Kadanoff-Baym approach to bound states in open quantum systems

TL;DR

The paper develops and applies the Kadanoff–Baym framework to open quantum systems with arbitrary baths and dimensions, enabling non-Markovian, memory-dependent dynamics of bound-state formation and thermalization. By expressing the full two-time KBEs in a discrete energy basis and solving for matrix Green-function elements with second-order self-energies, it reveals how bound states form, broaden, and regenerate under bath coupling, and how decoherence shapes relaxation times. The study systematically compares full KB dynamics with diagonal and quantum-kinetic master equation approximations, assessing spectral functions, entropy evolution, and KMS-consistency as evidence of equilibration to bath temperature. It demonstrates 1D and 3D bound-state formation (including deuteron-like states) and extends to open bosonic systems, highlighting the method's potential for describing bound-state dynamics in hot, dense environments relevant to heavy-ion physics and beyond, while noting substantial computational demands and the need for adaptive, distributed HPC approaches.

Abstract

In this paper, we extend the method of Kadanoff-Baym equations for open quantum systems to arbitrary kinds of systems and heat baths, either fermionic or bosonic. This includes three spacial dimensions and different potentials for the system-bath interaction or external traps. We study the quantum-mechanical formation of bound states in one and also in three dimensions with the full Kadanoff-Baym equations and compare them to more simplified approaches with and without memory effects. An in-depth examination of the thermodynamics of open systems is performed, showing perfect equilibration of the system's degrees of freedom along with a comprehensive investigation of the influence of the heat bath on the system's wave functions. The formation time, decay time and regeneration of bound states and their dependence on the temperature and coupling strength is explored We evaluate the non-equilibrium Kadanoff-Baym equations for the system particles, assuming that interactions are elastic two-particle collisions with the heat-bath particles. Finally, we describe in detail the method used to numerically solve the corresponding spatially heterogeneous integro-differential equations for the set of one-particle Green's functions.

Figures (24)

• Figure 1: The closed-time path $C$ with the times ordered as it is the case for $S^{<}$.
• Figure 2: Self energy diagrams for the scattering process in the open quantum system. The dotted lines are bath propagators, fixed to the free thermal real-time propgators, and full lines depict self-consistently evaluated system-particle propagators BINOSI200476.
• Figure 3: The basketball-diagram (right) and the diagram, that creates the tadpole (left), as a part of the $\Phi$-functional of the 2-PI effective action. The dotted lines denote bath propagators, affixed to the free thermal real-time propagators. The full lines depict system-particle propagators, which have been self-consistently evaluated BINOSI200476.
• Figure 4: The time evolution of the pure initial state \ref{['Formation_of_bound_states_2']} (full line) vs mixed state \ref{['Formation_of_bound_states_1']} (dashed line) plotted for the relevant matrix elements. In the top for the diagonal entries and in the bottom for the off-diagonal ones.
• Figure 5: The spectral functions of the states 0 (top), 12 (middle), and 24 (bottom) at $\bar{t}= 63 \; \mathrm{fm}$ of the full Kadanoff-Baym (full line) vs the diagonal approximation (dashed line) plotted for different states. The blue vertical line marks the bare on-shell energies of $h_0$, cf. \ref{['eigenfunction']}, and the black vertical line denotes the shifted peak energy for the diagonal/full Kadanoff-Baym equations.