Trapped Ions

Trapped-ion quantum computers encode qubits in internal states of individual ions confined in electromagnetic traps. Entangling operations are mediated by shared motional (phonon) modes of the ion chain.

IonChainSystem

Hamiltonian

A chain of $N$ ions, each with internal levels coupled to $M$ shared motional modes. In the rotating frame at frequency $\omega_{\text{frame}}$:

\[H = \sum_{i=1}^{N} (\omega_{q,i} - \omega_{\text{frame}})\, \sigma_i^+ \sigma_i^- \;+\; \sum_{m=1}^{M} \omega_m\, a_m^\dagger a_m \;+\; \sum_{i,m} \eta_{i,m}\, \sigma_i^x\,(a_m + a_m^\dagger) \;+\; \sum_i \bigl[ \Omega_{x,i}(t)\, \sigma_i^x + \Omega_{y,i}(t)\, \sigma_i^y \bigr]\]

where:

• $\sigma_i^{\pm}$ are raising/lowering operators for ion $i$
• $a_m, a_m^\dagger$ are phonon operators for motional mode $m$
• $\omega_{q,i}$ is the qubit transition frequency
• $\omega_m$ is the motional mode frequency
• $\eta_{i,m}$ is the Lamb-Dicke parameter coupling ion $i$ to mode $m$
• $\Omega_{x,i}(t), \Omega_{y,i}(t)$ are the laser drive amplitudes

The Hilbert space is $\mathcal{H} = \bigotimes_i \mathbb{C}^{d_{\text{ion}}} \otimes \bigotimes_m \mathbb{C}^{n_{\text{max}}}$, with total dimension $d_{\text{ion}}^N \times n_{\text{max}}^M$.

Construction

using Piccolo

# Two ions, one motional mode
sys_ion = IonChainSystem(
    N_ions = 2,
    N_modes = 1,
    mode_levels = 5,        # Fock space truncation
    ωq = 1.0,               # Qubit frequency (GHz)
    ωm = 0.1,               # Motional mode frequency (GHz)
    η = 0.1,                # Lamb-Dicke parameter
    drive_bounds = fill(0.5, 4),  # [Ωx₁, Ωy₁, Ωx₂, Ωy₂]
)
sys_ion.levels, sys_ion.n_drives

(20, 4)

Parameters

Example: Two-Ion Entangling Gate

T, N = 50.0, 100
times = collect(range(0, T, length = N))
pulse = ZeroOrderPulse(0.01 * randn(4, N), times)

# MS-type entangling gate: exp(-iπ/4 σx⊗σx) on ions ⊗ |0⟩ on mode
ψ_init = zeros(ComplexF64, sys_ion.levels)
ψ_init[1] = 1.0  # |00⟩ ⊗ |0⟩_phonon

# Target: Bell state in qubit subspace ⊗ ground motional state
MS(θ) = exp(-im * θ / 2 * kron(ComplexF64[0 1; 1 0], ComplexF64[0 1; 1 0]))
ψ_goal_qubit = MS(π / 2) * ComplexF64[1, 0, 0, 0]
ψ_goal = zeros(ComplexF64, sys_ion.levels)
ψ_goal[1:4] = ψ_goal_qubit

qtraj_ion = KetTrajectory(sys_ion, pulse, ψ_init, ψ_goal)

qcp_ion = SmoothPulseProblem(qtraj_ion, N; Q = 100.0)
solve!(qcp_ion; max_iter = 50, print_level = 1)
fidelity(qcp_ion)

0.9999994836977881

RadialMSGateSystem

Hamiltonian

Specialized system for the radial-mode Mølmer-Sørensen gate. In the interaction picture, the time-dependent Hamiltonian is:

\[H(t) = -\frac{i}{2} \sum_{i=1}^{N} \sum_{k=1}^{2N} \eta_{k,i}\, \Omega_i(t)\, \sigma_{x,i} \left( a_k\, e^{-i \delta_k t} - a_k^\dagger\, e^{i \delta_k t} \right)\]

where $k$ indexes the $2N$radial modes ($N$ modes along each transverse axis), $\Omega_i(t)$ is the Rabi frequency for ion $i$, and $\delta_k$ is the detuning from mode $k$.

This produces a time-dependent QuantumSystem where sys.H(u, t) evaluates the Hamiltonian at controls u and time t.

Construction

sys_ms = RadialMSGateSystem(
    N_ions = 2,
    mode_levels = 3,
    ωm_radial = [5.0, 5.0, 5.1, 5.1],  # 2N radial mode frequencies (GHz)
    δ = 0.2,                             # Detuning (GHz)
    η = 0.1,                             # Lamb-Dicke parameter
    drive_bounds = [1.0, 1.0],           # Per-ion amplitude bounds
)
sys_ms.levels, sys_ms.n_drives

(324, 2)

Parameters

RadialMSGateSystemWithPhase

Hamiltonian

Extends RadialMSGateSystem with phase controls$\phi_i(t)$ for AC Stark shift compensation:

\[H(t) = \frac{1}{2} \sum_{i,k} \eta_{k,i}\, \Omega_i(t) \left( \sigma_i^+ e^{i\phi_i(t)} + \sigma_i^- e^{-i\phi_i(t)} \right) \left( a_k\, e^{-i \delta_k t} + a_k^\dagger\, e^{i \delta_k t} \right)\]

Off-resonant coupling to spectator modes creates AC Stark shifts $\Delta E_{\text{Stark}} \sim \eta^2 \Omega^2 / \delta_{\text{spectator}}$. Modulating $\phi_i(t)$ cancels this time-varying phase accumulation, enabling high-fidelity gates.

Controls: $[\Omega_1, \phi_1, \Omega_2, \phi_2, \ldots]$ (interleaved amplitudes and phases).

Construction

sys_ms_phase = RadialMSGateSystemWithPhase(
    N_ions = 2,
    mode_levels = 3,
    ωm_radial = [5.0, 5.0, 5.1, 5.1],
    δ = 0.2,
    η = 0.1,
    amplitude_bounds = [1.0, 1.0],
    phase_bounds = [(-Float64(π), Float64(π)), (-Float64(π), Float64(π))],
)
sys_ms_phase.n_drives  # 4: [Ω₁, φ₁, Ω₂, φ₂]

4

Typical Parameters

Typical values for ${}^{171}$Yb$^+$ ions (e.g., Q-SCOUT platform):

Best Practices

1. Truncate Motional Modes Carefully

The phonon Fock space must be large enough to capture displaced states during the gate. Start with mode_levels = 5 and increase if dynamics require larger excursions.

2. Ensure Motional Closure

For MS gates, the motional state must return to vacuum at the end of the gate: $|\alpha_k(\tau)| \approx 0$ for all modes $k$. This is a necessary condition for separating the qubit and motional degrees of freedom.

3. Use Individual Ion Addressing

Each ion has independent $\Omega_{x,i}, \Omega_{y,i}$ controls, enabling individually optimized Rabi frequencies.

References

• Sørensen & Mølmer, "Quantum computation with ions in thermal motion," Phys. Rev. Lett.82, 1971 (1999)
• Sørensen & Mølmer, "Entanglement and quantum computation with ions in thermal motion," Phys. Rev. A62, 022311 (2000)
• Mizrahi et al., "Realization and Calibration of Continuously Parameterized Two-Qubit Gates on a Trapped-Ion Quantum Processor," IEEE TQE (2024)

See Also

• Quantum Systems Overview — General system API
• Pulses — GaussianPulse for analytical pulse shaping

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