# Bilateral photon emission from a vibrating mirror and multiphoton
entanglement generation

Alberto Mercurio<EMAIL_ADDRESS>Dipartimento di Scienze Matematiche
e Informatiche, Scienze Fisiche e Scienze della Terra, Università di Messina,
I-98166 Messina, Italy  Laboratory of Theoretical Physics of Nanosystems
(LTPN), Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
CH-1015 Lausanne, Switzerland  Center for Quantum Science and Engineering,
EPFL, CH-1015 Lausanne, Switzerland  Enrico Russo<EMAIL_ADDRESS>Departamento de Física Teórica de la Materia Condensada and Condensed Matter
Physics Center (IFIMAC), Universidad Autónoma de Madrid, E-28049 Madrid, Spain
Theoretical Quantum Physics Laboratory, RIKEN, Wako-shi, Saitama 351-0198,
Japan  Fabio Mauceri  Dipartimento di Scienze Matematiche e Informatiche,
Scienze Fisiche e Scienze della Terra, Università di Messina, I-98166 Messina,
Italy  Salvatore Savasta  Dipartimento di Scienze Matematiche e Informatiche,
Scienze Fisiche e Scienze della Terra, Università di Messina, I-98166 Messina,
Italy  Theoretical Quantum Physics Laboratory, RIKEN, Wako-shi, Saitama
351-0198, Japan  Franco Nori  Theoretical Quantum Physics Laboratory, RIKEN,
Wako-shi, Saitama 351-0198, Japan  RIKEN Center for Quantum Computing (RQC),
Wako-shi, Saitama 351-0198, Japan  Physics Department, The University of
Michigan, Ann Arbor, Michigan 48109-1040, USA  Vincenzo Macrì
<EMAIL_ADDRESS>Theoretical Quantum Physics Laboratory, RIKEN, Wako-
shi, Saitama 351-0198, Japan  Dipartimento di Ingegneria, Università degli
Studi di Palermo, Viale delle Scienze, 90128 Palermo, Italy  Rosario Lo Franco
Dipartimento di Ingegneria, Università degli Studi di Palermo, Viale delle
Scienze, 90128 Palermo, Italy

###### Abstract

Entanglement plays a crucial role in the development of quantum-enabled
devices. One significant objective is the deterministic creation and
distribution of entangled states, achieved, for example, through a mechanical
oscillator interacting with confined electromagnetic fields. In this study, we
explore a cavity resonator containing a two-sided perfect mirror. Although the
mirror separates the cavity modes into two independent confined
electromagnetic fields, the radiation pressure interaction gives rise to high-
order effective interactions across all subsystems. Depending on the chosen
resonant conditions, which are also related to the position of the mirror, we
study $2n$-photon entanglement generation and bilateral photon pair emission.
Demonstrating the non-classical nature of the mechanical oscillator, we
provide a pathway to control these phenomena, opening potential applications
in quantum technologies. Looking ahead, similar integrated devices could be
used to entangle subsystems across vastly different energy scales, such as
microwave and optical photons.

##  I Introduction

The ability to control quantum mechanical systems using radiation pressure has
given rise to the field of optomechanics Marquardt and Girvin (2009);
Nunnenkamp  _et al._ (2011); Aspelmeyer  _et al._ (2014); Barzanjeh  _et al._
(2022), an interesting platform for exploring the quantum properties of
mesoscopic objects Aspelmeyer  _et al._ (2010); Romero-Isart  _et al._ (2011);
Meystre (2013). Cavity optomechanical systems, which involve the interaction
between mechanical vibrations and electromagnetic fields, hold the potential
for observing quantized vibrational modes in macroscopic objects, even
reaching their ground state Schliesser  _et al._ (2009); Gröblacher  _et al._
(2009); Wilson-Rae  _et al._ (2007); Marquardt  _et al._ (2007); Elste  _et
al._ (2009); Teufel  _et al._ (2011); Chan  _et al._ (2011); Ojanen and Børkje
(2014). This opens the door to creating entangled and superposition
macroscopic states, paving the way for novel approaches to processing and
storing quantum information Marshall  _et al._ (2003); Pepper  _et al._
(2012); Stannigel  _et al._ (2012); Garziano  _et al._ (2015); Macrí  _et al._
(2016).

In general, when electromagnetic quantum fluctuations interact with a very
fast-oscillating boundary condition, pairwise real excitations can be created
from the vacuum of the electromagnetic field Moore (1970); Lambrecht  _et al._
(1996); Dodonov (2020). Such a purely quantum phenomenon is known as the
dynamical Casimir effect (DCE) Johansson  _et al._ (2009, 2010), which has
been experimentally realized in superconducting circuits Wilson  _et al._
(2011) and Josephson metamaterials Lähteenmäki  _et al._ (2013).

Cavity optomechanics involves the modulation of boundary conditions through a
mobile mirror, enabling the observation of the DCE. In this scenario, the
fundamental process involves the conversion of mechanical energy into photons
Macrì  _et al._ (2018). A detailed derivation of the optomechanical
Hamiltonian can be found in Ref. Law (1995). Subsequent advancements extended
this model to incorporate incoherent excitation of the mirror Settineri  _et
al._ (2018, 2019), and other works examined back-reaction and dissipation
effects within this framework Butera and Carusotto (2019); Ferreri  _et al._
(2022). Notably, investigations have expanded to consider a cavity with two
mobile mirrors Di Stefano  _et al._ (2019); Butera (2022); Fong  _et al._
(2019). In this case, the cavity field facilitates an effective interaction
between the two mirrors, resulting in phonon hopping. This broader exploration
adds depth to our understanding of the complex dynamics within optomechanical
systems.

The present work investigates a cavity resonator equipped with a two-sided
perfect mirror embedded within. This configuration corresponds to a tripartite
system, where two separated electromagnetic fields interact with the vibrating
mirror by radiation pressure (see Fig. 1). Although the mirror divides the
cavity modes into two separated confined electromagnetic fields, the
interaction of radiation pressure leads to high-order effective interactions
throughout all subsystems. Recently, this configuration has been studied,
shedding light on the dressed ground state and the correlation between the two
cavity modes Montalbano  _et al._ (2022), and on the optomechanically induced
two-photon hopping effect Russo  _et al._ (2023). Furthermore, path-entangled
microwave radiation was observed from strongly driven microwave resonators
Barzanjeh  _et al._ (2019). Here, instead, we propose a scheme to generate
$2n$-photon entanglement (e.g., two-, four-photon) and bilateral photon pair
emission (that we name Janus effect), which can be already achieved with few
photonic excitations. Each process is activated by a specific resonance
condition, which depends on the resonance frequency of the three subsystems,
and thus on the position of the mirror. Throughout our analysis, we carry out
analytical aspects and numerical simulations to delve into the resonant
dynamics and the interplay between the system’s parameters, including coupling
strengths, bare frequencies, and initial conditions.

Figure 1: Pictorial representation of our setup. Two electromagnetic cavities
separated by a movable two-sided perfect mirror.

The effects predicted in this work could be experimentally observed using
circuit-optomechanical systems, namely, employing mechanical micro- or nano-
resonators operating in the ultra-high-frequency range within the GHz spectral
domain O’Connell  _et al._ (2010); Primo  _et al._ (2023). Indeed, the
addition of artificial atoms in a superconducting microwave setup strengthens
the coupling with the mechanical resonator Heikkilä  _et al._ (2014);
Pirkkalainen  _et al._ (2015); Rouxinol  _et al._ (2016); Aporvari and Vitali
(2021); Manninen  _et al._ (2022), making it a very promising setup. A
valuable alternative approach would entail employing a quantum simulator
Johansson  _et al._ (2014); Kim  _et al._ (2015), wherein two LC circuits
emulate the cavities, and a superconducting quantum interference device
(SQUID) takes on the role of the high-frequency vibrating mirror.

The article is structured as follows: in Section II we introduce our quantum
model, analyzing in detail three specific resonance conditions: (i) two-photon
entanglement generation in Section II.1; (ii) four-photon entanglement
generation in Section II.2; (iii) bilateral photon emission, which we call
Janus effect in Section II.3. Finally, in Section III we offer concluding
remarks and outline potential trails for future research in this field. Some
details are left on the Appendices. More precisely, in Appendix A we employ
the Schrieffer-Wolff method to derive the effective Hamiltonians, and we also
show all the coefficients related to the Janus effective Hamiltonian.

##  II Quantum Model

Consider two non-interacting single-mode cavities separated by a vibrating
two-sided perfect mirror, as sketched in Fig. 1. The three bosons are
described by ladder operators
$\hat{a}(\hat{a}^{\dagger}),\hat{c}(\hat{c}^{\dagger})$ for the cavities and
$\hat{b}(\hat{b}^{\dagger})$ for the mirror, satisfying canonical commutation
relations. The Hamiltonian can be derived by quantizing the classical
Lagrangian description (see Appendix in Ref. Russo  _et al._ (2023)), and it
reads ($\hbar=1$)

$\begin{split}\hat{H}=&\
\omega_{a}\hat{a}^{\dagger}\hat{a}+\omega_{b}\hat{b}^{\dagger}\hat{b}+\omega_{c}\hat{c}^{\dagger}\hat{c}\\\
&-\frac{g}{2}\left[(\hat{a}+\hat{a}^{\dagger})^{2}-\Omega^{2}(\hat{c}+\hat{c}^{\dagger})^{2}\right](\hat{b}+\hat{b}^{\dagger})\,.\end{split}$
(1)

where $\omega_{a},\omega_{c}$ are the bare energies of the cavities,
$\omega_{b}$ is the bare energy of the mirror, $g$ is the coupling strength,
and the ratio $\Omega=\omega_{c}/\omega_{a}$ is related to the mirror
position. In the limit of large detuning $\omega_{b}\ll\omega_{a,c}$, the
rotating wave approximation can be applied, and the standard optomechanical
interaction term, proportional to the number operators
$\hat{a}^{\dagger}\hat{a}(\hat{c}^{\dagger}\hat{c})$, is obtained Aspelmeyer
_et al._ (2014). With this Hamiltonian, we will describe three peculiar
configurations. By employing the Schrieffer-Wolff approach (see Refs.
Schrieffer and Wolff (1966); Bravyi  _et al._ (2011); Girvin and Yang (2019)
and Appendix A), we obtain effective Hamiltonians which directly show the
high-order non-linear processes related to specific resonance conditions.

Figure 2: Two-photon entanglement generation. The time evolution of the mean
values of the dressed number operators for different trajectories and the
average over $1000$ trajectories. (a) A trajectory where the first jump occurs
with a phonon loss, locking the system into the state $\ket{0,1,0}$, until a
second quantum jump occurs. (b) A trajectory where the first jump occurs in
cavity $c$, locking the system into the state $\ket{0,0,1}$, as a clear
signature of an entangled state. (c) The average behavior shows the coherent
and dissipative energy exchange between the bare state $\ket{0,2,0}$ and the
entangled state $\ket{\psi_{+}}$. The used parameters are:
$\omega_{a}=\omega_{c}=\omega_{b}+g^{2}/\omega_{b}$, $g=0.05\omega_{b}$, and
$\gamma_{a}=\gamma_{b}=\gamma_{c}=5\times 10^{-4}\omega_{b}$.

Although we analytically characterize all these processes by using effective
Hamiltonians, all the simulations are carried out by employing the quantum
trajectory approach Mølmer  _et al._ (1993); Carmichael (2009), and using the
full Hamiltonian in Eq. 1. To explore the phenomenology, we numerically
calculate the time evolution of the mean values of the _dressed_ number
operators Ridolfo  _et al._ (2012); Le Boité (2020), i.e.,
$\langle\hat{\mathcal{X}}_{o}^{-}\hat{\mathcal{X}}_{o}^{+}\rangle$
($o\in\\{a,b,c\\}$). Each dressed operator is defined as

$\hat{\mathcal{X}}_{o}^{+}\equiv\sum_{j>k}\langle
k|(\hat{o}+\hat{o}^{\dagger})|j\rangle\outerproduct{k}{j}\,,\quad\hat{\mathcal{X}}_{o}^{-}=(\hat{\mathcal{X}}_{o}^{+})^{\dagger}\,,$
(2)

where $|j\rangle$ is the $j$-th eigenstate of the full Hamiltonian in Eq. 1.
This properly defines the jump operators, which by construction act like an
annihilation (creation) operator in the energy basis. In this dressed picture,
the quantum jumps are between the dressed states (the eigenstates of the full
Hamiltonian) which contain contributions from bare states with an arbitrary
number of excitations. With this notation, we refer to the means value of the
number operator of the single quantum trajectory, while average quantities
obtained over 1000 quantum trajectories are indicated as
$\overline{\langle\hat{\mathcal{X}}_{o}^{-}\hat{\mathcal{X}}_{o}^{+}}\rangle$.
The dissipation rates of the three subsystems are indicated as $\gamma_{a}$,
$\gamma_{b}$, and $\gamma_{c}$.

###  II.1 Two-photon entanglement generation

Let us consider the condition $\omega_{a}\simeq\omega_{b}\simeq\omega_{c}$.
The effective Hamiltonian up to the second order becomes (see Appendix A)

$\displaystyle\hat{H}_{\rm eff}=$ $\displaystyle\hat{H}_{0}+\hat{H}_{\rm
shift}+\hat{H}_{\rm hop}+\hat{H}_{\rm ent}+\mathcal{O}(g^{3})$ (3)
$\displaystyle\hat{H}_{\rm shift}=$
$\displaystyle-\frac{g^{2}}{3\omega_{b}}-\frac{4g^{2}}{3\omega_{b}}\left({\hat{a}^{\dagger}}\hat{a}+{\hat{c}^{\dagger}}\hat{c}\right)$
$\displaystyle-\frac{4g^{2}}{3\omega_{b}}\left({\hat{a}^{\dagger}}\hat{a}+1+{\hat{c}^{\dagger}}\hat{c}\right)\hat{b}^{\dagger}\hat{b}$
$\displaystyle-\frac{5g^{2}}{6\omega_{b}}\left({\hat{a}^{\dagger}}{}^{2}\hat{a}^{2}+{\hat{c}^{\dagger}}{}^{2}\hat{c}^{2}\right)+\frac{2g^{2}}{\omega_{b}}{\hat{a}^{\dagger}}\hat{a}{\hat{c}^{\dagger}}\hat{c}$
$\displaystyle\hat{H}_{\rm hop}=$
$\displaystyle-\frac{g^{2}}{6\omega_{b}}\left({\hat{a}^{\dagger}}{}^{2}\hat{c}^{2}+{\hat{c}^{\dagger}}{}^{2}\hat{a}^{2}\right)$
$\displaystyle\hat{H}_{\rm ent}=$
$\displaystyle-\frac{g^{2}}{\omega_{b}}\left(({\hat{a}^{\dagger}}{}^{2}+{\hat{c}^{\dagger}}{}^{2})\hat{b}^{2}+(\hat{a}^{2}+\hat{c}^{2}){\hat{b}^{\dagger}}{}^{2}\right)\,,$

where
$\hat{H}_{0}=\omega_{a}\hat{a}^{\dagger}\hat{a}+\omega_{b}\hat{b}^{\dagger}\hat{b}+\omega_{c}\hat{c}^{\dagger}\hat{c}$
is the same in any derivation of effective Hamiltonian.

The Hamiltonian term $\hat{H}_{\rm shift}$ contains only numbers operators
(and their powers) describing bare energy shift due to the perturbation. The
two effective interaction terms clarify an otherwise complex dynamic implicit
in Eq. 1. Indeed, $\hat{H}_{\rm hop}$ shows the two-photon hopping sub-process
Russo  _et al._ (2023) while $\hat{H}_{\rm ent}$ links the three sub-parts
together ultimately bringing to tripartite entanglement.

The effective Hamiltonian in Eq. 3 admits the lowest energy closed dynamics in
the sub-Hilbert space spanned by the states
$\\{\ket{2,0,0},\ket{0,2,0},\ket{0,0,2}\\}$. Here, the first and third entries
represent the $a$ and $c$ cavity excitations, respectively, while the second
represents the mirror ($b$) excitation. Under the resonance condition
$\omega_{a}=\omega_{c}=\omega$, we can define the ordered basis $\\{\lvert
0,2,0\rangle,\lvert\psi_{+}^{\rm(2e)}\rangle,\lvert\psi_{-}^{\rm(2e)}\rangle\\}$,
with $\lvert\psi_{\pm}^{\rm(2e)}\rangle=(\lvert 2,0,0\rangle\pm\lvert
0,0,2\rangle)/\sqrt{2}$ being the symmetric and anti-symmetric two-photon
maximally entangled states between the two cavities. Notice that these
entangled states have the structure of NOON states, which are typically
exploited in quantum-enhanced metrology Giovannetti  _et al._ (2011).
Therefore, the Hamiltonian takes the block form

$H=\begin{pmatrix}2\omega_{b}-\frac{3g^{2}}{\omega_{b}}&-\frac{2\sqrt{2}g^{2}}{\omega_{b}}&0\\\
-\frac{2\sqrt{2}g^{2}}{\omega_{b}}&2\omega-\frac{5g^{2}}{\omega_{b}}&0\\\
0&0&2\omega-\frac{13g^{2}}{3\omega_{b}}\end{pmatrix}\,,$ (4)

highlighting the fact that the dynamics occurs between the state $\lvert
0,2,0\rangle$ and $\lvert\psi_{+}^{\rm(2e)}\rangle$. In other words, two
phonons are exchanged with a symmetric entangled state of two totally
delocalized photons. The non-uniformity of the matrix elements in Eq. 4 stems
from the different coefficients appearing respectively in $\hat{H}_{\rm hop}$
and $\hat{H}_{\rm ent}$ as well as the different coefficients appearing in
$\hat{H}_{\rm shift}$. Indeed, in general, the eigenstates of the effective
Hamiltonian are written as a generic superposition of $\lvert 0,2,0\rangle$
and $\lvert\psi_{+}^{\rm(2e)}\rangle$. However, we can make the superposition
symmetric by choosing the condition $\omega=\omega_{b}+g^{2}/\omega_{b}$,
which makes the upper-left block a symmetric matrix with equal diagonal terms.
The same condition can be found by minimizing the difference of the two
eigenvalues in Eq. 4 as a function of $\omega$. Under this condition the
eigenstates of the upper-left block become

$\lvert\phi_{\pm}^{\mathrm{(2e)}}\rangle=(\ket{0,2,0}\pm\lvert\psi_{+}^{\rm(2e)}\rangle)/\sqrt{2}\,.$
(5)

In Fig. 2(a-b) we show two different trajectories, while Fig. 2(c) shows the
behavior master-equation-like that arises taking the average over $1000$
trajectories. Both the snapshots highlight a fascinating trapping effect that
occurs whenever one photon is detected in one of the cavities, or when a
phonon is detected in the mirror. For instance, in Fig. 2(a) we clearly see
that the first jump occurs with a phonon loss, locking the system to the state
$\ket{0,1,0}$, while the coherent dynamics is lost. After a certain time, a
second jump occurs leaving the system in its ground state. On the contrary, in
Fig. 2(b) the first jump occurs in cavity $c$, and the system is locked in the
state $\ket{0,0,1}$. Note that, when the cavity $c$ jumps, the number of
photons in cavity $a$ immediately goes to zero, as a clear signature of the
quantum correlations exhibited in the entangled state
$\lvert\psi_{+}^{\rm(2e)}\rangle$. Again, when the second jump occurs, the
system reaches its ground state.

This trapping effect occurs because the effective Hamiltonian does not contain
terms that allow the exchange of a single photon-phonon excitation, while the
act of measuring (losses) is modeled as a single boson detection. The
remaining excitation is localized in the subsystem where the first measurement
occurred until a second measurement occurs. In Fig. 2(c) (obtained by
averaging over 1000 trajectories) one clearly sees the coherent and
dissipative energy exchanging over time, between the bare state $\ket{0,2,0}$
and $\lvert\psi_{+}^{\rm(2e)}\rangle$, meaning that two-phonon generates two-
photon entanglement. Note that, the above-mentioned trapping effects are
washed out by the averaging of a master equation Macrì  _et al._ (2022);
Minganti  _et al._ (2021). The parameters used to reproduce Fig. 2 are:
$\omega_{a}=\omega_{c}=\omega_{b}+g^{2}/\omega_{b}$, $g=0.05\omega_{b}$, and
$\gamma_{a}=\gamma_{b}=\gamma_{c}=5\times 10^{-4}\omega_{b}$.

Figure 3: Four-photon entanglement generation. The time evolution of the mean
values of the dressed number operators for different trajectories and the
average over $1000$ trajectories. (a) A trajectory where the first jump occurs
with a photon loss from cavity $a$, projecting the dynamics into an incomplete
coherent process involving the two-photon hopping between $\ket{3,0,0}$ and
$\ket{1,0,2}$. A second jump puts the system into the state $\ket{2,0,0}$,
allowing this time a complete two-photon hopping process between the two
cavities. When a third jump occurs with a photon loss from cavity $a$, the
system jumps to the locked state $\ket{1,0,0}$, and the coherent dynamics is
lost. After a certain time, a fourth jump leaves the system in its ground
state. (b) The occurrence of a jump with a phonon loss brings the system
directly to its ground state, because of the lack of energy. (c) The average
behavior shows the coherent and dissipative energy exchange between the bare
state $\ket{0,1,0}$ and the entangled state $\lvert\psi_{-}^{\rm(4e)}\rangle$.
The used parameters are: $\omega_{a}=\omega_{c}\approx 0.2566\omega_{b}$,
$g=0.03\omega_{b}$, and $\gamma_{a}=\gamma_{b}=\gamma_{c}=2\times
10^{-5}\omega_{b}$.

###  II.2 Four-photon entanglement generation

Under the resonant conditions, $\omega_{b}\simeq 4\omega$
($\omega=\omega_{a}=\omega_{c}$) the effective Hamiltonian up to the third
order in the coupling constant becomes (see Appendix A)

$\displaystyle\hat{H}_{\rm eff}=$ $\displaystyle\hat{H}_{0}+\hat{H}_{\rm
shift}+\hat{H}_{\rm hop}+\hat{H}_{\rm ent}\,,$ $\displaystyle\hat{H}_{\rm
shift}=$
$\displaystyle-\frac{2g^{2}}{3\omega_{b}}-\frac{5g^{2}}{3\omega_{b}}\left(\hat{a}^{\dagger}\hat{a}+\hat{c}^{\dagger}\hat{c}+\hat{a}^{\dagger}{}^{2}\hat{a}^{2}+\hat{c}^{\dagger}{}^{2}\hat{c}^{2}\right)$
$\displaystyle+\frac{4g^{2}}{3\omega_{b}}\left(\hat{a}^{\dagger}\hat{a}+\hat{c}^{\dagger}\hat{c}+1\right)\hat{b}^{\dagger}\hat{b}+\frac{2g^{2}}{\omega_{b}}\hat{a}^{\dagger}\hat{a}\hat{c}^{\dagger}\hat{c}$
$\displaystyle\hat{H}_{\rm hop}=$
$\displaystyle\frac{2g^{2}}{3\omega_{b}}\left(\hat{a}^{\dagger}{}^{2}\hat{c}^{2}+\hat{a}^{2}\hat{c}^{\dagger}{}^{2}\right)$
$\displaystyle\hat{H}_{\rm ent}=$
$\displaystyle\frac{2g^{3}}{3\omega_{b}^{2}}\left[(\hat{a}^{4}-\hat{c}^{4})\hat{b}^{\dagger}+(\hat{a}^{\dagger}{}^{4}-\hat{c}^{\dagger}{}^{4})\hat{b}\right]\,.$
(6)

As we did in the previous case we can look for the simplest closed dynamics
and project the Hamiltonian in corresponding basis. This subspace is spanned
by the states $\\{\ket{0,1,0},\ket{2,0,2},\ket{4,0,0},\ket{0,0,4}\\}$, but due
to the form of interaction Hamiltonian part and resonant conditions one can
define the ordered basis $\\{\lvert
0,1,0\rangle,\lvert\psi_{-}^{\rm(4e)}\rangle,\lvert\psi_{+}^{\rm(4e)}\rangle,\lvert
2,0,2\rangle\\}$, where $\lvert\psi_{\pm}^{\rm(4e)}\rangle=(\lvert
4,0,0\rangle\pm\lvert 0,0,4\rangle)/\sqrt{2}$ being the symmetric and anti-
symmetric four-photon maximally entangled states between the two cavities.
Again, we highlight that these states have the structure of NOON states. Using
this basis, the Hamiltonian takes the block form

$\displaystyle
H=\begin{pmatrix}\omega_{b}+\frac{4g^{2}}{3\omega_{b}}&\frac{8g^{3}}{\sqrt{3}\omega^{2}_{b}}&0&0\\\
\frac{8g^{3}}{\sqrt{3}\omega^{2}_{b}}&4\omega-\frac{80g^{2}}{3\omega_{b}}&0&0\\\
0&0&4\omega-\frac{80g^{2}}{3\omega_{b}}&\frac{8g^{2}}{\sqrt{3}\omega_{b}}\\\
0&0&\frac{8g^{2}}{\sqrt{3}\omega_{b}}&4\omega-\frac{16g^{2}}{3\omega_{b}}\end{pmatrix}\,,$
(7)

revealing the two concurrent dynamics: the oscillation between $\lvert
0,1,0\rangle$ and $\lvert\psi_{-}^{\rm(4e)}\rangle$ and the one between
$\lvert 2,0,2\rangle$ and $\lvert\psi_{+}^{\rm(4e)}\rangle$. The latter,
namely, the lower-right part of the matrix describing the dynamics between the
states $\ket{2,0,2}$ and $\lvert\psi_{+}^{\rm(4e)}\rangle$, can be explained
in terms of the two-photon hopping terms Russo  _et al._ (2023). It originates
only from the third line in (II.2), and thus it does not introduce any new
effect. Since the mirror plays a role only in the first sub-dynamics, only the
upper-left block becomes important for the four-photon entanglement generation
process we are describing. To look for symmetric eigenstates, we can proceed
as in Section II.1, by choosing $\omega$ such that the diagonal terms are
equal. Note that, in this case, the same cannot be done for the lower-right
part because the non-linear $H_{\rm{shift}}$ acts differently on these states,
making the two dynamics mutually exclusive.

In particular, for $\omega=\frac{\omega_{b}}{4}+7\frac{g^{2}}{\omega_{b}}$,
the eigenstates take the form

$\lvert\phi_{\pm}^{\rm(4e)}\rangle=\frac{1}{\sqrt{2}}\left(\lvert
0,1,0\rangle\pm\lvert\psi_{-}^{\rm(4e)}\rangle\right)\,,$ (8)

in analogy to Section II.1. The above state again encompasses entangled
cavities but now with a higher number state.

Compared to the two-photon entanglement, we now get a third-order process,
which results in a greater sensitivity to finding the resonance point of
maximum interaction. This gives a small difference between the resonance
condition obtained analytically with the effective Hamiltonian and the one
required from the full Hamiltonian of Eq. 1. Therefore, to find the maximum
interaction point, we obtain $\omega$ through a numerical optimization
process, showing a very small difference from the analytical value, but large
enough to make this third-order process incomplete if using the analytical
point.

As can be seen from Fig. 3(a) the trapping effect is more cumbersome. Once a
measurement occurs in cavity $a$, it rips away one photon from a previously
four-photon entangled state. This measurement projects the dynamics into an
incomplete two-photon hopping process, between the states $\lvert
3,0,0\rangle$ and $\lvert 1,0,2\rangle$. Indeed, the specific resonance
condition we choose here does not match with that involving this subprocess,
and for this reason, we have an incomplete coherent dynamic. A second
measurement restores the inherent parity, washing away these spurious
beatings, and letting the two cavities interact under the influence of a
complete two-photon hopping interaction. When a third jump occurs with a
photon loss, from the cavity $a$ in our case, the system jumps to the locked
state $\ket{1,0,0}$, while the coherent dynamics is lost. After a certain
time, a fourth jump leaves the system in its ground state. In Fig. 3(b) we
show how the occurrence of a jump with a phonon loss brings the systems
directly to its ground state, because of the lack of energy. The average
behavior obtained by averaging over 1000 trajectories shows the coherent and
dissipative energy exchange between the bare state $\ket{0,1,0}$ and the
entangled state $\lvert\psi_{-}^{\rm(4e)}\rangle$. This is reported in Fig.
3(c). Note that, the averaging process hides the complex dynamics described
above. The parameters used to reproduce Fig. 3 are:
$\omega_{a}=\omega_{c}\approx 0.2566\omega_{b}$, $g=0.03\omega_{b}$, and
$\gamma_{a}=\gamma_{b}=\gamma_{c}=2\times 10^{-5}\omega_{b}$.

Figure 4: The Janus effect. Time evolution of the mean values of the dressed
number operators for both cavities and the mirror. In the $(a)$ panel a first
jump causes the initial state $\ket{2,0,2}$ to collapse to the state
$\ket{1,0,2}$ with consequent trapping. Indeed, from now on, there are no sub-
processes involved, and the dynamics becomes trivial. In panel $(b)$ the
initial dynamics is interrupted by the measurement of the phonon, leaving the
system in its ground state. Panel $(c)$ shows the average over 1000
trajectories. The used parameters are: $\omega_{a}=\omega_{b}/4+\varepsilon$,
$\varepsilon=\omega_{b}/15$, $\Omega\approx 0.6067$, $g=0.05\omega_{b}$,
$\gamma_{a}=\gamma_{b}=\gamma_{c}=2\times 10^{-5}\omega_{b}$.

###  II.3 Janus effect

We now move to an asymmetric case in which the resonance condition is
expressed as $\omega_{b}\simeq 2(\omega_{a}+\omega_{c})$ but
$\omega_{a}\neq\omega_{c}$ (i.e., the mirror not in the middle). Under this
condition the effective Hamiltonian in Appendix A up to the third order
becomes

$\displaystyle\hat{H}_{\rm eff}=$ $\displaystyle\hat{H}_{0}+\hat{H}_{\rm
shift}+\hat{H}_{\rm Jan}\,,$ (9) $\displaystyle\hat{H}_{\rm shift}=$
$\displaystyle\Omega_{a}\hat{a}^{\dagger}\hat{a}+\Omega_{b}\hat{b}^{\dagger}\hat{b}+\Omega_{c}\hat{c}^{\dagger}\hat{c}+\alpha_{a}\hat{a}^{\dagger}{}^{2}\hat{a}^{2}+\alpha_{c}\hat{c}^{\dagger}{}^{2}\hat{c}^{2}$
$\displaystyle+\alpha_{a,b}\hat{a}^{\dagger}\hat{a}\hat{b}^{\dagger}\hat{b}+\alpha_{a,c}\hat{a}^{\dagger}\hat{a}\hat{c}^{\dagger}\hat{c}+\alpha_{b,c}\hat{b}^{\dagger}\hat{b}\hat{c}^{\dagger}\hat{c}$
$\displaystyle\hat{H}_{\rm Jan}=$ $\displaystyle g_{\rm
eff}\left(\hat{a}^{\dagger}{}^{2}\hat{c}^{\dagger}{}^{2}\hat{b}+\hat{a}^{2}\hat{c}^{2}\hat{b}^{\dagger}\right)$

where the corresponding coefficients and also the coupling strength $g_{\rm
eff}$ are written in Appendix A. Note that, $g_{\rm eff}=0$ if
$\omega_{a}=\omega_{c}$. We call the last interaction Hamiltonian the Janus
interaction because the exchange is bilateral: for each phonon, two photons
are simultaneously generated in each cavity, and conversely. Now, the
projecting space for the first reduced dynamics is simply spanned by the
states $\\{\ket{0,1,0},\ket{2,0,2}\\}$, and the Hamiltonian takes the form

$H=\begin{pmatrix}\Omega_{b}&2g_{\rm eff}\\\ 2g_{\rm
eff}&2(\Omega_{a}+\Omega_{c}+\alpha_{a}+\alpha_{c}+2\alpha_{a,c})\end{pmatrix}\,.$
(10)

The point of maximum interaction can be found again by equating the two
diagonal terms. However, as for the four-photon entanglement, this is a third-
order process, and a more accurate resonance condition is found numerically.
Since we need $\omega_{a}\neq\omega_{c}$, we can fix
$\omega_{a}=\omega_{b}/4+\varepsilon$ and $\omega_{c}$ to satisfy the required
resonance condition.

Here we choose $\varepsilon=\omega_{b}/15$, and a numerical optimization
procedure allows us to find the value of $\omega_{c}$ (and so also $\Omega$)
to get symmetric eigenstates in terms of $\ket{0,1,0}$ and $\ket{2,0,2}$.
Under this condition, the two eigenstates are

$\lvert\psi_{\pm}^{\rm(Jan)}\rangle=\frac{1}{\sqrt{2}}\left(\ket{0,1,0}\pm\ket{2,0,2}\right)\,.$

As before, we employ a quantum trajectory approach to investigate how a
measurement affects the dynamics of the system. In Fig. 4(a) a single quantum
jump collapses the initial state $\ket{2,0,2}$ into $\ket{1,0,2}$. The latter
state has no sub-processes and exhibits trivial dynamics. Fig. 4(b) shows how
the phonon measurement changes the initial dynamics and leaves the system in
the ground state, in analogy to Section II.2. Fig. 4(c) represents the average
over 1000 trajectories. Note that, the “averaging” process hides the complex
dynamics described above. We used the following parameters for our simulation:
$\omega_{a}=\omega_{b}/4+\varepsilon$, $\varepsilon=\omega_{b}/15$,
$\Omega\approx 0.6067$, $g=0.05\omega_{b}$,
$\gamma_{a}=\gamma_{b}=\gamma_{c}=2\times 10^{-5}\omega_{b}$.

##  III Conclusions

In this work, we have investigated the quantum phenomena that arise from a
cavity resonator containing a two-sided perfect mirror, which acts as a
mechanical oscillator interacting with two separated electromagnetic fields by
radiation pressure. We have shown that, depending on the chosen resonant
conditions, this system can generate 2n-photon entanglement and bilateral
photon pair emission.

We have also explored the effects of the system’s parameters, such as the
coupling strengths, the bare frequencies, and the initial conditions, on the
entanglement and the photon emission. We have provided analytical and
numerical results to support our findings and to illustrate the feasibility of
observing these phenomena in realistic setups. Our work contributes to the
development of quantum-enabled devices that rely on the deterministic creation
and distribution of entangled states.

We have proposed circuit-optomechanical systems and quantum simulators as
possible platforms to implement our scheme, which could also be extended to
other physical systems and energy scales. Furthermore, our work opens up new
avenues for exploring quantum effects in tripartite systems. Our findings are
expected to stimulate further research in this direction and foster the
advancement of quantum technologies.

## Acknowledgements

F.N. is supported in part by: Nippon Telegraph and Telephone Corporation (NTT)
Research, the Japan Science and Technology Agency (JST) [via the Quantum Leap
Flagship Program (Q-LEAP), and the Moonshot R$\&$D Grant Number JPMJMS2061],
the Army Research Office (ARO) (Grant No. W911NF-18-1-0358), the Asian Office
of Aerospace Research and Development (AOARD) (via Grant No.
FA2386-20-1-4069), and the Foundational Questions Institute Fund (FQXi) via
Grant No. FQXi-IAF19-06. S.S. acknowledges the Army Research Office (ARO)
(Grant No. W911NF-19-1-0065). R.L.F. is supported by European Union (Next
Generation EU) via MUR D.M. Grant No. 737-2021.

##  Appendix A Effective Hamiltonian

The effective Hamiltonian, which shows in a direct way the high-order
processes, can be obtained through the Schrieffer-Wolff (SW) transformation
Schrieffer and Wolff (1966); Bravyi  _et al._ (2011); Girvin and Yang (2019).
We start by evaluating the following rotation

$\hat{H}_{\rm
eff}=e^{\lambda\hat{S}}\left(\hat{H}_{0}+\lambda\hat{H}_{I}\right)e^{-\lambda\hat{S}}\,,$
(11)

where

$\displaystyle\hat{H}_{0}=$
$\displaystyle\omega_{a}\hat{a}^{\dagger}\hat{a}+\omega_{b}\hat{b}^{\dagger}\hat{b}+\omega_{c}\hat{c}^{\dagger}\hat{c}$
(12) $\displaystyle\hat{H}_{I}=$
$\displaystyle\frac{g}{2}\left[\left(\hat{a}+\hat{a}^{\dagger}\right)^{2}-\Omega^{2}\left(\hat{c}+\hat{c}^{\dagger}\right)^{2}\right]\left(\hat{b}+\hat{b}^{\dagger}\right)\,,$

and $\lambda$ tracks how many times we apply the off-diagonal terms. Using the
Baker-Campbell-Hausdorff lemma

$\displaystyle e^{\hat{B}}\hat{A}e^{-\hat{B}}=$
$\displaystyle\hat{A}+\commutator{\hat{B}}{\hat{A}}+\frac{1}{2}\commutator{\hat{B}}{\commutator{\hat{B}}{\hat{A}}}+\ldots$
(13)
$\displaystyle+\frac{1}{n!}\underbrace{\left[\hat{B},\left[\hat{B},\left[\hat{B},\ldots\left[\hat{B}\right.\right.\right.\right.}_{n\
\text{times}},\left.\left.\left.\left.\hat{A}\right]\right]\right]\right]+\ldots\,,$

we have (up to the third order on the expansion)

$\displaystyle\hat{H}_{\rm eff}=$
$\displaystyle\hat{H}_{0}+\lambda\hat{H}_{I}+\lambda\left[\hat{S},\hat{H}_{0}+\lambda\hat{H}_{I}\right]$
$\displaystyle+\frac{\lambda^{2}}{2!}\left[\hat{S},\left[\hat{S},\hat{H}_{0}+\lambda\hat{H}_{I}\right]\right]$
$\displaystyle+\frac{\lambda^{3}}{3!}\commutator{\hat{S}}{\commutator{S}{\commutator{\hat{S}}{\hat{H}_{0}+\lambda\hat{H}_{I}}}}+\mathcal{O}(\lambda^{4})$
$\displaystyle=$
$\displaystyle\hat{H}_{0}+\lambda\left(\hat{H}_{I}+\commutator{\hat{S}}{\hat{H}_{0}}\right)$
$\displaystyle+\frac{\lambda^{2}}{2!}\left(2!\commutator{\hat{S}}{\hat{H}_{I}}+\commutator{\hat{S}}{\commutator{\hat{S}}{\hat{H}_{0}}}\right)$
$\displaystyle+\frac{\lambda^{3}}{3!}\left(\frac{3!}{2!}\commutator{\hat{S}}{\commutator{\hat{S}}{\hat{H}_{I}}}+\commutator{\hat{S}}{\commutator{\hat{S}}{\commutator{\hat{S}}{\hat{H}_{0}}}}\right)$
$\displaystyle+\mathcal{O}(\lambda^{4})\,.$

In order to cancel the linear term in $\lambda$, we now choose $\hat{S}$ such
that $[\hat{S},\hat{H}_{0}]=-\hat{H}_{I}$, and the total effective Hamiltonian
becomes

$\displaystyle\hat{H}_{\rm eff}\simeq$
$\displaystyle\hat{H}_{0}+\frac{\lambda^{2}}{2}\commutator{\hat{S}}{\hat{H}_{I}}+\frac{\lambda^{3}}{3}\commutator{\hat{S}}{\commutator{\hat{S}}{\hat{H}_{I}}}$
$\displaystyle=\hat{H}_{0}+\lambda^{2}\hat{H}_{\rm
eff}^{(2)}+\lambda^{3}\hat{H}_{\rm eff}^{(3)}\,.$ (14)

The most crucial step in doing SW transformation is to get the generator
$\hat{S}$, such that $[\hat{S},\hat{H}_{0}]=-\hat{H}_{I}$. In the following,
we apply a systematic method to obtain it.

We impose $\hat{S}=[\hat{H}_{0},\hat{H}_{I}]$, and, leaving the coefficients
undefined since they will be obtained using the condition
$[\hat{S},\hat{H}_{0}]=-\hat{H}_{I}$, we get

$\displaystyle\hat{S}=$
$\displaystyle\commutator{\hat{H}_{0}}{\hat{H}_{I}}=c_{0}\hat{b}+c_{1}{\hat{b}^{\dagger}}+c_{3}{\hat{c}^{\dagger}}{}^{2}\hat{b}+c_{4}{\hat{a}^{\dagger}}{\hat{b}^{\dagger}}\hat{a}$
$\displaystyle+c_{5}{\hat{b}^{\dagger}}{\hat{c}^{\dagger}}\hat{c}+c_{6}{\hat{c}^{\dagger}}\hat{b}\hat{c}+c_{7}{\hat{a}^{\dagger}}{}^{2}\hat{b}+c_{8}\hat{a}{}^{2}\hat{b}+c_{9}{\hat{a}^{\dagger}}{}^{2}{\hat{b}^{\dagger}}$
$\displaystyle+c_{10}{\hat{b}^{\dagger}}\hat{a}{}^{2}+c_{11}{\hat{a}^{\dagger}}\hat{a}\hat{b}+c_{12}{\hat{b}^{\dagger}}\hat{c}{}^{2}+c_{13}{\hat{b}^{\dagger}}{\hat{c}^{\dagger}}{}^{2}+c_{14}\hat{b}\hat{c}{}^{2}\,.$

By using $[\hat{S},\hat{H}_{0}]=-\hat{H}_{I}$, the generator becomes

$\displaystyle\hat{S}=$
$\displaystyle\frac{g\left(1-\Omega^{2}\right)}{2\omega_{b}}\left(\hat{b}-\hat{b}^{\dagger}\right)+\frac{g}{\omega_{b}}\hat{a}^{\dagger}\hat{a}\left(\hat{b}-\hat{b}^{\dagger}\right)$
$\displaystyle-\frac{\Omega^{2}g}{\omega_{b}}\hat{c}^{\dagger}\hat{c}\left(\hat{b}-\hat{b}^{\dagger}\right)-\frac{g}{4\omega_{a}-2\omega_{b}}\left(\hat{a}^{\dagger}{}^{2}\hat{b}-\hat{a}^{2}\hat{b}^{\dagger}\right)$
$\displaystyle+\frac{g}{4\omega_{a}+2\omega_{b}}\left(\hat{a}^{2}\hat{b}-\hat{a}^{\dagger}{}^{2}\hat{b}^{\dagger}\right)$
$\displaystyle+\frac{\Omega^{2}g}{4\omega_{c}-2\omega_{b}}\left(\hat{c}^{\dagger}{}^{2}\hat{b}-\hat{c}^{2}\hat{b}^{\dagger}\right)$
$\displaystyle-\frac{\Omega^{2}g}{4\omega_{c}+2\omega_{b}}\left(\hat{c}^{2}\hat{b}-\hat{c}^{\dagger}{}^{2}\hat{b}^{\dagger}\right)\,,$
(15)

and the perturbative Hamiltonians $\hat{H}_{\rm eff}^{(2)}$ and $\hat{H}_{\rm
eff}^{(3)}$ for the second and third order respectively can be obtained
following Appendix A.

The total effective Hamiltonian expressed in Appendix A describes all the
high-order processes up to the third order. By imposing a specific resonance
condition, one can make a process dominant over the others. By applying the
rotating wave approximation (RWA) to $\hat{H}_{\mathrm{eff}}$, we reduce it to
a specific effective one that describes that specific process. As done in the
main text, we want to explore the generation of $2n$-photon entanglement
(e.g., two-, four-photon) and the bilateral photon emission, namely, the Janus
effect. In particular, the two-photon entanglement is obtained by choosing
$\omega_{a}\approx\omega_{b}\approx\omega_{c}$, leading to different
oscillating terms, which can be neglected by applying the RWA. All the
remaining non-oscillating terms form the specific effective Hamiltonian,
expressed in Eq. 3. Similarly, the same procedure can be performed in the case
of the four-photon entanglement ($\omega_{b}\approx 4\omega$, with
$\omega=\omega_{a}=\omega_{c}$), leading to the specific effective Hamiltonian
in Section II.2, and in the case of the Janus effect ($\omega_{b}\approx
2(\omega_{a}+\omega_{c})$) in Eq. 9. In the case of the Janus effect, here we
show all the coefficients contained inside Eq. 9:

$\displaystyle\Omega_{a}=$
$\displaystyle\frac{g^{2}\left(\Omega^{3}+2\Omega^{2}-3\Omega-5\right)}{\omega_{b}\left(\Omega+2\right)}$
$\displaystyle\Omega_{b}=$
$\displaystyle\frac{g^{2}\left(\Omega^{8}+3\Omega^{7}+2\Omega^{6}+2\Omega^{2}+3\Omega+1\right)}{\Omega\omega_{b}\left(2\Omega^{2}+5\Omega+2\right)}$
$\displaystyle\Omega_{c}=$
$\displaystyle\frac{\Omega^{2}g^{2}\left(-5\Omega^{3}-3\Omega^{2}+2\Omega+1\right)}{\omega_{b}\left(2\Omega+1\right)}$
$\displaystyle\alpha_{a}=$
$\displaystyle\frac{g^{2}\left(-3\Omega^{2}-6\Omega-1\right)}{2\Omega\omega_{b}\left(\Omega+2\right)}$
$\displaystyle\alpha_{c}=$
$\displaystyle\frac{\Omega^{4}g^{2}\left(-\Omega^{2}-6\Omega-3\right)}{2\omega_{b}\left(2\Omega+1\right)}$
$\displaystyle\alpha_{a,b}=$
$\displaystyle\frac{2g^{2}\left(\Omega+1\right)}{\Omega\omega_{b}\left(\Omega+2\right)}$
$\displaystyle\alpha_{a,c}=$
$\displaystyle\frac{2\Omega^{2}g^{2}}{\omega_{b}}$
$\displaystyle\alpha_{b,c}=$
$\displaystyle\frac{2\Omega^{5}g^{2}\left(\Omega+1\right)}{\omega_{b}\left(2\Omega+1\right)}$
$\displaystyle g_{\rm eff}=$ $\displaystyle\frac{\Omega
g^{3}\left(2\Omega^{6}+5\Omega^{5}+4\Omega^{4}-4\Omega^{2}-5\Omega-2\right)}{2\omega_{b}^{2}\cdot\left(2\Omega^{2}+5\Omega+2\right)}\,.$

After the RWA, the result is equal to that obtained with other procedures,
such as the generalized James’ effective Hamiltonian method Shao  _et al._
(2017).

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