Step.m

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:

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

where \(\bf{L}\) is a Liouvillian and \(\bf{\rho }\) 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

A 90-degree pulse in X phase on protons:

   Lx=(operator(spin_system,'L+','1H')+...
       operator(spin_system,'L-','1H'))/2;
   rho=step(spin_system,Lx,rho,pi/2);

A 1 millisecond evolution period under a Hamiltonian H:

   rho=step(spin_system,H,rho,1e-3);

A 45-degree pulse with a 60-degree phase on carbon:

   Lx=(operator(spin_system,'L+','13C')+...
       operator(spin_system,'L-','13C'))/2;
   Lx=(operator(spin_system,'L+','13C')+...
       operator(spin_system,'L-','13C'))/2;
   rho=step(spin_system,cosd(60)*Lx+sind(60)*Ly,rho,pi/4);

See also 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