🎯 Trap, Image & Cool 02 · Single-atom Temperature

Single-atom Temperature

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

Thermometry workflow

Read the survival curve before worrying about the derivation.

Release-recapture is action-first thermometry: switch off the trap, vary release time, measure survival, and fit temperature. The physics boxes explain why the fitted curve changes shape.

01What the curve means

Hot atoms expand faster, so survival drops earlier as release time increases.

02Run the simulator

Set species, trap geometry, depth, waist, and temperature; compare the generated survival curve.

03Fit temperature

Use experimental survival points to infer a temperature for the assumed trap potential.

04Check failure modes

Bad waist, anharmonicity, pointing drift, and nonthermal distributions can bias the result.

What the Survival Curve Means
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):

$$\alpha(\omega) = \sum_j \frac{\Delta E_j\,|d_j|^2}{3(\Delta E_j^2 - \omega^2)} + \alpha_{\rm core}$$

where Ξ”Eβ±Ό = E_excited βˆ’ E_ground in Hartree, dβ±Ό is the reduced matrix element in units of eΒ·aβ‚€, and Ο‰ is in atomic units (Eβ‚•/ℏ). Converting to SI and computing the peak intensity:

$$U_0 = \frac{\alpha_{\rm SI}\,I_{\rm peak}}{c\,\varepsilon_0}, \qquad I_{\rm peak} = \frac{2SP}{\pi w_0^2}$$

where S is the Strehl ratio (1 = perfect focus). The trap frequency is then:

$$\omega_r = \sqrt{\frac{4U_0}{mw_0^2}}, \qquad \omega_z = \sqrt{\frac{2U_0}{mz_R^2}}$$

2. Thermal Initial Conditions

In the harmonic approximation of the Gaussian well, the thermal position distribution has widths:

$$\sigma_r = \sqrt{\dfrac{k_{\rm B}T}{m\omega_r^2}} = \sqrt{\dfrac{k_{\rm B}T\,w_0^2}{4U_0}}$$
$$\sigma_z = \sqrt{\dfrac{k_{\rm B}T}{m\omega_z^2}} = \sqrt{\dfrac{k_{\rm B}T\,z_R^2}{2U_0}}$$

Velocities are sampled from a Maxwell-Boltzmann distribution:

$$v_{x,y,z} \sim \mathcal{N}\!\left(0,\,\sqrt{k_{\rm B}T/m}\right)$$

3. Ballistic Free Flight

During the release time Ξ”t, atoms propagate freely under gravity (along zΜ‚):

$$x(\Delta t) = x_0 + v_x\,\Delta t$$
$$y(\Delta t) = y_0 + v_y\,\Delta t$$
$$z(\Delta t) = z_0 + v_z\,\Delta t - \tfrac{1}{2}g\Delta t^2$$

The z-velocity acquires a gravitational component:

$$v_z(\Delta t) = v_{z0} - g\,\Delta t$$

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):

$$E_{\rm total} = \tfrac{1}{2}m|\mathbf{v}_f|^2 + U(\mathbf{r}_f) < 0$$

where the Gaussian trap potential is:

$$U(\rho,z) = -U_0\!\left(\frac{w_0}{w(z)}\right)^{\!2}\exp\!\left(\frac{-2\rho^2}{w(z)^2}\right)$$
$$w(z) = w_0\sqrt{1 + z^2/z_R^2}$$

So the recapture condition becomes: Β½m|v_f|Β² < Uβ‚€Β·(wβ‚€/w(z_f))Β²Β·exp(βˆ’2ρ_fΒ²/w(z_f)Β²)

Run the Monte Carlo Thermometer
Port of the Python/NumPy vectorised simulation, runs in your browser

Tweezer Parameters

Configure parameters and press Run.
Species Polarizability Data
D1 + D2 transition data used for dynamic polarizability calculation
Species Ξ”E_D1 (Eβ‚•) |d_D1| (eaβ‚€) Ξ”E_D2 (Eβ‚•) |d_D2| (eaβ‚€) Ξ±_core (a.u.) Mass (u)
ΒΉΒ³Β³Cs0.0509324.48900.0534566.323817.35133
⁸⁷Rb0.0573144.2310.0583965.97710.5487
Β²Β³Na0.0772583.52460.0773364.98381.8623
⁢Li0.0679063.3170.0679074.6892.046

Polarizability Formula (full)

In atomic units, with Ο‰_au = Eβ‚•/ℏ β‰ˆ 4.134Γ—10¹⁢ rad/s and Ο‰_laser = 2Ο€c/Ξ»:

$$\alpha(\omega) = \sum_{j\in\{D1,D2\}} \frac{\Delta E_j\,|d_j|^2}{3\!\left(\Delta E_j^2 - (\omega/\omega_{\rm au})^2\right)} + \alpha_{\rm core}$$

SI conversion: Ξ±_SI = Ξ±_au Γ— 4πΡ₀aβ‚€Β³ = Ξ±_au Γ— 1.6488Γ—10⁻⁴¹ CΒ·mΒ²/V. Note: for wavelengths far from resonance (Ξ» ≫ D1, D2 wavelengths for blue detuning, or red-detuned for standard tweezers), Ξ± > 0 and the trap minimum is at maximum intensity.

Assumptions & Failure Modes
  • β–ΈHarmonic initial distribution: Atom positions are sampled from a Gaussian distribution matching the harmonic approximation of the Gaussian well. Valid when T β‰ͺ Uβ‚€/k_B.
  • β–ΈBallistic flight: No inter-atomic collisions (single-atom tweezer). Gravity acts in the βˆ’z direction. No light forces during release.
  • β–ΈFull Gaussian recapture: The recapture condition uses the exact Gaussian potential (not harmonic approx), including the z-dependence through w(z) = wβ‚€βˆš(1 + zΒ²/z_RΒ²).
  • β–ΈScalar polarizability: We use only the scalar component; vector/tensor contributions are neglected (valid for linearly polarized, far-detuned tweezers in the electronic ground state).
  • β–ΈSingle-transition approximation: Only D1 and D2 lines contribute (core correction Ξ±_core included). Higher excited states add <1% for Ξ» > 700 nm.
  • β–ΈRecapture efficiency: The simulation does not model atom re-heating during recapture, quantum tunnelling, or motional-state filtering. It predicts the classical recapture probability.
  • β–ΈThermometry, 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.
  • β–ΈStatistical noise: For N_MC = 600, standard error β‰ˆ 1/√600 β‰ˆ 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 Savard et al. (1997), PRA 56 R1095 β€” laser-noise-induced heating of trapped atoms; heating rate model from intensity fluctuations Gehm et al. (1998), PRA 58 3914 β€” dynamics of noise-induced heating in atom traps; parametric and linear heating Thompson et al. (2013), PRL 110 133001 β€” coherent single-atom sideband cooling to the motional ground state in a tweezer