Soft-pulse dynamical decoupling in a cavity

Dynamical decoupling is a coherent control technique where the intrinsic and extrinsic couplings of a quantum system are effectively averaged out by application of specially designed driving fields (refocusing pulse sequences). This entails pumping energy into the system, which can be especially dangerous when it has sharp spectral features like a cavity mode close to resonance. In this work we show that such an effect can be avoided with properly constructed refocusing sequences. To this end we construct the average Hamiltonian expansion for the system evolution operator associated with a single “soft” $\pi$ pulse. To second order in the pulse duration, we characterize a symmetric pulse shape by three parameters, two of which can be made zero by shaping. We express the effective Hamiltonians for several pulse sequences in terms of these parameters and use the results to analyze the structure of error operators for a controlled Jaynes-Cummings Hamiltonian. When errors are cancelled to second order, numerical simulations show excellent qubit fidelity with strongly suppressed oscillator heating.

Figure 1

(Color online) Evolution of the qubit fidelity and the number of quanta at the oscillator $⟨n⟩$ (minimized and maximized over initial qubit states, respectively) at the end of the refocusing interval. Length-4 sequence $\mathbf{4}\mathbf{p}$ with Gaussian pulses $G_{010}$ applied on resonance with the qubit, ${\omega }_0=0$ in Eq. (36). Coupling constant $g=0.1/{\tau }_p$. Left panels: oscillator restricted to $n=0,1$ states, which makes it effectively a qubit. Right panels: oscillator restricted to $n\le 8$ levels, which over the simulation time is equivalent to infinity. Bottom panels correspond to oscillator in resonance with the qubit, top panels correspond to oscillator frequency bias ${\omega }_r=0.02/{\tau }_p$. Solid lines correspond to control pulses applied along the $x$ and $y$ axes [sequence $\mathbf{4}\mathbf{p}\left(xy\right)$]; dashed lines correspond to control pulses applied along the $x$ and $z$ axes [sequence $\mathbf{4}\mathbf{p}\left(xz\right)$].

Figure 2

(Color online) As in Fig. 1 but with first-order self-refocusing pulses $S_1$. The corresponding curves for pulses $Q_1$ (not shown) are virtually identical with linear scale.

Figure 3

(Color online) As in Fig. 1 but for the sequence $\mathbf{8}\mathbf{a}$. The term $\propto s$ in Eq. (40) produces large errors comparable for those for $\mathbf{4}\mathbf{p}$ sequence in Fig. 1. This indicates that this sequence is less stable to pulse shape errors as compared to, e.g., the symmetric sequence $\mathbf{8}\mathbf{s}$.

Figure 4

(Color online) As in Fig. 1 but for the sequence $\mathbf{8}\mathbf{a}$ with pulses $S_1$. The corresponding curves for pulses $Q_1$ (not shown) are virtually identical with linear scale.

Figure 5

(Color online) As in Fig. 1 but for the sequence $\mathbf{8}\mathbf{s}$.

Figure 6

(Color online) As in Fig. 1 but for the sequence $\mathbf{8}\mathbf{s}$ with pulses $S_1$. Formally, the sequence is of the same order as with Gaussian pulses, see Fig. 5; the noticeable differences are due to error terms of higher order which we dropped in our calculations. The off-resonance refocusing is excellent (the fidelity and $⟨n⟩$ curves in the top right panel run along the corresponding axes), while the on-resonance performance is also good. The plots with second-order pulses $Q_1$ (not shown) look almost identical. Note also that with the linear scale of these plots, the curves here are almost indistinguishable from those in Fig. 4.

Figure 7

(Color online) Refocusing with finite-width resonance at zero frequency bias. Simulation parameters as in Figs. 1, 2, 3, 4, 5, 6 except that the oscillator frequency is taken to be purely imaginary, ${\omega }_r=-i\Gamma$. The curves show evolution of the qubit fidelity and the number of quanta at the oscillator $⟨n⟩$ (minimized and maximized over initial qubit states, respectively) at the end of the refocusing interval. Left panes from bottom to top: sequences $\mathbf{4}\mathbf{p}$, $\mathbf{8}\mathbf{a}$, $\mathbf{8}\mathbf{s}$ with Gaussian pulses $G_{010}$ (cf. Figs. 1, 3, 5, respectively). Right panes show the results of the simulation with pulses $S_1$ (cf. Figs. 2, 4, 6, respectively). Oscillator is restricted to $n\le 8$ levels, which over the simulation time is equivalent to infinity. Solid lines correspond to control pulses applied along the $x$ and $y$ axes [sequences $\mathbf{4}\mathbf{p}\left(xy\right)$, $\mathbf{8}\mathbf{a}\left(xy\right)$, $\mathbf{8}\mathbf{s}\left(xy\right)$]; dashed lines correspond to control pulses applied along the $x$ and $z$ axes [sequences $\mathbf{4}\mathbf{p}\left(xz\right)$, etc.].