evolve

Calculates time traces for 1D and 2D pulse EPR experiments.

evolve

Syntax

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

See also the examples.

Description

This function calculates time traces for 1D or 2D pulse EPR experiments with up to four swept intervals/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.

For example, IncScheme = [1 2 -2 1] indicates 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 second being incremented and the third being decremented.

The following table lists all possible values for IncScheme with some corresponding experiments.

`IncScheme`	#periods	#dims	experiments
`[1]`	1	1	simple FID, echo transient, 3-pulse ESEEM, DEFENCE
`[1 1]`	2	1	2-pulse ESEEM, CP, 3-pulse and 4-pulse RIDME
`[1 -1]`	2	1	3-pulse-DEER, 4-pulse DEER, PEANUT,5-pulse RIDME
`[1 2]`	2	2	3-pulse ESEEM echo transient, HYSCORE, DONUT-HYSCORE
`[1 1 2]`	3	2	2-pulse ESEEM etc. with echo transient
`[1 -1 2]`	3	2
`[1 2 1]`	3	2	2D-3-pulse ESEEM
`[1 1 2 2]`	4	2	2D refocused 2-pulse echo
`[1 2 2 1]`	4	2	2D-CP
`[1 2 -2 1]`	4	2	2D-PEANUT
`[1 -1 1 -1]`	4	1
`[1 1 -1 -1]`	4	1	SIFTER
`[1 -1 -1 1]`	4	1	7-pulse DEER
`[1 1 -1 -1 2]`	5	2
`[1 -1 -1 1 2]`	5	2

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 which governs the evolutions during the incrementation periods. For a 2D experiment, Ham can be a cell 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 propagation operator that is applied to propagate the density matrix from the end of the kth evolution period to the beginning of the next evolution period. Mix has to be specified for experiments with more than one incrementation period. For example, for a two-pulse ESEEM, with IncScheme=[1 1], Mix contains the propagation matrix that represents the pi pulse between the two evolution periods.

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.

n is a vector that gives the number of points along each dimension. If only a scalar is given, it is used for all dimensions.

All matrices have to be in the same basis. Expected 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

Given values for Sig, Det, Ham, and Mix, the script

n = 128;
dt = 0.01; % microseconds
Inc = [1 2]; % 2 periods, 2D 
td = evolve(Sig,Det,Ham,n,dt,Inc,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.