🧲 Build 03 · MOT & Magnetic Trap Designer

MOT & Magnetic Trap Designer

Full-parameter magneto-optical trap calculator β€” from laser detuning and field gradient to damping coefficient, spring constant, capture velocity, and scattering rate. Includes a live detuning optimization curve, species comparison, and the full cooling pathway.

A magneto-optical trap (MOT) is the first stage of every cold-atom experiment. Three pairs of counter-propagating red-detuned beams with a magnetic quadrupole field simultaneously cool and confine atoms via radiation pressure. Understanding Ξ± (damping), ΞΊ (spring constant), and vc (capture velocity) is essential for optimizing loading rates and temperature before transfer into tweezers, a lattice, or an evaporative cooling stage.
Model validity: this calculator uses a 1D, low-velocity, two-level scattering-force model. Real alkali MOTs are multilevel systems with Clebsch-Gordan factors, optical pumping, polarization gradients, beam imbalance, reabsorption, and sub-Doppler physics. Treat Ξ±, ΞΊ, and capture velocity as scaling estimates, not calibrated predictions for a loaded cloud.

01 MOT Calculator

Laser & Field Parameters
0.5Ξ“1Ξ“ (Ξ± opt)5Ξ“
0.11 (sat.)3
⁸⁷Rb D2: Ξ“/2Ο€ = 6.065 MHz Β· Ξ» = 780 nm Β· I_sat = 1.67 mW/cmΒ²
Damping coeff. Ξ±
β€”
10⁻²⁰ kg/s
F = βˆ’Ξ±v
Spring constant ΞΊ
β€”
10⁻¹⁷ N/m
F = βˆ’ΞΊr
MOT frequency
β€”
Hz
Ο‰ = √(ΞΊ/m)
Capture velocity
β€”
m/s
v_c β‰ˆ |Ξ”|/k
Doppler limit T_D
β€”
ΞΌK
ℏΓ/2k_B
Max acceleration
β€”
m/sΒ²
ℏkΞ“s/2m(1+s)
Scattering rate R_sc
β€”
10⁢ photons/s
Ξ“/2 Β· s/(1+s+(2Ξ”/Ξ“)Β²)
Damping time Ο„
β€”
ms
Ο„ = m/Ξ±
Cloud radius Οƒ
β€”
ΞΌm (1/e)
√(k_BT_D/κ)

MOT force in 1D (two-beam, low-velocity limit)

Two counter-propagating beams in a quadrupole B-field create both velocity-dependent damping and position-dependent restoring forces. The effective detuning for each beam shifts as Ξ”eff = Ξ” Β± kv Β± ΞΌeffB(r)/ℏ.

$$F_{\rm MOT} \approx -\alpha v - \kappa r$$ $$\alpha = \frac{8\hbar k^2 s|\Delta|/\Gamma}{(1+s+(2\Delta/\Gamma)^2)^2} \qquad \kappa = \frac{\mu_{\rm eff}}{\hbar k}\cdot\frac{dB}{dz}\cdot\alpha$$ Optimal damping: $|\Delta|_{\rm opt} \approx \Gamma/(2\sqrt{3}) \approx 0.29\,\Gamma$ (low-$s$ limit)
Live MOT Visualization  Β·  2D cross-section
Lab tip: when loading atoms, start with a large detuning (~2–3Ξ“) and high gradient for maximum capture volume, then compress (reduce detuning + increase gradient) for higher density before the sub-Doppler stage. This "cMOT" step is often 10–50 ms.

02 Detuning Optimization

Damping Ξ± and steady-state temperature both depend on |Ξ”|/Ξ“. The orange curve shows where your current detuning sits on the tradeoff β€” high Ξ± wants small detuning, low temperature wants |Ξ”|~Ξ“/2. Practical MOTs live at |Ξ”| = 1.5–2.5Ξ“, a compromise between the two.

Ξ± (% of max) and T_MOT / T_Doppler vs |Ξ”|/Ξ“ β€” live operating point shown
Damping Ξ± (% of Ξ±max) T_MOT / T_Doppler (Γ—100) Your operating point
Optimal Ξ± detuning
|Ξ”| β‰ˆ 0.29 Ξ“
Maximum damping at low s. Shifts to ~0.5Ξ“βˆš(1+s) with saturation. Rarely used in practice β€” capture velocity is too small.
Minimum temperature
|Ξ”| = Ξ“/2
Doppler limit T_D = ℏΓ/2k_B reached here at sβ†’0. At finite s, minimum shifts to |Ξ”| = (Ξ“/2)√(1+s).
Practical range
1.5 – 2.5 Ξ“
Balances adequate damping, reasonable capture velocity, and tolerable cloud temperature. Most alkali MOTs operate here.
Lab tip: if your MOT fluorescence is high but the cloud stays hot, try moving toward larger |Ξ”|. If loading is slow, try a smaller detuning to increase capture velocity β€” but watch for reduced damping and a hotter cloud.

03 Species at a Glance

Key MOT parameters for all five species at your current detuning and saturation settings. Highlighted row is your selected atom.

Atom Ξ» (nm) Ξ“/2Ο€ (MHz) T_Doppler (ΞΌK) T_recoil (nK) v_c at |Ξ”| (m/s) Ξ±/Ξ±_Rb
Li note: Li's nearly degenerate D1/D2 lines cause mixing in the MOT β€” gray molasses on D1 is the standard sub-Doppler step. K note: K39's small ground-state hyperfine splitting (~461 MHz) requires careful repumper alignment to avoid optical pumping out of the cycling transition.

04 From MOT to Experiment β€” The Cooling Cascade

The MOT is stage one. Here's how temperature, atom number, and density evolve from the atomic source all the way to a tweezer array or quantum gas.

πŸ”₯
Atomic Source
300 K β†’ ~100 mK
Oven / 2D MOT + Zeeman slower. Slows atoms from ~300 m/s to capture velocity.
β†’
🧲
3D MOT
100 – 300 ΞΌK
10⁷–10⁹ atoms. mm-scale cloud. Density ~10¹⁰/cmΒ³. Limited by re-absorption.
β†’
πŸ—œοΈ
cMOT
30 – 100 ΞΌK
Ramp: ↑ gradient, ↓ repumper, ↑ |Ξ”|. Higher density for better transfer. ~10–50 ms.
β†’
🌫️
Gray Molasses / PGC
2 – 20 ΞΌK
Sub-Doppler. Dark states (gray molasses) or Sisyphus (PGC). B-field zeroed to <50 mG.
β†’
🎯
Tweezer / Lattice
5 – 30 ΞΌK in trap
Single-atom loading (~50%). Site-resolved imaging. Sideband cool to nΜ„ < 0.1.
β†’
βš›οΈ
Experiment
< 1 ΞΌK
Rydberg gates, quantum simulation, BEC, or quantum gas microscopy.

Sub-Doppler: why it works

Polarization gradients create spatially varying light shifts. Atoms climbing a potential hill scatter a photon and are reborn at the bottom β€” net energy removed per cycle. For σ⁺σ⁻ beams (Sisyphus) this gives T ∝ Uβ‚€/k_B where Uβ‚€ β‰ͺ ℏΓ, reaching 1–20 ΞΌK. Gray molasses (D1 line) exploits dark states to further suppress heating.

Tweezer loading: the 50% barrier

Gaussian tweezers load atoms from the MOT stochastically. Parity projection β€” two atoms collide and both leave via light-assisted collisions β€” caps loading at ~50% per site. Blue-detuned light pulses or resolved sideband cooling can push fidelity above 90% by deterministic preparation after loading.

05 Physics Reference

Position- and velocity-dependent forces from two beams

For a moving atom at position r in the quadrupole field, the effective detuning seen by each beam is Ξ”eff = Ξ” Β± kv Β± ΞΌeffB(r)/ℏ. Expanding to first order in kv and ΞΌeffB/ℏ|Ξ”| gives the linear damping + spring-constant picture.

$$R_{\rm sc} = \frac{\Gamma}{2}\cdot\frac{s}{1+s+(2\Delta_{\rm eff}/\Gamma)^2}$$ Net force ($|kv|\ll|\Delta|$): $F = F_+ - F_- \approx -\alpha v - \kappa r$
$$\alpha = \frac{8\hbar k^2 s|\Delta|/\Gamma}{(1+s+(2\Delta/\Gamma)^2)^2} \quad \text{[1D, 2 beams]}$$ $$\kappa = \frac{\mu_{\rm eff}}{\hbar k}\cdot\frac{dB}{dz}\cdot\alpha$$ Optimal $\alpha$: $|\Delta| = \Gamma/(2\sqrt{3})\approx 0.29\,\Gamma$ (low-$s$ limit)
Sign convention: this page uses Ξ” = Ο‰_L βˆ’ Ο‰_0, so red detuning is Ξ” < 0. Some AMO texts use Ξ” = Ο‰_0 βˆ’ Ο‰_L, flipping the sign of every term in the damping algebra. Always confirm the convention before comparing formulas.

v_c sets the flux captured from the background vapor

Only atoms slower than v_c β‰ˆ |Ξ”|/k can be captured before they traverse the beam. The loading rate L scales strongly with v_c and beam area A: L ∝ n_bg Β· v_c⁴ Β· A (Maxwell-Boltzmann tail). Larger beams and larger detuning dramatically increase loading β€” at the cost of a hotter cloud.

Capture velocity: $v_c \approx |\Delta|/k$ (velocity at which Doppler shift = detuning) $$L \approx n_{\rm bg}\,\bar{v}\left(\frac{v_c}{\bar{v}}\right)^4 \cdot \pi w^2 \quad \bar{v}=\sqrt{8k_BT_{\rm oven}/\pi m}$$ Stopping distance: $d = v_c^2/(2a_{\rm max})$, must satisfy $d \lesssim w$ (beam radius)
Maximum deceleration: $a_{\rm max} = \hbar k\Gamma s/(2m(1+s))$

For Rb87 at Ξ” = βˆ’1.5Ξ“: v_c β‰ˆ 7 m/s and a_max β‰ˆ 10⁡ m/sΒ². Stopping distance ~0.25 mm, comfortably within a 10 mm beam. Background pressure of ~10⁻⁸ Torr gives loading rates of 10⁷–10⁸ atoms/s for a standard vapor-cell MOT.

Polarization gradients and dark states break the Doppler limit

The Doppler limit T_D = ℏΓ/2k_B assumes a two-level atom. Multi-level alkalis in polarization-gradient fields experience Sisyphus cooling (σ⁺σ⁻ geometry) or gray-molasses cooling (D1 line, dark states), reaching temperatures 10–100Γ— below T_D.

Sisyphus (PGC): $T_{\rm PGC} \approx \frac{U_0}{k_B} \propto \frac{\hbar\Omega_R^2}{\Gamma|\Delta|}$; requires $|\Delta|\gg\Gamma$, $B=0$
Gray molasses (D1): exploits $\Lambda$ dark states; $T\sim 5$–$20\,\mu$K for Rb, Cs, Li
Sideband cooling (tweezer): $T\sim \hbar\omega_{\rm trap}/k_B$; $\bar{n}<0.1$ achievable
Comparison (Rb87): $T_D = 146\,\mu$K $\to$ PGC $\sim10$–$30\,\mu$K $\to$ SBC $\sim1\,\mu$K

Key requirement for all sub-Doppler cooling: residual B-field <50–100 mG at the cloud. Eddy currents from switching MOT coils typically require 1–10 ms of wait time before the molasses phase begins.

Peak density is set by reabsorption, not atom number

Scattered photons from atoms near the cloud center are reabsorbed by outer atoms, creating an outward radiation pressure that limits peak density. This gives a density-independent equilibrium: adding more atoms increases cloud size, not density.

Peak density limit: $n_{\rm max} \approx \frac{\kappa}{\sigma_{\rm sc}\,\hbar k \Gamma / 2}$ (multiple-scattering limit)
Typical Rb87 MOT: $n_{\rm peak} \sim 10^{10}$–$10^{11}$ cm$^{-3}$
cMOT fix: reducing repumper intensity pumps atoms to dark $F=1$ ground state β†’ less reabsorption β†’ $5$–$10\times$ density gain
Dark-SPOT MOT: shadow mask blocks repumper at center; density $\sim 10^{12}$ cm$^{-3}$

06 References

πŸ“„ Metcalf & van der Straten β€” Laser Cooling and Trapping (textbook) textbook πŸ“„ Chu (1998) β€” Nobel Lecture: The manipulation of neutral particles peer-reviewed πŸ“„ Raab et al. (1987) β€” Trapping of neutral sodium atoms with radiation pressure peer-reviewed πŸ“„ Anderson et al. (1995) β€” Observation of BEC in a dilute atomic vapor peer-reviewed πŸ“„ Steck β€” Quantum and Atom Optics lecture notes (free, comprehensive) textbook

See Also

🌑️
TOF Thermometry
Temperature from time-of-flight expansion
 ❄️
Cooling Simulator
Doppler and Sisyphus cooling dynamics
 🌑️
Laser Cooling
Laser cooling physics and limits