Pulse Sequences

Pulse sequences are ordered lists of controlled rotations and free-evolution intervals. They are the working language of spin resonance, atomic clocks, trapped ions, superconducting qubits, color centers, quantum sensing, and many two-level calibration experiments.

This page is a reference guide to notation and standard sequence families. The physics of basic Rabi and Ramsey protocols is explained in Rabi and Ramsey Control. Average Hamiltonian Theory owns toggling-frame cycle engineering and commutator corrections; Dynamical Decoupling owns noise filters and protection protocols. The quantum-information deployment workflow owns choosing a licensed sequence for a declared control window, realizing it on the hardware timing grid, and validating task-facing benefit; this reference retains notation and sequence-family definitions.

Pulse Notation

In a rotating frame, a pulse with phase $\phi$ and area $\theta$ is commonly represented as

R_\phi(\theta) = \exp \left[ - \frac{i\theta}{2} \left( \cos\phi\,X+\sin\phi\,Y \right) \right].

The common shorthand is:

Symbol	Meaning
$X_\theta$	$R_0(\theta)$ , rotation about $+x$
$Y_\theta$	$R_{\pi/2}(\theta)$ , rotation about $+y$
$X_\pi$	nominal $\pi$ pulse about $x$
$X_{\pi/2}$	nominal $\pi/2$ pulse about $x$
$\tau$	free-evolution interval
$M_Z$	measurement in the $Z$ basis, when specified

Different communities put phases on the pulse, the state, or the readout reference. A pulse sequence is not fully specified unless it states the rotating frame, detuning convention, pulse phases, pulse areas, timing, and measurement axis.

Primitive Pulses

A $\pi$ pulse ideally flips the Bloch vector through angle $\pi$ about the chosen transverse axis. A $\pi/2$ pulse ideally moves a $Z$ eigenstate to the equator or maps equatorial phase into population.

For a rectangular resonant pulse with Rabi frequency $\Omega$ ,

\theta=\Omega t.

Thus

t_\pi=\frac{\pi}{\Omega}, \qquad t_{\pi/2}=\frac{\pi}{2\Omega}.

Finite pulse width matters. During a real pulse, the system may still experience detuning, relaxation, dephasing, leakage, and drive noise. Ideal instantaneous-pulse notation is a model, not a hardware description.

Standard Sequences

Sequence	Schematic form	Main use
Rabi pulse	$R_\phi(\theta)$	population transfer and pulse-area calibration
Ramsey	$X_{\pi/2}$ — $\tau$ — $X_{\pi/2}$ — $M_Z$	detuning and free-evolution coherence
Phase-scanned Ramsey	$X_{\pi/2}$ — $\tau$ — $R_\phi(\pi/2)$ — $M_Z$	fringe phase and contrast
Hahn echo	$X_{\pi/2}$ — $\tau$ — $X_\pi$ — $\tau$ — $X_{\pi/2}$ — $M_Z$	refocusing quasi-static dephasing
Carr–Purcell	$X_{\pi/2}$ — $(\tau$ — $X_\pi$ — $\tau)^N$ — readout	repeated refocusing
CPMG	phase-adjusted Carr–Purcell family	robust echo trains for selected spin component
XY family	alternating $X_\pi$ and $Y_\pi$ pulses	reducing sensitivity to some pulse errors
Composite pulse	several rotations replacing one intended pulse	robustness to amplitude or detuning errors

The table hides many convention choices. For example, some Hahn-echo sequences use a final $Y_{\pi/2}$ or absorb the final analysis pulse into the readout phase. Those choices change the displayed population fringe but not the underlying refocusing idea.

Ramsey Sequence

A Ramsey sequence uses two $\pi/2$ pulses separated by free evolution. In its simplest phase convention,

X_{\pi/2} \quad \tau \quad X_{\pi/2} \quad M_Z.

During the interval $\tau=T$ , detuning produces phase $\Delta T$ . The measured population oscillates as a fringe:

P_e(T) = \frac12 \left[ 1+C(T)\cos(\Delta T+\phi_0) \right],

where $C(T)$ is the contrast envelope. Scanning the second pulse phase $\phi$ is often cleaner than scanning the free-evolution time because it separates readout contrast from slow frequency drift.

Ramsey is highly sensitive to low-frequency detuning noise. That is why a Ramsey time is often written $T_2^*$ rather than $T_2$ .

Hahn Echo

The Hahn echo inserts a refocusing $\pi$ pulse halfway through the free-evolution interval:

X_{\pi/2} \quad \tau \quad X_\pi \quad \tau \quad X_{\pi/2} \quad M_Z.

For longitudinal detuning noise, the toggling function is

y(t) = \begin{cases} +1, & 0\lt t\lt T/2,\\ -1, & T/2\lt t\lt T. \end{cases}

If the detuning is constant during the sequence, the accumulated phase cancels:

\int_0^T dt\,y(t)\Delta = 0.

Echo therefore refocuses quasi-static dephasing, not arbitrary decoherence. It does not undo energy relaxation, spontaneous emission, measurement backaction, or noise that changes appreciably within the sequence.

Carr–Purcell and CPMG

Repeated echo pulses extend the refocusing idea. For $N$ equally spaced $\pi$ pulses over total time $T$ , one common placement is

t_j = \left(j-\frac12\right) \frac{T}{N}, \qquad j=1,\ldots,N.

The toggling function changes sign at each pulse. The resulting filter suppresses noise near zero frequency and samples noise near frequencies set by the pulse spacing. The scale is roughly

\omega \sim \frac{\pi N}{T}.

The Carr–Purcell–Meiboom–Gill refinement chooses pulse phases so that certain pulse errors do not accumulate for the targeted initial spin component. In practice, “CPMG” often names a family of robust echo trains rather than one universally fixed symbolic string.

XY Families

XY sequences alternate pulse axes, such as

X_\pi \quad Y_\pi \quad X_\pi \quad Y_\pi,

with specified delays between pulses. The purpose is to average some systematic pulse errors that a single-axis train would accumulate coherently. XY4, XY8, and related sequences are common in spin qubits and sensing because they combine decoupling with partial robustness to control imperfections.

The price is extra structure: phases, timing symmetry, finite-width corrections, and the initial state can all matter. A named XY sequence is not a substitute for specifying the actual pulse schedule.

Composite-Pulse Preview

A composite pulse replaces one target rotation with several rotations chosen so that leading calibration errors cancel. For example, an intended $X_\theta$ pulse might be replaced by

R_{\phi_3}(\theta_3) R_{\phi_2}(\theta_2) R_{\phi_1}(\theta_1),

where the phases and angles are chosen to reduce sensitivity to amplitude error, detuning error, or both. Composite pulses are not the same as dynamical decoupling. They make an intended control operation more robust, while decoupling reshapes the effect of environmental coupling during storage or sensing.

This page only gives the preview. A complete treatment requires specifying the error model and expanding the resulting unitary or channel in that error parameter.

Open-System Reading

Every pulse sequence defines more than an ideal unitary. It defines:

a time-dependent Hamiltonian during pulses;
a free-evolution Hamiltonian between pulses;
a toggling frame for system operators;
a filter for some noise components;
a measurement basis;
a sensitivity to pulse errors and finite bandwidth;
an exposure time to relaxation and dephasing.

For Gaussian dephasing noise, many sequences are summarized by

W(T)=e^{-\chi(T)},

with

\chi(T) = \frac12 \int_{-\infty}^{\infty} \frac{d\omega}{2\pi}\, S_{\xi\xi}(\omega) |Y(\omega,T)|^2,

under a particular two-sided spectral convention. Here $Y(\omega,T)$ is determined by the sequence’s modulation function. This formula is useful only when the noise model and convention match the experiment.

Common Mistakes

Writing “a $\pi$ pulse” without stating the axis or phase.
Treating an ideal sequence diagram as if pulses had zero duration and no errors.
Comparing Ramsey, echo, and CPMG decay times as if they measured the same quantity.
Calling a sequence “CPMG” without specifying pulse phases and timing.
Using a filter-function formula with a one-sided spectrum but a two-sided normalization.
Assuming composite pulses suppress environmental noise in the same way as dynamical decoupling.
Ignoring leakage when strong or short pulses have broad spectral width.
Forgetting that the final analysis pulse is part of the measurement convention.

Exercises

Phase of a transverse pulse

What rotation axis is generated by a pulse with phase $\phi=\pi/2$ ?

Solution

The axis is

\cos(\pi/2)X+\sin(\pi/2)Y = Y.

Thus $R_{\pi/2}(\theta)$ is a rotation about the $+y$ axis in this convention. This is why phase conventions must be stated: shifting the drive phase changes the rotation axis.

Echo cancellation of static detuning

For a Hahn echo, take $y(t)=+1$ for $0\lt t\lt T/2$ and $y(t)=-1$ for $T/2\lt t\lt T$ . Show that a constant detuning $\Delta$ produces zero net phase.

Solution

The accumulated phase is

\varphi = \Delta \int_0^T dt\,y(t).

The integral is

\int_0^{T/2}dt - \int_{T/2}^{T}dt = \frac{T}{2} - \frac{T}{2} = 0.

Therefore $\varphi=0$ . This cancellation is exact for static detuning in the ideal-pulse model; time-dependent noise and pulse errors require further analysis.

CPMG timing scale

An echo train uses $N=20$ equally spaced refocusing pulses over total time $T=200\,\mu\mathrm{s}$ . Estimate the angular frequency scale near which the filter has strong weight using $\omega\sim\pi N/T$ .

Solution

Use

\omega \sim \frac{\pi N}{T} = \frac{20\pi}{200\,\mu\mathrm{s}} = 0.1\pi\,\mu\mathrm{s}^{-1}.

Since $1\,\mu\mathrm{s}^{-1}=10^6\,\mathrm{s}^{-1}$ ,

\omega \sim 3.1\times10^5\,\mathrm{s}^{-1}.

As an ordinary frequency this is

f = \frac{\omega}{2\pi} \sim 50\,\mathrm{kHz}.

The estimate is only a scale; the exact filter shape depends on the full sequence and pulse model.

Cross-Links

References

E. L. Hahn, “Spin echoes,” Physical Review 80, 580–594 (1950).
H. Y. Carr and E. M. Purcell, “Effects of diffusion on free precession in nuclear magnetic resonance experiments,” Physical Review 94, 630–638 (1954).
S. Meiboom and D. Gill, “Modified spin-echo method for measuring nuclear relaxation times,” Review of Scientific Instruments 29, 688–691 (1958).
M. H. Levitt, “Composite pulses,” Progress in Nuclear Magnetic Resonance Spectroscopy 18, 61–122 (1986).
M. H. Levitt, Spin Dynamics: Basics of Nuclear Magnetic Resonance, 2nd ed., Wiley (2008).
L. Viola and S. Lloyd, “Dynamical suppression of decoherence in two-state quantum systems,” Physical Review A 58, 2733–2744 (1998).
G. de Lange, Z. H. Wang, D. Ristè, V. V. Dobrovitski, and R. Hanson, “Universal dynamical decoupling of a single solid-state spin from a spin bath,” Science 330, 60–63 (2010).