Universal stability of coherently diffusive one-dimensional systems with respect to decoherence

Static disorder in a three-dimensional crystal degrades the ideal ballistic dynamics until it produces a localized regime. This metal-insulator transition is often preceded by coherent diffusion. By studying three paradigmatic one-dimensional models, namely, the Harper-Hofstadter-Aubry-André and Fibonacci tight-binding chains, along with the power-banded random matrix model, we show that whenever coherent diffusion is present, transport is exceptionally stable against decoherent noise. This is completely at odds with what happens for coherently ballistic and localized dynamics, where the diffusion coefficient strongly depends on the environmental decoherence. A universal dependence of the diffusion coefficient on the decoherence strength is analytically derived: The diffusion coefficient remains almost decoherence independent until the coherence time becomes comparable to the mean elastic scattering time. Thus, systems with a quantum diffusive regime could be used to design robust quantum wires. Moreover, our results might shed light on the functionality of many biological systems, which often operate at the border between the ballistic and localized regimes.

Figure 1

(a) HHAA model. Horizontal lines are site energies, Lorentzian uncertainties indicate decoherence, and $J$ is the hopping amplitude. (b) Excitation spreading for the HHAA model with $N=10000$ for different $W$. Coherent dynamics are shown with solid curves, while symbols show decoherent dynamics with a fixed value of ${\gamma }_{\varphi }=0.01J$. Symbols and curves share the same color to denote a given $W$. The vertical arrow shows the mean elastic scattering time ${\tau }_W$ [Eq. (3)]. Black dashed lines are the analytical estimates for ${\gamma }_{\varphi }=0$ (see the text for details). The vertical dotted line shows the decoherence time ${\tau }_{\varphi }=\hslash /{\gamma }_{\varphi }$. (c) Scaled diffusion coefficient $D/a^2$ vs the decoherence strength ${\gamma }_{\varphi }/J$ for different $W$. Black solid curves result from Eq. (4). Different regimes are extended (green circles), critical (red squares), and localized (blue triangles). The horizontal black dotted line is the coherent theoretical estimate $D_0/a^2$, the black dashed line is the asymptotic $D/a^2\simeq 2J^2/\hslash {\gamma }_{\varphi }$, and the vertical dotted line is the characteristic decoherence ${\gamma }_{\varphi }^c=2\hslash /{\tau }_W$. Numerical data were obtained by the QD method (symbols) for $N=1000$. In all panels $q=(\sqrt{5}-1)/2, J=1$, and $\hslash =1$.

Figure 2

(a) Probability of finding the excitation in the initial site $P_{00}\left(t\right)$ for a HHAA chain for a system evolving with $\mathcal{L}$ until time ${\tau }_R=25$ (first vertical dashed line) when the sign of the Hamiltonian is inverted, i.e., it continues evolving with ${\mathcal{L}}^{†}$. The Loschmidt echo occurs at $P_{00}(t=2{\tau }_R)\equiv M(t={\tau }_R)$ (second vertical dashed line) and corresponds to the purity. Different colors distinguish decoherence strengths. For the stronger ones the echo is not evident and $P_{00}\left(t\right)$ approaches a diffusive dynamics (black curve). (b) Loschmidt echo decay $M\left(t\right)$ for different ${\gamma }_{\varphi }$ at the critical point computed with the QD method. The dashed line is a prediction based on the coherent diffusion coefficient resulting from the Hamiltonian dynamics. Vertical dotted lines show $t=\frac{4\hslash }{{\gamma }_{\varphi }}$. All data were obtained with $q=(\sqrt{5}-1)/2, J=1, \hslash =1, W=2J$, and $N=1000$.

Figure 3

Normalized diffusion coefficient $D/D_0$ vs renormalized decoherence strength ${\tau }_W/{\tau }_{\varphi } [D_0=D({\gamma }_{\varphi }=0$)]. Symbols represent results obtained from QD dynamics for the HHAA chain at criticality (closed red squares); the Fibonacci chain (closed dark red triangles); and the PBRM model in the extended phase (open green diamonds), at the critical point (open red squares), and in the localized phase (open blue circles). The solid curve is the universal Eq. (5) while the black dashed line is the limit of ${\tau }_W/{\tau }_{\varphi }>2$. The horizontal dotted line is $D=D_0$. For the HHAA and the Fibonacci chains, ${\tau }_W$ and $D_0$ were computed analytically [Eq. (3)]. For the PBRM model $b=0.01$ and $D_0$ results are from a fitted ${\tau }_W$.

Figure 4

(a) Steady-state current vs ${\gamma }_{\varphi }/j$ for the HHAA model for $N=100$ and $W=2J$ obtained with three different methods: the master equation [green (light gray) curve], the average transfer time method (red circles), and the heuristic expression (A7) [blue (dark gray) curve]. (b) Rescaled steady-state current $I_{\text{SS}}{N}^2$ as a function of dephasing (${\gamma }_{\varphi }/J$) in the extended, critical, and localized (colors) regime for different system's sizes $N=\{20,40,100\}$. The $I_{\text{SS}}{N}^2$ is calculated using the ME method and shown with different line types depending on $N$. The diffusion-coefficient-based (heuristic) estimation of the current for $N=1000$ is shown with yellow (light gray) dash–double-dotted curves.

Figure 5

Spreading of excitation for the HHAA model. The initial state is at a single site in the middle of the chain. Solid curves show the results of the pure dephasing Haken-Strobl model from a Lindbladian evolution (ME). Dashed curves show the results of the quantum-drift simulation. Colors and line types indicate different ${\gamma }_{\varphi }$ for the ME and QD data, respectively. The parameters are $N=100$ and (a) $W=0$, (b) $W=2J$, and (c) $W=20J$.

Figure 6

Diffusion coefficient $D/a^2$, calculated using the Green-Kubo approach [Eq. (D2)], vs ${\gamma }_{\varphi }/J$ for the HHAA model with Haken-Stobl dephasing for $N=\{40,100,1000,4000\}$ (yellow, green, red, and blue curves, respectively) and (a) $W/J=0$ (metallic regime), (b) $W/J=2.0$ (MIT regime), and (c) $W/J=20$ (insulator regime). Vertical dashed lines indicate the values of ${\gamma }_{\varphi }$ below which finite-size effects are relevant. The dependence of this value on $N$ is shown on top of the black arrow. (d) Diffusion coefficient $D/a^2$ vs $W/J$ for different ${\gamma }_{\varphi }$ (see the legend). Curves are calculated from Eq. (D2). Symbols show results obtained from the quantum-drift method (Appendix pp2). Here $N=1000$.

Figure 7

Diffusion coefficient vs the decoherence strength for the HHAA model. Symbols represent data obtained from the time evolution, dotted curves from Eq. (D2) (Green-Kubo expression), dash-dotted colored curves from Eq. (D16) ($\delta$ process), and black solid curves from Eq. (D8) (Poisson process). Also shown is the diffusion coefficient in the strong dephasing regime (black dashed line) and the diffusion coefficient in the absence of dephasing at the MIT ($D_0$) as a horizontal dotted line [see Eq. (C8)]. The yellow solid curve corresponds to Eq. (D12). (b) is the same as (a) but in lin-log scale and excluding the extended regime data.

Figure 8

Diffusion coefficient vs decoherence strength for the HHAA model. Symbols represent data obtained from the QD time evolution, dotted curves results from the Green-Kubo formula, and dashed lines the analytical estimations. (a) Extended phase. The analytical results correspond to Eq. (D18) (black dashed lines). (b) Localized phase. The analytical results correspond to Eq. (D21) (black dashed lines). The parameters are $N=1000, Q=(\sqrt{5}-1)/2, J=1$, and $\hslash =1$.

Figure 9

Time evolution of the spreading of an excitation in a HHAA chain with $q=q_g/m$ at criticality for (a) ${\gamma }_{\varphi }=0$ and (b) ${\gamma }_{\varphi }=0.02$. The vertical dashed lines show ${\tau }_W$ [Eq. (C7)], the crosswise dashed lines correspond to ${\sigma }^2\left(t\right)=2D_{0}t$ [Eq. (C8)], and the dotted line shows the initial ballistic evolution [Eq. (C3)]. (c) Diffusion coefficient as a function of the dephasing strength ${\gamma }_{\varphi }$. Here $D$ is calculated from the QD dynamics (symbols) and from Eq. (D2) (dashed lines). The horizontal dotted lines show $D_0$ [Eq. (C8)] and the vertical dashed lines ${\gamma }_{\varphi }^c=\frac{2\hslash }{{\tau }_W}$. (d) Diffusion coefficient and dephasing strength are rescaled by $D_0$ and ${\gamma }_{\varphi }^c$, respectively. Colors represent different values of $m$ as indicated in the legends. The parameters are $N=10000$ and $W=2J$.

Figure 10

(a) Time evolution of the spreading in the absence of dephasing for $W=\{1,3.15,5\}$ in the Fibonacci chain (lime green, maroon, and periwinkle solid curves, respectively). The black dotted line shows the initial ballistic spreading [Eq. (C3)], the vertical arrows show ${\tau }_W$ [Eq. (C6)], and the black dashed lines show the power-law behavior after ${\tau }_W$. (b) Diffusion coefficient as a function of the dephasing strength. The symbols are fitted directly from the QD dynamics in the presence of dephasing, the black solid curves show Eq. (D8) numerically integrated using a Poisson process, and the yellow (light gray) solid curves show the analytical expression (F3). The vertical arrows correspond to $2\hslash /{\tau }_W$, where the strong dephasing regime indicated by a black dotted line starts. The black dashed lines indicate the power dependence for small ${\gamma }_{\varphi }$. The simulations were done for a chain of length $N={10}^4$.

Figure 11

(a) Time evolution of the spreading of an initially localized excitation in the absence of dephasing in the PBRM model for $\mu =1, b=0.01$, and $N=\{100,1000,5000,10000\}$ (yellow, green, red, and blue curves, respectively). The vertical lines denote ${\tau }_W$ (black) and $t_s$ (colored), while the crosswise dashed line represents the theoretical diffusive spreading ${\sigma }_0^2\left(t\right)/v_0^2\approx \sqrt{2}t$. (b) Diffusion coefficient obtained through Eq. (D2) for (from top to bottom) $\mu =\{0.80,1.00,1.30\}$ (colored curves). The vertical black dashed line marks the characteristic dephasing where the dynamics start to be dominated by noise and the initial ballistic dynamics (strong Zeno regime). The vertical colored dotted lines show the values of ${\gamma }_{\varphi }=2\hslash /t_s$ below which the finite-size effect starts to be relevant. The dependence of the values on $N$ is indicated in each plot.

Figure 12

Time evolution of the purity (Loschmidt echo) $M\left(t\right)$ with different dephasing strength in a HHAA chain with $N=1000$ and (a) $W=J$ (extended phase, (b) $W=2J$ (MIT), and (c) $W=3J$ (localized phase). Colored curves represent different ${\gamma }_{\varphi }$ as indicated in the legends. The black dashed lines are theoretical predictions $M\left(t\right)\propto \frac{1}{\sqrt{Dt}}$, with $D$ obtained from (a) and (c) Eq. (D2) and (b) Eq. (C8).

Figure 13

(a) Probability of finding the particle in the initial site $P_{00}\left(t\right)$ for a HHAA chain with $W=2J$. Up to time ${\tau }_R$ (${\tau }_R=25$, first vertical dashed line) the system evolves with $\mathcal{L}$; thereafter the sign of the Hamiltonian is inverted (the system evolves with ${\mathcal{L}}^{†}$). The value of $P_{00}\left(t\right)$ at $2{\tau }_R$ (echo, second vertical dashed line) corresponds to the purity of the system at time ${\tau }_R \left[M\left({\tau }_R\right)\right]$. Colored curves represent different ${\gamma }_{\varphi }$ as indicated in the legend. (b) Purity (echo) at fixed time ${\tau }_R=\{10,25,100,500\}$ as a function of the dephasing time ${\tau }_{\varphi }=\hslash /{\gamma }_{\varphi }$ in a HHAA chain with $W=2J$. The vertical colored dashed lines mark ${\tau }_R/4$, while the colored dash-dotted lines show the analytical behavior for ${\tau }_{\varphi }\ll {\tau }_R$. (c) Time at which the variance of the wave packet reaches its minimum after a Hamiltonian inversion at ${\tau }_R$ in a HHAA chain with $W=2J$. Vertical colored dashed lines represent $2{\tau }_R$, while the horizontal lines stand for ${\tau }_R$.