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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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. 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.
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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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<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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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==
 
==Syntax==
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==Arguments==
 
==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  
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                    propagation; centre point piecewise-constant
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                    rule if one matrix is supplied, piecewise-
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                    linear rule if two matrices {left, right}
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                    are supplied, piecewise-quadratic if three
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                    matrices {left, midpoint, right} are given.
 
   
 
   
 
       rho        -  state vector or density matrix to be propagated
 
       rho        -  state vector or density matrix to be propagated
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''Version 2.5, authors: [[Ilya Kuprov]], [[Luke Edwards]]''
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''Version 2.7, authors: [[Ilya Kuprov]], [[Luke Edwards]], [[Anupama Acharya]]''

Revision as of 12:50, 1 November 2022

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. 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.

     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;
   Ly=(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 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 Liovillian again. Long-term propagation (trajectories, observables) under a static Liovillian 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 2.7, authors: Ilya Kuprov, Luke Edwards, Anupama Acharya

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