Finite Su-Schrieffer-Heeger chains coupled to a two-level emitter: Hybridization of edge and emitter states

The Hamiltonian for the one-dimensional Su-Schrieffer-Heeger (SSH) chain is one of the simplest Hamiltonians that supports topological states. This work considers between one and three finite SSH chains with open boundary conditions that either share a lattice site (or cavity), which—in turn—is coupled to a two-level emitter, or are coupled to the same two-level emitter. We investigate the system properties as functions of the emitter-cavity coupling strength $g$ and the detuning between the emitter energy and the center of the band gap. It is found that the energy scale introduced by the edge states that are supported by the uncoupled finite SSH chains leads to a $g$-dependent hybridization of the emitter and edge states that is unique to finite-chain systems. A highly accurate analytical three-state model that captures the band-gap physics of $k$-chain $(k\ge 1)$ systems is developed. To quantify the robustness of the topological system characteristics, the inverse participation ratio for the cavity-shared and emitter-shared systems consisting of $k$ chains is analyzed as a function of the on-site disorder strength. The $g$-dependent hybridization of the emitter and uncoupled edge states can be probed dynamically.

Figure 1

Illustration of the SSH chain. Each unit cell, marked by an orange box and labeled by $n$, contains a cavity that belongs to sublattice 1 (solid blue circle) and a cavity that belongs to sublattice 2 (solid red circle). The intraunit and interunit hopping energies are denoted by $u$ and $v$, respectively. The spacing between neighboring unit cells is $2a$.

Figure 2

Characteristics of single finite SSH chain (without the emitter) with open BCs, $N=31$, and $v/u=2$. The blue circles in the main panel show the eigenenergies as a function of the normalized eigenstate index. The eigenenergies are distributed symmetrically around zero energy. The spectrum features two nearly continuous energy bands and two states in the band gap with energy close to zero. The energy gap has an energy width of $2(v-u)$ while the bands each have a width of $2u$; correspondingly, the lowest and highest energy levels are separated by $2(u+v)$. For comparison, the solid black lines show the energy bands of the infinite chain with periodic BCs; the blue circles agree quite well with the solid lines. The eigenstates of the two edge states for finite $N$ are illustrated in the upper left and the lower right insets. The size of the solid circles is directly proportional to the square of the expansion coefficient of the site basis state $|n,j\rangle$, with the color marking the sign of the expansion coefficients (see text for details).

Figure 3

Schematic of emitter- and cavity-shared $k$-chain systems. The blue lines represent SSH chains; from the left to the right, the number of chains increases from one to three. Each chain is coupled to an emitter (orange circle with two vertical lines). The red line represents the emitter-cavity coupling. In the emitter-shared scenario (top middle and top right panels), chains do not share cavities and one cavity of each chain is coupled to the emitter; for $g=0$, the chains in the two- and three-chain cases are decoupled. In the cavity-shared scenario (bottom middle and bottom right panels), the two- and three-chain systems share the cavity that the emitter is coupled to. For the one-chain system, the number of sites is $2N$ (there exists no distinction between the emitter- and cavity-coupled one-chain systems). The number of sites is $4N$ and $6N$ for the emitter-shared two- and three-chain systems, respectively. The number of sites is $4N-1$ and $6N-2$ for the cavity-shared two- and three-chain systems, respectively. The number of edge and dark states reported next to the schematic applies to the cavity part of the Hilbert space, excluding the emitter; in the special case where the emitter detuning is zero, the coupled systems support one additional dark state (see Appendix pp1).

Figure 4

Static properties of the one-chain, cavity-shared two-chain, and cavity-shared three-chain systems with open BCs, $N=15$, $n^{*}=8$, and $v/u=2$ as a function of the detuning $\hslash {\omega }_e$; the figure zooms in on the physics in the vicinity of the middle of the gap. The solid, dashed, and dotted lines are for the one-chain, two-chain, and three-chain systems, respectively. The coupling $g/u$ increases from left to right; specifically, the first, second, third, and fourth columns are for $g/u={10}^{-4}$, ${10}^{-3}$, ${10}^{-2}$, and ${10}^{-1}$, respectively. The top row shows the three energy levels $E_l^{\text{gap}}$ located in the energy gap. The left axis reports the energy in units of $u$ ($u$ is much larger than the typical scale of the energies in the gap) while the right axis reports the energy in units of $E_{\text{edge}}$ ($E_{\text{edge}}\approx 4.58\times {10}^{-5}u$). The bottom row shows the overlap square ${\mathcal{O}}_l$. Note that the scale of the figures is the same for the first and second columns but changes for the third and fourth columns.

Figure 5

Contribution $|c_e{|}^2$ of the state $|\text{vac};e\rangle$ to the gap states as a function of $g/u$ for the one-chain system with $N=15$, $n^{*}=8$, $v/u=2$, and $\hslash {\omega }_e=0$; open and periodic BCs are considered. The lines show approximate analytical results while the symbols show results obtained by diagonalizing the full Hamiltonian. The red solid line and symbols show $|c_e{|}^2$ of the zero-energy state for open BCs. The inset in the lower left (red box) represents the corresponding eigenstate for $g/u={10}^{-4}$. For comparison, the green dashed line and symbols show $|c_e{|}^2$ for the zero-energy state for periodic BCs. The inset in the upper left (green box) represents the corresponding eigenstate for $g/u={10}^{-4}$. The blue dotted line and symbols show $|c_e{|}^2$ of the finite-energy states for open BCs. The insets in the lower right corner (blue boxes) represent the corresponding eigenstates for $g/u={10}^{-1}$. Note that the sketches of the photonic populations for $g/u={10}^{-4}$ are, to enhance readability, multiplied by ${10}^6$ (upper green box on the left) and ${10}^2$ (lower red box on the left) relative to those for $g/u={10}^{-1}$. The green cross in the insets marks the unit cell that the emitter is coupled to. As a reference point, the arrow marks the coupling strength for which $|G/E_{\text{edge}}|=1$.

Figure 6

Characteristics of the cavity-shared two-chain system (without the emitter) with open BCs, $N=15$, and $v/u=2$; the shared cavity belongs to sublattice 1 and is part of the $8\mathrm{th}$ unit cell (i.e., $n^{*}=8$). The main panel shows the eigenenergies as a function of the normalized eigenstate index. The energy spectrum features two nearly continuous energy bands, five states in the band gap with energy equal to or close to zero (the state with energy exactly equal to zero is a dark state), one nontopological bound state below the bottom of the lower energy band, and one nontopological bound state above the top of the upper energy band. Insets: [(a), (b)] The eigenstates of the two nontopological bound states are shown in the upper left and upper right insets; they have energy $\pm 3.516u$. (c) The edge state shown in the lower left inset has an energy of $-4.56\times {10}^{-5}u$. (d) The dark state shown in the lower right inset has an energy of zero. The first and second SSH chains are chosen to lie along the $x$ and $y$ axes (the axes are chosen arbitrarily). As in Fig. 2, positive and negative expansion coefficients are shown by blue and red circles, respectively, with the size of the circles being proportional to the square of the expansion coefficients. As opposed to offsetting the sites that belong to sublattices 1 and 2 (as in Fig. 2), the lighter blue and lighter red colors correspond to sites that belong to sublattice 1 while the darker blue and darker red colors correspond to sites that belong to sublattice 2. Note the different scales of the axes of the insets.

Figure 7

IPR [see Eq. (15)] for uniformly distributed on-site disorder as a function of $\mathrm{\Delta }/u$ (this quantity defines the scaled disorder strength window) for $\hslash {\omega }_e/u=-5\times {10}^{-5}$, $N=15$, $v/u=2$, $n^{*}=8$, open BCs, and [(ai)–(aiii)] $g/u={10}^{-4}$, [(bi)–(biii)] $g/u={10}^{-3}$, [(ci)–(ciii)] $g/u={10}^{-2}$, and [(di)–(diii)] $g/u={10}^{-1}$. The left, middle, and right columns are for the one-chain, cavity-shared two-chain, and emitter-shared two-chain systems, respectively. The IPR is calculated for the three eigenstates of the system with disorder that have the largest overlap with the gap states of the corresponding disorder-free system. The symbols and error bars are obtained by averaging over $5\times {10}^3$ disorder realizations. In panels (ai) and (aii) as well as in parts of panels (bi) and (ci), the blue and green symbols are essentially indistinguishable. In the large-disorder regime of panels (dii) and (diii), the green and red symbols are essentially indistinguishable.

Figure 8

Frequency of energy, color coded via the scale shown on the far right, for uniformly distributed on-site disorder (each panel uses $5\times {10}^3$ disorder realizations) as a function of $\mathrm{\Delta }/u$ for $\hslash {\omega }_e/u=-5\times {10}^{-5}$, $N=15$, $v/u=2$, $n^{*}=8$, open BCs, and [(ai)–(aiii)] $g/u={10}^{-4}$, [(bi)–(biii)] $g/u={10}^{-3}$, [(ci)–(ciii)] $g/u={10}^{-2}$, and [(di)–(diii)] $g/u={10}^{-1}$. The left, middle, and right columns are for the one-chain, cavity-shared two-chain, and emitter-shared two-chain systems, respectively.

Figure 9

Same as Fig. 8, but focusing on a smaller range of disorder strengths and a larger energy window.

Figure 10

Population dynamics of the [(a), (c)] one-chain system and [(b), (d)] cavity-shared two-chain system with $N=15$, $v/u=2$, $n^{*}=8$, $\hslash {\omega }_e/u=-5\times {10}^{-5}$, and open BCs. The populations of the site basis states $|n,j;\sigma \rangle$ (first chain of one- and two-chain systems) and $|n^{\prime },j;\sigma \rangle$ (second chain of two-chain system) are shown as a function of time; the color bar shown on the right applies to all panels. The top and bottom rows are for $g/u={10}^{-3}$ and $g/u={10}^{-1}$, respectively. For the one-chain system, the order of the basis states from left to right is $|1,1;g\rangle$, $|1,2;g\rangle$, $...$, $|N,1;g\rangle$, $|N,2;g\rangle$, and $|\text{vac};e\rangle$; the “white stripe” immediately to the left of the last basis state visually separates the basis states where the emitter is in $|g\rangle$ from those where the emitter is in $|e\rangle$. For the two-chain system, the order of the basis states from left to right is $|1,1;g\rangle$, $...$, $|N,2;g\rangle$, $|1^{\prime },1;g\rangle$, $...$, $|N^{\prime },2;g\rangle$, and $|\text{vac};e\rangle$; the first, second, and third “white stripes” (from left to right) visually separate the basis states belonging to the first and second chain, mark the sublattice 1 cavity of the second chain that is “missing” from the second chain (due to it being shared with the first chain), and separate basis states where the emitter is in $|g\rangle$ from those where the emitter is in $|e\rangle$, respectively. The red and white arrows highlight the presence and absence, respectively, of population in the cavities located at the ends of the chains.