a new random function restricted along the curve

Initialize the Cholesky decomposition of the covariance matrix of a random function.

Compute the characteristic functional of the process from the numerical quadrature of the covariance matrix. Using Simpson's rule by default.

Compute the final phase of the characteristic functional of the process from the numerical quadrature of the covariance matrix. Using Simpson's rule by default.

Covariance matrix of a Gaussian random field. When P₁=P₂, it is the auto-covariance matrix of a Gaussian random process. When P₁!=P₂, it is the cross-covariance matrix between two Gaussian random processes.

Arguments

• P₁::Vector{<:Point}: time-position array
• P₂::Vector{<:Point}: time-position array
• process::GaussianRandomField: a Gaussian random field

Auto-Covariance matrix of a Gaussian random process.

Arguments

• P::Vector{<:Point}: time-position array
• process::GaussianRandomField: a Gaussian random field

Returns

• Symmetric{Real}: auto-covariance matrix

Covariance function of Gaussian random field.

Arguments

• p₁::Point: time-position array
• p₂::Point: time-position array
• process<:GaussianRandomField: a Gaussian random field, e.g. OrnsteinUhlenbeckField or PinkLorentzianField