Abstract

We propose a setup allowing to entangle two directly non-interacting radiation modes applying four sequential pulsed quantum resonant interactions with a noisy vibrational mode of a mechanical oscillator which plays the role of the mediator. We analyze Gaussian entanglement of the radiation modes generated by the transducer and confirm that the noisy mechanical mode can mediate generation of entanglement. The entanglement, however, is limited if the interaction gains are not individually optimized. We prove the robustness of the transducer to optical losses and the influence of the mechanical bath and propose the ways to achieve maximal performance through the individual optimization.

© 2017 Optical Society of America

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2016 (6)

M. Asjad, P. Tombesi, and D. Vitali, “Feedback control of two-mode output entanglement and steering in cavity optomechanics,” Phys. Rev. A 94, 052312 (2016).
[Crossref]

F. Q. Lecocq, J. B. Clark, R. W. Simmonds, J. A. Aumentado, and J. D. Teufel, “Mechanically mediated microwave frequency conversion in the quantum regime,” Phys. Rev. Lett. 116, 043601 (2016).
[Crossref] [PubMed]

J. S. Bennett, K. Khosla, L. S. Madsen, M. R. Vanner, H. Rubinsztein-Dunlop, and W. P. Bowen, “A quantum optomechanical interface beyond the resolved sideband regime,” New J. Phys. 18, 053030 (2016).
[Crossref]

A. A. Rakhubovsky, N. Vostrosablin, and R. Filip, “Squeezer-based pulsed optomechanical interface,” Phys. Rev. A 93, 033813 (2016).
[Crossref]

H. Ogawa, H. Ohdan, K. Miyata, M. Taguchi, K. Makino, H. Yonezawa, J. Yoshikawa, and A. Furusawa, “Real-time quadrature measurement of a single-photon wave packet with continuous temporal-mode matching,” Phys. Rev. Lett. 116, 233602 (2016).
[Crossref] [PubMed]

K. Makino, Y. Hashimoto, J. Yoshikawa, H. Ohdan, T. Toyama, P. van Loock, and A. Furusawa, “Synchronization of optical photons for quantum information processing,” Science Advances 2, e1501772 (2016).
[Crossref] [PubMed]

2015 (6)

V. Kupčík and Radim Filip, “Continous-variable entanglement mediated by a thermal oscillator,” Phys. Rev. A 92, 022346 (2015).
[Crossref]

L. Tian, “Optoelectromechanical transducer: Reversible conversion between microwave and optical photons,” Annalen der Physik 527, 1–14 (2015).
[Crossref]

F. Lecocq, J. B. Clark, R. W. Simmonds, J. Aumentado, and J. D. Teufel, “Quantum nondemolition measurement of a nonclassical state of a massive object,” Phys. Rev. X 5, 041037 (2015)

S. M. Meenehan, J. D. Cohen, G. S. MacCabe, F. Marsili, M. D. Shaw, and O. Painter, “Pulsed excitation dynamics of an optomechanical crystal resonator near its quantum ground state of motion,” Phys. Rev. X 5, 041002 (2015).

C. Dong, V. Fiore, M. C. Kuzyk, L. Tian, and H. Wang, “Optical wavelength conversion via optomechanical coupling in a silica resonator,” Ann. Phys. (Berlin) 527, 100 (2015).
[Crossref]

M. Asjad, S. Zippilli, P. Tombesi, and D. Vitali, “Large distance continuous variable communication with concatenated swaps,” Physica Scripta 90, 7 (2015).
[Crossref]

2014 (7)

R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, “Bidirectional and efficient conversion between microwave and optical light,” Nature Physics 10, 321–326 (2014).
[Crossref]

T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J.M. Taylor, A. S. 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]

Y. Miwa, J.-I. Yoshikawa, N. Iwata, M. Endo, P. Marek, R. Filip, P. van Loock, and A. Furusawa, “Exploring a New Regime for Processing Optical Qubits: Squeezing and Unsqueezing Single Photons,” Phys. Rev. Lett. 113, 013601 (2014).
[Crossref] [PubMed]

K. Xia, M. R. Vanner, and J. Twamley, “An opto-magneto-mechanical quantum interface between distant superconducting qubits,” Sci. Rep. 4, 5571 (2014).
[Crossref] [PubMed]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391 (2014).
[Crossref]

M. Metcalfe, “Applications of cavity optomechanics,” Appl. Phys. Rev. 1, 031105 (2014).
[Crossref]

J. Suh, A. J. Weinstein, C. U. Lei, E. E. Wollman, S. K. Steinke, P. Meystre, A. A. Clerk, and K. C. Schwab, “Mechanically detecting and avoiding the quantum fluctuations of a microwave field,” Science 344 (6189), 1262–1265 (2014).
[Crossref] [PubMed]

2013 (8)

M. R. Vanner, J. Hofer, G. D. Cole, and M. Aspelmeyer, “Cooling-by-measurement and mechanical state tomography via pulsed optomechanics,” Nature Communications 4, 2295 (2013).
[Crossref] [PubMed]

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]

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]

S. Takeda, T. Mizuta, M. Fuwa, P. van Loock, and A. Furusawa, “Determenistic quantum teleportation of photonic quantum bits by a hybrid technique”, Nature 500, 315–318 (2013).
[Crossref] [PubMed]

S. A. McGee, D. Meiser, C. A. Regal, K. W. Lehnert, and M. J. Holland, “Mechanical resonators for storage and transfer of electrical and optical quantum states,” Phys. Rev. A 87, 053818 (2013).
[Crossref]

Y. Liu, M. Davanco, V. Aksyuk, and K. Srinivasan, “Electromagnetically induced transparency and wideband wavelength conversion in silicon nitride microdisk optomechanical resonators,” Phys. Rev. Lett. 110, 223603 (2013).
[Crossref] [PubMed]

J. Bochmann, A. Vainsencher, D. D. Awschalom, and A. N. Cleland, “Nanomechanical coupling between microwave and optical photons,” Nature Physics 9, 712–716 (2013).
[Crossref]

K. E. Khosla, M. R. Vanner, W. P. Bowen, and G. J. Milburn, “Quantum state preparation of a mechanical resonator using an optomechanical geometric phase,” New. J. Phys. 15, 043025 (2013).
[Crossref]

2012 (10)

G. Vacanti, R. Fazio, M. S. Kim, G. M. Palme, M. Paternostro, and V. Vedral, “Geometric phase kickback in a mesoscopic qubit-oscillator system,” Phys. Rev. A 85, 022129 (2012).
[Crossref]

I. Pikovsky, M. R. Vanner, M. Aspelmeyer, M. S. Kim, and Č. Brukner, “Probing Planck-scale physics with quantum optics,” Nature Phys. 8, 393–397 (2012).
[Crossref]

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
[Crossref]

J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 31196 (2012).
[Crossref] [PubMed]

Sh. Barzanjeh, M. Abdi, G. J. Milburn, P. Tombesi, and D. Vitali, “Reversible Optical-to-Microwave Quantum Interface,” Phys. Rev. Lett. 109, 130503 (2012).
[Crossref] [PubMed]

Y-D Wang and A. A. Clerk, “Using interference for high fidelity quantum state transfer in optomechanics,” Phys. Rev. Lett. 108, 153604 (2012).
[Crossref]

L. Tian, “Adiabatic state conversion and pulse transmission in optomechanical system,” Phys. Rev. Lett. 108, 153604 (2012).
[Crossref]

Y-D Wang and A. A. Clerk, “Using dark modes for high-fidelity optomechanical quantum state transfer,” New. J. Phys. 14, 105010 (2012).
[Crossref]

P. C. Maurer, G. Kucsko, C. Latta, L. Jiang, N.Y. Yao, S. D. Bennett, F. Pastawski, D. Hunger, N. Chisholm, M. Markham, D. J. Twitchen, J. I. Cirac, and M. D. Lukin, “Room-Temperature Quantum Bit Memory Exceeding One Second,” Science 336, 1283–1286 (2012).
[Crossref] [PubMed]

M. Hafezi, Z. Kim, S. L. Rolston, L. A. Orozco, B. L. Lev, and J. M. Taylor, “Atomic interface between microwave and optical photos,” Phys. Rev. A 85, 020302 (2012).
[Crossref]

2011 (5)

C. A. Regal and W. Lehnert, “From cavity electromechanics to cavity optomechanics,” Journal of Physics: Conference Series 264, 012025 (2011).

A. Huck, S. Kumar, A. Shakoor, and U. L. Andersen, “Controlled Coupling of a Single Nitrogen-Vacancy Center to a Silver Nanowire,” Phys. Rev. Lett. 106, 096801 (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]

M. Winger, T. D. Blasius, T. P. Mayer Alegre, A. H. Safavi-Naeini, S. Meenehan, J. Cohen, S. Stobbe, and O. Painter, “A chip-scale integrated cavity-electro-optomechanics platform,” Opt. Express 19, 24905–24921 (2011).
[Crossref]

M. R. Vanner, I. Pikovsky, G. D. Cole, M. S. Kim, Č. Brukner, K. Hammerer, G. J. Milburn, and M. Aspelmeyer, “Pulsed Quantum Optomechanics,” PNAS 108, 16182 (2011).
[Crossref] [PubMed]

2010 (3)

L. Tian and H. Wang, “Optical wavelength conversion of quantum states with optomechanics,” Phys. Rev. A. 82, 053806 (2010).
[Crossref]

P. Marek and R. Filip, “Noise-resilient quantum interface based on quantum nondemolition interactions,” Phys. Rev. A 81, 042325 (2010).
[Crossref]

D. Marcos, M. Wubs, J. M. Taylor, R. Aguado, M. D. Lukin, and A. S. Sørensen, “Coupling nitrogen-vacancy centers in diamond to superconducting flux qubits,” Phys. Rev. Lett. 105, 210501 (2010).
[Crossref]

2009 (4)

J. Verdú, H. Zoubi, Ch. Koller, J. Majer, H. Ritsch, and J. Schmiedmayer, “Strong magnetic coupling of an ultracold gas to a superconducting waveguide cavity,” Phys. Rev. Lett. 103, 043603 (2009).
[Crossref] [PubMed]

M. Wallquist, K. Hammerer, P. Rabl, M. Lukin, and P. Zoller, “Hybrid quantum devices and quantum engineering,” Phys. Scr. 2009, 014001 (2009).
[Crossref]

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]

K. Hammerer, M. Wallquist, C. Genes, M. Ludwig, F. Marquardt, P. Treutlein, P. Zoller, J. Ye, and H. J. Kimble, “Strong Coupling of a Mechanical Oscillator and a Single Atom,” Phys. Rev. Lett. 103, 063005 (2009).
[Crossref] [PubMed]

2008 (2)

A. A. Clerk, F. Marquardt, and K. Jacobs, “Back-action evasion ans squeezing of a mechanical resonator using a cavity detector,” New J. Phys. 10, 095010 (2008).
[Crossref]

H. J. Kimble, “The quantum internet,” Nature 453, 1023–1030 (2008).
[Crossref] [PubMed]

2003 (1)

D. Leibfried, B. DeMarco, V. Meyer, D. Lucas, M. Barett, J. Britton, W. M. Itano, B. Jelenković, C. Lange, T. Rosenband, and D. J. Wineland, “Experimental demonstration of a robust high-fidelity geometric two ion-qubit phase gate,” Nature 422, 412–415 (2003).
[Crossref] [PubMed]

2001 (2)

V. Giovannetti and D. Vitali, “Phase-noise measurement in a cavity with a movable mirror undergoing quantum Brownian motion,” Phys. Rev. A 63, 023812 (2001).
[Crossref]

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87, 050402 (2001).
[Crossref] [PubMed]

2000 (4)

L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84(12), 2722–2725 (2000).
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R. Simon, “Peres-Horodecki separability criterion for continuous variable systems,” Phys. Rev. Lett. 84(12), 2726–2729 (2000).
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A. S. Sørensen and K. Mølmer, “Entanglement and quantum computation with ions in thermal motion,” Phys. Rev. A. 62, 022311 (2000).
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G. J. Milburn, S. Schneider, and D. F. V. James, “Ion trap quantum computing with warm atoms,” Fortschr. Phys. 48, 801–810 (2000).
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C. K. Law, “Interaction between a moving mirror and radiation pressure: A Hamiltonian formulation,” Phys. Rev. A 51, 2537–2541 (1995).
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1987 (1)

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1407 (1987).
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1980 (1)

V. B. Braginsky, Y. I. Vorontsov, and K. S. Thorne, “Quantum Nondemolition Measurements,” Science 209, 547–557 (1980).
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1978 (1)

V. B. Braginsky, Y. I. Vorontsov, and F. Y. Khalili, “Optimal quantum measurement in detectors of gravitational radiation,” JETP Lett 27, 276 (1978).

Abdi, M.

Sh. Barzanjeh, M. Abdi, G. J. Milburn, P. Tombesi, and D. Vitali, “Reversible Optical-to-Microwave Quantum Interface,” Phys. Rev. Lett. 109, 130503 (2012).
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Aguado, R.

D. Marcos, M. Wubs, J. M. Taylor, R. Aguado, M. D. Lukin, and A. S. Sørensen, “Coupling nitrogen-vacancy centers in diamond to superconducting flux qubits,” Phys. Rev. Lett. 105, 210501 (2010).
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Aichele, T.

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87, 050402 (2001).
[Crossref] [PubMed]

Aksyuk, V.

Y. Liu, M. Davanco, V. Aksyuk, and K. Srinivasan, “Electromagnetically induced transparency and wideband wavelength conversion in silicon nitride microdisk optomechanical resonators,” Phys. Rev. Lett. 110, 223603 (2013).
[Crossref] [PubMed]

Andersen, U. L.

A. Huck, S. Kumar, A. Shakoor, and U. L. Andersen, “Controlled Coupling of a Single Nitrogen-Vacancy Center to a Silver Nanowire,” Phys. Rev. Lett. 106, 096801 (2011).
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Andrews, R. W.

R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, “Bidirectional and efficient conversion between microwave and optical light,” Nature Physics 10, 321–326 (2014).
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Appel, J.

T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J.M. Taylor, A. S. Sørensen, K. Usami, A. Schliesser, and E. S. Polzik, “Optical detection of radio waves through a nanomechanical transducer,” Nature 507, 81–85 (2014).
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M. Asjad, P. Tombesi, and D. Vitali, “Feedback control of two-mode output entanglement and steering in cavity optomechanics,” Phys. Rev. A 94, 052312 (2016).
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M. Asjad, S. Zippilli, P. Tombesi, and D. Vitali, “Large distance continuous variable communication with concatenated swaps,” Physica Scripta 90, 7 (2015).
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M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391 (2014).
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M. R. Vanner, J. Hofer, G. D. Cole, and M. Aspelmeyer, “Cooling-by-measurement and mechanical state tomography via pulsed optomechanics,” Nature Communications 4, 2295 (2013).
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I. Pikovsky, M. R. Vanner, M. Aspelmeyer, M. S. Kim, and Č. Brukner, “Probing Planck-scale physics with quantum optics,” Nature Phys. 8, 393–397 (2012).
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S. G. Hofer, W. Wieczorek, M. Aspelmeyer, and K. Hammerer, “Quantum entanglement and teleportation in pulsed cavity optomechanics,” Phys. Rev. A. 84, 052327 (2011).
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M. R. Vanner, I. Pikovsky, G. D. Cole, M. S. Kim, Č. Brukner, K. Hammerer, G. J. Milburn, and M. Aspelmeyer, “Pulsed Quantum Optomechanics,” PNAS 108, 16182 (2011).
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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]

Aumentado, J.

F. Lecocq, J. B. Clark, R. W. Simmonds, J. Aumentado, and J. D. Teufel, “Quantum nondemolition measurement of a nonclassical state of a massive object,” Phys. Rev. X 5, 041037 (2015)

Aumentado, J. A.

F. Q. Lecocq, J. B. Clark, R. W. Simmonds, J. A. Aumentado, and J. D. Teufel, “Mechanically mediated microwave frequency conversion in the quantum regime,” Phys. Rev. Lett. 116, 043601 (2016).
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J. Bochmann, A. Vainsencher, D. D. Awschalom, and A. N. Cleland, “Nanomechanical coupling between microwave and optical photons,” Nature Physics 9, 712–716 (2013).
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T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J.M. Taylor, A. S. 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]

Barett, M.

D. Leibfried, B. DeMarco, V. Meyer, D. Lucas, M. Barett, J. Britton, W. M. Itano, B. Jelenković, C. Lange, T. Rosenband, and D. J. Wineland, “Experimental demonstration of a robust high-fidelity geometric two ion-qubit phase gate,” Nature 422, 412–415 (2003).
[Crossref] [PubMed]

Barzanjeh, Sh.

Sh. Barzanjeh, M. Abdi, G. J. Milburn, P. Tombesi, and D. Vitali, “Reversible Optical-to-Microwave Quantum Interface,” Phys. Rev. Lett. 109, 130503 (2012).
[Crossref] [PubMed]

Bennett, J. S.

J. S. Bennett, K. Khosla, L. S. Madsen, M. R. Vanner, H. Rubinsztein-Dunlop, and W. P. Bowen, “A quantum optomechanical interface beyond the resolved sideband regime,” New J. Phys. 18, 053030 (2016).
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Bennett, S. D.

P. C. Maurer, G. Kucsko, C. Latta, L. Jiang, N.Y. Yao, S. D. Bennett, F. Pastawski, D. Hunger, N. Chisholm, M. Markham, D. J. Twitchen, J. I. Cirac, and M. D. Lukin, “Room-Temperature Quantum Bit Memory Exceeding One Second,” Science 336, 1283–1286 (2012).
[Crossref] [PubMed]

Benson, O.

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87, 050402 (2001).
[Crossref] [PubMed]

Berry, M. V.

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1407 (1987).
[Crossref]

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Bochmann, J.

J. Bochmann, A. Vainsencher, D. D. Awschalom, and A. N. Cleland, “Nanomechanical coupling between microwave and optical photons,” Nature Physics 9, 712–716 (2013).
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Bowen, W. P.

J. S. Bennett, K. Khosla, L. S. Madsen, M. R. Vanner, H. Rubinsztein-Dunlop, and W. P. Bowen, “A quantum optomechanical interface beyond the resolved sideband regime,” New J. Phys. 18, 053030 (2016).
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K. E. Khosla, M. R. Vanner, W. P. Bowen, and G. J. Milburn, “Quantum state preparation of a mechanical resonator using an optomechanical geometric phase,” New. J. Phys. 15, 043025 (2013).
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W. P. Bowen and G. J. Milburn, Quantum Optomechanics (CRC Press, 2015), Appendix A “Linear detection of optical fields”.

Braginsky, V. B.

V. B. Braginsky, Y. I. Vorontsov, and K. S. Thorne, “Quantum Nondemolition Measurements,” Science 209, 547–557 (1980).
[Crossref] [PubMed]

V. B. Braginsky, Y. I. Vorontsov, and F. Y. Khalili, “Optimal quantum measurement in detectors of gravitational radiation,” JETP Lett 27, 276 (1978).

Britton, J.

D. Leibfried, B. DeMarco, V. Meyer, D. Lucas, M. Barett, J. Britton, W. M. Itano, B. Jelenković, C. Lange, T. Rosenband, and D. J. Wineland, “Experimental demonstration of a robust high-fidelity geometric two ion-qubit phase gate,” Nature 422, 412–415 (2003).
[Crossref] [PubMed]

Brukner, C.

I. Pikovsky, M. R. Vanner, M. Aspelmeyer, M. S. Kim, and Č. Brukner, “Probing Planck-scale physics with quantum optics,” Nature Phys. 8, 393–397 (2012).
[Crossref]

M. R. Vanner, I. Pikovsky, G. D. Cole, M. S. Kim, Č. Brukner, K. Hammerer, G. J. Milburn, and M. Aspelmeyer, “Pulsed Quantum Optomechanics,” PNAS 108, 16182 (2011).
[Crossref] [PubMed]

Cerf, N. J.

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
[Crossref]

Chan, J.

J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 31196 (2012).
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Chisholm, N.

P. C. Maurer, G. Kucsko, C. Latta, L. Jiang, N.Y. Yao, S. D. Bennett, F. Pastawski, D. Hunger, N. Chisholm, M. Markham, D. J. Twitchen, J. I. Cirac, and M. D. Lukin, “Room-Temperature Quantum Bit Memory Exceeding One Second,” Science 336, 1283–1286 (2012).
[Crossref] [PubMed]

Cicak, K.

R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, “Bidirectional and efficient conversion between microwave and optical light,” Nature Physics 10, 321–326 (2014).
[Crossref]

Cirac, J. I.

P. C. Maurer, G. Kucsko, C. Latta, L. Jiang, N.Y. Yao, S. D. Bennett, F. Pastawski, D. Hunger, N. Chisholm, M. Markham, D. J. Twitchen, J. I. Cirac, and M. D. Lukin, “Room-Temperature Quantum Bit Memory Exceeding One Second,” Science 336, 1283–1286 (2012).
[Crossref] [PubMed]

L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84(12), 2722–2725 (2000).
[Crossref] [PubMed]

Clark, J. B.

F. Q. Lecocq, J. B. Clark, R. W. Simmonds, J. A. Aumentado, and J. D. Teufel, “Mechanically mediated microwave frequency conversion in the quantum regime,” Phys. Rev. Lett. 116, 043601 (2016).
[Crossref] [PubMed]

F. Lecocq, J. B. Clark, R. W. Simmonds, J. Aumentado, and J. D. Teufel, “Quantum nondemolition measurement of a nonclassical state of a massive object,” Phys. Rev. X 5, 041037 (2015)

Cleland, A. N.

J. Bochmann, A. Vainsencher, D. D. Awschalom, and A. N. Cleland, “Nanomechanical coupling between microwave and optical photons,” Nature Physics 9, 712–716 (2013).
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Clerk, A. A.

J. Suh, A. J. Weinstein, C. U. Lei, E. E. Wollman, S. K. Steinke, P. Meystre, A. A. Clerk, and K. C. Schwab, “Mechanically detecting and avoiding the quantum fluctuations of a microwave field,” Science 344 (6189), 1262–1265 (2014).
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Y-D Wang and A. A. Clerk, “Using dark modes for high-fidelity optomechanical quantum state transfer,” New. J. Phys. 14, 105010 (2012).
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Y-D Wang and A. A. Clerk, “Using interference for high fidelity quantum state transfer in optomechanics,” Phys. Rev. Lett. 108, 153604 (2012).
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A. A. Clerk, F. Marquardt, and K. Jacobs, “Back-action evasion ans squeezing of a mechanical resonator using a cavity detector,” New J. Phys. 10, 095010 (2008).
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Cohen, J. D.

S. M. Meenehan, J. D. Cohen, G. S. MacCabe, F. Marsili, M. D. Shaw, and O. Painter, “Pulsed excitation dynamics of an optomechanical crystal resonator near its quantum ground state of motion,” Phys. Rev. X 5, 041002 (2015).

Cole, G. D.

M. R. Vanner, J. Hofer, G. D. Cole, and M. Aspelmeyer, “Cooling-by-measurement and mechanical state tomography via pulsed optomechanics,” Nature Communications 4, 2295 (2013).
[Crossref] [PubMed]

M. R. Vanner, I. Pikovsky, G. D. Cole, M. S. Kim, Č. Brukner, K. Hammerer, G. J. Milburn, and M. Aspelmeyer, “Pulsed Quantum Optomechanics,” PNAS 108, 16182 (2011).
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Davanco, M.

Y. Liu, M. Davanco, V. Aksyuk, and K. Srinivasan, “Electromagnetically induced transparency and wideband wavelength conversion in silicon nitride microdisk optomechanical resonators,” Phys. Rev. Lett. 110, 223603 (2013).
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DeMarco, B.

D. Leibfried, B. DeMarco, V. Meyer, D. Lucas, M. Barett, J. Britton, W. M. Itano, B. Jelenković, C. Lange, T. Rosenband, and D. J. Wineland, “Experimental demonstration of a robust high-fidelity geometric two ion-qubit phase gate,” Nature 422, 412–415 (2003).
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C. Dong, V. Fiore, M. C. Kuzyk, L. Tian, and H. Wang, “Optical wavelength conversion via optomechanical coupling in a silica resonator,” Ann. Phys. (Berlin) 527, 100 (2015).
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L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84(12), 2722–2725 (2000).
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Endo, M.

Y. Miwa, J.-I. Yoshikawa, N. Iwata, M. Endo, P. Marek, R. Filip, P. van Loock, and A. Furusawa, “Exploring a New Regime for Processing Optical Qubits: Squeezing and Unsqueezing Single Photons,” Phys. Rev. Lett. 113, 013601 (2014).
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G. Vacanti, R. Fazio, M. S. Kim, G. M. Palme, M. Paternostro, and V. Vedral, “Geometric phase kickback in a mesoscopic qubit-oscillator system,” Phys. Rev. A 85, 022129 (2012).
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A. A. Rakhubovsky, N. Vostrosablin, and R. Filip, “Squeezer-based pulsed optomechanical interface,” Phys. Rev. A 93, 033813 (2016).
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Y. Miwa, J.-I. Yoshikawa, N. Iwata, M. Endo, P. Marek, R. Filip, P. van Loock, and A. Furusawa, “Exploring a New Regime for Processing Optical Qubits: Squeezing and Unsqueezing Single Photons,” Phys. Rev. Lett. 113, 013601 (2014).
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P. Marek and R. Filip, “Noise-resilient quantum interface based on quantum nondemolition interactions,” Phys. Rev. A 81, 042325 (2010).
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V. Kupčík and Radim Filip, “Continous-variable entanglement mediated by a thermal oscillator,” Phys. Rev. A 92, 022346 (2015).
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C. Dong, V. Fiore, M. C. Kuzyk, L. Tian, and H. Wang, “Optical wavelength conversion via optomechanical coupling in a silica resonator,” Ann. Phys. (Berlin) 527, 100 (2015).
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H. Ogawa, H. Ohdan, K. Miyata, M. Taguchi, K. Makino, H. Yonezawa, J. Yoshikawa, and A. Furusawa, “Real-time quadrature measurement of a single-photon wave packet with continuous temporal-mode matching,” Phys. Rev. Lett. 116, 233602 (2016).
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K. Makino, Y. Hashimoto, J. Yoshikawa, H. Ohdan, T. Toyama, P. van Loock, and A. Furusawa, “Synchronization of optical photons for quantum information processing,” Science Advances 2, e1501772 (2016).
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Y. Miwa, J.-I. Yoshikawa, N. Iwata, M. Endo, P. Marek, R. Filip, P. van Loock, and A. Furusawa, “Exploring a New Regime for Processing Optical Qubits: Squeezing and Unsqueezing Single Photons,” Phys. Rev. Lett. 113, 013601 (2014).
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S. Takeda, T. Mizuta, M. Fuwa, P. van Loock, and A. Furusawa, “Determenistic quantum teleportation of photonic quantum bits by a hybrid technique”, Nature 500, 315–318 (2013).
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Fuwa, M.

S. Takeda, T. Mizuta, M. Fuwa, P. van Loock, and A. Furusawa, “Determenistic quantum teleportation of photonic quantum bits by a hybrid technique”, Nature 500, 315–318 (2013).
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García-Patrón, R.

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
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Genes, C.

K. Hammerer, M. Wallquist, C. Genes, M. Ludwig, F. Marquardt, P. Treutlein, P. Zoller, J. Ye, and H. J. Kimble, “Strong Coupling of a Mechanical Oscillator and a Single Atom,” Phys. Rev. Lett. 103, 063005 (2009).
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P. Treutlein, C. Genes, K. Hammerer, M. Poggio, and P. Rabl, “Hybrid Mechanical Systems,” in Cavity Optomechanics, M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, eds., (Springer, 2014)

Giedke, G.

L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84(12), 2722–2725 (2000).
[Crossref] [PubMed]

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V. Giovannetti and D. Vitali, “Phase-noise measurement in a cavity with a movable mirror undergoing quantum Brownian motion,” Phys. Rev. A 63, 023812 (2001).
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M. Hafezi, Z. Kim, S. L. Rolston, L. A. Orozco, B. L. Lev, and J. M. Taylor, “Atomic interface between microwave and optical photos,” Phys. Rev. A 85, 020302 (2012).
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M. R. Vanner, I. Pikovsky, G. D. Cole, M. S. Kim, Č. Brukner, K. Hammerer, G. J. Milburn, and M. Aspelmeyer, “Pulsed Quantum Optomechanics,” PNAS 108, 16182 (2011).
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S. G. Hofer, W. Wieczorek, M. Aspelmeyer, and K. Hammerer, “Quantum entanglement and teleportation in pulsed cavity optomechanics,” Phys. Rev. A. 84, 052327 (2011).
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K. Hammerer, M. Wallquist, C. Genes, M. Ludwig, F. Marquardt, P. Treutlein, P. Zoller, J. Ye, and H. J. Kimble, “Strong Coupling of a Mechanical Oscillator and a Single Atom,” Phys. Rev. Lett. 103, 063005 (2009).
[Crossref] [PubMed]

M. Wallquist, K. Hammerer, P. Rabl, M. Lukin, and P. Zoller, “Hybrid quantum devices and quantum engineering,” Phys. Scr. 2009, 014001 (2009).
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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]

P. Treutlein, C. Genes, K. Hammerer, M. Poggio, and P. Rabl, “Hybrid Mechanical Systems,” in Cavity Optomechanics, M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, eds., (Springer, 2014)

Hansen, H.

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87, 050402 (2001).
[Crossref] [PubMed]

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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).
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K. Makino, Y. Hashimoto, J. Yoshikawa, H. Ohdan, T. Toyama, P. van Loock, and A. Furusawa, “Synchronization of optical photons for quantum information processing,” Science Advances 2, e1501772 (2016).
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J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 31196 (2012).
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Hofer, J.

M. R. Vanner, J. Hofer, G. D. Cole, and M. Aspelmeyer, “Cooling-by-measurement and mechanical state tomography via pulsed optomechanics,” Nature Communications 4, 2295 (2013).
[Crossref] [PubMed]

Hofer, S. G.

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

A. Huck, S. Kumar, A. Shakoor, and U. L. Andersen, “Controlled Coupling of a Single Nitrogen-Vacancy Center to a Silver Nanowire,” Phys. Rev. Lett. 106, 096801 (2011).
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Hunger, D.

P. C. Maurer, G. Kucsko, C. Latta, L. Jiang, N.Y. Yao, S. D. Bennett, F. Pastawski, D. Hunger, N. Chisholm, M. Markham, D. J. Twitchen, J. I. Cirac, and M. D. Lukin, “Room-Temperature Quantum Bit Memory Exceeding One Second,” Science 336, 1283–1286 (2012).
[Crossref] [PubMed]

Itano, W. M.

D. Leibfried, B. DeMarco, V. Meyer, D. Lucas, M. Barett, J. Britton, W. M. Itano, B. Jelenković, C. Lange, T. Rosenband, and D. J. Wineland, “Experimental demonstration of a robust high-fidelity geometric two ion-qubit phase gate,” Nature 422, 412–415 (2003).
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Figures (5)

Fig. 1
Fig. 1 (a) Schematic representation of the pulsed transducer interconnecting two radiation modes A and B which may be both optical as well as optical and microwave fields. The first amplitude-modulated pulse A (red) containing classical Stokes and anti-Stokes sidebands on ω cav A ± ω m and quantum fluctuations, which are in the vacuum state |0〉, on ω cav A, is sent to the first cavity to interact with mechanical mode M during the time τA. After the interaction is complete the pulse is sent to the delay line whereas the second pulse B (blue), being in the vacuum state |0〉 as well, interacts with M during τB within the second cavity. The interactions of the two pulses are then repeated one more time and both pulses are released to the outputs. The pulses do not overlap during the protocol. Radiation pulses are subject to losses Tls and the mechanical mode is coupled at rate γ to the mechanical bath with mean occupation number nth. (b) Block-diagram of the sequence of the interactions between the pulses of modes A and B with the mechanical mediator M.

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Fig. 2
Fig. 2 Logarithmic negativity EN as the function of QND coupling strength η = g 2 τ κ in the lossless adiabatic case (top purple curve) and in the case of radiation losses Tls present (lower solid curves). See definition of Tls in (7). Dotted lines correspond to the case of the optimal combination of the strengths of individual interactions. It demonstrates that radiation losses only partially limit generation of entanglement from the transducer. The optimization of gains is efficient only for large loss.

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Fig. 3
Fig. 3 Logarithmic negativity EN as the function of the QND coupling η = g 2 τ κ. For both top and bottom figures solid and dashed lines correspond to the non-optimized case with respectively g or τ varied, dotted lines with markers show the result of optimization. For the plots we used γ = 1.5 × 10−6 κ. (a) Lossless case in presence of mechanical bath. Parameters are varied in the following regions: 0 ≤ g ≤ 0.4κ (for fixed τ = 687/κ); 7 × 102/κ ≤ τ ≤ 9 × 104/κ (for fixed g = 0.03κ). This plot demonstrates that mechanical bath does not affect the entanglement drastically and the optimization is efficient for larger mechanical bath occupation numbers. (b) Mechanical bath and radiation losses. Parameters here are varied in the following regions: 0 ≤ g ≤ 0.4κ (for fixed τ = 687/κ); 7 × 102/κ ≤ τ ≤ 9 × 104/κ (for fixed g = 0.03κ). This plot demonstrates that the performance of the proposed transducer may be quite high. Even for large bath occupation nth = 200 the optimization helps to reach significant values of the entanglement.

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Fig. 4
Fig. 4 Optimized logarithmic negativity as the function of the radiation losses Tls for different values of the initial occupation number nth. This figure reflects the fact that the influence of the radiation losses on the performance of the setup is more significant than the one of the thermal state of the mechanical mediator. For high value of losses (Tls ≪ 1), independently from the value of initial occupation, the optimization does not help sufficiently.

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Fig. 5
Fig. 5 Logarithmic negativity EN as the function of QND coupling η ′ = 2 g A g B τ A τ B κ A κ B in the presence of mechanical bath and radiation losses for the case of asymmetric transducer. Solid lines correspond to the non-optimized cases with gA,B varied (for fixed τA = 690/κA and τB = 80/κB), dashed lines — for the same case with τA,B increased (for fixed gA = 0.03κA and gB = 0.03κB), dotted lines with markers stand for the optimized cases. Parameters are varied in the following regions: 0 ≤ gA ≤ 0.07κA; 0 ≤ gB ≤ 0.1κB; 2.2 × 102/κA ≤ τA ≤ 4.4 × 103/κA; 2.3/κB ≤ τB ≤ 113/κB with κB = 0.01 × κA and γ = 1.5 × 10−4κB. Brown dashed line is responsible for the changes in τB while changes in τA in the corresponding region does not lead to any entanglement appearance thus demonstrating the asymmetry of the system. This figure demonstrates that the proposed transducer is feasible to entangle optical and microwave fields with the state of the art experimental possibilities.

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Equations (28)

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ℋ = ℏ ω c 4 ( X 2 + Y 2 ) + ℏ ω m 4 ( p 2 + q 2 ) − ℏ g 0 4 q ( X 2 + Y 2 ) ,
ℋ i = ℏ ϰ i q Y A                   or                   ℋ j = ℏ ϰ j p X B ,
X ˙ A = − κ A X A + 2 κ A X A in + ϰ 1 q , X ˙ B = − κ B X B + 2 κ B X B in , Y ˙ A = − κ A Y A + 2 κ A Y A in , Y ˙ B = − κ B Y B + 2 κ B Y B in − ϰ 2 p , q ˙ = − γ 2 q + γ ξ x 1 , q ˙ = − γ 2 q + γ ξ x 2 + ϰ 2 X B , p ˙ = − γ 2 p + γ ξ p 1 − ϰ 1 Y A , p ˙ = − γ 2 p + γ ξ p 2 ,
Q out ( t ) = 2 κ Q ( t ) − Q in ( t ) ,
𝒳 A out = 𝒳 A in − η 2 𝒳 B in , 𝒳 B out = 𝒳 B in , q = q ( 0 ) , 𝒴 A out = 𝒴 A in , 𝒴 B out = 𝒴 B in + η 2 𝒴 A in , p = p ( 0 ) ,
ω m ≫ κ ≫ τ − 1 ≫ γ n th .
E N = max [ 0 , − log 2 ν − ] ,
ν − = 1 2 Σ ( V ) − Σ ( V ) 2 − 4 det V ,
Σ ( V ) = det 𝒱 1 + det 𝒱 2 − 2 det 𝒱 c ,
V = [ 𝒱 1 𝒱 c 𝒱 c T 𝒱 2 ] .
ν − 0 = 1 − 2 η 4 [ 1 + η − 4 − 1 ] .
𝒬 → T ls 𝒬 + 1 − T ls Q ls
ν − ≃ 1 − 1 2 ( 1 + T ls ) η 2 .
∂ E N ∂ η | η → 0 ≃ 2 ln 2 T ls η .
[ ξ x i , p i ( t ) , ξ x i , p i ( t ′ ) ] = 2 i δ ( t − t ′ ) ,
〈 ξ x i , p i ( t ) ξ x i , p i ( t ′ ) 〉 = ( 2 n th + 1 ) δ ( t − t ′ ) .
0 = − κ X A + 2 κ X A in + ϰ q ; q ˙ = 0 ; 0 = − κ Y A + 2 κ Y A in ; p ˙ = − ϰ Y A .
X A ( t ) = 2 κ X A in ( t ) + ϰ κ q ( t ) ,                     Y A ( t ) = 2 κ Y A in ( t ) .
q ( τ ) = q ( 0 ) ;                     p ( τ ) = p ( 0 ) − ϰ ∫ 0 τ Y A ( t ) d t = p ( 0 ) − ϰ 2 κ ∫ 0 τ Y A in ( t ) d t ≡ p ( 0 ) − ϰ 2 τ κ 𝒴 A in ,
X A out ( t ) = X A in ( t ) + ϰ 2 κ q ( 0 ) ,                     Y A out ( t ) = Y A in ( t ) .
X B out ( t ) = X B in ( t ) ,             Y B out ( t ) = Y B in ( t ) − ϰ 2 κ p ( τ ) ,             q ( 2 τ ) = q ( τ ) + ϰ 2 τ κ 𝒳 B in ,           p ( 2 τ ) = p ( τ ) .
X A in ( t ) = X A out ( t − 2 τ ) ,                   Y A in ( t ) = Y A out ( t − 2 τ ) ,               for     2 τ ≤ t ≤ 3 τ .
Y A ( t ) = 2 κ Y A in ( t ) = 2 κ Y A out ( t − 2 τ ) = 2 κ Y A in ( t − 2 τ ) ,                   for   2 τ ≤ t ≤ 3 τ ,
p ( 3 τ ) = p ( 2 τ ) + ϰ ∫ 2 τ 3 τ Y A ( t ) d t = p ( 0 ) − ϰ 2 κ [ ∫ 0 τ Y A in ( t ) d t − ∫ 2 τ 3 τ Y A in ( t − 2 τ ) d t ] = p ( 0 ) .
X A out ( t ) = X A in ( t ) − ϰ 2 κ q ( 2 τ ) = X A in ( t − 2 τ ) + ϰ 2 κ q ( 0 ) − ϰ 2 κ ( q ( 0 ) + ϰ 2 τ κ 𝒳 B in ) = X A in ( t − 2 τ ) − 2 ϰ 2 τ κ 𝒳 B in .
q ( 3 τ ) = q ( 2 τ ) ,           Y A out ( t ) = Y A in ( t − 2 τ ) .
𝒳 A out ≡ 1 τ ∫ 2 τ 3 τ d t X A out ( t ) = 1 τ ∫ 0 τ d t X A in ( t ) − 2 ϰ 2 τ κ 𝒳 B in ≡ 𝒳 A in − η 2 𝒳 B in , 𝒴 A out ≡ 1 τ ∫ 2 τ 3 τ d t Y A out ( t ) = 𝒴 A in .
𝒳 B out ≡ 1 τ ∫ 3 τ 4 τ d t X B out ( t ) = 𝒳 B in , 𝒴 B out ≡ 1 τ ∫ 3 τ 4 τ d t Y B out ( t ) = 𝒴 B in + η 2 𝒴 A in .

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