Properties of phonon modes of an ion-trap quantum computer in the Aubry phase

We study analytically and numerically the properties of phonon modes in an ion quantum computer. The ion chain is placed in a harmonic trap with an additional periodic potential of which dimensionless amplitude $K$ determines three main phases available for quantum computations: at zero $K$ we have the case of Cirac-Zoller quantum computer, below a certain critical amplitude $K<K_c$ the ions are in the Kolmogorov-Arnold-Moser (KAM) phase, with delocalized phonon modes and free chain sliding, and above the critical amplitude $K>K_c$ ions are in the pinned Aubry phase with a finite frequency gap protecting quantum gates from temperature and other external fluctuations. For the Aubry phase, in contrast to the Cirac-Zoller and KAM phases, the phonon gap remains independent of the number of ions placed in the trap keeping a fixed ion density around the trap center. We show that in the Aubry phase the phonon modes are much better localized compared to the Cirac-Zoller and KAM cases. Thus in the Aubry phase the recoil pulses lead to local oscillations of ions, while in the other two phases they spread rapidly over the whole ion chains, making them rather sensible to external fluctuations. We argue that the properties of localized phonon modes and phonon gap in the Aubry phase provide advantages for the ion quantum computations in this phase with a large number of ions.

Figure 1

(a) Spectrum of phonon excitations ${\omega }_k$ as a function of the scaled mode number $k=i/N$ ($i=0,...,N-1$). Here $N=50$, $\nu \approx {\nu }_g$, and ${\omega }_{\mathrm{tr}}=0.014$. (b) Evolution of the lowest phonon mode ${\omega }_0$ as the number of ions $N$ increases for $\nu \approx {\nu }_g$. For both panels blue dots, red squares, and black circles respectively correspond to the Cirac-Zoller case ($K=0$), the KAM case ($K=0.0154\approx K_c/3$), and the Aubry case ($K=0.1386\approx 3K_c$). The dashed line in panel (b) indicates the dependence ${\omega }_0=0.47/N^b$ with $b=1$; the fit of blue points for the Cirac-Zoller case gives the exponent $b=0.892\pm 0.00028$ [for the other cases the fit values of $b$ are $b=0.743\pm 0.023$ (KAM) and $b=0.0132\pm 0.0026$ (Aubry)].

Figure 2

Amplitude of eigenmodes ${\psi }_k$ as a function of ion positions $x_i$ for the KAM phase at $K=0.0154\approx K_c/3$ (first row) and the Aubry phase at $K=0.1386\approx 3K_c$ (second row), for $k=0$ (left column) and $k=1$ (right column). Here $N=50$, $\nu \approx {\nu }_g$, and ${\omega }_{\mathrm{tr}}=0.014$.

Figure 3

Inverse participation ratio ${\xi }_k$ as a function of mode index $k$ for $K=0$ (blue dots), $K=0.0154\approx K_c/3$ (red squares), and $3K_c=0.1386$ (black circles). Here $N=50$, $\nu \approx {\nu }_g$, and ${\omega }_{\mathrm{tr}}=0.014$.

Figure 4

Average inverse participation ratio $\overline{\xi }$ as a function of number of ions for the Cirac-Zoller case $K=0$ (blue dots), KAM case $K=0.0154\approx K_c/3$ (red squares), and the Aubry phase $K=0.1386\approx 3K_c$ (black circles). The algebraic fit $\overline{\xi }=a{N}^b$ for ${\omega }_{\mathrm{tr}}=0.014$ gives respectively the value of exponent: $b=0.796\pm 0.0029$ (Cirac-Zoller case), $b=0.696\pm 0.0049$ (KAM case), and $b=0.349\pm 0.021$ (Aubry case).

Figure 5

Recoil pulse spreading over ion chain in time. The vertical axis shows ion number $i$ and the horizontal axis shows the dimensionless time $t$. The initial momentum recoil $\delta P_0={10}^{-3}$ is given to ion $i=23$, for the Cirac-Zoller case $K=0$ (top panel), the Aubry phase at $K=0.1386\approx 3K_c$ with $\nu \approx {\nu }_g$ (middle panel), and with integer density $\nu \approx 1$ (bottom panel). Here $N=50$. The color gives the relative ion momentum $P_i/\delta P_0$.

Figure 6

Chain representation in the KAM phase ($K=0.0154\approx K_c/3$). The black line shows the potential $V\left(x\right)={\omega }_{\mathrm{tr}}^2{x}^2/2-Kcosx$ and the black dots show the position of ions $x_i$. On the first line the whole chain is represented while on the second we zoom on the 17 central ions. Here $N=50$, $\nu \approx {\nu }_g$, and ${\omega }_{\mathrm{tr}}=0.014$. We highlight the center of the harmonic trap with a blue vertical line.

Figure 7

Same as Fig. 6 for the Aubry phase ($K=0.1386\approx 3K_c$).

Figure 8

Functions related to the ground state for $K=0.0154$ (KAM phase at left column) and $K=0.1386$ (Aubry phase at right column). Top row: the kick function $g\left(x\right)=(p_{i+1}-p_i)/K$ obtained from the ion ground-state positions ${\left\{x_i\right\}}_{i=N/3,...,2N/3}$ (blue dots); the dashed curve shows the theoretical kick function $g\left(x\right)=-sinx$. Central row: phase-space portrait of the map (2); blue dots show the points obtained from the equilibrium ion positions ${\{p_i,x_i\}}_{i=N/3,...,2N/3}$. Bottom row: the hull function $h\left(x\right)$ defined as the positions of the ions ${\left\{x_i\right\}}_{i=N/3,...,2N/3}$ versus their positions at $K=0$. Here $N=50$, $\nu ={\nu }_g\approx 1.618$, and ${\omega }_{\mathrm{tr}}=0.014$.

Figure 9

Lowest phonon mode ${\omega }_0$ as the periodic potential amplitude $K$ increases for $N=50$ (blue dots) and $N=150$ (red squares). Here $\nu ={\nu }_g$.

Figure 10

Amplitude of eigenmodes ${\psi }_k\left(i\right)$ as a function of ion positions $x_i$ for KAM phase at $K=0.0154\approx K_c/3$ (first row) and Aubry phase at $K=0.1386\approx 3K_c$ (second row) for $k=2,3,4,5$ (columns left to right). Here $N=50$, $\nu \approx {\nu }_g$, and ${\omega }_{\mathrm{tr}}=0.014$.

Figure 11

Recoil pulse spreading over ion chain in time. The vertical axis shows ion number $i$ and the horizontal axis shows dimensionless time $t$. The initial momentum recoil $\delta P_0={10}^{-3}$ is given to ion $i=23$ in the KAM phase $K=0.0154$. Here $N=50$ and ${\omega }_{\mathrm{tr}}=0.014$. The color gives the relative ion momentum $P_i/\delta P_0$.

Figure 12

Distribution histogram of fractions of $N_s=200$ energy lowest stable ion configurations over their energy $E_s$ rescaled by the ground-state energy $E_g$ (shown in the vertical axis). Here ${\omega }_{\mathrm{tr}}=0.014$, $N=50$ (a) and ${\omega }_{\mathrm{tr}}=0.005$, $N=150$ (b); $K=0.1386$, $\nu \approx {\nu }_g$ in the central part of the ion chain.

Figure 13

Distribution of ${\omega }_0/{\omega }_g$ for the $N_s=200$ lowest-energy configurations (ordered by energy starting from the ground state), where ${\omega }_0$ is the lowest mode phonon frequency for the configurations and parameters of Fig. 12 and rescaling is done via the frequency ${\omega }_g=0.23969050023353308$ of the ground state obtained via minimization procedure described in [19]; dashed horizontal line marks the value of ${\omega }_0=0.2259555158207294$ for the lowest-energy configuration found by the gradual step $K$ increase minimization used in the main part of the article.