# Resolving Vacuum Fluctuations and Nonclassical States with Cavity-Enhanced
Tripartite Interactions

Jing Tang  Guangdong Provincial Key Laboratory of Quantum Metrology and
Sensing $\&$ Sc hool of Physics and Astronomy, Sun Yat-Sen University (Zhuhai
Campus), Zhuhai 519082, China  Yuangang Deng<EMAIL_ADDRESS>Guangdong Provincial Key Laboratory of Quantum Metrology and Sensing $\&$ Sc
hool of Physics and Astronomy, Sun Yat-Sen University (Zhuhai Campus), Zhuhai
519082, China

(August 31, 2024)

# Unveiling Vacuum Fluctuations and Nonclassical States with Cavity-Enhanced
Tripartite Interactions

Jing Tang  Guangdong Provincial Key Laboratory of Quantum Metrology and
Sensing $\&$ Sc hool of Physics and Astronomy, Sun Yat-Sen University (Zhuhai
Campus), Zhuhai 519082, China  Yuangang Deng<EMAIL_ADDRESS>Guangdong Provincial Key Laboratory of Quantum Metrology and Sensing $\&$ Sc
hool of Physics and Astronomy, Sun Yat-Sen University (Zhuhai Campus), Zhuhai
519082, China

(August 31, 2024)

###### Abstract

Enhancing and tailoring light-matter interactions offer remarkable nonlinear
resources with wide-ranging applications in various scientific disciplines. In
this study, we investigate the construction of strong and deterministic
tripartite ‘beamsplitter’ (‘squeeze’) interactions by utilizing cavity-
enhanced nonlinear anti-Stokes (Stokes) scattering within the spin-photon-
phonon degrees of freedom. We explore the 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 nonclassical quantum emitters are attributed to
decay-enhanced single-quanta blockade and the utilization of long-lived
motional phonons, resulting in strong nonlinearity. This work unveils
significant opportunities for studying hitherto unexplored physical phenomena
and provides novel perspectives on fundamental physics dominated by strong
tripartite interactions.

Introduction.—The interface between cavity and quantum matter offer a
prominent platform for harnessing the distinctive characteristics of different
degrees of freedom, facilitating the engineering of strongly correlated
quantum phases Ritsch  _et al._ (2013); Mivehvar  _et al._ (2021); Deng and Yi
(2023); Mivehvar  _et al._ (2018); Léonard  _et al._ (2017a); Kim  _et al._
(2018) and special nonclassical states with wide-ranging applications in
quantum technologies Raimond  _et al._ (2001); Hammerer  _et al._ (2010);
Reiserer and Rempe (2015); Reiserer (2022). Remarkable advancements of ground
states in low-energy macroscopic mechanical oscillators Aspelmeyer  _et al._
(2014); Marquardt  _et al._ (2007); Teufel  _et al._ (2011); Leibfried  _et
al._ (1996); Lv  _et al._ (2018), such as optomechanical and center-of-mass
motion, has opened up remarkable frontiers for quantum information and sensing
Bennett and DiVincenzo (2000); Forn-Díaz  _et al._ (2019); Ge  _et al._
(2019), quantum metrology Giovannetti  _et al._ (2006); Pezzè  _et al._
(2018); Degen  _et al._ (2017), and studying fundamental physics Wollman  _et
al._ (2015); Palomaki  _et al._ (2013); Meekhof  _et al._ (1996). Leveraging
their long coherence times, significant advancements have been achieved,
ranging from n-quanta bundle states Muñoz  _et al._ (2014); Chang  _et al._
(2016); Bin  _et al._ (2020, 2021); Deng  _et al._ (2021) to long-lived
phonon-to-optical quantum transducers Mirhosseini  _et al._ (2020); Briegel
_et al._ (1998); Forsch  _et al._ (2020); Bagci  _et al._ (2014); Bochmann
_et al._ (2013); Schuetz  _et al._ (2015); Noguchi  _et al._ (2017), violation
of Bell inequalities Mari and Eisert (2012); Arvidsson-Shukur  _et al._
(2020); Marinković  _et al._ (2018), and gravitational wave detection Miao
_et al._ (2015); Korobko  _et al._ (2017); Cataño Lopez  _et al._ (2020).
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 Kimble (2008)) and
long-lived mechanical modes Aspelmeyer  _et al._ (2014). The seminal
interactions are the generalized quantum Rabi model Rabi (1936); Jaynes and
Cummings (1963) and nonlinear optomechanical interactions Aspelmeyer  _et al._
(2014).

Meanwhile, the ability to manipulate optomechanical at the single (few)
quantum level O?Connell  _et al._ (2010); Chu  _et al._ (2018); Arrangoiz-
Arriola  _et al._ (2019); Sletten  _et al._ (2019) and measure high-order
phonon correlations Patil  _et al._ (2022); Riedinger  _et al._ (2016); Hong
_et al._ (2017); Riedinger  _et al._ (2018) has opened up new avenues for
experimental 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 Aspelmeyer
_et al._ (2014); Rabi (1936); Jaynes and Cummings (1963), constructing a
tripartite interactions involving diverse degrees of freedom holds significant
promise for investigating fundamental physics. Fascinating examples include
the investigation of high-fidelity tripartite entanglement Lance  _et al._
(2004); Jing  _et al._ (2003); Villar  _et al._ (2006); Dür  _et al._ (2000);
Bennett  _et al._ (1999), novel nonclassical squeezing states Kustura  _et
al._ (2022); Pontin  _et al._ (2014); Kienzler  _et al._ (2016), and hybrid
quantum networks Cirac  _et al._ (1997); Duan and Monroe (2010); Deng  _et
al._ (2017) 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 Cotrufo  _et al._ (2017a); Zhou  _et al._ (2022); Hei
_et al._ (2023a). 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 Letter, we present an experimental scheme to achieve a 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 of long-lived
center-of-mass motion, distinguishing from extensively studied single-atom
cavity QEDs Birnbaum  _et al._ (2005); Mücke  _et al._ (2010); Hamsen  _et
al._ (2017). Specifically, we realize 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 the
photon and phonon fields. Additionally, the high-quality single-photon and
-photon 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 strongly 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 of the tripartite spin-photon-phonon
interactions allows us to clearly encode complex quantum states into the long-
lived motional phonon, and pave the way for versatile applications in quantum
information and metrology Lecocq  _et al._ (2015); Wolf  _et al._ (2019); Ma
_et al._ (2021); Kozlov  _et al._ (2018); Pikovski  _et al._ (2012); Shomroni
_et al._ (2019).

Model.— As illustrated in Fig. 1, we consider a single alkaline-earth-metal
atom, such as, 88Sr, confined within a high-finesse cavity and a one-
dimensional harmonic trap. The relevant atomic levels consist of a ground
state $|g\rangle$ (${}^{1}S_{0}$) and a long-lived electronic orbital state
$|e\rangle$ (${}^{3}P_{1}$), corresponding to the narrow (7.5-kHz-wide)
dipole-forbidden transition. To engineer the dynamic spin-orbit coupling, the
atomic transitions $|g\rangle\leftrightarrow|e\rangle$ are resonant coupled by
a transverse pump field with Rabi frequency $\Omega$ and quantized standing
wave cavity field originated from the collective Bragg scattering. In the
Lamb-Dicke regime Vogel and Filho (1995), the tripartite spin-photon-phonon
Hamiltonian is given by SM

$\displaystyle\hat{\cal{H}}/\hbar$
$\displaystyle=\omega_{b}\hat{b}^{\dagger}\hat{b}+\Delta_{c}\hat{a}^{\dagger}\hat{a}+{\frac{\Delta}{2}}\sigma_{z}+\Omega\hat{\sigma}_{x}$
$\displaystyle+g(\hat{b}^{\dagger}+\hat{b})(\hat{a}^{\dagger}\hat{\sigma}_{-}+\hat{a}\hat{\sigma}_{+}),$
(1)

where $\hat{a}$ denotes the annihilation operator of single-mode cavity,
$\hat{b}$ denotes the annihilation operator of bosonic phonon for motional
mode with harmonic oscillator frequency $\omega_{b}$, $\hat{\sigma}_{x,y,z}$
are the Pauli matrices, $\Delta_{c}$ is the cavity-light detuning, $\Delta$ is
the atom-light detuning, and $g$ is the effective photon-phonon coupling
associated with the zero-point fluctuation of the trapped atom oscillator.
Apparently, Hamiltonian (1) captures the tripartite spin-magnon-
optomechanical-type interactions Zhou  _et al._ (2022); Lai  _et al._ (2022);
Cotrufo  _et al._ (2017b); Hei  _et al._ (2023b), which are fundamental
building blocks in quantum optics and photonics community, for example,
tripartite entanglement Lance  _et al._ (2004); Jing  _et al._ (2003); Villar
_et al._ (2006); Dür  _et al._ (2000); Bennett  _et al._ (1999).
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.

Figure 1: (color online). (a) Schematic diagram of cavity-enhanced tripartite
interactions. (b) Principle of realizing tripartite ‘beamsplitter’ and
‘squeeze’ interactions, depicting two distinct processes of red and blue
sideband transitions. (c) Scattering picture of the anti-Stokes (Stokes)
process with a photon leading to the annihilation (creation) of a motional
phonon.

To capture the underlying of physics, one can obtain a tripartite
‘beamsplitter’ interactions SM

$\displaystyle{\cal\hat{H}}_{\rm{B}}/\hbar$
$\displaystyle={\frac{\Delta}{2}}\sigma_{z}+\Omega\hat{\sigma}_{x}+g(\hat{a}^{\dagger}\hat{b}\hat{\sigma}_{-}+\hat{b}^{\dagger}\hat{a}\hat{\sigma}_{+}),$
(2)

under the rotating-wave approximation (RWA) with $g/\omega_{b}\ll 1$ and red-
sideband resonance $\Delta_{c}/\omega_{b}=1$. This nonlinear anti-Stokes
process denotes an annihilation of a motional phonon up converting a pump
photon into a cavity photon, corresponding to the coherent exchange among
quantum emitter, cavity, and phonon fields. For firstly neglecting the weak
pump field ($\Omega=0$), the system exhibits a ${\cal U}(1)$ symmetry
satisfying the commutation relation, $[{\cal
R}_{\theta},{\cal\hat{H}}_{\rm{B}}]=0$, where ${\cal
R}_{\theta}=\exp[-i\theta(2\hat{a}^{\dagger}\hat{a}+\hat{b}^{\dagger}\hat{b}+\sigma_{z}/2)]$
is the action of the operator. The occurrence of ${\cal U}(1)$ symmetry
breaking corresponds to single-atom superradiance Kim  _et al._ (2018).

Under the RWA ($g/\omega_{b}\ll 1$) and blue-sideband resonance
($\Delta_{c}/\omega_{b}\approx-1$), Hamiltonian (1) will reduce to tripartite
‘squeeze’ interactions in a rotating reference frame at the frequency
$\omega^{\prime}$ ($\approx\omega_{b}$)

$\displaystyle{\cal\hat{H}}_{\rm{S}}/\hbar$
$\displaystyle=\delta_{b}\hat{b}^{\dagger}\hat{b}+\delta_{a}\hat{a}^{\dagger}\hat{a}+{\frac{\delta_{a}-\delta}{2}}\sigma_{z}+\Omega\hat{\sigma}_{x}$
$\displaystyle+g(\hat{a}^{\dagger}\hat{b}^{\dagger}\hat{\sigma}_{-}+\hat{b}\hat{a}\hat{\sigma}_{+}),$
(3)

where $\delta_{b}=\omega_{b}-\omega^{\prime}$,
$\delta_{a}=\Delta_{c}+\omega^{\prime}$, and $\delta=\delta_{a}-\Delta$ is the
detuning of the cavity emission frequency from the atomic resonance. The
Hamiltonian (Resolving Vacuum Fluctuations and Nonclassical States with
Cavity-Enhanced Tripartite Interactions) 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 $\Omega=0$, these tripartite interactions also exhibit
another ${\cal U}(1)$ symmetry characterized by the operator, ${\cal
R}_{\theta}=\exp[-i\theta(\hat{a}^{\dagger}\hat{a}+\hat{b}^{\dagger}\hat{b}+\sigma_{z})]$,
satisfying ${\cal
R}_{\theta}^{\dagger}(\hat{a},\hat{b},\sigma_{-},\sigma_{+}){\cal
R}_{\theta}=(\hat{a}e^{-i\theta},\hat{b}e^{-i\theta},\sigma_{-}e^{-i2\theta},\sigma_{+}e^{i2\theta})$.
For single-atom superradiance, the strong nonclassical correlated photon-
phonon pairs in the 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 parametric
down-conversion process in bimodal cavities Peng and Deng (2022), our
hybridization system highlights the direct entering the strong-coupling regime
with the tripartite single-atom cooperativity ${\cal
C}=g^{3}/(\gamma\kappa_{a}\kappa_{b})\gg 1$, as both motional phonon and
atomic excited state possess long lifetimes. Here, $\gamma$ is atomic
spontaneous emission rate of the excited state, $\kappa_{a}$ and $\kappa_{b}$
are the decay rates of the photon and phonon, respectively. In our numerical
simulation, we adopt $\gamma=7.5\leavevmode\nobreak\ {\rm
kHz}\leavevmode\nobreak\ (2\pi)$ Ludlow  _et al._ (2015); Norcia and Thompson
(2016); Winchester  _et al._ (2017), $\kappa_{b}/\gamma=1$,
$\kappa_{a}/\gamma=10$ Norcia and Thompson (2016); Winchester  _et al._
(2017); Wolke  _et al._ (2012), and $\Omega/\gamma=0.2\kappa_{a}$. The single
atom-cavity coupling is $g/\gamma=40$, which has been demonstrated in the
state-of-the-art cavity QEDs experiments Hamsen  _et al._ (2017); Léonard  _et
al._ (2017a, b). The independent controlled parameters are $\omega_{b}$ and
$\Delta_{c}$.

Results.—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 SM . Figure 2
presents 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) illustrate the population of photons and motional phonons in
$t\Delta_{c}$ parameter plane. For the blue sideband focus
$\Delta_{c}/\omega_{b}\approx-1$, the periodic oscillations of the quantized
fields are dominated by the tripartite ‘squeeze’ interactions described by Eq.
(Resolving Vacuum Fluctuations and Nonclassical States with Cavity-Enhanced
Tripartite Interactions). These oscillations correspond to the vacuum Rabi
frequency $\sqrt{g^{2}+(\omega_{b}+\Delta_{c})^{2}/4}$. With time evolution,
the Rabi oscillation exhibits a clear pattern of complete collapse and
posterior revival with a gradually decreasing amplitude. The difference in
amplitude between the photon and phonon oscillations arises due to the
different decay rates $\kappa_{a}$ and $\kappa_{b}$. In Fig. 2(c), we perform
the evolution of spin excitation
$\langle\hat{\sigma}_{+}\hat{\sigma}_{-}\rangle$ for different phonon
frequencies $\omega_{b}$. In contrast to the Rabi oscillation observed for
$\omega_{b}/g\gg 1$, the average spin excitation
$\langle\hat{\sigma}_{+}\hat{\sigma}_{-}\rangle$ in the strong coupling regime
($\omega_{b}/g\leqslant 1$) does not display a complete collapse and revival.
Moreover, the total excitation
$\hat{N}_{e}=\langle\hat{\sigma}_{+}\hat{\sigma}_{-}\rangle+[\langle\hat{a}^{\dagger}\hat{a}\rangle+\langle\hat{b}^{\dagger}\hat{b}\rangle]/2$
clearly exceed 1 [Fig. 2(d)]. This departure clearly demonstrates that the
out-of-equilibrium dynamics go beyond the tripartite ‘squeeze’ interactions,
as RWA is breaks down for small phonon frequency ($\omega_{b}/g\leqslant 1$).
Indeed, the total excitation number remains conserved satisfying
$[\hat{N}_{e},{\cal\hat{H}}_{\rm{S}}]=0$ when the system dissipations are
neglected.

Figure 2: (color online). (a)-(b) Photon and phonon occupations response as a
function of $t$ and $\Delta_{c}$ with $\omega_{b}/g=10$. (c)-(d) Evolution of
$\langle\hat{\sigma}_{+}\hat{\sigma}_{-}\rangle$ and $\hat{N}_{e}$ for
different $\omega_{b}$ with $\Delta_{c}/\omega_{b}=-1$. (e)-(f) $\omega_{b}$
dependence of $\langle\hat{a}^{\dagger}\hat{a}\rangle$ and
$\langle\hat{b}^{\dagger}\hat{b}\rangle$ for different $\Delta_{c}$ at
$gt/\pi=0.5$. The other parameter is $\Delta=0$.

Figures 2(e) and 2(f) show the photon and phonon occupations as a function of
$\omega_{b}$ for different values of $\Delta_{c}$. Clearly,
$\langle\hat{a}^{\dagger}\hat{a}\rangle$ and
$\langle\hat{b}^{\dagger}\hat{b}\rangle$ both decrease, eventually saturating
at zero at the red sideband ($\Delta_{c}/\omega_{b}=1$). While for the blue
sideband ($\Delta_{c}/\omega_{b}=-1$), the average number of photon (phonon)
is gradually increasing (decreasing), reaching saturation values for
$\omega_{b}/g\gg 1$. Analogous to bipartite Rabi model Lv  _et al._ (2018),
the observed $\langle\hat{b}^{\dagger}\hat{b}\rangle>1$ is the hallmark for
entering the strong coupling regime in tripartite model. Remarkably, the
difference in cavity (phonon) occupancies between the blue and red sidebands
is approximately equal to one, extracted by full numerical simulations that
take into account the finite linewidth of each mode. This result unambiguously
demonstrates the vacuum fluctuations of the cavity and motional phonon
originally Heisenberg’s uncertainty principle.

To acquire more physical insight, we can study the dynamics in the Schrödinger
picture, describing the time evolution of photon and phonon amplitudes
excluding the dissipations of system. The time-dependent states after the
nonlinear Stokes and anti-Stokes processes read SM

$\displaystyle|\psi(t)\rangle_{S}$
$\displaystyle=\cos\theta_{S}|n,n,e\rangle-i\sin\theta_{S}|n+1,n+1,g\rangle,$
$\displaystyle|\psi(t)\rangle_{B}$
$\displaystyle=\cos\theta_{B}|n,n,e\rangle-i\sin\theta_{B}|n+1,n-1,g\rangle,$
(4)

where $n$ represents the initial excitation of quanta in the quantized fields,
$\theta_{S}=g(n+1)t$ and $\theta_{B}=g\sqrt{n(n+1)}t$ are accumulated phases
after tripartite ‘squeeze’ and ‘beamsplitter’ interactions, respectively. It
is evident that the average photon and phonon occupancies satisfy
$\langle\hat{a}^{\dagger}\hat{a}\rangle=\langle\hat{b}^{\dagger}\hat{b}\rangle=1$
at $t=\pi/2g$ 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
$\langle\hat{a}^{\dagger}\hat{a}\rangle=\langle\hat{b}^{\dagger}\hat{b}\rangle=0$.
Notably, the observed vacuum fluctuations reveal a striking signature of
quantum nature, associated with the commutation relation,
$[\hat{o},\hat{o}^{\dagger}]=1$. The advantage of the tripartite device is
that vacuum fluctuations can be directly obtained without relying on free
parameters. This is in contrast to pioneering experiment in bipartite
optomechanics, where vacuum fluctuations were resolved by normalizing the
ratio of the final cavity displacement to the initial mechanical displacement
Lecocq  _et al._ (2015).

Figure 3: (color online). (a)-(b) The evolution of $g^{(2)}_{aa}(0)$ and
$g^{(2)}_{bb}(0)$ for different $\omega_{b}$ with $\Delta_{c}/\omega_{b}=-1$.
(c)-(d) Wigner functions of the photon and phonon at $gt/\pi=0.5$ and
$\omega_{b}/g=10$. (g)-(h) The corresponding photon- and phonon-number
distributions $p(q)$. The remaining parameter is $\Delta=0$.

To gain further insight into the quantum statistics, we plot second-order
correlation function $g_{aa}^{(2)}(0)$ and $g_{bb}^{(2)}(0)$ for different
$\omega_{b}$ in the weak coupling regime, as displayed in Figs. 3(a) and 3(b).
We find that the value of $g_{aa}^{(2)}(0)$ for the cavity is independent on
large phonon frequency. Remarkably, we achieve the strong instantaneous photon
blockade with $g_{aa}^{(2)}(0)=2.5\times 10^{-5}$ and an occupation of
$\langle\hat{a}^{\dagger}\hat{a}\rangle=0.79$ at $gt/\pi=0.5$, which
demonstrates the potential for a high-quality single-photon source even with
weak single atom-cavity coupling ($g/\kappa_{a}=4$). Additionally, we check
that both photon and phonon occupations are robust against variations in the
large value of $\omega_{b}$ SM . The intuitive understanding is that RWA is
valid when $\omega_{b}/g>10$. Interestingly, the second-order correlation for
the motional phonon exhibits high sensitivity to $\omega_{b}$. The phonon
antibunching amplitude increases rapidly with $\omega_{b}$. For RWA
(equivalent to $\omega_{b}/g\sim\infty$), the high-quality single-phonon
source is realized with $g_{bb}^{(2)}(0)=2.3\times 10^{-5}$ and occupation
$\langle\hat{b}^{\dagger}\hat{b}\rangle=0.94$ at $gt/\pi=0.5$. These results
indicate that $g_{bb}^{(2)}(0)$ can not be exclusively determined by the
tripartite ‘squeeze’ Hamiltonian even for $\omega_{b}/g\gg 1$, when far-
resonance high-frequency terms
$g(e^{-i2\omega_{b}t}\hat{a}^{\dagger}\hat{b}\hat{\sigma}_{-}+{\rm H.c.})$ are
ignored. We note that the quantum statistics for the cavity photon and
motional phonon can be experimentally measured by utilizing Hanbury, Brown,
and Twiss interferometer Birnbaum  _et al._ (2005) and spin state-resolved
projective measurements Lv  _et al._ (2018); Wolf  _et al._ (2019); Patil  _et
al._ (2022).

Figure 4: (color online). (a)-(b) show the values of log$[g^{(2)}_{aa}(0)]$
and log$[g^{(2)}_{bb}(0)]$ on $\delta\delta_{a}$ parameter plane. (c)-(d) The
corresponding steady-state population $n_{a}^{(s)}$ and $n_{b}^{(s)}$. The red
dashed lines denote the analytic vacuum Rabi splittings in the upper and lower
branches. (e)-(f) $\tau$ dependence of $g^{(2)}_{aa}(\tau)$ and
$g^{(2)}_{bb}(\tau)$ with $\delta_{a}/g=(1+\sqrt{3})/2$ and $\delta/g=1$.
Inset displays the steady-state distribution $\tilde{p}(q)$. The other
parameters are $\delta_{a}/\delta_{b}=1$ and $\omega^{\prime}/g=100$

The concept of quasiprobability distribution, such as Wigner function, leads
to tremendously successful in modern quantum physics. In order to extract the
quantum nature of the realized single-quanta states, we calculate Wigner
functions $W(\alpha)$ and $W(\beta)$ in phase-space amplitudes $\alpha$ and
$\beta$ Royer (1977). The calculated results, shown in Figs. 3(c) and 3(d),
reveal the presence of negative values in Wigner function, unambiguously
demonstrating the nonclassical nature of cavity and motional phonon. Although
experimentally certifying nonclassicality remains a challenging task, the full
tomographic reconstruction of Wigner function and related phase-space
distributions can be extracted by performing correlation measurement
techniques of cavity Bohmann and Agudelo (2020); Biagi  _et al._ (2021) and
phonon number distribution in mechanical resonator Chu  _et al._ (2018).
Figures 3(e) and 3(f) depict the photon- and phonon-number distributions
$p(q)={\rm Tr}(|q\rangle\langle q|\rho)$, which provide a quantitative measure
of the quality of single-quanta emissions. Clearly, the probability of multi-
quanta excitations ($n\geqslant 2$) roughly zero reveals the single-quanta
nature for photons and phonons, which agrees well with second-order
correlation function $g_{aa}^{(2)}(0)$ and $g_{bb}^{(2)}(0)$ presented in
Figs. 3(a) and 3(b). Thus our system serves as the high-quality single-photon
and single-phonon source with a clear signature of nonclassicality.

Besides out-of-equilibrium dynamics, we further explore the steady-state
properties of the tripartite ‘squeeze’ interactions (Stokes process) with
$\omega^{\prime}/g=100$. We emphasize that both photon and phonon emissions
are independent of the initial state of the atomic field in a realistic
scenario. Figures 4(a) and 4(b) display the second-order correlation functions
in the $\delta\delta_{a}$ parameter plane. It is shown that the values of
$g^{(2)}_{aa}(0)$ for photons and $g^{(2)}_{bb}(0)$ for motional phonons vary
over a wide range. Remarkably, we achieve significant photon and phonon
blockades with $g^{(2)}_{aa}(0)\sim 10^{-3}$ and $g^{(2)}_{bb}(0)\sim
10^{-2}$, which represent the hallmark 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
$\delta^{(\pm)}_{a}=[\delta\pm\sqrt{2g^{2}+\delta^{2}}]/2$ derived from the
energy spectrum (red dashed lines) SM . We find that the energy spectrum
anharmonicity is enhanced by tuning $\delta$, 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 depicted in Figs.
4(c) and 4(d), corresponding to a minimum value of $\sqrt{2}g$ for both cavity
and motional phonon modes.

To further analyze the quantum coherence maintained within dissipations, we
plot interval $\tau$ dependence of $g^{(2)}_{aa}(\tau)$ and
$g^{(2)}_{bb}(\tau)$, 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 $\kappa_{a}^{-1}$, 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 $\tilde{p}(q)\equiv qp(q)/n^{(s)}$, which characterizes the
fraction of $q$-quanta states among the total excitations. Remarkably, we
observe strong photon- and phonon-blockade with the typical probability of
multiquanta excitations ($n>2$) being below $0.001\%$ and $0.07\%$,
respectively. We verify that a large decay rate corresponds to a strong
antibunching amplitude and a small occupation number of emissions SM . 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 Peng and Deng (2022). The mechanism of
decay-enhanced strong single-quanta blockade could facilitate the experimental
feasibility for engineering special nonclassical states beyond the limits of
strong atom-cavity coupling.

Conclusion.—We propose a novel architecture incorporates a single atom with
its long-lived motional degree of freedom within an optical cavity leveraging
state-of-the-art experimental techniques. By realizing two distinct types of
deterministic tripartite spin-photon-phonon interactions, we have realized an
interesting nonlinear resource for generating strong nonclassical quantum
emitters. Compared to the experimentally resolving vacuum fluctuations in
bipartite optomechanical system Lecocq  _et al._ (2015), our tripartite system
enables directly extract vacuum fluctuations inherently origin Heisenberg’s
uncertainty principle without free parameters. Moreover, the high-quality
single-quanta emissions are demonstrated that surpass the limits of strong
single atom-cavity coupling. This achievement is 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 for exploring the tripartite
entanglement Lance  _et al._ (2004); Jing  _et al._ (2003); Villar  _et al._
(2006); Dür  _et al._ (2000); Bennett  _et al._ (1999), decoherence mechanism
Arndt and Hornberger (2014), and distributed quantum information processing
Monroe  _et al._ (2014); Fitzsimons (2017); Buhrman  _et al._ (2010); Reiserer
(2022).

## ACKNOWLEDGMENTS

This work was supported by the NSFC (Grant No. 12274473 and Grant No.
12135018) and National Key R$\&$D Program of China (Grant No. 2018YFA0307500).

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Supplemental Material:
Unveiling Vacuum Fluctuations and Nonclassical States with Cavity-Enhanced
Tripartite Interactions

Jing Tang1 and Yuangang Deng1
1Guangdong Provincial Key Laboratory of Quantum Metrology and Sensing $\&$
School of Physics and Astronomy, Sun Yat-Sen University (Zhuhai Campus),
Zhuhai 519082, China

In this supplementary material, we provide additional details on (I) the
derivation of the tripartite Hamiltonian and calculation of the energy
spectrum, (II) numerical solution of the master equation, (III) unveiling
vacuum fluctuations from Schrödinger picture dynamics, and (IV) studying the
effect of motional phonon decay. These discussions contribute a comprehensive
understanding of our study on unveiling vacuum fluctuations and exploring
nonclassical states with cavity-enhanced tripartite interactions.

##  1 I. 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 depicted 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 $\omega_{c}$. For specificity, the two relevant energy levels for a
single 88Sr atom with one electronic ground state $|g\rangle$ (${}^{1}S_{0}$)
and one long-lived electronic orbital state $|e\rangle$ (${}^{3}P_{1}$) are
included. The atomic narrow (7.5-kHz-wide) dipole-forbidden transition
frequency between ${}^{1}S_{0}-^{3}P_{1}$ is $\omega_{e}$. This corresponds to
the wavelength of $\lambda=689$ nm and the wave vector $k_{L}=2\pi/\lambda$.
As can be seen, the atomic transitions is resonant illuminated by a classical
transverse pump field propagating perpendicular to cavity axis with frequency
$\omega_{p}$ and Rabi frequency $\Omega$. To engineer the dynamical spin-orbit
coupling, our proposal employsutilizes a single-mode optical cavity to couple
the atomic transitions between $|g\rangle\leftrightarrow|e\rangle$, 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.

Now, considering the level diagram depicted in Fig. 1 of the main text, we can
derive the Hamiltonian for the internal states under the rotating-wave
approximation (RWA). The Hamiltonian is given by

$\displaystyle\hat{\cal H}_{0}/\hbar$
$\displaystyle=\frac{{p}_{x}^{2}}{2M}+V(x)+{\frac{\Delta}{2}}\sigma_{z}+\Delta_{c}\hat{a}^{\dagger}\hat{a}+g_{0}\sin(k_{L}x)(\hat{a}^{\dagger}\hat{\sigma}_{-}+\hat{a}\hat{\sigma}_{+})+\Omega\hat{\sigma}_{x},$
(S1)

where $M$ is the mass of the atom, $\hat{a}$ is the annihilation operator of
the cavity field, $\sigma_{x,y,z}$ are Pauli matrices representing the
spin-$1/2$ system with $\sigma_{\pm}=(\sigma_{x}\pm i\sigma_{y})/2$,
$\Delta_{c}=\omega_{c}-\omega_{p}$ is the cavity-light detuning, and
$\Delta=\omega_{e}-\omega_{p}$ is the atom-light detuning. In our setup, we
consider the single 88Sr atom confined in a one-dimensional spin-independent
harmonic trap $V(x)=\frac{1}{2}M\omega_{b}^{2}x^{2}$, where $\omega_{b}$ is
the adjustable trap frequency. The trap frequency is respect to the zero-point
fluctuation amplitude of the trapped atom oscillator $x_{\rm
ZPF}=\sqrt{{\hbar}/{2M\omega_{b}}}$. In the Lamb-Dicke regime with
$k_{L}x_{\rm ZPF}\ll 1$ Vogel and Filho (1995), the Hamiltonian (S1) can be
simplified to

$\displaystyle\hat{\cal H}_{1}$
$\displaystyle=\frac{{p}_{x}^{2}}{2M}+V(x)+{\frac{\Delta}{2}}\sigma_{z}+\Delta_{c}\hat{a}^{\dagger}\hat{a}+g_{0}k_{L}x(\hat{a}^{\dagger}\hat{\sigma}_{-}+\hat{a}\hat{\sigma}_{+})+\Omega\hat{\sigma}_{x}.$
(S2)

To gain further insight, we can rewrite the single atom-cavity Hamiltonian
(S2) in terms of position-momentum representation

$\displaystyle
x=\sqrt{\frac{\hbar}{2M\omega_{b}}}(\hat{b}^{\dagger}+\hat{b}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
p_{x}=i\sqrt{\frac{{M\hbar\omega_{b}}}{2}}(\hat{b}^{\dagger}-\hat{b}),$ (S3)

where $\hat{b}$ is the annihilation operator of bosonic motional phonon for
harmonic oscillator frequency $\omega_{b}$. The operators of motional phonon
satisfies the commutation relations, $[\hat{b},\hat{b}^{\dagger}]=1$. With
these operators, we can derive a tripartite spin-photon-phonon Hamiltonian as
follows

$\displaystyle\hat{\cal{H}}_{2}/\hbar$
$\displaystyle=\omega_{b}\hat{b}^{\dagger}\hat{b}+\Delta_{c}\hat{a}^{\dagger}\hat{a}+{\frac{\Delta}{2}}\sigma_{z}+\Omega\hat{\sigma}_{x}+g(\hat{b}^{\dagger}+\hat{b})(\hat{a}^{\dagger}\hat{\sigma}_{-}+\hat{a}\hat{\sigma}_{+}),$
(S4)

where $g=g_{0}k_{L}x_{\rm ZPF}$ is the effective photon-phonon coupling.
Apparently, this cavity-enhanced tripartite Hamiltonian $\hat{\cal{H}}$
introduces a 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.

To capture the essence of physics, we introduce the unitary transformation in
the rotating frame, defined as

$\displaystyle{\cal U}$
$\displaystyle=\exp\\{-i[\Delta_{c}^{\prime}\hat{a}^{\dagger}\hat{a}+\omega^{\prime}\hat{b}^{\dagger}\hat{b}]t\\},$
(S5)

then tripartite spin-photon-phonon Hamiltonian (S4) reduces to

$\displaystyle{\cal\hat{H}}_{3}/\hbar$ $\displaystyle={\cal
U}^{\dagger}{\cal\hat{H}}_{2}{\cal U}-i{\cal
U}^{\dagger}\frac{\partial}{\partial{t}}{\cal U}$
$\displaystyle=\delta_{b}\hat{b}^{\dagger}\hat{b}+\delta_{a}\hat{a}^{\dagger}\hat{a}+{\frac{\delta_{a}-\delta}{2}}\sigma_{z}+\Omega\hat{\sigma}_{x}+g(\hat{b}^{\dagger}e^{i\omega^{\prime}t}+\hat{b}e^{-i\omega^{\prime}t})(\hat{a}^{\dagger}\hat{\sigma}_{-}e^{i\Delta_{c}^{\prime}t}+\hat{a}\hat{\sigma}_{+}e^{-i\Delta_{c}^{\prime}t}),$
(S6)

with $\delta_{b}=\omega-\omega^{\prime}$,
$\delta_{a}=\Delta_{c}-\Delta_{c}^{\prime}$, and $\delta=\delta_{a}-\Delta$
being the detuning of the cavity radiation frequency from the atomic
resonance.

Under the condition of red-sideband resonance
$\omega^{\prime}=\Delta_{c}^{\prime}\approx\omega_{b}$, the nonrotating wave
coupling terms with the high-frequency prefactor $e^{\pm
i(\omega^{\prime}+\Delta_{c}^{\prime})t}$ in Eq. (1) can be neglected using
the rotating wave approximation (RWA) with $g/\omega_{b}\ll 1$. This
simplification leads to a ‘beamsplitter’ interactions with respect to the
annihilation of a motional phonon up-convert into a quantized cavity photon

$\displaystyle{\cal\hat{H}}_{\rm{B}}/\hbar$
$\displaystyle=\delta_{b}\hat{b}^{\dagger}\hat{b}+\delta_{a}\hat{a}^{\dagger}\hat{a}+{\frac{\delta_{a}-\delta}{2}}\sigma_{z}+\Omega\hat{\sigma}_{x}+g(\hat{a}^{\dagger}\hat{b}\hat{\sigma}_{-}+\hat{b}^{\dagger}\hat{a}\hat{\sigma}_{+}),$
(S7)

This nonlinear anti-Stokes process represents the coherent exchange of cavity
and motional phonon. When the pump field is absent ($\Omega=0$), the
Hamiltonian (S7) possesses a ${\cal U}(1)$ symmetry characterized by the
action of the operator

$\displaystyle{\cal
R}_{\theta}=\exp[-i\theta(2\hat{a}^{\dagger}\hat{a}+\hat{b}^{\dagger}\hat{b}+\sigma_{z}/2)],$
(S8)

which transforms the operators as

$\displaystyle{\cal
R}_{\theta}^{\dagger}(\hat{a},\hat{b},\sigma_{-},\sigma_{+}){\cal
R}_{\theta}=(\hat{a}e^{-i2\theta},\hat{b}e^{-i\theta},\sigma_{-}e^{-i\theta},\sigma_{+}e^{i\theta}),$
(S9)

The breaking of the ${\cal U}(1)$ symmetry breaking is related to single-atom
superradiant phase transitions.

For blue-sideband resonance
$\omega^{\prime}=-\Delta_{c}^{\prime}\approx\omega_{b}$, the rotating wave
coupling terms in Eq. (1) with the high-frequency prefactor $e^{\pm
i(\omega^{\prime}-\Delta_{c}^{\prime})t}$ are safely neglected under RWA with
$g/\omega_{b}\ll 1$. This simplification leads to a ‘squeeze’
interactionsHamiltonian with respect to the pump photon is down-converted into
correlated photon-phonon pairs

$\displaystyle{\cal\hat{H}}_{\rm{S}}/\hbar$
$\displaystyle=\delta_{b}\hat{b}^{\dagger}\hat{b}+\delta_{a}\hat{a}^{\dagger}\hat{a}+{\frac{\delta_{a}-\delta}{2}}\sigma_{z}+\Omega\hat{\sigma}_{x}+g(\hat{a}^{\dagger}\hat{b}^{\dagger}\hat{\sigma}_{-}+\hat{b}\hat{a}\hat{\sigma}_{+}),$
(S10)

This nonlinear Stokes process denotes the amplification and entanglement of
the cavity and the motional phonon. In the absence of the pump field
($\Omega=0$), the Hamiltonian (S7) possesses another ${\cal U}(1)$ symmetry
characterized by the action of the operator

$\displaystyle{\cal
R}_{\theta}=\exp[-i\theta(\hat{a}^{\dagger}\hat{a}+\hat{b}^{\dagger}\hat{b}+\sigma_{z})]$
(S11)

which yields

$\displaystyle{\cal
R}_{\theta}^{\dagger}(\hat{a},\hat{b},\sigma_{-},\sigma_{+}){\cal
R}_{\theta}=(\hat{a}e^{-i\theta},\hat{b}e^{-i\theta},\sigma_{-}e^{-i2\theta},\sigma_{+}e^{i2\theta}).$
(S12)

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

Regarding the energy spectrum, the analytical results for the tripartite
‘squeeze’ interactions can be obtained by diagonalizing the Hamiltonian (S10)
for first ignoring the weak classical pump field. It is worth noting that the
difference in excitation number between the cavity mode and the motional
phonon is $zero$ due to the deterministic parametric down-conversion process.
Therefore the restricted Hilbert spaces for the single atom-cavity system are
$|n_{a}-1,n_{b}-1,e\rangle$ and $|n_{a},n_{b},g\rangle$, where $n_{a}$
($n_{b}$) denotes the photon (phonon) number excitation of the system.
Explicitly, the energy eigenvalues of system with $\delta_{a}=\delta_{b}$
fixed, are given by

$\displaystyle E_{n_{a},n_{b},\pm}$
$\displaystyle=(n_{a}+n_{b})\delta_{a}-\frac{\delta_{a}+\delta}{2}\pm\frac{1}{2}\sqrt{4g^{2}n_{a}n_{b}+(\delta_{a}+\delta)^{2}},$
(S13)

Assuming $n_{a}=n_{b}=n$ without loss of generality, the energy spectrum can
be expressed as

$\displaystyle E_{n,n,\pm}$
$\displaystyle=2n\delta_{a}-\frac{\delta_{a}+\delta}{2}\pm\frac{1}{2}\sqrt{4g^{2}n^{2}+(\delta_{a}+\delta)^{2}},$
(S14)

corresponding to the single-quanta resonance ($n=1$)

$\displaystyle\delta^{(\pm)}_{a}$
$\displaystyle=\frac{1}{2}\delta\pm\frac{1}{2}\sqrt{2g^{2}+\delta^{2}},$ (S15)

where $+$ ($-$) represents the higher (lower) branch. It is noteworthy that
the single-quanta resonance $\delta^{(\pm)}_{a}$ idepends on the value of
$\delta$, which is different from the realization of single-photon pairs for
bimodal cavities in Ref. Peng and Deng (2022) by utilizing the quantum
interference suppressed two-photon excitation. The detuning $\delta$-enhanced
vacuum Rabi splittings between the first pair of dressed states
$|1,1,\pm\rangle$ can significantly facilitate the realization of high-quality
single-quanta sources beyond the strong coupling regime in experiments.

##  2 II. Numerical solution of the master equation

To explore the quantum statistics of photon and phonon emissions, we
numerically solve the master equation in the dynamics and steady state using
the quantum optics toolbox Tan (1999). In the presence of the complete
dissipations of the system, the time evolution of the density matrix $\rho$
dictated by the full Hamiltonian (1) in the main text obeys the master
equation

$\frac{d{\rho}}{dt}=-i[{\cal{\hat{H}}},{\rho}]+\frac{\gamma}{2}\mathcal{\cal{D}}[{\hat{\sigma}}_{-}]\rho+\frac{\kappa_{a}}{2}\mathcal{\cal{D}}[\hat{a}]\rho+\frac{\kappa_{b}}{2}\mathcal{\cal{D}}[\hat{a}]\rho,$
(S16)

where $\gamma$ is atomic spontaneous emission rate of the long-lived excited
state, $\kappa_{a}$ and $\kappa_{b}$ is, respectively, the decay rate of the
photon and phonon, and
$\mathcal{D}[\hat{o}]\rho=2\hat{o}{\rho}\hat{o}^{\dagger}-\hat{o}^{\dagger}\hat{o}{\rho}-{\rho}\hat{o}^{\dagger}\hat{o}$
denotes the Lindblad type of dissipation. We should emphasize the standard
Lindblad equation (S16) is applicable for characterizing the strong coupling
tripartite spin-photon-phonon interaction Beaudoin  _et al._ (2011); Ridolfo
_et al._ (2012), since the Hamiltonian (1) is essentially originating from RWA
with the condition $g/\omega_{c}\ll 1$. We neglect the weak pure dephasing
effects in the long-lived atom and phonon fields.

To characterize the nonclassical correlations, we utilize the standard second-
order correlation function Glauber (1963)

$\displaystyle
g_{oo}^{(2)}(\tau)=\frac{\langle\hat{o}^{\dagger}(t)\hat{o}^{\dagger}(t+\tau)\hat{o}(t+\tau)\hat{o}(t)\rangle}{\langle\hat{o}^{\dagger}(t)\hat{o}(t)\rangle^{2}},$
(S17)

with $o=a,b$. For $\tau=0$, $g_{aa}^{(2)}(0)$ [$g_{bb}^{(2)}(0)$] represents
the cavity (phonon) mode and is a key physical quantity for assessing the
purity of single-quanta emissions Senellart  _et al._ (2017). The value of
$g_{oo}^{(2)}(0)$ should be less than 1 to ensure sub-Poissonian statistics.
Furthermore, for nonzero time interval, $g_{oo}^{(2)}(\tau)$ can be
straightforwardly calculated using the quantum regression theorem.
Experimental measurements of $g_{oo}^{(2)}(\tau)$ can be measured using
techniques such as Hanbury, Brown, and Twiss interferometer for cavity photons
Birnbaum  _et al._ (2005) and spin state-resolved projective measurements for
motional phonons Lv  _et al._ (2018); Wolf  _et al._ (2019); Patil  _et al._
(2022). Now, the quantum statistics for PB should satisfy two conditions:
$g_{oo}^{(2)}(0)<1$ to ensure the sub-Poissonian statistics and
$g_{oo}^{(2)}(0)<g_{oo}^{(2)}(\tau)$ to ensure antibunching with respect to
single photon or phonon.

##  3 III. Resolving Vacuum Fluctuations from Schrödinger picture dynamics

In order to gain further insight, we simplify the analytical model by
neglecting 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 $|n,n\rangle$, while the
single atom is in the excited state $|e\rangle|$.

###  3.1 The tripartite ‘squeeze’ interactions

For the resonance case with $\delta_{a}=\delta_{b}=\delta=0$, the tripartite
‘squeeze’ Hamiltonian simplifies to

$\displaystyle{\cal\hat{H}}_{\rm{S}}/\hbar$
$\displaystyle=g(\hat{a}^{\dagger}\hat{b}^{\dagger}\hat{\sigma}_{-}+\hat{b}\hat{a}\hat{\sigma}_{+}).$
(S18)

It is worth noting that the difference in excitation number between the photon
and phonon fields is zero due to the deterministic parametric down-conversion
process. As a result, the restricted Hilbert spaces for the single atom-cavity
system are $|n,n,e\rangle$ and $|n+1,n+1,g\rangle$. The energy eigenvalues of
Eq. (S18) are given by

$\displaystyle\lambda_{\pm}(n,n)/\hbar=\pm g(n+1),$ (S19)

and the corresponding eigenstates $|n,\pm\rangle$ associated with these energy
eigenvalues satisfy

$\displaystyle|n,n,+\rangle$
$\displaystyle=[|n,n,e\rangle+|n+1,n+1,g\rangle]/\sqrt{2}$
$\displaystyle|n,n,-\rangle$
$\displaystyle=[|n,n,e\rangle-|n+1,n+1,g\rangle]/\sqrt{2}.$ (S20)

It is now possible to obtain the dynamics of a general state by expanding it
on to the dressed eigenstates. We consider an initial state for the quantized
fields that satisfies $|\psi_{{\rm{field}}}(t=0)=|n,n\rangle$ and assume the
single atom is injected in the long-lived excited state. Thus the initial
state for the tripartite system reads

$\displaystyle|\psi(t=0)\rangle$
$\displaystyle=[|n,n,+\rangle+|n,n,-\rangle]/\sqrt{2},$ (S21)

Since $|n,\pm\rangle$ are stationary states of the tripartite spin-photon-
phonon system, the state vector with time development can be straightforwardly
calculated as follows

$\displaystyle|\psi(t)\rangle$
$\displaystyle=e^{-i{\cal\hat{H}}_{\rm{S}}/\hbar}|\psi(t=0)\rangle=[|n,n,+\rangle
e^{-i\lambda_{+}(n)t/\hbar}+|n,n,-\rangle
e^{-i\lambda_{-}(n)t/\hbar}]/\sqrt{2},$
$\displaystyle=\cos\theta_{S}|n,n,e\rangle-i\sin\theta_{S}|n+1,n+1,g\rangle,$
(S22)

where $\theta_{S}=g(n+1)t$ 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

$\displaystyle|\psi(t)\rangle=\cos(gt)|0,0,e\rangle-i\sin(gt)|1,1,g\rangle.$
(S23)

Clearly, the system undergoes Rabi oscillations between the vacuum state and a
state with one photon and one phonon.

###  3.2 The tripartite ‘beamsplitter’ interactions

For the resonance case with $\delta_{a}=\delta_{b}=\delta=0$, the tripartite
‘beamsplitter’ Hamiltonian reduce to

$\displaystyle{\cal\hat{H}}_{\rm{B}}/\hbar$
$\displaystyle={\frac{\Delta}{2}}\sigma_{z}+\Omega\hat{\sigma}_{x}+g(\hat{a}^{\dagger}\hat{b}\hat{\sigma}_{-}+\hat{b}^{\dagger}\hat{a}\hat{\sigma}_{+}),$
(S24)

assuming that the field modes are initially in Fock states $|n,n\rangle$.
Therefore the restricted Hilbert spaces for the single atom-cavity system are
$|n,n,e\rangle$ and $|n+1,n-1,g\rangle$. The energy eigenvalues of Eq. (S24)
are given by

$\displaystyle\lambda_{\pm}(n,n)/\hbar=\pm g\sqrt{n(n+1)},$ (S25)

and the eigenstates $|n,\pm\rangle$ associated with the energy eigenvalues
satisfy

$\displaystyle|n,n,+\rangle$
$\displaystyle=[|n,n,e\rangle+|n+1,n-1,g\rangle]/\sqrt{2}$
$\displaystyle|n,n,-\rangle$
$\displaystyle=[|n,n,e\rangle-|n+1,n-1,g\rangle]/\sqrt{2}.$ (S26)

It is now possible to obtain the dynamics of a general state by expanding it
on to the dressed eigenstates. We consider an initial state for the quantized
fields that satisfies $|\psi_{{\rm{field}}}(t=0)=|n,n\rangle$ and assume the
single atom is injected in the long-lived excited state. Thus the initial
state for the tripartite system reads

$\displaystyle|\psi(t=0)\rangle$
$\displaystyle=[|n,n,+\rangle+|n,n,-\rangle]/\sqrt{2},$ (S27)

Since $|n,\pm\rangle$ are stationary states of the tripartite spin-photon-
phonon system, the state vector with time development can be straightforwardly
calculated as follows

$\displaystyle|\psi(t)\rangle$
$\displaystyle=e^{-i{\cal\hat{H}}_{\rm{B}}/\hbar}|\psi(t=0)\rangle=[|n,n,+\rangle
e^{-i\lambda_{+}(n,n)t/\hbar}+|n,n,-\rangle
e^{-i\lambda_{-}(n,n)t/\hbar}]/\sqrt{2},$
$\displaystyle=\cos\theta_{B}|n,n,e\rangle-i\sin\theta_{B}|n+1,n-1,g\rangle,$
(S28)

associated with $\theta=g\sqrt{n(n+1)}t$ ($n>0$) being the accumulated phases
after tripartite ‘beamsplitter’ interactions. Interestingly, consider both
photon and motional phonon are initially in the vacuum states, the state of
the tripartite system as a function of time satisfies
$|\psi(t)\rangle=|0,0,e\rangle$, which corresponds to zero average photon and
phonon occupancies.

##  4 IV. 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
$g^{(2)}_{oo}(0)$ and the corresponding steady-state population $n^{(s)}_{o}$
with leaving the motional phonon decay rate $\kappa_{b}$ as a tunable
parameter. The results are shown in Figs. S1(a) and S1(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. As $\kappa_{b}$
increasing, the antibunching amplitude $g^{(2)}_{aa}(0)$ rapidly grow, albeit
the photon excitation $n^{(s)}_{a}$ remains roughly unchanged when
$\kappa_{b}/\kappa_{a}<1$. As for motional phonon, the value of
$g^{(2)}_{bb}(0)$ is insensitive to variations in $\kappa_{b}$, but the phonon
excitation $n^{(s)}_{b}$ rapidly decreases with increasing $\kappa_{b}$.
Compared to the long-lived motional phonon, the antibunching amplitude for
photon is significantly enhanced by over one orders of magnitude with up to
$g^{(2)}_{aa}(\tau)<10^{-3}$ when the decay rate ratio satisfies
$\kappa_{b}/\kappa_{a}\sim 0.01$. 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 a high-quality quantum
sources.

Figure S1: (color online). (a) and (b) shows the phonon decay $\kappa_{b}$
dependence of $g^{(2)}_{oo}(0)$ and $n^{(s)}_{o}$, respectively. (c)-(d) Time
interval $\tau$ dependence of $g^{(2)}_{aa}(\tau)$ and $g^{(2)}_{bb}(\tau)$
for different values of $\kappa_{b}$. Inset shows the steady-state
distribution $\tilde{p}(q)$ with $\kappa_{a}/\kappa_{b}=10$. The other
parameters are $\delta_{a}=\delta_{b}=g(1+\sqrt{3})/2$, $\delta/g=1$, and
$\omega^{\prime}/g=100$.

To further analyze the single-quanta blockade nature of photon and phonon
emissions, we plot the interval $\tau$ dependence of the second-order
correlation functions $g^{(2)}_{aa}(\tau)$ (c) and $g^{(2)}_{bb}(\tau)$ for
different values of $\kappa_{b}$, as shown in Figs. S1(c) and S1(d).
Interestingly, the decay times for both $g^{(2)}_{aa}(\tau)$ (c) and
$g^{(2)}_{bb}(\tau)$ are obviously longer than the typical time scale of
$\kappa_{a}^{-1}$. 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 $g^{(2)}_{oo}(\tau)>g^{(2)}_{oo}(0)$
decreases gradually as the decay ratio $\kappa_{a}/\kappa_{b}$ increases.

To characterize the single-quanta blockade nature of the photon and phonon
emissions, we examine the steady-state distribution $\tilde{p}(q)\equiv
qp(q)/n^{(s)}$ 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
$\kappa_{a}/\kappa_{b}=10$. These results demonstrate that the proposed
tripartite system can serve as a high-quality quantum source with strong
photon and phonon blockade.


# Max-Weight Scheduling and Quality-Aware Streaming for Device-to-Device Video
Delivery

Joongheon Kim, Andreas F. Molisch, Giuseppe Caire, J. Kim is with the
Department of Computer Science, University of Southern California, Los
Angeles, CA 90089, USA e-mail<EMAIL_ADDRESS>Caire and A.F. Molisch are
with the Department of Electrical Engineering, University of Southern
California, Los Angeles, CA 90089, USA e-mails: {caire<EMAIL_ADDRESS>
###### Abstract

We propose and analyze centralized and distributed algorithms for device-to-
device video scheduling and streaming. The proposed algorithms address jointly
the problems of device-to-device link scheduling and video quality adaptation
in streaming. Our simulations show that the proposed algorithms significantly
outperform conventional separated approaches that treat these two problems
independently.

##  I Introduction

Due to the increased video traffic, efficient video-aware transmission schemes
are of highest importance [1]. In particular, video throughput of wireless
networks can be enhanced by device-to-device (D2D) communications [5].

D2D video transmission needs to deal with two aspects: (i) link scheduling and
(ii) adaptive video quality in streaming. Traditionally these aspects have
been treated separately. In this paper we propose and analyze algorithms that
take the interrelationship between those tasks into account.

For centralized scheduling, the scheduling decisions can be made by a base
station, assuming it knows the channel states for all the possible D2D pairs.
We can formulate the scheduling problem as a max-weight independent set (MWIS)
problem which we solve (approximately) using a well-established message-
passing algorithm [3]. The weights for the MWIS problem arise from the
streaming (quality control) part.

Currently, the most well-known distributed D2D scheduler is FlashLinQ [2].
However, FlashLinQ does not incorporate naturally a video-quality-aware
mechanism, and therefore its suitability for D2D video streaming remains open.
We propose an improvement by incorporating weights.

Our streaming algorithm dynamically controls the quality mode of each chunk to
maximize the total quality subject to all data being supportable. It does so
via a stochastic formulation that can trade off queue stability with delay.

Obviously, higher video quality requires transmission of more bits, which
increases the probability of playback stalls. The tradeoff between video
quality and stall probability is a key measure for any transmission scheme,
and we use it to assess the performance of the algorithms in this paper. Our
simulations show that algorithms that exploit the interconnection between
scheduling and streaming provide a significantly better performance, i.e.,
better tradeoff between stall probability and video quality.

##  II The Proposed Algorithms

Each D2D TX has a queue whose length evolves as:

$Q_{i}(t+1)=\max\left[0,Q_{i}(t)-\mu_{i}(t)\right]+\lambda_{i}(t),l_{i}\in\mathcal{L}$
(1)

where $Q_{i}(t)$, $\mu_{i}(t)$, and $\lambda_{i}(t)$ stand for the queue
backlog size at the TX of the $i$-th D2D link $l_{i}$ at time $t$, the number
of bits leaving the queue of the TX of $l_{i}$, and the number of bits added
to the queue of the TX of $l_{i}$ (associated with the placement of chunks),
respectively.

Suppose that a chunk (the basic transmission unit, about 0.5 sec. of video)
contains $\mathcal{N}$ pixels and each chunk of each file $f$ is encoded at a
number of different quality modes $q\in M=\left\\{q_{1}\cdots q_{M}\right\\}$.
Let $\mathbb{P}_{f}(q,\tau)$ and $\mathcal{N}\mathbb{B}_{f}(q,\tau)$ denote
the video quality measure (e.g., peak-signal-to-noise-ratio (PSNR)) and the
number of bits for $f$ at chunk time $\tau$ with $q$, respectively. A network
controller chooses $q_{i}(\tau)$ for each $\tau$ for all requesting RXs $i$,
allocates the source coding rate $\mathbb{B}_{f_{i}}(q_{i}(\tau),\tau)$. The
TX of $l_{i}$ places the $\mathcal{N}\mathbb{B}_{f_{i}}(q_{i}(\tau),\tau)$
bits in its TX queue $Q_{i}(\tau)$.

###  II-A Max-Weight Scheduling

Centralized scheduling with MWIS formulation: The scheduling is based on a
link conflict graph such that the links scheduled to be active simultaneously
in any time slot must form an independent set. In the conflict graph where the
D2D links constitute the nodes, i.e., $l_{i}\in\mathcal{L},\forall
i\in\\{1,\cdots,|\mathcal{L}|\\}$ where $\mathcal{L}$ stands for the set of
D2D links. Thus, edge in the conflict graph between $l_{j}$ and $l_{k}$ is
denoted as $\mathcal{E}_{(j,k)}$, and $\mathcal{E}_{(j,k)}=1$ if $l_{j}$
interferes with $l_{k}$; otherwise, $\mathcal{E}_{(j,k)}=0$. Now, the
objective is to find the set of links that can maximize the sum of the weights
$w_{i}$ of each $l_{i}$. This problem is a MWIS, and can be mathematically
expressed as: maximizing $\sum_{\forall l_{i}\in\mathcal{L}}\nolimits
w_{i}\mathcal{I}_{i}$ subject to $\mathcal{I}_{j}+\mathcal{I}_{k}\leq 1$, if
$\mathcal{E}_{(j,k)}=1,\forall l_{j}\in\mathcal{L},\forall
l_{k}\in\mathcal{L}$ where $\mathcal{I}_{i}$ is a boolean index of $l_{i}$
which is $1$ if $l_{i}$ is scheduled (otherwise, $\mathcal{I}_{i}=0$). In our
case, the weights are updated at each $t$, such that $w_{i}=Q_{i}(t)r_{i}(t)$,
where $r_{i}(t)$ indicates the rate supported by link $\mathcal{l}_{i}$ at
$t$. In fact, it is well-known that such max-weight policy achieves strong
stability of the transmission queues, whenever the arrival rates are
stabilizable, i.e., they fall inside the ergodic achievable rate region of the
system [4]. The exact value of $r_{i}(t)$ of D2D link $l_{i}$ cannot be
obtained before a scheduling decision is made because the interference is
unknown. To circumvent this problem, $r_{i}(t)$ can be approximated as follows
[6]: $r_{i}(t)=\log_{2}\left(1+\frac{\mathcal{P}_{s_{i}\rightarrow
d_{i}}(t)\left\|h_{i\rightarrow i}\right\|^{2}}{\sigma^{2}+\gamma}\right)$
where $\mathcal{P}_{s_{i}\rightarrow d_{i}}(t)$ is the TX power from $s_{i}$
intended for $d_{i}$ at $t$, $h_{i\rightarrow i}$ is the (complex amplitude)
channel gain from $s_{i}$ to $d_{i}$, $\sigma$ is the standard deviation of
the (Gaussian) background noise, $\gamma$ is the interference thresholds,
i.e., the maximum admissible interference $\gamma$ from a single interferer
scheduled at the same time. For solving MWIS problem, various approximation
algorithms have been proposed because MWIS is NP-hard. Among them, we use
well-established message-passing that shows an excellent tradeoff between
performance and complexity [3].

Distributed max-weight scheduling: In FlashLinQ, lower-priority D2D links can
transmit only if they do not create significant interference to the higher-
priority D2D links [2]. With the concept of max-weight scheduling, we set the
priorities of D2D links as $U_{i}(t)\triangleq\left\\{r_{i}(t)\cdot
Q_{i}(t)\right\\}^{-1}$ which let each link wait $U_{i}(t)$ before
transmission.

TABLE I: Expected Num. of Stalls ($\alpha=2$, $\gamma=5$ dB) PBT | mpMWIS-QP: | FlashLinQ-Q: | FlashLinQ-QP:
---|---|---|---
7 sec. | 0.7 | 2.4 | 0.8
8 sec. |  0 [No Stalls] | 1.9 | 0.3
9 sec. |  0 [No Stalls] | 1.1 |  0 [No Stalls]
10 sec. |  0 [No Stalls] | 0.3 |  0 [No Stalls]

###  II-B Quality-Aware Streaming

Arrival process (placement of chunks): In each chunk time
$\tau\in\\{0,1,\cdots\\}$, the TX of every link should place chunks within its
queue. Different quality modes are available in each chunk, where a higher
mode requires more bits. We aim to maximize the total quality over all
scheduled links subject to stability of the scheduled TX queues. Let
$\mathbb{P}(t)=\sum_{l_{i}\in\mathcal{L}}\mathbb{P}_{f_{i}}\left(q_{i}(t),t\right)$.
Then the following formulation is our objective function where (3) means all
queues should fulfill rate stability [4]:

$\displaystyle\max$
$\displaystyle\lim_{\tau\rightarrow\infty}\frac{1}{\tau}\sum_{t=0}^{\tau-1}\nolimits\mathbb{E}\left[\mathbb{P}(t)\right]$
(2) subject to
$\displaystyle\lim_{\tau\rightarrow\infty}\frac{1}{\tau}\sum_{t=0}^{\tau-1}\nolimits\mathbb{E}\left[Q_{i}(t)\right]<\infty,\forall
l_{i}\in\mathcal{L}.$ (3)

Now, the quality control involves choosing $q_{i}(t)$, the quality mode at
chunk time $t$; as follows [4]:

$\arg\max_{q_{i}(t)\in
M}\left[\mathbb{P}_{f_{i}}\left(q_{i}(t),t\right)-\alpha\left\\{\mathcal{N}\mathbb{B}_{f_{i}}(q_{i}(t),t)\right\\}Q_{i}(t)\right].$
(4)

Thus, $\lambda_{i}(t)$ can be $\mathcal{N}\mathbb{B}_{f_{i}}(q_{i}(t),t)$ (if
$\tau\text{ mod }t=0$, otherwise, $\lambda_{i}(t)=0$) when the optimal $q$ is
determined $\forall l_{i}$.

Departure process (TX of bits): According to Shannon’s capacity equation,
$\mu_{i}(t)$ can be computed as:
$\mu_{i}(t)=\mathcal{B}\cdot\log_{2}\left[1+\frac{\mathcal{P}_{s_{i}\rightarrow
d_{i}}(t)\left\|h_{i\rightarrow i}\right\|^{2}}{\sigma^{2}+\sum_{j\neq
i}\mathcal{P}_{s_{j}\rightarrow d_{i}}(t)\left\|h_{j\rightarrow
i}\right\|^{2}}\right]$ where $\forall l_{i},\forall l_{j}\in\mathcal{L}^{*}$
where $\mathcal{L}^{*}$ is the set of simultaneously scheduled links, $i\neq
j$, $\mathcal{P}_{s_{a}\rightarrow d_{b}}(t)$ is the power transmitted by
$s_{a}$ intended for $d_{b}$, and $h_{j\rightarrow i}$ is the channel gain
from the TX of $l_{j}$ to the RX of $l_{i}$ at $t$, $\mathcal{B}$ is a
bandwidth. Depending on the distribution of used chunk sizes, the time scale
of departure process is set independent to the time scale of arrival process.

##  III Performance Evaluation

In our simulations, we used 2-hour 4 video traces with realistic D2D layout
and pathloss models [5]. Our proposed centralized and distributed algorithms
are denoted as mpMWIS-QP and FlashLinQ-QP. They are compared with the
FlashLinQ variant which controls the quality as in §II-B, but the scheduling
is randomized (named FlashLinQ-Q).

Pre-Buffering Time (PBT): Table I shows results when the PBT is from 7 to 10
(unit: sec). We can see that mpMWIS-QP and FlashLinQ-QP present no stalls when
PBTs=8 and PBT=9, i.e., mpMWIS-QP shows the best performance. But, FlashLinQ-Q
has stalls even if PBT=10.

Figure 1: Avg. Quality vs. Expected Num. of Stalls

Average Quality vs. Num. of Stalls: Changing $\alpha$ allows to trade off
quality with the number of stalls. For a given acceptable number of stalls,
Fig. 1 shows mpMWIS-QP provides the highest PSNR; and the performance of
FlashLinQ-QP is around $0.4$ dB lower than the one of mpMWIS-QP; FlashLinQ-Q
shows around $1.6$ dB lower PSNR.

##  IV Concluding Remarks

This paper proposed centralized quality-aware adaptive streaming and
scheduling algorithms which can be used for D2D video delivery applications.
For basestation-aided centralized scheduling, a message-passing algorithm is
used to obtain the solutions of the MWIS link scheduling problem.. For
distributed scheduling, we improved a FlashLinQ scheduler with the principle
of max-weight scheduling. For streaming, a quality-aware stochastic algorithm
is introduced.

## Acknowledgement

Full extended version of this abstract is under review to be published in
IEEE/ACM Transactions on Networking; and the title of the manuscript is
“Quality-Aware Streaming and Scheduling for Device-to-Device Video Delivery”.

## References

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  * [3] S. Sanghavi, D. Shah, and A.S. Willsky, “Message Passing for Maximum Weight Independent Set,” IEEE Transactions on Information Theory, 55(11):4822-4834, November 2009.
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