Efficient excitation transfer in an LH2-inspired nanoscale stacked ring geometry

TL;DR

We address the problem of achieving efficient inter-layer excitation transfer in LH2-inspired nanoscale ring geometries. The main approach uses a Markovian master equation for collectively interacting two-level emitters arranged in three-dimensional stacked rings, with a Bloch-mode decomposition by angular momentum $m$ to identify subradiant pathways. Key findings show that inter-layer transfer is maximized via dark subradiant modes and can reach near-unity values in nanoscale geometries when vertical separation is tuned, while asymmetric transfer favors sparse-to-dense configurations; disorder and dephasing can further modulate transport via noise-assisted mechanisms. The work provides design principles for biomimetic light-matter platforms and emitter arrays aimed at efficient energy transfer, with potential applications in quantum networks and sensing.

Abstract

Subwavelength ring-shaped structures of quantum emitters exhibit outstanding radiation properties and are useful for antennas, excitation transport, and storage. Taking inspiration from the oligomeric geometry of biological light-harvesting 2 (LH2) complexes, we study here generic examples and predict highly efficient excitation transfer in a three-dimensional (3D) subwavelength concentric stacked ring structure with a diameter of 400 $nm$, formed by two-level atoms. Utilizing the quantum optical open system master equation approach for the collective dipole dynamics, we demonstrate that, depending on the system parameters, our bio-mimicked 3D ring enables efficient excitation transfer between two ring layers. Our findings open prospects for engineering other biomimetic light-matter platforms and emitter arrays to achieve efficient energy transfer.

Figures (10)

• Figure 1: (a) Pigment arrangements of biological LH2 complex of purple photosynthetic bacteria Rbl. acidophilus (image taken from bujak:APL:2011 and flipped vertically for the analogy) with bacteriochlorophyll (BChl) (Green: BChl B800, Red: BChl B850) and carotenoids (in yellow). The diameter of the B800 ring is around 6 nm (parameters are taken from Ref. Montemayor:JPCB:2018 and listed in Appendix-\ref{['apen800-850']}). (b) Bio-mimicked enlarged stacked concentric nanoring structure formed of two-level emitters with 800 nm transition wavelength (parameters in Table-\ref{['tab2']}). We consider one BChl as a point-like emitter (blue circles) with fixed dipole orientation. Top ring ${R_1}$ is with $N_1 = 9$ emitters. Bottom ring ${R_2}$ has $N_2 = 18$ emitters. They are arranged in two concentric rings (${R_{2_a}}$ and ${R_{2_b}}$ with $N_{2_{a(b)}} = 9$ dipoles each). Small black-solid lines (or magenta-dotted lines) indicate the tangential (transverse) dipole orientation. The stacked geometry combines into nine unit cells. The orange-dashed contour '$\alpha$' indicates one unit cell, which contains $d = 3$ dipoles. (c) Angles ($\theta, \phi$) define the orientation of the $i^{th}$ dipole with dipole-moment $\boldsymbol{\mu}_i$. (d) Collective frequency shifts $(\Omega_m/\Gamma_0)$ and (e) decay rates $(\Gamma_m/\Gamma_0)$ for the eigenmodes with angular momentum $m$, corresponding to the geometry in (b). For each 800 nm dipole the calculated spontaneous emission rate is $\Gamma_0 \sim$ 25.7 MHz.
• Figure 2: (a) Bio-ring geometry (not to scale) with all 800 nm dipoles (blue-solid circles) and biological dipole orientations (indicated by small black-solid lines). Parameters are taken from Ref. Montemayor:JPCB:2018 and enlisted in Appendix-\ref{['apen800-850']}. (b) The collective energy shift ($\Omega_m/\Gamma_0$) and (c) collective decay rate ($\Gamma_m/\Gamma_0$) of the top-ring ${R_1}$ for different modes $m$. Blue-dotted circles denote the darkest mode $m=\pm 4$. Variation of $\langle\hat{\sigma}^{ee}_m (t)\rangle_{R_2}$ with scaled time $\Gamma_0t$ for $m=\pm 4$ (higher-transfer) and $0$ (less-transfer) at (d) $Z_1 = 16.5$ Å (biological vertical layer separation Montemayor:JPCB:2018) and (e) $Z'_1 = 8$ Å ($Z'_1 < Z_1$). For $Z'_1$ the maximum value of inter-ring excitation transfer ${R_1}\Rightarrow{R_2}$, can be boosted to 100% (e) from 21% (d), for the most sub-radiant mode $m = \pm 4$ (blue-solid curves) of ring ${R_1}$. Plots (f) and (g) show the quantitative estimations for the reverse process ${R_2}\Rightarrow{R_1}$ for the darkest mode of ring ${R_2}$. Those displays around 10% and 80% transfer to ring ${R_1}$, respectively.
• Figure 3: Inter-layer excitation transfer (${R_1}\Rightarrow{R_2}$) for nanoring [Fig. \ref{['fig1']}(b)]: with tunable dipole-orientations in $R_2$ (a)-(b), rotation of $R_2$ (c) and increase of ring-size (d). We consider the dipoles to be tangentially oriented in ring ${R_1}$. For different dipole orientations in ring ${R_2}$ (angles $\theta_2, \phi_2$), we show ${\rm Max}[\langle\hat{\sigma}^{ee}_m(t)\rangle_{R_2}] \in \{0,1\}$ for the anti-symmetric sub-radiant mode $m=\pm 4$ (a) and for the symmetric radiant mode $m=0$ (b) of sparse ring ${R_1}$. A few example snaps for dipole orientations in the ring layers are displayed for the mentioned ($\theta_2, \phi_2$) values in the plot (a). For certain dipole orientations in $R_2$ the $m=\pm 4$ exhibits a maximum of excitation transfer $\sim$ 99.5%, i.e., ${\rm Max} (\langle\hat{\sigma}^{ee}_m(t)\rangle_{R_2}) \sim 1$ (indicated by the blue arrow). In contrast, for the radiant mode, $m=0$ the excitation transfer is generally less (indicated by the green arrow), around 25% (b). We consider all dipoles to be in tangential orientation (in both layers) for plots (c) and (d). (c) With the rotation of ring ${R_2}$ ($\phi_{{rot}_2}$), we plot ${\rm Max} (\langle\hat{\sigma}^{ee}_m(t)\rangle_{R_2})$ for the above mentioned modes. Plot (d) shows the effect of increasing ring size (ring radius $r_1$ and vertical ring separation $Z_1 = r_1/(20/8)$ are changed together) on excitation transfer with $m=0, \pm 4$. The inset (e) shows the same variation (as in (d)), but for longer $r_1$, i.e., up to $3 \lambda$.
• Figure 4: Computed populations $\langle \hat{n} \rangle_i$ for the $i^{th}$ eigenstate of stacked LH2 rings. 28 eigenstates are indexed in $x$-axis. Plots (a) and (b) are with actual LH2 ring-layer separations, i.e., 16.5 Å and bio-dipole orientations. Plots (c),(d) with the above dipole orientation, but with decreased vertical inter-layer separation 8 Å; for all three rings, i.e., ${R_1}$ (in black), ${R_{2_a}}$ (in red) and ${R_{2_b}}$ (in blue), as displayed. In (c) and (d) some of the eigenstates (in ring $R_1$ and $R_3$ mostly, due to similar dipole orientations) exhibit shared populations in higher proportions, i.e., improved hybridization than the former case as in (a)-(b).
• Figure 5: Theoretical estimations with B800-B850 LH2 model $C_9$ rings (see Table-\ref{['tab1']} for parameters). (a) Collective energy shift ($\Omega_m/\Gamma_0$) and (b) collective decay rate ($\Gamma_m/\Gamma_0$) for angular momentum modes $m$. $\Gamma_0$ corresponds to the spontaneous emission rate for 800 nm dipole $\sim 25.7$ MHz. (c) Temporal evolution of the excitation energy transfer ${R_1}$$\Rightarrow$${R_2}$ for different eigenmode $m$ of ring ${R_1}$ with scaled time $\Gamma_0 t$ (here, $3\times 10^{-5}\Gamma_0 t \equiv 1.1~ps$).