How π-pulse sequences extend qubit coherence by engineering the spectral filter function —
from Hahn echo to XY-16, with live filter function and T₂ enhancement calculators.
Physics context: Rb87 neutral atom arrays (Manetsch 2025, T₂ = 12.6 s).
01 / The Core Problem
T₂* vs T₂: Why Coherence Decays
Three timescales govern qubit memory: energy relaxation T₁, true dephasing T₂, and
the free-induction-decay time T₂*. Understanding how they differ is the starting point
for dynamical decoupling.
Timescale Hierarchy
T₁ (energy relaxation): spontaneous emission, photon scattering, sets a hard ceiling. Always T₂ ≤ 2T₁.
T₂ ("true" T₂): dephasing from dynamic noise that changes during the experiment.
A spin-echo can refocus the static part; only the time-varying part survives.
T₂* (FID time): the apparent dephasing seen in free induction decay —
includes both static inhomogeneity across the ensemble AND dynamic noise.
Almost always T₂* ≪ T₂ ≤ 2T₁.
Static inhomogeneity: B-field gradient, differential light shifts, laser frequency drift
Dynamic noise: magnetic field fluctuations, motional dephasing, laser phase noise
In neutral-atom arrays: T₂* limited by inter-atom B-field gradient, T₁ can be ≫ T₂
Free Induction Decay Models
For Gaussian inhomogeneous broadening (dominant in most neutral atom experiments):
$$W_{\rm FID}(T) = e^{-(T/T_2^*)^2}$$
For Lorentzian broadening (Markovian / exponential noise):
$$W_{\rm FID}(T) = e^{-T/T_2^*}$$
Both collapse to a simple envelope, but the physical origin differs.
The key: T₂* is a measured number. T₂ requires refocusing (spin echo or DD).
Rb87 T₂*
~22 s
Manetsch 2025 (6100-atom array)
T₂ with XY-16
12.6 s
DD-measured T₂ (pulse error limited)
T₁ (Rb87)
≫ T₂
Spontaneous emission negligible
Fundamental limit
T₂ ≤ 2T₁
Irreducible energy relaxation
Note on Manetsch 2025: In this experiment T₂(XY-16) = 12.6 s is actually
shorter than T₂* ≈ 22 s. This is because at these timescales, coherent pulse-error
accumulation in the DD sequence dominates over the dephasing being suppressed.
For most AMO systems, T₂* ≪ T₂(DD) and DD gives a large improvement.
The General Framework: Phase Accumulation
A qubit in a fluctuating field $\delta\omega(t)$ accumulates random phase:
$$\varphi = \int_0^T y(t)\,\delta\omega(t)\,dt$$
where $y(t) = \pm 1$ is the switching function determined by the pulse sequence.
Coherence in the Gaussian noise approximation:
Here $S(\omega)$ is the one-sided noise power spectral density [rad²/s] and
$\tilde{Y}(\omega,T) = \int_0^T y(t)\,e^{i\omega t}\,dt$ is the
filter function. T₂ is defined by $\chi(T_2) = 1$,
i.e., $W(T_2) = 1/e$.
02 / Measuring T₂*
Ramsey Spectroscopy: The Free-Precession Experiment
Ramsey spectroscopy is the fundamental technique for measuring qubit coherence.
Before you can extend T₂, you need to measure T₂*, and Ramsey is how you do it.
It also reveals what is dephasing your qubit, pointing toward which DD sequence to use.
The Ramsey Sequence
The sequence is: π/2 pulse → free evolution T → π/2 pulse → measure.
First π/2 pulse: rotates qubit from |0⟩ to the equator of the Bloch sphere, creating superposition |+⟩ = (|0⟩ + |1⟩)/√2.
Free evolution T: the qubit precesses at its Larmor frequency. Any detuning δ from the drive frequency causes it to rotate in the equatorial plane. Noise $\delta\omega(t)$ causes random phase $\varphi = \int_0^T \delta\omega\, dt$.
Second π/2 pulse: maps the equatorial phase back to a population difference, the phase is now readable.
Measure: P(|0⟩) = (1 + cos(δT + φ))/2. For a single detuning δ, this oscillates as T increases ("Ramsey fringes"). For an ensemble with many δ values, the fringes wash out at rate 1/T₂*.
Ramsey sequence. The switching function y(t) = +1 throughout, phase from all fields (static AND dynamic) accumulates. The second π/2 converts the accumulated phase into a measurable population difference.
What You Measure: Ramsey Fringes & T₂*
If you apply a drive at detuning $\delta = \omega_{\rm drive} - \omega_{\rm qubit}$, the
qubit precesses at $\delta$ during $T$. Sweeping $T$ gives oscillating fringes:
Fringe frequency → detuning δ (how far off-resonance you are). Use this to lock to qubit frequency.
Fringe contrast decay → T₂* (dephasing time). The contrast |C(T)| = e^{-(T/T₂*)ⁿ} gives you T₂* directly from the decay envelope.
Fringe shape (Gaussian vs exponential): Gaussian decay → static inhomogeneity dominates (e.g., B-field gradient across array). Exponential decay → dynamic noise dominates (e.g., laser phase noise). This tells you whether DD will help dramatically (Gaussian case, where static gradient is the enemy) or only modestly (exponential case).
Multiple revivals: If fringe contrast dips then revives, there is a discrete noise source (e.g., a 50 Hz power line pickup).
In neutral atom arrays: Run a Ramsey experiment on all ~100–6100 atoms simultaneously.
Each atom is an independent clock. The ensemble-averaged signal washes out at T₂*.
But even single-atom Ramsey (no ensemble average) shows coherence decay, this is the true single-qubit T₂*.
Manetsch 2025 measured single-atom T₂* ≈ 22 s in their 6100-qubit Rb87 array using
Ramsey with XY-16 DD (i.e., they used DD to first measure the long-time coherence, then applied it operationally).
Ramsey vs Spin Echo: What Each Measures
Ramsey (FID): $\varphi = \int_0^T \delta\omega(t)\,dt$, integrates ALL noise (static + dynamic) with $y(t) = +1$.
Measures $T_2^*$.
Spin echo: $\varphi = \int_0^{T/2}\delta\omega\,dt - \int_{T/2}^T\delta\omega\,dt$, subtracts the two halves.
Static fields cancel; only noise that changes between the two halves survives. Measures a longer $T_2^{\rm echo} \geq T_2^*$.
The ratio $T_2^{\rm echo}/T_2^*$ tells you how much of the dephasing is static (large ratio → mostly static → gradient problem → DD is very effective) vs dynamic (ratio ≈ 1 → dynamic noise dominates → DD helps less).
In Rb87 arrays: ratio can be 10–100× showing static B-field gradient as dominant problem.
FID Dephasing vs CPMG Refocusing — side by side
Free Induction Decay (no DD)
CPMG-4 (4 π-pulses)
FID: spins fan out and never refocus. CPMG: π-pulses flip the fan, reversing dephasing.
03 / The Simplest Refocusing
Hahn Echo: One π-Pulse
The spin echo (Hahn 1950) uses a single π-pulse at t = T/2 to refocus all static
field inhomogeneity. It is the prototype for all dynamical decoupling.
How It Works
The π-pulse at t = T/2 flips the qubit, making $y(t) = +1$ for $t < T/2$ and
$y(t) = -1$ for $t > T/2$.
For a constant field $\delta\omega = \text{const}$:
Refocusing intuition: Imagine all atoms in an ensemble processing at slightly different
rates due to B-field gradients. At T/2, the π-pulse reverses their relative phases. At T,
all atoms realign, the "echo", regardless of their individual offset frequencies.
This works perfectly as long as the offsets don't change during [0,T].
03 / N-Pulse Sequences
CPMG: N Equally-Spaced π-Pulses
The Carr-Purcell-Meiboom-Gill (CPMG) sequence is the standard multi-pulse dynamical
decoupling protocol. N π-pulses, equally spaced, push the filter function peak to
higher and higher frequencies.
Pulse Timing
N π-pulses at positions:
$$t_k^{\rm CPMG} = \frac{(2k-1)\,T}{2N}, \quad k = 1,\ldots,N$$
The inter-pulse spacing is $\tau = T/N$. The first pulse is at $\tau/2$, then every $\tau$.
The switching function $y(t)$ alternates between $+1$ and $-1$ at each pulse,
starting at $+1$.
CPMG caveat: All pulses rotate around the same axis (x). If the π rotation angle
is slightly off by $\epsilon$, errors accumulate coherently: after N pulses the error
is $\sim N\epsilon$ (not $\sqrt{N}\epsilon$). This is fine for single-qubit experiments but
catastrophic for large arrays where different atoms see different pulse areas.
04 / Robust Sequences
XY-4, XY-8, XY-16: Cancelling Pulse Errors
XY sequences use the same timing as CPMG but alternate the rotation axis between x and y.
This causes pulse errors to cancel to progressively higher order.
XY-4
Four pulses in a cycle: $\pi_x\,\pi_y\,\pi_x\,\pi_y$.
Same spacing as CPMG-4, the filter function is identical for pure z-dephasing.
But: a small over-rotation error on $\pi_x$ is partially cancelled by the subsequent
$\pi_y$. First-order pulse errors cancel within one cycle.
XY-8 and XY-16
XY-8: $X\,Y\,X\,Y\,\bar{Y}\,\bar{X}\,\bar{Y}\,\bar{X}$
(where $\bar{X} = -\pi_x$, i.e., $\pi$ rotation about $-x$). Cancels both
first- and second-order errors (rotation angle AND axis tilt).
XY-16: Two concatenated XY-8 cycles. Third-order error suppression.
This is the sequence used in Manetsch 2025 for T₂ = 12.6 s in a 6100-atom Rb87 array.
Key distinction: For perfect pulses, CPMG-N = XY-4 = XY-8 = XY-16 give
identical coherence decay, the filter function is the same (it only depends on timing).
For imperfect pulses (real life), XY-16 >> CPMG because pulse errors cancel to higher order.
Sequence Comparison
Sequence
Pulses / cycle
Orders corrected
Filter function
Notes
CPMG-N
N
None (coherent accumulation)
Same as XY for z-noise
Simple; axis-error builds up linearly
XY-4
4
1st order
Identical to CPMG-4
Standard; good for 1D noise axis
XY-8
8
2nd order
Identical to CPMG-8
Compensates axis tilt + angle error
XY-16
16
3rd order
Identical to CPMG-16
Used in Manetsch 2025 (T₂ = 12.6 s)
KDD
16
Higher (multi-axis)
Similar shape
Knill DD; best for >1 noise axis
05 / The Key Insight
Filter Function Formalism
Every DD sequence acts as a bandpass filter on the noise PSD. Choose the sequence
whose filter has minimal overlap with where the noise actually lives.
$T_2$ is defined by $\chi(T_2) = 1$. To maximize $T_2$: engineer
$|\tilde{Y}(\omega,T)|^2$ so it has minimum overlap with $S(\omega)$.
1/f Noise ($S(\omega) \propto 1/\omega$)
Most AMO systems: B-field fluctuations, laser frequency noise.
Noise concentrated at low frequency.
Strategy: push filter peak to higher $\omega_c$ by adding more pulses.
Each doubling of N pushes $\omega_c$ up by 2×, cutting the overlap with 1/f noise.
Result: $T_2 \propto N^{0.5}$ (for $\alpha=1$)
White Noise ($S(\omega) = \text{const}$)
Markovian / shot-noise limited. Noise power flat across all frequencies.
DD does nothing for white noise, the integral $\int S\,|\tilde{Y}|^2\,d\omega$
is fixed by the total filter weight, not its peak location.
In this regime T₂ is pulse-error limited. Rabi driving better than DD.
Mechanical vibrations, AC power line (60 Hz), laser relaxation oscillation.
Strategy: tune $\tau = \pi/\omega_0$ to place a filter null
exactly at $\omega_0$. This is "resonance avoidance."
CPMG with specific N and T can null a specific noise frequency exactly.
$$\chi(T) = \frac{1}{2\pi}\int_0^\infty S(\omega)\,|\tilde{Y}(\omega,T)|^2\,d\omega, \qquad T_2 \text{ defined by } \chi(T_2) = 1$$
Physical picture: The switching function $y(t)$ is a square wave. Its Fourier transform
$\tilde{Y}(\omega,T)$ has power concentrated around $\omega_c = \pi N/T$. The dephasing $\chi(T)$
is literally a weighted inner product of the noise spectrum and the filter function —
minimizing their overlap is the design principle of every DD sequence.
06 / Interactive Calculator
Filter Function Viewer
Compare filter functions $|\tilde{Y}(\omega,T)|^2$ for FID, Hahn echo, CPMG-4, and CPMG-16
against the noise PSD. Adjust total time T and noise exponent α to see how overlap changes.
Filter Functions vs Noise PSD Log-Log
T = 0.10 s
α = 1.0 (1/f noise)
FID: χ = —
Hahn: χ = —
CPMG-4: χ = —
CPMG-16: χ = —
Solid lines: filter functions |Ỹ(ω,T)|². Dashed line: noise PSD S(ω) ∝ ω^−α (arb. units).
Chi values normalized so χ_FID(T) ≈ 1 at T = T₂*.
07 / T₂ Enhancement
T₂ vs N: Enhancement Factor Calculator
Compute T₂(N)/T₂* as a function of the number of CPMG pulses for different noise spectra.
Binary search finds T₂ for each N by requiring χ(T₂, N) = 1.
T₂ Enhancement Factor vs Number of Pulses Numerical
T₂* = 10.0 ms
α = 1.0
N=1: —
N=16: —
N=256: —
Scaling: T₂ ∝ N^?
Enhancement T₂(N)/T₂* vs N for CPMG sequences.
Expected scaling: α=0 → N^0 (no improvement), α=1 → N^0.5, α=2 → N^1.
Scaling law updates as you adjust the sliders.
08 / Noise Characterization
Noise Spectroscopy via DD
DD sequences don't just protect qubits, they can also measure the noise spectrum
that's causing decoherence (Biercuk et al., Nature 2011).
Protocol
Fix total time T; sweep N from 1 to ~1000.
For each N, measure the coherence W(T,N) and extract χ(T,N) = −ln W.
The filter function for CPMG-N has a sharp peak at $\omega_c = \pi N/T$.
The noise PSD at $\omega_c$ is proportional to χ at that peak:
$$S(\omega_c) \approx \frac{\chi(T,N)}{|\tilde{Y}(\omega_c,T)|^2 \Delta\omega_{\rm peak}}$$
By sweeping T as well, reconstruct $S(\omega)$ over many decades of frequency.
Manetsch 2025 Application
Used XY-16-based sequences on 6100-qubit Rb87 array to map out the noise spectrum.
Identified dominant sources:
This spectroscopy directly informs which DD sequence to use and how many pulses N
are needed to optimally suppress the noise.
Why this matters for large arrays
In a 6100-atom array, atoms at different positions see slightly different B-fields
(gradient). The noise spectroscopy reveals that the gradient fluctuates as a correlated
noise source, it's the same noise at every atom, just scaled by position.
This means all 6100 atoms can be simultaneously decoupled with the same pulse sequence
(XY-16) synchronized across the array.
If the noise were independent per atom (e.g., each atom's individual laser scatter),
DD would still work but would need to be applied on a per-atom basis.
09 / Experimental Context
Connection to Neutral Atom Experiments
How dynamical decoupling is used in current state-of-the-art neutral atom qubit experiments.
DD sequence: XY-16 chosen over CPMG for two reasons:
(1) atoms at different positions see different Rabi frequencies (Gaussian laser profile) →
spatially varying pulse errors → CPMG errors build up coherently;
(2) XY-16 cancels these to 3rd order.
Results: T₂* ≈ 22 s (free precession), T₂(XY-16) = 12.6(1) s.
The XY-16 T₂ is actually shorter than T₂* here, this is a regime where pulse error
accumulation from 12 seconds of continuous pulsing exceeds the dephasing being suppressed.
Still, XY-16 outperforms CPMG at all pulse numbers in this system.
Noise source: Primarily B-field gradient fluctuations
(dB/dx · x-position of atom). Secondary: differential AC Stark shift from
tweezers (light shift noise).
Yb171 (Ye, Thompson groups):
Uses $|I=1/2, m_I=\pm1/2\rangle$ nuclear spin qubit.
Magnetically insensitive to first order ($\sim 270\times$ less than Rb87).
T₂* already very long → less need for aggressive DD.
DD still used but marginal improvement.
General Principles for Large Arrays
As arrays scale to thousands of qubits, two effects make DD increasingly critical:
Spatial B-gradient noise: grows with array size →
T₂* shrinks as $\sim 1/\sqrt{N_{\rm atoms}}$ for random gradient.
Pulse inhomogeneity: atoms at edges of Gaussian beam
get systematically different pulse areas → XY-16 (not CPMG) becomes mandatory.
Noise spectroscopy: characterizing the noise spectrum
of the full array (not just single atoms) is needed to choose optimal DD parameters.
Array size
6,100
Rb87 atoms (Manetsch 2025)
DD sequence
XY-16
3rd-order pulse error cancellation
T₂* (free)
~22 s
FID coherence time
T₂ (XY-16)
12.6 s
Limited by pulse errors at this scale
Perspective: In most AMO systems (smaller arrays, less-perfect clocks states),
T₂* ≪ T₂ and DD gives 10–100× improvement. Manetsch 2025 is operating in an unusual regime
where T₂* is already extremely long (~22 s) due to very good B-field control, so the
remaining improvement from DD is marginal and actually competes with pulse-error accumulation.