Simulating the dynamics of open quantum systems is one of the most important and challenging problems in physics and chemistry [1,2]. The objective here is to develop an efficient method for computing the quantum dynamics of a set of relatively small number of ''interesting'' states, belonging to subspace Q, that are coupled to a much larger set of ''less interesting'' states, belonging to a (pre-diagonalized) subspace P. Such an approach is necessary because a full dimensionality treatment incorporating all states in Q and P is in general not feasible due to the formidable size of P.
In the past, the effect of the subspace P on Q was introduced in limited ways: by considering only short
time dynamics; by considering only a particular range of Q and P coupling strengths; or by restricting the choice of initial populations of Q and/or P [1,2]. Though it is possible to formulate the problem in general as a system of integro-differential equations (SIDEs), these equations suffer from poor numerical stability, with their numerical solution usually demanding vast computational resources.
We have recently developed a novel approach for coarse-grained open system quantum dynamics : The part Q, with Hamiltonian H_Q, of the system of interest, is defined; the rest part of the system P is described by the Hamiltonian H_P, and the time-independent interaction between Q and P is given by V. The eigenstates ö_k (k=1,..,m_Q) of H_Q and the eigenstates ÷_l (l=1,..,m_P) of H_P are obtained. A SIDEs of m_Q iequations is derived for the b_k(t) coefficients in the wave function expansion
Ø(t)=Ó_k b_k(t) ö_k + Ó_l c_l(t) ÷_l in Q and P. The effect of P on Q is included by the sum of M < m_P complex exponential terms, which approximate, exactly in the limit M-> m_P, the complex dynamical correlation functions in P. By further introducing M non-physical variables, so called ''effective modes'', the SIDE for the m_Q coefficients b_k(t), is transformed into a system of M(m_Q+1) ordinary differential equations, which can be readily solved by a single diagonalization of a general complex matrix. This approach is independent on any assumption on the spectrum of H_P, and can be combined with first-principle methods, for obtaining realistic data for defining the H_Q, H_P, and V terms. Most importantly, this approach is very computationally efficient, which is crucial for quantum dynamics studies on the nanoscale.
 H.-P. Breuer and F. Petrucione, The Theory of Open Quantum Systems, Clarendon, Oxford, 2006.
 U. Weiss, Quantum Dissipative Systems, World Scientific, Singapore, 1993.
 I. Thanopulos, P. Brumer, and M. Shapiro, J. Chem. Phys. 129, 194104 (2008).