This visualizer models a dual-species optical lattice elevator for continuous reloading of a neutral-atom quantum computer. Two counter-propagating laser beams (one tuned for Rb-87, one for Cs-133) form standing-wave optical lattices that transport atoms vertically via frequency chirping (Bang–Coast–Bang profile). A science array of qubits sits radially offset from the beam axis, within the field of view (FOV) of a high-NA objective.
The physics engine computes trap depths, scattering rates, transport trajectories, cross-talk, and coherence metrics in real time as parameters change. Six dimensionless figures of merit determine whether a design is viable.
| Symbol | Value | Description |
|---|---|---|
a0 | 5.29177 × 10−11 m | Bohr radius |
c | 2.99792 × 108 m/s | Speed of light |
kB | 1.38065 × 10−23 J/K | Boltzmann constant |
ℏ | 1.05457 × 10−34 J·s | Reduced Planck constant |
h | 6.62607 × 10−34 J·s | Planck constant |
g | 9.80665 m/s² | Gravitational acceleration |
mRb | 1.44316 × 10−25 kg | Rb-87 atomic mass |
mCs | 2.20695 × 10−25 kg | Cs-133 atomic mass |
ε0 | 8.85419 × 10−12 F/m | Vacuum permittivity |
| Species | Transition | λ (nm) | Γ/2π (MHz) | Isat (mW/cm²) |
|---|---|---|---|---|
| Cs-133 | D1 | 894.347 | 4.561 | 2.706 |
| Cs-133 | D2 | 852.347 | 5.223 | 1.654 |
| Rb-87 | D1 | 794.978 | 5.746 | 4.484 |
| Rb-87 | D2 | 780.241 | 6.065 | 2.503 |
Wavelengths: vacuum values. Sources: Steck, “Cesium D Line Data” & “Rubidium 87 D Line Data” (rev. 2021).
The scalar dynamic polarizability α0(ω) determines the AC Stark shift (trap depth) an atom experiences in a light field.
A pre-computed lookup table generated by the ARC (Alkali Rydberg Calculator) library, which sums over ~20 dipole-coupled excited states using quantum defect theory:
The table uses two resolutions to balance accuracy and size:
| Region | Range | Step | Δν Resolution |
|---|---|---|---|
| Background | 776–1100 nm | 0.20 nm | ≤100 GHz |
| Near Rb D2 (780.2 nm) | ±4 nm | 0.02 nm | ~10 GHz |
| Near Rb D1 (795.0 nm) | ±4 nm | 0.02 nm | ~9.5 GHz |
| Near Cs D2 (852.3 nm) | ±4 nm | 0.02 nm | ~8.3 GHz |
| Near Cs D1 (894.3 nm) | ±4 nm | 0.02 nm | ~7.5 GHz |
Total: 3226 wavelength points, merged and sorted. Interpolation uses binary search (O(log n)) to handle the non-uniform grid. ARC accuracy is typically ±1% for ground-state alkali polarizabilities.
ARC: Šibalić et al., Comput. Phys. Commun. 220, 319 (2017); Grimm et al. (2000).
where P is the total power, w0 the beam waist, zR the Rayleigh range, and r the radial distance from the beam axis.
Two counter-propagating beams with independent waists w01, w02 and focus positions z1, z2 form a standing wave. The lattice peak intensity on-axis is:
The factor of 4 arises from constructive interference at the antinodes of the standing wave: Imax = |E1 + E2|² = (√I1 + √I2)² ≈ 4√(I1I2) for similar beam powers.
Each species feels both lattices: Utotal = Uprimary + Ucross-talk. The minimum trap depth along z determines the bottleneck for transport.
Grimm et al. (2000), Eq. 4.
Off-resonant photon scattering heats atoms and limits transport survival and qubit coherence. The scattering rate from a single transition is:
The total scattering rate sums D1 and D2 contributions, and each species scatters from both lattice wavelengths. Intensities are in mW/cm² (Isat convention); the Gaussian beam calculation gives W/m², so a factor of 0.1 converts units (1 W/m² = 0.1 mW/cm²).
Scattering depends on the local intensity at the atom’s position, which differs between transport and science contexts:
| Context | Intensity used | Reason |
|---|---|---|
| Transport, same-species (e.g., Rb beam → Rb atom) | Iantinode = I1 + I2 + 2√(I1I2) | Atom trapped at standing-wave antinode (red-detuned lattice) |
| Transport, cross-species (e.g., Cs beam → Rb atom) | Iavg = I1 + I2 | Atom position uncorrelated with cross-species lattice (λRb ≠ λCs) |
| Science array (Ψ, DLS) | Iavg = I1 + I2 | Qubits in tweezers, not trapped in transport lattice |
Grimm et al. (2000), Eq. 8; Steck alkali D-line data.
Atoms are transported vertically by chirping the AOM driving frequency, making the lattice move. The Bang–Coast–Bang (BCB) profile is time-optimal for a given acceleration budget:
where L is the transport distance (default 26.7 cm) and aop = ζ · acrit is the operating acceleration with safety factor ζ.
The AOM frequency shift is Δν = 2v/λ, where v is the instantaneous lattice velocity. At peak velocity vmax, the peak shift Δνpeak = 2vmax/λ is typically 1–10 MHz. The AOM chirp rate during acceleration is d(Δν)/dt = 2aop/λ.
Safety factor ζ (SF): Sets the operating acceleration as a fraction of the critical (Landau–Zener) acceleration: aop = ζ × acrit. The critical acceleration acrit = πU0/(mλ/2) is the maximum before atoms escape the lattice. Typical values ζ = 0.10–0.20 give tunneling loss PLZ = exp(−ac/aop) < 10−5. The safety factor directly determines the one-way transport time τ and peak velocity vmax, and is reported as the scorecard metric ζ.
Circuit time τcircuit: The total experimental cycle time (in seconds) during which science-array qubits are exposed to stray transport beam light. This is a user-specified budget that encompasses the full experiment sequence: gate operations, mid-circuit measurements, error correction cycles, readout, and any idle time while the transport beams remain on. It does not model or include the loading process itself.
Chiu et al., “Continuous operation of a coherent 3,000-qubit system,” arXiv:2506.20660 (2025); Klostermann et al., Phys. Rev. A 105, 043319 (2022).
If the lattice accelerates too fast, atoms tunnel out of the lattice band. The survival probability per lattice period is:
The critical acceleration ac simplifies (using the expressions above) to:
This is the classical result: the maximum restoring force of a sinusoidal potential U0 cos²(kz) is F = πU0/d.
Peik et al., Phys. Rev. A 55, 2989 (1997); Morsch & Oberthaler, Rev. Mod. Phys. 78, 179 (2006).
The design scorecard evaluates six pass/fail criteria. All must pass for a viable design (green border).
| Symbol | Formula | Threshold | Physical Meaning |
|---|---|---|---|
| η | min(URb, UCs) / kBT |
> 4 | Trap depth vs. thermal energy. Boltzmann escape probability exp(−η) < 2% per oscillation. |
| Φ | max(Γ̅Rb·τ, Γ̅Cs·τ) |
< 0.1 | Expected scattered photons during transport. Survival = exp(−Φ). |
| ζ | aop / acrit |
< 0.15 | Safety factor. Keeps Landau–Zener tunneling loss PLZ < 10−5. |
| CT | min(|αRR/αRC|, |αCC/αCR|) |
> 5 | Cross-talk ratio. Ensures each lattice transports only its target species. |
| Ψ | max(Γsci,Rb, Γsci,Cs) · τcircuit |
< 0.1 | Stray-light scattering on science qubits during one loading cycle. |
| DLS | max(ηRbUstr,Rb, ηCsUstr,Cs) / h |
< 1 Hz | Differential light shift on clock qubits from stray beam intensity. |
Qubits are encoded in the clock states |F, mF=0⟩ ↔ |F′, mF=0⟩. Stray light from the transport beams at the science array produces an AC Stark shift on these states.
At second order in perturbation theory, the scalar polarizability α0 is identical for all hyperfine sublevels within a J=1/2 ground state. Both clock states have the same α0, so the light shift cancels and there is no differential shift at second order.
The DLS arises at third order (E1–hfs–E1): light virtually excites the atom to a P state, the hyperfine interaction in the excited P manifold mixes P1/2 and P3/2 (which have different hyperfine constants Ahfs), and the atom is de-excited back. This three-step process introduces an F-dependent correction:
where ΔFS is the fine-structure splitting of the excited P states.
| Shift Component | Contribution | Reason |
|---|---|---|
| Vector (rank 1) | 0 | Proportional to mF; vanishes for mF=0 |
| Tensor (rank 2) | 0 | Requires J ≥ 1; vanishes for J=1/2 |
| Scalar differential | ≠ 0 | Third-order E1–hfs–E1 perturbation |
where ηDLS = δαhfs/α0 is a dimensionless ratio that depends only on the atom:
| Species | ηDLS | k (Hz/(V/m)²) |
|---|---|---|
| Rb-87 | 1.3 × 10−3 | −1.234(22) × 10−10 |
| Cs-133 | 2.3 × 10−3 | −2.347(84) × 10−10 |
Arora, Sahoo & Safronova, Phys. Rev. A (2024); Rosenbusch et al., Phys. Rev. A 79, 013404 (2009).
Atoms arrive at the reservoir at a temperature Tfinal > T0 due to heating during transport. Two mechanisms contribute: photon recoil heating from off-resonant scattering, and parametric heating from laser intensity noise.
Each scattered photon imparts a momentum kick ℏk to the atom. The resulting energy increase per scatter is the recoil energy:
Averaging over random absorption and spontaneous emission directions in 3D, each photon heats the atom by (4/3) Tr. The scattering heating rate is:
where the sum runs over each lattice beam with its own wavelength λi and species-specific scattering rate Γi. The recoil momentum differs between the Rb and Cs lattice wavelengths, so same-species and cross-species scattering are tracked separately:
| Source | Recoil wavelength | Example (810 nm on Rb-87) |
|---|---|---|
| Same-species (Rb from Rb lattice) | λRb | Tr ≈ 170 nK |
| Cross-species (Rb from Cs lattice) | λCs | Tr ≈ 110 nK (at 920 nm) |
The scattering heating is additive: ΔTsc = Rsc · τ.
Fluctuations in the laser intensity modulate the trap depth, driving parametric excitation at twice the trap frequency. For a harmonic oscillator with frequency ν, intensity noise with one-sided power spectral density Sε(f) (fractional intensity noise, units 1/Hz) produces exponential energy growth:
The fractional intensity noise PSD is related to the RIN (relative intensity noise) specification of the laser:
The relevant trap frequency is the axial lattice frequency, computed from the lattice depth at the harmonic approximation:
where U0 is the minimum trap depth along the transport axis. Typical values are νax ∼ 100–300 kHz, so the critical noise frequency 2νax ∼ 200–600 kHz.
Both mechanisms act simultaneously during transport. The temperature evolves as:
where Rsc(z) is the position-dependent scattering heating rate (varies with local beam intensity along the lattice). The exact solution for constant Rsc and Γpara is:
This correctly reduces to T0 + Rscτ when Γp → 0 (no parametric heating), and to T0 exp(Γpτ) when Rsc → 0 (no scattering). The summary report integrates along the actual BCB trajectory using the position-dependent heating rate for higher accuracy.
| RIN (dBc/Hz) | Sε (1/Hz) | Typical laser | τheat at ν = 200 kHz |
|---|---|---|---|
| −150 | 10−15 | Stabilized Ti:Sapph | ∼ 105 s |
| −140 | 10−14 | Ti:Sapph / good fiber | ∼ 104 s |
| −130 | 10−13 | Standard fiber laser | ∼ 103 s |
| −120 | 10−12 | Noisy diode laser | ∼ 100 s |
| −100 | 10−10 | Very noisy source | ∼ 1 s |
Savard, O’Hara & Thomas, Phys. Rev. A 56, R1095 (1997); Gehm et al., Phys. Rev. A 58, 3914 (1998).
The optimizer searches for beam parameters (wavelength, power, waist, focus position) that minimize a chosen objective while satisfying physics constraints.
A derivative-free simplex method that works in normalized [0, 1] space (each parameter mapped from its user-set range).
Each iteration: (1) sort simplex, (2) compute centroid, (3) try reflection, expansion, contraction, or full shrink. All points are clamped to [0, 1].
To avoid local minima, the optimizer runs 50 independent starts from randomly sampled initial points within the unlocked parameter ranges. The best result across all starts is reported.
| Objective | Minimizes |
|---|---|
| Min Ψ (array scatter) | Science array scattering Ψ |
| Min Φ (transport scatter) | Transport scattering Φ |
| Min τ (transport time) | One-way transport time (ms) |
| Min P (total power) | Total optical power (W) |
| Max η (trap depth) | −η (maximizes trap depth) |
| Min Ψ + Φ (combined) | Sum of scattering metrics |
| Min DLS (light shift) | Differential light shift (Hz) |
Constraints are enforced via a penalty method. If a constraint is violated, a large penalty (106 × violation magnitude) is added to the objective:
Available constraints: η > 4, Φ < 0.1, ζ < 0.15, CT > 5, Ψ < 0.1, DLS < threshold. The primary objective’s own constraint is automatically excluded to avoid double-penalization.
Only parameters for active beams are included. If the Rb beam is off, Rb wavelength/power/waist are excluded from the search. The optimizer panel prunes stale parameters when beam configuration changes.
| # | Citation | Used For |
|---|---|---|
| 1 | Grimm, Weidemüller & Ovchinnikov, “Optical dipole traps for neutral atoms,” Adv. At. Mol. Opt. Phys. 42, 95 (2000) | Trap depth, polarizability, scattering rate formulas |
| 2 | Steck, “Cesium D Line Data” & “Rubidium 87 D Line Data” (rev. 2021) | D-line wavelengths, linewidths, saturation intensities |
| 3 | Peik et al., “Bloch oscillations of atoms, adiabatic rapid passage, and monokinetic atomic beams,” Phys. Rev. A 55, 2989 (1997) | Landau–Zener tunneling, critical acceleration |
| 4 | Morsch & Oberthaler, “Dynamics of Bose-Einstein condensates in optical lattices,” Rev. Mod. Phys. 78, 179 (2006) | Band structure, lattice dynamics |
| 5 | Chiu et al., “Continuous operation of a coherent 3,000-qubit system,” Nature (2025); arXiv:2506.20660 | BCB transport, continuous reloading architecture |
| 6 | Klostermann et al., Phys. Rev. A 105, 043319 (2022) | AOM chirp rates, lattice conveyor belt |
| 7 | Norcia et al., PRX Quantum 5, 030316 (2024) | Transport survival benchmarks (70% over 30 cm) |
| 8 | Arora, Sahoo & Safronova, Phys. Rev. A (2024) | Differential polarizability ηDLS values |
| 9 | Rosenbusch et al., Phys. Rev. A 79, 013404 (2009) | Hyperfine light shift theory |
| 10 | Saffman, Walker & Mølmer, Rev. Mod. Phys. 82, 2313 (2010) | Science array scattering threshold |
| 11 | Gyger et al., Phys. Rev. Research 6, 033104 (2024) | Continuous reloading with Sr-88 |
| 12 | Li et al., arXiv:2506.15633 (2025) | Yb-171 metastable qubit approach |
| 13 | Savard, O’Hara & Thomas, Phys. Rev. A 56, R1095 (1997) | Parametric heating from intensity noise |
| 14 | Gehm et al., Phys. Rev. A 58, 3914 (1998) | Photon recoil heating, noise-induced trap loss |