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calculateSignal_forHDCCE.m
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calculateSignal_forHDCCE.m
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% TO DO: Either remove HD-CCE functionality and "rm *_forHDCCE.m",
% or merge calculateSignal_forHDCCE() into calculateSignal()
% and then "rm *_forHDCCE.m".
function [Signal, AuxiliarySignal_1,AuxiliarySignal_2,AuxiliarySignal_3,...
AuxiliarySignal_4,Signals] ...
= calculateSignal_forHDCCE(System, Method, Nuclei,Clusters)
% Extract physical constants.
ge=System.ge; geff=System.gMatrix(3,3);
muB = System.muB; muN = System.muN; mu0 = System.mu0; hbar = System.hbar;
% Extract experimental parameters.
magneticField = System.magneticField;
nPoints = sum(System.nPoints);
dt = System.dt(1);
dt2 = System.dt(2);
N1 = System.nPoints(1);
B1x = System.RF.B1x;
B1y = System.RF.B1y;
nuRF = System.RF.nuRF;
doTR = false;
% ENUM
FID = 1; HAHN = 2; CPMG = 3; CPMG_CONST = 4; CPMG_2D = 5; HAHN_TR = 6;
Method_order = Method.order;
Method_useMultipleBathStates = Method.useMultipleBathStates;
Method_useInterlacedClusters = Method.useInterlacedClusters;
maxSize = 6;
% Define the theory to use at the given cluster size.
Theory = System.Theory;
theory = Theory(Method_order,:);
useMeanField = theory(10);
useMultipleBathStates = Method_useMultipleBathStates & useMeanField;
useInterlacedClusters = Method_useInterlacedClusters & useMeanField;
RandomSpinVector = Nuclei.RandomSpinVector;
maxSpinClusterSize = Nuclei.maxClusterSize;
methylTunnelingSplitting = Nuclei.methylTunnelingSplitting;
if Theory(Method_order,10)
Method_extraOrder = Method.extraOrder;
maxSuperclusterSize = Method_extraOrder;
else
Method_extraOrder = Method_order;
maxSuperclusterSize = Method_order;
end
if strcmp(Method.method,'HD-CCE')
doHDCCE = true;
else
doHDCCE = false;
end
if strcmp(Method.method,'OffsetCCE')
doOffsetCCE = true;
else
doOffsetCCE = false;
end
% Convert variable to gpu compatible forms
dimensionality = 1;
if strcmp(System.experiment,'FID')
EXPERIMENT = FID;
total_time = System.Time(end);
elseif strcmp(System.experiment,'Hahn')
EXPERIMENT = HAHN;
total_time = 2*System.Time(end);
elseif strcmp(System.experiment,'CPMG')
EXPERIMENT = CPMG;
total_time = 4*System.Time(end);
elseif strcmp(System.experiment,'CPMG-const')
EXPERIMENT = CPMG_CONST;
total_time = 4*System.Time(end);
elseif strcmp(System.experiment,'CPMG-2D')
EXPERIMENT = CPMG_2D;
total_time = 4*System.Time(end);
dimensionality = 2;
elseif strcmp(System.experiment,'Hahn-TR')
EXPERIMENT = HAHN;
total_time = 2*System.Time(end);
doTR = true;
B1x = System.RF.B1x(1);
B1y = System.RF.B1y(1);
nuRF = System.RF.nuRF(1);
B1x2 = System.RF.B1x(2);
B1y2 = System.RF.B1y(2);
nuRF2 = System.RF.nuRF(2);
else
error('The experiment ''%s'' is not supported.',System.experiment);
end
Nuclei_Coordinates = Nuclei.Coordinates;
Nuclei_Abundance = Nuclei.Abundance;
Nuclei_Spin = Nuclei.Spin;
Nuclei_g = Nuclei.Nuclear_g;
NumberStates = Nuclei.NumberStates;
state_multiplicity = Nuclei.StateMultiplicity;
ZeemanStates = Nuclei.ZeemanStates;
MeanFieldCoefficients = Nuclei.MeanFieldCoefficients;
MeanFieldTotal = Nuclei.MeanFieldTotal;
max_basis = max(NumberStates);
useThermalEnsemble = System.useThermalEnsemble;
States = zeros(max_basis,Nuclei.number);
betaT = 2*pi*System.hbar /System.kT; % 1/Hz.
Qtensors = Nuclei.Qtensor;
Atensors = Nuclei.Atensor;
FermiContact = Nuclei.FermiContact;
nStates0 =System.nStates;
nStates =System.nStates;
spinHalfOnly = System.spinHalfOnly;
% Unpack spin operators.
Op = Nuclei.SpinOperators;
SpinXiXjOps = Nuclei.SpinXiXjOperators;
% Unpack spin operators to individual variables.
[Spin2Op1,Spin3Op1,Spin4Op1,Spin2Op2,Spin3Op2,Spin4Op2, ...
Spin2Op3,Spin3Op3,Spin4Op3, Spin2Op4,Spin3Op4,Spin4Op4, ...
Spin2Op5,Spin3Op5,Spin4Op5, Spin2Op6,Spin3Op6,Spin4Op6, ...
SpinXiXjOp_1, SpinXiXjOp_2, SpinXiXjOp_3, ...
SpinXiXjOp_4, SpinXiXjOp_5, SpinXiXjOp_6]...
= initializeSpinOps(maxSpinClusterSize, Op, SpinXiXjOps);
Method_allowHDcoupling = Method.allowHDcoupling;
if Method_allowHDcoupling
SpinOperators_HD = Nuclei.SpinOperators_HD;
SpinOperators_DH = Nuclei.SpinOperators_DH;
SpinXiXjOperators_HD = Nuclei.SpinXiXjOperators_HD;
SpinXiXjOperators_DH = Nuclei.SpinXiXjOperators_DH;
else
SpinOperators_HD = [];
SpinOperators_DH = [];
SpinXiXjOperators_HD = [];
SpinXiXjOperators_DH = [];
end
numberClusters = Nuclei.numberClusters(1:maxSuperclusterSize);
nucleiList = 1:numberClusters(1);
maxNumberClusters = max(numberClusters(1:maxSuperclusterSize));
% CluserArray(iCluster,:,clusterSize) = nuclear indices.
ClusterArray = zeros(maxNumberClusters,Method_order,Method_order);
% Change data type of Clusters to a 3D array.
for isize = 1:Method_extraOrder
for ii = 1:Nuclei.numberClusters(isize)
ClusterArray(ii,1:isize,isize) = Clusters{isize}(ii,:);
end
end
lockRotors = System.Methyl.lockRotors;
methylMethod1 = System.Methyl.method==0;
% Initialize coherences and subcluster indices.
[Coherences_1, SubclusterIndices_2, Coherences_2, ...
SubclusterIndices_3, Coherences_3, SubclusterIndices_4, Coherences_4, ...
SubclusterIndices_5, Coherences_5, SubclusterIndices_6, Coherences_6, ...
SubclusterIndices_7, Coherences_7, SubclusterIndices_8, Coherences_8, ...
SubclusterIndices_9, Coherences_9, SubclusterIndices_10, Coherences_10, ...
SubclusterIndices_11, Coherences_11, SubclusterIndices_12, Coherences_12,...
rotationalMatrix_c1_m1, rotationalMatrix_d1_m1, ...
rotationalMatrix_c2_m1, rotationalMatrix_c2_m2, ...
rotationalMatrix_d2_m1, rotationalMatrix_d2_m2] ...
= initializeCoherences(Method_order, numberClusters, ...
Nuclei.rotationalMatrix,nPoints^dimensionality,methylMethod1,...
lockRotors,System.Methyl.methylMethylCoupling,maxSize);
% Methyl Groups
isMethylCarbon = strcmp(Nuclei.Type,'CH3');
isMethylHydron = strcmp(Nuclei.Type,'CH3_1H');
%-------------------------------------------------------------------------------
% gpu code
%
% At this point all varables should be in a gpu compatible form,
% and can be trasnfered over to the gpu.
%
% if useGPU
% dataOnGPU = trasferDataToGPU();
% end
%
% gpu code
% ISSUE NOTE: GPU not yet implemented.
dataOnGPU = false;
if ~dataOnGPU
warning(['The CluE GPU feature is not yet implemented. ',...
'CluE will run on the CPU.']);
end
%-------------------------------------------------------------------------------
for clusterSize = 1:Method_order
% Find coherences
numClusters = numberClusters(clusterSize);
for iCluster = numClusters:-1:1
Cluster = ClusterArray(iCluster,1:clusterSize,clusterSize);
if any(isMethylHydron(Cluster) )
error(['Error in calculateSignal_forHDCCE(): ',...
'A methyl hydron is misplaced.']);
end
if( length(unique(Cluster)) < length( Cluster ) )
error(['Error in calculateSignal_forHDCCE(): ',...
'Cluster constains duplicate indices.']);
end
% Check if iCluster is valid.
if Cluster(1,1) == 0, continue; end
% Determin the number of spins in the cluster.
thisClusterSize = clusterSize + 2*sum(isMethylCarbon(Cluster));
% Initialize cluster.
thisCluster = zeros(1,thisClusterSize);
% Initialize counters.
thisIndex = 0;
methyl_number = 0;
% Loop over the cluster to assign methyl rotors.
for ii = 1:clusterSize
thisIndex = thisIndex + 1;
if isMethylCarbon(Cluster(ii))
% Count the number of metyls.
methyl_number = methyl_number + 1;
% Unpack spins from methyl pseudo-particle.
thisCluster(thisIndex:thisIndex+2) = Cluster(ii) + [1,2,3];
thisIndex = thisIndex + 2;
else
thisCluster(thisIndex) = Cluster(ii);
end
end
if( length(unique(thisCluster)) < length( thisCluster ) )
error(['Error in calculateSignal_forHDCCE(): ',...
'This cluster constains duplicate indices.']);
end
if mod(sum(isMethylHydron(thisCluster) ),3) ~= 0
error(['Error in calculateSignal_forHDCCE(): ',...
'This cluster contains a partial methyl group.']);
end
if any(isMethylCarbon(thisCluster) )
error(['Error in calculateSignal_forHDCCE(): ',...
'A methyl carbon is misplaced.']);
end
% Generate methyl permutations.
cyclicPermutation = thisCluster;
if lockRotors || ~methylMethod1
nPerm = 1;
else
cyclicPermutation = getCyclicPermutations(isMethylCarbon(Cluster));
nPerm = size(cyclicPermutation,1);
cyclicPermutation = thisCluster(cyclicPermutation);
end
% Decide if the cluster should be skipped:
% check if all spin have the same I,
% and that if I >= 1 is enabled.
skipCluster = (spinHalfOnly && state_multiplicity(thisCluster(1)) > 2);
if ~ Method_allowHDcoupling
skipCluster = skipCluster || (~all(state_multiplicity(thisCluster) ...
== state_multiplicity(thisCluster(1))));
end
if skipCluster
switch clusterSize
case 1
Coherences_1(iCluster,:) = ones(size(Coherences_1(iCluster,:) ));
case 2
Coherences_2(iCluster,:) = ones(size(Coherences_2(iCluster,:) ));
case 3
Coherences_3(iCluster,:) = ones(size(Coherences_3(iCluster,:) ));
case 4
Coherences_4(iCluster,:) = ones(size(Coherences_4(iCluster,:) ));
case 5
Coherences_5(iCluster,:) = ones(size(Coherences_5(iCluster,:) ));
case 6
Coherences_6(iCluster,:) = ones(size(Coherences_6(iCluster,:) ));
end
continue;
end
% Set subcluster indices.
[SubclusterIndices_2, SubclusterIndices_3,SubclusterIndices_4,...
SubclusterIndices_5,SubclusterIndices_6, ...
SubclusterIndices_7, SubclusterIndices_8,...
SubclusterIndices_9,SubclusterIndices_10,...
SubclusterIndices_11,SubclusterIndices_12] ...
= setSubclusterIndices( ClusterArray,clusterSize,iCluster,...
SubclusterIndices_2, SubclusterIndices_3,SubclusterIndices_4,...
SubclusterIndices_5,SubclusterIndices_6,...
SubclusterIndices_7, SubclusterIndices_8,...
SubclusterIndices_9,SubclusterIndices_10,...
SubclusterIndices_11, SubclusterIndices_12);
% Select the appropriate spin operator.
[SpinOp,SpinXiXjOp] = getSpinOps( thisClusterSize,Nuclei_Spin(Cluster),...
Spin2Op1,Spin3Op1,Spin4Op1, Spin2Op2,Spin3Op2,Spin4Op2, ...
Spin2Op3,Spin3Op3,Spin4Op3, Spin2Op4,Spin3Op4,Spin4Op4, ...
Spin2Op5,Spin3Op5,Spin4Op5, Spin2Op6,Spin3Op6,Spin4Op6, ...
SpinXiXjOp_1, SpinXiXjOp_2, SpinXiXjOp_3, SpinXiXjOp_4, ...
SpinXiXjOp_5, SpinXiXjOp_6, SpinOperators_HD, SpinOperators_DH, ...
SpinXiXjOperators_HD,SpinXiXjOperators_DH);
if useInterlacedClusters(Method_order,clusterSize)
% Get union of all clusters that contain the primary cluster.
SuperclusterUnion = findSuperClusterUnion(...
ClusterArray,clusterSize,iCluster);
randStateIndex = randperm(3^numel(SuperclusterUnion));
% Get the set of nuclei not in the primary cluster
% that unions with the primary cluster to give SuperclusterUnion.
ClusterComplement = SuperclusterUnion( ...
~any(SuperclusterUnion==thisCluster',1) );
[tensors,zeroIndex] = pairwisetensors(Nuclei_g, Nuclei_Coordinates,...
thisCluster,Atensors,magneticField, ge,geff, muB, muN, mu0, hbar,...
theory,B1x,B1y,nuRF);
tensors0 = tensors;
if useThermalEnsemble
nStates(clusterSize) = max( 1, min(nStates0(clusterSize),prod(...
state_multiplicity(ClusterComplement))));
else
nStates(clusterSize) = max( 1, min(nStates0(clusterSize),prod(...
state_multiplicity(SuperclusterUnion))));
end
elseif useMultipleBathStates
ClusterComplement = nucleiList( ~any(nucleiList==thisCluster',1) );
[tensors,zeroIndex] = pairwisetensors(Nuclei_g, Nuclei_Coordinates,...
thisCluster,Atensors,magneticField, ge,geff, muB, muN, mu0, ...
hbar,theory,B1x,B1y,nuRF);
tensors0 = tensors;
end
H_alphaMF = 0;
H_betaMF = 0;
% density matrix size
SpinSpaceDim = prod(state_multiplicity(thisCluster));
HilbertSpaceDim = SpinSpaceDim*nPerm;
switch nPerm
case 1 % 0 rotors
Hb = zeros(HilbertSpaceDim);
case 3 % 1 rotor
switch SpinSpaceDim
case 8 % CH3
Hb = rotationalMatrix_c1_m1;
case 27 % CD3
Hb = rotationalMatrix_d1_m1;
case 16 % H, CH3
Hb = rotationalMatrix_c2_m1;
case 81 % D, CD3
Hb = rotationalMatrix_d2_m1;
end
case 9 % 2 rotors
switch SpinSpaceDim
case 64 % CH3, CH3
Hb = rotationalMatrix_c2_m2;
case 729 % CH3, CH3
Hb = rotationalMatrix_d2_m2;
end
otherwise
Hb = zeros(HilbertSpaceDim);
end
Ha = Hb;
for iave = 1:nStates(clusterSize)
% Loop over cyclic permutations.
for iPerm = 1:nPerm
% Permute rotor spins.
thisCluster = cyclicPermutation(iPerm,:);
% Get interaction tensors.
[tensors,zeroIndex] = pairwisetensors(Nuclei_g, ...
Nuclei_Coordinates,thisCluster,Atensors,magneticField, ge,geff,...
muB, muN, mu0, hbar,theory,B1x,B1y,nuRF);
qtensors = Qtensors(:,:,thisCluster);
if doTR
[tensors_TR,~] = pairwisetensors(Nuclei_g, ...
Nuclei_Coordinates,thisCluster,Atensors,magneticField, ge,geff,...
muB, muN, mu0, hbar,theory,B1x2,B1y2,nuRF2);
qtensors = Qtensors(:,:,thisCluster);
else
tensors_TR = [];
end
% Build Hamiltonians.
[H_alpha,H_beta] = ...
assembleHamiltonian(state_multiplicity(thisCluster),...
tensors,SpinOp,qtensors,SpinXiXjOp, theory,isMethylHydron(thisCluster),...
methylMethod1,methylTunnelingSplitting(thisCluster,thisCluster));
if ~isempty(tensors_TR)
[H_alpha_TR,H_beta_TR] = ...
assembleHamiltonian(state_multiplicity(thisCluster),...
tensors,SpinOp,qtensors,SpinXiXjOp, theory,isMethylHydron(thisCluster),...
methylMethod1, methylTunnelingSplitting(thisCluster,thisCluster));
else
H_alpha_TR = [];
H_beta_TR = [];
end
if useInterlacedClusters(Method_order,clusterSize)
if ~useThermalEnsemble
% density matrix size
dm_size = prod(state_multiplicity(thisCluster));
% Initialize density matrix.
densityMatrix = zeros(dm_size);
% Set cluster spin-state.
dm_index = mod(iave,dm_size) + 1;
densityMatrix(dm_index,dm_index) = 1;
iState = randStateIndex(iave); % iState = floor(iave/dm_size)+1;
else
densityMatrix = [];
% density matrix size
dm_size = prod(state_multiplicity(thisCluster));
iState = randStateIndex(iave);
end
% Adjust tensors for external fields.
if ~isempty(ClusterComplement)
if useInterlacedClusters(Method_order,clusterSize)
tensors = getInterlacedTensors(tensors0,zeroIndex, ...
thisCluster,ClusterComplement,iState,...
Nuclei_g,state_multiplicity,Nuclei_Coordinates, muN, mu0, hbar);
else
tensors = getMeanMagneticFieldTensors(tensors,...
RandomSpinVector, thisCluster,ClusterComplement,iState,...
Nuclei_g,Nuclei_Coordinates, muN, mu0, hbar);
end
%tensors = tensors0; % TEMPORARY
else
tensors = tensors0;
if iave >1
continue;
end
end
[H_alpha,H_beta] = ...
assembleHamiltonian(state_multiplicity(thisCluster),...
tensors,SpinOp,qtensors,SpinXiXjOp,...
theory,isMethylHydron(thisCluster),methylMethod1, ...
methylTunnelingSplitting(thisCluster,thisCluster));
elseif useMultipleBathStates
iState = iave;
tensors = getMeanMagneticFieldTensors(tensors,RandomSpinVector,...
thisCluster,ClusterComplement,iState,...
Nuclei_g,Nuclei_Coordinates, muN, mu0, hbar);
[H_alpha,H_beta] = ...
assembleHamiltonian(state_multiplicity(thisCluster),...
tensors,SpinOp,qtensors,SpinXiXjOp,theory,isMethylHydron(thisCluster),...
methylMethod1,methylTunnelingSplitting(thisCluster,thisCluster));
if useThermalEnsemble
densityMatrix = [];
else
densityMatrix = getDensityMatrix(ZeemanStates(iave,thisCluster),...
state_multiplicity(thisCluster),thisCluster);
end
else
if useThermalEnsemble
densityMatrix = [];
else
densityMatrix = getDensityMatrix(ZeemanStates(iave,thisCluster),...
state_multiplicity(thisCluster),thisCluster);
end
end
% Add mean field term to the Hamiltonian.
Halpha = H_alpha + H_alphaMF;
Hbeta = H_beta + H_betaMF;
% Determine which block in the rotational Hamiltonian
% this configuration is assigned to.
hRange = (iPerm-1)*SpinSpaceDim+(1:SpinSpaceDim);
% Add configuration Hamiltonian to the full Hamiltonian.
Hb(hRange,hRange) = Hbeta;
Ha(hRange,hRange) = Halpha;
if ~isempty(H_alpha_TR)
Hb_TR(hRange,hRange) = H_beta_TR;
Ha_TR(hRange,hRange) = H_alpha_TR;
else
Hb_TR = [];
Ha_TR = [];
end
end
% Calculate cluster coherence.
switch clusterSize
case 1
Coherences_1(iCluster,:) = Coherences_1(iCluster,:) + ...
propagate(total_time,nPoints,dt,dt2,N1,Hb,Ha,Hb_TR,Ha_TR, ...
EXPERIMENT,densityMatrix, useThermalEnsemble, betaT);
case 2
Coherences_2(iCluster,:) = Coherences_2(iCluster,:) + ...
propagate(total_time,nPoints,dt,dt2,N1,Hb,Ha,Hb_TR,Ha_TR, ...
EXPERIMENT,densityMatrix, useThermalEnsemble, betaT);
case 3
Coherences_3(iCluster,:) = Coherences_3(iCluster,:) + ...
propagate(total_time,nPoints,dt,dt2,N1,Hb,Ha,Hb_TR,Ha_TR, ...
EXPERIMENT,densityMatrix, useThermalEnsemble, betaT);
case 4
Coherences_4(iCluster,:) = Coherences_4(iCluster,:) + ...
propagate(total_time,nPoints,dt,dt2,N1,Hb,Ha,Hb_TR,Ha_TR, ...
EXPERIMENT,densityMatrix, useThermalEnsemble, betaT);
case 5
Coherences_5(iCluster,:) = Coherences_5(iCluster,:) + ...
propagate(total_time,nPoints,dt,dt2,N1,Hb,Ha,Hb_TR,Ha_TR, ...
EXPERIMENT,densityMatrix, useThermalEnsemble, betaT);
case 6
Coherences_6(iCluster,:) = Coherences_6(iCluster,:) + ...
propagate(total_time,nPoints,dt,dt2,N1,Hb,Ha,Hb_TR,Ha_TR, ...
EXPERIMENT,densityMatrix, useThermalEnsemble, betaT);
end
end
end
end
for clusterSize = 1:Method_order
switch clusterSize
case 1
unusedClusters = find(Coherences_1(:,1)==0)';
Coherences_1(unusedClusters,:) = 1;
case 2
unusedClusters = find(Coherences_2(:,1)==0)';
Coherences_2(unusedClusters,:) = 1;
case 3
unusedClusters = find(Coherences_3(:,1)==0)';
Coherences_3(unusedClusters,:) = 1;
case 4
unusedClusters = find(Coherences_4(:,1)==0)';
Coherences_4(unusedClusters,:) = 1;
case 5
unusedClusters = find(Coherences_5(:,1)==0)';
Coherences_5(unusedClusters,:) = 1;
case 6
unusedClusters = find(Coherences_6(:,1)==0)';
Coherences_6(unusedClusters,:) = 1;
end
end
% Calculate signal
%-------------------------------------------------------------------------------
if doHDCCE
Signal = ClusterArray;
AuxiliarySignal_1 = Coherences_1;
AuxiliarySignal_2 = Coherences_2;
AuxiliarySignal_3 = SubclusterIndices_2;
AuxiliarySignal_4 = dimensionality;
Signals = [];
return
end
if doOffsetCCE
[Signals, AuxiliarySignal_1,AuxiliarySignal_2,...
AuxiliarySignal_3,AuxiliarySignal_4] ...
= doOffsetClusterCorrelationExpansion_forHDCCE(Coherences_1,Coherences_2,...
Coherences_3,Coherences_4,Coherences_5,Coherences_6,ClusterArray, ...
SubclusterIndices_2,SubclusterIndices_3,SubclusterIndices_4,...
SubclusterIndices_5,SubclusterIndices_6,...
nPoints,dimensionality, Method_order,numberClusters, Nuclei_Abundance);
Signal = Signals(Method_order,:);
return
end
[Signals, AuxiliarySignal_1,AuxiliarySignal_2,...
AuxiliarySignal_3,AuxiliarySignal_4] ...
= doClusterCorrelationExpansion_forHDCCE(Coherences_1,Coherences_2,...
Coherences_3,Coherences_4,Coherences_5,Coherences_6,ClusterArray, ...
SubclusterIndices_2,SubclusterIndices_3,SubclusterIndices_4,...
SubclusterIndices_5,SubclusterIndices_6,...
nPoints,dimensionality, Method_order,numberClusters, Nuclei_Abundance);
Signal = Signals(Method_order,:);
%-------------------------------------------------------------------------------
% gpu code end
%
% At this point the data should be transfered back to the cpu.
%
% if useGPU
% transferDataToCPU();
% end
%
% gpu code end
%-------------------------------------------------------------------------------
% return; % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
end
% ==============================================================================
% Generate Density Matrix
% ==============================================================================
function densityMatrix = getDensityMatrix(states,multiplicities,Cluster)
clustersize = length(Cluster);
prod_state = states(clustersize);
offset_factor = multiplicities(clustersize);
for iSpin = clustersize-1:-1:1
prod_state = prod_state + offset_factor*(states(iSpin) - 1);
offset_factor = offset_factor*multiplicities(iSpin);
end
densityMatrix = zeros(offset_factor);
densityMatrix(prod_state,prod_state) = 1;
end
% ========================================================================
% Propagate Function
% ========================================================================
function Signal = propagate(total_time,nPoints,dt,dt2,N1,...
Hamiltonian_beta,Hamiltonian_alpha,...
Hamiltonian_beta_TR,Hamiltonian_alpha_TR ,...
EXPERIMENT, densityMatrix, useThermalEnsemble, betaT)
% ENUM
FID = 1; HAHN = 2; CPMG = 3; CPMG_CONST = 4; CPMG_2D = 5; HAHN_TR = 6;
% Ensure subHamiltinians are Hermitian.
Hamiltonian_beta =(Hamiltonian_beta+Hamiltonian_beta')/2;
Hamiltonian_alpha =(Hamiltonian_alpha+Hamiltonian_alpha')/2;
% Get density matrix.
if useThermalEnsemble
DensityMatrix = propagator_eig(...
(Hamiltonian_alpha+Hamiltonian_beta)/2,-1i*betaT,0);
else
DensityMatrix = densityMatrix;
end
% Vectorize density matrix.
vecDensityMatrixT = reshape(DensityMatrix.',1,[])/trace(DensityMatrix);
% Generate propagators for time element dt.
[dU_beta, dU_beta2] = propagator_eig(Hamiltonian_beta,dt,dt2);
[dU_alpha, dU_alpha2] = propagator_eig(Hamiltonian_alpha,dt, dt2);
% Find the dimensionality of the Hilbert space.
nStates = length(Hamiltonian_beta);
U_beta = eye(nStates);
U_alpha = eye(nStates);
doTR = ~isempty(Hamiltonian_beta_TR);
if doTR
% Ensure subHamiltinians are Hermitian.
Hamiltonian_beta_TR =(Hamiltonian_beta_TR+Hamiltonian_beta_TR')/2;
Hamiltonian_alpha_TR =(Hamiltonian_alpha_TR+Hamiltonian_alpha_TR')/2;
% Generate propagators for time element dt.
[dU_beta_TR, dU_beta2_TR] = propagator_eig(Hamiltonian_beta_TR,dt,dt2);
[dU_alpha_TR, dU_alpha2_TR] = propagator_eig(Hamiltonian_alpha_TR,dt, dt2);
end
% Get second experimental dimesion propagators.
if EXPERIMENT==CPMG
U_beta_2 = eye(nStates);
U_alpha_2 = eye(nStates);
end
% Get second experimental dimesion propagators.
if EXPERIMENT == CPMG_CONST
U_beta_2 = propagator_eig(Hamiltonian_beta,total_time);
U_alpha_2 = propagator_eig(Hamiltonian_alpha,total_time);
end
% Initialize signal.
v= ones(1 ,nPoints);
% Loop over time points.
for iTime = 1:nPoints
% Find the correct experiment
switch EXPERIMENT
case FID
% Generate time dependent detection operator.
U_ = U_beta'*U_alpha;
v(iTime) = vecDensityMatrixT*U_(:);
case HAHN
% Generate time dependent detection operator.
U_ = U_beta'*U_alpha'*U_beta*U_alpha;
v(iTime) = vecDensityMatrixT*U_(:);
case CPMG
% Generate time dependent detection operator.
U_ = U_alpha'*U_beta' * ...
(U_beta'*U_alpha' * U_beta*U_alpha) * U_alpha*U_beta;
v(iTime) = vecDensityMatrixT*U_(:);
case CPMG_CONST
% THIS NEEDS TO BE UPDATED TO USE dt2 WHEN iTime > N1.
[U_beta, U_beta_2] = propagator_eig(...
Hamiltonian_beta,(iTime-1)*dt, total_time/4-(iTime-1)*dt);
[U_alpha, U_alpha_2] = propagator_eig(...
Hamiltonian_alpha,(iTime-1)*dt, total_time/4-(iTime-1)*dt);
% Generate time dependent detection operator.
U_ = U_alpha_2'*U_beta_2' * ...
(U_beta'*U_alpha' * U_beta*U_alpha) * U_alpha_2*U_beta_2;
v(iTime) = vecDensityMatrixT*U_(:);
case CPMG_2D
U_beta_2 = eye(nStates);
U_alpha_2 = eye(nStates);
% Loop over second experimental dimension.
for jTime = 1:iTime %nPoints
% Generate time dependent detection operator.
U_ = U_alpha_2'*U_beta_2' * ...
(U_beta'*U_alpha' * U_beta*U_alpha) * U_alpha_2*U_beta_2;
v(iTime,jTime) = vecDensityMatrixT*U_(:);
v(jTime,iTime) = v(iTime,jTime)';
% Increment propagator.
if jTime < N1
U_beta_2 = dU_beta*U_beta_2;
U_alpha_2 = dU_alpha*U_alpha_2;
else
U_beta_2 = dU_beta2*U_beta_2;
U_alpha_2 = dU_alpha2*U_alpha_2;
end
end
case HAHN_TR
% Generate time dependent detection operator.
U_ = U_beta'*U_beta_TR'*U_alpha'*U_alpha_TR' ...
* U_beta_TR*U_beta*U_alpha_TR*U_alpha;
v(iTime) = vecDensityMatrixT*U_(:);
end
% Increment propagator.
if iTime< N1
U_beta = dU_beta*U_beta;
U_alpha = dU_alpha*U_alpha;
else
U_beta = dU_beta2*U_beta;
U_alpha = dU_alpha2*U_alpha;
end
if doTR
% Increment propagator.
if iTime< N1
U_beta_TR = dU_beta_TR*U_beta_TR;
U_alpha_TR = dU_alpha_TR*U_alpha_TR;
else
U_beta_TR = dU_beta2_TR*U_beta_TR;
U_alpha_TR = dU_alpha2_TR*U_alpha_TR;
end
end
end
% Return signal as a vector.
if EXPERIMENT == CPMG_2D
Signal = reshape(v.',1,[]);
else
Signal = v;
end
if any(abs(v) - 1 > 1e-9)
error('Coherence error: coherence cannot be larger than 1.');
end
end
% ========================================================================
% Subcluster Function
% ========================================================================
function Indices = findSubclusters(Clusters,clusterSize,iCluster,...
reference_clusterSize)
% Given the ith cluster of size clusterSize,
% Indices(jCluster,subCluster_size) = jth cluster of sizesubCluster_size
% that is a subcluster of the ith cluster of size clusterSize
%
% SubclusterIndices_clusterSize(jCluster,subCluster_size, iCluster) =
% the jth cluster of size subCluster_size that is a subcluster of
% the ith ccluster of size clusterSize.
% In a valid cluster, all indices must be natural numbers.
Indices = zeros(nchoosek(reference_clusterSize, ...
ceil(reference_clusterSize/2)),reference_clusterSize);
jCluster = 0;
% Each cluster is a subset of itself.
Indices(1,clusterSize)=iCluster;
if clusterSize==1
% All non-empty subsets have been found.
return;
end
% Loop over all cluster sizes up to clusterSize.
for index = 1:clusterSize
% CluserArray(iCluster,:,clusterSize) = nuclear indices.
Cluster = Clusters(iCluster, 1:clusterSize ,clusterSize);
% Cluster(Cluster==0) = [];
% Remove one element labeled by index from
% Clusters{clusterSize}(iCluster:) to get a sub-cluster with one less element.
SubCluster = [Cluster(1:index-1), Cluster(index+1:end)];
% Search for a valid cluster that equals the subcluster.
Search = Clusters(:, 1:clusterSize -1, clusterSize -1)==SubCluster;
% Locate where the subcluster is.
subclusterIndex = find( all(Search,2) );
% Skip if there is no subclustere.
if isempty(subclusterIndex)
continue;
end
jCluster = jCluster + 1;
Indices(jCluster,clusterSize -1) = subclusterIndex;
if clusterSize > 2
% Given the ith cluster of size clusterSize,
% Indices(jCluster,subCluster_size) = jth cluster of sizesubCluster_size
% that is a subcluster of the ith cluster of size clusterSize
% Recursively find smaller subclusters
Subindices = findSubclusters(Clusters,clusterSize -1, ...
subclusterIndex,reference_clusterSize);
% Assign subcluster indices.
Indices(:,1:clusterSize-1) = Subindices(:,1:clusterSize-1);
end
end
end
% ========================================================================
% Calculate propagator using diagonalization
% ========================================================================
function [U1,U2] = propagator_eig(Ham,t1,t2)
%Ham = (Ham+Ham')/2; % "hermitianize" Hamiltonian
[EigenVectors, EigenValues] = eig(Ham);
Udiag1 = exp(-2i*pi*diag(EigenValues)*t1);
Udiag2 = exp(-2i*pi*diag(EigenValues)*t2);
U1 = EigenVectors*diag(Udiag1)*EigenVectors';
U2 = EigenVectors*diag(Udiag2)*EigenVectors';
end
% ========================================================================
% New Function TEMPORARY
% ========================================================================
function Indices = findSubclusters0(Clusters,clusterSize,iCluster)
% Indices{size} = list of all jCluster such that Clusters{size}(jCluster,:)
% is a subcluster of Clusters{clusterSize}(iCluster,:)
% In a valid cluster, all indices must be natural numbers.
if Clusters{clusterSize}(iCluster,1) == 0
Indices = [];
return;
end
% Each cluster is a subset of itself.
Indices{clusterSize}=iCluster;
if clusterSize==1
% All non-empty subsets have been found.
return;
end
% Loop over all cluster sizes up to clusterSize.
for index = 1:clusterSize
% Remove one element labeled by index from Clusters{clusterSize}(iCluster:)
% to get a sub-cluster with one less element.
SubCluster = [Clusters{clusterSize}(iCluster,1:index-1), ...
Clusters{clusterSize}(iCluster,index+1:end)];
% Search for a valid cluster that equals the subcluster.
Search = Clusters{clusterSize -1}==SubCluster;
for ii = 2:size(Search,2)
Search(:,1) = Search(:,1).*Search(:,ii);
end
subclusterIndex = find(Search(:,1)==1);
if isempty(subclusterIndex)
continue;
end
Indices{clusterSize -1} = [Indices{clusterSize -1} , subclusterIndex];
if clusterSize > 2
Subindices = findSubclusters0(Clusters,clusterSize -1,subclusterIndex);
for isize = (clusterSize-2):-1:1
if ~isempty(Subindices{isize})
Indices{isize} = [Indices{isize},Subindices{isize}];
end
end
end
end
for isize = 1:clusterSize
Indices{isize} = unique(Indices{isize});
end
end
% ==============================================================================
% Generate cyclic permutation
% ==============================================================================
function cycPerm = getCyclicPermutations(boolCluster)
N = length(boolCluster);
numberRotors = sum(boolCluster);
% no methyls
if numberRotors == 0
cycPerm = 1:N;
return;
end
% 1-Clusters
if N ==1
cycPerm = [1,2,3;2,3,1;3,1,2];
return;
end
% 2-Clusters
if N == 2
if numberRotors == 2
cycPerm = [1,2,3,4,5,6;...
2,3,1,4,5,6; ...
3,1,2,4,5,6; ...
1,2,3,5,6,4;...
2,3,1,5,6,4; ...
3,1,2,5,6,4; ...
1,2,3,6,4,5;...
2,3,1,6,4,5; ...
3,1,2,6,4,5];
elseif boolCluster(1)