Non-Markovian qubit dynamics in the presence of $1/f$ noise

Within the lowest-order Born approximation, we calculate the exact dynamics of a qubit in the presence of $1/f$ noise without Markov approximation. We show that the non-Markovian qubit time-evolution exhibits asymmetries and beatings that can be observed experimentally and cannot be explained within a Markovian theory. The present theory for $1/f$ noise is relevant for both spin- and superconducting qubit realizations in solid-state devices, where $1/f$ noise is ubiquitous.

Figure 1

(Color online) Non-Markovian time-evolution of the unbiased $\left(ϵ=0\right)$ qubit (spin) $z$-component $⟨{\sigma }_z\left(t\right)⟩$, for $A/{\Delta }^2=0.05$ and ${\gamma }_0/\Delta =0.05$ (solid oscillating black line). The Markovian pole contribution ${⟨{\sigma }_z\left(t\right)⟩}_{\text{poles}}$ is plotted as a dashed line for comparison. The essential non-Markovian part is nonexponential and given by the branch cut contribution ${⟨{\sigma }_z\left(t\right)⟩}_{\text{bc}}$ (solid decreasing red line). Inset: Plot for $A/{\Delta }^2=0.005$ and ${\gamma }_0/\Delta ={10}^{-10}$. Here, the essential non-Markovian part is the long-time asymmetry which carries information about the initial state.

Figure 2

(Color online) Analytic structure of $⟨{\sigma }_z\left(s\right)⟩$ in the complex $s$ plane for (a) the unbiased case, $ϵ=0$, and (b) the biased case, $ϵ\ne 0$. Red dots denote poles and blue lines branch cuts.

Figure 3

(Color online) Branch cut integral function $I_1\left(a,b\right)$ (solid red line) and two asymptotes (black dashed and blue dotted lines).

Figure 4

(Color online) Shift and splitting of the poles of $⟨{\sigma }_z\left(s\right)⟩$ for the biased system ($ϵ=0.3$ and ${\gamma }_0=0.05$). Shown is the pole located at $s=iE$ for the undamped system $\left(A=0\right)$, indicated as a red dot (see Inset b). The pole at $s=-iE$ behaves similarly. With increasing $A$ the pole shifts toward the vicinity of the bp, where a second pole (orange dot) appears. Shown as red and orange dots are the two poles for $A=0.05$. The splitting of the poles leads to a beating in ${⟨{\sigma }_z\left(t\right)⟩}_{\text{poles}}$, see Fig. 5. Inset: analytic structure of $⟨{\sigma }_z\left(s\right)⟩$ for $ϵ\ne 0$, where red dots are poles and blue lines are branch cuts (see Fig. 2).

Figure 5

(Color online) Oscillation $⟨{\sigma }_z\left(t\right)⟩$ of the biased qubit for $ϵ/\Delta =0.3$, $A/{\Delta }^2=0.05$, and ${\gamma }_0/\Delta =0.05$. The beating due to the splitting of the poles at $\pm iE$ can be observed in ${⟨{\sigma }_z\left(t\right)⟩}_{\text{poles}}$.