Building Blocks for Hamiltonians
The Hamiltonian describing a Rydberg gate pulse on a group of atoms is block-diagonal, with one subsystem for every computational basis state. To determine the dynamics of a gate pulse, one can analyze the dynamics of each subsystem separately. A transformation to a convenient basis may reduce the dimensionality of a subsystem. Moreover, infinite Rydberg interaction strengths lead to a reduction of the relevant Hilbert space dimension.
Here, we provide the Hamiltonians for various numbers of atoms in \(|1\rangle\), arranged in various geometries, exhibiting interactions of infinite or finite strength when excited to \(|r\rangle\).
- H_k_atoms_perfect_blockade(Delta_1, Delta_r, Xi, Omega, decay, k)[source]
\(k\) atoms, infinite Rydberg interaction between all atoms:
- Parameters:
- Returns:
2-level system Hamiltonian.
- Return type:
Array
- H_2_atoms(Delta_1, Delta_r, Xi, Omega, decay, V)[source]
Two atoms, Rydberg interaction \(V\) between atoms:
- Parameters:
- Returns:
3-level system Hamiltonian.
- Return type:
Array
- H_3_atoms_inf_V(Delta_1, Delta_r, Xi, Omega, decay, V)[source]
Three atoms arranged in an isosceles triangle, infinite Rydberg interaction between nearest neighbours, Rydberg interaction \(V\) between next-nearest neighbours:
- Parameters:
- Returns:
4-level system Hamiltonian.
- Return type:
Array
- H_3_atoms_symmetric(Delta_1, Delta_r, Xi, Omega, decay, V)[source]
Three atoms arranged in an equilateral triangle, Rydberg interaction \(V\) between atoms:
- Parameters:
- Returns:
4-level system Hamiltonian.
- Return type:
Array
- H_3_atoms(Delta_1, Delta_r, Xi, Omega, decay, Vnn, Vnnn)[source]
Three atoms arranged in an isosceles triangle, Rydberg interaction \(V_{\mathrm{nn}}\) between nearest neighbours, Rydberg interaction \(V_{\mathrm{nnn}}\) between next-nearest neighbours:
- Parameters:
Delta_1 (float) – Laser detuning of the qubit state |1>.
Delta_r (float) – Laser detuning of the Rydberg state |r>.
Xi (float) – Laser phase.
Omega (float) – Rabi frequency amplitude.
decay (float) – Rydberg-decay rate.
Vnn (float) – Rydberg interaction strength between nearest neighbours.
Vnnn (float) – Rydberg interaction strength between next-nearest neighbours.
- Returns:
6-level system Hamiltonian.
- Return type:
Array
- H_4_atoms_inf_V(Delta_1, Delta_r, Xi, Omega, decay, V)[source]
Four atoms arranged in a pyramid, infinite Rydberg interaction between nearest neighbours, Rydberg interaction \(V\) between next-nearest neighbours:
- Parameters:
- Returns:
5-level system Hamiltonian.
- Return type:
Array
- H_4_atoms_symmetric(Delta_1, Delta_r, Xi, Omega, decay, V)[source]
Four atoms arranged in a tetrahedron, Rydberg interaction \(V\) between atoms:
- Parameters:
- Returns:
5-level system Hamiltonian.
- Return type:
Array
- H_4_atoms(Delta_1, Delta_r, Xi, Omega, decay, Vnn, Vnnn)[source]
Four atoms arranged in a pyramid, Rydberg interaction \(V_{\mathrm{nn}}\) between nearest neighbours, Rydberg interaction \(V_{\mathrm{nnn}}\) between next-nearest neighbours:
- Parameters:
Delta_1 (float) – Laser detuning of the qubit state |1>.
Delta_r (float) – Laser detuning of the Rydberg state |r>.
Xi (float) – Laser phase.
Omega (float) – Rabi frequency amplitude.
decay (float) – Rydberg-decay rate.
Vnn (float) – Rydberg interaction strength between nearest neighbours.
Vnnn (float) – Rydberg interaction strength between next-nearest neighbours.
- Returns:
8-level system Hamiltonian.
- Return type:
Array
- H_1_atom_general(Delta_1, Delta_r, Xi, Omega, decay, s1=1.0)[source]
One atom with arbitrary scaling of the Rabi frequency.
Basis ordering: \(|0\rangle, |1\rangle\).
- Parameters:
- Returns:
2-level system Hamiltonian.
- Return type:
Array
- H_2_atoms_general(Delta_1, Delta_r, Xi, Omega, decay, V12, s1=1.0, s2=1.0)[source]
Two atoms with arbitrary scaling of Rabi frequencies and arbitrary Rydberg interaction.
Basis ordering: \(|00\rangle, |01\rangle, |10\rangle, |11\rangle\).
- Parameters:
Delta_1 (float) – Laser detuning of the qubit state |1>.
Delta_r (float) – Laser detuning of the Rydberg state |r>.
Xi (float) – Laser phase.
Omega (float) – Rabi frequency amplitude.
decay (float) – Rydberg-decay rate.
V12 (float) – Rydberg interaction strength between atoms 1 and 2.
s1 (float) – Rabi frequency scaling factor for atom 1.
s2 (float) – Rabi frequency scaling factor for atom 2.
- Returns:
4-level system Hamiltonian.
- Return type:
Array
- H_3_atoms_general(Delta_1, Delta_r, Xi, Omega, decay, V12, V13, V23, s1=1.0, s2=1.0, s3=1.0)[source]
Three atoms with arbitrary scaling of Rabi frequencies and arbitrary Rydberg interactions.
Basis ordering: \(|000\rangle, |001\rangle, |010\rangle, |011\rangle, |100\rangle, |101\rangle, |110\rangle, |111\rangle\).
- Parameters:
Delta_1 (float) – Laser detuning of the qubit state |1>.
Delta_r (float) – Laser detuning of the Rydberg state |r>.
Xi (float) – Laser phase.
Omega (float) – Rabi frequency amplitude.
decay (float) – Rydberg-decay rate.
V12 (float) – Rydberg interaction strength between atoms 1 and 2.
V13 (float) – Rydberg interaction strength between atoms 1 and 3.
V23 (float) – Rydberg interaction strength between atoms 2 and 3.
s1 (float) – Rabi frequency scaling factor for atom 1.
s2 (float) – Rabi frequency scaling factor for atom 2.
s3 (float) – Rabi frequency scaling factor for atom 3.
- Returns:
8-level system Hamiltonian.
- Return type:
Array
- H_4_atoms_general(Delta_1, Delta_r, Xi, Omega, decay, V12, V13, V14, V23, V24, V34, s1=1.0, s2=1.0, s3=1.0, s4=1.0)[source]
Four atoms with arbitrary scaling of Rabi frequencies and arbitrary Rydberg interactions.
Basis ordering: \(|0000\rangle, |0001\rangle, |0010\rangle, |0011\rangle, |0100\rangle, \ldots, |1111\rangle\).
- Parameters:
Delta_1 (float) – Laser detuning of the qubit state |1>.
Delta_r (float) – Laser detuning of the Rydberg state |r>.
Xi (float) – Laser phase.
Omega (float) – Rabi frequency amplitude.
decay (float) – Rydberg-decay rate.
V12 (float) – Rydberg interaction strength between atoms 1 and 2.
V13 (float) – Rydberg interaction strength between atoms 1 and 3.
V14 (float) – Rydberg interaction strength between atoms 1 and 4.
V23 (float) – Rydberg interaction strength between atoms 2 and 3.
V24 (float) – Rydberg interaction strength between atoms 2 and 4.
V34 (float) – Rydberg interaction strength between atoms 3 and 4.
s1 (float) – Rabi frequency scaling factor for atom 1.
s2 (float) – Rabi frequency scaling factor for atom 2.
s3 (float) – Rabi frequency scaling factor for atom 3.
s4 (float) – Rabi frequency scaling factor for atom 4.
- Returns:
16-level system Hamiltonian.
- Return type:
Array