Qubit Coherence
& 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):

For Lorentzian broadening (Markovian / exponential noise):

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).

The Ramsey Sequence

The sequence is: π/2 pulse → free evolution T → π/2 pulse → measure.

1. First π/2 pulse: rotates qubit from |0⟩ to the equator of the Bloch sphere, creating superposition |+⟩ = (|0⟩ + |1⟩)/√2.
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 $\varphi = \int_0^T \delta\omega\, dt$.
3. Second π/2 pulse: maps the equatorial phase back to a population difference, the phase is now readable.
4. 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*.

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:

Reading the Ramsey Signal

• 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).

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 $\delta\omega = \text{const}$:

The static contribution cancels exactly. Only noise that changes between the two halves (i.e., noise at $\omega \sim 1/T$) survives.

Filter Function

The Hahn echo filter function:

Key properties:

• Zero at $\omega = 0$, static fields refocused
• Peak near $\omega \approx 2\pi/T$, sensitive to noise at the echo timescale
• High-pass character: suppresses $\omega \ll 1/T$ noise

Pulse Timing

N π-pulses at positions:

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 $+1$.

Filter Function

Peak location: $\omega_c = \pi/\tau = \pi N / T$. More pulses → peak shifts to higher frequency.

For $1/f$ noise ($S(\omega) \propto 1/\omega$): higher $\omega_c$ → less noise power → better coherence:

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 Cs-133 array.

Protocol

1. Fix total time T; sweep N from 1 to ~1000.
2. For each N, measure the coherence W(T,N) and extract χ(T,N) = −ln W.
3. The filter function for CPMG-N has a sharp peak at $\omega_c = \pi N/T$.
4. The noise PSD at $\omega_c$ is proportional to χ at that peak: $$S(\omega_c) \approx \frac{\chi(T,N)}{|\tilde{Y}(\omega_c,T)|^2 \Delta\omega_{\rm peak}}$$
5. By sweeping T as well, reconstruct $S(\omega)$ over many decades of frequency.

Manetsch 2025 Application

Used XY-16-based sequences on a 6100-atom Cs-133 array to characterize coherence and pulse-error limitations. Identified dominant sources:

• DC to ~1 Hz: magnetic field drift (1/f character)
• ~10 Hz–1 kHz: B-field gradient fluctuations (main T₂* limitation)
• >1 kHz: technical laser/RF noise floor

This spectroscopy directly informs which DD sequence to use and how many pulses N are needed to optimally suppress the noise.

Rb87 vs Yb171 Qubits

Rb87 (Lukin, Greiner, Kaufman groups): Uses $|F=1\rangle\leftrightarrow|F=2\rangle$ hyperfine qubit. First-order Zeeman sensitivity: $\sim 0.7\,\text{MHz/G}$ differential shift → magnetic noise couples directly to qubit → T₂* limited by B-field gradients → DD essential.

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.