Figure 1

(Color online) Fidelity of GHZ generation vs $N$ with and without gap protection. We assume a noise with steplike spectral density with amplitude $f=2\text{kHz}$ and cutoff frequency ${\omega }_c=6\text{kHz}$. The solid red and dashed green lines are for a protected system, $\lambda =\chi$, with $\chi =25$ and 10 kHz, respectively. The black dotted and blue dotted-dashed lines are for an unprotected system, $\lambda =0$, with the same parameters $\chi =25$ and 10 kHz, respectively. In the inset we show $⟨{\widehat{J}}_x^{\left(0\right)}\left(t\right)⟩/N$ for $N=50$ and the various curves are for the same parameters as those shown in the main plot. The horizontal axis in the inset is in units of $\hslash$.Reuse & Permissions

Figure 2

(Color online) Schematic representation of the energy levels of the $\chi {\widehat{J}}_z^{\left(0\right)2}$ Hamiltonian and the effect of the different types of noise. As states with different $J$ but equal $\left|M\right|$ are degenerate, in the presence of phase decoherence (${\sigma }_z$ noise) they are populated during the time evolution. ${\sigma }_{x,y}$ noises couple states which differ by $\pm 1$ units of $M$ and therefore the small energy gap between them, of the order of $\chi$, naturally protects the system from these type of processes.Reuse & Permissions

Figure 3

(Color online) Schematic representation of the energy levels of the ${\widehat{H}}_c$ Hamiltonian. ${\widehat{H}}_{\text{prot}}$ lifts the degeneracy of the different $J$ manifolds and suppresses (in the slow noise limit) couplings between them. So if a $t=0$ system is in the MPM it remains there.Reuse & Permissions

Figure 4

(Color online) Fidelity to create GHZ state as a function of $N$ for systems in the presence of ${\sigma }_y$ noise: without MPM (green dashed line), without MPM but with spin echo (dotted-dashed blue line), with MPM (red solid line), and with both MPM and echo (dotted black line). We also show the degradation caused by pure dephasing in an unprotected system (long dashed purple line) for comparison purposes. For simplicity we assumed infinite correlation time, $f\left(\omega \right)={\rm Y}\delta \left(\omega /\chi \right)$, ${\rm Y}=0.1\chi$. The echo technique consisted of a perfect sudden $\pi$ pulse around the $x$ direction at $\chi t=\pi /4$. Because the $y$ components of the noise do not commute with ${\widehat{J}}_z^{2\left(0\right)}$ the dynamics was solved numerically. For the unprotected system all the $2^N$ states must be considered and we must limit the particle number by $N=10$. On the other hand, the restriction of the dynamics to the MPM in the protected scheme allowed us to extend the calculation to larger $N$ values. For the ${\sigma }_y$ noise the fidelity with protection does not become $N$ independent, but instead scales as $e^{-0.043N^{0.44}}$ (see fitted red dots). Nevertheless we do gain a factor of order $\sqrt{N}$ with respect to the unprotected system. The plot also shows that the combination of MPM with spin echo provides the best protection.Reuse & Permissions

Figure 5

(Color online) Energy-level diagram for a pair of ions illuminated by a monochromatic beam with detuning $\delta$ from the atomic transition. The quantity $n$ denotes the quantum number of trap oscillations with frequency $\nu$. $\delta$ is assumed to be large compared to the linewidth of the resonance and consequently the dominant processes are two-photon transitions. These transitions lead to a collective Hamiltonian for the many-ion system of the form $\chi \left({\widehat{J}}^{\left(0\right)2}-{\widehat{J}}_z^{\left(0\right)2}\right)$.Reuse & Permissions

Figure 6

(Color online) Phase sensitivity in precision spectroscopy with trapped ions as a function of $N$. The vertical axis is in a logarithmic scale. We used the same parameters as Fig. 1 for the noise $\chi =25\text{kHz}$, $\Omega =0.05\nu$, and $\nu =2\pi \times 3.86\text{MHz}$. The blue line shows the phase sensitivity without protection [Eq. (20)] and the red dashed line with protection. Even with these reduced number of ions we observe up to a 30% improvement.Reuse & Permissions

Figure 7

(Color online) Raman transition to a third level with atomic decay. Here $\Delta$ is the detuning of the laser fields with frequencies ${\omega }_{e1,2}$ from the one-photon resonances with frequencies ${\nu }_{1,2}$, i.e., $\Delta ={\omega }_{e1}-{\nu }_1={\omega }_{e2}-{\nu }_2$, and ${\Omega }_{1,2}$ are laser Rabi frequencies.Reuse & Permissions

Figure 8

(Color online) Fidelity to generate a GHZ state vs $\overline{\lambda }/\overline{\chi }$. In the inset we show $⟨{\widehat{J}}_x^{\left(0\right)}\left(t\right)⟩$. The $x$ axis there is in units of $\hslash$. The blue dotted-dashed, dotted black, dashed green, and solid red lines correspond to $\overline{\lambda }=0,5,10,20$, respectively. The plots are obtained by numerical evolution of Eq. (37) for $N=10$.Reuse & Permissions

Figure 9

In the presence of phase decoherence the fidelity of the GHZ state preparation in the lattice is degraded as $N$ grows because in the lattice the gap decreases and the generation time increases with $N$. In this plot we assumed the limit $\overline{\chi }⪡\overline{\lambda }$, where Eq. (42) holds and used a system with ${\omega }_c=\overline{\chi }$, $\overline{\lambda }=100\overline{\chi }$, and $\Gamma =0.01\overline{\chi }$. At $N=90$, $\Delta E_g={\omega }_c$ and it explains the drop of $\mathcal{F}$ for $N>90$.Reuse & Permissions