Enhancing and tailoring light–matter interactions offer remarkable nonlinear resources with wide-ranging applications in various scientific disciplines. In this study, strong and deterministic tripartite “beamsplitter” (“squeeze”) interactions are constructed by utilizing cavity-enhanced nonlinear anti-Stokes (Stokes) scattering within spin–photon–phonon degrees of freedom. We explore exotic dynamical and steady-state properties associated with the confined motion of a single atom within a high-finesse optical cavity. Notably, we demonstrate the direct extraction of vacuum fluctuations of photons and phonons, which are inherent in Heisenberg’s uncertainty principle, without requiring any free parameters. Moreover, our approach enables the realization of high-quality single-quanta sources with large average photon (phonon) occupancies. The underlying physical mechanisms responsible for generating the nonclassical quantum emitters are attributed to the decay-enhanced single-quanta blockade and long-lived motional phonons, resulting in strong nonlinearity. This work unveils significant opportunities for hitherto studying unexplored physical phenomena and provides novel perspectives on fundamental physics dominated by strong tripartite interactions.

## I. INTRODUCTION

The interface between a cavity and quantum matter offers a prominent platform for harnessing the distinctive characteristics of different degrees of freedom, facilitating the engineering of strongly correlated quantum phases^{1–3} and special nonclassical states with wide-ranging applications in quantum technologies.^{4,5} Remarkable advancements of ground states in low-energy macroscopic mechanical oscillators,^{6–8} such as optomechanical and center-of-mass motion, have opened up remarkable frontiers for quantum information and sensing,^{9,10} quantum metrology,^{11} and studying fundamental physics.^{12,13} Leveraging their long coherence times, significant advancements have been achieved, ranging from n-quanta bundle states^{14–17} to long-lived phonon-to-optical quantum transducers,^{18,19} violation of Bell inequalities,^{20,21} and gravitational wave detection.^{22} These pioneering explorations primarily focus on the fundamental and ubiquitous physical processes of pairwise coherent light–matter interactions between optical cavity (ideal for quantum networking tasks^{23}) and long-lived mechanical modes.^{6} The seminal interactions are the generalized quantum Rabi model^{24,25} and nonlinear optomechanical interactions.^{6}

Meanwhile, the ability to manipulate optomechanical interaction at the single (few) quantum level^{26–28} and measure high-order phonon correlations^{29} has opened up new avenues for experimentally realizing novel types of nonlinear interactions, enabling the study of intriguing aspects of quantum mechanics in hybrid quantum systems. Compared to the well-established paradigm of bipartite interactions,^{6,24,25} constructing tripartite interactions involving diverse degrees of freedom holds significant promise for investigating fundamental physics. Fascinating examples include the investigation of high-fidelity tripartite entanglement,^{30–33} novel nonclassical squeezing states,^{34,35} and hybrid quantum networks^{36,37} by introducing additional nonlinearities, such as atom and long-live mechanical mode. Recent theoretical advancements have explored coherent tripartite interactions between a quantum emitter, photon(magnon), and a mechanical oscillator.^{38–40} Despite rapid experimental progress, the generation of strong tripartite interactions remains a challenge, and the corresponding exotic fundamental quantum phenomena are yet to be fully explored.

In this work, we present an experimental scheme to achieve tunable tripartite interactions using a single alkaline-earth-metal atom deeply trapped in a high-finesse optical cavity. Our architecture, incorporating three distinct degrees of freedom, offers the advantage of long-lived center-of-mass motion, distinguishing from extensively studied single-atom cavity QEDs.^{41–43} In particular, we realize the deterministic tripartite spin–photon–phonon “beamsplitter” (“squeeze”) interactions, leveraging the cavity-enhanced nonlinear anti-Stokes (Stokes) scattering. By performing the photon and phonon number distributions, we directly resolve the vacuum fluctuations of both photon and phonon fields. In addition, the high-quality single-photon and -phonon sources are achieved without requiring single-photon strong coupling. We demonstrate the nonclassical nature of cavity and motional phonon emissions by analyzing their Wigner functions. Notably, the antibunching amplitudes for both photon and phonon exhibit strong sensitivity to the dissipation of motional phonon. This result provides a novel method for monitoring the decoherence of hybrid systems via quantum statistics measurement. In perspective, the strong nonlinearity inherent in the tripartite spin–photon–phonon interactions enables the clear encoding of complex quantum states into the long-lived motional phonon and paves the way for versatile applications in quantum information and metrology.^{44,45} Remarkably, the strong deterministic tripartite “squeeze” interactions can be employed to generate exotic graph states,^{46} which is in contrast to the probabilistic generation of graph states comprising entangled photons via a spontaneous parametric down-conversion process.^{47–49}

## II. MODEL AND HAMILTONIAN

^{88}Sr, confined within a high-finesse cavity and a one-dimensional harmonic trap. The relevant atomic levels consist of a ground state |

*g*⟩ (

^{1}

*S*

_{0}) and a long-lived electronic orbital state |

*e*⟩ (

^{3}

*P*

_{1}), corresponding to the narrow (7.5-kHz-wide) dipole-forbidden transition. To engineer the dynamic spin–orbit coupling, the atomic transitions |

*g*⟩ ↔ |

*e*⟩ are resonantly coupled by a transverse pump field with a Rabi frequency Ω, along with a quantized standing wave cavity field originating from collective Bragg scattering. Then, the tripartite spin–photon–phonon Hamiltonian is given by (see Appendix A for further details)

*ω*

_{b}, $\sigma \u0302x,y,z$ are the Pauli matrices, Δ

_{c}is the cavity-light detuning, Δ is the atom-light detuning, and

*g*is the effective photon–phonon coupling associated with the zero-point fluctuation of the trapped atom oscillator. In contrast to the bipartite Jaynes–Cummings model paradigm, where the single atom is trapped at the antinode of the cavity,

^{41}Eq. (1) is engineered in the Lamb–Dicke regime.

^{50}This configuration liberates the long-lived motional degrees of freedom, thereby offering a novel quantum resource for exploring fundamental physics and enabling a wide range of applications in quantum technology.

Apparently, Hamiltonian (1) captures the tripartite spin-magnon-optomechanical-type interactions,^{38–40,51} which are fundamental building blocks in quantum optics and photonics community, for example, tripartite entanglement.^{30–33} Interestingly, this interaction allows swapping excitations among an optical mode, quantum emitter, and long-lived motional phonon. Our proposal will facilitate the engineering of hybrid spin–photon–phonon quantum transducers and bridge the different research areas of science.

*g*/

*ω*

_{b}≪ 1 and red-sideband resonance Δ

_{c}/

*ω*

_{b}= 1. The nonlinear anti-Stokes process signifies the annihilation of a motional phonon, resulting in the up-conversion of a pump photon into a cavity photon. This process corresponds to the coherent exchange among the quantum emitter, cavity, and phonon fields. For first neglecting the weak pump field (Ω = 0), the system exhibits a $U(1)$ symmetry satisfying the commutation relation, $[R\theta ,H\u0302B]=0$, where $R\theta =exp[\u2212i\theta (2a\u0302\u2020a\u0302+b\u0302\u2020b\u0302+\sigma z/2)]$ is the action of the operator. The occurrence of $U(1)$ symmetry breaking is with respect to single-atom super-radiant phase transition.

*g*/

*ω*

_{b}≪ 1) and blue-sideband resonance (Δ

_{c}/

*ω*

_{b}≈ − 1), Hamiltonian (1) reduces to a tripartite “squeeze” interactions in a rotating reference frame at the frequency

*ω*′ $(\u2248\omega b)$,

*δ*

_{b}=

*ω*

_{b}−

*ω*′,

*δ*

_{a}= Δ

_{c}+

*ω*′, and

*δ*=

*δ*

_{a}− Δ is the detuning of the cavity emission frequency from the atomic resonance. The Hamiltonian (3) captures the cavity-enhanced Stokes scattering, representing a deterministic parametric down-conversion process with respect to the pump photon down-converted into entanglement of cavity and motional phonon. For Ω = 0, these tripartite interactions also exhibit another $U(1)$ symmetry characterized by the operator, $R\theta =exp[\u2212i\theta (a\u0302\u2020a\u0302+b\u0302\u2020b\u0302+\sigma z)]$, satisfying $R\theta \u2020(a\u0302,b\u0302,\sigma \u2212,\sigma +)R\theta =(a\u0302e\u2212i\theta ,b\u0302e\u2212i\theta ,\sigma \u2212e\u2212i2\theta ,\sigma +ei2\theta )$. For single-atom super-radiance, the strong nonclassical correlated photon–phonon pairs at single-quanta level are expected since the emitted photon and phonon are simultaneously created or destroyed. In contrast to the generation of correlated single-photon pairs via a parametric down-conversion process in bimodal cavities,

^{52}our hybrid system highlights the direct entry into the strong-coupling regime with the tripartite single-atom cooperativity $C=g3/(\gamma \kappa a\kappa b)\u226b1$, which is facilitated by the long lifetimes of both the motional phonon and the atomic excited state.

## III. RESULTS AND DISCUSSIONS

### A. Unveiling vacuum fluctuations

To investigate the out-of-equilibrium dynamics, we analyze the quantum statistics of photon and phonon emissions by solving the master equation, considering the complete dissipations of the system (see Appendix B). Figure 2 shows the numerical results depicting the time-dependent occupations and correlation evolutions. The single atom is initially prepared in the excited state, while the cavity and phonon fields are in the vacuum state. Figures 2(a) and 2(b) show the population of photons and motional phonons in the *t*Δ_{c} parameter plane. For the blue sideband Δ_{c}/*ω*_{b} ≈ − 1, the periodic oscillations of the quantized fields are dominated by the tripartite “squeeze” interactions described by Eq. (3). These oscillations correspond to the vacuum Rabi frequency $g2+(\omega b+\Delta c)2/4$. With the evolution of time, the Rabi oscillation exhibits a clear pattern of complete collapse and posterior revival, accompanied by a gradual decrease in amplitude. The difference in amplitude between the photon and phonon oscillations arises to the different decay rates *κ*_{a} and *κ*_{b}. In Fig. 2(c), we perform the evolution of spin excitation $\u27e8\sigma \u0302+\sigma \u0302\u2212\u27e9$ for different phonon frequencies *ω*_{b}. Compared to the Rabi oscillation observed for *ω*_{b}/*g* ≫ 1, the average spin excitation $\u27e8\sigma \u0302+\sigma \u0302\u2212\u27e9$ in the strong coupling regime (*ω*_{b}/*g* ⩽ 1) does not display a complete collapse and revival. Moreover, the total excitation $N\u0302e=\u27e8\sigma \u0302+\sigma \u0302\u2212\u27e9+[\u27e8a\u0302\u2020a\u0302\u27e9+\u27e8b\u0302\u2020b\u0302\u27e9]/2$ clearly exceeds 1 [Fig. 2(d)]. This departure clearly demonstrates that the out-of-equilibrium dynamics go beyond the tripartite “squeeze” interactions, as RWA breaks down for a small phonon frequency (*ω*_{b}/*g* ⩽ 1). Indeed, the total excitation number remains conserved, satisfying $[N\u0302e,H\u0302S]=0$ when the system dissipations are neglected.

Figures 2(e) and 2(f) show the photon and phonon occupations, respectively, as a function of *ω*_{b} for different values of Δ_{c}. Clearly, both $\u27e8a\u0302\u2020a\u0302\u27e9$ and $\u27e8b\u0302\u2020b\u0302\u27e9$ decrease, eventually saturating at zero at the red sideband (Δ_{c}/*ω*_{b} = 1). While for the blue sideband (Δ_{c}/*ω*_{b} = −1), the average photon (phonon) number gradually increases (decreases), reaching saturation values for *ω*_{b}/*g* ≫ 1. Analogous to the bipartite quantum Rabi model,^{8} the observed $\u27e8b\u0302\u2020b\u0302\u27e9>1$ is the hallmark for entering the strong coupling regime in the tripartite model. Remarkably, the difference in cavity (phonon) occupancies between the blue and red sidebands is approximately equal to one, as extracted by the full numerical simulations that take into account the finite linewidth of each mode. This result unambiguously demonstrates that the vacuum fluctuations of the cavity and motional phonon originate from the Heisenberg’s uncertainty principle.

*n*represents the initial excitation of quanta in the quantized fields, and

*θ*

_{S}=

*g*(

*n*+ 1)

*t*and $\theta B=gn(n+1)t$ are accumulated phases after tripartite “squeeze” and “beamsplitter” interactions, respectively. It is evident that the average photon and phonon occupancies satisfy $\u27e8a\u0302\u2020a\u0302\u27e9=\u27e8b\u0302\u2020b\u0302\u27e9=1$ at

*t*=

*π*/2

*g*even when the quantized fields are prepared in vacuum (

*n*= 0). This parametric amplification proceeds by cavity-enhanced Stokes scattering. Conversely, the cooling process leads to $\u27e8a\u0302\u2020a\u0302\u27e9=\u27e8b\u0302\u2020b\u0302\u27e9=0$. Notably, the observed vacuum fluctuations reveal a striking signature of quantum nature, associated with the commutation relation $[o\u0302,o\u0302\u2020]=1$ for both massless cavity field and motional phonon of the object particle. The advantage of the tripartite device is that vacuum fluctuations can be directly obtained without relying on free parameters. This is in contrast to the pioneering experiment in optomechanics featuring bipartite couplings, where vacuum fluctuations were resolved by normalizing the ratio of the final cavity displacement to the initial mechanical displacement.

^{44}More importantly, the long-lived center-of-mass motion at single-quanta emissions significantly improves the measurement accuracy of vacuum fluctuations and mitigates laser-induced heating from the Stokes process.

### B. Strong single-quanta emissions

Now, we turn to the quantum statistics of the system. Figures 3(a) and 3(b) show the second-order correlation function $gaa(2)(0)$ and $gbb(2)(0)$ for different *ω*_{b} in the weak coupling regime. Clearly, the value of $gaa(2)(0)$ for the cavity remains independent on the large phonon frequency. Remarkably, we achieve a strong instantaneous photon blockade with $gaa(2)(0)=2.5\xd710\u22125$ and an occupation of $\u27e8a\u0302\u2020a\u0302\u27e9=0.79$ at *gt*/*π* = 0.5, which demonstrates the potential for a high-quality single-photon source even with weak single atom-cavity coupling (*g*/*κ*_{a} = 4). In addition, we check that both photon and phonon occupations are robust against variations in the large value of *ω*_{b} (see Appendix D). The intuitive understanding is that RWA is valid when *ω*_{b}/*g* > 10. Interestingly, the second-order correlation for the motional phonon exhibits high sensitivity to *ω*_{b}. The amplitude of phonon antibunching increases rapidly with *ω*_{b}. Under RWA (equivalent to *ω*_{b}/*g* ∼ *∞*), the high-quality single-phonon source is realized with $gbb(2)(0)=2.3\xd710\u22125$ and occupation $\u27e8b\u0302\u2020b\u0302\u27e9=0.94$ at *gt*/*π* = 0.5. These results indicate that $gbb(2)(0)$ cannot be exclusively determined by the tripartite “squeeze” Hamiltonian even for *ω*_{b}/*g* ≫ 1, where far-resonance high-frequency terms $g(e\u2212i2\omega bta\u0302\u2020b\u0302\sigma \u0302\u2212+H.c.)$ are ignored. We note that the quantum statistics for the cavity photon and motional phonon can be experimentally measured using a Hanbury Brown and Twiss interferometer^{41} and spin state-resolved projective measurements.^{8,29}

The concept of quasiprobability distribution, such as Wigner function, has proven to be tremendously successful in modern quantum physics. To extract the quantum nature of the realized single-quanta states, we calculate Wigner functions *W*(*α*) and *W*(*β*) in phase-space amplitudes *α* and *β*. The calculated results, as shown in Figs. 3(c) and 3(d), unveil the existence of negative values in the Wigner function, providing clear evidence of the nonclassical nature of both the cavity and motional phonon. Although experimentally certifying nonclassicality remains a challenging task, the full tomographic reconstruction of the Wigner function and related phase-space distributions can be extracted by performing correlation measurement techniques of cavity^{53} and phonon number distribution in a mechanical resonator.^{26} Figures 3(e) and 3(f) show the photon- and phonon-number distributions, denoted as *p*(*q*) = Tr(|*q*⟩⟨*q*|*ρ*), which offer a quantitative measure of the quality of single-quanta emissions. Clearly, the probability of multiquanta excitations (*n* ⩾ 2) being roughly zero reveals the single-quanta nature for photons and phonons, which agrees well with the second-order correlation function $gaa(2)(0)$ and $gbb(2)(0)$ shown in Figs. 3(a) and 3(b). Therefore, our system can serve as the high-quality single-photon and single-phonon source with a clear signature of nonclassicality.

In addition to out-of-equilibrium dynamics, we explore the steady-state properties of the tripartite “squeeze” interactions (Stokes process) with *ω*′/*g* = 100. We emphasize that both photon and phonon emissions are independent on the initial state of the atomic field in a realistic scenario. Figures 4(a) and 4(b) show the second-order correlation functions in the *δδ*_{a} parameter plane. It is shown that the values of $gaa(2)(0)$ for photons and $gbb(2)(0)$ for motional phonons vary over a wide range. Of particular interest, we achieve significant photon and phonon blockades with $gaa(2)(0)\u223c10\u22123$ and $gbb(2)(0)\u223c10\u22122$, which are hallmarks of strong photon and phonon antibunching. As expected, strong antibunching occurs at the single-quanta resonance of upper and lower branches, which is in excellent agreement with the analytic vacuum Rabi splittings satisfying $\delta a(\xb1)=[\delta \xb12g2+\delta 2]/2$ derived from the energy spectrum (red dashed lines) (see Appendix D). We find that the energy spectrum anharmonicity is enhanced by tuning *δ*, resulting in significantly amplified antibunching amplitudes for the quantized fields. The distinct avoided crossing features for photon and phonon emissions are clear signatures of cavity-atom and phonon-atom normal-mode splittings, as shown in Figs. 4(c) and 4(d), corresponding to a minimum value of $2g$ for both cavity and motional phonon modes.

To deeply understand the quantum coherence maintained within dissipations, we plot interval *τ* dependence of $gaa(2)(\tau )$ and $gbb(2)(\tau )$, as shown in Figs. 4(e) and 4(f). Interestingly, the decay times for both photon and phonon are obviously longer than the typical time scale of $\kappa a\u22121$, thanks to the advantageous properties of long-lived motional phonons in the tripartite spin–photon–phonon system. The single-quanta blockade nature is further confirmed through the steady-state distribution $p\u0303(q)\u2261qp(q)/n(s)$, which characterizes the fraction of *q*-quanta states among the total excitations. Notably, we observe strong photon and phonon blockade, with the typical probability of multiquanta excitations (*n* ≥ 2) being below 0.001% for photons and 0.07% for phonons, respectively. We verify that a large decay rate corresponds to a strong antibunching amplitude and a small occupation number of emissions (see Appendix D). This distinct generation of strong single-quanta blockade differs from the study of single-photon pairs in bimodal cavities by utilizing the quantum interference suppressed two-photon excitations.^{52} The mechanism of decay-enhanced strong single-quanta blockade could facilitate the experimental feasibility of engineering specific nonclassical states beyond the limits of strong atom–cavity coupling.

## IV. CONCLUSIONS

We propose an exotic architecture that incorporates a single atom with its long-lived motional degree of freedom within an optical cavity, leveraging state-of-the-art experimental techniques. By constructing two distinct types of deterministic tripartite spin–photon–phonon interactions, we realized an interesting nonlinear resource for generating strong nonclassical quantum emitters. Compared to the experimentally resolving vacuum fluctuations in the bipartite optomechanical system,^{44} our tripartite system enables the direct extraction of vacuum fluctuations for photon and motional phonon fields that inherently originate from Heisenberg’s uncertainty principle without free parameters. Remarkably, the long-lived motional phonon could facilitate precision measurements of fundamental commutation relations in quantum mechanics. Moreover, the demonstrated high-quality single-quanta emissions surpass the limits of strong single atom-cavity coupling. This achievement can be attributed to the mechanism of decay-enhanced single-quanta blockade. By combining additional strong nonlinearity and exquisite manipulation of individual photon and long-lived mechanical mode, our results offer a platform for integrating the advantageous features of atom, optical cavity, and low-energy center-of-mass motion. This integrated approach fosters tremendous progress in exploring the tripartite entanglement,^{30–33} decoherence mechanism,^{54} and distributed quantum information processing.^{5} With a modified cavity featuring a movable end mirror,^{6,55} our proposal can be extended to a new system of motional phonon-coupled cavity optomechanics, presenting intriguing opportunities for studying phonon heat transfer via quantum fluctuations^{56} and phonon-mirror entanglement mediated by the dynamical Casimir effect.^{57,58}

## ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China (Grant Nos. 12135018, 12374365, and 12274473) and the Fundamental Research Funds for the Central Universities (Grant No. 24qnpy120).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Jing Tang**: Investigation (equal); Writing – original draft (equal). **Yuangang Deng**: Supervision (lead); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

### APPENDIX A: THE TRIPARTITE HAMILTONIAN

In this appendix, we present the details on the derivation of the tripartite spin–photon–phonon Hamiltonian for the specific laser configurations and level diagram shown in Fig. 1 of the main text. We focus on a single alkaline-earth-metal atom confined within a single-mode high-finesse cavity with a bare frequency *ω*_{c}. For specificity, the two relevant energy levels for a single ^{88}Sr atom with one electronic ground state |*g*⟩ (^{1}*S*_{0}) and one long-lived electronic orbital state |*e*⟩ (^{3}*P*_{1}) are included. The atomic narrow (7.5-kHz-wide) dipole-forbidden transition frequency between ^{1}*S*_{0} − ^{3}*P*_{1} is *ω*_{e}. This corresponds to the wavelength of *λ* = 689 nm and the wave vector *k*_{L} = 2*π*/*λ*. As can be seen, the atomic transitions are resonant, illuminated by a classical transverse pump field propagating perpendicular to a cavity axis with frequency *ω*_{p} and Rabi frequency Ω. To engineer the dynamical spin–orbit coupling, our proposal utilizes a single-mode optical cavity to couple the atomic transitions between |*g*⟩ ↔ |*e*⟩, which arises from the collective Bragg scattering. This coupling is characterized by a spatially dependent single atom-cavity coupling *g*_{0} sin(*k*_{L}*x*) with *g*_{0} being the maximum scattering rate of the single-atom cavity coupling.

*M*is the mass of the atom, $a\u0302$ is the annihilation operator of the cavity field,

*σ*

_{x,y,z}are Pauli matrices representing the spin-1/2 system with

*σ*

_{±}= (

*σ*

_{x}±

*iσ*

_{y})/2, Δ

_{c}=

*ω*

_{c}−

*ω*

_{p}is the cavity-light detuning, and Δ =

*ω*

_{e}−

*ω*

_{p}is the atom-light detuning. In our setup, we consider the single

^{88}Sr atom confined in a one-dimensional spin-independent harmonic trap $V(x)=12M\omega b2x2$, where

*ω*

_{b}is the adjustable trap frequency. The trap frequency is with respect to the zero-point fluctuation amplitude of the trapped atom oscillator $xZPF=\u210f/2M\omega b$. In the Lamb–Dicke regime with

*k*

_{L}

*x*

_{ZPF}≪ 1,

^{50}the Hamiltonian (A1) can be simplified to

*ω*

_{b}. The operators of motional phonon satisfy the commutation relations, $[b\u0302,b\u0302\u2020]=1$. With these operators, we can derive a tripartite spin–photon–phonon Hamiltonian as follows:

*g*=

*g*

_{0}

*k*

_{L}

*x*

_{ZPF}is the effective photon–phonon coupling. Apparently, this cavity-enhanced tripartite Hamiltonian $H\u0302$ introduces novel three-body interactions, which hold significant importance and interest, especially for an essential building block in quantum information processing and photonics community. Notably, this tripartite Hamiltonian enables swapping the excitations among the three distinct degrees of freedom in quantum systems. This capability opens up exciting possibilities for manipulating and exchanging quantum information in integrated quantum systems.

*δ*

_{b}=

*ω*−

*ω*′, $\delta a=\Delta c\u2212\Delta c\u2032$, and

*δ*=

*δ*

_{a}− Δ being the detuning of the cavity radiation frequency from the atomic resonance.

*g*/

*ω*

_{b}≪ 1. This simplification leads to “beamsplitter” interactions with respect to the annihilation of a motional phonon up-convert into a quantized cavity photon,

The breaking of the $U(1)$ symmetry breaking is related to single-atom super-radiant phase transitions.

*g*/

*ω*

_{b}≪ 1. This simplification leads to a “squeeze” interactions Hamiltonian with respect to the pump photon down-converted into correlated photon–phonon pairs,

For single-atom super-radiance, strong nonclassical correlated photon–phonon pairs at the single-quanta level are expected, as the emitted photon and phonon are simultaneously created or destroyed.

*zero*due to the deterministic parametric down-conversion process. Therefore, the restricted Hilbert spaces for the single atom-cavity system are |

*n*

_{a}− 1,

*n*

_{b}− 1,

*e*⟩ and |

*n*

_{a},

*n*

_{b},

*g*⟩, where

*n*

_{a}(

*n*

_{b}) denotes the photon (phonon) number excitation of the system. Explicitly, the energy eigenvalues of systems with

*δ*

_{a}=

*δ*

_{b}fixed, are given by

*n*

_{a}=

*n*

_{b}=

*n*without loss of generality, the energy spectrum can be expressed as

*n*= 1),

*δ*, which is different from the realization of single-photon pairs for bimodal cavities in Ref. 52 by utilizing the quantum interference suppressed two-photon excitation. The detuning

*δ*-enhanced vacuum Rabi splittings between the first pair of dressed states |1, 1, ±⟩ can significantly facilitate the realization of high-quality single-quanta sources beyond the strong coupling regime in experiments.

### APPENDIX B: NUMERICAL SOLUTION OF THE MASTER EQUATION

^{59}In the presence of the complete dissipations of the system, the time evolution of the density matrix

*ρ*dictated by the full Hamiltonian (1) in the main text obeys the master equation,

*γ*is the atomic spontaneous emission rate of the long-lived excited state,

*κ*

_{a}and

*κ*

_{b}are the decay rates of the photon and phonon, respectively, and $D[o\u0302]\rho =2o\u0302\rho o\u0302\u2020\u2212o\u0302\u2020o\u0302\rho \u2212\rho o\u0302\u2020o\u0302$ denotes the Lindblad type of dissipation. We should emphasize that the standard Lindblad equation (B1) is applicable for characterizing the strong coupling tripartite spin–photon–phonon interaction,

^{60,61}since the Hamiltonian (1) is essentially originating from RWA under the condition

*g*/

*ω*

_{c}≪ 1. We neglect the weak pure dephasing effects in the long-lived atom and phonon fields.

^{62}

^{,}

*o*=

*a*,

*b*. For

*τ*= 0, $gaa(2)(0)$ $[gbb(2)(0)]$ represents the cavity (phonon) mode and is a key physical quantity for assessing the purity of single-quanta emissions.

^{63}The value of $goo(2)(0)$ should be less than 1 to ensure sub-Poissonian statistics. Furthermore, for nonzero time interval, $goo(2)(\tau )$ can be straightforwardly calculated using the quantum regression theorem. Experimental measurements of $goo(2)(\tau )$ can be measured using techniques such as the Hanbury Brown and Twiss interferometer for cavity photons

^{41}and spin state-resolved projective measurements for motional phonons.

^{8,29,64}Now, the quantum statistics for PB should satisfy two conditions: $goo(2)(0)<1$ to ensure the sub-Poissonian statistics and $goo(2)(0)<goo(2)(\tau )$ to ensure antibunching with respect to single photon or phonon.

### APPENDIX C: RESOLVING VACUUM FLUCTUATIONS FROM SCHRÖDINGER PICTURE DYNAMICS

In order to gain further insight, we simplify the analytical model by neglecting the system damping and the weak pump field. This allows us to resolve the vacuum fluctuations of the atom oscillator’s motional phonon and the quantized cavity field from the Schrödinger picture dynamics. We consider both the photon and phonon fields to be in Fock states |*n*, *n*⟩, while the single atom is in the excited state |*e*⟩|.

#### 1. The tripartite “squeeze” interactions

*δ*

_{a}=

*δ*

_{b}=

*δ*= 0, the tripartite “squeeze” Hamiltonian simplifies to

*n*,

*n*,

*e*⟩ and |

*n*+ 1,

*n*+ 1,

*g*⟩. The energy eigenvalues of Eq. (C1) are given by

*n*, ±⟩ associated with these energy eigenvalues satisfy

*ψ*

_{field}(

*t*= 0) = |

*n*,

*n*⟩ and assume the single atom is injected in the long-lived excited state. Thus, the initial state for the tripartite system reads

*n*, ±⟩ are stationary states of the tripartite spin–photon–phonon system, the state vector with time development can be straightforwardly calculated as follows:

*θ*

_{S}=

*g*(

*n*+ 1)

*t*represents the accumulated phases after the tripartite “squeeze” interactions. It is clear that the state vector exhibits Rabi oscillations. Furthermore, if both the photon and motional phonon are initially in vacuum states, the state of the tripartite system as a function of time can be expressed as

#### 2. The tripartite “beamsplitter” interactions

*δ*

_{a}=

*δ*

_{b}=

*δ*= 0, the tripartite “beamsplitter” Hamiltonian reduces to

*n*,

*n*⟩. Therefore, the restricted Hilbert spaces for the single atom-cavity system are |

*n*,

*n*,

*e*⟩ and |

*n*+ 1,

*n*− 1,

*g*⟩. The energy eigenvalues of Eq. (C7) are given by

*n*, ±⟩ associated with the energy eigenvalues satisfy

*ψ*

_{field}(

*t*= 0) = |

*n*,

*n*⟩ and assume that the single atom is injected in the long-lived excited state. Thus, the initial state for the tripartite system reads

*n*, ±⟩ are stationary states of the tripartite spin–photon–phonon system, the state vector with time development can be straightforwardly calculated as follows:

*n*> 0) being the accumulated phases after tripartite “beamsplitter” interactions. Interestingly, considering that both photon and motional phonon are initially in the vacuum states, the state of the tripartite system as a function of time satisfies |

*ψ*(

*t*)⟩ = |0, 0,

*e*⟩, which corresponds to zero average photon and phonon occupancies.

### APPENDIX D: THE EFFECT OF MOTIONAL PHONON DECAY

In this section, we study the effect of motional phonon decay on the quantum coherence and emission properties of the tripartite spin–photon–phonon system. We numerically calculate the second-order correlation function $goo(2)(0)$ and the corresponding steady-state population $no(s)$ with leaving the motional phonon decay rate *κ*_{b} as a tunable parameter. The results are shown in Figs. 5(a) and 5(b). We observe that a larger phonon decay rate leads to a stronger antibunching amplitude [*g*^{(2)}*aa*(0)] and a smaller occupation number of emissions (*n*^{(s)}*a*). It is clear that the decay rate of the motional phonon significantly affects both the photon and phonon emissions. With increasing *κ*_{b}, the antibunching amplitude $gaa(2)(0)$ rapidly grow, albeit the photon excitation $na(s)$ which remains roughly unchanged when *κ*_{b}/*κ*_{a} < 1. As for motional phonon, the value of $gbb(2)(0)$ is insensitive to variations in *κ*_{b}, but the phonon excitation $nb(s)$ rapidly decreases with increasing *κ*_{b}. Compared to the long-lived motional phonon, the antibunching amplitude for photon is significantly enhanced by over one orders of magnitude with up to $gaa(2)(\tau )<10\u22123$ when the decay rate ratio satisfies *κ*_{b}/*κ*_{a} ∼ 0.01. Of importance, the proposed tripartite system exhibiting both strong photon- and phonon-blockade and corresponding to a large population of steady state can be used as high-quality quantum sources.

To further analyze the single-quanta blockade nature of photon and phonon emissions, we plot the interval *τ* dependence of the second-order correlation functions $gaa(2)(\tau )$ (c) and $gbb(2)(\tau )$ for different values of *κ*_{b}, as shown in Figs. 5(c) and 5(d). Interestingly, the decay times for both $gaa(2)(\tau )$ (c) and $gbb(2)(\tau )$ are obviously longer than the typical time scale of $\kappa a\u22121$. This behavior is a result of the advantages of the long-lived motional phonon in the tripartite spin–photon–phonon system. We show that the decay of antibunching with $goo(2)(\tau )>goo(2)(0)$ decreases gradually as the decay ratio *κ*_{a}/*κ*_{b} increases.

To characterize the single-quanta blockade nature of the photon and phonon emissions, we examine the steady-state distribution $p\u0303(q)\u2261qp(q)/n(s)$, which represents the fraction of *q*-quanta states among the total excitations of the steady state. Clearly, the strong photon and phonon blockade with the typical probability of multiquanta excitations (*n* > 2) being below 0.001% and 0.07% are achieved, respectively, for *κ*_{a}/*κ*_{b} = 10. These results demonstrate that the proposed tripartite system can serve as a high-quality quantum source with a strong photon and phonon blockade.

## REFERENCES

*n*-phonon bundle emission via the Stokes process