# shaped_pulse_xy.m

Shaped pulse function using Cartesian coordinates. Applies a user-specified pulse shape on user-specified operators while the rest of the drift Liouvillian continues to affect the spin system. The pulse is assumed to be piecewise-constant and should be supplied with sufficiently fine time discretization to properly reproduce the waveform.

## Syntax

    [rho,P]=shaped_pulse_xy(spin_system,drift,controls,...
amplitudes,slice_durs,rho,method)


## Arguments

      drift - the drift Liouvillian, the part of the Liouvillian that
should continue running in the background. This should
include the pulse transmitter offset, if any.

controls - a cell array of control operators corresponding to each
channel, this may include operators for spatial degrees
of freedom, such as gradients and diffusion.

amplitudes - a cell array of control amplitude vectors in angular
frequency units.

slice_durs - a vector containing the duration of each pulse slice,
seconds.

rho - state vector of the initial state, or a stack thereof.

method - propagation method, 'expv' for Krylov propagation,
'expm' for exponential propagation, 'evolution' for
Spinach evolution function


## Outputs

        rho - state vector for the final state, or a stack thereof

P - pulse propagator (expensive)


## Examples

An example of a Veshtort-Griffin shaped excitation pulse applied to a system with 31 J-coupled protons (examples/nmr_liquids/shaped_pulse_3.m):

## Notes

Of the three propagation methods, 'expv' is recommended because it runs Krylov propagation that avoids explicit matrix exponentiation. The 'expm' option forces Matlab's very inefficient matrix exponentiation path and should only be used for debugging pourposes. In very anomalous cases (long pulses, large state vector stacks, very large state spaces), the 'evolution' option might become necessary.