Storing entanglement of nuclear spins via Uhrig dynamical decoupling

Stroboscopic spin flips have already been shown to prolong the coherence times of quantum systems under noisy environments. Uhrig’s dynamical decoupling scheme provides an optimal sequence for a quantum system interacting with a dephasing bath. Several experimental demonstrations have already verified the efficiency of such dynamical decoupling schemes in preserving single-qubit coherences. In this work we describe the experimental study of Uhrig’s dynamical decoupling in preserving two-qubit entangled states using an ensemble of spin-1/2 nuclear pairs in solution state. We find that the performance of odd-order Uhrig sequences in preserving entanglement is superior to both even-order Uhrig sequences and periodic spin-flip sequences. We also find that there exists an optimal order of the Uhrig sequence in which a singlet state can be stored at high correlation for about 30 seconds.

Figure 1

The ${}^1\mathrm{H}$ NMR spectrum and the molecular structure of 5-chlorothiophene-2-carbonitrile.

Figure 2

NMR pulse sequence to study dynamical decoupling on Bell states. An incoherent mixture of singlet and triplet states is prepared, which under spin lock purifies to a singlet state. The resulting singlet state can be converted to other Bell states. Then a dynamical decoupling sequence can be applied and the performance of the sequence can be studied by characterizing the residual state using density matrix tomography.

Figure 3

Pulse sequences for various orders of Uhrig dynamical decoupling. Note that both UDD-1 and UDD-2 are equivalent to the CPMG sequence. The time instants are calculated according to Eq. (2), with $N$ being the order of UDD and the total period $T=N\times 4.0272$ ms.

Figure 4

Experimental correlations (circles) of the singlet state as a function of decoupling duration of various orders of UDD. Also shown in the top left figure is the correlation decay under no dynamical decoupling (squares).

Figure 5

The number of time instants at which the correlation exceeded 0.9 for various orders of UDD-$N$.

Figure 6

The decay of the singlet spin order measured by converting it into observable single quantum magnetizations. The decay was studied under the CPMG sequence (squares) as well as under the Uhrig sequence (filled circles). The dashed and the solid line correspond to the exponential fits for CPMG and UDD-7 data points respectively.

Figure 7

Experimental correlations of the product state and various Bell states as a function of duration under (i) no decoupling (open squares), (ii) the CPMG sequence (filled circles), and (iii) UDD-7 (open circles).