Tunable chiral spin–spin interactions in a spin-mechanical hybrid system: application to causal-effect simulation

Long-range chiral interactions are very attractive due to their potential applications in quantum simulation and quantum information processing. Here we propose and analyze a novel spin-mechanical hybrid quantum device for designing and engineering chiral spin–spin interactions by integrating spin qubits into a programmable mechanical chain. After mapping the Hamiltonian of the mechanical lattice to the Su–Schrieffer–Heeger model, we find that chiral spin–phonon bound states and spin–spin coupling interactions can be achieved. Specifically, the range and strength of chiral spin–spin couplings can be tuned in situ by the on-chip manipulation voltages. We further employ this setup to simulate the causal effects in long-range chiral-coupling systems, showing that the correlation functions propagate individually in two sublattices. These phenomena are very different from the situations in the conventional long-range coupling quantum systems without chiral symmetry.


Introduction
Causality is an important theme in fundamental science study [1][2][3].In quantum systems the cause effects are usually quantified by correlation functions, and the maximum speed with which correlations can propagate has profound implications in condensed matter physics and quantum information processing [4].It is well known that information could only propagate with a finite speed under the framework of relativity, but the situation is not so clear in non-relativistic quantum systems.Due to its unique features, it is a great challenge to establish causal models in quantum mechanics [5].For instance, it is not easy to describe the cause-effect relationship between two entangled objects.Thus far we believe that if a system is modeled by local Hamiltonians, an effective 'light cone' with exponentially decaying tails can be induced to bound the transfer of information [6][7][8].In recent years the research of chiral interactions injects new vitality into quantum optics and quantum information processing [9][10][11][12][13].For instance, chiral spin-spin interactions provide a promising route to quantum communication [14].However, little is known about the propagation of correlations in chiral-coupling systems, especially when long-range interactions are involved.As it is a enormous challenge to analytically solve the dynamical evolution of correlation functions in long-range chiral-coupling systems, a critical pathway is to perform quantum simulations in a tunable quantum device.
One of the most promising platforms for simulating the chiral-coupling system is the spin-mechanical hybrid device involving both mechanical resonators and spin qubits [15,16].In this type of quantum system the mechanical resonators are employed as quantum data bus to bridge the spin qubits [17].A typical example of spin qubits is the nitrogen-vacancy (NV) centers in diamond, for which the fast microwave manipulation, optical preparation and detection [18,19], and long coherence time [20] have been experimentally demonstrated.Thus far the spin-mechanical hybrid quantum devices have been investigated to entangle spin qubits [21], realize quantum logic gates [22], detect week magnetic fields [23][24][25], and perform quantum simulations [26,27].However, due to the small size of high-frequence resonator, the numbers of spin qubits integrated in these systems are limited.
Strong couplings between adjacent mechanical resonators play the central role in improving the scalability of spin-mechanical hybrid system.We stress that these mechanical-mechanical couplings can be achieved in different ways.For instance, strain force could induce mechanical-mechanical couplings [28][29][30].However these coupling interactions are rigid due to the intrinsically fixed geometry.To structure a spin-mechanical hybrid device with good controllability, here we consider the mechanical-mechanical couplings through three major routes.Firstly, the electrostatic coupling interactions between nearest-neighboring resonators have been experimentally demonstrated [31,32].With these mechanical-mechanical couplings, programmable mechanical resonator arrays have been realized to investigate the soliton edge states and phase transitions in 1D topological systems [33,34].Secondly, according to the achievement on nanoelectromechanical spin transducer by the M. D. Lukin group, the tunable mechanical-mechanical coupling interations may also be induced by capacitances [35].Finally, the strong coupling interactions between a mechanical resonator and a LC resonator have been experimentally demonstrated to reach ∼1M Hz.Using the LC resonator as a quantum data bus, we could also realize the mechanical-mechanical couplings indirectly [36,37].
In this work, we present a novel spin-mechanical hybrid quantum device for engineering and manipulating chiral spin-spin interactions.This setup composes of mechanical resonators and spin qubits, and combines strong mechanical-mechanical couplings with strong spin-mechanical couplings by positioning spin qubits near a magnetic mechanical chain.After tuning the mechanical chain to the Su-Schrieffer-Heeger (SSH) Hamiltonian, it finds that tunable chiral spin-phonon bound states and spin-spin interactions could be induced.We utilize this setup to simulate the causal effects in long-range chiral-coupling systems, showing that unusual propagations of correlation functions can be observed.The benefits of this spin-mechanical hybrid device are quite diverse.On the one hand, because the mechanical-mechanical couplings could be controlled by external voltages, and the spin-mechanical couplings can be tuned by external magnetic fields, one could design the Hamiltonian of the system as needed.On the other hand, this setup combines tunable ground states and long coherent time of NV spin qubits with good scalablity and controllability of mechanical chains, thus provides a very promising platform for designing unconventional spin-spin interactions [38,39].
The paper is divided into sections as follows: In section 2, we describe the setup, showing how the spin qubits can be integrated into a tunable mechanical chain.In section 3, we illustrate how the proposed physical system could induce chiral spin-phonon bound states and chiral spin-spin interactions.Next, in section 4, we apply this setup to simulate the causal effects in long-range chiral-coupling systems, using numerical simulation results with up to 12 spins.In section 5, we discuss the experimental feasibility of our scheme.Section 6 summarizes.

Description of the setup
As shown in figure 1(a), the spin-mechanical hybrid quantum device under consideration is realized by integrating spin qubits into a programmable mechanical chain.The basic structural unit of the setup consists of a doubly clamped silicon nitride (Si 3 N 4 ) resonator, a magnetic tip and a NV center spin qubit, see figure 1(b).A 15 nm Au layer is evaporated on the surface of silicon nitride, allowing the resonator connects with on-chip electrodes [32].The sharp magnetic tip is attached on the middle of the doubly clamped resonator, and the spin qubit is positioned above the magnetic tip at a very tiny distance.
The key to structuring the setup is the integration of strong mechanical-mechanical couplings and strong spin-mechanical couplings.As have been posted before, the tunable mechanical-mechanical couplings could be achieve through different routes.Here we employ the electrostatic coupling scheme, according to the recent achievement on programmable mechanical lattice by the J. F. Du gruop [32][33][34].We first consider an infinite mechanical chain with position index n ∈ {1, 2, 3, . ..}, and drive the mechanical resonators by applying voltages on the Au layer under 1 T static magnetic field along y-axis, using the standard magneto-motive technique [40].As illustrate in figure 1(c), we apply a voltage V n p (t) = V n dc + V n ac cos ω n p t between each adjacent resonators n and n + 1, and let the dc voltage V n dc much larger than the ac voltage V n ac , i.e.V n dc >> V n ac .Specifically, when the frequency of the ac field ω n p is equal to the frequency difference of the two resonators, i.e. ω n p = |ω n+1 m − ω n m |, a coherent coupling interaction between nearest-neighboring resonators is induced, see [32][33][34] and appendix A for more details.As the mechanical-mechanical coupling strength could be precisely controlled via the gate voltage, i.e.J ∝ CV dc V ac , it is easy for us to map the mechanical chain to the SSH model.It is known that there are two sites in each unit cell of the SSH model, accordingly, we now use the cell index j =int[0.5 * (n + 1)] to replace the position index n.Then the free Hamiltonian of the mechanical resonators reads Ĥm free = , and the SSH Hamiltonian of the mechanical chain can be expressed as (let h = 1): where âj and bj denote the annihilation operators of the mechanical modes in the {a j , b j } sublattices, ω m a j (ω m b j ) denotes the fundamental frequencies of the mechanical resonators with cell index j and sublattice index a (b), and J 1 = J(1 − δ) and J 2 = J(1 + δ) denote the intracell and intercell coupling strengths, respectively.
To investigate the collective modes of the mechanical chain, we turn to the momentum space by applying the Fourier transformation âk = 1 √ Nc ∑ j âj e ikj and bk = 1 where k = 2π j/N c is the wave number in the first Brillouin zone.After defining vector ⃗ V = (â k , bk ) in the momentum space, the Hamiltonian of the mechanical chain can be expressed as where the kernel It noted that the Hamiltonian Ĥm SSH (k) can be diagonalized as Ĥm , with an energy gap 4|Jδ| in the middle, see figure 1(d).
For each mechanical resonator there is a sharp magnet tip attached on the middle, and a NV center spin qubit is positioned above the magnet tip, see figure 1(b).When the mechanical resonator is vibrating, the magnetic field near the spin qubit is changed, resulting in a magnetic coupling between the resonator and the spin qubit.As illustrated in figure 1(e), the electronic ground state of the NV center is a spin-1 triplet, denoted as |m s ⟩ with m s = {0, ±1}.The resonance transition frequency between the state |0⟩ and the twofold degenerate state | ± 1⟩ is ω d ∼ 2π × 2.87 GHz.On the other hand, the fundamental eigenfrequency of the mechanical resonator works in the range of ω m a j /b j ∼ 2π × 2.5 M Hz.To overcome the energy mismatch, we apply a z-axis static magnetic field B z to remove the degeneracy of the states | − 1⟩ and | + 1⟩, and apply two microwave fields ⃗ B dr = Ω ± cos υ ± t⃗ e x to drive Rabi oscillations between |0⟩ ↔ | − 1⟩ and |0⟩ ↔ |1⟩.Then the spin qubits could be prepared to the three-level dressed states, denoted as |g⟩, |d⟩ and |e⟩, see figure 1(f).By tuning the external magnetic fields, we further let the transition frequency between sates |d⟩ and |e⟩ to match the fundamental frequency of mechanical resonator.As a result the coherent coupling interaction between mechanical resonator and spin qubit can be achieved, see [41] and appendix B for more details.We define spin operators as σz † and α ∈ {a, b}, then the free Hamiltonian of the spin qubits can be defined as where ω q α j denotes the frequency of spin qubit encoded in two-level states {|d⟩, |e⟩}.When considering one single qubit with cell index j and sublattice index α is coupled to the mechanical chain, the whole Hamiltonian of the system can be expressed as Ĥ( 1) Here Hamiltonian Ĥα j int corresponds to the coupling interacion between single spin qubit and mechanical resonator, which takes the form as Specifically, in the momentum space the Hamiltonian of spin-mechanical couplings reads

Chiral spin-phonon bound states
We now consider the case where one single spin qubit is coupled to the mechanical chain.For the 1D mechanical bath with SSH Hamiltonian, there always exist three bound states in the infinite condition [38].These bound states are combinations of spin and mechanical excitations.Instead of decaying into the collective mechanical modes, the spin qubit is partly dressed, and the mechanical modes are localized around the spin emitter [39].In the momentum space, the spin-phonon bound states can be expressed as where c q is the long-time population of the spin qubit, c a k (c b k ) is the probability amplitude for finding a phononic excitation with wave vector k in the {a k , b k } subspaces, |g, vac⟩ is the ground state of the system.If we first prepare the spin qubit to the excited state |e⟩, and prepare the mechanical resonators to the ground sate |vac⟩, then the long-time population of spin qubit can be estimated as c q = c q (t)| t→∞ = ⟨e, vac|e −i Ĥ(1) sys is the whole Hamiltonian of the system [38].The eigen energies of the bound states lie within the gaps of the mechanical bath.To further explore, we write the the Green function of the spin qubit where z denotes arbitrarily plural, ∆ α j = ω m α j − ω q α j denotes the detuning between the qubit and the nearby mechanical resonator, ∑ q (z) denotes the self energy of the spin qubit, see [42,43] and appendix C for more details.The eigen energies E bs of the bound states correspond to the Singular points of the Green function [44,45], namely With coupling parameters J = 1, δ = 0.1 and g = 0.6, we numerically solve equation ( 7) in three cases with ∆ 0,±1 α j = {0, ±2.3J}, and obtain the eigen energies of the bound states E 0,±1 bs = {0, ±2.532J}, see the red dots in figure 1(d).
With the eigenvalues E bs , we could further investigate the wave function of the bound states in the momentum space by solving the stationary Schrodinger equation Putting equations ( 4), ( 5) and ( 8) together, the wave function of the bound states could be analytically solved [38,39].Specifically, the wave function distributions bs ⟩ = Γ|ψ −1 bs ⟩.One of the most appealing ideas in SSH model is the robust confined edge states.There is no edge effect in the infinite condition.However, when one spin qubit is coupled to the mechanical chain, it 'cuts' the chain into two sections, breaking the inversion symmetry of the bath.In figure 2(e), we observe that the zero-energy (E 0 bs = 0) bound state localizes only on one side of the spin qubit, which is very similar to the topological soliton edge states in 1D systems [47].This is because the topological features of the SSH model are imprinted into the qubit-bath coupling interactions.After performing Fourier transform to equation ( 9), the real-space wave function of the zero-energy bound states can be analytically solved.We let the single spin qubit couple to the mechanical resonator with cell index j and sublattice index a, and consider the system in the case δ > 0, then the zero-energy bound-state wave function ∑ j {c a j , c b j } in the sublattice space {a j , b j } can be expressed as c a j = 0, and where λ = (1 − |δ|)/(1 + |δ|) denotes the decay length of the zero-energy bound state, and j = {0, ±1, ±2, ..} denotes the cell index.From equation (10), we see that the wave function is strictly zero when j < 0, and decays when j > 0. This result fits well with figure 2(e).As a comparison, in the case δ < 0 the wave function is strictly zero when j > 0, and decays when j < 0.

Tunable chiral spin-spin interactions
We now discuss the long-range spin-spin interactions mediated by the spin-phonon bound states.It noted that the phonon mediated spin-spin interactions have been investigated in a number of physical systems, including trapped ions [48], spin-optomechanical crystal [49], as well as spin-mechanical hybrid system [35].Specifically, the theoretical model of the spin-spin interactions mediated by the SSH bath has been presented in [38,39].In our system the mechanical lattice works as a phononic bath, therefore the long-range spin-spin couplings could be induced by exchange of virtual phonons.Here the phonons correspond to the phononic parts of the bound sates, see figures 2(d)-(f).We turn to considering the case displayed in see figure 1(a), i.e. for every mechanical resonator there is a spin qubit coupled magnetically.In this situation the whole Hamiltonian of the system reads In the Markovian regime (i.e. the band gap is much larger than the spin-mechanical coupling strength), the mechanical modes can be adiabatically eliminated, and the coupling interactions between spin qubits can be described by an effective Hamiltonian where J αβ x denotes the coupling strength between spin qubits σz α i and σz β j , i and j denote the cell index, x = i − j and α, β ∈ {a, b}.For the 1D mechanical bath with SSH Hamiltonian, J αβ x is associated to the self energy of the collective spin operators (see [38,39] and appendix C for more details), and can be expressed as All of the three bound states could induce the long-range spin-spin interactions.However hereafter we suppose all the spin qubits are in resonance with the nearby mechanical resonators (∆ α j = 0), and focus on the indirect spin-spin couplings bridged by the zero-energy (E bs = 0) bound states only.In this situation the topological features of the bound states (see figure 2(e)) will imprint into the spin-spin interactions.From equation (13) we observe that if two qubits belong to the same type sublattice (αβ ∈ {aa, bb}), there is no coupling interaction, i.e.J aa/bb x (0, x) = 0. Specifically, if they belong to different types of sublattices (αβ ∈ {ab, ba}), the coherent coupling strength can be expressed as We note that equations ( 12) and ( 14) define a spin chain with long-range chiral coupling interactions.Here the chiral symmetry operator is defined as the similar way as in the mechanical SSH model.Considering a spin chain Hamiltonian (12) with coupling strength equation ( 14), we can easily verify that Γ Ĥq eff Γ † = − Ĥq eff .From equation ( 14) it finds that λ (the decay length of the zero-energy bound states) could also be used to denote the effective length of the long-range spin-spin interactions.In figure 3(a), we display a model diagram of this long-range chiral-coupling spin chain by considering 10 spin qubits.In figures 3(b)-(d), we show that the decay length λ and coupling strength J ab x (0, x) can be tuned by the couping parameters {δ, g} of the system.

Simulation of causal effects
Correlation function is one of the most important parameters to quantify cause effects.On the one hand, two variables will not be correlated if there is no causal relationship between them.On the other hand, according to Reichenbach's principle [4,50], if two physical variables X and Y are statistically dependent, the causal relationship between them should be either: ( ) X and Y have a common cause Z, but there is no causal link between them, (4) X and Y have a common cause Z, and X is a cause of Y, or (5) X and Y have a common cause Z, and Y is a cause of X.
After mapping the two energy levels {|d⟩, |e⟩} of the NV centers to spin-1 2 particles, the correlation function between arbitrary two spin qubits can be expressed as where σz The propagation of correlations is an important topic in causal-effect study.For a quautum many-body system with short-range interactions only, Lieb-Robinson model shows that there is a constant velocity to limit the propagation of correlations [6].However, the situation is not so clear in the case with long-range interactions, especially when chiral symmetry is involved.In this spin-mechanical hybrid device, the range and strength of the indirect spin-spin couplings could be tuned in situ by the coupling parameters {δ, g}.Therefore, it provides a powerful platform for simulating the cause effects in long-range chiral-coupling systems.
To verify the effectiveness of our scheme, we consider a 12-qubit chiral-coupling spin chain with an effective Hamiltonian (12) and coupling parameters (14), and study the dynamical evolution of correlation functions numerically by solving the master equation.The coupling and decay (dephasing) parameters are set to be J ab 0 = 1, δ = 0.1, g = 0.4, γ d = 0 and γ s = 0. We investigate the effects of coupling range λ and initial state |ψ p/q ⟩ to the correlation functions, and display the results in figures 4(a1)-(a4).In particular, we prepare the system to initial state , and to initial state ).We denote the maximum correlations as c max 1,2,3,4 , find that c max 1,2,3,4 = {0.365,0.288, 0.199, 0.530} could be reached in figures 4(a1)-(a4), respectively.In general our conclusion can be summarized as follows: (i) The correlation functions are affected by the chiral (sublattice) symmetry of the system.On the other words, we say that the correlation functions c aa 1,j and c ab 1,j individually evolve and propagate in two sublattices {a j , b j }.Specifically, in figures 4(a1)-(a3) it is shown that the correlations c aa 1,j are much stronger than c ab 1,j .This implies that the correlations are inversely proportional to the strength of spin-spin interactions, i.e. c aa/ab 1,j ∝ 1/J aa/ab 1,j .(ii) The range of spin-spin interactions has significant effect on correlation functions.After comparing figures 4(a1)-(a3), it is easy to see that as the increasing of spin-spin interaction range λ, the strengths and propagation velocities of correlation functions are also increased.(iii) The correlation functions c aa/ab 1,j also depend on the initial states of the system, see figures 4(a1) and (a4).
It is instructive to compare the propagation of correlation functions in the spin-spin coupling systems with and without chiral (sublattice) symmetry.Before proceeding we recall that unidirectional propagations of correlations have been observed in the conventional long-range Ising and XY models [51,52].Here we define the fixed correlation amplitudes as C fix 1,2,3,4 = 0.1C , where the unit is 1/J ab 0 .We find that the propagation velocities v aa 1,j and v ab 1,j could be either positive or negative, indicating that the correlations could propagate bi-directionally.
The decay (γ d ) and dephasing (γ s ) processing may also effect the strength and propagation of correlation functions.In particular, here we investigate the evolution of correlation functions with different decay and dephasing rates in a 12-qubit system, where the coupling parameters and initial state are set as the same as in figure 4(a1).We first display the evolution of correlation functions C aa/ab 1,j in the ideal case where γ s = γ d = 0, showing that a maximum correlation C a max = {0.365}can be obtained, see figure 5(a).Then we study the effect of relaxation rate by considering the cases where γ s = 0 and γ d = {0.1,0.7, 1.5}, and find that maximum correlations could reach C b max = {0.338,0.196, 0.105}, see figures 5(b1)-(b3).It is observed that the correlations decrease on increasing the decay rates of the spin qubits.We also investigate the effect of dephasing rate, showing that the maximal correlations could reach C c max = {0.326,0.166, 0.082} when γ d = 0 and γ s = {0.1,0.7, 1.5}, see figures 5(c1)-(c3).Finally, we illustrate the evolution of correlation functions in the case where γ d = 0.1 and γ s = 0.1 in figure 5(d), and find that a maximal correlation C d max = {0.297}can be obtained.Above all, we conclude that strong correlations could only be observed in the strong coupling region where J ab 0 > {γ s , γ d }.

Experimental feasibility
Let us discuss the experimental feasibility of our scheme.We note that the voltage-controlled coupling interactions between adjacent mechanical resonators could be achieved in several ways, one may choose either of them for the implementation.On the one hand, utilizing the electrostatic parametric coupling scheme, the programmable mechanical lattices have been experimentally realized [32][33][34].Thus far, the mechanical-mechanical couplings are still weak in these devices, however we could adjust the distance between adjacent mechanical resonators to ∼50 nm, then the nearest-neighboring mechanical-mechanical coupling interactions could reach J ∼ 50 kHz.On the other hand, using the LC cavity as a quantum interface, two mechanical resonators could be coupled indirectly [35,37].Considering the coupling interactions between a LC cavity and a mechanical resonator could easily reach ∼1 M Hz [36], the mechanical-mechanical couplings in the indirect scheme can be estimated as J ∼ 100 kHz.
We now consider the spin-mechanical couplings.Before proceeding, we note that strong spin-mechanical coupling interactions have been investigated with several types of mechanical resonators, including clamped cantilevers [41,53], torsional cantilevers [54], suspended carbon nanotubes [55,56], as well as trapped nanodiamond [57][58][59].For a doubly clamped mechanical resonator, it is known that the fundamental vibration frequency ω m α j is depending on the geometrical parameters (l, w, h), i.e. ω m α j ≃ h l 2 (1.03 [40], where E ∼ 310 Gap denotes Young's modulus and ρ ∼ 3.12 × 10 3 kg m −3 denotes the density of silicon nitride.When considering a doubly clamped mechanical resonator with size (l, w, h) ≃ (20, 0.2, 0.1) µm, the fundamental vibrational mode has a frequency ω m α j ≃ 2.5 M Hz, and a zero point fluctuation amplitude In the magnetic field gradient G m ≃ 10 7 T/m, the spin-mechanical coupling strength could reach g = g e µ M G m z 0 ≃ 30 kHz, where g e ≃ 2 is the NV landé factor, and µ B ≃ 14 GHZ T −1 is the Bohr magneton.As for the decay and dephasing rates of spin qubits, here we take γ d = 10 Hz and γ s = 1 kHz.Noted that the NV centers in our system could also be replaced by other types of solid defects with longer coherence times, such as divacancy centers in silicon carbide [60] or rare-earth spin qubits [61].

Conclusion
In summary, we present a promising spin-mechanical hybrid quantum device with good scalablity, controllability as well as long coherence time, and apply it to simulate the causal effects in long-range chiral-coupling systems.In this setup, the parametric couplings between adjacent resonators are realized through the electrostatic forces, and the spin-mechanical couplings are induced by gradient magnetic field.We design the Hamiltonian of the mechanical chain to the SSH model, showing that long-range chiral coupling interactions could be induced by the mechanical bath.Specifically, the range and strength of the indirect spin-spin interactions could be tuned in situ by the coupling parameters of the system.
We employ this spin-mechanical quantum simulator to investigate the causal effects in long-range chiral-coupling systems, and find that the correlation functions are propagating bi-directionally in two sublattices.These phenomena are very different from the cases in the conventional long-range spin-spin coupling systems without chiral symmetry [51,52] and therefore deserve special attention.In addition, we also investigate the influences of other physical parameters to the correlation functions, including the range of spin-spin couplings, the initial states of the system, as well as the decaying and dephasing rates of the spin qubits.This spin-mechanical hybrid architecture combines long coherence time of spin qubits with good controllability and scalablity of mechanical chains, which may further be applied in quantum simulations [62][63][64], quantum computation [22], and quantum information processing with chiral interactions [65].case γ n = γ n+1 = γ m .After quantization, the free Hamiltonian of the two adjacent mechanical resonators reads Specifically, the coupling interactions between adjacent mechanical resonators can be described by Hamiltonian For the sake of simplicity, the decay of the mechanical resonators is ignored in the main text.In additional, a detailed description of the main parameters related to the mechanical chain is illustrated in table A1. operators of the mechanical resonator, and α ∈ {a, b}.This magnetic field induces the coupling interactions between the mechanical resonator and the spin qubit, described by Hamiltonian where g 0 = g e µ B Gz 0 denotes the spin-mechanical coupling strength.
To further explore suitable conditions for realizing the coherent coupling interactions between mechanical resonator and single spin qubit, we consider the Hamiltonian of the NV centers in the basis defined by the eigenstates of Ŝz , i.e.Ŝz |m s ⟩ = m s |m s ⟩, m s = {0, ±1}.In a frame rotating with the microwave frequencies, we have where ∆ ± = ω ± − ν ± denote the detunings of the microwave driving.For convenience we limit our discussion to symmetric condition where ∆ + = ∆ − = ∆ and Ω + = Ω − = Ω.We define a 'bright' superposition spin state |b⟩ = 1 , as displayed in figure 1(f).In the dressed-state basis, the spin-mechanical Hamiltonian can be rewritten as where g d = −g 0 sin(α) and g = g 0 cos(α).We further adjust the values of {B z , Ω ± , υ ± } to drive the system to the resonance condition ω ed ≃ ω m α j << ω dg , then the far-off-resonance state |g⟩ can be neglected.Under the rotating-wave approximation, the spin-mechanical coupling dynamics can be described by the Jaynes-Cummings (JC) Hamiltonian where σz α j = |e α j ⟩⟨e α j | − |d α j ⟩⟨d α j |, σ− α j = |d α j ⟩⟨e α j | and σ+ α j = |e α j ⟩⟨d α j | denote the spin operators of the effective two-level system, and ω q α j = ω ed denotes the corresponding transition frequency.Specifically, a description of the parameters of NV centers and spin-mechanical couplings can be found in table B1.

Appendix C. Self energy of spin operators
When one single spin qubit is coupled to the mechanical lattice, it gains and losses energies, leading to the coherent and dissipative dynamics.These dynamics can be analysed by the self energy of the spin qubit.For the SSH-like mechanical bath, both the lower and upper bounds contribute to the coupling interaction.In the Markovian regime, the self energy of single qubit can be expressed as We let ∑ q (z) = δ q − i 2 Γ q , and consider the dynamics of the excited spin qubit using perturbation theory and Fermi's golden rule.It is known that δ q corresponds to the energy shift, and Γ q Corresponds to the decay rate.More detailed discussion about the qubit-bath dynamics in non-Markovian regime can be found in [42,43].
We now turn to considering the case where two spin qubits σz α i and σz β j are coupled to the mechanical bath, where α, β ∈ {a, b} and i, j ∈ {1, 2, 3, ..N c }.When the band gap is much larger than the spin-mechanical coupling strength, the mechanical bath can be effectively traced out within the Markovian approximation, and the indirect coupling interactions between two spin qubits can be described by an effective Hamiltonian where αβ ∈ {aa, ab, ba, bb} and x = i − j.Noted that the spin chain Hamiltonian Ĥq eff in equation ( 12) is the summation of Ĥdd .
To investigate the spin-spin coupling strength, we first define the collective spin operators as σ+ (±) = 1 √ 2 (σ + α i ± σ+ β j ).Then the self energy of σ+ (±) can be expressed as where C α i and C β j denote the coefficients of σ+ α i and σ+ β j .From equations (C.4), we conclude that coherent oscillations between σ+ α i and σ+ β j can be observed in two conditions: (1) Γ M,+ = Γ M,− = 0 and (2) J M,+ ̸ = J M,− .Here the effective coupling strength reads J αβ x = |J M,+ − J M,− | [42].Finally, a detailed description of the parameters in self energy and spin-spin couplings is displayed in table C1.Detuning between the spin qubit and mechanical resonator with cell index j and sublattice index α Gq(z) Green function of the spin qubit with arbitrarily plural z ∑ q (z) Self energy of the spin qubit with arbitrarily plural z J αβ x Coherent coupling strength between spin qubits σz α i and σz β j , where α, β ∈ {a, b} x The difference between cell indexes i and j, x = i − j σ+ Collective spin operators defined between two spin qubits σ+ α i and σ+

Figure 1 .
Figure 1.Scheme diagrams.(a) Schematic of a spin-mechanical hybrid device for integrating spin qubits into a tunable mechanical chain.(b) The base unit of the setup.A NV spin qubit is coupled to a doubly clamped mechanical resonator through a magnetic tip.(c) The inside coupling strength between adjacent mechanical resonators is controlled by the outside manipulation voltage V n p (t) = V n dc + V n ac cos ω n p t.(d) The dispersion relations of the mechanical bath with SSH Hamiltonian, where the grey region denotes the energy gaps, the red dots denote the eigenvalues of the spin-phonon bound states, and the blue and gree regions denote the upper and lower energy bands, respectively.(e) Level diagram of the NV centers in the electronic ground state.(f) Dressed spin states in the presence of external driving magnetic fields.

Figure 2 .
Figure 2. Spin-phonon bound states of the system where J = 1, g = 0.6 and δ = 0.1.(a)-(c) Bound state wave function distributions {c a k , c b k } in the momentum space with eigen energies E −1 bs = −2.532J,E 0 bs = 0 and E +1 bs = 2.532J, respectively.(d)-(f) The corresponding bound state wave function distributions ∑ j {c a j , c b j } in the real space with the same eigen energies in (a)-(c).

Figure 3 .
Figure 3. Tunable chiral spin-spin interactions mediated by the zero-energy bound states.(a) Model diagram of the long-range chiral-coupling spin chain, where the black and red dots denote the σz aj and σz bj spin qubits, respectively.(b) Effective coupling range λ as a function of δ.(c) Absolute value of J ab x (0, x) as a function of coupling parameter δ, where x = {0, 1, 2, 3}.(d) Absolute value of J ab x (0, x) as a function of spin-mechanical coupling strength g.All the coupling strengths are normalized.

Table A1 .
Description of parameters of the programmable mechanical chain.Fundamental frequency of the mechanical resonator with cell index j and sublattice index α, α ∈ {a, b} âj Annihilation operator of the mechanical mode with cell index j and sublattice index a bj Annihilation operator of the mechanical mode with cell index j and sublattice index b

Table B1 .
Description of parameters of NV centers and spin-mechanical couplings.Frequency of the spin qubit with cell index j and sublattice index α, α ∈ {a, b} σ+ Lowering operator of spin qubit with cell index j and sublattice index α, α ∈ {a, b} σz Pauli Matrice σz of spin qubit with cell index j and sublattice index α, α ∈ {a, b} γ d Decay reate of the spin qubits γs Dephasing reate of the spin qubits |g, vac⟩The spin is in the ground state and the mechanical chain is in the vacuum state |e, vac⟩The spin is in the excited state and the mechanical chain is in the vacuum state