1. Dynamic Polarizability
The trap depth of an optical tweezer is set by the AC Stark shift. For an alkali atom with D1 and D2 resonances, the scalar dynamic polarizability is (in atomic units):
α(ω) = Σⱼ ΔEⱼ|dⱼ|² / [3(ΔEⱼ² − ω²)] + α_corewhere ΔEⱼ = E_excited − E_ground in Hartree, dⱼ is the reduced matrix element in units of e·a₀, and ω is in atomic units (Eₕ/ℏ). Converting to SI and computing the peak intensity:
U₀ = α_SI · I_peak / (c·ε₀), I_peak = S·2P/(π·w₀²)where S is the Strehl ratio (1 = perfect focus). The trap frequency is then:
ωᵣ = √(4U₀/mw₀²), ωᵤ = √(2U₀/mzᴿ²)2. Thermal Initial Conditions
In the harmonic approximation of the Gaussian well, the thermal position distribution has widths:
σᵣ = √(k_BT/mωᵣ²) = √(k_BTw₀²/4U₀) σᵤ = √(k_BT/mωᵤ²) = √(k_BTzᴿ²/2U₀)Velocities are sampled from a Maxwell-Boltzmann distribution:
vₓ,ᵧ,ᵤ ~ 𝒩(0, √(k_BT/m))3. Ballistic Free Flight
During the release time Δt, atoms propagate freely under gravity (along ẑ):
x(Δt) = x₀ + vₓΔt y(Δt) = y₀ + vᵧΔt z(Δt) = z₀ + vᵤΔt − ½gΔt²The z-velocity acquires a gravitational component:
vᵤ(Δt) = vᵤ₀ − gΔt4. Recapture Condition
After the tweezer is re-switched on, an atom at position (x_f, y_f, z_f) with velocity v_f is recaptured if its total energy is negative (bound state):
E_total = ½m|v_f|² + U(r_f) < 0where the Gaussian trap potential is:
U(ρ,z) = −U₀ · (w₀/w(z))² · exp(−2ρ²/w(z)²) w(z) = w₀√(1 + z²/zᴿ²)So the recapture condition becomes: ½m|v_f|² < U₀·(w₀/w(z_f))²·exp(−2ρ_f²/w(z_f)²)
Trajectory Sketch
Tweezer Parameters
| Species | ΔE_D1 (Eₕ) | |d_D1| (ea₀) | ΔE_D2 (Eₕ) | |d_D2| (ea₀) | α_core (a.u.) | Mass (u) |
|---|---|---|---|---|---|---|
| ¹³³Cs | 0.050932 | 4.4890 | 0.053456 | 6.3238 | 17.35 | 133 |
| ⁸⁷Rb | 0.057314 | 4.231 | 0.058396 | 5.977 | 10.54 | 87 |
| ²³Na | 0.077258 | 3.5246 | 0.077336 | 4.9838 | 1.86 | 23 |
| ⁶Li | 0.067906 | 3.317 | 0.067907 | 4.689 | 2.04 | 6 |
Polarizability Formula (full)
In atomic units, with ω_au = Eₕ/ℏ ≈ 4.134×10¹⁶ rad/s and ω_laser = 2πc/λ:
α(ω) = Σⱼ∈{D1,D2} ΔEⱼ · |dⱼ|² / [3(ΔEⱼ² − (ω/ω_au)²)] + α_coreSI conversion: α_SI = α_au × 4πε₀a₀³ = α_au × 1.6488×10⁻⁴¹ C·m²/V. Note: for wavelengths far from resonance (λ ≫ D1, D2 wavelengths for blue detuning, or red-detuned for standard tweezers), α > 0 and the trap minimum is at maximum intensity.
- ▸Harmonic initial distribution: Atom positions are sampled from a Gaussian distribution matching the harmonic approximation of the Gaussian well. Valid when T ≪ U₀/k_B.
- ▸Ballistic flight: No inter-atomic collisions (single-atom tweezer). Gravity acts in the −z direction. No light forces during release.
- ▸Full Gaussian recapture: The recapture condition uses the exact Gaussian potential (not harmonic approx), including the z-dependence through w(z) = w₀√(1 + z²/z_R²).
- ▸Scalar polarizability: We use only the scalar component; vector/tensor contributions are neglected (valid for linearly polarized, far-detuned tweezers in the electronic ground state).
- ▸Single-transition approximation: Only D1 and D2 lines contribute (core correction α_core included). Higher excited states add <1% for λ > 700 nm.
- ▸Recapture efficiency: The simulation does not model atom re-heating during recapture, quantum tunnelling, or motional-state filtering. It predicts the classical recapture probability.
- ▸Statistical noise: For N_MC = 600, standard error ≈ 1/√600 ≈ 4%. Increase to 1500 for smoother curves.