TIME-OF-FLIGHT THERMOMETRY

TOF Thermometry

Ballistic expansion calculator for ultracold atom clouds — extract temperature from σ² vs t² fits, compute de Broglie wavelength, phase-space density, and compare to fundamental cooling limits.

σ²(t)Expansion law
TTemperature fit
λ_dBde Broglie wavelength
PSDPhase-space density

01 Atom & Temperature

Atom Species
1 nK1 μK1 mK
1 μm50 μm500 μm
101 k10 M
⁸⁷Rb: mass = 86.909 u · D2 line Γ/2π = 6.065 MHz · λ = 780 nm
Temperature
—
—
Thermal velocity v_rms
—
mm/s
√(k_BT/m) per axis
de Broglie λ
—
—
h / √(2πmk_BT)
Phase-space density
—
n_peak × λ_dB³
—
Peak density n_peak
—
atoms / cm³
N / (2π)^(3/2) σ₀³
Distance to cooling limits
Doppler limit T_D —
Recoil limit T_rec —
Bar shows T / T_limit (capped at 100%). Below Doppler = sub-Doppler cooled.

02 Cloud Expansion σ(t)

TOF Range

Ballistic expansion law

After release from a trap, the atom cloud expands freely. For a thermal gas with 1D width σ₀ at t=0:

$$\sigma^2(t) = \sigma_0^2 + \frac{k_{\rm B} T}{m}\,t^2$$ $$\Rightarrow T = \frac{m\,(\sigma^2(t) - \sigma_0^2)}{k_{\rm B}\,t^2}$$ For harmonic trap ($\omega$): $\sigma_0 = \sqrt{k_{\rm B}T/(m\omega^2)}$ $$\Rightarrow \sigma^2(t) = \frac{k_{\rm B}T}{m}\left(\frac{1}{\omega^2} + t^2\right)$$
The slope of σ² vs t² gives k_BT/m directly — plot σ²(t²) and fit a line. Intercept = σ₀², slope = k_BT/m.
σ(t) — cloud radius vs time of flight
σ(t) — full expansion Free expansion only (σ₀=0) σ₀ (in-situ limit)

03 Fit Temperature from Data

Enter measured cloud widths σ at known TOF times t. The tool performs a linear least-squares fit of σ² vs t² to extract T and σ₀. You need at least 2 points.

Measured Data
# t (ms) σ (μm)
Fit Results
Add at least 2 data points to compute a fit.
σ² vs t² — linear fit

04 Cooling Limit Reference

T_Doppler = ℏΓ / (2k_B)

In a 1D optical molasses, the equilibrium between laser cooling and heating from photon recoil occurs at the Doppler temperature. It depends only on the natural linewidth Γ and is independent of atomic mass. Typical values: Rb 145 μK, Cs 125 μK, Na 235 μK.

$$T_{\rm Doppler} = \frac{\hbar\Gamma}{2k_{\rm B}}$$ Example ($^{87}$Rb, $\Gamma/2\pi = 6.065$ MHz): $T_D = 145.6\,\mu$K

T_rec = ℏ²k² / (m k_B)

Each absorbed photon imparts a momentum kick ℏk. The temperature equivalent of one recoil is the recoil temperature — the minimum achievable by any scheme that relies on spontaneous emission (e.g. Sisyphus cooling reaches ~T_rec). Sub-recoil cooling (VSCPT, Raman cooling, EIT) can go below this limit.

$$T_{\rm rec} = \frac{\hbar^2 k^2}{m k_{\rm B}} = \frac{E_{\rm rec}}{k_{\rm B}}$$ Example ($^{87}$Rb, $\lambda = 780$ nm): $E_{\rm rec}/k_{\rm B} = 362$ nK

PSD = n × λ_dB³

Phase-space density quantifies the occupation of quantum states. When PSD ≈ 2.612 (the Riemann ζ(3/2) value), a Bose gas in a uniform box reaches Bose-Einstein condensation. In a harmonic trap the critical condition is (k_BT_c/ℏω) = (N/ζ(3))^(1/3).

$$\lambda_{\rm dB} = \frac{h}{\sqrt{2\pi m k_{\rm B} T}}$$ $${\rm PSD} = n_{\rm peak}\,\lambda_{\rm dB}^3 = \frac{N}{(2\pi)^{3/2}\sigma^3}\cdot\left(\frac{h}{\sqrt{2\pi m k_{\rm B} T}}\right)^3$$ BEC threshold (uniform 3D box): PSD $= 2.612$
BEC critical $T$ (harmonic trap): $$T_c = \frac{\hbar\omega}{k_{\rm B}}\left(\frac{N}{\zeta(3)}\right)^{1/3} \qquad \zeta(3) \approx 1.202$$

05 References

📄 Metcalf & van der Straten (1999) — Laser Cooling and Trapping (Rev. Mod. Phys.) 📄 Ketterle, Durfee & Stamper-Kurn (1999) — Making, probing, and understanding BECs 📄 Bloch, Dalibard & Zwerger (2008) — Many-body physics with ultracold gases (Rev. Mod. Phys.) 📊 Steck — Alkali Data Sheets (mass, Γ, λ, I_sat for all species) 📄 Anderson et al. (1995) — Observation of Bose-Einstein Condensation (first BEC, Science)