Higher order approximations in quantum speed limit time of a qubit coupled to XXZ spin chain environment

The quantum speed limit (QSL) time of a single qubit coupled to Heisenberg XXZ spin bath is investigated. The time convolutionless (TCL) projection operators technique is followed for investigating the second (TCL2), third (TCL3) and fourth (TCL4) order approximations in QSL time of centrally interacting qubit. The results are discussed up to fourth order of perturbation expansions of the non-Markovian master equation of motion. Behavior of QSL time is studied against temperature, interaction time and coupling strength for second, third and fourth orders of master equation using TCL technique. The behavior of QSL time becomes less perturbative for third and fourth order approximations as compared to second order. It is investigated that QSL time of qubit becomes constant very quickly in the low temperature limit for third and fourth order approximation, which means evolution speed of qubit is less affected for higher orders of approximations. QSL time for third and fourth orders of TCL gives much better results at high temperature and also behave well at short times. It is observed that, as the order of the TCL master equation increases, the accuracy of the approximation improves, but the complexity of the equation also increases due to involvement of more and more interaction terms in the TCL master equation.