🎯 Tool 08 · Tweezer Thermometry (R&R)

Tweezer Thermometry (R&R)

Monte Carlo thermometry for optical tweezers. Release atoms, let them fly ballistically under gravity, then recapture, the survival curve encodes the temperature.

Release-recapture is a quick way to estimate the temperature of a single atom in an optical tweezer once the trap depth and waist are known. The trap is briefly switched off, the atom flies freely for a variable hold time τ, then the trap is restored. Hotter atoms expand faster and are less likely to be recaptured, so fitting the survival curve gives a thermometry signal without requiring Raman sideband spectroscopy.

Physics of Release-Recapture

How thermal atoms escape and the survival probability encodes temperature

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_core

where 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 − ½gΔt²

The z-velocity acquires a gravitational component:

vᵤ(Δt) = vᵤ₀ − gΔt

4. 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) < 0

where 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

Monte Carlo Simulator

Port of the Python/NumPy vectorised simulation, runs in your browser

Tweezer Parameters

Atom species

Wavelength (nm)

Power (mW)

Beam waist w₀ (μm)

Strehl ratio S

MC samples N

Max release time (ms)

Time steps

Temperatures to simulate (μK), click × to remove

Configure parameters and press Run.

Species Polarizability Data

D1 + D2 transition data used for dynamic polarizability calculation

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_core

SI 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.

Notes & Assumptions

▸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.
▸Thermometry, not direct trap-frequency metrology: Release-recapture primarily extracts temperature for a known potential. Trap frequencies enter through the assumed waist, depth, and Gaussian geometry; use parametric heating or sideband spectroscopy for a direct frequency measurement.
▸Statistical noise: For N_MC = 600, standard error ≈ 1/√600 ≈ 4%. Increase to 1500 for smoother curves.

Sources

Tuchendler et al. (2008), energy distribution and thermometry of a single atom in a tweezer Kaufman et al. (2012), single-atom tweezer cooling and thermometry context Steck alkali data sheets for line and polarizability inputs