Product Operators of COSY Experiment

s₀ = I_1z + I_2z
-----p/2(I_1x + I_2x)----->
s₁ = -I_1y - I_2y
-----W₁t₁I_1z + W₂t₁I_2z----->
s₂ = -I_1y cos(W₁t₁) - I_2y cos(W₂t₁) + I_1x sin(W₁t₁) + I_2x sin(W₂t₁)
-----pJ₁₂t₁(2I_1zI_2z)----->
s₃ = [-I_1y cos(W₁t₁) - I_2y cos(W₂t₁) + I_1x sin(W₁t₁) + I_2x sin(W₂t₁)] cos(pJ₁₂t₁) + [I_1xI_2z cos(W₁t₁) + I_1zI_2x cos(W₂t₁) + I_1yI_2z sin(W₁t₁) + I_1zI_2y sin(W₂t₁)] sin(pJ₁₂t₁)
-----p/2(I_1x + I_2x)----->
s₄ = [-I_1z cos(W₁t₁) - I_2z cos(W₂t₁) + I_1x sin(W₁t₁) + I_2x sin(W₂t₁)] cos(pJ₁₂t₁) + [-I_1xI_2y cos(W₁t₁) - I_1yI_2x cos(W₂t₁) - I_1zI_2y sin(W₁t₁) - I_1yI_2z sin(W₂t₁)] sin(pJ₁₂t₁)
-----during acquisition----->
s₅ = [-I_1z cos(W₁t₁) - I_2z cos(W₂t₁)] cos(pJ₁₂t₁)
+ [I_1x sin(W₁t₁) cos(W₁t₂) + I_2x sin(W₂t₁) cos(W₂t₂) + I_1y sin(W₁t₁) sin(W₁t₂) + I_2y sin(W₂t₁) sin(W₂t₂)] cos(pJ₁₂t₁) cos(pJ₁₂t₂) + [I_1yI_2z sin(W₁t₁) cos(W₁t₂) + I_1zI_2y sin(W₂t₁) cos(W₂t₂) - I_1xI_2z sin(W₁t₁) sin(W₁t₂) - I_1zI_2x sin(W₂t₁) sin(W₂t₂)] cos(pJ₁₂t₁) sin(pJ₁₂t₂)
+ {expansion of [-I_1xI_2y cos(W₁t₁) - I_1yI_2x cos(W₂t₁)] sin(pJ₁₂t₁) leads to 32 unobservable terms}
+ [-I_1zI_2y sin(W₁t₁) cos(W₂t₂) - I_1yI_2z sin(W₂t₁) cos(W₁t₂) +I_1zI_2x sin(W₁t₁) sin(W₂t₂) + I_1xI_2z sin(W₂t₁) sin(W₁t₂)] sin(pJ₁₂t₁) cos(pJ₁₂t₂) + [I_2x sin(W₁t₁) cos(W₂t₂) + I_1x sin(W₂t₁) cos(W₁t₂) + I_2y sin(W₁t₁) sin(W₂t₂) + I_2y sin(W₂t₁) sin(W₁t₂)] sin(pJ₁₂t₁) sin(pJ₁₂t₂)

Nomenclature of product operators

I_1z

longitudinal magnetization of spin 1

I_1x

in-phase x-magnetization of spin 1

I_1y

in-phase y-magnetization of spin 1

2I_1xI_2z

antiphase x-magnetization of spin 1, or more specifically, x-magnetization of spin 1 antiphase with respect to spin 2

2I_1zI_2z

longitudinal two-spin order of spins 1 and 2

2I_1xI_2x, 2I_1yI_2y, 2I_1xI_2y, 2I_1yI_2x

two-spin coherence of spins 1 and 2

1/2(2I_1xI_2x - I_1yI_2y) or 1/2(2I_1xI_2y + 2I_1yI_2x)

double quantum coherence

1/2(2I_1xI_2x + I_1yI_2y) or 1/2(2I_1xI_2y - 2I_1yI_2x)

zero quantum coherence