Quantum limit of mechanical occupation number

CLASSIFICATION OF OPTOMECHANICAL INTERACTION AND THE DISCOVERY OF COHERENT COUPLING

2.9 Appendix: Optomechanical cooling limit in the ring cavity system .1 Coupled optical and mechanical equations of motion.1Coupled optical and mechanical equations of motion

2.9.3 Quantum limit of mechanical occupation number

We then calculate the occupation number limit. In the case where the mechanical object is a high-Q-oscillator, we represent the mechanical oscillation in terms of mechanical creation and annihilation operators ˜𝑚^† and ˜𝑚 in the rotating frame of mechanical resonant frequencyΩ𝑚:

ˆ

𝑥 =𝑥_ZPF(𝑚 𝑒˜ ^{−𝑖Ω𝑚𝑡}+𝑚˜^†𝑒^{𝑖Ω𝑚𝑡}), (2.104) where𝑥_ZPF = p

~/2𝑚Ω𝑚 is the zero-point fluctuation of the mechanical oscillator.

Using𝜔 to represent the sideband frequency of mechanical oscillator around Ω𝑚, the Fourier components of ˜𝑚 and ˜𝑚^†read:

˜ 𝑚(𝑡) =

∫ +∞

−Ω𝑚

𝑑 𝜔 2𝜋

˜

𝑚(𝜔)𝑒^{−𝑖𝜔𝑡}, (2.105a)

˜ 𝑚^†(𝑡) =

∫ Ω𝑚

−∞

𝑑 𝜔 2𝜋

˜

𝑚^†(−𝜔)𝑒^{−𝑖𝜔𝑡}. (2.105b) Although the upper and lower limits for both integrals in Eq. (2.105) can be extended to infinity under condition𝜔 Ω𝑚, we keep this rigorous form for clearer future reference. The average mechanical occupation number is defined as [79]:

h𝑛(𝑡)i ≡

𝑚˜(𝑡)𝑚˜^†(𝑡) +𝑚˜^†(𝑡)𝑚˜(𝑡)

−1

2 . (2.106)

According to Eq. (2.105) ˜𝑚(𝑡)𝑚˜^†(𝑡) and ˜𝑚^†(𝑡)𝑚˜(𝑡)can be expressed as:

˜

𝑚(𝑡)𝑚˜^†(𝑡) =

∬ +∞

−Ω𝑚

𝑑 𝜔 𝑑 𝜔⁰

(2𝜋)² 𝑚˜(𝜔)𝑚˜^†(𝜔⁰)𝑒^𝑖(𝜔

0−𝜔)𝑡

, (2.107a)

˜

𝑚^†(𝑡)𝑚˜(𝑡) =

∬ +∞

−Ω𝑚

𝑑 𝜔 𝑑 𝜔⁰

(2𝜋)² 𝑚˜^†(𝜔⁰)𝑚˜(𝜔)𝑒^𝑖(𝜔

0−𝜔)𝑡

. (2.107b)

To obtain the mechanical occupation number, we need to calculate the second-order correlation function of mechanical operators

˜

𝑚(𝜔)𝑚˜^†(𝜔⁰) and

˜

𝑚^†(𝜔⁰)𝑚˜(𝜔) . Therefore, we need to obtain equations of motion of ˜𝑚,𝑚˜^†by rephrasing those of

ˆ

𝑥 ,𝑝ˆin Sec. 2.9.1 and Sec. 2.9.2.

Quoting Eq. (2.102) and ignoring the static force by letting𝐺_eff = 0, we obtain the second-order equation of motion of ˆ𝑥:

𝑚𝑥¥ˆ =𝐹ˆ_bafl−𝑚Ω²_𝑚𝑥ˆ−𝑚 𝛾_eff𝑥 .¤ˆ (2.108)

Transferred into frequency domain, Eq. (2.108) becomes:

𝑚(Ω²_𝑚 −Ω²−𝑖Ω𝛾_eff)𝑥ˆ(Ω) =𝐹ˆ_bafl(Ω), (2.109) which can be factorized under condition𝛾_eff Ω𝑚 as:

h𝛾_eff

2 −𝑖(Ω−Ω𝑚) i h𝛾_eff

2 −𝑖(Ω+Ω𝑚) i

ˆ

𝑥(Ω) = 𝐹ˆ_bafl(Ω) 𝑚

. (2.110)

According to Eqs. (2.104) and (2.105), the Fourier Transformation of ˆ𝑥reads:

ˆ 𝑥(Ω) ≡

∫ +∞

−∞

𝑑 𝑡𝑥ˆ(𝑡)𝑒^𝑖Ω𝑡

=𝑥_ZPF

∫ +∞

−∞

𝑑 𝑡

𝑒^{𝑖(Ω−Ω𝑚)𝑡}

∫ +∞

−Ω𝑚

𝑑 𝜔 2𝜋

˜

𝑚(𝜔)𝑒^{−𝑖𝜔𝑡} +𝑒^{𝑖(Ω+Ω𝑚)𝑡}

∫ Ω𝑚

−∞

𝑑 𝜔 2𝜋

˜

𝑚^†(−𝜔)𝑒^{−𝑖𝜔𝑡}

=𝑥_ZPF

˜

𝑚(Ω−Ω𝑚)𝜃(Ω) +𝑚^†(−Ω−Ω𝑚)𝜃(−Ω) ,

(2.111) where 𝜃(Ω) is the Heaviside step function and 𝛿(Ω) is the Dirac delta function.

Plugging Eq. (2.111) into Eq. (2.110) and considering the thermal force ˆ𝐹_th, we obtain the equations of motion for ˜𝑚^†,𝑚˜ in their frequency domain:

h𝛾_𝑚 +𝛾_opt

2 −𝑖𝜔

i

˜

𝑚(𝜔) = 𝑖𝑥_ZPF

~

h𝐹ˆ_bafl(Ω𝑚 +𝜔) +𝐹ˆ_th(Ω𝑚 +𝜔)i

, (2.112a) h𝛾_𝑚+𝛾_opt

2 +𝑖𝜔

i

˜

𝑚^†(𝜔) =−𝑖𝑥_ZPF

~

h𝐹ˆ_bafl(−(Ω𝑚 +𝜔)) +𝐹ˆ_th(−(Ω𝑚 +𝜔))i . (2.112b) The fluctuating back-action force on the mechanical oscillator in the frequency domain reads:

ˆ

𝐹_bafl(Ω) =2𝑖𝜔_𝑠~𝑘_𝑝

𝐶₊^∗𝑐ˆ₋(Ω) −𝐶₊𝑐ˆ^†₋(−Ω) −𝐶₋^∗𝑐ˆ₊(Ω) +𝐶₋𝑐ˆ^†₊(−Ω)

, (2.113) and satisfies the relation ˆ𝐹^†

bafl(Ω) = 𝐹ˆ_bafl(−Ω). The spectrum 𝑆_𝐹(Ω) of the back- action force ˆ𝐹_bafl(Ω)is defined as:

𝐹ˆ_bafl(Ω)𝐹ˆ_bafl(Ω⁰)

=2𝜋 𝛿(Ω⁰+Ω)𝑆_𝐹(Ω) (2.114) and takes the following expression:

𝑆_𝐹(Ω) =8𝐴²

in𝜔²

𝑠~²𝑘²

𝑝𝛾²

1

𝛾² 𝛾²+ (Ω−2𝜔_𝑠)² + 1 𝛾²+Ω²

𝛾²+4𝜔²

𝑠

. (2.115) Also, the thermal force ˆ𝐹_th has white spectrum:

𝐹ˆ_th(Ω)𝐹ˆ_th(Ω⁰)

=2𝜋 𝛿(Ω⁰+Ω)2𝑚 𝑘_𝐵𝑇 𝛾_𝑚. (2.116)

Thus, the second-order correlation function of mechanical operators can be calcu- lated by:

𝑚˜(𝜔)𝑚˜^†(𝜔⁰)

=2𝜋 𝛿(𝜔−𝜔⁰)𝑆₊(𝜔), (2.117a) 𝑚˜^†(𝜔⁰)𝑚˜(𝜔)

=2𝜋 𝛿(𝜔−𝜔⁰)𝑆₋(𝜔), (2.117b) with𝑆₊(𝜔)and𝑆₋(𝜔)defined as:

𝑆₊(𝜔) ≡ 𝑥²

ZPF/~²[𝑆_𝐹(Ω𝑚 +𝜔) +2𝑚 𝑘_𝐵𝑇 𝛾_𝑚] _𝛾

𝑚+𝛾_opt 2

2

+𝜔²

, (2.118a)

𝑆₋(𝜔) ≡ 𝑥²

ZPF/~²[𝑆_𝐹(−(Ω𝑚 +𝜔)) +2𝑚 𝑘_𝐵𝑇 𝛾_𝑚] _𝛾

𝑚+𝛾_opt 2

2

+𝜔²

. (2.118b)

According to Eq. (2.107), the time-domain mechanical correlation functions can be calculated as follows:

𝑚˜(𝑡)𝑚˜^†(𝑡)

=

∫ +∞

−Ω𝑚

𝑆₊(𝜔)𝑑 𝜔 2𝜋

= 4𝐴²

in𝛾²𝑘²

𝑝~𝜔²

𝑠

𝑚Ω𝑚 𝛾_𝑚 +𝛾_opt

1

𝛾² 𝛾²+ (Ω𝑚 −2𝜔_𝑠)² + 1 4Ω²𝑚𝜔²_𝑠

+ 𝛾_𝑚 𝛾_𝑚 +𝛾_opt

𝑘_𝐵𝑇 Ω𝑚~

, (2.119a) 𝑚˜^†(𝑡)𝑚˜(𝑡)

=

∫ +∞

−Ω𝑚

𝑆₋(𝜔)𝑑 𝜔 2𝜋

= 4𝐴²

in𝛾²𝑘²

𝑝~𝜔²

𝑠

𝑚Ω𝑚 𝛾_𝑚 +𝛾_opt

1

𝛾²(Ω𝑚+2𝜔_𝑠)² + 1 4Ω²𝑚𝜔²

𝑠

+ 𝛾_𝑚 𝛾_𝑚+𝛾_opt

𝑘_𝐵𝑇 Ω𝑚~

.

(2.119b)

Based on all derivation above, under conditionΩ𝑚 𝛾, the mechanical occupa- tion number defined in Eq. (2.106) can be expressed as:

h𝑛ˆi=

𝛾_opt 𝛾_𝑚 +𝛾_opt

1 2

𝛾² 4Ω²𝑚

− 𝛾_𝑚 𝛾_opt

+ 𝛾_𝑚 𝛾_𝑚 +𝛾_opt

𝑘_𝐵𝑇 Ω𝑚~

. (2.120)

Under further condition𝛾_opt 𝛾_m, we can rewrite the expression above to get the ultimate cooling limit:

h𝑛ˆi= 𝛾_𝑚 𝛾_𝑚+𝛾_opt

𝑛_th+

𝛾_opt 𝛾_𝑚 +𝛾_opt

𝑛_ba ≈𝑛_ba, (2.121) where𝑛_th = 𝑘_𝐵𝑇/Ω𝑚~is the thermal occupation number and𝑛_ba =𝛾²/8Ω²𝑚 is the back-action limited occupation number.

References

[1] Tobias J Kippenberg and Kerry J Vahala. Cavity optomechanics: back-action at the mesoscale. Science, 321(5893):1172–1176, 2008.

[2] Ivan Favero and Khaled Karrai. Optomechanics of deformable optical cavities.

Nature Photonics, 3(4):201, 2009.

[3] Markus Aspelmeyer, Tobias J Kippenberg, and Florian Marquardt. Cavity optomechanics. Reviews of Modern Physics, 86(4):1391, 2014.

[4] Warwick P Bowen and Gerard J Milburn. Quantum optomechanics. CRC press, 2015.

[5] M Bhattacharya and Pierre Meystre. Trapping and cooling a mirror to its quantum mechanical ground state. Physical Review Letters, 99(7):073601, 2007.

[6] Liu Yong-Chun, Hu Yu-Wen, Wong Chee Wei, and Xiao Yun-Feng. Review of cavity optomechanical cooling. Chinese Physics B, 22(11):114213, 2013.

[7] Andreas Sawadsky, Henning Kaufer, Ramon Moghadas Nia, Sergey P Tarabrin, Farid Ya Khalili, Klemens Hammerer, and Roman Schnabel. Ob- servation of generalized optomechanical coupling and cooling on cavity reso- nance. Physical Review Letters, 114(4):043601, 2015.

[8] Thomas P Purdy, P L Yu, R W Peterson, N S Kampel, and C A Regal. Strong optomechanical squeezing of light. Physical Review X, 3(3):031012, 2013.

[9] Andreas Kronwald, Florian Marquardt, and Aashish A Clerk. Dissipative optomechanical squeezing of light. New Journal of Physics, 16(6):063058, 2014.

[10] Nancy Aggarwal, Torrey Cullen, Jonathan Cripe, Garrett D Cole, Robert Lanza, Adam Libson, David Follman, Paula Heu, Thomas Corbitt, and Nergis Mavalvala. Room temperature optomechanical squeezing. arXiv preprint arXiv:1812.09942, 2018.

[11] Roman Schnabel. Squeezed states of light and their applications in laser interferometers, sec. 5.6. Physics Reports, 684:1–51, 2017.

[12] S Bose, K Jacobs, and P L Knight. Preparation of nonclassical states in cavities with a moving mirror. Physical Review A, 56(5):4175, 1997.

[13] Sougato Bose, Kurt Jacobs, and Peter L Knight. Scheme to probe the decoher- ence of a macroscopic object. Physical Review A, 59(5):3204, 1999.

[14] David Vitali, Sylvain Gigan, Anderson Ferreira, H R Böhm, Paolo Tombesi, Ariel Guerreiro, Vlatko Vedral, Anton Zeilinger, and Markus Aspelmeyer.

Optomechanical entanglement between a movable mirror and a cavity field.

Physical Review Letters, 98(3):030405, 2007.

[15] Haixing Miao, Stefan Danilishin, and Yanbei Chen. Universal quantum en- tanglement between an oscillator and continuous fields. Physical Review A, 81(5):052307, 2010.

[16] Stefano Mancini, Vittorio Giovannetti, David Vitali, and Paolo Tombesi. En- tangling macroscopic oscillators exploiting radiation pressure. Physical Re- view Letters, 88(12):120401, 2002.

[17] M Bhattacharya, P-L Giscard, and Pierre Meystre. Entangling the rovibrational modes of a macroscopic mirror using radiation pressure. Physical Review A, 77(3):030303(R), 2008.

[18] Michael J Hartmann and Martin B Plenio. Steady state entanglement in the mechanical vibrations of two dielectric membranes. Physical Review Letters, 101(20):200503, 2008.

[19] Roman Schnabel. Einstein-podolsky-rosen–entangled motion of two massive objects. Physical Review A, 92(1):012126, 2015.

[20] P D Nation. Nonclassical mechanical states in an optomechanical micromaser analog. Physical Review A, 88(5):053828, 2013.

[21] Matteo Brunelli, Oussama Houhou, Darren W Moore, Andreas Nunnenkamp, Mauro Paternostro, and Alessandro Ferraro. Unconditional preparation of nonclassical states via linear-and-quadratic optomechanics. Physical Review A, 98(6):063801, 2018.

[22] Emily J Davis, Zhaoyou Wang, Amir H Safavi-Naeini, and Monika H Schleier- Smith. Painting nonclassical states of spin or motion with shaped single photons. Physical Review Letters, 121(12):123602, 2018.

[23] Yanbei Chen. Macroscopic quantum mechanics: theory and experimental concepts of optomechanics. Journal of Physics B: Atomic, Molecular and Optical Physics, 46(10):104001, 2013.

[24] Tom P Purdy, Robert W Peterson, and C.A. Regal. Observation of radiation pressure shot noise on a macroscopic object. Science, 339(6121):801–804, 2013.

[25] Mateusz Bawaj, Ciro Biancofiore, Michele Bonaldi, Federica Bonfigli, An- tonio Borrielli, Giovanni Di Giuseppe, Lorenzo Marconi, Francesco Marino, Riccardo Natali, Antonio Pontin, et al. Probing deformed commutators with macroscopic harmonic oscillators. Nature Communications, 6:7503, 2015.

[26] Jie Li, Stefano Zippilli, Jing Zhang, and David Vitali. Discriminating the effects of collapse models from environmental diffusion with levitated nanospheres.

Physical Review A, 93(5):050102(R), 2016.

[27] Alessio Belenchia, Dionigi M T Benincasa, Stefano Liberati, Francesco Marin, Francesco Marino, and Antonello Ortolan. Tests of quantum-gravity- induced nonlocality via optomechanical experiments. Physical Review D, 95(2):026012, 2017.

[28] Lin Tian and Hailin Wang. Optical wavelength conversion of quantum states with optomechanics. Physical Review A, 82(5):053806, 2010.

[29] Jeff T Hill, Amir H Safavi-Naeini, Jasper Chan, and Oskar Painter. Coherent optical wavelength conversion via cavity optomechanics.Nature Communica- tions, 3:1196, 2012.

[30] C F Ockeloen-Korppi, Erno Damskägg, J M Pirkkalainen, T T Heikkilä, Francesco Massel, and M A Sillanpää. Low-noise amplification and frequency conversion with a multiport microwave optomechanical device. Physical Re- view X, 6(4):041024, 2016.

[31] F Lecocq, J B Clark, R W Simmonds, J Aumentado, and J D Teufel. Me- chanically mediated microwave frequency conversion in the quantum regime.

Physical Review Letters, 116(4):043601, 2016.

[32] JG Huang, Y Li, Lip Ket Chin, H Cai, YD Gu, Muhammad Faeyz Karim, JH Wu, TN Chen, ZC Yang, YL Hao, et al. A dissipative self-sustained optome- chanical resonator on a silicon chip. Applied Physics Letters, 112(5):051104, 2018.

[33] Marcelo Wu, Aaron C Hryciw, Chris Healey, David P Lake, Harishankar Jayakumar, Mark R Freeman, John P Davis, and Paul E Barclay. Dissipative and dispersive optomechanics in a nanocavity torque sensor. Physical Review X, 4(2):021052, 2014.

[34] David E McClelland, Nergis Mavalvala, Yanbei Chen, and Roman Schnabel.

Advanced interferometry, quantum optics and optomechanics in gravitational wave detectors. Laser & Photonics Reviews, 5(5):677–696, 2011.

[35] Benjamin P Abbott et al. Observation of gravitational waves from a binary black hole merger. Physical Review Letters, 116(6):061102, 2016.

[36] Benjamin P Abbott et al. Prospects for observing and localizing gravitational- wave transients with advanced ligo, advanced virgo and kagra.Living Reviews in Relativity, 21(1):3, 2018.

[37] Gregory M Harry. Advanced ligo: the next generation of gravitational wave detectors. Classical and Quantum Gravity, 27(8):084006, 2010.

[38] J Aasi et al. Advanced ligo. Classical and quantum gravity, 32(7):074001, 2015.

[39] F Acernese et al. Advanced virgo: a second-generation interferometric gravi- tational wave detector. Classical and Quantum Gravity, 32(2):024001, 2014.

[40] F Acernese et al. The advanced virgo detector. InJournal of Physics: Confer- ence Series, volume 610, page 012014. IOP Publishing, 2015.

[41] Harald Lück and the GEO600 Team. The geo600 project. Classical and quantum gravity, 14(6):1471, 1997.

[42] C Affeldt, K Danzmann, KL Dooley, H Grote, M Hewitson, S Hild, J Hough, J Leong, H Lück, M Prijatelj, et al. Advanced techniques in geo 600.Classical and quantum gravity, 31(22):224002, 2014.

[43] Yoichi Aso, Yuta Michimura, Kentaro Somiya, Masaki Ando, Osamu Miyakawa, Takanori Sekiguchi, Daisuke Tatsumi, and Hiroaki Yamamoto.

Interferometer design of the kagra gravitational wave detector. Physical Re- view D, 88(4):043007, 2013.

[44] Kentaro Somiya. Detector configuration of kagra–the japanese cryogenic gravitational-wave detector. Classical and Quantum Gravity, 29(12):124007, 2012.

[45] Dustin Kleckner, William Marshall, Michiel J A de Dood, Khodadad Nima Dinyari, Bart-Jan Pors, William T M Irvine, and Dirk Bouwmeester. High fi- nesse opto-mechanical cavity with a movable thirty-micron-size mirror. Phys- ical Review Letters, 96(17):173901, 2006.

[46] André Xuereb, Roman Schnabel, and Klemens Hammerer. Dissipative op- tomechanics in a michelson-sagnac interferometer. Physical Review Letters, 107(21):213604, 2011.

[47] Sergey P Vyatchanin and Andrey B Matsko. Quantum speed meter based on dissipative coupling. Physical Review A, 93(6):063817, 2016.

[48] Farid Ya Khalili, Sergey P Tarabrin, Klemens Hammerer, and Roman Schnabel. Generalized analysis of quantum noise and dynamic backaction in signal-recycled michelson-type laser interferometers. Physical Review A, 94(1):013844, 2016.

[49] André Xuereb, Peter Horak, and Tim Freegarde. Amplified optomechanics in a unidirectional ring cavity. Journal of Modern Optics, 58(15):1342–1348, 2011.

[50] Stefano Chesi, Ying-Dan Wang, and Jason Twamley. Diabolical points in multi-scatterer optomechanical systems. Scientific Reports, 5:7816 EP –, 01 2015.

[51] Arzu Yilmaz, Simon Schuster, Philip Wolf, Dag Schmidt, Max Eisele, Claus Zimmermann, and Sebastian Slama. Optomechanical damping of a nanomem- brane inside an optical ring cavity.New Journal of Physics, 19(1):013038, jan 2017.

[52] CK Law. Interaction between a moving mirror and radiation pressure: A hamiltonian formulation. Physical Review A, 51(3):2537, 1995.

[53] Sina Khorasani. Higher-order interactions in quantum optomechanics: Revis- iting theoretical foundations. Applied Sciences, 7(7):656, 2017.

[54] Belinda Heyun Pang. Theoretical Foundations for Quantum Measurement in a General Relativistic Framework. PhD thesis, California Institute of Technol- ogy, 2018.

[55] JD Thompson, BM Zwickl, AM Jayich, Florian Marquardt, SM Girvin, and JGE Harris. Strong dispersive coupling of a high-finesse cavity to a microme- chanical membrane. Nature, 452(7183):72, 2008.

[56] M Karuza, M Galassi, C Biancofiore, C Molinelli, R Natali, P Tombesi, G Di Giuseppe, and D Vitali. Tunable linear and quadratic optomechanical coupling for a tilted membrane within an optical cavity: theory and experiment.

Journal of Optics, 15(2):025704, 2012.

[57] Hong Xie, Chang-Geng Liao, Xiao Shang, Ming-Yong Ye, and Xiu-Min Lin.

Phonon blockade in a quadratically coupled optomechanical system. Physical Review A, 96(1):013861, 2017.

[58] J D P Machado, R J Slooter, and Ya M Blanter. Quantum signatures in quadratic optomechanics. Physical Review A, 99(5):053801, 2019.

[59] Haixing Miao, Chunnong Zhao, L Ju, and David G Blair. Quantum ground- state cooling and tripartite entanglement with three-mode optoacoustic inter- actions. Physical Review A, 79(6):063801, 2009.

[60] VB Braginsky, SE Strigin, and S Pr Vyatchanin. Parametric oscillatory in- stability in fabry–perot interferometer. Physics Letters A, 287(5-6):331–338, 2001.

[61] Matthew Evans, Slawek Gras, Peter Fritschel, John Miller, Lisa Barsotti, Denis Martynov, Aidan Brooks, Dennis Coyne, Rich Abbott, Rana X Adhikari, et al. Observation of parametric instability in advanced ligo. Physical Review Letters, 114(16):161102, 2015.

[62] Haixing Miao, Stefan Danilishin, Thomas Corbitt, and Yanbei Chen. Standard quantum limit for probing mechanical energy quantization. Physical Review Letters, 103(10):100402, 2009.

[63] Yiqiu Ma, Shtefan L Danilishin, Chunnong Zhao, Haixing Miao, W Zach Korth, Yanbei Chen, Robert L Ward, and David G Blair. Narrowing the filter-cavity bandwidth in gravitational-wave detectors via optomechanical in- teraction. Physical Review Letters, 113(15):151102, 2014.

[64] André Xuereb, Peter Horak, and Tim Freegarde. Amplified optomechanics in a unidirectional ring cavity. Journal of Modern Optics, 58(15):1342–1348, 2011.

[65] Stefano Chesi, Ying-Dan Wang, and Jason Twamley. Diabolical points in multi-scatterer optomechanical systems. Scientific reports, 5:7816, 2015.

[66] Arzu Yilmaz, Simon Schuster, Philip Wolf, Dag Schmidt, Max Eisele, Claus Zimmermann, and Sebastian Slama. Optomechanical damping of a nanomem- brane inside an optical ring cavity. New Journal of Physics, 19(1):013038, 2017.

[67] Boris Nagorny, Th Elsässer, and A Hemmerich. Collective atomic motion in an optical lattice formed inside a high finesse cavity. Physical Review Letters, 91(15):153003, 2003.

[68] D Kruse, Ch von Cube, C Zimmermann, and Ph W Courteille. Observa- tion of lasing mediated by collective atomic recoil. Physical Review Letters, 91(18):183601, 2003.

[69] Th Elsässer, B Nagorny, and A Hemmerich. Optical bistability and collec- tive behavior of atoms trapped in a high-q ring cavity. Physical Review A, 69(3):033403, 2004.

[70] Julian Klinner, Malik Lindholdt, Boris Nagorny, and Andreas Hemmerich.

Normal mode splitting and mechanical effects of an optical lattice in a ring cavity. Physical Review Letters, 96(2):023002, 2006.

[71] S Slama, S Bux, G Krenz, C Zimmermann, and Ph W Courteille. Superradiant rayleigh scattering and collective atomic recoil lasing in a ring cavity.Physical Review Letters, 98(5):053603, 2007.

[72] Helmut Ritsch, Peter Domokos, Ferdinand Brennecke, and Tilman Esslinger.

Cold atoms in cavity-generated dynamical optical potentials.Reviews of Mod- ern Physics, 85(2):553, 2013.

[73] D Schmidt, H Tomczyk, S Slama, and C Zimmermann. Dynamical instability of a bose-einstein condensate in an optical ring resonator. Physical Review Letters, 112(11):115302, 2014.

[74] Farokh Mivehvar, Stefan Ostermann, Francesco Piazza, and Helmut Ritsch.

Driven-dissipative supersolid in a ring cavity. Physical Review Letters, 120(12):123601, 2018.

[75] H K Cheung and C K Law. Nonadiabatic optomechanical hamiltonian of a moving dielectric membrane in a cavity. Physical Review A, 84(2):023812, 2011.

[76] Marlan O. Scully and M. Suhail Zubairy. Quantum Optics. Cambridge Uni- versity Press, 1997.

[77] William Hvidtfelt Padkær Nielsen, Yeghishe Tsaturyan, Christoffer Bo Møller, Eugene S Polzik, and Albert Schliesser. Multimode optomechanical system in the quantum regime. Proceedings of the National Academy of Sciences, 114(1):62–66, 2017.

[78] Haixing Miao, Yiqiu Ma, Chunnong Zhao, and Yanbei Chen. Enhancing the bandwidth of gravitational-wave detectors with unstable optomechanical filters. Physical review letters, 115(21):211104, 2015.

[79] Jun John Sakurai and Jim Napolitano. Modern quantum mechanics; 2nd ed.

Addison-Wesley, San Francisco, CA, 2011.

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PT-SYMMETRIC AMPLIFIER: BROADBAND SENSITIVITY