New method for directly computing reduced density matrices

We demonstrate the power of a first principle-based and practicable method that allows for the perturbative computation of reduced density matrix elements of an open quantum system without making use of any master equations. The approach is based on techniques from nonequilibrium quantum field theory such as thermo field dynamics, the Schwinger-Keldsyh formalism, and the Feynman-Vernon influence functional. It does not require the Markov approximation and is essentially a Lehmann-Szymanzik-Zimmermann-like reduction. To illustrate this method, we consider a real scalar field as an open quantum system interacting with an environment comprising another real scalar field. We give a general formula that allows for the perturbative computation of density matrix elements for any number of particles in a momentum basis. Finally, we consider a simple toy model and use this formula to obtain expressions for some of the system’s reduced density matrix elements.

Figure 1

Schematic depiction of the closed-time path for a density matrix $\rho$ evolving from an initial time 0 to a final time $t$ and backwards.

Figure 2

Diagrammatic representation of the terms contributing to $\aleph (t)$, as given in Eq. (73); solid/dashed lines represent $\varphi /\chi$ propagators.

Figure 3

Diagrammatic representation of the terms contributing to ${\rho }_{0;0}(t)$ if only ${\rho }_{1;1}(0)$ was initially nonvanishing, as given in Eq. (75); a crossed box depicts an insertion of an initial density matrix element, here of the matrix ${\rho }_{1;1}(0)$.

Figure 4

Diagrammatic representation of the terms contributing to ${\rho }_{1;0}(t)$ if (a) ${\rho }_{0;0}(0)$; (b), (c) ${\rho }_{1;0}(0)$; (d), (e) ${\rho }_{0;1}(0)$; and (f) ${\rho }_{1;1}(0)$ were initially nonvanishing, as given in Eq. (78).

Figure 5

Diagrammatic representation of the terms contributing to ${\rho }_{1;1}(t)$ if only ${\rho }_{1;1}(0)$ was initially nonvanishing, as given in Eq. (81).