Chiral spin-phonon bound states and spin-spin interactions with phononic lattices

When the hopping rates are spatially designed with a periodicity of $2$, the phononic crystal behaves as a dimer lattice, which is the phononic analog of the Su-Schrieffer-Heeger (SSH) model and allows for topological phonons at the quantum level. The Hamiltonian in Eq. (1) can be rewritten as

	$\displaystyle\hat{H}_{\text{ph}}$	$\displaystyle=$	$\displaystyle\omega_{m}\sum_{n}(\hat{a}_{n}^{\dagger}\hat{a}_{n}+\hat{b}_{n}^{\dagger}\hat{b}_{n})$		(3)
		$\displaystyle-$	$\displaystyle\sum_{n}(J_{1}\hat{a}_{n}^{\dagger}\hat{b}_{n}+J_{2}\hat{b}_{n}^{\dagger}\hat{a}_{n+1}+\text{H.c.}),$

where $\hat{a}_{n}$ and $\hat{b}_{n}$ are respectively the annihilation operator of phonon at sublattice $A$ and sublattice $B$ of the $n$th unit cell. The hopping rates can be replaced by $J_{1}=J(1+\delta)$, and $J_{2}=J(1-\delta)$, with $\delta$ the dimerization strength. To transform the Hamiltonian into momentum space, we impose periodic boundary conditions and introduce the Fourier transformation

$\displaystyle\hat{d}_{k}=\frac{1}{\sqrt{N}}\sum_{n=1}^{N}e^{-iknd_{0}}\hat{d}_{n}.$

(4)

For simplicity, we take the lattice constant $d_{0}=1$ below. Then the Hamiltonian of the phononic bath is transformed into $\hat{H}_{\text{ph}}=\sum_{k}\hat{V}_{k}^{\dagger}\hat{H}(k)\hat{V}_{k}$, with $\hat{V}_{k}^{\dagger}=(\hat{a}_{k}^{\dagger},\hat{b}_{k}^{\dagger})$ and the kernel

$\displaystyle\hat{H}(k)={\left(\begin{array}[]{ccc}\omega_{m}&-J_{1}-J_{2}e^{-ik}\\ -J_{1}-J_{2}e^{ik}&\omega_{m}\end{array}\right)}.$

(7)

Diagonalizing the Hamiltonian and taking $\omega_{m}$ as the energy reference, we can get the eigenenergies, that is, the dispersion relation

	$\displaystyle\omega_{\alpha/\beta}(k)$	$\displaystyle=$	$\displaystyle\pm\sqrt{J_{1}^{2}+J_{2}^{2}+2J_{1}J_{2}\cos k}$		(8)
		$\displaystyle=$	$\displaystyle\pm J\sqrt{2(1+\delta^{2})+2(1-\delta^{2})\cos k}.$

The two dispersions are symmetric with respect to the zero energy due to chiral symmetry and we set $\omega_{\alpha}(k)=-\omega_{\beta}(k)\equiv\omega(k)$ below. The band structure is shown in Fig. 2(c), where a middle bandgap with size $4J|\delta|$ is opened. In particular, for $\delta<0$, it corresponds to a topological phase of sound and there exists a pair of phononic edge states exponentially localized at the two ends when the system is finite [79].

We now consider the spins are coupled to the topological phonons in dimer lattices. The interaction Hamiltonian of the spins and bath in $k$-space is given by

	$\displaystyle\hat{H}_{\text{int}}$	$\displaystyle=$	$\displaystyle\frac{g}{\sqrt{2N}}\sum_{k,m=A}e^{ikx_{m}}\hat{\sigma}_{+}^{m}(\hat{\alpha}_{k}-\hat{\beta}_{k})$		(9)
		$\displaystyle+$	$\displaystyle\frac{g}{\sqrt{2N}}\sum_{k,m=B}e^{i(kx_{m}-\phi(k))}\hat{\sigma}_{+}^{m}(\hat{\alpha}_{k}+\hat{\beta}_{k})+\text{H.c.},$

with $\hat{\alpha}_{k}$ and $\hat{\beta}_{k}$ the eigenoperators in the diagonalization basis, $x_{m}$ the unit cell position and $\phi(k)\equiv\text{arg}(-J_{1}-J_{2}e^{-ik})$. The first (second) term represents the interactions between the phonons and spins located at the $A$ ($B$) sublattice. The single spins can acquire extra energy due to their interaction with the quantized phononic field, which is the self-energy expressed as

$\displaystyle\Sigma_{e}(z)=\sum_{k,i=j}\sum_{s=\alpha,\beta}\frac{\langle 0|\hat{\sigma}_{-}^{j}\hat{H}_{\text{int}}\hat{s}_{k}^{\dagger}|0\rangle\langle 0|\hat{s}_{k}\hat{H}_{\text{int}}\hat{\sigma}_{+}^{i}|0\rangle}{z-\omega_{s}(k)},$

(10)

with $|0\rangle=|g\rangle|\text{vac}\rangle$. In particular, within the Markovian approximation, $-2\text{Im}\Sigma_{e}(z)$ describes the decay rate and $\text{Re}\Sigma_{e}(z)$ represents the energy shift. The interaction with the phononic bath can also lead to collective self-energy $\Sigma_{ij}(z)$ of the spins, which can be obtained by simply replacing $i\neq j$ in the summation of $\Sigma_{e}$. Since the phononic dimer lattice has sublattice (chiral) symmetry, the pairwise collective self-energy can be classified into two categories: one from the spins in the same sublattice and the other from the spins in the different sublattice

		$\displaystyle\Sigma$	${}_{ij}^{AA/BB}(z)=\pm\frac{-g^{2}zy_{\text{min}}^{|x_{ij}|}}{\sqrt{z^{4}-4J^{2}(1+\delta^{2})z^{2}+16J^{4}\delta^{2}}}$		(11)
		$\displaystyle\Sigma$	${}_{ij}^{AB}(z)=\pm\frac{g^{2}J[(1+\delta)y_{\text{min}}^{|x_{ij}|}+(1-\delta)y_{\text{min}}^{|x_{ij}+1|}]}{\sqrt{z^{4}-4J^{2}(1+\delta^{2})z^{2}+16J^{4}\delta^{2}}},$		(12)

where $x_{ij}=x_{j}-x_{i}$ is the cell distance between two spins, $y_{\text{min}}=min(y_{+},y_{-})$ with respect to their absolute values and,

$\displaystyle y_{\pm}=\frac{z^{2}-2J^{2}(1+\delta^{2})\pm\sqrt{z^{4}-4J^{2}(1+\delta^{2})z^{2}+16J^{4}\delta^{2}}}{2J^{2}(1-\delta^{2})}.$

(13)

The sign in these two self-energy expressions is positive when $y_{\text{min}}=y_{+}$ and negative when $y_{\text{min}}=y_{-}$. The self-energy $\Sigma_{e}(z)=\Sigma_{i=j}^{AA/BB}(z)$ of single spins is also given here.

When the quantum emitters are coupled to this bath, the related topological features will be imprinted into the emitter-bath interaction. Specially, if the emitters’ frequency is chosen to be strictly equal to the zero-energy modes, the emitters can act as an effective edge of the lattice and localize the excitation on only one side [8, 9]. In this section, we show the formation of chiral spin-phonon bound states in this SiV-phononic crystal model.

We consider single spins coupled to the phononic dimer lattice, whose frequency lies within the bandgap. The spin no longer decays into the propagating modes, but is dressed by the acoustic modes close to the bandedges. The stationary state wavefunction in the single-excitation subspace can be expanded by

$\displaystyle|\psi\rangle=(C_{e}\hat{\sigma}_{+}+\sum_{k}\sum_{s=a,b}C_{k,s}\hat{s}_{k}^{\dagger})|g\rangle|\text{vac}\rangle,$

(14)

with $C_{e}$ the probability amplitude of the spin being in the upper state and $C_{k,s}$ the probability amplitude for finding a phononic excitation in sublattice $A/B$ at the wavevector $k$. The coefficients can be obtained by solving the secular equation $\hat{H}_{\text{tot}}|\psi\rangle=E_{\text{BS}}|\psi\rangle$, yielding

	$\displaystyle C_{k,a}^{A(B)}$	$\displaystyle=$	$\displaystyle\frac{gC_{e}E_{\text{BS}}}{E_{\text{BS}}^{2}-\omega^{2}(k)}\quad\bigg{(}\frac{gC_{e}\omega(k)e^{i\phi(k)}}{E_{\text{BS}}^{2}-\omega^{2}(k)}\bigg{)}$		(15)
	$\displaystyle C_{k,b}^{A(B)}$	$\displaystyle=$	$\displaystyle\frac{gC_{e}\omega(k)e^{-i\phi(k)}}{E_{\text{BS}}^{2}-\omega^{2}(k)}\quad\bigg{(}\frac{gC_{e}E_{\text{BS}}}{E_{\text{BS}}^{2}-\omega^{2}(k)}\bigg{)},$		(16)

where the superscript in the wavefunction labels the sublattice (at the $j=0$ unit cell) to which the spin is coupled. Doing the Fourier transformation, we can obtain the corresponding spatial distribution in sublattice $A/B$.

In this work, we mainly focus on the bound states at zero energy. A direct observation of the expressions shows that, when setting $E_{\text{BS}}=0$, $C_{j,a}^{A}=C_{j,b}^{B}=0$, with $j$ the unit cell index. The other two spatial distributions can be given analytically, similar to the calculation of the self-energy. For the case of $\delta>0$ and the spin at the $j=0$ unit cell,

$\displaystyle C_{j,b}^{A}=\left\{\begin{aligned} &\frac{gC_{e}(-1)^{j}}{J(1+\delta)}\bigg{(}\frac{1-\delta}{1+\delta}\bigg{)}^{j},&j\geq 0\\ &0,&j<0\end{aligned}\right.$

(17)

and $C_{j,a}^{B}=C_{-j,b}^{A}$, which indicates that the left (right) bound state vanishes when the spin is coupled to $A$ ($B$) sublattice. The chirality of the bound states arises from the inversion symmetry breaking of the phononic bath with respect to the coupling point, and reaches its maximum at $\omega_{\sigma}=\omega_{m}$. These two perfect chiral bound states are shown in Fig. 3(a) and 3(b), with numerical integrations of Eq. (15) and Eq. (16). Note that the number of chiral bound states is equal to that of the edge states.

We next consider the robustness of this chiral spin-phonon bound states in the dimer lattice. A finite chain with $40$ phononic cavities is used to simulate the SSH bath. The chiral bound states in the presence of disorder in the phonon hopping strength are depicted in Fig. 3(c), with adding the random terms $\hat{H}_{d}=\sum_{n}(\xi_{n}\hat{d}_{n}^{\dagger}\hat{d}_{n+1}+\text{H.c.})$ to the total Hamiltonian. The disorder strength $\xi_{n}$ is chosen within the range $[-W/2,W/2]$ for the $n$th lattice site. We show that the chiral behavior of the bound states is protected, where the chiral symmetry persists. As a contrast, the diagonal disorder is also taken into account, which is shown in Fig. 3(d). We find that the middle bound state now has distributions in the A/B sublattices and both sides of the spin. Besides, the energy of the bound state is no longer strictly fixed at zero.

Though the main topological feature is most pronounced in the zero-energy modes, the dynamics of spins in the phononic dimer lattice is different from that in the standard waveguide. In the weak coupling limit, the spin dynamics is subject to Eq. (11) and Eq. (12), therein the imaginary part represents the decay rate. From Eq. (11) (in the case $i=j$), we show that the decay rate of the spins is sublattice-independent, which can be roughly understood from the viewpoint that the spins couple to the two topology-different semi-infinite waveguides. Without loss of generality, we consider two spins resonant with the phononic band. There are three cases: i) the spins are coupled to $AA/BB$ sublattice; ii) the spins are coupled to $AB$ sublattice (from left to right); iii) the spins are coupled to $BA$ sublattice. In these three situations, the waveguide structures between the spins are standard-like, trivial and topological, respectively. As a result, the phonon-induced collective decay rate is different for the three cases. More interestingly, case II and case III are interchangeable by only varying the sign of the parameter $\delta$. Therefore, the collective radiation is topology-dependent.

The collective decay rate $-2\text{Im}\Sigma_{ij}$ can also be calculated by using the identical relation $\lim_{y\rightarrow 0^{+}}1/(x\pm iy)=P(1/x)\mp i\pi\delta(x)$, which can provide a more intuitive insight to the features above. In the standard waveguide, the collective decay rate oscillates versus the emitter’s relative position, i.e., $\Gamma_{ij}=\Gamma_{e}\cos(kx_{ij})$. In contrast, the collective decay rate in the SSH bath is given by

	$\displaystyle\Gamma_{ij}^{\text{AA/BB}}(\omega)$	$\displaystyle=$	$\displaystyle\text{sign}(\omega)\Gamma_{e}(\omega)\cos(k(\omega)x_{ij})$		(18)
	$\displaystyle\Gamma_{ij}^{\text{AB/BA}}(\omega)$	$\displaystyle=$	$\displaystyle\text{sign}(\omega)\Gamma_{e}(\omega)\cos(k(\omega)x_{ij}\mp\phi(k)).$		(19)

Here, $x_{ij}$ is the cell-distance. Obviously, the topology-dependent parameter (phase) $\phi(k)$ enters into Eq. (19). In Fig. 4, we plot $\Gamma_{ij}^{\text{AB}}(\omega)$ as a function of spin frequency $\omega_{\sigma}$ inside the upper band, with setting $x_{ij}=2$ and $\delta=\pm 0.3$. The case is opposite for $\omega_{\sigma}$ inside the lower band, i.e., $\Gamma_{ij}^{\text{AB}}(\omega)=-\Gamma_{ij}^{\text{AB}}(-\omega)$. Perfect super or subradiance occurs when $\Gamma_{ij}^{\text{AB}}=\pm\Gamma_{e}$. In particular, we find that the number of phononic super or subradiant points depends on the sign of $\delta$: In the trivial phase ($\delta>0$), there are $x_{ij}-1$ perfect superradiant points inside the two passbands, while in the topological phase ($\delta<0$), there are $x_{ij}$ frequency points.

The chiral spin-phonon bound states discussed above can mediate spin-spin interactions through the exchange of virtual phonons. As a result, the spin-spin interactions will inherit the features appearing in the bound states: i) decay exponentially with the spins’ distance; ii) the spin located in sublattice $A$ ($B$) only interacts with spin located in sublattice $B$ ($A$) on the one side; iii) robust against the off-diagonal disorder. For $\delta>0$, the interaction in the Markovion limit is given as

$\displaystyle\hat{H}_{s-s}=\sum_{i<j}J_{ij}(\hat{\sigma}_{+}^{i}\hat{\sigma}_{-}^{j}+\hat{\sigma}_{+}^{j}\hat{\sigma}_{-}^{i})$

(20)

with the coupling strength $J_{ij}^{AA/BB}=0$ and

$\displaystyle J_{ij}^{AB}=\left\{\begin{aligned} &\frac{g^{2}(-1)^{x_{ij}}}{J(1+\delta)}\bigg{(}\frac{1-\delta}{1+\delta}\bigg{)}^{x_{ij}},&x_{ij}\geq 0\\ &0,&x_{ij}<0\end{aligned}\right..$

(21)

To numerically show these chiral spin-spin interactions, we use a finite lattice chain with $12$ phononic cavities and consider three spins coupled to the second, third and $8$th cavity (see Fig. 5(a)). The off-diagonal disorder is also taken into account. The quantum dynamics of the three spins is plot in Fig. 5(b), with the initial state $|\psi_{0}\rangle=|1\rangle_{1}|0\rangle_{2}|0\rangle_{3}$. We find that the population of spin 3 is always zero and the population is oscillating between spin 1 and spin 2 with an amplitude approaching one, which reflects the directionality and robustness of the spin-spin interactions. Similarly, the quantum dynamics of spins in the presence of diagonal disorder is plot in Fig. 5(c) for comparison. Apart from extra population in spin 3, we find that the population of spin 1 can not reach zero, since the energy value of the bound state is no longer pinned at zero in this case.

In the finite system, edge effects inevitably occur especially in the topological phase, where a pair of edge states exponentially localize at the two ends of the lattice chain. The phononic edge states in a phononic crystal can be controlled with the addition of solid-state spins. In that case, the excitation can be transferred between spins and phononic edge states, with eigenenergies $\pm\epsilon$ and eigenstates $|E_{\pm}\rangle$. The dynamics is govern by the effective Hamiltonian

	$\displaystyle\hat{H}_{\text{eff}}$	$\displaystyle=$	$\displaystyle\sum_{\pm}\pm\epsilon|E_{\pm}\rangle\langle E_{\pm}|+(\omega_{\sigma}-\omega_{m})|e\rangle\langle e|$		(22)
		$\displaystyle+$	$\displaystyle\sum_{\pm}g_{\pm}(|e\rangle\langle E_{\pm}|+|E_{\pm}\rangle\langle e|),$

with $g_{\pm}=g\langle E_{\pm}|\hat{d}_{x_{0}}^{\dagger}|\text{vac}\rangle$ and $g_{+}=g_{-}$ ($g_{+}=-g_{-}$) when $\hat{d}_{x_{0}}^{\dagger}=\hat{b}_{x_{0}}^{\dagger}$ ($\hat{d}_{x_{0}}^{\dagger}=\hat{a}_{x_{0}}^{\dagger}$). When setting $\omega_{\sigma}=\omega_{m}$ and applying the Schrödinger equation, we can obtain the coupled equations of the coefficients

	$\displaystyle i\dot{C}_{e}(t)$	$\displaystyle=$	$\displaystyle g_{+}C_{+}(t)+g_{-}C_{-}(t)$		(23)
	$\displaystyle i\dot{C}_{+}(t)$	$\displaystyle=$	$\displaystyle\epsilon C_{+}(t)+g_{+}C_{e}(t)$		(24)
	$\displaystyle i\dot{C}_{-}(t)$	$\displaystyle=$	$\displaystyle-\epsilon C_{-}(t)+g_{-}C_{e}(t),$		(25)

with $C_{e}$ and $C_{\pm}$ the wavefuncions in the states $|e\rangle$ and $|E_{\pm}\rangle$, respectively. Solving these equations, the spin’s time evolution obeys

$\displaystyle C_{e}(t)=\frac{\epsilon^{2}+2g_{+}^{2}\cos(\omega_{0}t)}{\epsilon^{2}+2g_{+}^{2}},$

(26)

with $\omega_{0}=\sqrt{\epsilon^{2}+2g_{+}^{2}}$. In the case of a large size cavity chain, $\epsilon\simeq 0$ and the spin exchanges energies between the left or right phononic edge state, which are symmetry and antisymmetry superpositions of the states $|E_{\pm}\rangle$, respectively.

Concretely, we consider a single spin coupled to the $5$th phononic cavity of a $12$ cavity chain designed with staggered hopping strengths and being in the topological phase. The system’s dynamics is shown in Fig. 6. We show that the excitation is mainly transferred between the spin and the left phononic edge state, in line with the prediction based on Eq. (26).

Similar to the dimer lattice, a trimer lattice is modeled with alternating hopping rates and having a periodicity of 3. The Hamiltonian of the phononic waveguide is rewritten as

	$\displaystyle\hat{H}_{\text{ph}}$	$\displaystyle=$	$\displaystyle\omega_{m}\sum_{n}(\hat{a}_{n}^{\dagger}\hat{a}_{n}+\hat{b}_{n}^{\dagger}\hat{b}_{n}+\hat{c}_{n}^{\dagger}\hat{c}_{n})$		(27)
		$\displaystyle-$	$\displaystyle\sum_{n}(J_{a}\hat{a}_{n}^{\dagger}\hat{b}_{n}+J_{b}\hat{b}_{n}^{\dagger}\hat{c}_{n}+J_{c}\hat{c}_{n}^{\dagger}\hat{a}_{n+1}+\text{H.c.}),$

with $\hat{a}_{n}$, $\hat{b}_{n}$ and $\hat{c}_{n}$ the annihilation operators for the $A$, $B$ and $C$ phononic modes at the $n$th unit cell. In a common trimer lattice, the phonon tunneling strengths $J_{a}$, $J_{b}$ and $J_{c}$ are unequal and there is no symmetry. The inversion symmetry only recovers in the special case of $J_{a}=J_{b}\neq J_{c}$. Performing the same operation as in Sec. III.A, the Hamiltonian can be transformed into momentum space, with the kernel of Hamiltonian

$\displaystyle\hat{H}(k)={\left(\begin{array}[]{ccc}\omega_{m}&-J_{a}&-J_{c}e^{-ik}\\ -J_{a}&\omega_{m}&-J_{b}\\ -J_{c}e^{ik}&-J_{b}&\omega_{m}\end{array}\right)}.$

(31)

Unlike the dimer lattice where the analytical solution to the dispersion relation is given, here we introduce a unitary matrix $M=(V_{1},V_{2},V_{3})$ to diagonalize the kernel of Hamiltonian as $\hat{D}(k)=M^{\dagger}\hat{H}(k)M=\text{diag}(\omega_{\alpha},\omega_{\beta},\omega_{\gamma})$, with $V_{i}$ ($i\!=\!1,2,3$) the eigenvectors and $\omega_{s}$ ($s\!=\!\alpha,\beta,\gamma$) the eigenvalues. The eigenoperators are given by $(\hat{\alpha}_{k}\,\,\hat{\beta}_{k}\,\,\hat{\gamma}_{k})^{T}=M^{\dagger}(\hat{a}_{k}\,\,\hat{b}_{k}\,\,\hat{c}_{k})^{T}$. Then the Hamiltonian of the phononic trimer lattice in $k$-space reads

$\displaystyle\hat{H}_{\text{ph}}=\sum_{k}\sum_{s=\alpha,\beta,\gamma}\omega_{s}(k)\hat{s}_{k}^{\dagger}\hat{s}_{k}.$

(32)

The band structure is numerically plotted in Fig. 2(d), with $J_{a}=J$, $J_{b}=4J$ and $J_{c}=3J$.

Working in $k$-space, the interaction Hamiltonian for the spins and phonon modes in the trimer lattice becomes

$\displaystyle\hat{H}_{\text{int}}=\frac{g}{\sqrt{N}}\sum_{k}\sum_{m,s}e^{ikx_{m}}M(m,s)\hat{\sigma}_{+}^{m}\hat{s}_{k}+\text{H.c.},$

(33)

where the interaction is projected to the new basis. The self-energy of the spins is expressed as

$\displaystyle\Sigma_{e}(z)=\frac{g^{2}}{2\pi}\int_{-\pi}^{\pi}dk\sum_{m=n,s}\frac{\hat{M}(n,s)\hat{M}^{-1}(s,m)}{z-\omega_{s}(k)}.$

(34)

The collective self-energy $\Sigma_{ij}$ of the spins can be obtained by simply replacing $m\neq n$ in the summation of $\Sigma_{e}$ and adding a propagation factor $e^{ik(x_{n}-x_{m})}$ to the integral.

When the spin’s frequency lies within the phononic bandgap, there exist spin-phonon bound states in the single-excitation subspace. The spatial distribution of the phononic part of the bound state can be calculated by solving the stationary Schrödinger equation, yielding

$\displaystyle C_{j,n}^{m}=\frac{gC_{e}}{2\pi}\int_{-\pi}^{\pi}dke^{ikj}\sum_{s}\frac{\hat{M}(n,s)\hat{M}^{-1}(s,m)}{E_{BS}-\omega_{s}(k)}.$

(35)

Here, $C_{j,n}^{m}$ is the wavefunction at $n=a/b/c$ sublattice of the $j$ unit cell when the spin is coupled to the $m=A/B/C$ sublattice at the $j=0$ unit cell.

Inspired by the fact that chiral spin-phonon bound states appear at the zero-energy modes in a dimer lattice, we start to find the chiral bound states in a trimer lattice by setting the energy of the bound states equal to that of the chiral phononic edge states. For a finite system with hopping rates $(J_{a},J_{b},J_{c})=(1,4,3)J$, there exist two edge states localized at the right end with energies $\pm 4J$. Thus, we plot the phononic part of the bound states in Fig. 7(a,d) with setting $E_{\text{BS}}=\pm 4J$ in Eq. (35). We find the spin-phonon bound state is chiral only when the spin is coupled to the $A$ sublattice, i.e., $C_{j,n}^{A}=0$ for $j\geq 0$ and $C_{j,n}^{A}\neq 0$ for $j<0$. Also, these bound states have no components in the $A$ sublattice $C_{j,a}^{A}=0$, while the amplitudes in the $B$ and $C$ sublattices are equal $|C_{j,b}^{A}|=|C_{j,c}^{A}|$. Furthermore, the bound states are fractionally chiral when the spin is coupled to the $B$ and $C$ sublattices, which is reflected in the components on the $A$ sublattice, $C_{j,a}^{B/C}=0$ for $j\leq 0$ and $C_{j,a}^{B/C}\neq 0$ for $j>0$.

However, the edge modes change when varying the boundary of the trimer lattice. When the finite system is constructed by $(J_{a},J_{b},J_{c})=(3,1,4)J$, there are two pairs of edge states localized at the boundary, with energies $\pm 3J$ and $\pm J$. While for the finite system constructed by $(J_{a},J_{b},J_{c})=(4,3,1)J$, there is no edge mode. The bulk properties of the two cases are the same as that of $(J_{a},J_{b},J_{c})=(1,4,3)J$. Thus, we turn to study the spin-phonon bound states at these frequencies. In Fig. 7(b,e) and 7(c,f), we show two pairs of chiral bound states located on the right/left side of the spin with $E_{\text{BS}}=\pm 3J/\pm J$, when the spin is coupled to the $B/C$ sublattice. Note that the six chiral bound states are one-to-one corresponding to the six edge states.

We now consider the robustness of the chiral bound states in the trimer lattice. As an example, we add disorder to the finite system with $20$ unit cells, by setting $(J_{a},J_{b},J_{c})=(1,4,3)J$, where a spin is coupled to the $B$ sublattice with energy $\omega_{\sigma}=\omega_{m}+3J$. In Fig. 7(g), we allow disorder in the intracell hopping $J_{a}$ and $J_{b}$ independently. While in Fig. 7(h), disorder in the intercell hopping $J_{c}$ is plotted. On the other hand, bound states with on-site disorder acting in $\omega_{m}$ are shown in Fig. 7(i). We observe several features from the numerical results. First, the directionality is robust to disorder in the intracell hopping. Second, the bound state has weight in each sublattices and both sides while the eigenenergies have a small deviation, in the presence of disorder in the intercell hopping or on-site disorder. Generally, the directional bound state is still robust when the disorder strength is on the order of the coupling/hopping strength. All these results indicate that the directional spin-phonon bound states can inherit the robustness of chiral edge states in the trimer lattice.

From the discussion of chiral spin-phonon bound states in both lattices, we can conclude that the spin indeed acts as an effective boundary of the two semi-infinite waveguide structures. In this case, the system can be divided into three parts: the spin and two semi-infinite phononic waveguides. Contrary to what happens in the dimer lattice where the two semi-infinite structures are the same for spin at the $A/B$ sublattice, there are six different semi-infinite structures in the trimer lattice when the spin is respectively coupled to the $A/B/C$ sublattice. Thus, we predict that $\Sigma_{e}^{A}\neq\Sigma_{e}^{B}\neq\Sigma_{e}^{C}$ in the inversion-symmetry breaking phase of the trimer lattice, while $\Sigma_{e}^{A}=\Sigma_{e}^{C}\neq\Sigma_{e}^{B}$ in the inversion-symmetric phase of the trimer model. We note that the real part of the self-energy is the energy shift, which is small compared to the resonance frequency. Furthermore, the shift is zero when the spin’s frequency lies within the band. Therefore, we mainly focus on the spin relaxation when the frequency is resonant with the phononic waveguide.

Without loss of generality, we consider a single spin with frequency within the upper band, which is coupled to the $m\!=\!A/B/C$ sublattice, respectively. In the Markovian regime, the spin’s decay rate $\Gamma_{e}^{m}(\omega_{\sigma})=-2\text{Im}\Sigma_{e}^{m}$ can be written as

$\displaystyle\Gamma_{e}^{m}(\omega_{\sigma})=\frac{2g^{2}\hat{M}(m,0)\hat{M}^{-1}(0,m)}{v_{g}(\omega_{\sigma})},$

(36)

with $v_{g}(\omega_{\sigma})=\partial\omega/\partial k|_{\omega=\omega_{\sigma}}$ the group velocity. In Fig. 8, we numerically plot the decay rate of the spin coupled to different sublattices when the spin’s frequency lies within the upper band. The hopping rates are $(J_{a},J_{b},J_{c})=(1,4,3)J$, $(J_{a},J_{b},J_{c})=(1,1,3)J$ and $J_{a}\!\!=\!\!J_{b}\!\!=\!\!J_{c}\!\!=\!\!J$ in Fig. 8(a), 8(b) and 8(c), respectively. We find $\Gamma_{e}^{A}\!\neq\!\Gamma_{e}^{B}\!\neq\!\Gamma_{e}^{C}$ for $J_{a}\!\neq\!J_{b}\neq\!J_{c}$. In the special case $J_{a}\!=\!J_{b}\!\neq\!J_{c}$ where inversion symmetry recovers, $\Gamma_{e}^{A}=\Gamma_{e}^{C}\neq\Gamma_{e}^{B}$. Moreover, the standard 1D band-edge divergence of $\Gamma_{e}^{B}$ is canceled in the inversion-symmetric phase of the trimer model, when $\omega_{\sigma}$ approaches the lower band-edge of $\omega_{\alpha}(k)$. For comparison, the situation that the spin interacts with a normal coupled cavity waveguide is shown in Fig. 8(c), which predicts a decay rate $\Gamma_{e}=g^{2}/|\sin(k)|$. In general, the spin relaxation in a trimer lattice is sublattice-dependent, which is consistent with the discussion above.

We now discuss the spin-spin interactions mediated by the chiral spin-phonon bound states appearing in the trimer lattice. In the weak coupling limit, the interaction strength between two spins is

$\displaystyle J_{ij}^{mn}=\frac{g^{2}}{2\pi}\int_{-\pi}^{\pi}dke^{ik(j-i)}\sum_{s}\frac{\hat{M}(n,s)\hat{M}^{-1}(s,m)}{E_{BS}-\omega_{s}(k)},$

(37)

with $i,j$ the cell index, and $m,n$ the sublattice index. As an example, we take advantage of the bound state shown in Fig. 7(b) involving three spins with one at the $B$ sublattice and two neighbouring spins at the $A/C$ sublattices. When the spins at the $A/C$ sublattices are placed on the left side of the spin $B$, as shown in Fig. 9(a), the two neighbouring spins are decoupled to the spin $B$ due to the directionality of the bound state and the population can be transferred between the two neighbouring spins. For the case of the spins at the $A/C$ sublattices on the right side of the spin $B$, as shown in Fig. 9(b), the excitation is exchanged between the spin at the $B$ sublattice and the antisymmetry superposition of the two neighbouring spins. The dynamics of the three spins can be modeled with an effective Hamiltonian in the interaction picture as

	$\displaystyle\hat{H}_{\text{eff}}$	$\displaystyle=$	$\displaystyle J_{B}(\hat{\sigma}_{+}^{B}\hat{\sigma}_{-}^{C}-\hat{\sigma}_{+}^{B}\hat{\sigma}_{-}^{A}+\text{H.c.})+J_{AC}(\hat{\sigma}_{+}^{A}\hat{\sigma}_{-}^{C}+\text{H.c.})$		(38)
		$\displaystyle=$	$\displaystyle\sqrt{2}J_{B}(\hat{\sigma}_{+}^{B}\hat{\sigma}_{-}^{\text{at}}+\text{H.c.})+J_{AC}(\hat{\sigma}_{+}^{\text{st}}\hat{\sigma}_{-}^{\text{st}}-\hat{\sigma}_{+}^{\text{at}}\hat{\sigma}_{-}^{\text{at}}),$

with $\hat{\sigma}_{-}^{\text{st/at}}=(\hat{\sigma}_{-}^{C}\pm\hat{\sigma}_{-}^{A})/\sqrt{2}$ and $J_{B}=J_{ij}^{BC}=J_{ij+1}^{BC}$. In the first situation $J_{B}=0$ for $j\geq i$, while in the second situation $J_{B}\neq 0$ for $j<i$. This reflects the directionality of the spin-spin interaction when the spin is coupled to the $B$ sublattice. When focusing on the case $J_{B}\neq 0$, we find that the transition is not resonant due to the interaction between the neighbouring spins. Note that the detuning (the last term in the second line of Eq. (38)) can be removed by external driving fields; thus the antisymmetric combinations of spin $A$ and spin $C$ acting as a single spin can be resonantly coupled to spin $B$. Finally, we emphasize that the spin-spin interaction is directional only when one of the spins is at the $B$ sublattice.

We consider a resonator chain composed of $12$ lattice sites to numerically examine the two cases, as shown in Fig. 9. The disorder in the intracell hopping rate $J_{a}$ and $J_{b}$ is also taken into account and the spin’s frequency is $\omega_{\sigma}\simeq 3J$ in both cases. In the first case, the three spins are placed at the third, $4$th and $5$th lattice sites and the Rabi oscillation between spin $A$ and spin $C$ is observed in Fig. 9(c), with the initial state $|\psi_{0}\rangle=|e\rangle_{A}|g\rangle_{B}|g\rangle_{C}$. We show the population in spin $B$ keeps zero. In the second case, the three spins are placed at the $5$th, $9$th and $10$th lattice sites and the population is transferred between spin $B$ and spin $A/C$ with the initial state $|\psi_{0}\rangle=|g\rangle_{A}|e\rangle_{B}|g\rangle_{C}$, as shown in Fig. 9(d). Though the energy of the bound state formed by spin $B$ is strictly pinned under disorders ($E_{\text{BS}}^{B}=3J$), the energy of the bound states formed by spin $A$ and spin $C$ are not strictly pinned ($E_{\text{BS}}^{A/C}\approx 3J$). Therefore, the population of the spin $B$ can not reach zero with the time evolution.

Because the edge modes in the trimer lattice remain in the bandgap, the quantum control on these modes becomes possible with the addition of spins. Furthermore, since the edge states in the inversion-symmetry breaking phase of the trimer lattice have no degeneracy, we can directly manipulate one of the phononic edge states on the boundary of the phononic crystal with the addition of a single spin. The resonance transfer can occur via tuning the frequency of the spin, with the interaction Hamiltonian

$\displaystyle\hat{H}_{e-s}=g_{e-s}(|E\rangle\langle e|+|e\rangle\langle E|),$

(39)

where $g_{e-s}\!=\!g\langle E|\hat{d}_{x_{0}}^{\dagger}|\text{vac}\rangle$ is the coupling rate. As an example, we consider the situation that a single spin is placed at the $5$th cavity of a $12$ cavity chain, with alternating hopping strengths $(J_{a},J_{b},J_{c})=(1,4,3)J$. There exist two edge states localized on the right side of the system with energy $E_{\text{edge}}=\pm 4J$. In Fig. 10, we plot the time evolution of the system and show a transition with the amplitude approaching one between the spin and one of the chiral phononic edge states.