Abstract

We investigate a hybrid quantum system combining cavity quantum electrodynamics and optomechanics, where a photon mode is coupled to a four-level tripod atom and to a mechanical mode via radiation pressure. We find that within the single-photon optomechanics and Lamb-Dicke limit, the presence of the tripod atom alters the optical properties of the cavity radiation field drastically, and gives rise to completely quantum destructive interference effects in the optical scattering. The heating rate can be dramatically suppressed via utilizing the completely destructive interference involving atom, photon and phonon, and the obtained result is analogous to that of the resolved sideband regime. The heating process is only connected to the scattering of cavity damping path, which is also far-off resonance. Meanwhile, the cooling rate assisted by the atomic transitions can be significantly enhanced, where the cooling process occurs through the cavity and atomic dissipation paths. Finally, the ground-state cooling of the movable mirror is achievable and even more robust to heating process and thermal noise.

© 2014 Optical Society of America

1. Introduction

Cavity optomechanics explores the radiation pressure interaction between an optical cavity field and a macroscopic mechanical objects, for example, coated atomic force microscopy cantilevers [1], SiN3 membrane dispersively coupled to an optical cavity [2], and superconducting microwave resonators coupled to a nanomembrane beam [3]. In the last decades, tremendous efforts were devoted to investigate optomechanical cooling since the achievement of quantum ground state is a crucial step towards quantum control of macroscopic mechanical systems [4,5]. For a typical optomechanical system, the cavity field is pumped by a strong laser to enhance the optomechanical coupling [68]. Recently, with the rapid technical advancement, there are several experiments [913], such as bilayer photonic crystal [12] and optomechanical disk resonators [13], currently approaching to the single-photon strong coupling regime, where the presence of a single photon displaces the mechanical oscillator by more than its zero-point uncertainty. And recent theoretical investigations show that optomechanical single-photon nonlinear effects can be enhanced due to an increased photon-phonon coupling in multimode optical setups [14,15], or in a driven optomechanical system where a resonance enhances interactions between polaritons [1618]. The optomechanical cooling behavior in the single-photon strong-coupling regime has been investigated and different phenomena are revealed in [19].

Nowadays in experiment the movable membrane can be cooled to its ground state by utilizing the resolved-sideband scheme in the cavity optomechanical system [20]. However, due to the existence of blue-sideband heating mechanism and the crucial requirements of experimental setup, it is difficult to achieve a lower temperature in a fast rate. There have been theoretical investigations to break through this limit by exploring the quantum interference effects in quantum dot system [21] and two-mode optical cavity optomechanical system [8]. Inspired with the recent success in the well-developed techniques in cavity quantum electrodynamics (CQED), a hybrid atom-optomechanical system may be employed to improve the cooling performance of the mechanical oscillator. It is because that the radiation pressure depends on the number statistics of photons, for example, using type I optical parametric amplifier or Kerr medium placed inside a cavity [22, 23], and the photon statistics can be dramatically altered by utilizing the robust quantum interference of transitions between the states of atom-cavity system [2427]. Moreover, there have been many attractive proposals to experimentally realize such hybrid atom-optomechanical system that uses ultracold atoms as quantum “handle” on the mechanics [28]. Thus, it may be promising to realize such a hybrid atom-optomechanical device combining CQED and optomechanics that can exert the quantum interference where (artificial) atoms, photons and phonons are involved to provide better performance on the cooling of membrane motion.

In this work we consider a hybrid atom-optomechanical system where the cavity interacting with a single four-level tripod atom system and close to the regime of single-photon nonlinear optomechanical effects becoming significant. We focus on the strong coupling between the atomic transitions and cavity field and neglect the external motion of the atom. Possible realizations of such a setup could be a single atom tightly trapped inside an optical resonator with pendular end mirror, an artificial atom inside an optomechanical photonic crystal cavity, or a photonic crystal in diamond coupled to mechanical motion, where a nitrogen-vacancy (NV) center realizes the tripod system. Similar hybrid optomechanical setups composed of a two-level (artificial) atom inside the optomechanical system have been investigated [29, 30], where the quantum interference allows one to suppress the Stokes component of the scattered light. However, it is impossible to completely suppress the heating processes via cavity damping or atomic dissipation paths, which may hinder better cooling performance. In contrast, with use of a four-level tripod atom in our setup, the heating process via atomic dissipation path can be completely suppressed by using double completely quantum destructive interferences: a cavity-mediated three-photon resonance to prohibit the carrier transition [31] and another resonance involving mechanical phonon, photon and atom to prohibit the phonon increasing transitions. Besides the cancelation of the heating process via atomic dissipation path, the resulting heating coefficient is analogous to that of resolved-sideband regime [20], and the heating process is only connected to the scattering of cavity damping path, which is also far-off resonance. In a good cavity limit, the heating process can be further suppressed. Meanwhile, the cooling process assisted by the atomic transition can be notably enhanced. On one side, due to additional path mediated by the atom attributed to the photon transition, the cooling process via the cavity damping path is largely enhanced. On the other side, the cooling process can also occur via the atomic dissipation path to further enlarge the cooling rate. As a consequence, the ground-state cooling of movable mirror is achievable, and the enhanced cooling rate can make the cooled movable oscillator more robust against heating process and thermal noise.

The paper is organized as follows: The hybrid atom-optomechanical system is introduced in Sec. 2, and the master equation for the movable mirror is derived in Sec. 3. In Sec. 4 we obtain the explicit expressions for the heating and cooling coefficients via perturbation in Ωp, and finally the conclusion is driven. In the appendix we present some details of calculations.

2. Description of the hybrid atom-optomechanical system

The model under consideration consists of a four-level atom in tripod configuration, fixed at a position and coupled to the field of a single mode of an optomechanical cavity by dipole interaction, as shown in Fig. 1. The topology of such an optomechanical setup is formed by a Fabry-Pérot cavity with a moving end mirror. The movable mirror is treated as a quantum-mechanical harmonic oscillator with effective mass m, frequency ν and energy decay rate γm. The photons in the cavity will exert a radiation pressure force on the movable mirror due to the momentum transfer. Meanwhile, the atom in tripod configuration comprised of one excited state |e〉 and three ground states |gi〉(i = 1, 2, 3) with energy frequencies ωe and ωgi respectively, is placed inside the cavity. Two laser fields with frequencies ωL j(j = 1, 2) are applied to drive dipole-allowed transitions |e〉 ↔ |gj〉 with Rabi frequencies Ωj, respectively, and the third transition |g3〉 ↔ |e〉 is coupled to the optomechanical cavity field of frequency ωc with the coupling strength g. We focus here on the internal degrees of freedom of the atom and neglect its center-of-mass motion. The optomechanical cavity is weakly driven by a laser field of frequency ωp and the driving strength is Ωp. The atom and cavity excitations decay into the vacuum reservoir with rates γ and κ, respectively.

 

Fig. 1 Sketch of the hybrid atom-optomechanical setup, consisting of a cavity coupled to a mechanical oscillator by radiation pressure with an atom in tripod configuration placed inside the cavity.

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Thus the system can be described by the Hamiltonian,

H^=H^osc+H^at+H^cav+H^cav-at+H^L-at+H^L-cav+F^x^,
which is written in a rotating frame of the lasers’ frequency. The Hamiltonian of free movable mirror
H^osc=νb^b^
accounts for the energy of the mechanical oscillator with frequency ν and mass m, and the Hamiltonians
H^at=δc3|ee|+j(δjδc3)|gjgj|+Δ|g3g3|,H^cav=Δa^a^,H^cav-at=g(|eg3|a^+|g3e|a^),H^L-at=jΩj2(|egj|+|gje|),H^L-cav=Ωp2(a^+a^)
describe the energies of the internal atomic states, the cavity field, the coupling interaction between the cavity mode and the atomic transition |e〉 ↔ |g3〉, the drive of laser fields on the transitions |e〉 ↔ |gj〉 and the coherent driving of the cavity by the weak pump laser, respectively. Here δc3 = ωc − (ωeωg3) is the detuning of the cavity frequency ωc to the atomic transition |e〉 ↔ |g3〉, δj = ωLj − (ωeωgj) indicates the detuning of the j-th laser frequency ωLj to the corresponding atomic dipole transition |e〉 ↔ |gj〉, and Δ = ωpωc describes the detuning between the cavity and the probe fields. The operator â is the photon annihilation operator for the cavity mode, and the annihilation operator is connected with the oscillator’s position and momentum operators,
x^=ξ(b^+b^),p^=12iξ(b^b^),
where ξ=1/2mν is the width of the oscillator’s ground-state wave packet. Because the effective cavity length is slightly changed by the small displacements of the mechanical oscillator, we can take the mechanical action linearly in the oscillator’s displacement and it is represented by the term F̂x̂, which has the form
F^x^=χa^a^(b^+b^),
and introduces an interaction between the cavity and the mechanical oscillator with the coupling strength characterized by χ.

The time-dependent density operator ρ̂ of this system obeys the master equation

ddtρ^=i[H^,ρ^]+^ρ^+𝒦^ρ^+^mρ^,
where the superoperators ℒ̂ρ̂ and 𝒦̂ρ̂ written as
^ρ^=iγi2(2|gie|ρ^|egi||ee|ρ^ρ^|ee|),𝒦^ρ^=κ2(2a^ρ^a^a^a^ρ^ρ^a^a^),
describe the dipole spontaneous emission and cavity decay, and ℒ̂mρ̂ describes the damping of the movable mirror coupled to thermal bath
^mρ^=γm2(n¯th+1)(2b^ρ^b^b^b^ρ^ρ^b^b^)+γm2n¯th(2b^ρ^b^b^b^ρ^ρ^b^b^),
with th the thermal mean phonon number.

3. Derivation of the master equation for the movable mirror

For the single-photon optomechanics, we make use of the typical unitary transformation given by Û = exp(−Ŝ) with Ŝ = ηââ() and η = χ/ν [32]. The sum of two Hamiltonians Ĥosc and F̂x̂ will be diagonalized in the form

H^osc+F^x^=νb^b^ηχ(a^a^)2,
while the Hamiltonians Ĥcav-at and ĤL-cav become
H^cav-at=g[|eg3|a^eη(b^b^)+|g3e|a^eη(b^b^)],H^L-cav=Ωp2[a^eη(b^b^)+a^eη(b^b^)].
The other Hamiltonians Ĥat, Ĥcav and ĤL-at in Eq. (1) remain unchanged. Meanwhile, the superoperator 𝒦̂ρ̂ describing cavity decay becomes
𝒦^ρ^=κ2[2a^eη(b^b^)ρ^a^eη(b^b^)a^a^ρ^ρ^a^a^].
For the transformation of the term ℒ̂mρ̂ describing the mechanical damping, we need to substitute the operator and in Eq. (8) by ( + ηââ) and ( + ηââ), respectively. Because the conditions ηγmthωm and η2γmthκ are satisfied, it is appropriate to get rid of the off-resonate terms, i.e. the first-order terms of η, and the second-order terms of η, which is much smaller than the other terms of ℒ̂mρ̂. Therefore, the form of ℒ̂mρ̂ in Eq. (8) keeps unchanged in the later discussion.

Because the cavity is sufficiently weakly driven, which means that the average photon number of the cavity mode |ε|2|Ωp/2Δ+iκ/2|2 is much smaller than unity, we can truncate the Hilbert space of the system. Hence we investigate the cooling dynamics in the subspace comprised of at most one excitation of the cavity mode, i.e. the relevant Hilbert space is spanned by the states

{|e,0,g1,0|,|g2,0,g3,0|,|g3,1}.
We write |e, 0〉 ≡ |e〉, |g1, 0〉 ≡ |g1〉, |g2, 0〉 ≡ |g2〉, |g3, 0〉 ≡ |g3〉 and |g3, 1〉 ≡ |1〉 for later convenience. Within the subspace, the Hamiltonians in Eqs. (9) and (10) and the superoperator in Eq. (11) correspondingly become
H^osc+F^x^=νb^b^ηχ|11|,H^cav-at=g[|e1|eη(b^b^)+|1e|eη(b^b^)],H^L-cav=Ωp2[|g31|eη(b^b^)+|1g3|eη(b^b^)],
and
𝒦^ρ^=κ2[2|g31|eη(b^b^)ρ^|1g3|eη(b^b^)|11|ρ^ρ^|11|].
The form of term ĤL-at is unchanged and the form for the sum of Ĥat and Ĥcav turns to be the same as Ĥat in Eq. (3). The term ℒ̂ρ̂ describing the dipole spontaneous emission and the term ℒ̂mρ̂ describing the mechanical damping are also unchanged in this subspace.

In the following, we assume that the optomechanical coupling is in the Lamb-Dicke limit, i.e., η ≪ 1, where the transitions that change motional quantum number of movable mirror by more than one are strongly supressed. A Taylor expansion of exponential operator is possible,

eη(b^b^)=1+η(b^b^)+O(η2).
Then we can follow the same procedure as taken in ground-state cooling of a trapped ion or atom [31, 33], where the Hamiltonian of the system is written into two parts: zeroth- and first-order Hamiltonians in η, i.e., Ĥ = Ĥ0 + Ĥ1. In the rotating frame of ηχ, with the redefined parameters δc3δc3ηχ, Δ → Δ + ηχ, the Hamiltonian in zeroth-order of η contains the free evolution of the movable mirror Ĥosc and the atom-cavity system H^freea-c, which is given by
H^freea-c=δc3|ee|+j(δjδc3)|gjgj|+Δ|g3g3|+jΩj2(|egj|+|gje|)+g(|e1|+|1e|)+Ωp2(|g31|+|1g3|),
and the Hamiltonian in first-order of η that represents the interaction between the movable mirror and atom-cavity system is Ĥ1 = iV̂1(), with
V^1=iη[g|e1|+Ωp2|g31|]+H.c..

When the movable mirror is weakly coupled to the atom-cavity system, we can follow the procedure proposed by Cirac et al. [34] by use of the second-order perturbation theory with respect to the Lamb-Dicke parameter η to obtain the master equation of mirror motion. We should divide the evolution of the system into three parts: the free evolution of atom-cavity and mirror systems, and the interaction between them, which are represented by the superoperators 𝒮̂0, 𝒮̂2 and 𝒮̂1 respectively. Thus the master equation is written in the form

ddtρ^=(𝒮^0+𝒮^1+𝒮^2)ρ^.
The superoperator 𝒮̂0ρ̂ for the free atom-cavity system is given by
𝒮^0ρ^=i[H^freea-c,ρ^]+^ρ^+𝒦^0ρ^,
with
𝒦^0ρ^=κ2(2|g31|ρ^|1g3||11|ρ^ρ^|11|).
The superoperator 𝒮̂2ρ̂ for the free movable mirror is given by
𝒮^2ρ^=i[H^osc,ρ^]+^mρ^.
The interaction between the atom-cavity system and the movable mirror is given by
𝒮^1ρ^=i[H^1,ρ^]+𝒦^2ρ^,
where 𝒦̂2ρ̂ is the diffusion caused when one expands the superoperator (14) describing the cavity decay, and given in the form
𝒦^2ρ^=κ|g31|[η(b^b^)ρ^+η22(b^ρ^b^+b^ρ^b^b^b^ρ^b^b^ρ^)+H.c.]|1g3|.
Here the operator is in the representation of polaron transformation, and Eq. (23) is generated by the polaron transformation. In order to obtain the realistic steady-state mean phonon number, we should return to the original representation, where the cavity damping is represented by Eq. (7). Then in the following we should ignore the term [19, 32].

Tracing over the atom-cavity variables and keeping the motion equation up to the second order of η, we have the master equation for the movable mirror

ddtμ^=Tra-c{[𝒫^𝒮^2𝒫^+𝒫^𝒮^1(𝒮^0)1𝒮^1𝒫^]ρ^},
where μ̂ is the reduced density operator for the movable mirror and 𝒫̂ is the projection operator on the subspace with zero eigenvalue of 𝒮̂0 [34]. After some direct calculations we obtain the explicit master equation of movable mirror in the rotating frame of oscillating frequency ν
ddtμ^=S(ν)(b^μ^b^b^b^μ^)+S(ν)(b^μ^b^b^b^μ^)+H.c.+^mμ^,
where the spectral of two-time correlation function S(ν) of atomic and cavity operators is expressed as
S(ν)=0dteiνtV^1(t)V^1(0)st.
Here the transition operators σmn = |m〉 〈n| are expressed in terms of a complete set of states {|m〉} = {|e〉, |g1〉, |g2〉, |g3〉, |1〉}.

Then we can directly derive the rate equation for the mean phonon number 〈〉, namely

n^˙=(AA++γm)n^+A++γmn¯th,
where
A±=2Re{S(ν)}
are the heating (A+) and cooling (A) coefficients. Then the steady-state mean phonon number 〈st and the cooling rate W can be easily obtained as
n^st=A++γmn¯thAA++γm,W=AA++γm.
We assume that there only exists the spontaneous decay along the transition path |e〉 → |g3〉, which will simplify the analytical expressions for heating and cooling coefficients in the following section, and even when the other dissipations are nonzero, they will not qualitatively affect the cooling dynamics [31]. Hence in the following we will denote γ3 = γ.

4. Achieving explicit expressions for the heating and cooling coefficients via perturbation in Ωp

The cooling and heating rates A± are related to the spectral of two-time correlation function S(ν) in Eq. (28). According to the quantum regression theorem [35], the two-time correlation functions 〈σmn(t)σkl(0)〉st obey the same equations as ρnm(t), which are governed by the superoperator (19), and together with the initial values 〈σmn(0)σkl(0)〉st = δnkρlm(∞), where ρlm(∞) are the steady-state solutions, the spectral of two-time correlation S(ν) is achievable. The concrete evolution equations for time-dependent density matrix elements ρmn(t) derived from Eq. (19) are given as

ρ˙ee=γρee+i(Ω12ρeg1+Ω22ρeg2+gρe1)+c.c.,ρ˙gjgj=γjρeeiΩj2(ρegjρgje),ρ˙11=κρ11+i(Ωp2ρ1g3gρe1)+c.c.,ρ˙eg1=(iδ1γ2)ρeg1+iΩ12(ρeeρg1g1)iΩ22ρg2g1igρ1g1,ρ˙eg2=(iδ2γ2)ρeg2+iΩ22(ρeeρg2g2)iΩ12ρg1g2igρ1g2,ρ˙e1=(iδc3γ+κ2)ρe1+ig(ρeeρ11)ijΩj2ρgj1+iΩp2ρeg3,ρ˙g1g2=i(δ2δ1)ρg1g2iΩ12ρeg2+iΩ22ρg1e,ρ˙g11=[i(δc3δ1)κ2]ρg11iΩ12ρe1+igρg1e+iΩp2ρg1g3,ρ˙g21=[i(δc3δ2)κ2]ρg21iΩ22ρe1+igρg2e+iΩp2ρg2g3,ρ˙eg3=[i(δc3+Δ)γ2]ρeg3ijΩj2ρgjg3igρ1g3+iΩp2ρe1,ρ˙g1g3=i(δc3+Δδ1)ρg1g3iΩ12ρeg3+iΩp2ρg11,ρ˙g2g3=i(δc3+Δδ2)ρg2g3iΩ22ρeg3+iΩp2ρg21,ρ˙1g3=(iΔκ2)ρ1g3igρeg3iΩp2(ρg3g3ρ11).

However, it is difficult to obtain the explicit expressions of the two-time correlations from the complex equations (30). Because of the small laser-cavity coupling strength Ωp, it offers the opportunity to achieve the analytical expressions by employing the second-order perturbation method on Ωp, which is feasible to describe the physics within the mechanical cooling dynamics [30, 31]. In the following we denote ρmn(0), ρmn(1) and ρmn(2) to represent the density matrix elements in orders of Ωp0, Ωp1 and Ωp2, respectively, and resort to the method of Laplace transform of the evolution equations to obtain the spectral of two-time correlations.

According to the perturbation method with respect to Ωp, we can obtain the i-th(i = 0, 1, 2) order steady-state density matrix elements in appendix A. With the steady-state solutions and Laplace transforms of the density matrix elements given in appendix A, and applying the quantum regression theorem, after some calculations we obtain the form of the analytical expressions for the heating and cooling coefficients

A±/η2=γ|𝒯2±|2+κ|g𝒯2±+g𝒯1+Ωp/2Δν+iκ/2|2,
which describe absorption of a photon of the probe laser pumping the cavity, followed by emission into the modes of the external field either by atomic or by cavity decay, and can be written as the sum of various contributions of different physical origin [31]. The transition amplitudes 𝒯1 and 𝒯2± connected to the emission or absorption of a vibrational phonon are expressed as
𝒯1=gΩp/2f(Δ)(δc3+Δδ1)(δc3+Δδ2),𝒯2±=±νgΩp/2f(Δ)f(Δν)γ(Δ)(δc3+Δνδ1)(δc3+Δνδ2),
where the functions are
f(Δ)=(Δ+iκ/2)γ(Δ)g2(δc3+Δδ1)(δc3+Δδ2),γ(Δ)=(δc3+Δδ1)(δc3+Δδ2)(δc3+Δ+iγ/2)(Ω1/2)2(δc3+Δδ2)(Ω2/2)2(δc3+Δδ1).

4.1. Suppression of heating process via completely quantum destructive interference

In order to achieve a lower cooling temperature, we should suppress the heating processes and enhance the cooling processes. The heating processes are shown in Fig. 2, where |n〉 is phonon number state, the green lines indicate the carrier transition into atomic excited state |e, n〉 and the blue lines indicate the mechanical phonon-increased transitions. From appendix A, we can verify that the carrier transition to the state |e, n〉 becomes zero, i.e., ρee(2)()=0 when the detuning δ1 or δ2 satisfies the relation

δc3+Δδ1=0orδc3+Δδ2=0.
Without loss of generality, we will take the condition δc3 + Δ − δ1 = 0, which corresponds to the completely quantum destructive interference between the transitions |g3, n〉 ↔ |1, n〉 ↔ |e, n〉 and |g1, n〉 ↔ |e, n〉. A cavity photon and two laser photons are involved in the interference and these three photons are resonant in the atom-cavity system. The destructive interference completely prohibits the carrier transition in |e, n〉, and will also lead to the transition amplitude 𝒯1 equal to zero.

 

Fig. 2 Sketch of the suppression of heating processes. The phonon number states are |n〉 in (a) and |n + 1〉 in (b). Through the completely quantum destructive interference between excitation paths |g3, n〉 → |1, n〉 → |e, n〉 and |g1, n〉 → |e, n〉 shown in (a) by the green lines, the carrier transition into |e, n〉 is prohibited. By use of the completely quantum destructive interferences between excitation paths |g2, n + 1〉 → |e, n + 1〉 and |g3, n〉 → |1, n〉 → |e, n + 1〉 shown in (b) by the solid blue lines, and between excitation paths |g2, n + 1〉 → |e, n + 1〉 and |g3, n〉 → |1, n + 1〉 → |e, n + 1〉 shown in (b) by the dashed blue lines, the phonon-increased transitions are significantly suppressed.

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Further, one can find that when the detuning δ2 fulfills the relation

δc3+Δνδ2=0,
the phonon-increased transition amplitude 𝒯2+ in Eq. (32) also becomes zero, because the blue-sideband transition into state |e, n + 1〉 is completely prohibited via the destructive quantum interference, and the further transition |e, n + 1〉 ↔ |1, n + 1〉 with the atom-cavity coupling strength g is also prohibited. Thus 𝒯2+ which appears twice in Eq. (31) is cancelled.

The heating scattering processes are explicitly indicated in Fig. 2, where these mechanical phonon-involved transitions are denoted by the blue lines. The two completely quantum destructive interferences between |g3, n〉 ↔ |1, n〉 ↔ |e, n + 1〉 and |e, n + 1〉 ↔ |g2, n + 1〉, and between |g3, n〉 ↔ |1, n + 1〉 ↔ |e, n + 1〉 and |e, n + 1〉 ↔ |g2, n + 1〉 can simultaneously occur under the condition of (35). By utilizing the two completely destructive quantum interferences, the heating coefficient in Eq. (31) can be simplified into the form

A+/η2=κ(Ωp/2)2(Δν)2+(κ/2)2,
which indicates the heating caused by the phonon-increased transition from |g3, n〉 to |1, n + 1〉, which is analogous to the mechanism of single-photon cooling of mechanical oscillator in the Lamb-Dicke limit, i.e., χ/ν ≪ 1 [19]. Obviously, the heating process connected to the atomic decay completely disappears, and in a good optical cavity limit with a weak pumping strength and appropriate detuning Δ, the heating coefficient in Eq. (36) can be drastically suppressed.

Compared with the hybrid optomechanical system comprised of a two-level atom inside the cavity [30], where quantum interference is not complete since the heating scattering processes via dipole dissipation and cavity damping paths both exist, here the heating process through the dipole dissipation path is completely suppressed with the use of double completely quantum destructive interferences. While the cooling process via the dipole dissipation path still exists and even can be strongly enhanced in the limit κγ, to make the cooled mirror more robust to the heating processes and thermal noise.

4.2. Analysis of cooling coefficient while heating coefficient is suppressed

To obtain the ground-state cooling of the movable mirror, which is also robust to thermal noise, we should enhance the cooling coefficient besides the well-suppressed heating coefficient. Under the conditions of Eqs. (34) and (35), the transition amplitude 𝒯1 is zero and phonon-decreased amplitude 𝒯2 becomes the form

𝒯2=νgε(Δ+ν+iκ2)(iγ2+δ1+νΩ124νΩ228ν)g2.
The structure of the transition amplitude is mainly determined by the denominator, where the real and imaginary parts are (Δ+ν)(δ1+νΩ124νΩ228ν)κγ/4g2 and κ/2(δ1+νΩ124νΩ228ν)+γ/2(Δ+ν) respectively. In the limit κγ and single atom cooperativity C = 4g2/κγ ≫ 1, the real part dominates over the imaginary part. The amplitude becomes maximal when the real part of denominator is zero, which corresponds to the minimal of the energy defect of the intermediate scattering states [31]. Therefore we choose the detuning δ1 as
δ1=Ω124ν+Ω228ν+g2+κγ/4Δ+νν,
and the transition amplitude 𝒯2 becomes
𝒯2=iνgεκ2g2+κγ/4Δ+ν+γ2(Δ+ν),
where ε=Ωp/2Δ+iκ/2. To concentrate to physics, we consider the good cavity limit and strong atom-cavity coupling regime, i.e., κ ≪ Δ, the amplitude 𝒯2 is approximately an imaginary number and the cooling rate is
A/η2=κ(Ωp/2)2+g2|𝒯2|2(Δ+ν)2+(κ/2)2+γ|𝒯2|2.
Obviously, when the detuning Δ is negative, the cooling rate dominates over heating rate, and the ground-state cooling is achievable.

To numerically show the achievement of the ground-state cooling, we choose Δ = −2ν, Ωp = ν, where the single photon assumption is accredited because the photon number of the cavity is about 0.065, well below unity. The steady-state phonon number 〈st and cooling rate W/η2 as functions of δ1 and δ2 are plotted in Fig. 3, where we have ignored the thermal noise here and will take it into account in the following discussion. In the figure the completely destructive quantum interference to suppress the carrier transition in Eq. (34) is employed, and the movable mirror is cooled down to the ground state with a fast cooling rate. The high cooling efficiency occurs almost along the diagonal line, i.e., δ2 = δ1ν, which is just the condition of the second completely destructive quantum interference involving mechanical phonon-increased transitions in Eq. (35). The minimal of steady-state phonon number is about 0.003, and the corresponding detuning δ1 is about −24ν determined from the Eq. (38).

 

Fig. 3 Steady-state phonon number 〈st and cooling rate W/η2 as functions of detunings δ1 and δ2 with parameters in units of ν: Ω1 = Ω2 = 6ν, κ = 0.1ν, Δ = −2ν, γ = 10ν, Ωp = ν, g = 6ν, δc3 = δ1 − Δ.

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In order to obtain the intuitive understanding of the role of the atom-cavity interaction in the mechanical cooling, we work out in the limit κγ and large atom cooperativity C ≫ 1 with Δ = −2ν. The cooling and heating rates of the mechanical motion are approximately

A/η24g2|ε|2/γ,A+/η24κ|ε|2/9,
and therefore the cooling limit is
n^A+/A49C,
where C measures the strength of the coherent atom-cavity coupling. Here C = 144 with the parameters in Fig. 3 and phonon number 49C is about 0.003, which is consistent with the numerical result.

When there is no atom presented in the optomechanical system, i.e., atom-cavity interaction strength g = 0, the heating and cooling coefficients become

A±/η2=κ(Ωp/2)2(Δν)2+(κ/2)2,
which are in the same form as cooling in the single-photon regime of cavity optomechanics in the Lamb-Dicke limit, i.e., χ/ν ≪ 1 [19]. It should be noted that the detuning Δ here has been rotated in frame of single-photon induced frequency shift ηχ. The optimal cooling occurs at the red sideband Δ = −ν, and the cooling rate is A=η2κΩp2/κ2. In order to fulfill the weak driving approximation, the strength ηΩp should be much smaller than κ to validate the perturbation approximation. For example, when ηΩp = 0.1, the cooling rate A is 0.001ν with κ = 0.1ν in the resolved-sideband regime. In the realistic condition, the mechanical oscillator suffers to the thermal heating, and we should take into account of intrinsic damping of the mechanical oscillator induced by its coupling to thermal noise. In a cryogenic environment the mechanical quality factor Q = ν/γm can reach 106 or even 107 with current [10, 36] or near-future technology [37]. When the initial thermal occupation th is 1000 with mechanical quality factor Q = 106, the thermal heating rate γmth leads to the final mean phonon number 〈st ≈ 1.

A promising avenue to circumvent the limitations inherent to optomechanical cooling is the use of an auxiliary quantum system that can enhance the effective optomechanical response of the resonantor, such as hybrid optomechanical systems [3841]. In our present scheme, there exists an extra cooling path via the atomic decay, and the large atomic decay rate can relax the requirement of the weak optomechanical coupling limit, i.e., the effective optomechanical coupling strength beyond the cavity width, since it can significantly decrease the timescale of the system to reach the steady state. With the increased coupling strength and enlarged decay rate, we can achieve the faster and more robust cooling dynamics. For example, in this paper we can choose η = 0.1, where the exponential expansion and the weak-coupling approximation can be both satisfied, and from the numerical results in Fig. 3 the optimal cooling rate becomes W ≈ 0.0068ν. With the same thermal occupation and quality factor given above, the final mean phonon number is 〈〉 ≈ 0.14, which is much lower than unity. Moreover, when the initial thermal occupation th is below 6800 with mechanical quality factor Q = 106, the thermal heating rate γmth is smaller than the cooling rate induced by atom-cavity system, and the ground-state cooling of mechanical oscillator is achievable.

When compared with the optomechanical cooling assisted by two-level atom, we set the driving strength of the additional transitions Ω1 and Ω2 zero, the transition amplitudes 𝒯1 and 𝒯2± are simplified as

𝒯1=gΩp/2D(Δ),𝒯2±=±νgΩp/2(δc3+Δ+iγ/2)D(Δ)D(Δν),
where
D(Δ)=(Δ+iκ/2)(δc3+Δ+iγ/2)g2.
The heating and cooling coefficients becomes
A±/η2=|Ωp/2(δc3+Δ+iγ/2)D(Δ)D(Δν)|2[γν2g2+κ|D(Δ)ν(Δ+iκ/2)|2],
Obviously, the destructive interference is not complete due to the final existence of heating processes related to the cavity and atomic decay, and the minimal cooling limit 〈stA+/A = 0.015 with the parameters in Fig. 3 through choosing the optimal detuning δc3. However, with use of the double completely destructive interferences in the hybrid optomechanical system assisted by tripod atom, the heating rate in Eq. (36) is only connected to cavity decay rate κ. The heating process related to atomic decay is completely suppressed, while the cooling processes related to both atomic and cavity decay exist. In the limit κγ, it is easier to achieve lower cooling limit and faster cooling rate through tuning the parameters. Since the heating rate is strongly suppressed in the good-cavity limit, we can focus on the enlargement of atomic decay to enhance the cooling rate to realize ground-state cooling of mechanical motion. In Fig. 3, the final mean phonon number without taking into account of thermal noise is 0.003.

Thus, with use of the four-level tripod atom within the optomechanical system and exploiting the completely quantum destructive interferences, the heating process becomes analogous to that of the cooling in single-photon regime in the Lamb-Dicke limit while the cooling rate assisted by the atomic degrees of freedom can be remarkably enhanced, leading the cooled mirror robust against the thermal noise. Notable signatures of quantum interference involving the atomic, photonic, and mechanical quantum degrees of freedom should be detectable in the cooling dynamics, in such a way demonstrating the coherent dynamics of the composite system.

5. Conclusion

To summarize, we have investigated a hybrid atom-optomechanical quantum system comprised of a four-level tripod atom within an optomechanical resonator, where a photon mode is coupled to the tripod atom and to the mechanical mode via radiation pressure. We find that within the single-photon nonlinear optomechanics and Lamb-Dicke limit, the presence of the tripod atom alters the optical properties of the cavity drastically, and gives rise to completely destructive interference effects in the optical scattering. The heating process can be drastically suppressed by utilizing two completely destructive quantum interferences that are mediated by the atom, leading to the heating process via dipole dissipation path canceled. The obtained heating coefficient is analogous to that of cooling of single-photon regime in Lamb-Dicke limit and far-off resonance, which can be further suppressed in a good cavity limit. Meanwhile, the cooling rate can be notably enhanced assisted by the atomic degrees of freedom, since an additional transition mediated by the atom attributes to the cooling process through the cavity damping path and the cooling process through the atomic dissipation path also occurs. Thus, the cooling rate can be much enhanced compared with that of single-photon regime, and the enlarged cooling rate can make the ground-state cooling of mirror achievable and the cooled mirror more robust to heating process and thermal noise.

A. Derivation of two-time correlation function with use of the perturbation method

To obtain the analytical expressions of heating and cooling rates A± in Eq. (28), we make use of the perturbation method with respect to Ωp. In the following we denote ρmn(0), ρmn(1) and ρmn(2) to indicate the density matrix elements in the orders of Ωp0, Ωp1 and Ωp2, respectively. To achieve the solutions of the time-dependent density matrix elements we resort to Laplace transform of the evolution equations, which is defined as

ρ(i)(s)=0estρ(i)(t)dt,(i=0,1,2).

A.1. The density matrix in the zeroth order of Ωp

The evolution equation of density matrix elements in zeroth order of Ωp can be obtained from Eq. (30), which is a separate equation

ρ˙g3g3(0)=γρee(0).
Obviously, the steady-state solutions for the density matrix elements are ρg3g3(0)()=1, and the other terms are zero.

A.2. The density matrix in the first order of Ωp

To achieve the correction of density matrix in the first order of Ωp, we substitute the zeroth-order terms into the right hand side (RHS) of Eq. (30), and find out that the evolution equations for the nonzero density matrix elements are given as

ρ˙eg3(1)=[i(δc3+Δ)γ2]ρeg3(1)ijΩj2ρgjg3(1)igρ1g3(1),ρ˙g1g3(1)=i(δc3+Δδ1)ρg1g3(1)iΩ12ρeg3(1),ρ˙g2g3(1)=i(δc3+Δδ2)ρg2g3(1)iΩ22ρeg3(1),ρ˙1g3(1)=(iΔκ2)ρ1g3(1)igρeg3(1)iΩp2ρg3g3(0),
while the other density matrix elements are zero in the steady state. The steady-state solutions for these density matrix can be easily obtained by substituting ρg3g3(0)()=1 into the equations, which are denoted as ρ(1)(∞). The Laplace transform of these density matrix elements are given in the form
ρ(1)(s)=1sL(1)ρ(1)(),
where the Liouvillian operator L(1) describes the evolution of ρ(1) determined by Eq. (49).

A.3. The density matrix in the second order of Ωp

The correction of density matrix in the second-order Ωp can be derived following the same procedure. By substituting the first-order terms into the RHS of the evolution equations in Eq. (30), we obtain a group of nonzero density matrix elements on the second order forming a complete set of evolution equations, which are given as

ρ˙ee(2)=γρee(2)+i(Ω12ρeg1(2)+Ω22ρeg2(2)+gρe1(2))+c.c.,ρ˙gjgj(2)=γjρee(2)iΩj2(ρegj(2)ρgje(2)),ρ˙11(2)=κρ11(2)+i(Ωp2ρ1g3(1)gρe1(2))+c.c.,ρ˙eg1(2)=(iδ1γ2)ρeg1(2)+iΩ12(ρee(2)ρg1g1(2))iΩ22ρg2g1(2)igρ1g1(2),ρ˙eg2(2)=(iδ2γ2)ρeg2(2)+iΩ22(ρee(2)ρg2g2(2))iΩ12ρg1g2(2)igρ1g2(2),ρ˙e1(2)=(iδc3γ+κ2)ρe1(2)+ig(ρee(2)ρ11(2))ijΩj2ρgj1(2)+iΩp2ρeg3(1),ρ˙g1g2(2)=i(δ2δ1)ρg1g2(2)iΩ12ρeg2(2)+iΩ22ρg1e(2),ρ˙g11(2)=[i(δc3δ1)κ2]ρg11(2)iΩ12ρe1(2)+igρg1e(2)+iΩp2ρg1g3(1),ρ˙g21(2)=[i(δc3δ2)κ2]ρg21(2)iΩ22ρe1(2)+igρg2e(2)+iΩp2ρg2g3(1),
It should be noted that the average values of the operators that appear in in Eq. (17) are contained in this group of equations. The steady-state solution of density matrix in the second-order of Ωp can be calculated by substituting the first-order results ρ(1)(∞) into Eq. (51), which are denoted as ρ(2)(∞). For example, ρee(2)() that is connected to mechanical effects on the movable oscillator from atomic spontaneously emitted photon is given as
ρee(2)()=(Ωp/2)2|f(Δ)|2g2(δc3+Δδ1)2(δc3+Δδ2)2,
which is inversely proportional to
f(Δ)=(iκ2+Δ)[(δc3+Δδ1)(δc3+Δδ2)(δc3+Δ+iγ2)(δc3+Δδ2)Ω12/4(δc3+Δδ1)Ω22/4]g2(δc3+Δδ1)(δc3+Δδ2).

The Laplace transform of the second-order density matrix elements ρ(2)(s) is given by

ρ(2)(s)=1sL(2)[ρ(2)()+1sL(1)ρ(1)()],
where L(2) is the Liouvillian operator that describes the evolution of ρ(2) in Eq. (51). The spectral of two-time correlation function S(ν) in Eq. (26) can be calculated from 〈1(s)1(0)〉s by substituting s with −. By applying the quantum regression theorem, 〈1(s)1(0)〉s can be derived from the single-time average V^1(s)s=iηg[ρ1e(2)(s)ρe1(2)(s)]+iηΩp2[ρ1g3(1)(s)ρg31(1)(s)] [42], which is achievable in Eqs. (50) and (54).

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant No. 61275123), the National Basic Research Program of China (Grant No. 2012CB921602), and the Key Laboratory of Advanced Micro-Structure of Tongji University.

References and links

1. D. Kleckner and D. Bouwmeester, “Sub-kelvin optical cooling of a micromechanical resonator,” Nature 444, 75–78 (2006). [CrossRef]   [PubMed]  

2. J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008). [CrossRef]   [PubMed]  

3. C. A. Regal, J. D. Teufel, and K. W. Lehnert, “Measuring nanomechanical motion with a microwave cavity interferometer,” Nat. Phys. 4, 555–560 (2008). [CrossRef]  

4. J. Zhang, Y.-X. Liu, and F. Nori, “Cooling and squeezing the fluctuations of a nanomechanical beam by indirect quantum feedback control,” Phys. Rev. A 79, 052102 (2009). [CrossRef]  

5. Y. Li, L.-A. Wu, and Z. D. Wang, “Fast ground-state cooling of mechanical resonators with time-dependent optical cavities,” Phys. Rev. A 83, 043804 (2011). [CrossRef]  

6. F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007). [CrossRef]   [PubMed]  

7. L. H. Sun, G. X. Li, and Z. Ficek, “First-order coherence versus entanglement in a nanomechanical cavity,” Phys. Rev. A 85, 022327 (2012). [CrossRef]  

8. W. J. Gu and G. X. Li, “Quantum interference effects on ground-state optomechanical cooling,” Phys. Rev. A 87, 025804 (2013). [CrossRef]  

9. J. Teufel, T. Donner, Dale Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011). [CrossRef]   [PubMed]  

10. A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. Mayer Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108, 033602 (2012). [CrossRef]   [PubMed]  

11. S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010). [CrossRef]   [PubMed]  

12. Y.-G. Roh, T. Tanabe, A. Shinya, H. Taniyama, E. Kuramochi, S. Matsuo, T. Sato, and M. Notomi, “Strong optomechanical interaction in a bilayer photonic crystal,” Phys. Rev. B 81, 121101(R) (2010). [CrossRef]  

13. L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “Wavelength-sized GaAs optomechanical resonators with gigahertz frequency,” Appl. Phys. Lett. 98, 113108 (2011). [CrossRef]  

14. K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109, 013603 (2012). [CrossRef]   [PubMed]  

15. M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, “Enhanced quantum nonlinearities in a two-mode optomechanical system,” Phys. Rev. Lett. 109, 063601 (2012). [CrossRef]   [PubMed]  

16. M. A. Lemonde, N. Didier, and A. A. Clerk, “Nonlinear interaction effects in a strongly driven optomechanical cavity,” Phys. Rev. Lett. 111, 053602 (2013). [CrossRef]   [PubMed]  

17. K. Børkje, A. Nunnenkamp, J. D. Teufel, and S. M. Girvin, “Signatures of nonlinear cavity optomechanics in the weak coupling regime,” Phys. Rev. Lett. 111, 053603 (2013). [CrossRef]   [PubMed]  

18. Y. C. Liu, Y. F. Xiao, Y. L. Chen, X. C. Yu, and Q. H. Gong, “Parametric down-conversion and polariton pair generation in optomechanical systems,” Phys. Rev. Lett. 111, 083601 (2013). [CrossRef]   [PubMed]  

19. A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Cooling in the single-photon strong-coupling regime of cavity optomechanics,” Phys. Rev. A 85, 051803(R) (2012). [CrossRef]  

20. I. Wilso-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007). [CrossRef]  

21. J. P. Zhu and G. X. Li, “Ground-state cooling of a nanomechanical resonator with a triple quantum dot via quantum interference,” Phys. Rev. A 86, 053828 (2012). [CrossRef]  

22. S. Huang and G. S. Agarwal, “Enhancement of cavity cooling of a micromechanical mirror using parametric interactions,” Phys. Rev. A 79, 013821 (2009). [CrossRef]  

23. T. Kumar, A. B. Bhattacherjee, and ManMohan, “Dynamics of a movable micromirror in a nonlinear optical cavity,” Phys. Rev. A 81, 013835 (2010). [CrossRef]  

24. S. Rebić, A. S. Parkins, and S. M. Tan, “Photon statistics of a single-atom intracavity system involving electromagnetically induced transparency,” Phys. Rev. A 65, 063804 (2002). [CrossRef]  

25. M. Mücke, E. Figueroa, J. Bochmann, C. Hahn, K. Murr, S. Ritter, C. J. Villas-Boas, and G. Rempe, “Electromagnetically induced transparency with single atoms in a cavity,” Nature 465, 755–758(2010). [CrossRef]   [PubMed]  

26. Y. Wu and X. X. Yang, “Algebraic method for solving a class of coupled-channel cavity QED models,” Phys. Rev. A 63, 043816 (2001). [CrossRef]  

27. S. Zhang, Q. H. Duan, C. Guo, C. W. Wu, W. Wu, and P. X. Chen, “Cavity-assisted cooling of a trapped atom using cavity-induced transparency,” Phys. Rev. A 89, 013402 (2014). [CrossRef]  

28. A. Schliesser and T. J. Kippenberg, “Hybrid atom-optomechanics,” Physics 4, 97 (2011). [CrossRef]  

29. J. Restrepo, C. Ciuti, and I. Favero, “Single-polariton optomechanics,” Phys. Rev. Lett. 112, 013601 (2014). [CrossRef]   [PubMed]  

30. D. Breyer and M. Bienert, “Light scattering in an optomechanical cavity coupled to a single atom,” Phys. Rev. A 86, 053819 (2012). [CrossRef]  

31. M. Bienert and G. Morigi, “Cavity cooling of a trapped atom using electromagnetically induced transparency,” New J. Phys. 14, 023002 (2012). [CrossRef]  

32. A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107, 063602 (2011). [CrossRef]   [PubMed]  

33. Z. Yi, G. X. Li, and Y. P. Yang, “Cavity-mediated cooling of a trapped -type three-level atom using a standing-wave laser field,” Phys. Rev. A 87, 053408 (2013). [CrossRef]  

34. J. I. Cirac, R. Blatt, P. Zoller, and W. D. Phillips, “Laser cooling of trapped ions in a standing wave,” Phys. Rev. A 46, 2668–2681 (1992). [CrossRef]   [PubMed]  

35. J. S. Peng and G. X. Li, Introduction to Modern Quantum Optics (World Scientific) (1998).

36. J. Hofer, A. Schliesser, and T. J. Kippenberg, “Cavity optomechanics with ultrahigh-Q crystalline microresonators,” Phys. Rev. A 82, 031804 (2010). [CrossRef]  

37. K.-K. Ni, R. Norte, D. J. Wilson, J. D. Hood, D. E. Chang, O. Painter, and H. J. Kimble, “Enhancement of mechanical Q factors by optical trapping,” Phys. Rev. Lett. 108, 214302 (2012). [CrossRef]   [PubMed]  

38. C. Genes, H. Ritsch, and D. Vitali, “Micromechanical oscillator ground-state cooling via resonant intracavity optical gain or absorption,” Phys. Rev. A 80, 061803(R) (2009). [CrossRef]  

39. C. Genes, H. Ritsch, M. Drewsen, and A. Dantan, “Atom-membrane cooling and entanglement using cavity electromagnetically induced transparency,” Phys. Rev. A 84, 051801(R) (2011). [CrossRef]  

40. F. Bariani, S. Singh, L. F. Buchmann, M. Vengalattore, and P. Meystre, “Hybrid optomechanical cooling by atomic Λ systems,” arXiv:1407.1073.

41. B. Rogers, N. Lo Gullo, G. De Chiara, G. M. Palma, and M. Paternostro, “Hybrid optomechanics for quantum technologies,” arXiv:1402.1195.

42. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1997). [CrossRef]  

References

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  1. D. Kleckner and D. Bouwmeester, “Sub-kelvin optical cooling of a micromechanical resonator,” Nature 444, 75–78 (2006).
    [Crossref] [PubMed]
  2. J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
    [Crossref] [PubMed]
  3. C. A. Regal, J. D. Teufel, and K. W. Lehnert, “Measuring nanomechanical motion with a microwave cavity interferometer,” Nat. Phys. 4, 555–560 (2008).
    [Crossref]
  4. J. Zhang, Y.-X. Liu, and F. Nori, “Cooling and squeezing the fluctuations of a nanomechanical beam by indirect quantum feedback control,” Phys. Rev. A 79, 052102 (2009).
    [Crossref]
  5. Y. Li, L.-A. Wu, and Z. D. Wang, “Fast ground-state cooling of mechanical resonators with time-dependent optical cavities,” Phys. Rev. A 83, 043804 (2011).
    [Crossref]
  6. F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
    [Crossref] [PubMed]
  7. L. H. Sun, G. X. Li, and Z. Ficek, “First-order coherence versus entanglement in a nanomechanical cavity,” Phys. Rev. A 85, 022327 (2012).
    [Crossref]
  8. W. J. Gu and G. X. Li, “Quantum interference effects on ground-state optomechanical cooling,” Phys. Rev. A 87, 025804 (2013).
    [Crossref]
  9. J. Teufel, T. Donner, Dale Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
    [Crossref] [PubMed]
  10. A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. Mayer Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108, 033602 (2012).
    [Crossref] [PubMed]
  11. S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
    [Crossref] [PubMed]
  12. Y.-G. Roh, T. Tanabe, A. Shinya, H. Taniyama, E. Kuramochi, S. Matsuo, T. Sato, and M. Notomi, “Strong optomechanical interaction in a bilayer photonic crystal,” Phys. Rev. B 81, 121101(R) (2010).
    [Crossref]
  13. L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “Wavelength-sized GaAs optomechanical resonators with gigahertz frequency,” Appl. Phys. Lett. 98, 113108 (2011).
    [Crossref]
  14. K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109, 013603 (2012).
    [Crossref] [PubMed]
  15. M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, “Enhanced quantum nonlinearities in a two-mode optomechanical system,” Phys. Rev. Lett. 109, 063601 (2012).
    [Crossref] [PubMed]
  16. M. A. Lemonde, N. Didier, and A. A. Clerk, “Nonlinear interaction effects in a strongly driven optomechanical cavity,” Phys. Rev. Lett. 111, 053602 (2013).
    [Crossref] [PubMed]
  17. K. Børkje, A. Nunnenkamp, J. D. Teufel, and S. M. Girvin, “Signatures of nonlinear cavity optomechanics in the weak coupling regime,” Phys. Rev. Lett. 111, 053603 (2013).
    [Crossref] [PubMed]
  18. Y. C. Liu, Y. F. Xiao, Y. L. Chen, X. C. Yu, and Q. H. Gong, “Parametric down-conversion and polariton pair generation in optomechanical systems,” Phys. Rev. Lett. 111, 083601 (2013).
    [Crossref] [PubMed]
  19. A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Cooling in the single-photon strong-coupling regime of cavity optomechanics,” Phys. Rev. A 85, 051803(R) (2012).
    [Crossref]
  20. I. Wilso-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
    [Crossref]
  21. J. P. Zhu and G. X. Li, “Ground-state cooling of a nanomechanical resonator with a triple quantum dot via quantum interference,” Phys. Rev. A 86, 053828 (2012).
    [Crossref]
  22. S. Huang and G. S. Agarwal, “Enhancement of cavity cooling of a micromechanical mirror using parametric interactions,” Phys. Rev. A 79, 013821 (2009).
    [Crossref]
  23. T. Kumar, A. B. Bhattacherjee, and ManMohan, “Dynamics of a movable micromirror in a nonlinear optical cavity,” Phys. Rev. A 81, 013835 (2010).
    [Crossref]
  24. S. Rebić, A. S. Parkins, and S. M. Tan, “Photon statistics of a single-atom intracavity system involving electromagnetically induced transparency,” Phys. Rev. A 65, 063804 (2002).
    [Crossref]
  25. M. Mücke, E. Figueroa, J. Bochmann, C. Hahn, K. Murr, S. Ritter, C. J. Villas-Boas, and G. Rempe, “Electromagnetically induced transparency with single atoms in a cavity,” Nature 465, 755–758(2010).
    [Crossref] [PubMed]
  26. Y. Wu and X. X. Yang, “Algebraic method for solving a class of coupled-channel cavity QED models,” Phys. Rev. A 63, 043816 (2001).
    [Crossref]
  27. S. Zhang, Q. H. Duan, C. Guo, C. W. Wu, W. Wu, and P. X. Chen, “Cavity-assisted cooling of a trapped atom using cavity-induced transparency,” Phys. Rev. A 89, 013402 (2014).
    [Crossref]
  28. A. Schliesser and T. J. Kippenberg, “Hybrid atom-optomechanics,” Physics 4, 97 (2011).
    [Crossref]
  29. J. Restrepo, C. Ciuti, and I. Favero, “Single-polariton optomechanics,” Phys. Rev. Lett. 112, 013601 (2014).
    [Crossref] [PubMed]
  30. D. Breyer and M. Bienert, “Light scattering in an optomechanical cavity coupled to a single atom,” Phys. Rev. A 86, 053819 (2012).
    [Crossref]
  31. M. Bienert and G. Morigi, “Cavity cooling of a trapped atom using electromagnetically induced transparency,” New J. Phys. 14, 023002 (2012).
    [Crossref]
  32. A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107, 063602 (2011).
    [Crossref] [PubMed]
  33. Z. Yi, G. X. Li, and Y. P. Yang, “Cavity-mediated cooling of a trapped -type three-level atom using a standing-wave laser field,” Phys. Rev. A 87, 053408 (2013).
    [Crossref]
  34. J. I. Cirac, R. Blatt, P. Zoller, and W. D. Phillips, “Laser cooling of trapped ions in a standing wave,” Phys. Rev. A 46, 2668–2681 (1992).
    [Crossref] [PubMed]
  35. J. S. Peng and G. X. Li, Introduction to Modern Quantum Optics (World Scientific) (1998).
  36. J. Hofer, A. Schliesser, and T. J. Kippenberg, “Cavity optomechanics with ultrahigh-Q crystalline microresonators,” Phys. Rev. A 82, 031804 (2010).
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  37. K.-K. Ni, R. Norte, D. J. Wilson, J. D. Hood, D. E. Chang, O. Painter, and H. J. Kimble, “Enhancement of mechanical Q factors by optical trapping,” Phys. Rev. Lett. 108, 214302 (2012).
    [Crossref] [PubMed]
  38. C. Genes, H. Ritsch, and D. Vitali, “Micromechanical oscillator ground-state cooling via resonant intracavity optical gain or absorption,” Phys. Rev. A 80, 061803(R) (2009).
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  39. C. Genes, H. Ritsch, M. Drewsen, and A. Dantan, “Atom-membrane cooling and entanglement using cavity electromagnetically induced transparency,” Phys. Rev. A 84, 051801(R) (2011).
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  40. F. Bariani, S. Singh, L. F. Buchmann, M. Vengalattore, and P. Meystre, “Hybrid optomechanical cooling by atomic Λ systems,” arXiv:1407.1073.
  41. B. Rogers, N. Lo Gullo, G. De Chiara, G. M. Palma, and M. Paternostro, “Hybrid optomechanics for quantum technologies,” arXiv:1402.1195.
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2014 (2)

J. Restrepo, C. Ciuti, and I. Favero, “Single-polariton optomechanics,” Phys. Rev. Lett. 112, 013601 (2014).
[Crossref] [PubMed]

S. Zhang, Q. H. Duan, C. Guo, C. W. Wu, W. Wu, and P. X. Chen, “Cavity-assisted cooling of a trapped atom using cavity-induced transparency,” Phys. Rev. A 89, 013402 (2014).
[Crossref]

2013 (5)

Z. Yi, G. X. Li, and Y. P. Yang, “Cavity-mediated cooling of a trapped -type three-level atom using a standing-wave laser field,” Phys. Rev. A 87, 053408 (2013).
[Crossref]

W. J. Gu and G. X. Li, “Quantum interference effects on ground-state optomechanical cooling,” Phys. Rev. A 87, 025804 (2013).
[Crossref]

M. A. Lemonde, N. Didier, and A. A. Clerk, “Nonlinear interaction effects in a strongly driven optomechanical cavity,” Phys. Rev. Lett. 111, 053602 (2013).
[Crossref] [PubMed]

K. Børkje, A. Nunnenkamp, J. D. Teufel, and S. M. Girvin, “Signatures of nonlinear cavity optomechanics in the weak coupling regime,” Phys. Rev. Lett. 111, 053603 (2013).
[Crossref] [PubMed]

Y. C. Liu, Y. F. Xiao, Y. L. Chen, X. C. Yu, and Q. H. Gong, “Parametric down-conversion and polariton pair generation in optomechanical systems,” Phys. Rev. Lett. 111, 083601 (2013).
[Crossref] [PubMed]

2012 (9)

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Cooling in the single-photon strong-coupling regime of cavity optomechanics,” Phys. Rev. A 85, 051803(R) (2012).
[Crossref]

A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. Mayer Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108, 033602 (2012).
[Crossref] [PubMed]

L. H. Sun, G. X. Li, and Z. Ficek, “First-order coherence versus entanglement in a nanomechanical cavity,” Phys. Rev. A 85, 022327 (2012).
[Crossref]

D. Breyer and M. Bienert, “Light scattering in an optomechanical cavity coupled to a single atom,” Phys. Rev. A 86, 053819 (2012).
[Crossref]

M. Bienert and G. Morigi, “Cavity cooling of a trapped atom using electromagnetically induced transparency,” New J. Phys. 14, 023002 (2012).
[Crossref]

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109, 013603 (2012).
[Crossref] [PubMed]

M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, “Enhanced quantum nonlinearities in a two-mode optomechanical system,” Phys. Rev. Lett. 109, 063601 (2012).
[Crossref] [PubMed]

J. P. Zhu and G. X. Li, “Ground-state cooling of a nanomechanical resonator with a triple quantum dot via quantum interference,” Phys. Rev. A 86, 053828 (2012).
[Crossref]

K.-K. Ni, R. Norte, D. J. Wilson, J. D. Hood, D. E. Chang, O. Painter, and H. J. Kimble, “Enhancement of mechanical Q factors by optical trapping,” Phys. Rev. Lett. 108, 214302 (2012).
[Crossref] [PubMed]

2011 (6)

C. Genes, H. Ritsch, M. Drewsen, and A. Dantan, “Atom-membrane cooling and entanglement using cavity electromagnetically induced transparency,” Phys. Rev. A 84, 051801(R) (2011).
[Crossref]

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107, 063602 (2011).
[Crossref] [PubMed]

A. Schliesser and T. J. Kippenberg, “Hybrid atom-optomechanics,” Physics 4, 97 (2011).
[Crossref]

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “Wavelength-sized GaAs optomechanical resonators with gigahertz frequency,” Appl. Phys. Lett. 98, 113108 (2011).
[Crossref]

J. Teufel, T. Donner, Dale Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[Crossref] [PubMed]

Y. Li, L.-A. Wu, and Z. D. Wang, “Fast ground-state cooling of mechanical resonators with time-dependent optical cavities,” Phys. Rev. A 83, 043804 (2011).
[Crossref]

2010 (5)

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref] [PubMed]

Y.-G. Roh, T. Tanabe, A. Shinya, H. Taniyama, E. Kuramochi, S. Matsuo, T. Sato, and M. Notomi, “Strong optomechanical interaction in a bilayer photonic crystal,” Phys. Rev. B 81, 121101(R) (2010).
[Crossref]

T. Kumar, A. B. Bhattacherjee, and ManMohan, “Dynamics of a movable micromirror in a nonlinear optical cavity,” Phys. Rev. A 81, 013835 (2010).
[Crossref]

M. Mücke, E. Figueroa, J. Bochmann, C. Hahn, K. Murr, S. Ritter, C. J. Villas-Boas, and G. Rempe, “Electromagnetically induced transparency with single atoms in a cavity,” Nature 465, 755–758(2010).
[Crossref] [PubMed]

J. Hofer, A. Schliesser, and T. J. Kippenberg, “Cavity optomechanics with ultrahigh-Q crystalline microresonators,” Phys. Rev. A 82, 031804 (2010).
[Crossref]

2009 (3)

C. Genes, H. Ritsch, and D. Vitali, “Micromechanical oscillator ground-state cooling via resonant intracavity optical gain or absorption,” Phys. Rev. A 80, 061803(R) (2009).
[Crossref]

S. Huang and G. S. Agarwal, “Enhancement of cavity cooling of a micromechanical mirror using parametric interactions,” Phys. Rev. A 79, 013821 (2009).
[Crossref]

J. Zhang, Y.-X. Liu, and F. Nori, “Cooling and squeezing the fluctuations of a nanomechanical beam by indirect quantum feedback control,” Phys. Rev. A 79, 052102 (2009).
[Crossref]

2008 (2)

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref] [PubMed]

C. A. Regal, J. D. Teufel, and K. W. Lehnert, “Measuring nanomechanical motion with a microwave cavity interferometer,” Nat. Phys. 4, 555–560 (2008).
[Crossref]

2007 (2)

F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
[Crossref] [PubMed]

I. Wilso-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
[Crossref]

2006 (1)

D. Kleckner and D. Bouwmeester, “Sub-kelvin optical cooling of a micromechanical resonator,” Nature 444, 75–78 (2006).
[Crossref] [PubMed]

2002 (1)

S. Rebić, A. S. Parkins, and S. M. Tan, “Photon statistics of a single-atom intracavity system involving electromagnetically induced transparency,” Phys. Rev. A 65, 063804 (2002).
[Crossref]

2001 (1)

Y. Wu and X. X. Yang, “Algebraic method for solving a class of coupled-channel cavity QED models,” Phys. Rev. A 63, 043816 (2001).
[Crossref]

1992 (1)

J. I. Cirac, R. Blatt, P. Zoller, and W. D. Phillips, “Laser cooling of trapped ions in a standing wave,” Phys. Rev. A 46, 2668–2681 (1992).
[Crossref] [PubMed]

Agarwal, G. S.

S. Huang and G. S. Agarwal, “Enhancement of cavity cooling of a micromechanical mirror using parametric interactions,” Phys. Rev. A 79, 013821 (2009).
[Crossref]

Allman, M. S.

J. Teufel, T. Donner, Dale Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[Crossref] [PubMed]

Arcizet, O.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref] [PubMed]

Baker, C.

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “Wavelength-sized GaAs optomechanical resonators with gigahertz frequency,” Appl. Phys. Lett. 98, 113108 (2011).
[Crossref]

Bariani, F.

F. Bariani, S. Singh, L. F. Buchmann, M. Vengalattore, and P. Meystre, “Hybrid optomechanical cooling by atomic Λ systems,” arXiv:1407.1073.

Bennett, S. D.

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109, 013603 (2012).
[Crossref] [PubMed]

Bhattacherjee, A. B.

T. Kumar, A. B. Bhattacherjee, and ManMohan, “Dynamics of a movable micromirror in a nonlinear optical cavity,” Phys. Rev. A 81, 013835 (2010).
[Crossref]

Bienert, M.

D. Breyer and M. Bienert, “Light scattering in an optomechanical cavity coupled to a single atom,” Phys. Rev. A 86, 053819 (2012).
[Crossref]

M. Bienert and G. Morigi, “Cavity cooling of a trapped atom using electromagnetically induced transparency,” New J. Phys. 14, 023002 (2012).
[Crossref]

Blatt, R.

J. I. Cirac, R. Blatt, P. Zoller, and W. D. Phillips, “Laser cooling of trapped ions in a standing wave,” Phys. Rev. A 46, 2668–2681 (1992).
[Crossref] [PubMed]

Bochmann, J.

M. Mücke, E. Figueroa, J. Bochmann, C. Hahn, K. Murr, S. Ritter, C. J. Villas-Boas, and G. Rempe, “Electromagnetically induced transparency with single atoms in a cavity,” Nature 465, 755–758(2010).
[Crossref] [PubMed]

Børkje, K.

K. Børkje, A. Nunnenkamp, J. D. Teufel, and S. M. Girvin, “Signatures of nonlinear cavity optomechanics in the weak coupling regime,” Phys. Rev. Lett. 111, 053603 (2013).
[Crossref] [PubMed]

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Cooling in the single-photon strong-coupling regime of cavity optomechanics,” Phys. Rev. A 85, 051803(R) (2012).
[Crossref]

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107, 063602 (2011).
[Crossref] [PubMed]

Bouwmeester, D.

D. Kleckner and D. Bouwmeester, “Sub-kelvin optical cooling of a micromechanical resonator,” Nature 444, 75–78 (2006).
[Crossref] [PubMed]

Breyer, D.

D. Breyer and M. Bienert, “Light scattering in an optomechanical cavity coupled to a single atom,” Phys. Rev. A 86, 053819 (2012).
[Crossref]

Buchmann, L. F.

F. Bariani, S. Singh, L. F. Buchmann, M. Vengalattore, and P. Meystre, “Hybrid optomechanical cooling by atomic Λ systems,” arXiv:1407.1073.

Chan, J.

A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. Mayer Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108, 033602 (2012).
[Crossref] [PubMed]

Chang, D. E.

K.-K. Ni, R. Norte, D. J. Wilson, J. D. Hood, D. E. Chang, O. Painter, and H. J. Kimble, “Enhancement of mechanical Q factors by optical trapping,” Phys. Rev. Lett. 108, 214302 (2012).
[Crossref] [PubMed]

Chen, J. P.

F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
[Crossref] [PubMed]

Chen, P. X.

S. Zhang, Q. H. Duan, C. Guo, C. W. Wu, W. Wu, and P. X. Chen, “Cavity-assisted cooling of a trapped atom using cavity-induced transparency,” Phys. Rev. A 89, 013402 (2014).
[Crossref]

Chen, Y. L.

Y. C. Liu, Y. F. Xiao, Y. L. Chen, X. C. Yu, and Q. H. Gong, “Parametric down-conversion and polariton pair generation in optomechanical systems,” Phys. Rev. Lett. 111, 083601 (2013).
[Crossref] [PubMed]

Cicak, K.

J. Teufel, T. Donner, Dale Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[Crossref] [PubMed]

Cirac, J. I.

J. I. Cirac, R. Blatt, P. Zoller, and W. D. Phillips, “Laser cooling of trapped ions in a standing wave,” Phys. Rev. A 46, 2668–2681 (1992).
[Crossref] [PubMed]

Ciuti, C.

J. Restrepo, C. Ciuti, and I. Favero, “Single-polariton optomechanics,” Phys. Rev. Lett. 112, 013601 (2014).
[Crossref] [PubMed]

Clerk, A. A.

M. A. Lemonde, N. Didier, and A. A. Clerk, “Nonlinear interaction effects in a strongly driven optomechanical cavity,” Phys. Rev. Lett. 111, 053602 (2013).
[Crossref] [PubMed]

F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
[Crossref] [PubMed]

Dantan, A.

C. Genes, H. Ritsch, M. Drewsen, and A. Dantan, “Atom-membrane cooling and entanglement using cavity electromagnetically induced transparency,” Phys. Rev. A 84, 051801(R) (2011).
[Crossref]

De Chiara, G.

B. Rogers, N. Lo Gullo, G. De Chiara, G. M. Palma, and M. Paternostro, “Hybrid optomechanics for quantum technologies,” arXiv:1402.1195.

Deléglise, S.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref] [PubMed]

Didier, N.

M. A. Lemonde, N. Didier, and A. A. Clerk, “Nonlinear interaction effects in a strongly driven optomechanical cavity,” Phys. Rev. Lett. 111, 053602 (2013).
[Crossref] [PubMed]

Ding, L.

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “Wavelength-sized GaAs optomechanical resonators with gigahertz frequency,” Appl. Phys. Lett. 98, 113108 (2011).
[Crossref]

Donner, T.

J. Teufel, T. Donner, Dale Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[Crossref] [PubMed]

Drewsen, M.

C. Genes, H. Ritsch, M. Drewsen, and A. Dantan, “Atom-membrane cooling and entanglement using cavity electromagnetically induced transparency,” Phys. Rev. A 84, 051801(R) (2011).
[Crossref]

Duan, Q. H.

S. Zhang, Q. H. Duan, C. Guo, C. W. Wu, W. Wu, and P. X. Chen, “Cavity-assisted cooling of a trapped atom using cavity-induced transparency,” Phys. Rev. A 89, 013402 (2014).
[Crossref]

Ducci, S.

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “Wavelength-sized GaAs optomechanical resonators with gigahertz frequency,” Appl. Phys. Lett. 98, 113108 (2011).
[Crossref]

Favero, I.

J. Restrepo, C. Ciuti, and I. Favero, “Single-polariton optomechanics,” Phys. Rev. Lett. 112, 013601 (2014).
[Crossref] [PubMed]

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “Wavelength-sized GaAs optomechanical resonators with gigahertz frequency,” Appl. Phys. Lett. 98, 113108 (2011).
[Crossref]

Ficek, Z.

L. H. Sun, G. X. Li, and Z. Ficek, “First-order coherence versus entanglement in a nanomechanical cavity,” Phys. Rev. A 85, 022327 (2012).
[Crossref]

Figueroa, E.

M. Mücke, E. Figueroa, J. Bochmann, C. Hahn, K. Murr, S. Ritter, C. J. Villas-Boas, and G. Rempe, “Electromagnetically induced transparency with single atoms in a cavity,” Nature 465, 755–758(2010).
[Crossref] [PubMed]

Gavartin, E.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref] [PubMed]

Genes, C.

C. Genes, H. Ritsch, M. Drewsen, and A. Dantan, “Atom-membrane cooling and entanglement using cavity electromagnetically induced transparency,” Phys. Rev. A 84, 051801(R) (2011).
[Crossref]

C. Genes, H. Ritsch, and D. Vitali, “Micromechanical oscillator ground-state cooling via resonant intracavity optical gain or absorption,” Phys. Rev. A 80, 061803(R) (2009).
[Crossref]

Girvin, S. M.

K. Børkje, A. Nunnenkamp, J. D. Teufel, and S. M. Girvin, “Signatures of nonlinear cavity optomechanics in the weak coupling regime,” Phys. Rev. Lett. 111, 053603 (2013).
[Crossref] [PubMed]

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Cooling in the single-photon strong-coupling regime of cavity optomechanics,” Phys. Rev. A 85, 051803(R) (2012).
[Crossref]

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107, 063602 (2011).
[Crossref] [PubMed]

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref] [PubMed]

F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
[Crossref] [PubMed]

Gong, Q. H.

Y. C. Liu, Y. F. Xiao, Y. L. Chen, X. C. Yu, and Q. H. Gong, “Parametric down-conversion and polariton pair generation in optomechanical systems,” Phys. Rev. Lett. 111, 083601 (2013).
[Crossref] [PubMed]

Gu, W. J.

W. J. Gu and G. X. Li, “Quantum interference effects on ground-state optomechanical cooling,” Phys. Rev. A 87, 025804 (2013).
[Crossref]

Guo, C.

S. Zhang, Q. H. Duan, C. Guo, C. W. Wu, W. Wu, and P. X. Chen, “Cavity-assisted cooling of a trapped atom using cavity-induced transparency,” Phys. Rev. A 89, 013402 (2014).
[Crossref]

Habraken, S. J. M.

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109, 013603 (2012).
[Crossref] [PubMed]

Hahn, C.

M. Mücke, E. Figueroa, J. Bochmann, C. Hahn, K. Murr, S. Ritter, C. J. Villas-Boas, and G. Rempe, “Electromagnetically induced transparency with single atoms in a cavity,” Nature 465, 755–758(2010).
[Crossref] [PubMed]

Harlow, J. W.

J. Teufel, T. Donner, Dale Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[Crossref] [PubMed]

Harris, J. G. E.

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref] [PubMed]

Hill, J. T.

A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. Mayer Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108, 033602 (2012).
[Crossref] [PubMed]

Hofer, J.

J. Hofer, A. Schliesser, and T. J. Kippenberg, “Cavity optomechanics with ultrahigh-Q crystalline microresonators,” Phys. Rev. A 82, 031804 (2010).
[Crossref]

Hood, J. D.

K.-K. Ni, R. Norte, D. J. Wilson, J. D. Hood, D. E. Chang, O. Painter, and H. J. Kimble, “Enhancement of mechanical Q factors by optical trapping,” Phys. Rev. Lett. 108, 214302 (2012).
[Crossref] [PubMed]

Huang, S.

S. Huang and G. S. Agarwal, “Enhancement of cavity cooling of a micromechanical mirror using parametric interactions,” Phys. Rev. A 79, 013821 (2009).
[Crossref]

Jayich, A. M.

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref] [PubMed]

Kimble, H. J.

K.-K. Ni, R. Norte, D. J. Wilson, J. D. Hood, D. E. Chang, O. Painter, and H. J. Kimble, “Enhancement of mechanical Q factors by optical trapping,” Phys. Rev. Lett. 108, 214302 (2012).
[Crossref] [PubMed]

Kippenberg, T. J.

A. Schliesser and T. J. Kippenberg, “Hybrid atom-optomechanics,” Physics 4, 97 (2011).
[Crossref]

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref] [PubMed]

J. Hofer, A. Schliesser, and T. J. Kippenberg, “Cavity optomechanics with ultrahigh-Q crystalline microresonators,” Phys. Rev. A 82, 031804 (2010).
[Crossref]

I. Wilso-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
[Crossref]

Kleckner, D.

D. Kleckner and D. Bouwmeester, “Sub-kelvin optical cooling of a micromechanical resonator,” Nature 444, 75–78 (2006).
[Crossref] [PubMed]

Komar, P.

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109, 013603 (2012).
[Crossref] [PubMed]

Krause, A.

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K. Børkje, A. Nunnenkamp, J. D. Teufel, and S. M. Girvin, “Signatures of nonlinear cavity optomechanics in the weak coupling regime,” Phys. Rev. Lett. 111, 053603 (2013).
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J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
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Appl. Phys. Lett. (1)

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “Wavelength-sized GaAs optomechanical resonators with gigahertz frequency,” Appl. Phys. Lett. 98, 113108 (2011).
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Nat. Phys. (1)

C. A. Regal, J. D. Teufel, and K. W. Lehnert, “Measuring nanomechanical motion with a microwave cavity interferometer,” Nat. Phys. 4, 555–560 (2008).
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Nature (4)

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Figures (3)

Fig. 1
Fig. 1 Sketch of the hybrid atom-optomechanical setup, consisting of a cavity coupled to a mechanical oscillator by radiation pressure with an atom in tripod configuration placed inside the cavity.
Fig. 2
Fig. 2 Sketch of the suppression of heating processes. The phonon number states are |n〉 in (a) and |n + 1〉 in (b). Through the completely quantum destructive interference between excitation paths |g3, n〉 → |1, n〉 → |e, n〉 and |g1, n〉 → |e, n〉 shown in (a) by the green lines, the carrier transition into |e, n〉 is prohibited. By use of the completely quantum destructive interferences between excitation paths |g2, n + 1〉 → |e, n + 1〉 and |g3, n〉 → |1, n〉 → |e, n + 1〉 shown in (b) by the solid blue lines, and between excitation paths |g2, n + 1〉 → |e, n + 1〉 and |g3, n〉 → |1, n + 1〉 → |e, n + 1〉 shown in (b) by the dashed blue lines, the phonon-increased transitions are significantly suppressed.
Fig. 3
Fig. 3 Steady-state phonon number 〈st and cooling rate W/η2 as functions of detunings δ1 and δ2 with parameters in units of ν: Ω1 = Ω2 = 6ν, κ = 0.1ν, Δ = −2ν, γ = 10ν, Ωp = ν, g = 6ν, δc3 = δ1 − Δ.

Equations (54)

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H ^ = H ^ osc + H ^ at + H ^ cav + H ^ cav-at + H ^ L-at + H ^ L-cav + F ^ x ^ ,
H ^ osc = ν b ^ b ^
H ^ at = δ c 3 | e e | + j ( δ j δ c 3 ) | g j g j | + Δ | g 3 g 3 | , H ^ cav = Δ a ^ a ^ , H ^ cav-at = g ( | e g 3 | a ^ + | g 3 e | a ^ ) , H ^ L-at = j Ω j 2 ( | e g j | + | g j e | ) , H ^ L-cav = Ω p 2 ( a ^ + a ^ )
x ^ = ξ ( b ^ + b ^ ) , p ^ = 1 2 i ξ ( b ^ b ^ ) ,
F ^ x ^ = χ a ^ a ^ ( b ^ + b ^ ) ,
d d t ρ ^ = i [ H ^ , ρ ^ ] + ^ ρ ^ + 𝒦 ^ ρ ^ + ^ m ρ ^ ,
^ ρ ^ = i γ i 2 ( 2 | g i e | ρ ^ | e g i | | e e | ρ ^ ρ ^ | e e | ) , 𝒦 ^ ρ ^ = κ 2 ( 2 a ^ ρ ^ a ^ a ^ a ^ ρ ^ ρ ^ a ^ a ^ ) ,
^ m ρ ^ = γ m 2 ( n ¯ th + 1 ) ( 2 b ^ ρ ^ b ^ b ^ b ^ ρ ^ ρ ^ b ^ b ^ ) + γ m 2 n ¯ th ( 2 b ^ ρ ^ b ^ b ^ b ^ ρ ^ ρ ^ b ^ b ^ ) ,
H ^ osc + F ^ x ^ = ν b ^ b ^ η χ ( a ^ a ^ ) 2 ,
H ^ cav-at = g [ | e g 3 | a ^ e η ( b ^ b ^ ) + | g 3 e | a ^ e η ( b ^ b ^ ) ] , H ^ L-cav = Ω p 2 [ a ^ e η ( b ^ b ^ ) + a ^ e η ( b ^ b ^ ) ] .
𝒦 ^ ρ ^ = κ 2 [ 2 a ^ e η ( b ^ b ^ ) ρ ^ a ^ e η ( b ^ b ^ ) a ^ a ^ ρ ^ ρ ^ a ^ a ^ ] .
{ | e , 0 , g 1 , 0 | , | g 2 , 0 , g 3 , 0 | , | g 3 , 1 } .
H ^ osc + F ^ x ^ = ν b ^ b ^ η χ | 1 1 | , H ^ cav-at = g [ | e 1 | e η ( b ^ b ^ ) + | 1 e | e η ( b ^ b ^ ) ] , H ^ L-cav = Ω p 2 [ | g 3 1 | e η ( b ^ b ^ ) + | 1 g 3 | e η ( b ^ b ^ ) ] ,
𝒦 ^ ρ ^ = κ 2 [ 2 | g 3 1 | e η ( b ^ b ^ ) ρ ^ | 1 g 3 | e η ( b ^ b ^ ) | 1 1 | ρ ^ ρ ^ | 1 1 | ] .
e η ( b ^ b ^ ) = 1 + η ( b ^ b ^ ) + O ( η 2 ) .
H ^ free a-c = δ c 3 | e e | + j ( δ j δ c 3 ) | g j g j | + Δ | g 3 g 3 | + j Ω j 2 ( | e g j | + | g j e | ) + g ( | e 1 | + | 1 e | ) + Ω p 2 ( | g 3 1 | + | 1 g 3 | ) ,
V ^ 1 = i η [ g | e 1 | + Ω p 2 | g 3 1 | ] + H . c . .
d d t ρ ^ = ( 𝒮 ^ 0 + 𝒮 ^ 1 + 𝒮 ^ 2 ) ρ ^ .
𝒮 ^ 0 ρ ^ = i [ H ^ free a-c , ρ ^ ] + ^ ρ ^ + 𝒦 ^ 0 ρ ^ ,
𝒦 ^ 0 ρ ^ = κ 2 ( 2 | g 3 1 | ρ ^ | 1 g 3 | | 1 1 | ρ ^ ρ ^ | 1 1 | ) .
𝒮 ^ 2 ρ ^ = i [ H ^ osc , ρ ^ ] + ^ m ρ ^ .
𝒮 ^ 1 ρ ^ = i [ H ^ 1 , ρ ^ ] + 𝒦 ^ 2 ρ ^ ,
𝒦 ^ 2 ρ ^ = κ | g 3 1 | [ η ( b ^ b ^ ) ρ ^ + η 2 2 ( b ^ ρ ^ b ^ + b ^ ρ ^ b ^ b ^ b ^ ρ ^ b ^ b ^ ρ ^ ) + H . c . ] | 1 g 3 | .
d d t μ ^ = Tr a-c { [ 𝒫 ^ 𝒮 ^ 2 𝒫 ^ + 𝒫 ^ 𝒮 ^ 1 ( 𝒮 ^ 0 ) 1 𝒮 ^ 1 𝒫 ^ ] ρ ^ } ,
d d t μ ^ = S ( ν ) ( b ^ μ ^ b ^ b ^ b ^ μ ^ ) + S ( ν ) ( b ^ μ ^ b ^ b ^ b ^ μ ^ ) + H . c . + ^ m μ ^ ,
S ( ν ) = 0 d t e i ν t V ^ 1 ( t ) V ^ 1 ( 0 ) st .
n ^ ˙ = ( A A + + γ m ) n ^ + A + + γ m n ¯ th ,
A ± = 2 Re { S ( ν ) }
n ^ st = A + + γ m n ¯ th A A + + γ m , W = A A + + γ m .
ρ ˙ e e = γ ρ e e + i ( Ω 1 2 ρ e g 1 + Ω 2 2 ρ e g 2 + g ρ e 1 ) + c . c . , ρ ˙ g j g j = γ j ρ e e i Ω j 2 ( ρ e g j ρ g j e ) , ρ ˙ 11 = κ ρ 11 + i ( Ω p 2 ρ 1 g 3 g ρ e 1 ) + c . c . , ρ ˙ e g 1 = ( i δ 1 γ 2 ) ρ e g 1 + i Ω 1 2 ( ρ e e ρ g 1 g 1 ) i Ω 2 2 ρ g 2 g 1 i g ρ 1 g 1 , ρ ˙ e g 2 = ( i δ 2 γ 2 ) ρ e g 2 + i Ω 2 2 ( ρ e e ρ g 2 g 2 ) i Ω 1 2 ρ g 1 g 2 i g ρ 1 g 2 , ρ ˙ e 1 = ( i δ c 3 γ + κ 2 ) ρ e 1 + i g ( ρ e e ρ 11 ) i j Ω j 2 ρ g j 1 + i Ω p 2 ρ e g 3 , ρ ˙ g 1 g 2 = i ( δ 2 δ 1 ) ρ g 1 g 2 i Ω 1 2 ρ e g 2 + i Ω 2 2 ρ g 1 e , ρ ˙ g 1 1 = [ i ( δ c 3 δ 1 ) κ 2 ] ρ g 1 1 i Ω 1 2 ρ e 1 + i g ρ g 1 e + i Ω p 2 ρ g 1 g 3 , ρ ˙ g 2 1 = [ i ( δ c 3 δ 2 ) κ 2 ] ρ g 2 1 i Ω 2 2 ρ e 1 + i g ρ g 2 e + i Ω p 2 ρ g 2 g 3 , ρ ˙ e g 3 = [ i ( δ c 3 + Δ ) γ 2 ] ρ e g 3 i j Ω j 2 ρ g j g 3 i g ρ 1 g 3 + i Ω p 2 ρ e 1 , ρ ˙ g 1 g 3 = i ( δ c 3 + Δ δ 1 ) ρ g 1 g 3 i Ω 1 2 ρ e g 3 + i Ω p 2 ρ g 1 1 , ρ ˙ g 2 g 3 = i ( δ c 3 + Δ δ 2 ) ρ g 2 g 3 i Ω 2 2 ρ e g 3 + i Ω p 2 ρ g 2 1 , ρ ˙ 1 g 3 = ( i Δ κ 2 ) ρ 1 g 3 i g ρ e g 3 i Ω p 2 ( ρ g 3 g 3 ρ 11 ) .
A ± / η 2 = γ | 𝒯 2 ± | 2 + κ | g 𝒯 2 ± + g 𝒯 1 + Ω p / 2 Δ ν + i κ / 2 | 2 ,
𝒯 1 = g Ω p / 2 f ( Δ ) ( δ c 3 + Δ δ 1 ) ( δ c 3 + Δ δ 2 ) , 𝒯 2 ± = ± ν g Ω p / 2 f ( Δ ) f ( Δ ν ) γ ( Δ ) ( δ c 3 + Δ ν δ 1 ) ( δ c 3 + Δ ν δ 2 ) ,
f ( Δ ) = ( Δ + i κ / 2 ) γ ( Δ ) g 2 ( δ c 3 + Δ δ 1 ) ( δ c 3 + Δ δ 2 ) , γ ( Δ ) = ( δ c 3 + Δ δ 1 ) ( δ c 3 + Δ δ 2 ) ( δ c 3 + Δ + i γ / 2 ) ( Ω 1 / 2 ) 2 ( δ c 3 + Δ δ 2 ) ( Ω 2 / 2 ) 2 ( δ c 3 + Δ δ 1 ) .
δ c 3 + Δ δ 1 = 0 or δ c 3 + Δ δ 2 = 0 .
δ c 3 + Δ ν δ 2 = 0 ,
A + / η 2 = κ ( Ω p / 2 ) 2 ( Δ ν ) 2 + ( κ / 2 ) 2 ,
𝒯 2 = ν g ε ( Δ + ν + i κ 2 ) ( i γ 2 + δ 1 + ν Ω 1 2 4 ν Ω 2 2 8 ν ) g 2 .
δ 1 = Ω 1 2 4 ν + Ω 2 2 8 ν + g 2 + κ γ / 4 Δ + ν ν ,
𝒯 2 = i ν g ε κ 2 g 2 + κ γ / 4 Δ + ν + γ 2 ( Δ + ν ) ,
A / η 2 = κ ( Ω p / 2 ) 2 + g 2 | 𝒯 2 | 2 ( Δ + ν ) 2 + ( κ / 2 ) 2 + γ | 𝒯 2 | 2 .
A / η 2 4 g 2 | ε | 2 / γ , A + / η 2 4 κ | ε | 2 / 9 ,
n ^ A + / A 4 9 C ,
A ± / η 2 = κ ( Ω p / 2 ) 2 ( Δ ν ) 2 + ( κ / 2 ) 2 ,
𝒯 1 = g Ω p / 2 D ( Δ ) , 𝒯 2 ± = ± ν g Ω p / 2 ( δ c 3 + Δ + i γ / 2 ) D ( Δ ) D ( Δ ν ) ,
D ( Δ ) = ( Δ + i κ / 2 ) ( δ c 3 + Δ + i γ / 2 ) g 2 .
A ± / η 2 = | Ω p / 2 ( δ c 3 + Δ + i γ / 2 ) D ( Δ ) D ( Δ ν ) | 2 [ γ ν 2 g 2 + κ | D ( Δ ) ν ( Δ + i κ / 2 ) | 2 ] ,
ρ ( i ) ( s ) = 0 e s t ρ ( i ) ( t ) d t , ( i = 0 , 1 , 2 ) .
ρ ˙ g 3 g 3 ( 0 ) = γ ρ e e ( 0 ) .
ρ ˙ e g 3 ( 1 ) = [ i ( δ c 3 + Δ ) γ 2 ] ρ e g 3 ( 1 ) i j Ω j 2 ρ g j g 3 ( 1 ) i g ρ 1 g 3 ( 1 ) , ρ ˙ g 1 g 3 ( 1 ) = i ( δ c 3 + Δ δ 1 ) ρ g 1 g 3 ( 1 ) i Ω 1 2 ρ e g 3 ( 1 ) , ρ ˙ g 2 g 3 ( 1 ) = i ( δ c 3 + Δ δ 2 ) ρ g 2 g 3 ( 1 ) i Ω 2 2 ρ e g 3 ( 1 ) , ρ ˙ 1 g 3 ( 1 ) = ( i Δ κ 2 ) ρ 1 g 3 ( 1 ) i g ρ e g 3 ( 1 ) i Ω p 2 ρ g 3 g 3 ( 0 ) ,
ρ ( 1 ) ( s ) = 1 s L ( 1 ) ρ ( 1 ) ( ) ,
ρ ˙ e e ( 2 ) = γ ρ e e ( 2 ) + i ( Ω 1 2 ρ e g 1 ( 2 ) + Ω 2 2 ρ e g 2 ( 2 ) + g ρ e 1 ( 2 ) ) + c . c . , ρ ˙ g j g j ( 2 ) = γ j ρ e e ( 2 ) i Ω j 2 ( ρ e g j ( 2 ) ρ g j e ( 2 ) ) , ρ ˙ 11 ( 2 ) = κ ρ 11 ( 2 ) + i ( Ω p 2 ρ 1 g 3 ( 1 ) g ρ e 1 ( 2 ) ) + c . c . , ρ ˙ e g 1 ( 2 ) = ( i δ 1 γ 2 ) ρ e g 1 ( 2 ) + i Ω 1 2 ( ρ e e ( 2 ) ρ g 1 g 1 ( 2 ) ) i Ω 2 2 ρ g 2 g 1 ( 2 ) i g ρ 1 g 1 ( 2 ) , ρ ˙ e g 2 ( 2 ) = ( i δ 2 γ 2 ) ρ e g 2 ( 2 ) + i Ω 2 2 ( ρ e e ( 2 ) ρ g 2 g 2 ( 2 ) ) i Ω 1 2 ρ g 1 g 2 ( 2 ) i g ρ 1 g 2 ( 2 ) , ρ ˙ e 1 ( 2 ) = ( i δ c 3 γ + κ 2 ) ρ e 1 ( 2 ) + i g ( ρ e e ( 2 ) ρ 11 ( 2 ) ) i j Ω j 2 ρ g j 1 ( 2 ) + i Ω p 2 ρ e g 3 ( 1 ) , ρ ˙ g 1 g 2 ( 2 ) = i ( δ 2 δ 1 ) ρ g 1 g 2 ( 2 ) i Ω 1 2 ρ e g 2 ( 2 ) + i Ω 2 2 ρ g 1 e ( 2 ) , ρ ˙ g 1 1 ( 2 ) = [ i ( δ c 3 δ 1 ) κ 2 ] ρ g 1 1 ( 2 ) i Ω 1 2 ρ e 1 ( 2 ) + i g ρ g 1 e ( 2 ) + i Ω p 2 ρ g 1 g 3 ( 1 ) , ρ ˙ g 2 1 ( 2 ) = [ i ( δ c 3 δ 2 ) κ 2 ] ρ g 2 1 ( 2 ) i Ω 2 2 ρ e 1 ( 2 ) + i g ρ g 2 e ( 2 ) + i Ω p 2 ρ g 2 g 3 ( 1 ) ,
ρ e e ( 2 ) ( ) = ( Ω p / 2 ) 2 | f ( Δ ) | 2 g 2 ( δ c 3 + Δ δ 1 ) 2 ( δ c 3 + Δ δ 2 ) 2 ,
f ( Δ ) = ( i κ 2 + Δ ) [ ( δ c 3 + Δ δ 1 ) ( δ c 3 + Δ δ 2 ) ( δ c 3 + Δ + i γ 2 ) ( δ c 3 + Δ δ 2 ) Ω 1 2 / 4 ( δ c 3 + Δ δ 1 ) Ω 2 2 / 4 ] g 2 ( δ c 3 + Δ δ 1 ) ( δ c 3 + Δ δ 2 ) .
ρ ( 2 ) ( s ) = 1 s L ( 2 ) [ ρ ( 2 ) ( ) + 1 s L ( 1 ) ρ ( 1 ) ( ) ] ,

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