Protecting dissipative quantum state preparation via dynamical decoupling

We show that dissipative quantum state preparation processes can be protected against qubit dephasing by interlacing the state preparation control with dynamical decoupling (DD) control consisting of a sequence of short $\pi$ pulses. The inhomogeneous broadening can be suppressed to second order of the pulse interval, and the protection efficiency is nearly independent of the pulse sequence but determined by the average interval between pulses. The DD protection is numerically tested and found to be efficient against inhomogeneous dephasing on two exemplary dissipative state preparation schemes that use collective pumping to realize many-body singlets and linear cluster states, respectively. Numerical simulation also shows that the state preparation can be efficiently protected by $\pi$ pulses with completely random arrival time. Our results make possible the application of these state preparation schemes in inhomogeneously broadened systems. DD protection of state preparation against dynamical noises is also discussed using the example of Gaussian noise with a semiclasscial description.

Figure 1

Schematics of interlacing DD controls with the dissipative quantum state preparations. In the toggling frame, the noise has a sign change between odd and even time intervals and the noise effects average out over time. The effects of the dissipative preparation channel in the odd and even time intervals constructively add. In the laboratory frame, the dissipative preparation channel shall then take two different forms for the odd and even time intervals in general.

Figure 2

Schematics of DD control, formed by repetitions of a basic unit with duration $t_p$. The basic unit consists of an $N$-pulse sequence which can take various designs, including the CPMG, concatenated, and Uhrig sequences.

Figure 3

Steady state population of many-body singlet $P(J=0)$ versus the strength of inhomogeneous broadening $\Delta$, when the preparation process is protected using different DD pulse sequences. The ensemble consists of six qubits, with inhomogeneously broadened energies ${\omega }_i=\Delta (7-2i)/5$, $i=1,2,...,6$. The collective pumping rates are chosen to be ${\Lambda }_h=10{\Lambda }_i$ (see text). The red solid, blue dashed, brown dotted, orange dashed, and light yellow dotted curves show numerical results by solving the exact master equation, where the basic unit of the DD control consists of the CPMG, CDD3, CDD4, UDD5, and UDD10 pulse sequences, respectively. For the simulation presented in panel (a), the duration of the basic unit $t_p={10}^{-3}{\Lambda }_i^{-1}$ for all the DD controls compared, as shown in panel (c). In panel (b), the duration of the basic unit $t_p={10}^{-4}{\Lambda }_i^{-1}N$, $N$ being the number of pulses in the basic unit, and the average pulse interval $\overline{\tau }\equiv t_p/N$ is taken to be the same for all the DD controls compared, as shown in panel (d).

Figure 4

Steady state population of many-body singlet $P(J=0)$ as function of the inhomogeneous broadening $\Delta$ and the average pulse density $\overline{n}=N/t_p$, where the basic unit of DD control uses various $N$-pulse sequences. The ensemble consists of six qubits, with inhomogeneously broadened energies ${\omega }_i=\Delta (7-2i)/5$, $i=1,2,...,6$. The collective pumping rates are chosen to be ${\Lambda }_h=100{\Lambda }_i$.

Figure 5

Convergence of the Magnus expansion. Solid and dot-dashed curves show the steady state population of many-body singlet $P(J=0)$ obtained by exactly solving the master equation. Dashed and dotted curves are steady state $P(J=0)$ obtained from the Magnus expansion Eq. (3) where we only keep the leading higher-order Magnus term (see text). The ensemble consists of six qubits, with inhomogeneously broadened energies ${\omega }_i=\Delta (7-2i)/5$, $i=1,2,...,6$. The collective pumping rates are chosen to be ${\Lambda }_h=100{\Lambda }_i$. The blue, orange, red and green solid lines represent numerical results when the basic unit of DD control uses the CPMG, UDD3, CDD6, and UDD42 pulse sequences, respectively. The average pulse density is chosen to be $\overline{n}\equiv N/t_p={10}^{2.75}{\Lambda }_i$.

Figure 6

Steady state population of the many-body singlet $P(J=0)$ as a function of the inhomogeneous broadening $\Delta$ and the average pulse density $\overline{n}=N/t_p$ when the DD control uses a completely random pulse sequence. The ensemble consists of six qubits, with inhomogeneously broadened energies ${\omega }_i=\Delta (7-2i)/5$, $i=1,2,...,6$. The collective pumping rates are chosen to be ${\Lambda }_h=100{\Lambda }_i$.

Figure 7

Level scheme for realizing the stabilizer $S_3={\widehat{s}}_1^z{\widehat{s}}_2^z{\widehat{s}}_3^z$. In the notations such as $|RLL0\rangle$, the first three letters denote the state of qubits 1, 2, and 3, respectively, and the last integer denotes the number of cavity photons. The four involved states in the excited state manifold are $|1\rangle \equiv 1/\sqrt{2}\left(\right|RLL1\rangle +|ELL0\rangle )$, $|2\rangle \equiv 1/\sqrt{2}|RLR1\rangle -1/2\left(\right|ELR0\rangle +|RLE0\rangle )$, $|3\rangle \equiv 1/\sqrt{2}|RRL1\rangle -1/2\left(\right|ERL0\rangle +|REL0\rangle )$, and $|4\rangle \equiv 1/\sqrt{2}|RRR1\rangle -1/\sqrt{6}\left(\right|ERR0\rangle +|RER0\rangle +|RRE0\rangle )$. The wavy lines denote the spontaneous emission. Optical fields resonantly pump the four transitions denoted by the double-head solid lines. The eigenstate of stabilizer $S_3$ is realized as the steady state of the pumping. Panels (a) and (b) correspond, respectively, to pumping controls in the laboratory frame before and after a $\pi$ pulse is applied. In panel (a) the cavity couples to the $|R\rangle \leftrightarrow |E\rangle$ transitions with strength $g^R$, while in panel (b) the cavity couples to the the $|L\rangle \leftrightarrow |E\rangle$ transitions with strength $g^L$ [46]. In the toggling frame, the two pumping controls realize the same dissipative channel.

Figure 8

Fidelity between the steady state and the target four-qubit linear cluster state under the DD protection where the basic unit uses the UDD3 sequence. The inhomogeneously broadened qubit resonances are ${\omega }_i=\Delta (5-2i)/3$, $i=1,2,3,4$. The parameters for the preparation channel are ${\Omega }_j^L={\Omega }_j^R=0.01g$ and ${\Gamma }_j^L={\Gamma }_j^R=0.01g$. See text.