Thus, we shall not provide any further discussion along this line here, but one can obtain theoretical expressions for these four terms when the system–bath interaction and perturbation Hamiltonians are specified for a given molecular system of interest. 3 First-Order Transition between Population and Coherence Lastly, if the initial state is prepared on a coherence, rmn (t0 ) (for m ≠ n) at t0, and if the final state is a population, rmm, the coherence-to-population transition probability within the first-order perturbation theory is (1) TPmm ,mn (t ) = Tr [| m >< m | V1 (t , t0 ) | m >< n |].

107) 30 Two-Dimensional Optical Spectroscopy Note that the initial ket vector |m> does not undergo any quantum transition so that we need to consider the first-order perturbation-induced n → m transition. 107 can be diagrammatically represented as (1) (t ) = < | m >< m | TPmn ,mn | m >< n | > . 108) Only when the above expectation value of the population operator | m >< m | does not vanish, the transition from the coherence rmn (t0 ) to the population on the mth state becomes allowed and finite.

23 is in phase with the external field, and the coefficient is determined by the real part of the susceptibility. On the other hand, the second term, whose magnitude is determined by the imaginary (dissipative) part of the susceptibility, is out of phase with the external field. This explains why the imaginary part of the susceptibility is related to the absorption of light by matter. 2 System–bath Interaction and Line Broadening In this book, we shall consider a few model systems to provide detailed descriptions of relationships between the linear and nonlinear spectroscopic observables and molecular properties.