NMR Quantum Simulation of Localization Effects Induced by Decoherence

The loss of coherence in quantum mechanical superposition states limits the time for which quantum information remains useful. Similarly, it limits the distance over which quantum information can be transmitted. Here, we investigate in a nuclear spin-based quantum simulator, the localization of the size of spin clusters that are generated by a Hamiltonian driving the transmission of information, while a variable-strength perturbation counteracts the spreading. We find that the system reaches a dynamic equilibrium size, which decreases with the square of the perturbation strength.

Figure 1

NMR sequence for the quantum simulations. An unperturbed evolution is achieved when ${\tau }_{\Sigma }=0$. The effective Hamiltonian ${\widehat{\mathcal{H}}}_0$ is generated by the sequence of $\pi /2$ pulses shown in the upper part of the figure.

Figure 2

Time evolution of the cluster size of correlated spins with the unperturbed Hamiltonian ${\widehat{\mathcal{H}}}_0$. Distributions of the squared amplitudes $A_M$ of density operator components as a function of the coherence order $M$ are shown for two different cluster sizes. The latter are obtained from the half-width $2\sqrt{\ln (2K)}$ of the distribution function $A_M$.

Figure 3

(a) Time evolution of the cluster size. The black squares represent the unperturbed time evolution and the other symbols correspond to different perturbation strengths according to the legend. (b),(c) Distributions of the amplitudes $A_M(N{\tau }_c)$ for unperturbed dynamics [(b), $p=0$] and a perturbed evolution [(c), $p=0.108$], respectively. The perturbed evolution in (c) shows localization at a cluster size $K_{\mathrm{loc}}\simeq 56$ spins.

Figure 4

(a) NMR pulse sequence for preparing different initial cluster sizes and subsequently evolving them in the presence of a perturbation. (b) Time evolution of the correlated cluster size starting from different initial sates. Filled symbols are evolutions from an uncorrelated initial state for two different perturbation strengths given in the legend. Empty symbols start from an initial state with $K_0$ correlated spins.

Figure 5

Localized cluster size $K_{\mathrm{loc}}$ (square symbols) of correlated spins versus the perturbation strength $p$. Three dynamical regimes for the evolution of the cluster size are identified depending on the number of correlated spins compared with the perturbation dependent localization value: (a) a cluster size decreases, (b) a cluster size increases, (c) stationary regime.