Readout of quasiperiodic systems using qubits

We develop a theoretical scheme to perform a readout of the properties of a quasiperiodic system by coupling it to one or two qubits. We show that the decoherence dynamics of a single qubit coupled via a pure dephasing type term to a one-dimensional quasiperiodic system with a potential given by the André-Aubry-Harper (AAH) model and its generalized versions (GAAH model) is sensitive to the nature of the single-particle eigenstates (SPEs). More specifically, we can use the non-Markovianity of the qubit dynamics as quantified by the backflow of information to clearly distinguish the localized, delocalized, and mixed regimes with a mobility edge of the AAH and GAAH models and evidence the transition between them. By attaching two qubits at distinct sites of the system, we demonstrate that the transport property of the quasiperiodic system is encoded in the scaling of the threshold time to develop correlations between the qubits with the distance between the qubits. This scaling can also be used to distinguish and infer different regimes of transport such as ballistic, diffusive, and no transport engendered by SPEs that are delocalized, critical, and localized respectively. In addition, the localization length of the SPEs can also be gleaned from the exponential decay of correlations at long times as a function of distance between qubits. When there is a mobility edge allowing the coexistence of different kinds of SPEs in the spectrum, such as the coexistence of localized and delocalized states in the GAAH models, we find that the transport behavior and the scaling of the threshold time with qubit separation are governed by the fastest spreading states.

Figure 1

Schematic representation of the readout of a quasiperiodic system using one or two qubits. (a) A single qubit (represented by the red circle) with energy splitting ${\omega }_A$ between its two levels is coupled to the $i$ th site of an one-dimensional quasiperiodic system (the black lines represent the on-site potential of the lattice). (b) Two qubits with splitting energies ${\omega }_A$ and ${\omega }_B$ are coupled to two different sites $i$ and $j$ of the quasiperiodic system respectively.

Figure 2

(a) Ordered ($\lambda =0$, blue) and quasiperiodic AAH on-site potential ($\lambda =0.5$, orange) [see Eq. (1)] as a function of site index. (b) Vacuum decoherence factor for a single qubit as a function of the site to which it is coupled reflecting the underlying on-site potential (total number of sites = 55). Vertical lines denote the Fibbonacci numbers at which point the potential and decoherence factors almost repeat themselves, as expected for the irrational value $b=\frac{\sqrt{5}-1}{2}$.

Figure 3

(a) Vacuum decoherence factor as a function of time for a 610-site regular AAH model ($\alpha =0$) with the qubit coupled to site $i=72$. The dynamics goes from damped to oscillatory as $\lambda$ is tuned over delocalized ($\lambda =0$ and $\lambda =0.5$), critical ($\lambda =1.0$) and localized ($\lambda =1.2$) regimes of the AAH model. (b) Inverse participation ratio of eigenmodes at the qubit coupling site $i, P_i$ (see text for definition) as a function of the number of sites $N$. $P_i$ scales as $1/N$ in the delocalized regime, as $N^0$ for the localized case, and as $N^{-b}$ with $0<b<1$ in the critical regime.

Figure 4

Backflow of information $\mathcal{N}$ measuring the non-Markovianity in the decoherence dynamics of a qubit coupled to different sites (see legend) of a quasiperiodic lattice (regular AAH model $\alpha =0$) as a function of $\lambda$ (number of sites $N=2584$). Clearly the non-Markovianity measure is sensitive to the localization-delocalization transition at $\lambda =1$.

Figure 5

(a) Fraction of localized states as a function of $\alpha \text{−}\lambda$ for the GAAH model. The critical lines in purple and black separate the region with a mobility edge from the delocalized and localized regimes respectively. Backflow of information $\mathcal{N}$ as a function of $\alpha \text{−}\lambda$ for a GAAH lattice with number of sites $N=2584$ with a qubit coupled to site $i=1$ (b) and $i=19$ (c). When there is a mobility edge, $\mathcal{N}$ becomes dependent on the site to which the qubit is coupled.

Figure 6

Dynamics of correlation between two qubits coupled to sites $i$ and $j$ of the regular AAH model, as measured by the magnitude of the covariance, is shown for the delocalized regime ($\lambda <1$) in (a) and the critical regime in (b) ($\lambda =1$). In both cases the correlations start building in a significant manner after a threshold time $t^{*}$. While $t^{*}$ scales linearly with the separation $|i-j|$ in the delocalized regime as shown in (c), in the critical regime it displays quadratic scaling; see (d). Here, we have fixed $i=N/4, j=3N/4$ and varied the system size as $N=400$, 800, and 1200 to change $|i-j|$.

Figure 7

Dynamics of correlation between two qubits coupled to sites $i$ and $j$ of the open regular AAH model with system size $N=400$, as measured by the magnitude of the covariance, for the localized regime with $\lambda =1.2$. (a) Exponentially decaying magnitude of correlations build up when the two sites are comparable to the localization length (blue curve). Here, we have taken $i=170$ and $j=230$. When the separation is much larger than the localization length no correlations build even after long time (yellow curve). Here, we have chosen $i=140$ and $j=260$. (b) Here, we have shown that the long-time ($\sim {10}^4$) correlation decays exponentially with the distance between two qubits in the localized regime.

Figure 8

Magnitude of covariance at long times ($\sim {10}^4$) is plotted as a function the distance $i-j$ between two qubits in the localized regime of the regular AAH model with $\lambda =1.2$ (blue dots), 1.3 (orange squares), and 1.4 (green squares). Clearly the exponential decay of the covariance agrees with the straight lines, which represent an exponential decay of the form $exp(-|i-j|ln[\lambda \left]\right)$ expected from the behavior of localized SPEs.

Figure 9

Correlation strength in the long-time limit for two qubits coupled to sites $i$ and $j$ of the GAAH model in the $\alpha \text{−}\lambda$ plane. In (a) $[i=290,j=690]$ and (b) $[i=233,j=610]$ the correlation strength is large (small) in the fully delocalized (localized) regimes with no mobility edge. In contrast, the correlation shows dependence on the site to which the qubits couple when there is a mobility edge. In (a) and (b) system size $N=2584$. In (c) [(d)] we show the linear [quadratic] scaling of the threshold time $t^{*}$ as a function of qubit separation for parameters above the black critical line [on the black critical line]. Here, we have fixed $i=N/4, j=3N/4$ and varied the system size as $N$ to change $|i-j|$.