MOT Designer — AMO Career

01 Magneto-Optical Trap (MOT)

Laser & Field Parameters

Atom species

Detuning |Δ|/Γ (red-detuned): 1.5 × Γ

0.5Γ−Γ (opt. Doppler)5Γ

Saturation parameter s = I/I_sat: 1.0

0.11 (sat.)3

Beam 1/e² radius w (mm): 10 mm

Axial B-field gradient dB/dz (G/cm): 15 G/cm

Damping coeff. α

—

10⁻²⁰ kg/s (1D, 2 beams)

F = −αv

Spring constant κ

—

10⁻¹⁷ N/m

F = −κr (radial)

MOT trap frequency

—

Hz

ω = √(κ/m)

Capture velocity

—

m/s

v_cap ≈ |Δ|/k

Doppler temperature

—

μK

ℏΓ/2k_B (limit)

Max acceleration

—

m/s²

a_max = ℏkΓs/(2m(1+s))

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

Each beam exerts a scattering force ℏk × R_sc on the atom. Two counter-propagating beams with the quadrupole field create both a velocity-dependent damping force and a position-dependent restoring force. In the low-saturation, low-velocity limit:

$$F_{\rm MOT} \approx -\alpha v - \kappa r$$ $$\alpha = \frac{8\hbar k^2 s |\Delta|/\Gamma}{(1 + s + (2\Delta/\Gamma)^2)^2} \quad \text{[damping]}$$ $$\kappa = \alpha \cdot \frac{\mu_{\rm eff}}{\hbar k} \cdot \frac{dB}{dz} \quad \text{[spring via B-field gradient]}$$ $\mu_{\rm eff} = g_F m_F \mu_B$ (effective magnetic moment)

02 Ioffe-Pritchard Magnetic Trap

IP Trap Field Parameters

Trapping state (g_F × m_F): +1.0 μ_B

Radial gradient B′ (G/cm): 150 G/cm

10 G/cm200 G/cm500 G/cm

Bias field B₀ (G): 5 G

1 G20 G100 G

Axial curvature B″ (G/cm²): 100 G/cm²

Atom temperature T (μK): 100 μK

Radial freq ω_r /2π

—

Hz

√(μ_eff B′² / m B₀)

Axial freq ω_z /2π

—

Hz

√(μ_eff B″ / m)

Geometric mean freq

—

Hz = (ω_r² ω_z)^(1/3) / 2π

relevant for BEC T_c

Trap depth U₀

—

μK

μ_eff × B₀ / k_B

η parameter

—

Evaporation efficiency: η = U₀ / (k_B T)

η=1 (poor)η=5 (good)η=10+ (excellent)

03 Trap Potential Profile

U(r) = μ_eff · |B(r)| — radial cut through IP trap

U(r) — radial potential k_B T (atom energy) Trap depth U₀ = μ_eff B₀

04 Physics Reference

MOT force: damping and spring constant ▶

Radiation pressure and position-dependent detuning

In a MOT the two counter-propagating beams create opposing radiation pressure forces. For a moving atom at position r in the quadrupole field, the effective detuning for each beam is Δ_eff = Δ ± kv ± μ_eff B(r)/ℏ. This creates both velocity (Doppler) and position (Zeeman) dependent forces.

$$R_{\rm sc} = \frac{\Gamma}{2} \cdot \frac{s}{1 + s + (2\Delta_{\rm eff}/\Gamma)^2}$$ Net force (two beams, $|kv| \ll |\Delta|$): $F = F_+ - F_- \approx -\alpha v - \kappa r$
Damping (1D, 2 beams): $$\alpha = \frac{8\hbar k^2 s|\Delta|/\Gamma}{(1 + s + (2\Delta/\Gamma)^2)^2}$$ Spring constant: $$\kappa = \frac{\mu_{eff}}{\hbar k}\cdot\frac{\partial B}{\partial r}\cdot\alpha = \mu_{eff}\cdot\frac{dB}{dz}\cdot\frac{8k^2 s|\Delta|/\Gamma}{(1+s+(2\Delta/\Gamma)^2)^2}$$ Optimal damping: $\Delta = -\Gamma/(2\sqrt{3}) \approx -0.29\Gamma \Rightarrow \alpha_{max}$

Ioffe-Pritchard trap — field geometry ▶

Non-zero minimum prevents Majorana losses

An IP trap combines a radial quadrupole field (wires or Ioffe bars: gradient B′) with axial pinch coils that create a non-zero minimum field B₀ and axial curvature B″. The non-zero field bottom eliminates Majorana spin-flip losses that plague quadrupole traps.

Field magnitude near trap center ($\rho \ll B_0/B'$): $$|B(\rho,z)| \approx B_0 + \left(\frac{B'^2}{2B_0} - \frac{B''}{2}\right)\rho^2 + \frac{B''}{2}z^2$$ Trap frequencies: $$\omega_r = \sqrt{\frac{\mu_{\rm eff}(B'^2/B_0 - B'')}{m}} \quad \text{[radial]}, \qquad \omega_z = \sqrt{\frac{\mu_{\rm eff} B''}{m}} \quad \text{[axial]}$$ Stability: $B'^2/B_0 > B''$ (radial trap not cancelled by curvature)
Trap depth: $U_0 \approx \mu_{eff} \cdot B_0$

QUIC trap (Esslinger group) and cloverleaf trap (Ketterle group) are two common IP-type implementations. Typical: B₀ = 1–10 G, B′ = 100–300 G/cm, B″ = 50–200 G/cm².

Evaporative cooling — η parameter and runaway evaporation ▶

η = U₀ / (k_B T) — truncation parameter

Evaporative cooling removes the hottest atoms (tail of the Maxwell-Boltzmann distribution) by RF or microwave induced spin-flip at a tunable cutoff energy η × k_B T. As hot atoms leave and the cloud rethermalizes via elastic collisions, the temperature drops.

$$\frac{dT}{T} = -\frac{\eta - 4}{\eta - 3 + \ldots}\cdot\frac{dN}{N}$$ "Runaway evaporation": $\gamma_{\rm el}/\gamma_{\rm inel} > 150$ (Rb-87 3-body limit)
Good: $\eta \gtrsim 5$; optimal: $\eta \approx 8$–$10$ (BEC achievable); insufficient: $\eta < 4$
RF evaporation ramp: $\nu_{\rm RF}(t)$ decreases from $\nu_0$ to $\nu_f$ $$E_{cut} = h\nu_{RF} - m_F g_F \mu_B B_0$$

Majorana losses in quadrupole traps ▶

Spin-flip at the zero of the field

A pure quadrupole trap has a zero-field point at the center. Atoms that pass near this point can undergo a non-adiabatic Majorana spin-flip from a weak-field-seeking to a strong-field-seeking state, ejecting them from the trap. The rate increases as the cloud cools and atoms approach the zero-field region.

$$\Gamma_M \approx \omega_r \cdot \frac{\hbar}{m\omega_r\langle r^2\rangle} \approx \omega_r \cdot \frac{T_{\rm rec}}{T}$$ $$\omega_r = \sqrt{\frac{\mu_B B'^2}{m B_0}} \quad \text{[for quadrupole: } B_0 \to 0\text{]}$$ For quadrupole ($B_0 = 0$): $\Gamma_M \to \infty$ — catastrophic losses
Fix: add bias field (IP trap, TOP trap) or optical plug (Rb BEC 1995).
Lifetime in quadrupole: $\tau_M \sim m\bar{v}/(\mu_B B')$ where $\bar{v} = \sqrt{2k_{\rm B}T/m}$

05 References

📄 Metcalf & van der Straten — Laser Cooling and Trapping (textbook) 📄 Petrich et al. (1995) — Stable, Tightly Confining Magnetic Trap (TOP trap, first Rb BEC paper) 📄 Anderson et al. (1995) — Observation of BEC in dilute Rb gas (Science) 📄 Ketterle, Durfee & Stamper-Kurn — Making, probing, and understanding BECs 📄 Esslinger et al. (1998) — Bose-Einstein condensation in a QUIC trap