Phonon-assisted coherent transport of excitations in Rydberg-dressed atom arrays

Polarons, which arise from the self-trapping interaction between electrons and lattice distortions in a solid, have been known and extensively investigated for nearly a century. Nevertheless, the study of polarons continues to be an active and evolving field, with ongoing advancements in both fundamental understanding and practical applications. Here, we present a microscopic model that exhibits a diverse range of dynamic behavior, arising from the intricate interplay between two excitation-phonon coupling terms. The derivation of the model is based on an experimentally feasible Rydberg-dressed system with dipole-dipole interactions, making it a promising candidate for realization in a Rydberg atoms quantum simulator for excitation dynamics interacting with optical phonons. Remarkably, our analysis reveals a growing asymmetry in Bloch oscillations, leading to a macroscopic transport of nonspreading excitations under a constant force. Finally, we demonstrate the robustness of our findings against on-site random potential.

Figure 1

Schematic illustration of all processes in the effective Hamiltonian ${\widehat{H}}_{\text{eff}}$ in Eq. (16), describing the dynamics of a single excitation in a one-dimensional array of Rydberg atoms located at $x_0(j+u_j)$, with $u_j$ being a dimensionless distortion from an equilibrium position. ${\widehat{H}}_{\text{ex}}$ describes a bare hopping (in the limit $u_j\to 0$) of an excitation at site $j$ to its neighboring sites with amplitude $J_0$ in the presence of a constant force $F$ and on-site disorder ${\epsilon }_j$, see Eq. (17). The effective hopping and on-site potential is further modified by the phonon couplings $g_J$ and $g_W$, respectively, see Eq. (19).

Figure 2

The top panel illustrates the center of mass motion $x\left(t\right)$ of an excitation (dashed blue line) under a constant, external force $F=0.2$, and the corresponding participation ratio $R_P\left(t\right)$ (solid green line), evaluated at the final evolution time $t_f$, and plotted as functions of the dimensionless parameter $\kappa$ [see Eq. (16) and definitions below it]. In the bottom panel, the ratio $\xi =\left|x\right|/R_P$ is shown. The peaks in the plot correspond to parameter regimes where a well-localized excitation is transferred under the influence of a constant force $F$. All physical parameters have been chosen with careful consideration of their experimental relevance, as discussed in the main text. The time evolution range is $t\in [0,t_f]$, where the final time $t_f$ is chosen as $t_f=2.1T_B=4.2\pi /F\approx 66$.

Figure 3

The panels illustrate the diverse dynamic behaviors observed in our study. The (a)–(d) first column showcases the time evolution of the excitation density $|{\psi }_j{\left(t\right)|}^2$ (color encoded), while (e)–(h) the second column, panels illustrates the corresponding temporal changes in the center-of-mass position $x\left(t\right)$ (dashed blue lines) and the participation ratio $R_P\left(t\right)$ (solid green lines). In the first row ($\kappa =0.8$), near-perfect Bloch oscillations are observed. However, upon closer examination, a subtle yet discernible asymmetry becomes apparent, as indicated by the nonzero value of $x\left(t\right)$. This asymmetry becomes more pronounced in the second row for a higher $\kappa$ value of 0.83. The subsequent rows of the figure provide insights into the time evolution of the excitation density for the specific cases of a well-localized wave function ($\kappa =0.834$) and a spreading wave function ($\kappa =0.86$). These distinct parameter regimes highlight the contrasting behavior and spatial characteristics of the excitations. All physical parameters are the same as in Fig. 2.

Figure 4

Color-encoded participation ratio $R_P\left(t_f\right)$, Eq. (26), at the final evolution time for a broad range of coupling strengths, $g_J\in [0,45]$ and $g_W\in [-20,20]$, in the presence of optical (left column, $\eta =0$) and acoustic phonons (right column, $\eta =1$). Each row corresponds to a specific value of the effective mass $m_{\mathrm{eff}}$. Panels (a),(b) correspond to $m_{\mathrm{eff}}=2$, panels (c),(d) correspond to $m_{\mathrm{eff}}=1$, and panels (e),(f) correspond to $m_{\mathrm{eff}}=0.5$. Across all panels, we observe a mixture of extended states (warm colors) and well-localized, nonspreading wave solutions (dark blue colors). Moreover, decreasing effective mass $m_{\mathrm{eff}}$ narrows the delocalized phase. Despite some differences, both types of phonons exhibit qualitatively similar behavior, see the discussion in the main text. The remaining parameters used for this analysis are ${\omega }_{\mathrm{eff}}=10, J_0=1$.

Figure 5

The influence of a constant force on the propagation of nonspreading solutions. Panels (a),(b) depict the color-encoded participation ratio $R_P\left(t_f\right)$ for optical and acoustic phonons, respectively, showing a shift in the boundary between extended and localized states due to the applied force. Panels (c),(d) display color-encoded $\xi \left(t_f\right)$, a measure for selecting well-localized solutions propagating in a single direction. Stable transport islands of such solutions are observed, indicated by warm colors. Panel (c) corresponds to optical phonons, while panel (d) corresponds to acoustic phonons. The remaining parameters used for this analysis are $F=0.2, m_{\mathrm{eff}}=0.5, {\omega }_{\mathrm{eff}}=10, J_0=1$. While comparing with Fig. 4 mind a shifted colorscale.

Figure 6

Transport of an excitation coupled to phonons under an external force $F=0.2$. Panels (a),(c) present color-encoded time evolution of the excitation density $|{\psi }_i{|}^2$, while panels (b),(d) present color-encoded time evolution of the phonon field $|u_i|$. The top row corresponds to optical phonons ($g_W=16, g_J=7$), while the bottom row corresponds to acoustic phonons ($g_W=17,g_J=10$).

Figure 7

Robustness of nondispersive moving solutions against on-site disorder, ${\epsilon }_j\in [-W/2,W/2]$. Panels (a),(b) display color-encoded time evolution of the excitation density $|{\psi }_i{|}^2$ for optical ($g_W=16, g_J=7$) and acoustic ($g_W=17, g_J=10$) phonons, respectively, disorder strength $W=0.6$. Panels (c),(d) show the center-of-mass position $x\left(t_f\right)$ and the participation ratio $R_P\left(t_f\right)$ at the final evolution time $t_f$, plotted as functions of the disorder amplitude $W$ [top lines (circles) correspond to acoustic phonons, while bottom lines (diamonds) to optical phonons]. Although the center-of-mass positions decreases, a substantial macroscopic transport remains visible. The remaining parameters used for this analysis are $F=0.2, m_{\mathrm{eff}}=0.5, {\omega }_{\mathrm{eff}}=10, J_0=1$.