Performance comparison of dynamical decoupling sequences for a qubit in a rapidly fluctuating spin bath

Avoiding the loss of coherence of quantum mechanical states is an important prerequisite for quantum information processing. Dynamical decoupling (DD) is one of the most effective experimental methods for maintaining coherence, especially when one can access only the qubit system and not its environment (bath). It involves the application of pulses to the system whose net effect is a reversal of the system-environment interaction. In any real system, however, the environment is not static, and therefore the reversal of the system-environment interaction becomes imperfect if the spacing between refocusing pulses becomes comparable to or longer than the correlation time of the environment. The efficiency of the refocusing improves therefore if the spacing between the pulses is reduced. Here, we quantify the efficiency of different DD sequences in preserving different quantum states. We use ${}^{13}\mathrm{C}$ nuclear spins as qubits and an environment of ${}^1\mathrm{H}$ nuclear spins as the environment, which couples to the qubit via magnetic dipole-dipole couplings. Strong dipole-dipole couplings between the proton spins result in a rapidly fluctuating environment with a correlation time of the order of 100 $\mu$s. Our experimental results show that short delays between the pulses yield better performance if they are compared with the bath correlation time. However, as the pulse spacing becomes shorter than the bath correlation time, an optimum is reached. For even shorter delays, the pulse imperfections dominate over the decoherence losses and cause the quantum state to decay.

Figure 1

Schematic representation of dynamical decoupling. The solid boxes represents the control pulses.

Figure 2

Schemes of dynamical decoupling pulse sequences. Empty and solid rectangles represent $\pi /2$ and $\pi$ pulses, respectively. $M$ represents the number of iterations of the cycle. (a) Initial state preparation before application of the DD sequence. (b) Hahn spin-echo sequence. (c) CPMG (${\varphi }_2={\varphi }_1$) and CPMG-2 (${\varphi }_2={\varphi }_1+\pi$) sequences. (d) PDD sequence. (e) CDD sequence of order $n$, CDD${}_n=C_n$. (f) UDD sequence scheme with four pulses, i.e., UDD of order 4, UDD${}_4$.

Figure 3

Evolution of the normalized spin correlation functions for the bath spins (protons). The solid line represents the proton FID signal [$i_x(t)$] and the dashed line the numerically simulated $i_z(t)$.

Figure 4

Survival probability of the $S$-spin under free evolution (${}^{13}\mathrm{C}$ FID) and after a Hahn-echo sequence. An initial condition ${\mathrm{Ŝ}}_x$ was prepared.

Figure 5

Magnetization evolution for the CPMG sequence. From top to bottom the initial conditions are ${\mathrm{Ŝ}}_y$, ${\mathrm{Ŝ}}_x$, and ${\mathrm{Ŝ}}_z$. The left-hand panels represent the ${}^{13}\mathrm{C}$ magnetization as a function of the total evolution time while right-hand panels show its evolution as a function of the number of applied pulses. The legend at the bottom gives the delays $\tau$ between successive pulses.

Figure 6

Magnetization evolution for the CPMG-2 sequence. From top to bottom the initial conditions are ${\mathrm{Ŝ}}_x,$ ${\mathrm{Ŝ}}_y$, and ${\mathrm{Ŝ}}_z$. The left-hand panels represent the ${}^{13}\mathrm{C}$ magnetization as a function of the total evolution time, while the right-hand panels show its evolution as a function of the number of applied pulses. The legend at the bottom gives the delays $\tau$ between successive pulses.

Figure 7

Signal decay of the initial state of the qubit for the PDD sequence. From top to bottom the initial conditions are ${\mathrm{Ŝ}}_x,$ ${\mathrm{Ŝ}}_y$, and ${\mathrm{Ŝ}}_z$. The left-hand panels represent the decay as a function of the total evolution time while the right-hand panels show the decay as a function of the number of applied pulses. The legend at the bottom gives the delays $\tau$ between successive pulses.

Figure 8

Relaxation times of different initial conditions under DD conditions as a function of the delay between pulses $\tau$ (left panels) and the cycle time ${\tau }_c$ (right panels). From top to bottom the initial condition is given by ${\mathrm{\rho ̂}}_0={\mathrm{Ŝ}}_x$, ${\mathrm{Ŝ}}_y$, and ${\mathrm{Ŝ}}_z$ respectively. An optimal $\tau$ and consequently ${\tau }_c$ is observed for each sequence. The reduction of the relaxation time to the right side of the optimal value is due to the shifting environment: in this regime the cycle time is longer than the correlation time of the bath, ${\tau }_c>{\tau }_B$. The reduction for short cycle times indicates that in this regime, accumulated pulse errors dominate.

Figure 9

CDD order $n$ as a function of its optimal delay between pulses ${\tau }_{\mathrm{opt}}$ to reduce decoherence. The experimental square points seem to satisfy a relation given by $n=c-b\ln ({\tau }_{\mathrm{opt}}/{\tau }_B)$. The solid (red) line shows a fitting curve with parameters $c=(0.9\pm 0.2)$ and $b=(-0.9\pm 0.1)$.

Figure 10

Time evolution of the survival probability for the optimal delay between pulses of the different DD sequences. From top to bottom the initial states are ${\mathrm{Ŝ}}_x,$ ${\mathrm{Ŝ}}_y$, and ${\mathrm{Ŝ}}_z$. The optimal delays $\tau$ are given in the legends.

Figure 11

Signal decay of the initial state of the qubit for the CDD${}_2$ sequence. From top to bottom the initial conditions are ${\mathrm{Ŝ}}_x,$ ${\mathrm{Ŝ}}_y$, and ${\mathrm{Ŝ}}_z$. The left-hand panels represent the decay as a function of the total evolution time while the right-hand panels show the decay as a function of the number of applied pulses. The legend at the bottom gives the delays $\tau$ between successive pulses.

Figure 12

Signal decay of the initial state of the qubit for the CDD${}_3$ sequence. From top to bottom the initial conditions are ${\mathrm{Ŝ}}_x,$ ${\mathrm{Ŝ}}_y$, and ${\mathrm{Ŝ}}_z$. The left-hand panels represent the decay as a function of the total evolution time while right-hand panels the decay as a function of the number of applied pulses. The legend at the bottom gives the delays $\tau$ between successive pulses.

Figure 13

Signal decay of the initial state of the qubit for the CDD${}_4$ sequence. From top to bottom the initial conditions are ${\mathrm{Ŝ}}_x,$ ${\mathrm{Ŝ}}_y$, and ${\mathrm{Ŝ}}_z$. The left-hand panels represent the decay as a function of the total evolution time while the right-hand panels show the decay as a function of the number of applied pulses. The legend at the bottom gives the delays $\tau$ between successive pulses.