Limitations on the maximal level of entanglement of two singlet-triplet qubits in GaAs quantum dots

We analyze in detail a procedure of entangling of two singlet-triplet $S$-$T₀$ qubits operated in a regime when energy associated with the magnetic field gradient, ΔB_z, is an order of magnitude smaller than the exchange energy, J, between singlet and triplet states [Shulman M. et al., Science 336, 202 (2012)]. We have studied theoretically a single S-T₀ qubit in free induction decay and spin echo experiments. We have obtained analytical expressions for time dependence of components of its Bloch vector for quasistatical fluctuations of ΔB_z and quasistatical or dynamical 1/f^β-type fluctuations of J. We have then considered the impact of fluctuations of these parameters on the efficiency of the entangling procedure which uses an Ising-type coupling between two S-T₀ qubits. Particularly, we have obtained an analytical expression for evolution of two qubits affected by 1/f^β-type fluctuations of J. This expression indicates the maximal level of entanglement that can be generated by performing the entangling procedure. Our results deliver also an evidence that in the above-mentioned experiment, the S-T₀ qubits were affected by uncorrelated 1/f^β charge noises.