Three fiber types appear in a typical lab:
- Single-mode (SM) — supports only the fundamental HE₁₁ mode; any higher-order input content is rejected. Essential wherever spatial coherence matters.
- Polarisation-maintaining (PM) — SM fiber with stress rods that introduce birefringence; preserves the input polarisation when aligned to the principal axis. Every beam requiring a defined polarisation at the atom must travel on PM fiber.
- Multimode (MM) — large core, easy to couple; used for wavemeter pick-offs, diagnostics, and detection paths where polarisation is unimportant.
A lens of focal length f converts an input Gaussian beam of 1/e² radius w_in into a focused waist:
The coupling condition is w₀ = w_f, where w_f = MFD/2 is the fiber mode radius. Solving for the required focal length:
Trade-off: shorter f → tighter focus, higher peak efficiency, stricter alignment tolerances. Longer f → more robust over temperature drifts.
🔧 Mode-matching calculator
Formula: f = π w_in w_f / λ · Verify with VNA or throughput measurement
- Use a PBS to prepare clean linear polarisation upstream.
- Rotate a HWP after the PBS to align the polarisation to the fiber's keyed (slow/fast) axis.
- Optimise coupling for maximum throughput.
- Measure extinction ratio (ER) at the output through an analyser PBS.
- Iterate HWP angle to maximise ER.
The solution is double-passing: retroreflect the first-order beam back through the same crystal. The frequency shift doubles to 2 f_RF and the angular deviations cancel exactly — the output beam direction is independent of f_RF. Every tunable beam in the experiment uses a double-pass AOM.
The most compact double-pass geometry:
- AOM crystal → first-order beam deflected by Bragg angle
- Lens (f = one focal length from crystal) collimates and focuses
- QWP + mirror at one focal length from lens — retroreflects beam
- QWP traversed twice → polarisation rotated 90°
- Second pass through AOM → frequency shifted by another f_RF
- PBS transmits the 2f_RF output (orthogonal polarisation) and rejects zero-order
The cat's-eye geometry makes retroreflection insensitive to mirror tilt, which is why it is preferred over simpler flat-mirror configurations.
| Property | Longitudinal | Shear-wave |
|---|---|---|
| Acoustic mode | Atoms ∥ propagation | Atoms ⊥ propagation |
| Sound velocity | ~4–6 km/s | ~1–2 km/s |
| Deflection angle | Larger (AOD use) | Smaller |
| RF bandwidth | Broader | Narrower |
| Diffraction eff. | Lower at peak | Higher at peak |
🔧 AOM switching-time calculator
Formula: t_rise = d_beam / v_s · Focus tighter to get faster switching
Stokes parameters
The complete polarisation state is described by four intensity measurements:
| Parameter | Meaning |
|---|---|
| S₀ = I_total | Total power |
| S₁ = I_H − I_V | Horizontal vs. vertical linear |
| S₂ = I₊₄₅ − I₋₄₅ | Diagonal linear |
| S₃ = I_RCP − I_LCP | Right vs. left circular |
P = degree of polarisation (1 = fully polarised)
Rotating-QWP measurement method
Place a QWP on a rotation stage before a PBS analyser. As QWP angle θ is swept, transmitted intensity follows:
S₀ = A − C | S₁ = 2C | S₂ = 2D | S₃ = B
Use 50–100 evenly spaced angles for reliable results.
QWP retardance calibration
Measure between two PBS ports with linear input. Any deviation from 90° is corrected in the fitting procedure. Calibrate every QWP in optical-pumping and imaging paths.
Saturated-Absorption Spectroscopy (SAS) Locking
Two counter-propagating beams through a heated vapour cell create a Doppler-free Lamb dip: only atoms with near-zero velocity see both beams resonantly, so a narrow (~linewidth) feature appears on the Doppler-broadened background.
Error signal generation: FM dithering (current modulation or EOM sideband) + phase-sensitive detection (lock-in) converts the Lamb dip into an error signal that crosses zero at line centre.
Not suitable for: Cs 685 nm quadrupole line (I_sat ≈ 2.3 W/cm², impractical for SAS) → use PDH cavity lock instead.
Beat-Note (Offset) Locking
When two lasers need a fixed frequency difference, overlap them on a fast photodiode and compare the heterodyne beat-note to an RF reference.
Example: Li D2 laser offset phase-locked to Li D1 laser (Vescent electronics), inheriting the D1 absolute reference while maintaining ~10–100 Hz relative linewidth for coherent Raman processes.
- Tunable offset (change RF reference frequency)
- Fast relative linewidth (~Hz level)
- No vacuum infrastructure required
- One well-stabilised master laser as absolute reference
Cs D2 (SAS locked, absolute) ──── beat-lock ──→ Cs repumper
Vescent D2-135 offset PLL · Analog Devices HMC984 PFD · Mini-Circuits RF mixers
Pound-Drever-Hall (PDH) Cavity Locking
For wavelengths without accessible atomic references, or when sub-kHz linewidth is required, lock to a high-finesse Fabry-Pérot cavity.
Principle: Phase-modulate the input beam (EOM at Ω) → detect reflected light on fast photodiode → demodulate at Ω → antisymmetric error signal that crosses zero exactly at cavity resonance.
→ mode-matching telescope → ULE cavity (vacuum, mK temp. stabilisation)
Two-loop feedback: Fast path to laser current (MHz BW) + slow path to PZT (kHz BW). Cavity drift ≈ 2.5 kHz per 0.01°C temperature excursion.
Black (2001) — PDH tutorial (Am. J. Phys.) · Ludlow et al. (2015) — Optical atomic clocks
Lock hierarchy — how all lasers are referenced
The three methods form a complementary hierarchy:
| Method | Absolute ref? | Linewidth | Infrastructure |
|---|---|---|---|
| SAS | ✅ Yes (atomic line) | ~MHz | Vapour cell + lock-in |
| Beat-note | Via master laser | ~Hz (relative) | Fast PD + RF electronics |
| PDH cavity | Via cavity (ULE) | ~kHz | Vacuum + thermal control |
└── Li D2 (671 nm) ─────── beat lock (Vescent D2-135)
Cs D2 (852 nm) ─── SAS locked ──────────── absolute reference
└── Cs repumper ──────── beat lock
Cs 685 nm ──────── PDH (ULE cavity) ──── <1 kHz linewidth
1064 nm tweezer ─── free-running (stable Nd:YAG/fiber laser)
- State preparation — put the atom in a single, well-defined |F, m_F⟩ state before any coherent manipulation.
- Detection preparation — define the initial condition for state-selective fluorescence imaging.
Imperfect optical pumping is a direct source of systematic error in lifetime measurements, qubit state detection, and gate fidelity.
Cesium: pumping to |F=4, m_F=+4⟩
Scheme: σ⁺-polarised light driving F=4 → F′=4 has no allowed absorption for an atom already in m_F=+4 (which would require Δm_F=+1, but no m′_F=+5 exists in the excited state). That state is dark; all other m_F sublevels are continuously depopulated until the atom accumulates in m_F=+4.
Repumper: A beam resonant with F=3→F′=4 returns any population that decayed to F=3.
Why the magnetic-field direction matters: The quantisation axis is defined by the local B-field, not the laser k-vector. If the bias field is misaligned with the beam, the σ⁺ in the lab frame decomposes into σ⁺, π, and σ⁻ in the atom's frame. Apply a bias field of ~6 G along the optical axis of the pumping beam.
Diagnostic: the depumping-ratio test
- Drive F=4→F′=4 without the repumper — atoms in F=4 scatter photons and heat out of the trap.
- Under aligned B-field: atoms in the dark state |4,+4⟩ survive for ~1 ms (off-resonant scattering only).
- Under deliberately misaligned B-field (~45°): σ⁺ acquires σ⁻ component → atoms depumped in ~10 μs.
- Target depumping ratio > 100 (aligned/misaligned survival times).
Adjustment knobs: laser frequency (exact line centre), QWP orientation (polarisation purity), bias field direction. Together these achieve >99% pumping fidelity.
✅ ER of pumping beam fiber > 20 dB
✅ Bias field coil along pump beam axis
✅ Pump laser on F=4→F′=4 (not F=3→F′=4)
✅ Repumper on F=3→F′=4
✅ Depumping ratio > 100
✅ Pump pulse duration > 5 × (1/Γ_scatter)
Lithium: D2-line optical pumping
Procedure
- Coarse beam alignment to MOT on the diagonal camera.
- Fine alignment on a single trapped atom.
- Verify resonance by scanning laser frequency over the atom-loss signal.
- Characterise fidelity with the depumping-ratio test (same as Cs).
Loop antenna for Li hyperfine transitions
A single-turn loop radiates primarily through its magnetic dipole field. The radiation resistance of a small loop of area A at frequency f is:
For any loop that fits near a vacuum cell, R_rad ≪ 50 Ω. Efficient power delivery from a 50 Ω source therefore requires an impedance-matching network.
Three approaches tested (with VNA)
- Capacitive loading — series or parallel capacitor shifts resonance. Result: parallel ~47 pF proved most effective for 76 MHz, compact geometry.
- Transmission-line stub matching — moves impedance on Smith chart. Works in principle, but spurious resonances can complicate things.
- Discrete LC networks — more design freedom but more components.
Achieved: reflection minimum ~10 dB, sufficient B-field at the atom with ~100 W amplifier.
Lessons learned
- Lead length (connector to loop) contributes parasitic inductance at 100 MHz — non-negligible.
- Simulate with SimSmith before building every iteration.
- The resonant frequency scales inversely with circumference at fixed inductance.
🔧 Loop antenna quick estimate
Approximate formulas — always verify with VNA measurement.
Siglent SVA1015X VNA · SimSmith (free Smith chart simulator) · Mini-Circuits RF amplifiers
Feshbach coil safety interlocks
Feshbach coils produce fields of order 1000 G by carrying large DC currents. Sudden current interruption generates inductive voltage spikes that can damage power supplies and coils.
Hardware interlock logic (essential rules)
- Monitor: coil temperature, current level, supply voltage
- On threshold exceeded: ramp down smoothly, do not switch off abruptly
- All interlock logic implemented in relay hardware, independent of computer control
- Interlock circuit must be untriggerable by software bugs
Kepco BOP bipolar · iSeg precision current sources · AMETEK Programmable Power
How it works
Gaussian-Process Regression (GPR): a non-parametric Bayesian model that maintains a probabilistic map of the response surface (e.g. atom survival vs. laser detuning + power). At each step it returns a predicted value and an uncertainty — unexplored regions have high uncertainty.
Acquisition function: selects the next measurement point by trading off exploitation (sample near current optimum) and exploration (reduce uncertainty in poorly sampled regions).
| Acquisition function | When to use |
|---|---|
| Expected Improvement (EI) | Default; works well near optimum |
| Upper Confidence Bound (UCB) | When signal is absent, need broad search |
| Probability of Improvement (PI) | Conservative; avoids risk |
Workflow
- Collect 5–20 initial points from Latin hypercube or random design
- Train GPR model; inspect posterior mean + uncertainty
- Select next point by maximising EI/UCB
- Run experiment, append data, retrain, repeat
scikit-learn GPR · BayesianOptimization (Python) · Meta Ax platform · Frazier (2018) — BO tutorial
Grey dotted = true function · Purple band = GP ±2σ · Yellow = observations · Green dashed = next query point
Physical model
- H_internal: all hyperfine and Zeeman sublevels (typically 12–24 states for D1 of an alkali)
- H_HO: harmonic oscillator, truncated at N_HO = 10–20 Fock states
- Total dimension: d_int × N_HO (e.g. 12 × 12 = 144 for Li D1)
Key feature: the full matrix exponential is used for the recoil operator R̂ = e^(iη(â+â†)) (not the Lamb-Dicke expansion), so the code is valid beyond the strict Lamb-Dicke regime.
3 validation cross-checks
- Fock distribution P(n) — fit to Boltzmann to extract T_eff
- Excited-state fraction p_e — compare to measured photon rate
- Temperature minimum — verify location in 2D parameter scans
Code structure
# Entry points:
QAtomTweezer.py
QAtomTweezer_SingleLevel.py
# Main callable:
SteadyStateTweezer(
x, # [δ1, δ2, Ω1, Ω2, φ1, φ2]
wh, # trap freq in units of Γ
Nh, # HO truncation
atom, # AtomSettings object
eta, # Lamb-Dicke parameter
pol, # polarisation config
)
# Returns: ⟨n⟩, P(n), p_e
Performance
QuTiP documentation · joblib (parallelism) · Johansson et al. (2013) — QuTiP 2 paper
A bare semiconductor diode lases on multiple longitudinal modes. Two tuning mechanisms in a bare diode:
- Injection current: ∂ν/∂I ~ 1–3 GHz/A (fast but noisy)
- Temperature: ∂ν/∂T ~ −20 to −40 GHz/K (slow, hysteretic)
Neither alone provides the narrow linewidth (<100 kHz) or mode-hop-free tuning range (>1 GHz) required.
The Littrow ECDL solution
- Holographic grating at Littrow angle feeds first-order diffraction back into the diode
- Selects one longitudinal mode of the extended cavity
- Grating angle (PZT-tuned) + injection current → mode-hop-free tuning over 1–2 GHz
- AR coating on front facet suppresses internal Fabry-Pérot resonances
Common failure modes
- Mode hops: usually from temperature drift; cure with better temperature control
- Reduced output: check AR coating; diodes degrade with age and excessive current
- Multiple modes: grating feedback misaligned; realign while monitoring on wavemeter
- Linewidth broadening: current noise from noisy driver; use low-noise current source
Alignment tips
- Set temperature for approximate target wavelength
- Coarsely align grating to first-order feedback with IR card
- Monitor wavelength on wavemeter; find single-mode region
- Maximise mode-hop-free range by co-scanning PZT + current (feed-forward)
D1 (670.992 nm) and D2 (670.977 nm)
MOT (D2), Zeeman slower (D2), Λ-GM cooling and tweezer imaging (D1)
Γ/2π = 5.87 MHz (D1 and D2 nearly identical)
D1 and D2 lines separated by only ~10 GHz — both needed simultaneously. D2 laser offset-locked to D1. Lightest alkali: recoil temperature 3.54 μK, largest recoil in the alkalis.
D1: SAS locked to vapour cell. D2: beat-note locked to D1 (Vescent D2-135).
Toptica or home-built ECDL at 671 nm; TA amplifier often needed for MOT power
D2 (852.347 nm); D1 at 894.6 nm also used in some labs
MOT, fluorescence imaging, repumper
Γ/2π = 5.23 MHz
Large hyperfine splitting (9.193 GHz) means cooler and repumper must be offset by ~9.2 GHz — use AOM chain or separate laser with beat lock.
SAS locked directly to Cs D2 line in vapour cell.
6S₁/₂ → 5D₅/₂ (electric quadrupole, Γ/2π ≈ 117.6 kHz = 3.5 Hz natural linewidth)
Narrow-line sideband cooling of Cs in tweezer; excited-state lifetime measurement
Γ/2π = 117.6 kHz (3.5 Hz natural; resolved sidebands at typical trap frequency)
Forbidden E2 transition → I_sat ≈ 2.3 W/cm² (much higher than D-lines). No SAS possible. Requires PDH lock to ULE cavity for <1 kHz linewidth. Astigmatism correction needed (prism pair before cavity).
PDH locked to ULE cavity (L=77.5 mm, F≈15 000, linewidth ~100 kHz). Laser linewidth ~1 kHz.
Thorlabs L685P010 (AR-coated front facet strongly preferred for stable ECDL operation)
Far-detuned (no resonant absorption); acts as conservative dipole trap potential
Creates the optical tweezer potential; all atoms trapped in the 1064 nm focus
Intensity noise, not frequency noise. Intensity noise at trap frequencies (kHz) causes parametric heating.
< −130 dBc/Hz at trap sidebands. Needs intensity stabilisation (AOM servo on pick-off PD).
Free-running (stable Nd:YAG or fiber laser); intensity servo via AOM feedback.
Coherent Mephisto · NKT Photonics Koheras · Azurlight fiber amplifier · Thorlabs Nd:YAG
📚 References & Further Reading
Phatak (2025) — PhD Thesis, Purdue (Chapter 6)
"Cooling Lithium and Cesium Single Atoms in Optical Tweezers"
Metcalf & van der Straten — Laser Cooling and Trapping (1999)
Foot — Atomic Physics (OUP, 2005)
Saleh & Teich — Fundamentals of Photonics (Wiley)
Pozar — Microwave Engineering (Wiley, 4th ed.)