We　consider　the　unitary　time　evolution　of　the　joint　state　of　a　complex　quantum　system,　composed　of　two　atomic　qubits　and　two　sets　of　Lorentzian　reservoirs;　two　atomic　qubits　are　initially　in　a　maximally　entangled　state.　We　study　the　pure　geometric　phase　for　the　joint　system　with　the　cases　of　both　Markovian　and　non-Markovian　Lorentzian　reservoirs　as　the　system　undergoes　parallel　transport,　and　find　that　the　geometric　phase　is　related　to　the　effective　coupling　between　the　entangled　qubits　and　the　reservoirs.　This　distinct　result,　as　compared　to　that　found　in　the　previous　studies,　is　due　to　the　entanglement　between　the　qubits　and　the　reservoirs.　Finally,　the　relation　between　entanglement　and　the　geometric　phase　is　discussed.　(C)　2017　Elsevier　Inc.　All　rights　reserved.