Response function

The ResponseFunction object in Quanty defines (linear) response functions. For Hamiltonian $H$, ground-state $| \psi_0 \rangle$ and a list of transition operators $T_i$ with $i \in [1,n]$ we define the response function $G(\omega,\Gamma)$ as giving an $n$ by $n$ matrix for each value of $\omega$ and $\Gamma$. The elements of this matrix are given as $$G_{i,j}(\omega,\Gamma) = \left\langle \psi_0 \middle| T^{\dagger}_i \frac{1}{\omega - H + \mathrm{i} \Gamma/2 + E_0} T^{\phantom{\dagger}}_j \middle| \psi_0 \right\rangle,$$ with $E_0 = \left\langle \psi_0 \middle| H \middle| \psi_0 \right\rangle$. One can calculate response functions using the Quanty function CreateSpectra. This function returns two objects. At the first position a spectra object that contains the intensity for given values of $\omega$ and one specific value of $\Gamma$ on a grid. At the second position CreateSpectra returns a ResponseFunction object.

ResponseFunctions are objects that can be evaluated at any frequency or imaginary onset. For example:

Example.Quanty

H   = Matrix.ToOperator( Matrix.Diagonal({1,2,3,4,5}) )
psi = NewWavefunction(5,0,{{"00000",1}})
T = {}
for i=0,4 do
  T[i+1] = NewOperator(5,0,{{i,1}})
end
S, G = CreateSpectra(H,T,psi)
omega = 1.1
gamma = 0.01
print(G[1](omega,gamma))

returns

(9.9750623441396 - 0.49875311720698 I)

i.e. the value of the response function for the first transition operator defined at $\omega=1.1$ and $\Gamma=0.01$.

Besides single complex valued functions we can generate a response function that returns a matrix for each value of $\omega$. For example

Example.Quanty

H   = Matrix.ToOperator( Matrix.Diagonal({1,2,3,4,5}) )
psi = NewWavefunction(5,0,{{"00000",1}})
T = {}
for i=0,4 do
  T[i+1] = NewOperator(5,0,{{i,1}})
end
S, G = CreateSpectra(H,T,psi,{{"Tensor",true}})
omega = 1.1
gamma = 0.01
print(G(omega,gamma))

returns

{ { (9.9750623441396 - 0.49875311720698 I) , 0 , 0 , 0 , 0 } , 
  { 0 , (-1.1110768186167 - 0.0061726489923151 I) , 0 , 0 , 0 } , 
  { 0 , 0 , (-0.52631214465274 - 0.0013850319596125 I) , 0 , 0 } , 
  { 0 , 0 , 0 , (-0.34482656115767 - 0.00059452855372011 I) , 0 } , 
  { 0 , 0 , 0 , 0 , (-0.25640983496082 - 0.00032873055764208 I) } }

i.e. a 5 by 5 matrix with matrix elements $G_{i,j}(\omega,\gamma)$. (In this case the response function is diagonal as the Hamiltonian is diagonal.)