Strong many-particle localization and quantum computing with perpetually coupled qubits

We demonstrate the onset of strong on-site localization in a many-particle system, with effective localization length smaller than the intersite distance. The localization is obtained by constructing a bounded one-parameter sequence of on-site energies that eliminates resonant hopping between both nearest and remote sites. This sequence leads to quasiexponential decay of the single-particle transition amplitude. It also leads to on-site localization of stationary many-particle states in a finite-length chain. For an infinite many-particle system, we study the time during which all states remain on-site localized. We show that, for any number of particles, this time scales as a high power of the ratio of the bandwidth of on-site energies to the hopping integral. The proposed energy sequence is robust with respect to small errors. The formulation applies to fermions as well as perpetually coupled qubits. The results show viability of quantum computing with time-independent qubit coupling.

Figure 1

(Color online) The energies ${\epsilon }_n∕h$ for $\alpha =0.3$. The left panel shows ${\epsilon }_n∕h$ for the sites $n=1,2,\mathrm{\dots },50$. Sites with close on-site energies are spatially separated. The right panel shows ${\epsilon }_n∕h$ for a much longer array, $n=1,\mathrm{\dots },2000$. The energy spectrum displays a multisubband structure, with clearly identifiable 16 subbands in this case.

Figure 2

(Color online). The exponent $\nu$ of the $\alpha$ dependence of the transition amplitude $K_n\left(m\right)$ for the efficient distance $m=200$ (crosses) and $m=1000$ (circles) as a function of site number $n$. The dashed lines show the analytical limits on $\nu$.

Figure 3

(Color online). The mean single-particle inverse participation ratio $⟨I_1⟩$ vs $\alpha$ for $h∕J=20$ (upper panel) and $h∕J=10$ (lower panel). The data refer to three values of the chain length $L$. The vertical dashed lines show the analytical estimate for the threshold of strong localization. The insets show the maximal IPR over all eigenstates, $I_{1\max }\equiv {\max }_{\mathrm{\lambda }}{I}_{1\mathrm{\lambda }}$. It sharply decreases with the increasing $\alpha$. The peak of $I_{1\max }$ for $L=300$ near $\alpha =0.1$ is due to the boundary. Near the minimum over $\alpha$, we have $I_{1\max }\approx 1.01$ for $h∕J=10$, and $I_{1\max }\approx 1.003$ for $h∕J=20$. This demonstrates strong on-site single-particle localization.

Figure 4

(Color) The IPR for six excitations on the first 12 sites of the chain (5). The reduced bandwidth of the energy spectrum is $h∕J=20$ (top panel) and $h∕J=10$ (lower panel). The purple, red, green, and blue curves refer to the coupling parameter $\mathrm{\Delta }=0$, 0.3, 1, and 3, respectively. The peaks of $⟨I_6⟩$ for $\mathrm{\Delta }=0$ are a single-particle boundary effect. The insets show the maximal $I_6$ for $\mathrm{\Delta }\ne 0$. Sharp isolated peaks of $I_{6\max }$ vs $\alpha$ result from the hybridization of resonating on-site many-particle states, see Appendix .

Figure 5

(Color online) Time evolution of the squared amplitude $\mid A{\mid }^2$ of the on-site state $\mid \mathrm{\Phi }(416,419,420,422,423,424)⟩$ in a 12-site section of the chain between $n=415$ and $n=426$. The oscillating line refers to the original sequence (5) with $\alpha =0.25$. The nearly constant line refers to the modified sequence (20) with $\alpha =0.25$, ${\alpha }^{\prime }=0.22$. In both cases $h∕J=20$ and $\mathrm{\Delta }=1$.

Figure 6

(Color online) The low-energy part of the two-particle energy differences $\delta {\epsilon }_n∕h$ (16) for all transitions with $\varkappa ⩽5$; $n$ is the smallest site number involved in the transition, $n>2$. The data refer to $\alpha =0.25$. The left panel corresponds to the sequence (5). The right panel refers to the modified sequence (20) with ${\alpha }^{\prime }=0.22$ and shows the zero-energy gap in $\delta \epsilon$.

Figure 7

(Color online) Upper panels: all energy differences $\mathrm{\delta }{\epsilon }_n^{\mathrm{err}}∕h=\mid {\epsilon }_n^{\mathrm{err}}+{\epsilon }_{n_1}^{\mathrm{err}}-{\epsilon }_{n_2}^{\mathrm{err}}-{\epsilon }_{n_3}^{\mathrm{err}}\mid ∕h$ for the transitions (19, 21, 22) that correspond to the number of intermediate steps $\varkappa ⩽5$. The data refer to $\alpha =0.25$, ${\alpha }^{\prime }=0.22$, and to a specific realization of the random numbers $r_n$ in Eq. (24). The boxes from left to right correspond to the values of the noise intensity $D={\alpha }^s$ in Eq. (24) with $s=5$,4, and 3. Lower panel: the scaled minimal gap $R$ (25) as a function of the exponent $s=\ln D∕\ln \alpha$ averaged over 10 realizations of noise. Error bars show the standard deviation of $R$.

Figure 8

(Color online) The on-site energies ${\epsilon }_n^{\mathrm{PDC}}∕h$ (26) for $\alpha =0.3$. The left panel shows the energies for the sites $n=1,2,\mathrm{\dots },50$. States with close on-site energies are spatially separated. The right panel shows ${\epsilon }_n^{\mathrm{PDC}}∕h$ for a much longer array, $n=1,\mathrm{\dots },2000$. The energy spectrum displays a multisubband structure. Because of the symmetry of sequence (26), the number of points in each subband is approximately the same.

Figure 9

(Color online) The normalized distributions of the IPR $P\left(I\right)$ for the random energy sequence (27). The upper and lower panels refer to the many- and single-particle IPR’s: six excitations with $\mathrm{\Delta }=1$ in the chain of length $L=12$ and one excitation in the chain of $L=300$, with 2000 and 80 realizations, respectively. The left and right panels refer to the overall bandwidth of the on-site energy spectrum $W∕J=13$ and 26. The narrow columns at $I=1$ showed with dotted lines refer to sequence (5), with $h∕J=10$ and 20 for the left and right panels, respectively. The values of $\alpha$ in Eq. (5) were chosen so as to have the same bandwidths of on-site energies as the corresponding random sequences: in the lower panels $\alpha \approx (W-h)∕W\approx 0.231$ (as in an infinite chain), whereas in the upper panels $\alpha =0.274$ to allow for a comparatively small number of sites. The insets show $\ln P$ vs $I$.

Figure 10

(Color online). The dependence of $\alpha$ on $h∕J$ as given by the condition $⟨I_1⟩=\mathrm{const}$ for different $⟨I_1⟩$ in the chain with $L=300$. The lines in the lower panel listed from down upward (thin solid, dotted, dashed, long-dashed, and dot-dashed) correspond to $⟨I_1⟩=95$, 50, 10, 2, and 1.2, respectively. The lines in the upper panel listed from down upwards (long-dashed, dotted, dashed, and dot-dashed) correspond to comparatively small $⟨I_1⟩=2$, 1.6, 1.4, and 1.2, respectively. The bold lines $\alpha =J^2∕16h^2$ and $\alpha =J∕h$ display the asymptotic behavior of $\alpha$ for large and small $⟨I_1⟩$; the line $\alpha =J∕2h$ corresponds to $\alpha ={\alpha }_{\mathrm{th}}$.