Ground-state cooling of an oscillator in a hybrid atom-optomechanical system.

Ground-state cooling of an oscillator in a hybrid atom-optomechanical system Zhen Yi,1 Gao-xiang Li,1,∗ Shao-ping Wu,1 and Ya-ping Yang2 1 Department

of Physics, Huazhong Normal University, Wuhan 430079, China of Physics, Tongji University, Shanghai 200092, China

2 Department

∗ [email protected]

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 LambDicke 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 OCIS codes: (220.4880) Optomechanics; (140.3320) Laser cooling.

References and links 1. D. Kleckner and D. Bouwmeester, “Sub-kelvin optical cooling of a micromechanical resonator,” Nature 444, 75–78 (2006). 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). 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). 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). 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). 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). 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). 8. W. J. Gu and G. X. Li, “Quantum interference effects on ground-state optomechanical cooling,” Phys. Rev. A 87, 025804 (2013). 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). 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).

#212864 - $15.00 USD Received 26 May 2014; revised 12 Jul 2014; accepted 29 Jul 2014; published 12 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020060 | OPTICS EXPRESS 20060

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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 22. S. Huang and G. S. Agarwal, “Enhancement of cavity cooling of a micromechanical mirror using parametric interactions,” Phys. Rev. A 79, 013821 (2009). 23. T. Kumar, A. B. Bhattacherjee, and ManMohan, “Dynamics of a movable micromirror in a nonlinear optical cavity,” Phys. Rev. A 81, 013835 (2010). 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). 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). 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). 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). 28. A. Schliesser and T. J. Kippenberg, “Hybrid atom-optomechanics,” Physics 4, 97 (2011). 29. J. Restrepo, C. Ciuti, and I. Favero, “Single-polariton optomechanics,” Phys. Rev. Lett. 112, 013601 (2014). 30. D. Breyer and M. Bienert, “Light scattering in an optomechanical cavity coupled to a single atom,” Phys. Rev. A 86, 053819 (2012). 31. M. Bienert and G. Morigi, “Cavity cooling of a trapped atom using electromagnetically induced transparency,” New J. Phys. 14, 023002 (2012). 32. A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107, 063602 (2011). 33. Z. Yi, G. X. Li, and Y. P. Yang, “Cavity-mediated cooling of a trapped -type three-level atom using a standingwave laser field,” Phys. Rev. A 87, 053408 (2013). 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). 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). 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). 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). 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). 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).


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 [6–8]. Recently, with the rapid technical advancement, there are several experiments [9–13], 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 zeropoint 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 [16–18]. 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 [24–27]. 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 twolevel (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 Ω p , 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 quantummechanical 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 |ei and three ground states |gi i (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 |ei ↔ g j with Rabi frequencies Ω j , respectively, and the third transition |g3 i ↔ |ei 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 ω p and the driving strength is Ω p . The atom and cavity excitations decay into the vacuum reservoir with rates γ and κ, respectively.

 p

 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.

Thus the system can be described by the Hamiltonian, Hˆ = Hˆ osc + Hˆ at + Hˆ cav + Hˆ cav-at + Hˆ L-at + Hˆ L-cav + Fˆ x, ˆ

(1)


which is written in a rotating frame of the lasers’ frequency. The Hamiltonian of free movable mirror Hˆ osc = ν bˆ † bˆ (2) accounts for the energy of the mechanical oscillator with frequency ν and mass m, and the Hamiltonians Hˆ at = −δc3 |ei he| + ∑ j (δ j − δc3 ) g j g j + ∆ |g3 i hg3 | , Hˆ cav = −∆aˆ† a, ˆ Hˆ cav-at = g(|ei hg3 | aˆ + |g3 i he| aˆ† ),

Ωj Hˆ L-at = ∑ j (|ei g j + g j he|), 2 Ω p Hˆ L-cav = (aˆ + aˆ† ) 2

(3)

describe the energies of the internal atomic states, the cavity field, the coupling interaction between the cavity mode and the atomic transition |ei ↔ |g3 i, the drive of laser fields on the transitions |ei ↔ g j and the coherent driving of the cavity by the weak pump laser, respectively. Here δc3 = ωc − (ωe − ωg3 ) is the detuning of the cavity frequency ωc to the atomic transition |ei ↔ |g3 i, δ j = ωL j − (ωe − ωg j ) indicates the detuning of the j-th laser frequency ωL j to the corresponding atomic dipole transition |ei ↔ |g j i, and ∆ = ω p − ωc describes the detuning between the cavity and the probe fields. The operator aˆ is the photon annihilation operator for the cavity mode, and the annihilation operator bˆ is connected with the oscillator’s position and momentum operators, xˆ = ξ (bˆ + bˆ † ),

pˆ =

1 ˆ ˆ† (b − b ), 2iξ

(4)

p 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 xˆ and it is represented by the term Fˆ x, ˆ which has the form Fˆ xˆ = −χ aˆ† a( ˆ bˆ + bˆ † ),

(5)

and introduces an interaction between the cavity and the mechanical oscillator with the coupling strength characterized by χ. The time-dependent density operator ρˆ of this system obeys the master equation d ˆ ρˆ + Lˆ ρˆ + Kˆ ρˆ + Lˆm ρ, ˆ ρˆ = −i H, dt

(6)

where the superoperators Lˆ ρˆ and Kˆ ρˆ written as γi Lˆ ρˆ = ∑i (2 |gi i he| ρˆ |ei hgi | − |ei he| ρˆ − ρˆ |ei he|) , 2 κ ˆ K ρˆ = 2aˆρˆ aˆ† − aˆ† aˆρˆ − ρˆ aˆ† aˆ , 2

(7)

describe the dipole spontaneous emission and cavity decay, and Lˆm ρˆ describes the damping of the movable mirror coupled to thermal bath γm ˆ + γm n¯ th (2bˆ † ρˆ bˆ − bˆ bˆ † ρˆ − ρˆ bˆ bˆ † ), Lˆm ρˆ = (n¯ th + 1)(2bˆ ρˆ bˆ † − bˆ † bˆ ρˆ − ρˆ bˆ † b) 2 2

(8)

with n¯ th the thermal mean phonon number. #212864 - $15.00 USD Received 26 May 2014; revised 12 Jul 2014; accepted 29 Jul 2014; published 12 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020060 | OPTICS EXPRESS 20064

3.

Derivation of the master equation for the movable mirror

For the single-photon optomechanics, we make use of the typical unitary transformation given ˆ and η = χ/ν [32]. The sum of two Hamiltonians Hˆ osc ˆ with Sˆ = η aˆ† a( by Uˆ = exp(−S) ˆ bˆ † − b) ˆ and F xˆ will be diagonalized in the form 2 Hˆ osc + Fˆ xˆ = ν bˆ † bˆ − η χ aˆ† aˆ , while the Hamiltonians Hˆ cav-at and Hˆ L-cav become h i ˆ† ˆ ˆ ˆ† Hˆ cav-at = g |ei hg3 | ae ˆ η(b −b) + |g3 i he| aˆ† eη(b−b ) , i Ω p h η(bˆ † −b) ˆ ˆ ˆ† Hˆ L-cav = ae ˆ + aˆ† eη(b−b ) . 2

(9)

(10)

The other Hamiltonians Hˆ at , Hˆ cav and Hˆ L-at in Eq. (1) remain unchanged. Meanwhile, the superoperator Kˆ ρˆ describing cavity decay becomes i κ h η(bˆ † −b) ˆ ˆ ˆ† Kˆ ρˆ = 2ae ˆ ρˆ aˆ† eη(b−b ) − aˆ† aˆρˆ − ρˆ aˆ† aˆ . (11) 2 For the transformation of the term Lˆm ρˆ describing the mechanical damping, we need to substitute the operator bˆ and bˆ † in Eq. (8) by (bˆ + η aˆ† a) ˆ and (bˆ † + η aˆ† a), ˆ respectively. Because the 2 conditions ηγm n¯ th ωm and η γm n¯ th κ are satisfied, it is appropriate to get rid of the offresonate terms, i.e. the first-order terms of η, and the second-order terms of η, which is much ˆ Therefore, the form of Lˆm ρˆ in Eq. (8) keeps unchanged smaller than the other terms of Lˆm ρ. in the later discussion. Because the cavity is sufficiently weakly driven, which means that the average photon num Ω p /2 2 ber of the cavity mode |ε|2 ≡ ∆+iκ/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, 0i , |g1 , 0i , |g2 , 0i , |g3 , 0i , |g3 , 1i} .

(12)

We write |e, 0i ≡ |ei, |g1 , 0i ≡ |g1 i, |g2 , 0i ≡ |g2 i, |g3 , 0i ≡ |g3 i and |g3 , 1i ≡ |1i for later convenience. Within the subspace, the Hamiltonians in Eqs. (9) and (10) and the superoperator in Eq. (11) correspondingly become Hˆ osc + Fˆ xˆ = ν bˆ † bˆ − η χ |1i h1| , h i ˆ† ˆ ˆ ˆ† Hˆ cav-at = g |ei h1| eη(b −b) + |1i he| eη(b−b ) , i Ωp h ˆ† ˆ ˆ ˆ† |g3 i h1| eη(b −b) + |1i hg3 | eη(b−b ) , Hˆ L-cav = 2

(13)

and κ ˆ† ˆ ˆ ˆ† Kˆ ρˆ = 2 |g3 i h1| eη(b −b) ρˆ |1i hg3 | eη(b−b ) − |1i h1| ρˆ − ρˆ |1i h1| . 2

(14)

The form of term Hˆ L-at is unchanged and the form for the sum of Hˆ at and Hˆ cav turns to be the same as Hˆ at in Eq. (3). The term Lˆ ρˆ describing the dipole spontaneous emission and the term Lˆm ρˆ 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, ˆ ˆ†

eη(b−b ) = 1 + η(bˆ − bˆ † ) + O(η 2 ).

(15)

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 firstorder Hamiltonians in η, i.e., Hˆ = Hˆ 0 + Hˆ 1 . In the rotating frame of η χ, with the redefined parameters δc3 → δc3 − η χ, ∆ → ∆ + η χ, the Hamiltonian in zeroth-order of η contains the a-c , which is given by free evolution of the movable mirror Hˆ osc and the atom-cavity system Hˆ free a-c Hˆ free = − δc3 |ei he| + ∑ j (δ j − δc3 ) g j g j + ∆ |g3 i hg3 |

Ωj Ωp +∑j (|ei g j + g j he|) + g(|ei h1| + |1i he|) + (|g3 i h1| + |1i hg3 |), (16) 2 2 and the Hamiltonian in first-order of η that represents the interaction between the movable mirror and atom-cavity system is Hˆ 1 = iVˆ1 (bˆ − bˆ † ), with Ωp |g3 i h1| + H.c.. Vˆ1 = iη g |ei h1| + (17) 2 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 Sˆ0 , Sˆ2 and Sˆ1 respectively. Thus the master equation is written in the form d ˆ ρˆ = Sˆ0 + Sˆ1 + Sˆ2 ρ. dt

(18)

The superoperator Sˆ0 ρˆ for the free atom-cavity system is given by a-c ˆ ˆ Sˆ0 ρˆ = −i[Hˆ free , ρ] + Lˆ ρˆ + Kˆ0 ρ,

(19)

κ Kˆ0 ρˆ = (2 |g3 i h1| ρˆ |1i hg3 | − |1i h1| ρˆ − ρˆ |1i h1|). 2

(20)

with

The superoperator Sˆ2 ρˆ for the free movable mirror is given by ˆ Sˆ2 ρˆ = −i Hˆ osc , ρˆ + Lˆm ρ.

(21)

The interaction between the atom-cavity system and the movable mirror is given by ˆ Sˆ1 ρˆ = −i Hˆ 1 , ρˆ + Kˆ2 ρ,

(22)

where Kˆ2 ρˆ is the diffusion caused when one expands the superoperator (14) describing the cavity decay, and given in the form i h 2 ˆ ρˆ + η (bˆ ρˆ bˆ † + bˆ † ρˆ bˆ − bˆ bˆ † ρˆ − bˆ † bˆ ρ) ˆ + H.c. |1i hg3 | . Kˆ2 ρˆ = κ |g3 i h1| η(bˆ † − b) 2

(23)


Here the operator bˆ 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 d µˆ = Tra-c Pˆ Sˆ2 Pˆ + Pˆ Sˆ1 (−Sˆ0 )−1 Sˆ1 Pˆ ρˆ , dt

(24)

where µˆ is the reduced density operator for the movable mirror and Pˆ is the projection operator on the subspace with zero eigenvalue of Sˆ0 [34]. After some direct calculations we obtain the explicit master equation of movable mirror in the rotating frame of oscillating frequency ν d ˆ + S(−ν)(bˆ † µˆ bˆ − bˆ bˆ † µ) ˆ + H.c. + Lˆm µ, ˆ µˆ = S(ν)(bˆ µˆ bˆ † − bˆ † bˆ µ) dt

(25)

where the spectral of two-time correlation function S(ν) of atomic and cavity operators is expressed as Z ∞

S(ν) = 0

dteiνt hVˆ1 (t)Vˆ1 (0)ist .

(26)

Here the transition operators σmn = |mi hn| are expressed in terms of a complete set of states {|mi} = {|ei , |g1 i , |g2 i , |g3 i , |1i}. Then we can directly derive the rate equation for the mean phonon number hni, ˆ namely ˙ˆ = −(A− − A+ + γm )hni hni ˆ + A+ + γm n¯ th ,

(27)

where A± = 2Re{S(∓ν)}

(28)

are the heating (A+ ) and cooling (A− ) coefficients. Then the steady-state mean phonon number hni ˆ st and the cooling rate W can be easily obtained as hni ˆ st =

A+ + γm n¯ th , A− − A+ + γm

W = A− − A+ + γm .

(29)

We assume that there only exists the spontaneous decay along the transition path |ei → |g3 i, 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 Ω p

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 hσmn (t)σkl (0)ist obey the same equations as ρnm (t), which are governed by the superoperator (19), and together with the initial values hσmn (0)σkl (0)ist = δnk ρlm (∞), 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 Ω2 Ω1 ρeg1 + ρeg2 + gρe1 + c.c., ρ˙ ee = −γρee + i 2 2 Ωj ρ˙ g j g j = γ j ρee − i (ρeg j − ρg j e ), 2 Ωp ρ1g3 − gρe1 + c.c., ρ˙ 11 = −κρ11 + i 2 γ Ω1 Ω2 ρ˙ eg1 = iδ1 − ρeg1 + i (ρee − ρg1 g1 ) − i ρg2 g1 − igρ1g1 , 2 2 2 Ω2 γ Ω1 ρeg2 + i (ρee − ρg2 g2 ) − i ρg1 g2 − igρ1g2 , ρ˙ eg2 = iδ2 − 2 2 2 Ωj Ωp γ +κ ρ˙ e1 = iδc3 − ρe1 + ig(ρee − ρ11 ) − i ∑ j ρg 1 + i ρeg3 , 2 2 j 2 Ω1 Ω2 ρ˙ g1 g2 = i(δ2 − δ1 )ρg1 g2 − i ρeg2 + i ρg1 e , 2 2 h Ωp κi Ω1 ρ˙ g1 1 = i(δc3 − δ1 ) − ρg1 1 − i ρe1 + igρg1 e + i ρg1 g3 , 2 2 2 h Ωp Ω2 κi ρg2 1 − i ρe1 + igρg2 e + i ρg2 g3 , ρ˙ g2 1 = i(δc3 − δ2 ) − 2 2 2 i h Ωj Ωp γ ρg g − igρ1g3 + i ρe1 , ρ˙ eg3 = i(δc3 + ∆) − ρeg3 − i ∑ j 2 2 j 3 2 Ωp Ω1 ρ˙ g1 g3 = i(δc3 + ∆ − δ1 )ρg1 g3 − i ρeg3 + i ρg1 1 , 2 2 Ωp Ω2 ρ˙ g2 g3 = i(δc3 + ∆ − δ2 )ρg2 g3 − i ρeg3 + i ρg2 1 , 2 2 Ωp κ ρ1g3 − igρeg3 − i (ρg3 g3 − ρ11 ). ρ˙ 1g3 = i∆ − (30) 2 2 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 Ω p , it offers the opportunity to achieve the analytical expressions by employing the second-order perturbation method on Ω p , which is feasible to describe the physics within the mechanical cooling (0) (1) (2) dynamics [30, 31]. In the following we denote ρmn , ρmn and ρmn to represent the density ma0 1 2 trix elements in orders of Ω p , Ω p and Ω p , 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 Ω p , 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 2 gT ± + gT1 + Ω p /2 2 , A± /η 2 = γ T2± + κ 2 ∆ ∓ ν + iκ/2

(31)

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 #212864 - $15.00 USD Received 26 May 2014; revised 12 Jul 2014; accepted 29 Jul 2014; published 12 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020060 | OPTICS EXPRESS 20068

T1 and T2± connected to the emission or absorption of a vibrational phonon are expressed as gΩ p /2 (δc3 + ∆ − δ1 )(δc3 + ∆ − δ2 ), f (∆) ±νgΩ p /2 T2± = Fγ (∆)(δc3 + ∆ ∓ ν − δ1 )(δc3 + ∆ ∓ ν − δ2 ), f (∆) f (∆ ∓ ν) T1 =

(32)

where the functions are f (∆) = (∆ + iκ/2) Fγ (∆) − g2 (δc3 + ∆ − δ1 )(δc3 + ∆ − δ2 ), Fγ (∆) =(δc3 + ∆ − δ1 )(δc3 + ∆ − δ2 ) (δc3 + ∆ + iγ/2) − (Ω1 /2)2 (δc3 + ∆ − δ2 ) − (Ω2 /2)2 (δc3 + ∆ − δ1 ). 4.1.

(33)

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 |ni is phonon number state, the green lines indicate the carrier transition into atomic excited state |e, ni and the blue lines indicate the mechanical phonon-increased transitions. From appendix A, we can (2) verify that the carrier transition to the state |e, ni becomes zero, i.e., ρee (∞) = 0 when the detuning δ1 or δ2 satisfies the relation δc3 + ∆ − δ1 = 0

or δc3 + ∆ − δ2 = 0.

(34)

Without loss of generality, we will take the condition δc3 +∆ −δ1 = 0, which corresponds to the completely quantum destructive interference between the transitions |g3 , ni ↔ |1, ni ↔ |e, ni and |g1 , ni ↔ |e, ni. 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, ni, and will also lead to the transition amplitude T1 equal to zero. Further, one can find that when the detuning δ2 fulfills the relation δc3 + ∆ − ν − δ2 = 0,

(35)

the phonon-increased transition amplitude T2+ in Eq. (32) also becomes zero, because the bluesideband transition into state |e, n + 1i is completely prohibited via the destructive quantum interference, and the further transition |e, n + 1i ↔ |1, n + 1i 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 , ni ↔ |1, ni ↔ |e, n + 1i and |e, n + 1i ↔ |g2 , n + 1i, and between |g3 , ni ↔ |1, n + 1i ↔ |e, n + 1i and |e, n + 1i ↔ |g2 , n + 1i 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+ /η 2 = κ

(Ω p /2)2 , (∆ − ν)2 + (κ/2)2

(36)

which indicates the heating caused by the phonon-increased transition from |g3 , ni to |1, n + 1i, 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 #212864 - $15.00 USD Received 26 May 2014; revised 12 Jul 2014; accepted 29 Jul 2014; published 12 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020060 | OPTICS EXPRESS 20069

(a)

|e,n+1Ò

(b)

|e,nÒ

|g1,n+1Ò |g1,nÒ

|g2,n+1Ò |g2,nÒ

|1,n+1Ò |1,nÒ |g3,n+1Ò

|g3,nÒ

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

decay completely disappears, and in a good optical cavity limit with a weak pumping strength and appropriate detuning ∆, 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 κ γ, 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 (∆ + ν

+ i κ2 )(i 2γ

+ δ1 + ν −

Ω21 4ν

−

Ω22 2 8ν ) − g

.

(37)

The structure of the transition amplitude is mainly determined by the denominator, where the real and imaginary parts are (∆ + ν)(δ1 + ν − Ω22 8ν ) + γ/2(∆ + ν) respectively. In the limit κ

Ω21 4ν

−

Ω22 2 8ν ) − κγ/4 − g

and κ/2(δ1 + ν −

Ω21 4ν

−

and single atom cooperativity C = 4g2 /κγ

γ 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 =

Ω21 Ω22 g2 + κγ/4 + + − ν, 4ν 8ν ∆+ν

(38)


and the transition amplitude T2− becomes T2− = i

νgε κ g2 +κγ/4 2 ∆+ν

+ 2γ (∆ + ν)

,

(39)

Ω /2

p where ε = ∆+iκ/2 . To concentrate to physics, we consider the good cavity limit and strong atom-cavity coupling regime, i.e., κ ∆, the amplitude T2− is approximately an imaginary number and the cooling rate is 2 − 2 (Ω p /2)2 + g2 T2− 2 T . + γ (40) A− /η = κ 2 (∆ + ν)2 + (κ/2)2

Obviously, when the detuning ∆ is negative, the cooling rate dominates over heating rate, and the ground-state cooling is achievable. 〈n〉st

−24.5

0.1 0.08

δ2

0.06

−25 0.04 0.02

−25.5 −24.5

−24.3

−24.5

−24.1

−23.9

δ1 W/η2

−23.7

−23.5

0.65 0.6

δ2

0.55

−25

0.5 0.45 −25.5 −24.5

−24.3

−24.1

−23.9

−23.7

−23.5

δ1 Fig. 3. Steady-state phonon number hni ˆ st and cooling rate W /η 2 as functions of detunings δ1 and δ2 with parameters in units of ν: Ω1 = Ω2 = 6ν, κ = 0.1ν, ∆ = −2ν, γ = 10ν, Ω p = ν, g = 6ν, δc3 = δ1 − ∆.

To numerically show the achievement of the ground-state cooling, we choose ∆ = −2ν, Ω p = 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 hni ˆ st and cooling rate W /η 2 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 = δ1 − ν, which is just the condition of the second completely destructive quantum interference involving mechanical phonon-increased #212864 - $15.00 USD Received 26 May 2014; revised 12 Jul 2014; accepted 29 Jul 2014; published 12 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020060 | OPTICS EXPRESS 20071

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). 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 κ γ and large atom cooperativity C 1 with ∆ = −2ν. The cooling and heating rates of the mechanical motion are approximately A− /η 2 ≈ 4g2 |ε|2 /γ,

A+ /η 2 ≈ 4κ |ε|2 /9,

(41)

and therefore the cooling limit is hni ˆ ∞ ≈ A+ /A− ≈

4 , 9C

(42)

where C measures the strength of the coherent atom-cavity coupling. Here C = 144 with the pa4 is about 0.003, which is consistent with the numerical rameters in Fig. 3 and phonon number 9C 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± /η 2 = κ

(Ω p /2)2 , (∆ ∓ ν)2 + (κ/2)2

(43)

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 ∆ here has been rotated in frame of single-photon induced frequency shift η χ. The optimal cooling occurs at the red sideband ∆ = −ν, and the cooling rate is A− = η 2 κΩ2p /κ 2 . In order to fulfill the weak driving approximation, the strength ηΩ p should be much smaller than κ to validate the perturbation approximation. For example, when ηΩ p /κ = 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 n¯ th is 1000 with mechanical quality factor Q = 106 , the thermal heating rate γm n¯ th leads to the final mean phonon number hni ˆ 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 [38–41]. 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 hni ˆ ≈ 0.14, which is much lower than unity. Moreover, when the initial thermal occupation n¯ th is below 6800 with mechanical quality factor Q = 106 , the thermal heating rate γm n¯ th 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 #212864 - $15.00 USD Received 26 May 2014; revised 12 Jul 2014; accepted 29 Jul 2014; published 12 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. 17 | DOI:10.1364/OE.22.020060 | OPTICS EXPRESS 20072

T2± are simplified as T1 =

gΩ p /2 , D(∆)

T2± =

±νgΩ p /2(δc3 + ∆ + iγ/2) , D(∆)D(∆ ∓ ν)

(44)

where D(∆) = (∆ + iκ/2)(δc3 + ∆ + iγ/2) − g2 .

(45)

The heating and cooling coefficients becomes i Ω p /2(δc3 + ∆ + iγ/2) 2 h 2 2 γν g + κ |D(∆) ∓ ν(∆ + iκ/2)|2 , A± /η = D(∆)D(∆ ∓ ν) 2

(46)

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 hni ˆ st ≈ A+ /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 κ γ, 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 (0) (1) (2) of the perturbation method with respect to Ω p . In the following we denote ρmn , ρmn and ρmn to 0 1 2 indicate the density matrix elements in the orders of Ω p , Ω p and Ω p , 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) =

Z ∞

e−st ρ (i) (t)dt,

(i = 0, 1, 2).

(47)

0

A.1.

The density matrix in the zeroth order of Ω p

The evolution equation of density matrix elements in zeroth order of Ω p can be obtained from Eq. (30), which is a separate equation (0) (0) ρ˙ g3 g3 = γρee .

(48) (0)

Obviously, the steady-state solutions for the density matrix elements are ρg3 g3 (∞) = 1, and the other terms are zero. A.2.

The density matrix in the first order of Ω p

To achieve the correction of density matrix in the first order of Ω p , we substitute the zerothorder 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 h Ω j (1) γ i (1) (1) (1) ρg g − igρ1g3 , ρ˙ eg3 = i(δc3 + ∆) − ρeg3 − i ∑ j 2 2 j 3 Ω1 (1) (1) (1) ρ˙ g1 g3 = i(δc3 + ∆ − δ1 )ρg1 g3 − i ρeg3 , 2 Ω2 (1) (1) (1) ρ˙ g2 g3 = i(δc3 + ∆ − δ2 )ρg2 g3 − i ρeg3 , 2 Ω p (0) κ (1) (1) (1) ρ˙ 1g3 = i∆ − ρ − igρeg3 − i ρg3 g3 , 2 1g3 2

(49)

while the other density matrix elements are zero in the steady state. The steady-state solutions (0) for these density matrix can be easily obtained by substituting ρg3 g3 (∞) = 1 into the equations, (1) which are denoted as ρ (∞). The Laplace transform of these density matrix elements are given in the form ρ (1) (s) =

1 ρ (1) (∞), s − L(1)

(50)

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 Ω p

The correction of density matrix in the second-order Ω p 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 Ω1 (2) Ω2 (2) (2) (2) (2) ρ˙ ee = −γρee + i ρeg1 + ρeg2 + gρe1 + c.c., 2 2 Ω j (2) (2) (2) (2) ρeg j − ρg j e , ρ˙ g j g j = γ j ρee − i 2 Ω p (1) (2) (2) (2) ρ˙ 11 = −κρ11 + i ρ1g3 − gρe1 + c.c., 2 γ Ω1 (2) Ω2 (2) (2) (2) (2) (2) ρeg1 + i (ρee − ρg1 g1 ) − i ρg2 g1 − igρ1g1 , ρ˙ eg1 = iδ1 − 2 2 2 γ (2) Ω2 (2) Ω1 (2) (2) (2) (2) ρ˙ eg2 = iδ2 − ρeg2 + i (ρee − ρg2 g2 ) − i ρg1 g2 − igρ1g2 , 2 2 2 Ω j (2) Ω p (1) γ +κ (2) (2) (2) (2) ˙ ρe1 + ig(ρee − ρ11 ) − i ∑ j ρe1 = iδc3 − ρ + i ρeg3 , 2 2 g j1 2 Ω1 (2) Ω2 (2) (2) (2) ρ˙ g1 g2 = i(δ2 − δ1 )ρg1 g2 − i ρeg2 + i ρg1 e , 2 2 h Ω p (1) κ i (2) Ω1 (2) (2) (2) ρ˙ g1 1 = i(δc3 − δ1 ) − ρ − i ρe1 + igρg1 e + i ρg1 g3 , 2 g1 1 2 2 h Ω p (1) κ i (2) Ω2 (2) (2) (2) ρ − i ρe1 + igρg2 e + i ρg2 g3 , ρ˙ g2 1 = i(δc3 − δ2 ) − (51) 2 g2 1 2 2 It should be noted that the average values of the operators that appear in Vˆ in Eq. (17) are contained in this group of equations. The steady-state solution of density matrix in the secondorder of Ω p can be calculated by substituting the first-order results ρ (1) (∞) into Eq. (51), which (2) are denoted as ρ (2) (∞). For example, ρee (∞) that is connected to mechanical effects on the movable oscillator from atomic spontaneously emitted photon is given as (2)

ρee (∞) =

(Ω p /2)2 2 g (δc3 + ∆ − δ1 )2 (δc3 + ∆ − δ2 )2 , | f (∆)|2

(52)

which is inversely proportional to h γ κ f (∆) =(i + ∆) (δc3 + ∆ − δ1 )(δc3 + ∆ − δ2 )(δc3 + ∆ + i ) − (δc3 + ∆ − δ2 )Ω21 /4 2 2 i − (δc3 + ∆ − δ1 )Ω22 /4 − g2 (δc3 + ∆ − δ1 )(δc3 + ∆ − δ2 ).

(53)

The Laplace transform of the second-order density matrix elements ρ (2) (s) is given by 1 1 (2) (2) (1) ρ (s) = ρ (∞) + ρ (∞) , (54) s − L(2) s − L(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 hVˆ1 (s)Vˆ1 (0)is by substituting s with −iν. By applying the quantum regression theorem, hVˆ1 (s)Vˆ1 (0)is can Ω (2) (2) (1) be derived from the single-time average hVˆ1 (s)is = iηg[ρ (s) − ρ (s)] + iη p [ρ (s) − 1e

e1

2

1g3

(1)

ρg3 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.


Strain-mediated coupling in a quantum dot-mechanical oscillator hybrid system.

Cavity cooling of an ensemble spin system.

Erratum: Quantum enhanced feedback cooling of a mechanical oscillator using nonclassical light.

A brain-cooling system in mammals.

How to make an oscillator.

Rhythmogenesis in a hybrid system--interconnecting an olivary neuron to an analog network of coupled oscillators.

High peak-power mid-infrared ZnGeP₂ optical parametric oscillator pumped by a Tm:fiber master oscillator power amplifier system.

Time dependent quantum thermodynamics of a coupled quantum oscillator system in a small thermal environment.

Characterization of Francisella species isolated from the cooling water of an air conditioning system.

Feedback cooling of an atomic spin ensemble.

Multirhythmicity in an optoelectronic oscillator with large delay.

Solar hydrogen from an aqueous, noble-metal-free hybrid system in a continuous-flow sampling reaction system.

Spatial arrangement of legionella colonies in intact biofilms from a model cooling water system.

Glucose Oscillations Can Activate an Endogenous Oscillator in Pancreatic Islets.

Rapid on-site monitoring of Legionella pneumophila in cooling tower water using a portable microfluidic system.

An injured brain needs cooling down: yes.

Implementing a universal gate set on a logical qubit encoded in an oscillator.

Effects of Normothermic Conditioned Microwave Irradiation on Cultured Cells Using an Irradiation System with Semiconductor Oscillator and Thermo-regulatory Applicator.

Computer simulation of a cytosolic calcium oscillator.

Energy relaxation of a dissipative quantum oscillator.

An arc control and protection system for the JET lower hybrid antenna based on an imaging system.

Cerenkov-light hybrid imaging system.

In vitro performance of lipid-PLGA hybrid nanoparticles as an antigen delivery system: lipid composition matters.

A hybrid dielectrophoretic system for trapping of microorganisms from water.