Abstract

We discuss the generation of strong stationary mechanical squeezing and entanglement in the modulated two-and three-mode optomechanics. Following the reservoir engineering scheme, the beam-splitter and parametric optomechanical interactions can be simultaneously achieved through appropriately choosing the modulation frequency on mechanical motion, which is essential to strong squeezing and entanglement. In the two-mode modulated optomechanics, squeezing is tunable by the relative ratio of parametric and beam-splitter couplings, and also robust to thermal noise due to the simultaneously optically induced cooling process. In the three-mode modulated optomechanics, strong EPR-type entanglement is also attainable, which can surpass the 3dB limit of nondegenerate parametric interaction. However, the ideal entanglement is impossible since only one of mechanical Bogoliubov modes is cooled by the cavity mode, which also makes the entanglement fragile to the mechanical noise.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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    [Crossref] [PubMed]
  3. A. Schliesser, O. Arcizet, R. Rivière, G. Anetsberger, and T. J. Kippenberg, “Resolved-sideband cooling and position measurement of a micromechanical oscillator close to the Heisenberg uncertainty limit,” Nat. Physics 5, 509–514 (2009).
    [Crossref]
  4. J. C. Sankey, C. Yang, B. M. Zwickl, M. Jayich, and J. G. E. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nature Phys 6, 707–712 (2010).
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  39. L. Tian, “Robust photon entanglement via quantum interference in optomechanical interfaces,” Phys. Rev. Lett. 110, 233602 (2013).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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2018 (1)

H. Fu, Z. C. Gong, L. P. Yang, T. H. Mao, C. P. Sun, S. Yi, Y. Li, and G. Y. Cao, “Coherent optomechanical switch for motion tranduction based on dynamically localized mechanical modes,” Phys. Rev. Appl. 9, 054024 (2018).
[Crossref]

2017 (4)

J. F. Huang, J. Q. Liao, L. Tian, and L. M. Kuang, “Manipulating counter-rotating interactions in the quantum Rabi model via modulation of transition frequency of the two-level system,” Phys. Rev. A 96, 043849 (2017).
[Crossref]

M. P. Silveri, J. A. Tuorila, E. V. Thuneberg, and G. S. Paraoanu, “Quantum systems under frequency modulation,” Rep. Prog. Phys. 80, 056002 (2017).
[Crossref] [PubMed]

H. Tan, Y. Wei, and G. Li, “Building mechanical Greenberger-Horne-Zeilinger and cluster states by harnessing optomechanical quantum steerable correlations,” Phys. Rev. A 96, 052331 (2017).
[Crossref]

A. Passian and G. Siopsis, “Quantum state atomic force microscopy,” Phys. Rev. A 95, 043812 (2017).
[Crossref]

2016 (4)

A. Pontin, M. Bonaldi, A. Borrielli, L. Marconi, F. Marino, G. Pandraud, G. A. Prodi, P. M. Sarro, E. Serra, and F. Marin, “Dynamical two-mode squeezing of thermal fluctuations in a cavity optomechanical system,” Phys. Rev. Lett. 116, 103601 (2016).
[Crossref] [PubMed]

J. Q. Liao, J. F. Huang, and L. Tian, “Generation of macroscopic Schrödinger-cat states in qubit-oscillator systems,” Phys. Rev. A 93, 033853 (2016).
[Crossref]

P. Z. G. Fonseca, E. B. Aranas, J. Millen, T. S. Monteiro, and P. F. Barker, “Nonlinear dynamics and strong cavity cooling of levitated nanoparticles,” Phys. Rev. Lett. 117, 173602 (2016).
[Crossref] [PubMed]

R. W. Peterson, T. P. Purdy, N. S. Kampel, R. W. Andrews, P. L. Yu, K. W. Lehnert, and C. A. Regal, “Laser cooling of a micromechanical membrane to the quantum backaction limit,” Phys. Rev. Lett. 116, 063601 (2016).
[Crossref] [PubMed]

2015 (7)

J. Millen, P. Z. G. Fonseca, T. Mavrogordatos, T. S. Monteiro, and P. F. Barker, “Cavity cooling a single charged levitated nanosphere,” Phys. Rev. Lett. 114, 123602 (2015).
[Crossref] [PubMed]

Y. S. Patil, S. Chakram, L. Chang, and M. Vengalattore, “Thermalmehanical two-mode squeezing in an ultrahigh-Q membrane resonator,” Phys. Rev. Lett. 115, 017202 (2015).
[Crossref]

M. Bienert and P. Barberis-Blostein, “Optomechanical laser cooling with mechanical modulations,” Phys. Rev. A 91, 023818 (2015).
[Crossref]

E. E. Wollman, C. U. Lei, A. J. Weinstein, J. Suh, A. Kronwald, F. Marquardt, A. A. Clerk, and K. C. Schwab, “Quantum squeezing of motion in a mechanical resontor,” Science 349, 952–955 (2015).
[Crossref] [PubMed]

J. M. Pirkkalainen, E. Damskägg, M. Brandt, F. Massel, and M. A. Sillanpää, “Squeezing of quantum noise of motion in a micromechanical resonator,” Phys. Rev. Lett. 115, 243601 (2015).
[Crossref] [PubMed]

C. Dong, Y. Wang, and H. Wang, “Optomechanical interfaces for hybrid quantum networks,” National Science Review 2, 510–519 (2015).
[Crossref]

A. Jöckel, A. Faber, T. Kampschulte, M. Korppi, M. T. Rakher, and P. Treutlein, “Sympathetic cooling of a membrane oscillator in a hybrid mechanical-atomic system,” Nat. Nanotechnol. 10, 55–59 (2015).
[Crossref]

2014 (4)

A. Pontin, M. Bonaldi, A. Borrielli, F. S. Cataliotti, F. Marino, G. A. Prodi, E. Serra, and F. Marin, “Squeezing a thermal mechanical oscillator by stabilized parametric effect on the optical spring,” Phys. Rev. Lett. 112, 023601 (2014).
[Crossref] [PubMed]

A. B. Shkarin, N. E. Flowers-Jacobs, S. W. Hoch, A. D. Kashkanova, C. Deutsch, J. Reichel, and J. G. E. Harris, “Optically mediated hybridization between two mechanical modes,” Phys. Rev. Lett. 112, 013602 (2014).
[Crossref] [PubMed]

I. Mahboob, H. Okamoto, K. Onomitsu, and H. Yamaguchi, “Two-mode thermal noise squeezing in an electromechanical resonator,” Phys. Rev. Lett. 113, 167203 (2014).
[Crossref]

Q. He and Z. Ficek, “Einstein-Podolsky-Rosen paradox and quantum steering in a three-mode optomechanical system,” Phys. Rev. A 89, 022332 (2014).
[Crossref]

2013 (8)

Y. D. Wang and A. A. Clerk, “Reservoir-engineered entanglement in optomechanical systems,” Phys. Rev. Lett. 110, 253601 (2013).
[Crossref] [PubMed]

L. Tian, “Robust photon entanglement via quantum interference in optomechanical interfaces,” Phys. Rev. Lett. 110, 233602 (2013).
[Crossref] [PubMed]

H. Tan, G. Li, and P. Meystre, “Dissipation-diven two-mode mechnaical squeezed states in optomechanical systems,” Phys. Rev. A 87, 033829 (2013).
[Crossref]

H. Tan, L. F. Buchmann, H. Seok, and G. Li, “Achieving steady-state entanglement of remote micromechanical oscillators by cascaded cavity coupling,” Phys. Rev. A 87, 022318 (2013).
[Crossref]

A. Bassi, K. Lochan, S. Satin, T. P. Singh, and H. Ulbricht, “Models of wave-function collapse, underlying theories, and experimental tests,” Rev. Mod. Phys. 85, 471–527 (2013).
[Crossref]

A. Szorkovszky, G. A. Brawley, A. C. Doherty, and W. P. Bowen, “Strong thermomechanical squeezing via weak measurement,” Phys. Rev. Lett. 110, 184301 (2013).
[Crossref] [PubMed]

A. Vinante and P. Falferi, “Feedback-enhanced parametric squeezing of mechanical motion,” Phys. Rev. Lett. 111, 207203 (2013).
[Crossref] [PubMed]

T. A. Palomaki, J. W. Harlow, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, “Coherent state transfer between itinerant microwave fields and a mechanical oscillator,” Nature 495, 210–214 (2013).
[Crossref] [PubMed]

2012 (2)

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]

A. Farace and V. Giovannetti, “Enhancing quantum effects via periodic modulations in optomechanical systems,” Phys. Rev. A 86, 013820 (2012).
[Crossref]

2011 (5)

H. Krauter, C. A. Muschik, K. Jensen, W. Wasilewski, J. M. Petersen, J. I. Cirac, and E. S. Poizik, “Entanglement generated by dissipation and steady state entanglement of two macroscopic objects,” Phys. Rev. Lett. 107, 080503 (2011).
[Crossref] [PubMed]

C. A. Muschik, E. S. Polzik, and J. I. Cirac, “Dissipatively driven entanglement of two macroscopic atomic ensembles,” Phys. Rev. A 83, 052312 (2011).
[Crossref]

S. Camerer, M. Korppi, A. Jöckel, D. Hunger, T. W. Hänsch, and P. Treutlein, “Realization of an optomechanical interface between ultracold atoms and a membrane,” Phys. Rev. Lett. 107, 223001 (2011).
[Crossref] [PubMed]

J. D. Teufel, T. Donner, D. 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]

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
[Crossref] [PubMed]

2010 (1)

J. C. Sankey, C. Yang, B. M. Zwickl, M. Jayich, and J. G. E. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nature Phys 6, 707–712 (2010).
[Crossref]

2009 (2)

S. Groblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460, 724–727 (2009).
[Crossref] [PubMed]

A. Schliesser, O. Arcizet, R. Rivière, G. Anetsberger, and T. J. Kippenberg, “Resolved-sideband cooling and position measurement of a micromechanical oscillator close to the Heisenberg uncertainty limit,” Nat. Physics 5, 509–514 (2009).
[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. Wilson-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] [PubMed]

2004 (1)

P. Rabl, A. Shnirman, and P. Zoller, “Generation of squeezed states of nanomechanical resontors by reservoir engineering,” Phys. Rev. B 70, 205304 (2004).
[Crossref]

2003 (1)

W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, “Towards quantum superpositions of a mirror,” Phys. Rev. Lett. 91, 130401 (2003).
[Crossref] [PubMed]

2000 (1)

L. M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (2000).
[Crossref] [PubMed]

1999 (1)

S. Bose, K. Jacobs, and P. L. Knight, “Scheme to probe the decoherence of a macroscopic object,” Phys. Rev. A 59, 3204–3210 (1999).
[Crossref]

1991 (1)

D. Rugar and P. Grütter, “Mechanical parametric amplification and thermomechanical noise squeezing,” Phys. Rev. Lett. 67, 699–702 (1991).
[Crossref] [PubMed]

1987 (1)

E. X. DeJesus and C. Kaufman, “Routh-Hurwitz crierion in the examination of eigenvalues of a system of nonlinear ordinary differential equations,” Phys. Rev. A 35, 5288–5290 (1987).
[Crossref]

Alegre, T. P. M.

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
[Crossref] [PubMed]

Allman, M. S.

J. D. Teufel, T. Donner, D. 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]

Andrews, R. W.

R. W. Peterson, T. P. Purdy, N. S. Kampel, R. W. Andrews, P. L. Yu, K. W. Lehnert, and C. A. Regal, “Laser cooling of a micromechanical membrane to the quantum backaction limit,” Phys. Rev. Lett. 116, 063601 (2016).
[Crossref] [PubMed]

Anetsberger, G.

A. Schliesser, O. Arcizet, R. Rivière, G. Anetsberger, and T. J. Kippenberg, “Resolved-sideband cooling and position measurement of a micromechanical oscillator close to the Heisenberg uncertainty limit,” Nat. Physics 5, 509–514 (2009).
[Crossref]

Aranas, E. B.

P. Z. G. Fonseca, E. B. Aranas, J. Millen, T. S. Monteiro, and P. F. Barker, “Nonlinear dynamics and strong cavity cooling of levitated nanoparticles,” Phys. Rev. Lett. 117, 173602 (2016).
[Crossref] [PubMed]

Arcizet, O.

A. Schliesser, O. Arcizet, R. Rivière, G. Anetsberger, and T. J. Kippenberg, “Resolved-sideband cooling and position measurement of a micromechanical oscillator close to the Heisenberg uncertainty limit,” Nat. Physics 5, 509–514 (2009).
[Crossref]

Aspelmeyer, M.

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
[Crossref] [PubMed]

S. Groblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460, 724–727 (2009).
[Crossref] [PubMed]

Barberis-Blostein, P.

M. Bienert and P. Barberis-Blostein, “Optomechanical laser cooling with mechanical modulations,” Phys. Rev. A 91, 023818 (2015).
[Crossref]

Barker, P. F.

P. Z. G. Fonseca, E. B. Aranas, J. Millen, T. S. Monteiro, and P. F. Barker, “Nonlinear dynamics and strong cavity cooling of levitated nanoparticles,” Phys. Rev. Lett. 117, 173602 (2016).
[Crossref] [PubMed]

J. Millen, P. Z. G. Fonseca, T. Mavrogordatos, T. S. Monteiro, and P. F. Barker, “Cavity cooling a single charged levitated nanosphere,” Phys. Rev. Lett. 114, 123602 (2015).
[Crossref] [PubMed]

Bassi, A.

A. Bassi, K. Lochan, S. Satin, T. P. Singh, and H. Ulbricht, “Models of wave-function collapse, underlying theories, and experimental tests,” Rev. Mod. Phys. 85, 471–527 (2013).
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E. E. Wollman, C. U. Lei, A. J. Weinstein, J. Suh, A. Kronwald, F. Marquardt, A. A. Clerk, and K. C. Schwab, “Quantum squeezing of motion in a mechanical resontor,” Science 349, 952–955 (2015).
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Other (1)

X. Li, Y. Ma, J. Han, T. Chen, Y. Xu, W. Cai, H. Wang, Y. P. Song, Z. Y. Xue, Z. Q. Yin, and L. Sun, “Perfect remote quantum state transfer in a superconducting qubit chain with parametrically tunable couplings,” https://arxiv.org/abs/1806.03886. 03886.

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Figures (4)

Fig. 1
Fig. 1 Schematic of the two-mode optomechanical system consisted of cavity mode and the frequency-modulated mechanical oscillator. The cavity mode a ^ is driven by an external laser field with amplitude E and frequency ωL , and couples to the modulated oscillator via radiation pressure with the strength g.
Fig. 2
Fig. 2 Mechanical squeezing (dB) versus controllable parameter λ (G1/G0) with different initial thermal occupations. The other parameters are: γm = 2π ×0.18Hz, G0 = 2π ×20kHz.
Fig. 3
Fig. 3 Schematic of the three-mode optomechanical system consisted of a cavity mode and two frequency-modulated mechanical oscillators. The cavity mode a ^ is driven by an external laser field with amplitude E and frequency ωL , and couples to two modulated oscillators via radiation pressure with the strengths g1 and g2.
Fig. 4
Fig. 4 Entanglement indicated by degree of two-mode squeezing V versus controllable parameter G2/G1 with different initial thermal occupations. The other parameters are: γm = 2π × 0.18Hz, G1 = 2π × 20kHz.

Equations (42)

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H ^ = ω c a ^ a ^ + [ ω m + ϵ sin ( v t + φ ) ] b ^ b ^ g 0 a ^ a ^ ( b ^ + b ^ ) + i ( E a ^ e i ω L t E * a ^ e i ω L t ) ,
H ^ = Δ a ^ a ^ + [ ω m + ϵ sin ( v t + φ ) ] b ^ b ^ g 0 a ^ a ^ ( b ^ + b ^ ) + i ( E a ^ E * a ^ ) ,
U ^ ( t ) = exp [ i η cos ( v t + φ ) b ^ b ^ ]
H ^ ˜ = U ^ ( t ) H ^ U ^ ( t ) + i d U ^ ( t ) d t U ^ ( t ) ,
H ^ ˜ = Δ a ^ a ^ + ω m b ^ b ^ g 0 a ^ a ^ [ b ^ e i η cos ( v t + φ ) + b ^ e i η cos ( v t + φ ) ] + i ( E a ^ E * a ^ ) .
exp [ i η cos ( v t + φ ) ] = n = i n J n ( η ) exp [ i n ( v t + φ ) ] ,
H ^ ˜ = Δ c ^ c ^ + ω m d ^ d ^ g ( c ^ + c ^ ) n = i n e i n ( v t + φ ) J n ( η ) [ d ^ + ( 1 ) n d ^ ] ,
H ^ ˜ = n = i n e i n ( v t + φ ) g ˜ n [ c ^ d ^ e i ( ω m Δ ) t + ( 1 ) n c ^ d ^ e i ( ω m + Δ ) t ] + H .c ,
H ^ ˜ eff = g ˜ 0 ( c ^ d ^ + d ^ c ^ ) g ˜ 1 ( c ^ d ^ + c ^ d ^ ) .
d d t c ^ = κ 2 c ^ + i g ˜ 0 d ^ + i g ˜ 1 d ^ + κ c ^ in , d d t d ^ = γ m 2 d ^ + i g ˜ 0 c ^ + i g ˜ 1 c ^ + γ m d ^ in ,
c ^ in ( t ) c ^ in ( t ) = δ ( t t ) , d ^ in ( t ) d ^ in ( t ) = ( n th + 1 ) δ ( t t ) .
c ^ i g ˜ 0 κ / 2 d ^ + i g ˜ 1 κ / 2 d ^ + 2 κ c ^ in .
d d t d ^ = ( G 1 2 G 0 2 γ m 2 ) d ^ + i G 0 c ^ in + i G 1 c ^ in + γ m d ^ in , d d t d ^ = ( G 1 2 G 0 2 γ m 2 ) d ^ i G 0 c ^ in i G 1 c ^ in + γ m d ^ in ,
d ^ ( ω ) = 1 2 π d ^ ( t ) e i ω t d t , d ^ ( ω ) = 1 2 π d ^ ( t ) e i ω t d t ,
d ^ ( ω ) = i G 0 c ^ in ( ω ) + i G 1 c ^ in ( ω ) + γ m d ^ in ( ω ) i ω + ( G 1 G 0 γ m ) / 2 , d ^ ( ω ) = i G 0 c ^ in ( ω ) + i G 1 c ^ in ( ω ) γ m d ^ in ( ω ) i ω + ( G 1 G 0 γ m ) / 2 .
X ^ θ ( ω ) = 1 2 [ e i θ d ^ ( ω ) + e i θ d ^ ( ω ) ] .
S X θ ( ω ) = X ^ θ ( ω ) X ^ θ ( ω ) δ ( ω + ω ) = 1 2 G 0 + G 1 2 G 0 G 1 cos 2 θ + γ m ( 2 n th + 1 ) ω 2 + ( G 1 G 0 γ m ) 2 / 4 ,
X ^ θ 2 = 1 2 π S X θ ( ω ) d ω .
X ^ θ 2 = G 1 + γ m n th G 0 G 1 cos 2 θ G 0 G 1 + γ m + 1 2 ,
X ^ θ 2 = G 1 + γ m n th G 0 G 1 G 0 G 1 + γ m + 1 2 .
X ^ 0 2 λ 1 + λ + γ m n th G 0 ( 1 λ ) + 1 2 ,
λ = 1 + γ m n th G 0 γ m 2 n th 2 G 0 2 + 2 γ m n th G 0 .
H ^ = ω c a ^ a ^ + j = 1 2 { [ ω m ( j ) + ϵ j sin ( v j t + φ j ) ] b ^ j b ^ j g 0 ( j ) a ^ a ^ ( b ^ j + b ^ j ) } + i ( E a ^ e i ω L t E * a ^ e i ω L t ) ,
U ^ ( t ) = j = 1 2 exp [ i η j cos ( v j t + φ j ) b ^ j b ^ j ]
H ^ ˜ = Δ a ^ a ^ + j = 1 2 { ω m ( j ) b ^ j b ^ j g 0 ( j ) a ^ a ^ [ b ^ j e i η j cos ( v j t + φ j ) + b ^ j e i η j cos ( v j t + φ j ) ] } + i ( E a ^ E * a ^ ) ,
H ^ ˜ = Δ c ^ c ^ + j = 1 2 { ω m ( j ) d ^ j d ^ j g j ( c ^ + c ^ ) × n = i n e i n ( v j t + φ j ) J n ( η j ) [ d ^ j + ( 1 ) n d ^ j ] } .
H ^ ˜ int = j = 1 2 n = g j i n J n ( η j ) e i n ( v j t + φ j ) [ c ^ d ^ j e i ( Δ ω m ( j ) ) t + ( 1 ) n c ^ d ^ j e i ( Δ + ω m ( j ) ) t ] + H . c .
H ^ ˜ int = g ˜ 1 ( c ^ d ^ 1 + d ^ 1 c ^ ) g ˜ 2 ( c ^ d ^ 2 + c ^ d ^ 2 ) ,
d d t c ^ = κ 2 c ^ + i g ˜ 1 d ^ 1 + i g ˜ 2 d ^ 2 + κ c ^ in , d d t d ^ 1 = γ m ( 1 ) 2 d ^ 1 + i g ˜ 1 c ^ + γ m ( 1 ) d ^ 1,in , d d t d ^ 2 = γ m ( 2 ) 2 d ^ 2 i g ˜ 2 c ^ + γ m ( 2 ) d ^ 2,in ,
c ^ i g ˜ 1 κ / 2 d ^ 1 + i g ˜ 2 κ / 2 d ^ 2 + 2 κ c ^ in .
d d t d ^ 1 = ( G 1 2 + γ m ( 1 ) 2 ) d ^ 1 G 1 G 2 2 d ^ 2 + i G 1 c ^ in + γ m ( 1 ) d ^ 1 , in , d d t d ^ 2 = ( G 2 2 γ m ( 2 ) 2 ) d ^ 2 + G 1 G 2 2 d ^ 1 i G 2 c ^ in + γ m ( 2 ) d ^ 2 , in ,
G 1 > G 2 γ m ( 2 ) γ m ( 1 ) , G 1 > G 2 γ m ( 1 ) / γ m ( 2 ) γ m ( 1 ) .
d ^ 1 ( ω ) = χ 1 c ( ω ) c ^ in ( ω ) + χ 11 ( ω ) d ^ 1,in ( ω ) + χ 12 ( ω ) d ^ 2 , in ( ω ) , d ^ 2 ( ω ) = χ 2 c ( ω ) c ^ in ( ω ) + χ 21 ( ω ) d ^ 1,in ( ω ) + χ 22 ( ω ) d ^ 2 , in ( ω ) ,
χ 1 c ( ω ) = i G 1 ( i ω + γ m ( 2 ) / 2 ) / D ( ω ) , χ 11 ( ω ) = γ m ( 1 ) [ i ω ( G 2 γ m ( 2 ) ) / 2 ] / D ( ω ) , χ 12 ( ω ) = γ m ( 2 ) G 1 G 2 / 2 D ( ω ) , χ 2 c ( ω ) = G 2 ( i ω γ m ( 1 ) / 2 ) / D ( ω ) , χ 21 ( ω ) = γ m ( 1 ) G 1 G 2 / 2 D ( ω ) , χ 22 ( ω ) = γ m ( 2 ) [ i ω + ( G 1 + γ m ( 1 ) ) / 2 ] / D ( ω ) , D ( ω ) = [ i ω G 1 + γ m ( 1 ) 2 ] [ i ω + G 2 γ m ( 2 ) 2 ] + G 1 G 2 4 .
d ^ 1 d ^ 1 ( ω ) = | χ 11 ( ω ) | 2 n th ( 1 ) + | χ 12 ( ω ) | 2 ( n th ( 2 ) + 1 ) , d ^ 2 d ^ 2 ( ω ) = | χ 21 ( ω ) | 2 ( n th ( 1 ) + 1 ) + | χ 22 ( ω ) | 2 n th ( 2 ) + | χ 2 c ( ω ) | 2 , d ^ 1 d ^ 2 ( ω ) = χ 11 ( ω ) χ 21 ( ω ) ( n th ( 1 ) + 1 ) + χ 12 ( ω ) χ 22 ( ω ) n th ( 2 ) + χ 1 c ( ω ) χ 2 c ( ω ) .
d ^ 1 d ^ 1 = [ ( G 2 γ m ) ( G 2 2 γ m ) + G 1 γ m ] n th ( 1 ) + G 1 G 2 ( n th ( 2 ) + 1 ) ( G 1 G 2 + γ m ) ( G 1 G 2 + 2 γ m ) , d ^ 2 d ^ 2 = G 1 G 2 ( n th ( 1 ) + 1 ) + [ ( G 1 + γ m ) ( G 1 + 2 γ m ) G 2 γ m ] n th ( 2 ) ( G 1 + G 2 + γ m ) ( G 1 G 2 + 2 γ m ) + G 2 G 1 G 2 + γ m , d ^ 1 d ^ 2 = G 1 G 2 [ ( G 2 γ m ) n th ( 1 ) + ( G 1 + γ m ) ( n th ( 2 ) + 1 ) ] ( G 1 G 2 + γ m ) ( G 1 G 2 + 2 γ m ) .
d ^ 1 d ^ 1 = γ m n th ( 1 ) G 1 + γ m , d ^ 2 d ^ 2 = n th ( 2 ) , d ^ 1 d ^ 2 = 0.
V = 1 2 [ ( X ^ 1 + X ^ 2 ) 2 + ( P ^ 1 P ^ 2 ) 2 ] ,
X ^ j = 1 2 ( d ^ j + d ^ j ) , P ^ j = 1 2 ( d ^ j d ^ j ) .
V = d ^ 1 d ^ 1 + d ^ 2 d ^ 2 + 2 d ^ 1 d ^ 2 + 1.
V = G 2 n th ( 1 ) + G 1 n th ( 2 ) G 2 ( 2 G 1 + G 2 ) ( G 1 + G 2 ) 2 + 1.
w ^ = G 1 G 1 G 2 d ^ 1 + G 2 G 1 G 2 d ^ 2 , u ^ = G 2 G 1 G 2 d ^ 1 + G 1 G 1 G 2 d ^ 2 ,