Evolution.m

Time evolution function optimised for extended evolution periods with multiple time steps, such as trajectory calculation and detection stages of time-domain experiments. For brief one-off evolution events (hard pulses, slices of shaped pulses) or time-dependent Liouvillians use step.m instead.

This function runs trajectory level state space analysis (https://doi.org/10.1016/j.jmr.2008.08.008), partitions the state space into the smallest possible independently evolving subspaces, and runs exponential time propagation in each subspace:

\({\bf{\rho }}\left( {t + \Delta t} \right) = \exp \left[ { - i{\bf{L}}\Delta t} \right]{\bf{\rho }}\left( t \right)\)

Such analysis is only warranted if the propagation time is long and the Liouvillian matrix is large - this module should not be used for one-off events, such as hard pulses. It should also not be used with small spin systems or time-dependent Hamiltonians because the expensive preliminary analysis would in that case dominate the calculation time.

The default propagation mechanism inside evolution.m is matrix exponentiation with the propagators obtained by calling propagator.m. When this function detects that the dimension of the Liouvillian is too big for exponentiation, it calls krylov.m with the same parameters. If the function detects that the Liouvillian is too small (fewer then five spins) to bother with any sophisticated state space reduction, it simply calls Matlab's expm function.

Syntax

    answer=evolution(spin_system,L,coil,rho,timestep,nsteps,output,destination)

Parameters

Arguments for Liouville space calculations:

     L      - the Liouvillian to be used during evolution

     rho    - the initial state vector or a horizontal stack thereof

     output - a string giving the type of evolution that is required

               'final' - returns the final state vector or a horizontal
                         stack thereof.

               'trajectory' - returns the stack of state vectors giving
                              the trajectory of the system starting from
                              rho with the user-specified number of steps
                              and step length.

               'total'   - returns the integral of the observable trace
                           from the simulation start to infinity. This
                           option requires the presence of relaxation.

               'refocus' - evolves the first vector for zero steps,
                           second vector for one step, third vector for
                           two steps, etc., consistent with the second
                           stage of evolution in the indirect dimension
                           after a refocusing pulse.

               'observable' - returns the time dynamics of an observable
                              as a vector (if starting from a single ini-
                              tial state) or a matrix (if starting from a
                              stack of initial states).

               'multichannel' - returns the time dynamics of several
                                observables as rows of a matrix. Note
                                that destination state screening may be
                                less efficient when there are multiple
                                destinations to screen against.

     coil   - the detection state, used when 'observable' is specified as
              the output option. If 'multichannel' is selected, the coil
              should contain multiple columns corresponding to individual
              observable vectors.

     destination - (optional) the state to be used for destination state
                   screening.

Arguments for Hilbert space calculations:

      L         - Hamiltonian matrix

      coil      - observable operator (if any)

      rho       - initial density matrix

      timestep  - duration of a single time step (seconds)

      nsteps    - number of steps to take

      output    - a string giving the type of evolution that is required

               'final' - returns the final density matrix.

               'trajectory' - returns a cell array of density matrices
                              giving the trajectory of the system star-
                              ting from rho with the user-specified num-
                              ber of steps and step length.

               'refocus' - evolves the first matrix for zero steps,
                           second matrix for one step, third matrix for
                           two steps, etc., consistent with the second
                           stage of evolution in the indirect dimension
                           after a refocusing pulse.

               'observable' - returns the time dynamics of an observable
                              as a vector.

      destination - this argument is ignored.

Outputs

    answer - a vector, a matrix, or a cell array or matrices,
             depending on the options set during the call

Notes

1. Calculation of final states and observables in Hilbert space is parallelised (http://dx.doi.org/10.1063/1.3679656) and tested all the way to 128-core (16 nodes, 8 cores each) configuations. Parallelization of the trajectory calculation does not appear to yield any benefits due to large amount of inter-thread communication.
2. One-off propagation events (e.g. a hard pulse) should be computed with step.m, which is more efficient in those cases.
3. For large Liouvillians that cannot be exponentiated, use krylov.m, which avoids matrix exponentials.
4. For shaped pulses, use shaped_pulse_af.m and shaped_pulse_xy.m
5. GPUs are supported, add 'gpu' to sys.enable array during calculation setup.

See also

step.m, krylov.m, reduce.m, propagator.m, shaped_pulse_af.m, shaped_pulse_xy.m, isergen.m, iserstep.m, steady.m, Kernel_functions

Version 2.3, authors: Ilya Kuprov, Luke Edwards, Ohad Levinkron