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. For toggling-frame cycle expansions and commutator corrections, see Average Hamiltonian Theory. Dynamical Decoupling explains noise filters, and the quantum-information treatment discusses implementing and testing sequences on hardware.

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 rotates the Bloch vector by $\pi$ about its axis: the component parallel to that axis is unchanged, while the two perpendicular components reverse sign. 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	$X_{\pi/2}$ — $(\tau$ — $Y_\pi$ — $\tau)^N$ — readout	refocusing with reduced pulse-area error for the prepared $y$ 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 at fixed $T$ estimates the fringe contrast and phase independently of the assumed time-envelope model. Drift during the scan can still bias both; interleaving phase settings can help.

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 refocuses static detuning exactly in this ideal-pulse model and suppresses a band of slowly varying noise. Faster fluctuations are filtered according to their frequency; some can be enhanced rather than suppressed. It does not reverse an unobserved spontaneous-emission event or erase information already carried away by a measured bath.

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. With the preparation $X_{\pi/2}$ applied to a ground state at $-z$, the magnetization points along $+y$. The $Y_\pi$ refocusing pulses then rotate about that component, so a small area error perturbs it less than in an $X_\pi$ train. This protection is state-dependent; it is not a universal robust memory for every Bloch vector.

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.

The rightmost operator in this product acts first. To establish robustness, specify the error model and expand the total rotation in that error parameter; adding pulses without such cancellation merely adds duration and new error opportunities.

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

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

for zero-mean stationary classical frequency noise and the two-sided spectrum defined in Noise Spectra. For ideal transverse $\pi$ pulses, $Y(\omega,T)$ follows the sign-changing longitudinal coupling. During finite pulses the rotated coupling generally has several Pauli components, so one scalar switching function is insufficient. This formula is useful only when the noise model and convention match the experiment.

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 with a finite refocusing pulse

A static detuning $\delta$ acts throughout two equal free intervals of length $\tau$ and a rectangular refocusing pulse. During the pulse,

$$H_p=\frac{\hbar}{2}(\Omega X+\delta Z),\qquad t_p=\frac{\pi}{\Omega},\qquad \Omega>0.$$

There is no dissipation in this calibration-error example. With $F=\exp(-i\delta\tau Z/2)$, the full operation is $U=F\exp(-iH_pt_p/\hbar)F$. Compare it with the ideal echo operation $U_0=-iX$, which an instantaneous perfect pulse produces for any $\delta\tau$. Find the average gate fidelity, using

$$F_{\mathrm{avg}}(U,U_0) =\frac{2+|\operatorname{Tr}(U_0^\dagger U)|^2}{6}.$$

For qubits this also equals the mean output overlap on the six Pauli eigenstates, as discussed in Optimal Control; the general fidelity relation is given by Nielsen (2002).

Solution

Set $\epsilon=\delta/\Omega$, $r=\sqrt{1+\epsilon^2}$, and $a=\pi r/2$. The pulse is

$$P=\cos a\,I-i\frac{\sin a}{r}(X+\epsilon Z).$$

Since $FXF=X$ while $FIF$ and $FZF$ contain only $I$ and $Z$, the coefficient of $X$ in $U=FPF$ is $-i\sin a/r$. Consequently,

$$\operatorname{Tr}(U_0^\dagger U)=\frac{2\sin a}{r},\qquad F_{\mathrm{avg}} =\frac{1}{3}\left[1+ \frac{2\sin^2\!\left(\frac{\pi}{2}\sqrt{1+\epsilon^2}\right)}{1+\epsilon^2}\right].$$

The leading infidelity is $1-F_{\mathrm{avg}}=2\epsilon^2/3+O(\epsilon^4)$. Equal free intervals still cancel the ideal static phase, but the detuning tilts the axis during the finite pulse. Increasing the free delay does not remove this error.

The delay changes which input states are most affected even though this average is independent of $\tau$. Indeed, with $\varphi=\delta\tau$,

$$U_0^\dagger U =I-i\epsilon(\sin\varphi\,X+\cos\varphi\,Y)+O(\epsilon^2).$$

This expansion keeps $\varphi$ explicit and does not require $|\delta\tau|\ll1$. A single-state echo contrast can therefore conceal a coherent gate error. Time-varying detuning, unequal free intervals, pulse shaping, or relaxation during the sequence require their own calculation.

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.

References

• H. Y. Carr and E. M. Purcell, “Effects of diffusion on free precession in nuclear magnetic resonance experiments,” Physical Review 94, 630–638 (1954).
• 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).
• E. L. Hahn, “Spin echoes,” Physical Review 80, 580–594 (1950).
• 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).
• M. A. Nielsen, “A simple formula for the average gate fidelity of a quantum dynamical operation,” Physics Letters A 303, 249–252 (2002). arXiv:quant-ph/0205035.
• S. Meiboom and D. Gill, “Modified spin-echo method for measuring nuclear relaxation times,” Review of Scientific Instruments 29, 688–691 (1958).
• L. Viola and S. Lloyd, “Dynamical suppression of decoherence in two-state quantum systems,” Physical Review A 58, 2733–2744 (1998).