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. In Liouville space, this function calculates the action by a matrix exponential on a vector without computing the matrix exponential. This 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; centre point piecewise-constant
                   rule if one matrix is supplied, piecewise-
                   linear rule if two matrices {left, right}
                   are supplied, piecewise-quadratic if three
                   matrices {left, midpoint, right} are given.

                   State-dependent evolution generators are
                   supported: if L{1} is a function handle (see
                   iserstep.m documentation), L{2} is current
                   time, and L{3} is the method (see iserstep.m
                   documentation), the problem is routed to a
                   an appropriate Lie group solver.

     rho        -  state vector or density matrix to be propagated

     time_step  -  length of the time step to take

Note: we initially had a faithful implementation of the Krylov process
      here - subspace, orthogonalisation, projection, etc., but in all
      our testing it was much inferior to the reordered Taylor process
      that is currently implemented below.

Note: the peculiar sequence of algebraic operations in the code below
      is designed to minimise the memory footprint in large cases.

ilya.kuprov@weizmann.ac.il
ledwards@cbs.mpg.de
a.acharya@soton.ac.uk
c.musselwhite@soton.ac.uk

Outputs

rho - state vector or density matrix

Examples

A 90-degree pulse in X phase on protons:

   Lx=operator(spin_system,'Lx','1H');
   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,'Lx','13C');
   Ly=operator(spin_system,'Ly','13C');
   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 function 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 Liouvillian again. Long-term propagation (trajectories, observables) under a static Liouvillian should be handled with evolution.m or krylov.m functions instead.

See also

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

Version 2.8, authors: Ilya Kuprov, Luke Edwards, Anupama Acharya