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:

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

Matrix exponentials are computed explicitly. When this function detects that the dimension of the Liouvillian is too big for exponentiation, it calls krylov.m with the same parameters.

Arguments

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.