  We study the generation of quantum entanglement between two giant atoms
coupled to a common one-dimensional waveguide. Here each giant atom interacts
with the waveguide at two separate coupling points. Within the Wigner-Weisskopf
framework for single coupling points, we obtain the time-delayed quantum master
equations governing the evolution of the two giant atoms for three different
coupling configurations: separated, braided, and nested couplings. For each
coupling configuration, we consider both the Markovian and non-Markovian
entanglement dynamics of the giant atoms, which are initially in two different
separable states: single- and double-excitation states. Our results show that
the generated entanglement depends on the phase shift, time delay, atomic
initial state, and the coupling configuration. For the single-excitation
initial state, there exists the steady-state entanglement for each coupling in
both the Markovian and non-Markovian regimes due to the appearance of the dark
state. For the double-excitation initial state, we observe entanglement sudden
birth via adjusting the phase shift in both regimes. In particular, the
maximally achievable entanglement for the nested coupling is about one order of
magnitude larger than those of separate and braided couplings. We also find
that the maximal entanglement for these three coupling configurations can be
enhanced in the case of small time delays. This work can be utilized for the
generation and control of entanglement in quantum networks based on giant-atom
waveguide-QED systems, which have wide potential applications in quantum
information processing.
