evolve
Introduced in 1.0
Synopsis

Calculates time traces for 1D and 2D ESEEM experiments.

td = evolve(Sig,Det,Ham,n,dt)
td = evolve(Sig,Det,Ham,n,dt,IncScheme)
td = evolve(Sig,Det,Ham,n,dt,IncScheme,Mix)
Description

This function calculates time traces for 1D or 2D ESEEM experiments with up to four swept invervals/lengths.

The row vector IncScheme specifies the incrementation scheme of the ESEEM experiment to simulate. It can contain only 1, -1, 2 and -2. 1 symbolises the t1 dimension (along columns in the output), 2 is the t2 dimension (along rows). The sign determines whether the evolution period is incremented or decremented along the given dimension. IncScheme = [1 2 -2 1] means that the first and fourth evolution period are incremented together yielding the t1 dimension, whereas the second and third period give the t2 dimension with the third being decremented.

There is only a limited set of incrementation schemes supported by evolve. The following table lists all possible values for IncScheme with some corresponding ESEEM experiments.

IncSchemeESEEM experiments
[1]simple FID, 3-pulse, DEFENCE
[1 1]2-pulse, CP
[1 -1]PEANUT
[1 2]Hyscore, DONUT-Hyscore
[1 2 1]2D-3-pulse ESEEM
[1 2 2 1]2D-CP
[1 2 -2 1]2D-PEANUT

Sig is the density matrix at the start of the first evolution period and doesn't have to be a thermal equilibrium density. It can be a density prepared to a non-equilibrium state by a preparation sequence.

Det is the detection operator used in calculating the actual signal. It, too, can be a matrix describing a complete detection sequence. It can be non-Hermitian like [eqn], so that a complex time-domain signal is returned in td.

Ham is the Hamilton operator with governs the evolutions. For a 2D experiment Ham can be a 3D array, in which case Ham(:,:,1) will be used for the first dimension and Ham(:,:,2) for the second. Otherwise the same Hamiltonian is used for both dimensions.

If Mix is given, it is assumed to be a cell array containing all unitary matrices representing the mixing sequences sandwiched between the swept periods. Mix{k} is the mixer after the kth evolution period. Mix has to be specified for experiments with more than one sweep period.

dt gives the time increment for the evolution period. For a 2D experiment, it can be a 2-vector containing the increments for the different axes. If only a scalar is given, it is used for all dimensions. The same applies to n, which gives the number of points in each dimension.

All matrices have to be in the same basis. Units are Megahertz for Ham and Det, and microseconds for dt. In each dimension the first point in td contains the signal arising from the initial density matrix.

Examples

The line

td = evolve(Sig,Det,Ham,128,0.01,[1 2],Mix);

simulates a HYSCORE or a DONUT-HYSCORE spectrum with 128x128 points and a step time of 10 ns in both dimensions.

Algorithm

The function uses the standard equations [eqn] and evaluates them in the eigenbasis of the propagator [eqn] after transformation to Liouville space. The propagation superoperator is then diagonal as well, and the density can be evolved by simply multiplying it with the diagonal of the superpropagator element-by-element. The state space trace is evaluated in a similar way.

See also

nucfr2d, propint, sigeq