VAMPIRE

Auto-Correlation

Z-Gradient

1. SHG

   \(\frac{dS_{mk}}{dE_n^*} = 0\)

   \(\frac{dS_{mk}}{dE_n} = D_{kn}\left(A_{mk}+A_{mk}'+E_k\cdot e^{-i\tau_m\omega_n}\cdot(e^{-i\tau_0\omega_n} + e^{i\phi_n})\right)\)

2. THG

   \(\frac{dS_{mk}}{dE_n^*} = 0\)

   \(\frac{dS_{mk}}{dE_n} = D_{kn}(A_{mk}+A_{mk}')\cdot\left(A_{mk}+A_{mk}'+2E_k\cdot e^{-i\tau_m\omega_n}\cdot(e^{-i\tau_0\omega_n} + e^{i\phi_n})\right)\)

3. PG

   \(\frac{dS_{mk}}{dE_n^*} = D_{nk}E_k(A_{mk}+A_{mk}')\cdot e^{i\tau_m\omega_n}(e^{i\tau_0\omega_n}+e^{-i\phi_n})\)

   \(\frac{dS_{mk}}{dE_n} = D_{kn}(A_{mk}+A_{mk}')^*\cdot\left(A_{mk}+A_{mk}'+E_k e^{-i\tau_m\omega_n}(e^{-i\tau_0\omega_n}+e^{i\phi_n})\right)\)

4. SD

   \(\frac{dS_{mk}}{dE_n^*} = 2 D_{nk}E_k (A_{mk}+A_{mk}')^*\cdot e^{i\tau_m\omega_n}(e^{i\tau_0\omega_n}+e^{-i\phi_n})\)

   \(\frac{dS_{mk}}{dE_n} = D_{kn}\left((A_{mk}+A_{mk}')^*\right)^2\)

5. nth-HG

   \(\frac{dS_{mk}}{dE_n^*} = 0\)

   \(\frac{dS_{mk}}{dE_n} = D_{kn}(A_{mk}+A_{mk}')^{n-2}\cdot\left(A_{mk}+A_{mk}'+(n-1)E_k\cdot e^{-i\tau_m\omega_n}\cdot(e^{-i\tau_0\omega_n} + e^{i\phi_n})\right)\)

Z-Pseudo-Hessian

1. SHG

   \(V_{zz}=0\)

2. THG

   \(V_{zz}=0\)

3. PG

   \(\frac{d}{dE_p}\left(\frac{dS_{mk}}{dE_n}\right)^* = D_{nk}D_{kp}\cdot\left((E_k'+A_{mk}'')^*\cdot e^{-i\tau_m\omega_p}(e^{-i\tau_0\omega_p} + e^{i\phi_p}) + E_k^*\cdot e^{i\tau_m\omega_n}(e^{-i\tau_0\omega_n} + e^{i\phi_n})^*e^{-i\tau_m\omega_p}(e^{-i\tau_0\omega_p} + e^{i\phi_p})\right)\)

   \(\frac{d}{dE_p^*}\left(\frac{dS_{mk}}{dE_n^*}\right)^* = D_{pk}D_{kn}\cdot\left((E_k'+A_{mk}'')^*\cdot e^{-i\tau_m\omega_n}(e^{-i\tau_0\omega_n} + e^{i\phi_n}) + E_k^*\cdot e^{-i\tau_m\omega_n}(e^{-i\tau_0\omega_n} + e^{i\phi_n}) e^{i\tau_m\omega_p}(e^{-i\tau_0\omega_p} + e^{i\phi_p})^*\right)\)

4. SD

   \(\frac{d}{dE_p}\left(\frac{dS_{mk}}{dE_n}\right)^* = D_{nk}D_{kp}\cdot 2(E_k'+A_{mk}'')e^{i\tau_m\omega_p}(e^{-i\tau_0\omega_p} + e^{i\phi_p})^*\)

   \(\frac{d}{dE_p^*}\left(\frac{dS_{mk}}{dE_n^*}\right)^* = D_{pk}D_{kn}\cdot 2(E_k'+A_{mk}'')e^{-i\tau_m\omega_n}(e^{-i\tau_0\omega_n} + e^{i\phi_n})\)

5. nth-HG

   \(V_{zz} = 0\)

Cross-Correlation

Z-Gradient (with respect to pulse)

Same for all nonlinear methods

\(\frac{dS_{mk}}{dE_n^*} = 0\)

\(\frac{dS_{mk}}{dE_n} = D_{kn}G_{mk}\)

Z-Gradient (with respect to gate-pulse)

1. SHG

   \(\frac{dS_{mk}}{dA_n^*} = 0\)

   \(\frac{dS_{mk}}{dA_n} = D_{kn}E_k e^{-i\tau_m\omega_n}(e^{-i\tau_0\omega_n}+e^{i\phi_n})\)

2. THG

   \(\frac{dS_{mk}}{dA_n^*} = 0\)

   \(\frac{dS_{mk}}{dA_n} = 2 D_{kn}E_k (A_{mk}+A_{mk}')\cdot e^{-i\tau_m\omega_n}(e^{-i\tau_0\omega_n}+e^{i\phi_n})\)

3. PG

   \(\frac{dS_{mk}}{dA_n^*} = D_{nk}E_k(A_{mk}+A_{mk}')\cdot e^{i\tau_m\omega_n}(e^{i\tau_0\omega_n}+e^{-i\phi_n})\)

   \(\frac{dS_{mk}}{dA_n} = D_{kn}E_k(A_{mk}+A_{mk}')^*\cdot e^{-i\tau_m\omega_n}(e^{-i\tau_0\omega_n}+e^{i\phi_n})\)

4. SD

   \(\frac{dS_{mk}}{dA_n^*} = 2 D_{nk}E_k (A_{mk}+A_{mk}')^*\cdot e^{i\tau_m\omega_n}(e^{i\tau_0\omega_n}+e^{-i\phi_n})\)

   \(\frac{dS_{mk}}{dA_n} = 0\)

5. nth-HG

   \(\frac{dS_{mk}}{dE_n^*} = 0\)

   \(\frac{dS_{mk}}{dE_n} = (n-1) D_{kn}E_k (A_{mk}+A_{mk}')^{n-2}\cdot e^{-i\tau_m\omega_n}(e^{-i\tau_0\omega_n}+e^{i\phi_n})\)

Z-Pseudo-Hessian (with respect to pulse)

Same for all nonlinear methods

\(V_{zz} = 0\)

Z-Pseudo-Hessian (with respect to gate-pulse)

1. SHG

   \(V_{zz}=0\)

2. THG

   \(V_{zz}=0\)

3. PG

   \(\frac{d}{dA_p}\left(\frac{dS_{mk}}{dA_n}\right)^* = D_{nk}D_{kp}\cdot e^{i\tau_m\omega_n}(e^{-i\tau_0\omega_n} + e^{i\phi_n})^* e^{-i\tau_m\omega_p}(e^{-i\tau_0\omega_p} + e^{i\phi_p})\cdot E_k^*\)

   \(\frac{d}{dA_p^*}\left(\frac{dS_{mk}}{dA_n^*}\right)^* = D_{kn}D_{pk}\cdot e^{-i\tau_m\omega_n}(e^{-i\tau_0\omega_n} + e^{i\phi_n})e^{i\tau_m\omega_p}(e^{-i\tau_0\omega_p} + e^{i\phi_p})^*\cdot E_k^*\)

4. SD

   \(V_{zz}=0\)

5. nth-HG

   \(V_{zz} = 0\)