  Applications of Time-Fractional Schrodinger Equations (TFSEs) to quantum
processes are instructive for understanding and describing the time behavior of
real physical systems. By applying three popular TFSEs, namely Naber's TFSE I,
Naber's TFSE II, and XGF's TFSE, to a basic open system model of a two-level
system (qubit) coupled resonantly to a dissipative environment, we solve
exactly for Time-Fractional Single Qubit Open Systems (TFSQOSs). However, the
three TFSEs perform badly for the following reasons. On the other hand, in the
respective frameworks of the three TFSEs, the total probability for obtaining
the system in a single-qubit state is not equal to one with time at fractional
order, implying that time-fractional quantum mechanics violates quantum
mechanical probability conservation. On the other hand, the latter two TFSEs
are not capable of describing the non-Markovian dynamics of the system at all
fractional order, only at some fractional order. To address this, we introduce
a well-performed TFSE by constructing a new analytic continuation of time
combined with the conformable fractional derivative, in which for all
fractional order, not only does the total probability for the system equal one
at all times but also the non-Markovian features can be observed throughout the
time evolution of the system. Furthermore, we study the performances of the
four TFSEs applying to an open system model of two isolated qubits each locally
interacting with its dissipative environment. By deriving the exact solutions
for time-fractional two qubits open systems, we show that our TFSE still
possesses the above two advantages compared with the other three TFSEs.
