step.m

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Time propagation function optimised for one-off calls, such as hard pulses or slices of shaped pulses. For trajectory calculation and detection periods of time-domain experiments, use evolution.m instead.

This function calculates the action by a matrix exponential on a vector without computing the matrix exponential. The actual implementation is more sophisticated, but the principle becomes apparent from the following equation:

[math]\exp \left[ { - i{\bf{L}}\Delta t} \right]{\bf{\rho }} = \sum\limits_{n = 0}^\infty {\frac{{{{\left( { - i\Delta t} \right)}^n}}}{{n!}}{\bf{L}}\left( {...\left( {{\bf{L}}\left( {{\bf{L\rho }}} \right)} \right)} \right)}[/math]

where [math]\bf{L}[/math] is a Liouvillian and [math]\bf{\rho }[/math] is a state vector. This operation is cheaper than matrix exponentiation, but only when it is performed once. If many time steps are required, it is cheaper to pre-compute the exponential, which is what evolution.m does.

Syntax

    rho=step(spin_system,L,rho,time_step)

Arguments

     L          -  Liouvillian or Hamiltonian to be used for propagation

     rho        -  state vector or density matrix to be propagated

     time_step  -  length of the time step to take

Outputs

     rho        -  state vector or density matrix

Examples

See the source code of shaped_pulse_xy.m and most NMR pulse sequences (cosy.m, hsqc.m, and others) for examples of this function being used.

Notes

  1. The sequence is programmed with a rather peculiar order of algebraic operations. This was carefully optimised to ensure best possible performance under a variety of scenarios (parallelisation, GPUs, large sparse arrays) in Matlab.
  2. Only use this function for short one-off events where you do not expect to see the same Liovillian again. Long-term propagation (trajectories, observables) under a static Liovillian should be handled with evolution.m or krylov.m functions instead.

See also

evolution.m, krylov.m, propagator.m, shaped_pulse_xy.m, shaped_pulse_af.m


Version 2.3, authors: Ilya Kuprov, Luke Edwards