🌡️ Trap, Image & Cool 06 · Laser Cooling Simulator

Laser Cooling Simulator

From room temperature to quantum ground state. Explore Doppler cooling, sub-Doppler gray molasses, and resolved-sideband cooling: the three stages that bring atoms to the quantum limit.

Cooling landscape

Separate free-space cooling from trapped-atom cooling.

This page starts with velocity-space Doppler cooling, then shows why sub-Doppler and sideband methods are needed as experiments move toward optical tweezers and Lamb-Dicke physics.

01Doppler damping

Use detuning and saturation to see a thermal velocity distribution narrow.

02Sub-Doppler mechanisms

Connect gray molasses and Sisyphus cooling to dark states, light shifts, and recoil scales.

03Sideband cooling

Move from velocity damping to motional-state cooling in a harmonic trap.

04Choose a method

Compare linewidth, recoil, trap frequency, and required laser complexity.

Live Doppler Cooling Simulator

Drag the sliders — watch the velocity distribution narrow as atoms cool

Detuning δ/Γ −0.5

Saturation s₀ 0.5

T = — μK ▏= Doppler limit ← hot | velocity | hot →

Temperature Ladder

From room temperature to motional ground state, 9 orders of magnitude

Why Cold Atoms for Quantum Computing?

Gate fidelity for a two-qubit Rydberg entangling gate scales as

$$\mathcal{F} \approx 1 - \alpha\langle n\rangle - \beta/\Omega_R$$

where ⟨n⟩ is the mean vibrational quantum number and Ω_R is the Rabi frequency. Each cooling stage suppresses thermal motion, reducing state-preparation errors from >10% to <0.5%. Resolved-sideband cooling to ⟨n⟩ < 0.1 is now routine in tweezer arrays.

Radiation Pressure Force

A two-level atom moving with velocity v along a laser beam of detuning δ = ω_L − ω_0 experiences a Doppler-shifted resonance at ω_0 − kv. The scattering rate

$$\Gamma_{\rm sc} = \frac{\Gamma}{2}\cdot\frac{s_0}{1 + s_0 + (2\delta\'/\Gamma)^2}$$

where δ' = δ + kv is the effective detuning and s₀ = I/I_sat. With counter-propagating beams the net force becomes velocity-dependent:

$$F(v) \approx -\alpha v, \qquad \alpha = \hbar k^2 \cdot \frac{4|\delta|\,s_0\,\Gamma}{(1 + s_0 + (2\delta/\Gamma)^2)^2}$$

This viscous force (α > 0 for red detuning δ < 0) cools atoms. The minimum Doppler temperature is set by the balance between velocity damping and momentum diffusion:

$$T_D = \frac{\hbar\Gamma}{2k_{\rm B}}$$

Force Curve F(v), interactive

Detuning δ/Γ −0.5

Saturation s₀ 0.5

α (damping) = —

T_Doppler vs Detuning

Minimum T_D = ℏΓ/(2k_B) at δ = −Γ/2

Doppler Limit in Practice

For Cs D2 line (Γ/2π = 5.234 MHz): T_D = 125 μK. For ⁶Li D2 (Γ/2π = 5.87 MHz): T_D = 141 μK. Sub-Doppler effects can break this limit, reaching the single-photon recoil scale E_r/k_B = ℏ²k²/(2mk_B).

A 3D molasses (6 beams ±x, ±y, ±z) provides isotropic cooling. Adding a magnetic field gradient creates a magneto-optical trap (MOT) that both cools and traps atoms.

Doppler Cooling Parameters by Species

⁶Li (671 nm)

MOT → cMOT → GM

Γ/2π = 5.87 MHz. T_D = 141 μK. Unresolved excited hyperfine structure, GM essential for sub-Doppler cooling. Single recoil E_r/k_B = 3.5 μK; one scatter cycle is roughly 7.1 μK without cooling.

⁸⁷Rb (780 nm)

MOT → PGC → evaporation

Γ/2π = 6.065 MHz. T_D = 146 μK. Well-resolved F levels. Sub-Doppler polarisation-gradient cooling (PGC) to ~4 μK.

¹³³Cs (852 nm)

MOT → cMOT → GM → RSB

Γ/2π = 5.234 MHz. T_D = 125 μK. Strong Sisyphus cooling. Tweezer arrays with RSB cooling to ⟨n⟩ < 0.1.

²³Na (589 nm)

Zeeman slower → MOT → PGC

Γ/2π = 9.795 MHz. T_D = 235 μK. Yellow D2 line. First atom laser (MIT 1997). High evaporation efficiency.

⁸⁸Sr (689 nm)

Blue MOT → Red MOT → magic tweezer

Narrow-line intercombination: Γ/2π = 7.5 kHz. T_D = 180 nK. Direct sub-μK MOT. Magic wavelength at 813 nm for optical lattice clock.

¹⁷⁴Yb (556 nm)

Blue MOT → Green MOT → lattice

Narrow ¹S₀→³P₁: Γ/2π = 182 kHz. T_D = 4.4 μK. Nuclear spin I=0. Ideal for SU(N) fermions (¹⁷¹Yb) and optical clocks.

Sisyphus Cooling (Polarisation-Gradient)

In a standing wave with spatially varying polarisation (lin⊥lin), the light-shifted sublevel energies vary with position on the scale of λ/4. An atom in |m_F = +F⟩ moving in the +x direction climbs a potential hill, losing kinetic energy. At the top it optically pumps to |m_F = −F⟩ (lowest potential at that location), and the cycle repeats, Sisyphus loses kinetic energy each half-cycle.

$$T_{\rm Sisyphus} \propto \frac{U_0^2}{E_r} \propto \frac{I^2}{\delta^2\,E_r}$$

The single-photon recoil energy E_r = ℏ²k²/(2m) sets the microscopic scale for random kicks. For Cs D2, E_r/k_B ≈ 99 nK. In practice, polarization-gradient cooling reaches μK-scale temperatures, often tens of recoil energies rather than a universal 2–4 recoil limit.

Λ-Enhanced Gray Molasses (GM) for ⁶Li

⁶Li has an unresolved excited-state hyperfine structure (Δ_HF < Γ), making Sisyphus PGC inefficient. Instead, a Λ-system gray molasses (D1 line at 671 nm) exploits a dark state between |F=1/2⟩ and |F=3/2⟩:

$$|\mathrm{dark}\rangle = \cos\theta\,|F\!=\!1/2\rangle - \sin\theta\,|F\!=\!3/2\rangle$$

Dark-state atoms are velocity-selected: slow atoms stay dark (low scatter), fast atoms couple to the bright state and get cooled. GM achieves T ≈ 40–60 μK for ⁶Li, well below T_D = 141 μK.

Our experiment: GM on the D1 line at δ/Γ ≈ +2, reaching T ≈ 50 μK before loading into the optical tweezer array.

Sisyphus Potential (lin⊥lin)

Sublevel energies U±(x) and optical pumping at crests

Λ System, Gray Molasses Level Diagram

D1 Λ-system: dark state protected from scattering

⁶Li Gray Molasses: Experimental Result

By applying a D1 gray molasses (Ω/2π ≈ 3 Γ, δ/2π ≈ +2 Γ) for 3 ms after the MOT switch-off, we achieved T ≈ 50 μK from T_MOT ≈ 200 μK, a 4× improvement in temperature that directly reduces the motional-state occupation after loading into the tweezer: ⟨n⟩_initial ∝ T/ω_trap.

Harmonic Trap & Sideband Structure

An atom in a harmonic potential with trap frequency ω_trap/2π has quantised motional states |n⟩. The motional sidebands appear at:

$$\omega_{\rm carrier} \pm n\cdot\omega_{\rm trap}/2\pi$$

In the Lamb-Dicke regime (η² = E_r/ℏω ≪ 1), transition rates on the red sideband (RSB, ω − ω_trap) go as η²n, while the blue sideband (BSB) goes as η²(n+1).

$$\eta = k\sqrt{\dfrac{\hbar}{2m\omega_{\rm trap}}} = \sqrt{\dfrac{E_r}{\hbar\omega_{\rm trap}}}$$

Resolved Sideband Cooling Cycle

Drive the RSB: |g,n⟩ → |e,n−1⟩. Spontaneous emission returns mostly to |g,n−1⟩ (Lamb-Dicke suppression). Each cycle removes one vibrational quantum:

$$\langle\dot{n}\rangle = -\Gamma_{\rm cool}\cdot n + \Gamma_{\rm heat}$$

Steady-state: ⟨n⟩_ss = (Γ_heat/Γ_cool) ≈ η²Γ²/(4ω²) for weak sideband. In practice ⟨n⟩_ss < 0.1 is achievable.

EIT-Assisted Cooling

Electromagnetically Induced Transparency (EIT) cooling uses a probe + coupling beam to engineer a narrow absorption feature at the RSB frequency. The dark state suppresses carrier scattering while the RSB absorption is enhanced, achieving faster cooling rates than conventional RSB cooling.

$$\langle n\rangle_{\rm EIT} \approx \frac{1}{4}\!\left(\frac{\gamma_{\rm dark}}{\omega_{\rm trap}}\right)^{\!2}$$

EIT is especially useful for motional frequencies ω_trap < Γ (unresolved sideband regime), enabling ground-state cooling in shallower traps.

Sideband Spectrum, interactive

⟨n⟩ 5

η (Lamb-Dicke) 0.15

RSB/BSB asymmetry ≈ ⟨n⟩/(⟨n⟩+1), thermometry probe

⟨n⟩(t) Cooling Evolution

Initial ⟨n⟩ 20

Γ_cool/ω 0.05

⟨n⟩(t) = ⟨n⟩_ss + (n₀ − ⟨n⟩_ss)e^{−Γ_cool t}

All Cooling Methods at a Glance

Method	Stage	T_min	Limit	Typical atoms	Notes
Zeeman Slower	Pre-cooling	~10 mK	Doppler	Most alkalis, Sr, Yb	Slows hot atomic beam before MOT
3D MOT	Stage 1	~100–200 μK	Doppler	All	Combines cooling + 3D spatial trapping
Compressed MOT (cMOT)	Stage 1b	~50 μK	Doppler / PGC	Cs, Rb, Li	Ramp B-field + detuning before sub-Doppler
Polarisation-Gradient (PGC)	Stage 2	2–10 μK	Photon recoil	Rb, Cs, K, Na	lin⊥lin or σ+σ− standing waves, no B-field
D1 Gray Molasses (GM)	Stage 2	~40–60 μK	Photon recoil	⁶Li, K (unresolved HF)	Λ-system dark state, works on D1 line
Narrow-line MOT (red)	Stage 2	~1 μK	Photon recoil	Sr (689 nm), Yb (556 nm)	Single-photon recoil kicks visible; direct sub-μK
Resolved Sideband (RSB)	Stage 3	⟨n⟩ < 0.1	Lamb-Dicke, η	Cs, Rb, Ca⁺, Mg⁺ in tweezers/lattices	Requires resolved ω_trap > Γ; closes on RSB
EIT Cooling	Stage 3	⟨n⟩ < 0.1	Dark-state linewidth	Ca⁺, Mg⁺, Sr, neutral atoms	Dark resonance engineered at RSB; faster rate

Temperature Scale Comparison (log₁₀ T / K)

Doppler Limit

T_D = ℏΓ/(2k_B). Set by the balance between laser cooling force and random recoil kicks from spontaneous emission. Scales with linewidth.

Recoil Limit

Single recoil: E_r/k_B = ℏ²k²/(2mk_B). One absorption plus spontaneous emission gives a two-recoil heating scale ≈2E_r/k_B. For Cs D2: E_r/k_B ≈ 99 nK and 2E_r/k_B ≈ 198 nK.

Lamb-Dicke Limit

In a trap: ⟨n⟩_ss → η²Γ²/(4ω²). Deep into Lamb-Dicke regime (η → 0) and resolved sidebands (ω ≫ Γ), ⟨n⟩ → 0.

pylcp, Laser Cooling Physics

pylcp (Python Laser Cooling Physics) is an open-source package for simulating laser-atom interactions. It supports rate-equation, optical Bloch equation, and Hund's case (a) Hamiltonian approaches.

import pylcp
import numpy as np

# Define laser beams (3D molasses)
laser_beams = pylcp.laserBeams([
  {'kvec': np.array([1,0,0]), 's': 0.5, 'delta': -0.5},
  {'kvec': np.array([-1,0,0]), 's': 0.5, 'delta': -0.5},
  # ... ±y, ±z beams
], beam_type=pylcp.infinitePlaneWaveBeam)

# Define atom (e.g., Rb F=2 → F'=3)
atom = pylcp.hamiltonians.singleF(
    F=2, Fp=3, gamma=1, dipole_matrix_element=1
)

# Integrate force equation
eqn = pylcp.rateeq(laser_beams, magField, atom)
eqn.set_initial_pop(np.array([1/5]*5))
sol = eqn.equilibrium_populations(r=np.zeros(3), v=np.array([0.1,0,0]))

arc, Alkali Rydberg Calculator

arc provides atomic structure data, dipole matrix elements, and Rydberg state properties for alkali atoms. Useful for computing polarizabilities, C₆ coefficients, and Rydberg blockade radii.

from arc import Caesium

atom = Caesium()

# Get natural linewidth of D2 line
gamma = atom.getTransitionRate(6,0,0.5, 6,1,1.5)
print(f"Γ/2π = {gamma/2/np.pi/1e6:.3f} MHz")

# Rydberg C6 coefficient for 70S1/2
C6 = atom.getC6term(70,0,0.5, 70,0,0.5)
R_b = (abs(C6) / (hbar * 2*np.pi*1e6)) ** (1/6)
print(f"Blockade radius = {R_b*1e6:.1f} μm")

QuTiP, Quantum Toolbox

For master equation simulation of the density matrix ρ under sideband cooling (Lindblad form):

$$\dot{\rho} = -i[H_{\rm eff},\rho] + \sum_k\!\left(L_k\rho L_k^\dagger - \tfrac{1}{2}\{L_k^\dagger L_k,\rho\}\right)$$

import qutip as qt
N = 30  # Fock space truncation
a = qt.destroy(N)
H = eta * Omega * (a + a.dag())  # RSB drive
c_ops = [np.sqrt(gamma) * a]     # decay
result = qt.mesolve(H, rho0, tlist, c_ops,
                    e_ops=[a.dag()*a])

Additional Libraries

AtomicUnits.jl (Julia), unit conversions for atomic physics. QuantumOptics.jl, fast master-equation solvers in Julia. MOLSCAT, molecular scattering for Feshbach resonances. COMSOL, FEM for magnetic trap geometry and electrode design.

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