Rydberg Tools

rydberg_hamiltonian_symmetric_blockaded(ϕ::Real, ϵ::Real, δ::Real)

Constructs the Hamiltonian for a symmetric Rydberg-blockaded two-atom system.

Basis

|00⟩, |01⟩, |11⟩, |0r⟩, |W⟩ where |W⟩ = (|1r⟩ + |r1⟩)/√2

Parameters

• ϕ::Real: Phase of the driving field
• ϵ::Real: Relative amplitude deviation parameter
• δ::Real: Detuning of the Rydberg state

Mathematical form

\[H = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{(1+\epsilon)e^{-i\phi}}{2} & 0 \\ 0 & 0 & 0 & 0 & \frac{(1+\epsilon)e^{-i\phi}}{\sqrt{2}} \\ 0 & \frac{(1+\epsilon)e^{i\phi}}{2} & 0 & \delta & 0 \\ 0 & 0 & \frac{(1+\epsilon)e^{i\phi}}{\sqrt{2}} & 0 & \delta \end{pmatrix}\]

Returns

• Matrix representing the Hamiltonian in the symmetric basis described above

rydberg_hamiltonian_full_blockaded(ϕ::Real, ϵ::Real, δ::Real)

Constructs the Hamiltonian for a fully-described Rydberg-blockaded two-atom system.

Basis

|00⟩, |01⟩, |10⟩, |11⟩, |0r⟩, |r0⟩, |W'⟩ where |W'⟩ = (|1r⟩ + |r1⟩)/√2

Parameters

• ϕ::Real: Phase of the driving field
• ϵ::Real: Relative amplitude deviation parameter
• δ::Real: Detuning of the Rydberg state

Mathematical form

\[H = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{(1+\epsilon)e^{-i\phi}}{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{(1+\epsilon)e^{-i\phi}}{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{(1+\epsilon)e^{-i\phi}}{\sqrt{2}} \\ 0 & \frac{(1+\epsilon)e^{i\phi}}{2} & 0 & 0 & \delta & 0 & 0 \\ 0 & 0 & \frac{(1+\epsilon)e^{i\phi}}{2} & 0 & 0 & \delta & 0 \\ 0 & 0 & 0 & \frac{(1+\epsilon)e^{i\phi}}{\sqrt{2}} & 0 & 0 & \delta \end{pmatrix}\]

Returns

• Matrix representing the Hamiltonian in the basis described above

rydberg_hamiltonian_full(ϕ::Real, Ω1::Real, Ω2::Real, δ1::Real, δ2::Real, B::Real)

Constructs the full Hamiltonian for a two-atom Rydberg system without symmetry constraints.

Basis

|00⟩, |01⟩, |10⟩, |11⟩, |0r⟩, |r0⟩, |1r⟩, |r1⟩, |rr⟩

Parameters

• ϕ::Real: Phase of the driving field
• Ω1::Real: Rabi frequency for the first atom
• Ω2::Real: Rabi frequency for the second atom
• δ1::Real: Detuning for the first atom
• δ2::Real: Detuning for the second atom
• B::Real: Rydberg-Rydberg blockade shift

Mathematical form

\[H = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{\Omega_1 e^{-i\phi}}{2} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{\Omega_2 e^{-i\phi}}{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{\Omega_1 e^{-i\phi}}{2} & \frac{\Omega_2 e^{-i\phi}}{2} & 0 \\ 0 & \frac{\Omega_1 e^{i\phi}}{2} & 0 & 0 & \delta_1 & 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{\Omega_2 e^{i\phi}}{2} & 0 & 0 & \delta_2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{\Omega_1 e^{i\phi}}{2} & 0 & \delta_1 & 0 & \frac{\Omega_2 e^{-i\phi}}{2} \\ 0 & 0 & 0 & 0 & 0 & \frac{\Omega_2 e^{i\phi}}{2} & 0 & \delta_2 & \frac{\Omega_1 e^{-i\phi}}{2} \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{\Omega_2 e^{i\phi}}{2} & \frac{\Omega_1 e^{i\phi}}{2} & \delta_1 + \delta_2 + B \end{pmatrix}\]

Returns

• Matrix representing the Hamiltonian in the full basis described above

cz_with_1q_phase_symmetric(θ::Real)

Constructs the CZ gate with additional single-qubit phase in the symmetric subspace.

Basis

|00⟩, |01⟩, |11⟩, |0r⟩, |W⟩ (same as in rydberg_hamiltonian_symmetric_blockaded)

Parameters

• θ::Real: Single-qubit phase parameter

Mathematical form

\[U = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & e^{i\theta} & 0 & 0 & 0 \\ 0 & 0 & e^{i(2\theta+\pi)} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix}\]

The diagonal structure encodes a CZ gate with additional single-qubit phase rotations.

Returns

• Diagonal matrix representing the CZ gate with phase rotations

cz_with_1q_phase_full(θ::Real; rydberg_dimension::Int = 5)

Constructs the CZ gate with additional single-qubit phase in the full computational basis.

Basis

|00⟩, |01⟩, |10⟩, |11⟩, |0r⟩, |r0⟩, |1r⟩, |r1⟩, |rr⟩ (same as in rydberg_hamiltonian_full)

Parameters

• θ::Real: Single-qubit phase parameter
• rydberg_dimension::Int=5: Dimension of the Rydberg subspace (optional, default: 5)

Mathematical form

\[U = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & e^{i\theta} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & e^{i\theta} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & e^{i(2\theta+\pi)} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}\]

The diagonal structure encodes a CZ gate with additional single-qubit phase rotations.

Returns

• Diagonal matrix representing the CZ gate with phase rotations in the full basis

unwrap_phase(ϕ)

Unwraps a sequence of phase values by removing jumps greater than π.

Adjusts phase values to maintain continuity across the 2π boundary, eliminating artificial discontinuities in phase data while preserving the actual phase evolution. Particularly useful for plotting phase values to avoid discontinuous jumps.

Parameters

• ϕ: Array of phase values to unwrap

Returns

• An array of unwrapped phase values with the same length as the input