Separation of scales: dynamical approximations for composite quantum systems*

Abstract

We consider composite quantum-dynamical systems that can be partitioned into weakly interacting subsystems, similar to system–bath type situations. Using a factorized wave function ansatz, we mathematically characterize dynamical scale separation. Specifically, we investigate a coupling régime that is partially flat, i.e. slowly varying with respect to one set of variables, for example, those of the bath. Further, we study the situation where one of the sets of variables is semiclassically scaled and derive a quantum–classical formulation. In both situations, we propose two schemes of dimension reduction: one based on Taylor expansion (collocation) and the other one based on partial averaging (mean-field). We analyze the error for the wave function and for the action of observables, obtaining comparable estimates for both approaches. The present study is the first step towards a general analysis of scale separation in the context of tensorized wavefunction representations.

Dynamical approximations for composite quantum systems: Assessment of error estimates for a separable ansatz

“…In line with our previous work [3], we consider a two-dimensional model system which is of system-bath type and exhibits anharmonicities both within the system subspace and in the system-bath coupling. Specifically, a double-well potential is chosen in the system subspace, which is coupled to a harmonic bath coordinate via a cubic (quadratic times linear) coupling term.…”

“…Specifically, a double-well potential is chosen in the system subspace, which is coupled to a harmonic bath coordinate via a cubic (quadratic times linear) coupling term. As pointed out in our previous analysis [3], a cubic coupling is a non-trivial case which is relevant for the description of vibrational dephasing [11,9] and Fermi resonances [2] in a molecular physics context.…”

“…x, y), and differs from the true Hamiltonian only with respect to the effective coupling potential (3) w eff u (t, x, y) = w φ (t, x) + w χ (t, y) − w u (t), so the above effective Hamiltonians can be rewritten as…”

“…Despite the importance of the TDH ansatz, an explicit error analysis of this approach is not often reported in the literature. In a recent formal paper [3], we therefore presented error estimates for the time propagation of composite quantum systems within the TDH approximation. We also compared different types of approximate product wavefunctions, i.e., based on Taylor expansion (collocation) or else on the TDH mean-field approach, and we further considered a semiclassical approximation within a quantumclassical type treatment.…”

“…As in Ref. [3], a cubic coupling is considered (i.e., linear in the tunneling coordinate and quadratic in the harmonic coordinate). Numerical TDH calculations for different parameter regimes are compared with correlated MCTDH calculations that can be considered as numerically exact for the present system.…”