Instead of watching the spin evolve, drive it deliberately
A resonant RF or microwave field can rotate a spin state in a controlled way. The language of pulses—\(\pi/2\), \(\pi\), phase, detuning and echo—comes from solving a driven two-level problem and then using those rotations as building blocks for spectroscopy and quantum control.
Calculate Rabi oscillations and the duration of ideal π/2 and π pulses.
Use the rotating-frame picture to understand resonance, detuning and pulse bandwidth.
Explain what an echo refocuses and what irreversible decoherence it cannot reverse.
Driven spin
Add an oscillating transverse field
The drive has two independent physical knobs: strength and frequency mismatch
What it is: The transverse oscillating field couples the two spin states; its amplitude sets the on-resonance Rabi frequency.
What it changes: It controls how quickly the Bloch vector rotates during a pulse and therefore the pulse area.
What you observe: The period of Rabi oscillations and the required duration of \(\pi/2\) and \(\pi\) pulses.
What it is: The difference between the applied drive frequency and the spin's resonance frequency.
What it changes: It tilts the effective field in the rotating frame and reduces the maximum achievable population transfer for a rectangular pulse.
What you observe: Off-resonance excitation, phase errors and frequency-selective pulse profiles.
For a spin-\(\tfrac12\) in a static field \(B_0\hat z\), add an oscillating field \(B_1(t)\) transverse to \(B_0\). In a rotating frame and under the rotating-wave approximation, a useful frequency-unit Hamiltonian is
where \(\Delta\nu\) is the detuning from resonance and \(\nu_1\) is the on-resonance Rabi frequency in cycles per second.
Rabi oscillations
On resonance, pulse duration becomes a rotation angle
Starting from one basis state, the driven transition probability is
On resonance, a \(\pi\) pulse requires \(t_\pi=1/(2\nu_1)\), while a \(\pi/2\) pulse requires \(t_{\pi/2}=1/(4\nu_1)\).
Pulse area and detuning
The pulse is resonant and has exactly the area of a π rotation.
The model assumes a coherent two-level system, a rectangular pulse and the rotating-wave approximation. Real pulse excitation profiles are modified by relaxation, inhomogeneity, additional levels and pulse shape.
Rotating frame
Make the fast Larmor motion disappear
The rotating frame is a change of viewpoint, not an extra physical force
What it is: A coordinate frame chosen to rotate near the microwave/RF frequency so the rapid laboratory-frame precession is factored out.
What it changes: A time-dependent drive becomes approximately a static effective field under the rotating-wave approximation.
What you observe: Simpler pulse trajectories and the intuitive effective-field picture used throughout magnetic resonance.
What it is: An approximation that neglects the counter-rotating drive component when the drive is near resonance and weak compared with the carrier frequency.
What it changes: It reduces the driven problem to slow dynamics around an effective field.
What you observe: Accurate standard pulse behaviour in its regime; systematic deviations for very strong or ultrabroadband driving.
In the laboratory frame the spin precesses rapidly around \(B_0\) while the microwave field oscillates. Transforming into a frame rotating near the drive frequency converts that problem into precession around an effective static field.
On resonance the effective field lies in the transverse plane. Off resonance it tilts toward \(z\), which is why a pulse of the same duration no longer performs the intended rotation.
Pulse bandwidth
Short pulses are spectrally broad
Time resolution and frequency selectivity are Fourier partners
What it is: The time over which the coherent drive is applied.
What it changes: Shortening the pulse broadens its frequency spectrum, while lengthening it makes excitation more selective.
What you observe: How much of an inhomogeneously broadened spectrum is rotated by the pulse.
What it is: The range of resonance offsets for which the pulse produces substantial rotation.
What it changes: It determines whether spins with different \(g\)-values, hyperfine shifts or orientations are excited uniformly.
What you observe: Frequency/field-selective pulse profiles and orientation selection in anisotropic EPR.
What it is: The time dependence of amplitude, phase and sometimes carrier frequency during the pulse.
What it changes: Shaping redistributes spectral power and can make rotations more robust to detuning or \(B_1\) inhomogeneity.
What you observe: Broader or more selective excitation, reduced pulse errors and improved echo/control fidelity.
A rectangular pulse of finite duration cannot be perfectly frequency selective. Its Fourier spectrum has a sinc-like envelope with characteristic width of order \(1/t_p\). Short, strong pulses therefore excite a broader range of resonance offsets; long, weak pulses are more selective.
This time–frequency tradeoff is central in magnetic resonance: pulse length, \(B_1\), spectral bandwidth and relaxation cannot be optimized independently.
Echoes
A \(\pi\) pulse can refocus static frequency offsets
An echo reverses reversible phase dispersion, not irreversible decoherence
What it is: Different members of an ensemble precess at slightly different static frequencies.
What it changes: The ensemble transverse signal cancels even though individual spins can remain coherent.
What you observe: A short apparent \(T_2^*\) that can be refocused by a pulse sequence.
What it is: A \(\pi/2-\tau-\pi-\tau\) sequence that reverses static phase accumulation caused by frequency offsets.
What it changes: It rephases spins at the echo time while leaving truly irreversible stochastic decoherence unrecovered.
What you observe: An echo amplitude whose decay reports homogeneous coherence loss more directly than a free-induction signal.
In a Hahn echo, an initial \(\pi/2\) pulse creates transverse coherence. Different members of an ensemble then accumulate different phases. A \(\pi\) pulse reverses the effect of static frequency offsets, producing an echo after the same free-evolution delay.
Irreversible dephasing is not refocused. That distinction is why echo experiments can separate homogeneous coherence decay from static inhomogeneous broadening.
Beyond rectangular pulses
Pulse shape and phase are control variables
Sequences of rotations with different phases can compensate systematic pulse errors.
Amplitude and frequency are varied smoothly to improve bandwidth or robustness.
Repeat the experiment with controlled pulse phases to select pathways and reject unwanted signals.
Numerically optimize waveform parameters to achieve a target state transfer under realistic constraints.
Radical pairs & RYDMR
Coherent driving can be read out chemically
In conventional EPR the pulse sequence is detected through magnetization or an echo. In a radical-pair experiment, the same resonant driving can instead change singlet–triplet dynamics and therefore a chemical reaction yield.
This is the connection to reaction-yield detected magnetic resonance: coherent control acts on the spin system, while chemistry provides the detector.
Selected reading
Examples from my work
Reaction-yield detected magnetic resonance spectroscopy of radical pairs in cryptochrome-4a: a computational study
Resonant driving of radical-pair spin dynamics with chemical-yield detection.
Free Radic. Biol. Med. (2026) →Weak Radiofrequency Field Effects on Biological Systems Mediated through the Radical Pair Mechanism
How oscillating fields interact with radical-pair dynamics across different regimes.
Chemical Reviews (2025) →Key external literature
Where to read next
These are deliberately selected from outside my own work: foundational papers or reviews that are especially useful for this topic.
Space Quantization in a Gyrating Magnetic Field
I. I. Rabi · Physical Review (1937). The classic analysis underlying resonantly driven angular-momentum transitions and Rabi oscillations.
Open DOI →Spin Echoes
E. L. Hahn · Physical Review (1950). The foundational pulse experiment establishing spin echoes and refocusing of static frequency dispersion.
Open DOI →