Pulse Simulation

Pulse simulation maps a specified waveform to a quantum trajectory and a measured outcome. A pulse is not fully defined by its duration and peak amplitude: carrier phase, envelope interpolation, truncation, timing, and filtering can change its action. This page treats forward simulation; optimal control treats choosing a waveform by optimization.

Helpful background. Driven two-level systems fixes Rabi and rotating-frame conventions.

Waveforms are part of the Hamiltonian

Write the model explicitly, for example

$$H(t)=H_0+\sum_a u_a(t)H_a.$$

Specify whether stored samples represent values at bin centers, endpoints, or coefficients of a piecewise-constant hold. Linear, cubic, and constant interpolation define different Hamiltonians between samples. Cubic interpolation can overshoot amplitude bounds; a visually smooth line does not guarantee a physically admissible pulse.

For a carrier-modulated drive, distinguish the real laboratory waveform from a complex rotating-frame envelope. State the sign of the carrier phase and the frame transformation. A sample rate that resolves an envelope need not resolve the laboratory carrier.

A Gaussian envelope $u(t)=u_0e^{-(t-t_c)^2/(2\sigma_t^2)}$ has infinite support. Truncating it to a finite interval changes its area and introduces endpoint behavior; record the truncation and any smooth window.

Pulse area as an exact commuting benchmark

For the resonant Hamiltonian $H(t)=\hbar u(t)\sigma_x/2$, all times commute and

$$U(T)=e^{-i\Theta\sigma_x/2}, \qquad \Theta=\int_0^T u(t)\,dt.$$

For an untruncated Gaussian on the whole line, $\Theta=u_0\sigma_t\sqrt{2\pi}$. A symmetric truncation at $t_c\pm L$ instead gives

$$\Theta_L=u_0\sigma_t\sqrt{2\pi}\, \operatorname{erf}\!\left(\frac{L}{\sqrt2\sigma_t}\right).$$

Comparing the simulated rotation with the integral of the actual interpolated waveform tests quadrature and timing. Comparing with the infinite Gaussian formula without including its omitted tails tests the wrong pulse.

Pulse area ceases to characterize the full action when detuning, changing control axes, or additional levels introduce noncommuting Hamiltonians. Equal areas can then produce different final states.

Ordering, discontinuities, and filters

For consecutive pulses, the earliest propagator appears on the right: $U_{\rm total}=U_N\cdots U_2U_1$. An $x$ rotation followed by a $y$ rotation generally differs from the reverse sequence. A two-pulse test with a phase-sensitive observable detects a reversed multiplication order that population-only tests may miss.

End an integration step exactly at each discontinuity of a piecewise pulse. High-order methods assume smoothness within a step, and adaptive error estimators may overlook a jump if no stage samples it. Use a time grid fine enough to represent the waveform independently of the internal solver tolerance.

If hardware filtering is included, simulate $u_{\rm applied}(t)=\int g(t-s)u_{\rm command}(s)\,ds$ with the stated impulse response. Include delay, finite bandwidth, and ring-down in the simulation interval. Normalizing a filtered pulse to restore its area does not automatically restore its noncommuting action.

Validation and uncertainty in the model

Refine waveform sampling and propagation tolerance separately. Compare an analytic rectangular or commuting Gaussian pulse before introducing detuning and extra levels. Check norm, final state fidelity, and observables during the sequence.

Repeat with small timing, amplitude, detuning, and phase changes when sensitivity matters. Distinguish numerical error from uncertainty in these model parameters. A two-level truncation can miss leakage into higher states even with a perfect integrator; enlarge the represented Hilbert space and monitor leakage probabilities.

For finite pulse trains, record whether phase is continuous across gaps or reset for each pulse. The distinction changes Ramsey interference and cannot be inferred from a plotted envelope alone.

Exercises

Two resonant rectangular pulses about the same axis have areas $\pi/3$ and $2\pi/3$. What is their total action, and why does the answer change for different axes?

Solution

For the same axis, the generators commute and areas add to $\pi$, giving $U=e^{-i\pi\sigma_x/2}=-i\sigma_x$. For different axes, the Pauli matrices do not commute, so the product cannot in general be replaced by one rotation whose angle is the sum of the two areas.

References

• Allen, L., and J. H. Eberly, Optical Resonance and Two-Level Atoms, Wiley, 1975 — pulse areas, phases, and coherent sequences.
• Khaneja, N., T. Reiss, C. Kehlet, T. Schulte-Herbrüggen, and S. J. Glaser, “Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms,” Journal of Magnetic Resonance 172, 296–305 (2005), doi:10.1016/j.jmr.2004.11.004 — discretized controls and propagation-based objectives.