Environment-Assisted Quantum Transport in a 10-qubit Network

The way in which energy is transported through an interacting system governs fundamental properties in nature such as thermal and electric conductivity or phase changes. Remarkably, environmental noise can enhance the transport, an effect known as environment-assisted quantum transport (ENAQT). In this Letter, we study ENAQT in a network of coupled spins subject to engineered static disorder and temporally varying dephasing noise. The interacting spin network is realized in a chain of trapped atomic ions, and energy transport is represented by the transfer of electronic excitation between ions. With increasing noise strength, we observe a crossover from coherent dynamics and Anderson localization to ENAQT and finally a suppression of transport due to the quantum Zeno effect. We find that in the regime where ENAQT is most effective, the transport is mainly diffusive, displaying coherences only at very short times. Further, we show that dephasing characterized by non-Markovian noise can maintain coherences longer than white noise dephasing, with a strong influence of the spectral structure on the transport efficiency. Our approach represents a controlled and scalable way to investigate quantum transport in many-body networks under static disorder and dynamic noise.

Figure 1

Main graph: Transport efficiency ${\eta }_8$ to the target (ion 8) under different strengths of static disorder (blue: $B_{\max }=0.5J_{\max }$; red: $B_{\max }=2.5J_{\max }$) and Markovian-like dephasing with rate $\gamma$. Experimental points (shown as dark squares and triangles) result from averaging over 20–40 random realizations of disorder and noise, with 25 experimental repetitions each. Error bars are derived via bootstrapping, based on 1000 samples (see [29] for details). The regimes of localization, ENAQT, and the quantum Zeno effect are indicated in gray. The data agree well with theoretical simulations of the coin-tossing random process (light bullets) realized in the experiment, while simulations with ideal Markovian white noise (lines) underestimate ENAQT at large $\gamma$. The simulation averages over 300 random realizations. Inset (a): Sketch of the transport network. The ions experience a long-range coupling, with darker and thicker connections indicating higher coupling strengths. The green arrows denote the source (3) and the target (8) for the excitation in the ion network. Inset (b): Sketch of the ion chain representing interacting spin-$½$ particles as blue circles, with the spin states denoted by black arrows. The ions are subject to random static and dynamic on-site excitation energies, indicated by $B_i$ and $W_i(t)$.

Figure 2

Excitation probability at ion 8 as a function of time, for strong static disorder $B_{\max }=2.5J_{\max }$ and increasing dephasing rate, from (a) to (d), $\gamma /J_{\max }=0|0.23|1|3.9$. Each data set (red to magenta triangles) results from averaging over 20–40 random realizations, with 25 repetitions each. Error bars are derived with bootstrapping [29], based on 1000 samples. With increasing $\gamma$, coherent oscillations damp out, and the data converge towards a model following diffusive, classical rate equations (blue solid line). This theoretical approximation is valid for times $t\gg 1/\gamma$. (The crossover $t_c=1/\gamma$ is illustrated by a blue dashed line.) Shaded areas show the time evolution of a theoretical model with ideal Markovian noise, averaged over 100 random realizations.

Figure 3

Left panels: Single-ion resolved excitation dynamics $p_i(t)$ for (a) the unperturbed system (no static disorder and no noise), (b) static disorder $B_{\max }=2.5J_{\max }$ with dephasing $\gamma =J_{\max }$, and (c) static disorder $B_{\max }=2.5J_{\max }$ with $\gamma =18.4J_{\max }$. The orange dotted line shows the maximum speed at which an excitation spreads (see Supplemental Material [42]) in order to estimate until when the reflection from the left boundary can be neglected (blue arrows). Right panels: Spatial width of the excitation wave packet ${\sigma }_{\mathrm{WP}}(t)$, calculated from the data in the left panels. Blue solid lines are fits of the form ${\sigma }_{\mathrm{WP}}=A{t}^C$ (fits from the respective other panels are included as dashed lines for comparison). The vertical black line at ${\sigma }_{\max }=5.3$ is the expected maximum of the wave-packet width, for a single excitation distributed equally over all ions. (a) $A=6.5\pm 0.5,C=1.05\pm 0.06$, (b) $A=3.2\pm 0.3,C=0.74\pm 0.17$, (c) $A=1.7\pm 0.1$, $C=0.44\pm 0.02$. All error bars are derived via bootstrapping [29], based on 100 samples.

Figure 4

Comparison of different noise models. (a) Spectral density functions $S(\omega )$ of (1) white noise and (2)–(6) non-Markovian noise models of Lorentzian shape. The curves are averaged over about 30 random realizations, each generated by a Gaussian random process based on 600 sampling points. The inset shows a zoom into the low-frequency domain. Vertical grey lines denote the difference frequencies between all eigenenergies of the disordered system. (b) Comparison of ${\eta }_{\text{target}}$ to target ions 8–10 subject to strong static disorder and the noise models shown in (a), as indicated by corresponding colors and numbering. Noise model zero indicates bare static disorder without noise. While broadband noise in the correct frequency range generically enhances transport efficiencies, for narrow-band noise the enhancement depends on the source and target ions. Each data point results from averaging over 25–30 random realizations, with 15 repetitions each. Error bars are derived with bootstrapping [29], based on 1000 samples. (c,d) Excitation probability of target ion 9 as a function of time under strong static disorder and different noise models. Panel (c) shows the result for the Markovian noise model (1). Oscillations, indicating coherent dynamics, are strongly damped. Panel (d) shows the effects of a narrow-band Lorentzian noise model covering only a few eigenstates (model 6). Here, coherences are more strongly maintained and are clearly discriminable from measurement errors up to about 30 ms.