Decay of Rabi Oscillations by Dipolar-Coupled Dynamical Spin Environments

We study the Rabi oscillations decay of a spin decohered by a spin bath whose internal dynamics is caused by dipolar coupling between the bath spins. The form and rate of decay as a function of the intrabath coupling is obtained analytically, and confirmed numerically. The complex form of decay smoothly varies from power law to exponential, and the rate changes nonmonotonically with the intrabath coupling, decreasing for both slow and fast baths. The form and rate of Rabi oscillations decay can be used to experimentally determine the intrabath coupling strength for a broad class of solid-state systems.

Figure 1

Exact simulations for $N=15$ bath spins. (a) Correlation function $C(t)=(1/b^2)⟨B(0)B(t)⟩$ (normalized by $b^2$) obtained from direct simulations (black), see also 31. Its fitting (magenta or gray) determines the bath parameters $b_1$, $b_2$, and $R$ for each value of the intrabath coupling scale $E_B$ (here $E_B=1$, $b_1=0.62$, $b_2=0.58$, $R=0.095$). (b) Rabi oscillations decay for $h_x=14.14$, $b=0.85$, and $E_B=1$; individual oscillations are not resolved. Numerics (black) agrees well with analytics (magenta, only the envelope shown). (c) Envelopes of simulated Rabi decay for $E_B=1$ (blue or dark gray), 0.1 (green or light gray), and 0 (static bath, red or gray); corresponding individual oscillations near $t=300$ are shown on panel (d) by the same colors. Analytical results practically coincide with simulations, and are not shown. The decay rate changes nonmonotonically with $E_B$: the decay is slower for $E_B=1$ (blue) and 0 (red) in comparison with $E_B=0.1$ (green). At $t=300$, on panel (d), the oscillation amplitude for $E_B=0.1$ (green) is twice smaller than that for $E_B=0$.

Figure 2

Numerical simulations for $N=159$ bath spins. (a) Correlation function $C(t)=(1/b^2)⟨B(0)B(t)⟩$ (normalized by $b^2$) obtained from $P$-representation sampling simulations (black); its fitting (magenta or gray) determines the parameter $R$ for each value of the intrabath coupling scale $E_B$ (here $E_B=1$ and $R=0.097$). In spite of noticeable statistical fluctuations [e.g., $C(0)$ is slightly larger than 1 due to them], the parameter $R$ is determined precisely enough to be used in Eq. (5). (b) Rabi oscillations decay for $h_x=15.0$, $b=1.39$, and $E_B=0.1$; individual oscillations are not resolved. Numerics (black) agrees well with analytics (magenta, only the envelope shown). (c) Envelopes of simulated Rabi decay for $E_B=0.1$ (blue or dark gray), 1 (green or light gray), and 10 (red or gray); the longitudinal decay for $E_B=10$ is taken into account. Analytical results practically coincide with simulations, and are not shown. The decay rate decreases for faster baths: the decay is slowest for $E_B=10$ (red). Individual oscillations for $E_B=10$ near $t=20$ are shown on panel (d) in red: their frequency is shifted by $b^2/2h_x=0.064$ from the value $h_x=15.0$ (oscillations with frequency $h_x=15.0$ are shown by dotted black line to demonstrate the phase difference accumulated since $t=0$).