Abstract

We propose a scheme for generation of the stationary continuous-variable entanglement and Einstein-Podolsky-Rosen (EPR) steering between an optical cavity mode and a nanomechanical resonator (NMR) mode. The cavity and the NMR are commonly coupled with two separated quantum dots (QDs), where the two QDs are driven simultaneously by a strong laser field. By adjusting the frequency of the strong laser field, the two QDs are nearly trapped on different dressed states, which is helpful to generate the entanglement between the cavity mode and the NMR mode. Due to the combined resonant interaction of the two QDs with the NMR-cavity subsystem, the photon and the phonon created and (or) annihilated are correlated. In this regime, the optimal entanglement of the two modes is obtained and the purity of the state of the NMR-cavity subsystem is near to 1. Furthermore, the coupling strength between the cavity and two QDs is different from the dot-NMR coupling strength, which leads to the different mean occupation numbers of the cavity and the NMR. In this case, one-way EPR steering is observed. In addition, through analyzing the purity, we find the conditions of the existence for the different types of EPR steering.

© 2015 Optical Society of America

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References

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2015 (3)

I. Kogias, A. R. Lee, S. Ragy, and G. Adesso, “Quantification of Gaussian quantum steering,” Phys. Rev. Lett. 114, 060403 (2015).
[Crossref] [PubMed]

S. Armstrong, M. Wang, R. Y. Teh, Q. H. Gong, Q. Y. He, J. Janousek, H. A. Bachor, M. D. Reid, and P. K. Lam, “Multipartite Einstein-Podolsky-Rosen steering and genuine tripartite entanglement with optical networks,” Nature Phys. 11, 167–172 (2015).
[Crossref]

H. T. Tan, X. C. Zhang, and G. X. Li, “Steady-state one-way Einstein-Podolsky-Rosen steering in optomechanical interfaces,” Phys. Rev. A 91, 032121 (2015).
[Crossref]

2014 (9)

J. Bowles, T. V’ertesi, M. T. Quintino, and N. Brunner, “One-way Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 112, 200402 (2014).
[Crossref]

P. Chowdhury, T. Pramanik, A. S. Majumdar, and G. S. Agarwal, “Einstein-Podolsky-Rosen steering using quantum correlations in non-Gaussian entangled states,” Phys. Rev. A 89, 012104 (2014).
[Crossref]

Y. Y. Liu, K. D. Petersson, J. Stehlik, J. M. Taylor, and J. R. Petta, “Photon emission from a cavity-coupled double quantum dot,” Phys. Rev. Lett. 113, 036801 (2014).
[Crossref] [PubMed]

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

G. H. Hovsepyan, A. R. Shahinyan, and G. Yu, Kryuchkyan, “Multiphoton blockades in pulsed regimes beyond stationary limits,” Phys. Rev. A 90, 013839 (2014).
[Crossref]

H. Wang, X. Gu, Y. X. Liu, A. Miranowicz, and F. Nori, “Optomechanical analog of two-color electromagnetically induced transparency: photon transmission through an optomechanical device with a two-level system,” Phys. Rev. A 90, 023817 (2014).
[Crossref]

T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J. M. Taylor, A. Sørensen, K. Usami, A. Schliesser, and E. S. Polzik, “Optical detection of radio waves through a nanomechanical transducer,” Nature 507, 81–85 (2014).
[Crossref] [PubMed]

M. J. Wolley and A. A. Clerk, “Two-mode squeezed states in cavity optomechanics via engineering of a single reservoir,” Phys. Rev. A 89, 063805 (2014).
[Crossref]

J. R. Johansson, N. Lambert, I. Mahboob, H. Yamaguchi, and F. Nori, “Entangled-state generation and Bell inequality violations in nanomechanical resonators,” Phys. Rev. B 90, 174307 (2014).
[Crossref]

2013 (12)

T. A. Palomaki, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, “Entangling mechanical motion with microwave fields,” Science 342, 710–713 (2013).
[Crossref] [PubMed]

X. W. Xu, Y. J. Zhao, and Y. X. Liu, “Entangled-state engineering of vibrational modes in a multimembrane optomechanical system,” Phys. Rev. A 88, 022325 (2013).
[Crossref]

Z. L. Xiang, S. Ashhab, J. Q. You, and F. Nori, “Hybrid quantum circuits: superconducting circuits interacting with other quantum system,” Rev. Mod. Phys. 85, 623 (2013).
[Crossref]

G. Z. Cohen and M. D. Ventra, “Reading, writing, and squeezing the entangled states of two nanomechanical resonators coupled to a SQUID,” Phys. Rev. B 87, 014513 (2013).
[Crossref]

A. Voje, A. Isacsson, and A. Croy, “Nonlinear-dissipation-induced entanglement of coupled nonlinear oscillators,” Phys. Rev. A 88, 022309 (2013).
[Crossref]

T. V. Gevorgyan and G. Yu. Kryuchkyan, “Parametrically driven nonlinear oscillator at a few-photon level,” J. Mod. Opt. 60, 860–868 (2013).
[Crossref]

A. Miranowicz, M. Paprzycka, Y. X. Liu, J. Bajer, and F. Nori, “Two-photon and three-photon blockades in driven nonlinear systems,” Phys. Rev. A 87, 023809 (2013).
[Crossref]

Q. Y. He and M. D. Reid, “Einstein-Podolsky-Rosen paradox and quantum steering in pulsed optomechanics,” Phys. Rev. A 88, 052121 (2013).
[Crossref]

M. R. Delbecq, L. E. Bruhat, J. J. Viennot, S. Datta, A. Cottet, and T. Kontos, “Photon-mediated interaction between distant quantum dot circuits,” Nat. Commun. 4, 1400 (2013).
[Crossref] [PubMed]

S. Das and M. A. Macovei, “Collective quantum dot inversion and amplification of photon and phonon waves,” Phys. Rev. B 88, 125306 (2013).
[Crossref]

J. J. Li and K. D. Zhu, “All-optical mass sensing with coupled mechanical resonator systems,” Phys. Rep. 525, 223–254 (2013).
[Crossref]

P. Grünwald and W. Vogel, “Optimal squeezing in the resonance fluorescence of single-photon emitters,” Phys. Rev. A 88, 023837 (2013).
[Crossref]

2012 (8)

O. Basarir, S. Bramhavar, and K. L. Ekinci, “Monolithic integration of a nanomechanical resonator to an optical microdisk cavity,” Opt. Express 20(4), 4272–4279 (2012).
[Crossref] [PubMed]

P. Grünwald and W. Vogel, “Optimal squeezing in resonance fluorescence via atomic-state purification,” Phys. Rev. Lett. 109, 013601 (2012).
[Crossref] [PubMed]

V. Haändchen, T. Eberle, S. Steinlechner, A. Samblowski, T. Franz, R. F. Werner, and R. Schnabel, “Observation of one-way Einstein-Podolsky-Rosen steering,” Nat. Photon. 6, 596–599 (2012).
[Crossref]

D. H. Smith, G. Gillett, M. P. D. Almeida, C. Branciard, A. Fedrizzi, T. J. Weinhold, A. Lita, B. Calkins, T. Gerrits, H. M. Wiseman, S. W. Nam, and A. G. White, “Conclusive quantum steering with superconducting transition-edge sensors,” Nat. Commun. 3, 625 (2012).
[Crossref] [PubMed]

A. Kowalewska-Kudłaszyk, W. Leoński, and J. Peřina, “Generalized Bell states generation in a parametrically excited nonlinear coupler,” Phys. Scr. T147, 014016 (2012).
[Crossref]

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]

N. Bergeal, F. Schackert, L. Frunzio, and M. H. Devoret, “Two-mode correlation of microwave quantum noise generated by parametric down-conversion,” Phys. Rev. Lett. 108, 123902 (2012).
[Crossref] [PubMed]

S. Rips, M. Kiffner, I. Wilson-Rae, and M. J. Hartmann, “Steady-state negative Wigner functions of nonlinear nanomechanical oscillators,” New J. Phys. 14, 023042 (2012).
[Crossref]

2011 (4)

A. Eichler, J. Moser, J. Chaste, M. Zdrojek, I. Wilson-Rae, and A. Bachtold, “Nonlinear damping in mechanical resonators made from carbon nanotubes and graphene,” Nature Nano. 6, 339–342 (2011).
[Crossref]

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

S. G. Hofer, W. Wieczorek, M. Aspelmeyer, and K. Hammerer, “Quantum entanglement and teleportation in pulsed cavity optomechanics,” Phys. Rev. A 84, 052327 (2011).
[Crossref]

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]

2010 (3)

A. Majumdar, N. Manquest, A. Faraon, and J. Vučković, “Theory of electro-optic modulation via a quantum dot coupled to a nano-resonator,” Opt. Express 18(5), 3974–3984 (2010).
[Crossref] [PubMed]

A Kowalewska-Kudłaszyk and W Leoński, “Squeezed vacuum reservoir effect for entanglement decay in the nonlinear quantum scissor system,” J. Phys. B: At. Mol. Opt. Phys. 43, 205503 (2010).
[Crossref]

D. J. Saunders, S. J. Jones, H. M. Wiseman, and G. J. Pryde, “Experimental EPR-steering using Bell-local states,” Nature Phys. 6, 845–849 (2010).
[Crossref]

2009 (2)

K. Hammerer, M. Aspelmeyer, E. S. Polzik, and P. Zoller, “Establishing Einstein-Poldosky-Rosen channels between nanomechanics and atomic ensembles,” Phys. Rev. Lett. 102, 020501 (2009).
[Crossref] [PubMed]

S. Zippilli, G. Morigi, and A. Bachtold, “Cooling carbon nanotubes to the phononic ground state with a constant electron current,” Phys. Rev. Lett. 102, 096804 (2009).
[Crossref] [PubMed]

2008 (2)

C. L. Degen, M. Poggio, H. J. Mamin, and D. Rugar, “Nuclear spin relaxation induced by a mechanical resonator,” Phys. Rev. Lett. 100, 137601 (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)

M. Macovei and G. X. Li, “Entangled light via nonlinear vacuum-multiparticle interactions,” Phys. Rev. A 76, 023818 (2007).
[Crossref]

S. C. Masmanidis, R. B. Karabalin, I. De Vlaminck, G. Borghs, M. R. Freeman, and M. L. Roukes, “Multifunctional nanomechanical systems via tunably coupled piezoelectric actuation,” Science 317, 780–783 (2007).
[Crossref] [PubMed]

2006 (1)

G. X. Li, T. H. Tan, and S. S. Ke, “Quantum-feedback-induced enhancement of continuous-variable entanglement in a self-phase-locked type-II nondegenerate optical parameter oscillator,” Phys. Rev. A 74, 012304 (2006).
[Crossref]

2005 (1)

K. C. Schwab and M.L. Roukes, “Putting mechanics into quantum mechanics,” Phys. Today 58(7), 36–42 (2005).
[Crossref]

2004 (1)

L. Tian and P. Zoller, “Coupled ion-nanomechanical systems,” Phys. Rev. Lett. 93, 266403 (2004).
[Crossref]

2000 (1)

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

1997 (1)

M. Löffler, D. E. Nikonov, O. A. Kocharovskaya, and M. O. Scully, “Strong-field index enhancement via selective population of dressed states,” Phys. Rev. A 56, 5014 (1997).
[Crossref]

1992 (1)

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein-Podolsky-Rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663 (1992).
[Crossref] [PubMed]

1989 (1)

M. D. Reid, “Demonstration of the Einstein-Podolsky-Rosen paradox using nondegenerate parametric amplification,” Phys. Rev. A 40, 913 (1989).
[Crossref] [PubMed]

Adesso, G.

I. Kogias, A. R. Lee, S. Ragy, and G. Adesso, “Quantification of Gaussian quantum steering,” Phys. Rev. Lett. 114, 060403 (2015).
[Crossref] [PubMed]

Agarwal, G. S.

P. Chowdhury, T. Pramanik, A. S. Majumdar, and G. S. Agarwal, “Einstein-Podolsky-Rosen steering using quantum correlations in non-Gaussian entangled states,” Phys. Rev. A 89, 012104 (2014).
[Crossref]

Alegre, T. P. M.

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Groblacher, 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).
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Figures (5)

Fig. 1
Fig. 1 (a) Schematic plot of two QDs that adsorb on the surface of the NMR inside a driven cavity. (b) Energy levels of the coupled QD-cavity system where the QD is red (the 1st QD) and blue (the 2nd QD) detuned from the laser.
Fig. 2
Fig. 2 The steady entanglement ϒ and the purity P characterizing the entanglement as a function of ωm for g 1 =4, λ 1 =2.5, Ω =50, Δ1 =98, κa =0.4, κb =0.001, n ¯ = 0.5 when only one QD is in the cavity.
Fig. 3
Fig. 3 The steady variance ϒ and the purity P characterizing the entanglement as a function of ωm for g 2 =1.1, λ 2 =1.6, Ω =50, Δ2 =−98, κa =0.01 when two QDs are in the cavity. The other parameters are the same with that in Fig. 2.
Fig. 4
Fig. 4 (a) The mean photon 〈a a〉 of the cavity mode and the mean phonon 〈b b〉 of the NMR mode as the functions of ωm. (b) The one-way EPR steering parameters V a|b and V b|a characterizing EPR steering as a function of ωm. The parameters are estimated follows: g 1 =3.8, g 2 =1.2, λ 2 =1.8, κa =0.1. The other parameters are the same with that in Fig. 3.
Fig. 5
Fig. 5 (a) The functions 1 P a + 1 P b and 1 P + 1 of three purities depend on ωm. The inset is quantity ϒ characterizing the entanglement versus ωm. (b)the global purity P and the marginal purities Pa , Pb as the functions of ωm. The parameters are the same with that in Fig. 4.

Equations (42)

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H = H 0 + H I ,
H 0 = j = 1 , 2 [ Δ j S z j + Ω ( S j + S + j ) ] Δ c a a + ω m b b ,
H I = j = 1 , 2 [ g j ( a S j + a S + j ) + λ j S z j ( b + b ) ] ,
| + j = s j | g j + c j | e j , | j = c j | g j s j | e j ,
S + j = s j c j R z j + c j 2 R + j s j 2 R + , S z j = 1 2 ( c j 2 s j 2 ) R z j s j c j ( R + j + R + j ) ,
H 0 = j = 1 , 2 Ω ¯ j R z j ω m a a + ω m b b .
V I = e i H 0 t H I e i H 0 t ,
V I = j = 1 , 2 ( { [ ( g j c j 2 a λ j s j c j b ) e i ( 2 Ω ¯ j + ω m ) t ( g j s j 2 a + λ j s j c j b ) e i ( 2 Ω ¯ j ω m ) t ] R + j + { [ λ j s j c j 2 a + λ j ( c j 2 s j 2 ) 2 b ] e i ω m t + h . c . } R z ) .
ρ ˙ = i [ V I , ρ ] + ρ ,
ρ = κ a D ( a ) + κ b ( n ¯ + 1 ) D ( b ) + κ b n ¯ D ( b ) + j = 1 , 2 [ γ j 0 D ( R + j ) + γ j D ( R + j ) ] ,
d d t ρ c = A 1 D ( a ) + A 2 D ( a ) + B 1 D ( b ) + B 2 D ( b ) + i F [ a b + a b , ρ ] + [ D 1 ( 2 a ρ b b a ρ ρ b a ) + D 2 ( 2 b ρ a a b ρ ρ a b ) + h . c . ] ,
d d t a a = x 1 a a y * a b + A 1 + c . c . ,
d d t b b = x 2 b b + y a b + B 1 + c . c . ,
d d t a b = ( x 1 + x 2 ) a b + y * ( a a + 1 ) y b b 2 D 1 ,
Re [ x 1 x 2 + ( x 1 x 2 ) 2 4 y 2 ] < 0 ,
M i j = T r [ ρ ( Δ ξ ^ 1 Δ ξ ^ 1 + Δ ξ ^ 1 Δ ξ ^ 1 ) / 2 ] = ( ξ ^ 1 ξ ^ 1 + ξ ^ 1 ξ ^ 1 ) / 2 ,
M = ( n 1 0 c 0 0 n 1 0 c c 0 n 2 0 0 c 0 n 2 ) .
Σ = 2 n 1 ς 2 + 2 n 2 ς 2 4 | c | < ς 2 + 1 ς 2 .
ϒ = Σ ς 2 1 ς 2 = 2 n 1 ς 2 + 2 n 2 ς 2 4 | c | ς 2 1 ς 2 < 0.
P = Tr ( ρ 2 ) = 1 4 ( n 1 n 2 c 2 ) .
V I 1 = [ ( g 1 s 1 2 a + λ 1 s 1 c 1 b ) R + 1 + h . c . ] .
V ˜ I 1 = S ( r ) V I 1 S ( r ) = A ( a ˜ R + 1 + h . c . ) .
V ˜ I 12 = [ a ˜ R + 1 + b ˜ R + 2 + h . c . ] .
Δ inf 2 X 1 Δ inf 2 P 1 < 1 4 ,
Δ inf 2 X 1 = ( X 1 k 1 X 2 ) 2 ,
Δ inf 2 P 1 = ( P 1 k 2 P 2 ) 2 ,
k 1 = X 1 X 2 X 2 2 , k 2 = P 1 P 2 P 2 2 .
V a | b = a a ( b b + 1 2 ) | a b | < 0.
V b | a = b b ( a a + 1 2 ) | a b | < 0.
a a b b | a b | < 0.
P a = Tr a ( ρ a 2 ) = 2 2 n 1 , P b = Tr b ( ρ b 2 ) = 1 2 n 2 .
1 P a + 1 P b > 1 P + 1 ,
P b < P < P a .
P a < P < 1 2 ( n 2 n 1 ) + 1 .
P + j = s j 4 s j 4 + c j 4 , P j = c j 4 s j 4 + c j 4 .
S 1 ( ω m ) = 0 δ R + j ( τ ) δ R + j ( 0 ) e i ω m τ d τ = 4 γ j + Γ j ( Γ j + i ω m ) ,
S 2 ( ω m ) = 0 δ R + j ( τ ) δ R + j ( 0 ) e i ω m τ d τ = 4 γ j Γ j ( Γ j + i ω m ) ,
S 3 ( ω m ) = 0 δ R z j ( τ ) δ R z j ( 0 ) e i ω m τ d τ = 64 γ j + γ j Γ j 2 ( Γ j + i ω m ) ,
A 1 j = g j 2 [ 4 s j 2 c j 2 P j + P j Γ j Γ j 2 + ω m 2 + s j 4 P j Γ j Γ j 2 + ( ω m 2 Ω j ) 2 + c j 4 P j + Γ j Γ j 2 + ( ω m + 2 Ω j ) 2 ] ,
B 1 j = λ j 2 [ ( c j 2 s j 2 ) 2 P j + P j Γ j Γ j 2 + ω m 2 + s j 2 c j 2 P j + Γ j Γ j 2 + ( ω m 2 Ω ¯ j ) 2 + s j 2 c j 2 P j Γ j Γ j 2 + ( ω m + 2 Ω ¯ j ) 2 ] + κ b n ¯ ,
D 1 j = g j λ j s j c j [ 2 ( c j 2 s j 2 ) P j + P j Γ j Γ j 2 + ω m 2 + s j 2 P j + Γ j Γ j 2 + ( ω m 2 Ω ¯ j ) 2 c j 2 P j Γ j Γ j 2 + ( ω m + 2 Ω ¯ j ) 2 ] ,
F 1 j = g j λ j s j c j [ s j 2 P j + ( ω m 2 Ω ¯ j ) Γ j 2 + ( ω m 2 Ω ¯ j ) 2 c j 2 P j ( ω m + 2 Ω ¯ j ) Γ j 2 + ( ω m + 2 Ω ¯ j ) 2 ] .

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