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 [3]: 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.

[1] H.-P. Breuer and F. Petrucione, The Theory of Open Quantum Systems, Clarendon, Oxford, 2006.
[2] U. Weiss, Quantum Dissipative Systems, World Scientific, Singapore, 1993.
[3] I. Thanopulos, P. Brumer, and M. Shapiro, J. Chem. Phys. 129, 194104 (2008).