Difference between revisions of "Step.m"

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{{DISPLAYTITLE:step.m}} __NOTOC__
 
{{DISPLAYTITLE:step.m}} __NOTOC__
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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.
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Propagation step function. Computes the action by a matrix exponential without computing that exponential. Supports one-, two-, and three-point product quadratures.
  
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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:
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==Syntax==
  
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<center><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></center>
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    rho=step(spin_system,L,rho,time_step)
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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.
 
  
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==Syntax==
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==Parameters==
  
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    rho=step(spin_system,L,rho,time_step)
 
  
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==Arguments==
 
  
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       L          -  Liouvillian or Hamiltonian to be used for propagation
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       L          -  Liouvillian or Hamiltonian to be used for propagation;
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                    centre point piecewise-constant rule if one matrix is
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                    supplied, piecewise-linear rule if two matrices {left,
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                    right} are supplied, and piecewise-quadratic rule if
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                    three matrices {left, midpoint, right} are supplied.
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                    State-dependent evolution generators are also supported:
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                    if L{1} is a function handle, L{2} is current time, and
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                    L{3} is the method, the problem is routed to iserstep.m
 
   
 
   
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       rho        -  state vector or density matrix to be propagated
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       rho        -  state vector or density matrix
 
   
 
   
 
       time_step  -  length of the time step to take
 
       time_step  -  length of the time step to take
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==Examples==
 
==Examples==
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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.
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A 90-degree pulse in X phase on protons:
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    Lx=operator(spin_system,'Lx','1H');
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    rho=step(spin_system,Lx,rho,pi/2);
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A 1 millisecond evolution period under a Hamiltonian H:
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    rho=step(spin_system,H,rho,1e-3);
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A 45-degree pulse with a 60-degree phase on carbon:
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    Lx=operator(spin_system,'Lx','13C');
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    Ly=operator(spin_system,'Ly','13C');
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    rho=step(spin_system,cosd(60)*Lx+sind(60)*Ly,rho,pi/4);
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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==
 
==Notes==
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#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.
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#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.
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This function originally used a faithful Krylov process, but testing found it inferior to the reordered Taylor process now used in the implementation. The current algebraic operation order is designed to minimise memory footprint in large cases.
  
 
==See also==
 
==See also==
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[[evolution.m]], [[krylov.m]], [[shaped_pulse_xy.m]], [[shaped_pulse_af.m]]
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[[evolution.m]], [[krylov.m]], [[propagator.m]], [[shaped_pulse_xy.m]], [[shaped_pulse_af.m]], [[cosy.m]], [[hsqc.m]], [[isergen.m]], [[iserstep.m]], [[steady.m]], [[Kernel_functions]]
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''Version 2.3, authors: [[Ilya Kuprov]], [[Luke Edwards]]''
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''Version 2.8, authors: [[Ilya Kuprov]], [[Luke Edwards]], [[Anupama Acharya]]''

Latest revision as of 19:41, 6 June 2026

Propagation step function. Computes the action by a matrix exponential without computing that exponential. Supports one-, two-, and three-point product quadratures.

Syntax

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

Parameters

     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, and piecewise-quadratic rule if
                   three matrices {left, midpoint, right} are supplied.
                   State-dependent evolution generators are also supported:
                   if L{1} is a function handle, L{2} is current time, and
                   L{3} is the method, the problem is routed to iserstep.m

     rho        -  state vector or density matrix

     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,'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

This function originally used a faithful Krylov process, but testing found it inferior to the reordered Taylor process now used in the implementation. The current algebraic operation order is designed to minimise memory footprint in large cases.

See also

evolution.m, krylov.m, propagator.m, shaped_pulse_xy.m, shaped_pulse_af.m, cosy.m, hsqc.m, isergen.m, iserstep.m, steady.m, Kernel_functions

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

Retrieved from "https://spindynamics.org/wiki/index.php?title=Step.m&oldid=9556"