Rydberg Atoms

Rydberg atom arrays use highly excited atomic states with strong van der Waals interactions to implement quantum gates and many-body simulations. The atoms are individually trapped (e.g., in optical tweezers) with global laser drives.

Hamiltonian

For $N$ atoms in a 1D chain with spacing $d$, the Rydberg Hamiltonian in the spin basis $|g\rangle = |0\rangle$, $|r\rangle = |1\rangle$ is:

\[H = \frac{\Omega(t)}{2} \sum_i \sigma_i^x \;-\; \Delta(t) \sum_i n_i \;+\; \sum_{i < j} \frac{C_6}{|r_i - r_j|^6}\, n_i\, n_j\]

where:

• $\Omega(t)$ is the global Rabi frequency (laser drive)
• $\Delta(t)$ is the global detuning
• $n_i = |r\rangle\langle r|_i$ is the Rydberg population operator
• $C_6$ is the van der Waals coefficient ($\approx 862\,690 \times 2\pi$ MHz·μm$^6$ for rubidium)

The interaction $C_6 / |r_i - r_j|^6$ gives rise to the Rydberg blockade: within the blockade radius $r_b = (C_6 / \Omega)^{1/6}$, double excitation is energetically forbidden.

If the Y drive is included (ignore_Y_drive = false), an additional term $\frac{\Omega_y(t)}{2} \sum_i \sigma_i^y$ appears, enabling full phase control of the drive.

Construction

using Piccolo

sys_rydberg = RydbergChainSystem(
    N = 3,                     # Number of atoms
    C = 862690 * 2π,           # van der Waals coefficient (MHz·μm⁶)
    distance = 8.7,            # Atom spacing (μm)
    cutoff_order = 1,          # 1 = nearest-neighbor only
    drive_bounds = [1.0, 1.0, 1.0],  # [Ωx, Ωy, Δ]
)
sys_rydberg.levels, sys_rydberg.n_drives

(8, 3)

The Hilbert space is $(\mathbb{C}^2)^{\otimes N}$ with dimension $2^N$.

Parameters

Interaction Range

The cutoff_order parameter controls which interaction terms are included:

For $C_6 / d^6 \gg \Omega$, the nearest-neighbor approximation is often sufficient. The ratio $V_{\text{nn}} / V_{\text{nnn}} = 2^6 = 64$.

# All-to-all interactions (default)
sys_all = RydbergChainSystem(N = 4, all2all = true, drive_bounds = [1.0, 1.0, 1.0])

# Nearest-neighbor only
sys_nn = RydbergChainSystem(
    N = 4,
    all2all = false,
    cutoff_order = 1,
    drive_bounds = [1.0, 1.0, 1.0],
)

sys_all.levels, sys_nn.levels

(16, 16)

Example: GHZ State Preparation

N_atoms = 3
sys = RydbergChainSystem(N = N_atoms, drive_bounds = [1.0, 1.0, 1.0])

T, N_steps = 5.0, 80
times = collect(range(0, T, length = N_steps))
pulse = ZeroOrderPulse(0.05 * randn(3, N_steps), times)

# GHZ state: (|000⟩ + |111⟩)/√2
ψ_init = zeros(ComplexF64, 2^N_atoms)
ψ_init[1] = 1.0  # |000⟩

ψ_ghz = zeros(ComplexF64, 2^N_atoms)
ψ_ghz[1] = 1 / √2       # |000⟩
ψ_ghz[end] = 1 / √2     # |111⟩

qtraj_rydberg = KetTrajectory(sys, pulse, ψ_init, ψ_ghz)

qcp_rydberg = SmoothPulseProblem(qtraj_rydberg, N_steps; Q = 100.0)
solve!(qcp_rydberg; max_iter = 100, print_level = 1)
fidelity(qcp_rydberg)

0.5106270056458393

Typical Parameters

Best Practices

1. Check the Blockade Regime

Ensure $V_{\text{nn}} = C_6 / d^6 \gg \Omega$ for the blockade mechanism to be effective. If $V \sim \Omega$, the system operates in the intermediate (anti-blockade) regime.

2. Start with Nearest-Neighbor Interactions

Use cutoff_order = 1 or all2all = false for initial explorations. Add longer-range interactions (all2all = true) when needed for accuracy.

3. Scaling

The Hilbert space grows as $2^N$, so direct Piccolo optimization is practical for $N \leq 8-10$ atoms.

References

• Saffman, Walker & Mølmer, "Quantum information with Rydberg atoms," Rev. Mod. Phys.82, 2313 (2010)
• Bernien et al., "Probing many-body dynamics on a 51-atom quantum simulator," Nature551, 579 (2017)

See Also

• Quantum Systems Overview — General system API
• Trajectories — KetTrajectory for state preparation

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