Rotation Operator: Interaction between Two-Level Atom in {|g>, |e>} Basis

[zDrajCa]

Hi, I'm working on the interaction between a two level atom (|g>, |e>)
In my exercise we have to use the rotation operator :

R(t)=exp[i(σ_z+1)ωt/2]

with σ_z the pauli matrix which is in the {|g>,|e>} basis :
(1 0)
(0 -1)

If i want to represent my rotation operator in the {|g>,|e>} basis. Then i can do:
σ_z +1 = (1 is the identity matrix)
( 2 0)
( 0 0)

Do my operator is :
(exp(iwt) 0 )
( 0 exp(0) )

Thanks for your answers.

 

[scottdave]

I'm not familiar with taking exponential of a matrix. But if you had exp(0) in the lower right position, would it also be exp(0) in the upper right and lower left?

 

[zDrajCa]

i have read that if a matrix was diagonal (my case right) ,then the exponential of the matrix is the exponential of his diagonal term

 

[TeethWhitener]

I'm not familiar with taking exponential of a matrix. But if you had exp(0) in the lower right position, would it also be exp(0) in the upper right and lower left?

https://en.wikipedia.org/wiki/Matrix_exponential
It can be expanded in a Taylor series.

the exponential of the matrix is the exponential of his diagonal term

Yes, this is right.