User:Fanchen Kong/sandbox

Multidimensional Spectroscopy
main article: Stochastic Liouville, Langevin, Fokker–Planck, and Master Equation Approaches to Quantum Dissipative Systems

In order to calculate a third-order two-dimensional (2D) spectra for a two-level system, the quantum Fokker–Planck equation with low-temperature correction terms(QFP-LTC) equation for the reduce density is given by
 * $$\frac{\partial \hat{\rho}(\mathbf{j};t)}{\partial t}=-i\hat{\mathcal{L}}s\hat{\rho}\left( \mathbf{j};t \right)-\left[ \sum_{k=0}^K j_k\nu_k + \hat{\Xi}\right]\hat{\rho}\left( \mathbf{j};t \right)-\sum_{k=0}^K \hat{\Phi}\hat{\rho}\left( \mathbf{j}_{k+};t \right)-\sum_{k=0}^K j_k\nu_k\hat{\Theta}_k\hat{\rho}\left( \mathbf{j}_{k-};t \right)$$

Define $${\nu}_0 \equiv \gamma $$ and introduce the (K+1)-dimensional vectors consisting of nonnegative integers, $$\mathbf{j}\equiv\left( j_0,j_1,j_2\ldots ,j_K \right)$$ and $$\mathbf{j}_{k\pm }\equiv\left( j_0,\ldots ,j_{k}\pm1,\ldots j_K \right)$$

The bath-induced relaxation operators are defined by

$$\hat{\Phi}\equiv i\hat{V}^\times $$

$$\hat{\Theta}_0\equiv i\frac{\zeta }{\beta \hbar \omega_0}\left[ z\cot z\hat{V}^\times -iz\hat{V}^\circ \right]$$

$$\hat{\Theta}_k\equiv i\frac{\zeta }{\beta \hbar \omega_0} \frac{2z^2}{\pi^2k^2-z^2}\hat{V}^\times \left( k\geqslant 1 \right)$$

and

$$\hat{\Xi}\equiv \sum_{k=1}^K \hat{\Phi}\hat{\Psi}_k+\frac{\zeta }{\beta \hbar \omega_0}\left( 1-z\cot z \right)\hat{V}^\times \hat{V}^\times $$

with $$z \equiv \frac{\beta \hbar \gamma}{2}$$

The third-order nonlinear response function is expressed as

$$R^{\left(3\right)} \equiv \left( \frac{i}{\hbar} \right)^3Tr_{tot}\left\{ \mu\left( t \right)\mu\left( t_3 \right)^\times \mu\left( t_2 \right)^\times \mu\left( t_1 \right)^\times \hat{\rho}_{tot}\left( t_0 \right)\right\}$$