Abstract

We investigated the estimation of an unknown Gaussian process (containing displacement, squeezing and phase-shift) applied to a matter system. The state of the matter system is not directly measured; instead, we measure an optical mode which interacts with the system. We propose an interferometric setup exploiting a beam-splitter-type of light-matter interaction with homodyne detectors and two methods of estimation. We demonstrate the superiority of the interferometric setup over alternative non-interferometric schemes. Importantly, we show that even limited coupling strength and a noisy matter system are sufficient for very good estimation. Our work opens the way to many future investigations of light-matter interferometry for experimental platforms in quantum metrology of matter systems.

© 2017 Optical Society of America

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References

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    [Crossref]
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2017 (1)

L. Ruppert and R. Filip, “Estimation of nonclassical independent Gaussian processes by classical interferometry,” Sci. Rep. 7, 39641 (2017).
[Crossref] [PubMed]

2016 (4)

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

R. Riedinger, S. Hong, R. A. Norte, J. A. Slater, J. Shang, A. G. Krause, V. Anant, M. Aspelmeyer, and S. Gröblacher, “Non-classical correlations between single photons and phonons from a mechanical oscillator,” Nature 530, 313–316 (2016).
[Crossref] [PubMed]

Y. Xuan, Y. Liu, L. T. Varghese, A. J. Metcalf, X. Xue, P. H. Wang, K. Han, J. A. Jaramillo-Villegas, A. Al Noman, C. Wang, S. Kim, M. Teng, Y. J. Lee, B. Niu, L. Fan, J. Wang, D. E. Leaird, A. M. Weiner, and M. Qi, “High-Q silicon nitride microresonators exhibiting low-power frequency comb initiation,” Optica 3(11), 1171–1180 (2016).
[Crossref]

M. Jachura, R. Chrapkiewicz, R. Demkowicz-Dobrzański, W. Wasilewski, and K. Banaszek, “Mode engineering for realistic quantum-enhanced interferometry,” Nature Comm. 7, 11411 (2016).
[Crossref]

2015 (5)

W. Wieczorek, S. G. Hofer, J. Hoelscher-Obermaier, R. Riedinger, K. Hammerer, and M. Aspelmeyer, “Optimal State Estimation for Cavity Optomechanical Systems,” Phys. Rev. Lett. 114, 223601 (2015).
[Crossref] [PubMed]

J.P. Dowling and K.P. Seshadreesan, “Quantum Optical Technologies for Metrology, Sensing, and Imaging,” J. Lightwave Tech. 33, 2359–2370 (2015).
[Crossref]

R. Demkowicz-Dobrzański, M. Jarzyna, and J. Kołodyński, “Quantum Limits in Optical Interferometry,” Progress in Optics 60, 345–435 (2015).
[Crossref]

D. Šafránek, A. R. Lee, and I. Fuentes, “Quantum parameter estimation using multi-mode Gaussian states,” New J. Phys. 17, 073016 (2015).
[Crossref]

G. Kurizki, P. Bertetb, Y. Kubob, K. Molmer, D Petrosyan, P. Rabl, and J. Schmiedmayer, “Quantum technologies with hybrid systems,” PNAS 112, 3866–3873 (2015).
[Crossref] [PubMed]

2014 (3)

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

N. Otterstrom, R. C. Pooser, and B. J. Lawrie, “Nonlinear optical magnetometry with accessible in situ optical squeezing,” Opt. Lett. 39(22), 6533–6536 (2014).
[Crossref] [PubMed]

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 Phys. 10, 321–326 (2014).
[Crossref]

2013 (5)

O. Pinel, P. Jian, N. Treps, C. Fabre, and D. Braun, “Quantum parameter estimation using general single-mode Gaussian states,” Phys. Rev. A 88, 040102(2013).
[Crossref]

S. A. Haine, “Information-Recycling Beam Splitters for Quantum Enhanced Atom Interferometry,” Phys. Rev. Lett. 110, 053002 (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]

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

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

2012 (3)

E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature 482, 63–67 (2012).
[Crossref] [PubMed]

O. Pinel, J. Fade, D. Braun, P. Jian, N. Treps, and C. Fabre, “Ultimate sensitivity of precision measurements with intense Gaussian quantum light: A multimodal approach,” Phys. Rev. A 85, 010101 (2012).
[Crossref]

X.-H. Bao, A. Reingruber, P. Dietrich, J. Rui, A. Dück, T. Strassel, L. Li, N. Liu, B. Zhao, and J.-W. Pan, “Efficient and long-lived quantum memory with cold atoms inside a ring cavity,” Nature Phys. 8, 517–521 (2012).
[Crossref]

2011 (3)

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nature Photon. 5, 222 (2011).
[Crossref]

M. Napolitano, M. Koschorreck, B. Dubost, N. Behbood, R. J. Sewell, and M. W. Mitchell, “Interaction-based quantum metrology showing scaling beyond the Heisenberg limit,” Nature 471, 486–489 (2011).
[Crossref] [PubMed]

M. Hosseini, B. M. Sparkes, G. Campbell, P.K. Lam, and B.C. Buchler, “High efficiency coherent optical memory with warm rubidium vapour,” Nature Commun. 2, 174 (2011).
[Crossref]

2010 (5)

M.P. Hedges, J.J. Longdell, Y. Li, and M.J. Sellars, “Efficient quantum memory for light,” Nature 465, 1052–1056 (2010).
[Crossref] [PubMed]

Y. W. Cho and Y. H. Kim, “Atomic vapor quantum memory for a photonic polarization qubit,” Opt. Express 18(25), 25786–25793 (2010).
[Crossref] [PubMed]

K. Hammerer, A. S. Sørensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. 82, 1041 (2010).
[Crossref]

W. Wasilewski, K. Jensen, H. Krauter, J. J. Renema, M. V. Balabas, and E. S. Polzik, “Quantum noise limited and entanglement-assisted magnetometry,” Phys. Rev. Lett. 104, 133601 (2010).
[Crossref] [PubMed]

F. Wolfgramm, A. Cere, F.A. Beduini, A. Predojević, M. Koschorreck, and M. W. Mitchell, “Squeezed-light optical magnetometry,” Phys. Rev. Lett. 105, 053601 (2010).
[Crossref] [PubMed]

2009 (6)

R. Gaiba and M. G. Paris, “Squeezed vacuum as a universal quantum probe,” Phys. Lett. A 373, 934–939 (2009).
[Crossref]

A. D. Cronin, J. Schmiedmayer, and D. E. Pritchard, “Optics and interferometry with atoms and molecules,” Rev. Mod. Phys. 81, 1051 (2009).
[Crossref]

M. Wallquist, K. Hammerer, P. Rabl, M. Lukin, and P. Zoller, “Hybrid quantum devices and quantum engineering,” Phys. Scripta 2009(T137), 014001 (2009).
[Crossref]

M. Hosseini, B.M. Sparkes, G. Hétet, J.J. Longdel, P.K. Lam, and B.C. Buchler, “Coherent optical pulse sequencer for quantum applications,” Nature 461, 241–245 (2009).
[Crossref] [PubMed]

B. Zhao, Y.-A. Chen, X.-H. Bao, T. Strassel, Ch.-S. Chuu, X.-M. Jin., J. Schmiedmayer, Z.-S. Yuan, S. Chen, and J.-W. Pan, “A millisecond quantum memory for scalable quantum networks,” Nature Phys. 5, 95–99 (2009).
[Crossref]

R. Zhao, Y. O. Dudin, S. D. Jenkins, C. J. Campbell, D. N. Matsukevich, T. A. B. Kennedy, and A. Kuzmich, “Long-lived quantum memory,” Nature Phys. 5, 100–104 (2009).
[Crossref]

2008 (1)

J. Appel, E. Figueroa, D. Korystov, M. Lobino, and A. I. Lvovsky, “Quantum memory for squeezed light,” Phys. Rev. Lett. 100, 093602 (2008).
[Crossref] [PubMed]

2007 (2)

J. Simon, H. Tanji, J.K. Thompson, and V. Vuletic, “Interfacing collective atomic excitations and single photons,” Phys. Rev. Lett. 98, 183601 (2007).
[Crossref] [PubMed]

X. B. Wang, T. Hiroshima, A. Tomita, and M. Hayashi, “Quantum information with Gaussian states,” Phys. Rep. 448(1), 1–111 (2007).
[Crossref]

Al Noman, A.

Anant, V.

R. Riedinger, S. Hong, R. A. Norte, J. A. Slater, J. Shang, A. G. Krause, V. Anant, M. Aspelmeyer, and S. Gröblacher, “Non-classical correlations between single photons and phonons from a mechanical oscillator,” Nature 530, 313–316 (2016).
[Crossref] [PubMed]

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 Phys. 10, 321–326 (2014).
[Crossref]

Appel, J.

J. Appel, E. Figueroa, D. Korystov, M. Lobino, and A. I. Lvovsky, “Quantum memory for squeezed light,” Phys. Rev. Lett. 100, 093602 (2008).
[Crossref] [PubMed]

Ashhab, S.

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

Aspelmeyer, M.

R. Riedinger, S. Hong, R. A. Norte, J. A. Slater, J. Shang, A. G. Krause, V. Anant, M. Aspelmeyer, and S. Gröblacher, “Non-classical correlations between single photons and phonons from a mechanical oscillator,” Nature 530, 313–316 (2016).
[Crossref] [PubMed]

W. Wieczorek, S. G. Hofer, J. Hoelscher-Obermaier, R. Riedinger, K. Hammerer, and M. Aspelmeyer, “Optimal State Estimation for Cavity Optomechanical Systems,” Phys. Rev. Lett. 114, 223601 (2015).
[Crossref] [PubMed]

Asplemeyer, M.

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

Awschalom, D.D.

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

Balabas, M. V.

W. Wasilewski, K. Jensen, H. Krauter, J. J. Renema, M. V. Balabas, and E. S. Polzik, “Quantum noise limited and entanglement-assisted magnetometry,” Phys. Rev. Lett. 104, 133601 (2010).
[Crossref] [PubMed]

Banaszek, K.

M. Jachura, R. Chrapkiewicz, R. Demkowicz-Dobrzański, W. Wasilewski, and K. Banaszek, “Mode engineering for realistic quantum-enhanced interferometry,” Nature Comm. 7, 11411 (2016).
[Crossref]

Bao, X.-H.

X.-H. Bao, A. Reingruber, P. Dietrich, J. Rui, A. Dück, T. Strassel, L. Li, N. Liu, B. Zhao, and J.-W. Pan, “Efficient and long-lived quantum memory with cold atoms inside a ring cavity,” Nature Phys. 8, 517–521 (2012).
[Crossref]

B. Zhao, Y.-A. Chen, X.-H. Bao, T. Strassel, Ch.-S. Chuu, X.-M. Jin., J. Schmiedmayer, Z.-S. Yuan, S. Chen, and J.-W. Pan, “A millisecond quantum memory for scalable quantum networks,” Nature Phys. 5, 95–99 (2009).
[Crossref]

Beduini, F.A.

F. Wolfgramm, A. Cere, F.A. Beduini, A. Predojević, M. Koschorreck, and M. W. Mitchell, “Squeezed-light optical magnetometry,” Phys. Rev. Lett. 105, 053601 (2010).
[Crossref] [PubMed]

Behbood, N.

M. Napolitano, M. Koschorreck, B. Dubost, N. Behbood, R. J. Sewell, and M. W. Mitchell, “Interaction-based quantum metrology showing scaling beyond the Heisenberg limit,” Nature 471, 486–489 (2011).
[Crossref] [PubMed]

Bertetb, P.

G. Kurizki, P. Bertetb, Y. Kubob, K. Molmer, D Petrosyan, P. Rabl, and J. Schmiedmayer, “Quantum technologies with hybrid systems,” PNAS 112, 3866–3873 (2015).
[Crossref] [PubMed]

Bochmann, J.

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

Braun, D.

O. Pinel, P. Jian, N. Treps, C. Fabre, and D. Braun, “Quantum parameter estimation using general single-mode Gaussian states,” Phys. Rev. A 88, 040102(2013).
[Crossref]

O. Pinel, J. Fade, D. Braun, P. Jian, N. Treps, and C. Fabre, “Ultimate sensitivity of precision measurements with intense Gaussian quantum light: A multimodal approach,” Phys. Rev. A 85, 010101 (2012).
[Crossref]

Buchler, B.C.

M. Hosseini, B. M. Sparkes, G. Campbell, P.K. Lam, and B.C. Buchler, “High efficiency coherent optical memory with warm rubidium vapour,” Nature Commun. 2, 174 (2011).
[Crossref]

M. Hosseini, B.M. Sparkes, G. Hétet, J.J. Longdel, P.K. Lam, and B.C. Buchler, “Coherent optical pulse sequencer for quantum applications,” Nature 461, 241–245 (2009).
[Crossref] [PubMed]

Campbell, C. J.

R. Zhao, Y. O. Dudin, S. D. Jenkins, C. J. Campbell, D. N. Matsukevich, T. A. B. Kennedy, and A. Kuzmich, “Long-lived quantum memory,” Nature Phys. 5, 100–104 (2009).
[Crossref]

Campbell, G.

M. Hosseini, B. M. Sparkes, G. Campbell, P.K. Lam, and B.C. Buchler, “High efficiency coherent optical memory with warm rubidium vapour,” Nature Commun. 2, 174 (2011).
[Crossref]

Cere, A.

F. Wolfgramm, A. Cere, F.A. Beduini, A. Predojević, M. Koschorreck, and M. W. Mitchell, “Squeezed-light optical magnetometry,” Phys. Rev. Lett. 105, 053601 (2010).
[Crossref] [PubMed]

Chen, S.

B. Zhao, Y.-A. Chen, X.-H. Bao, T. Strassel, Ch.-S. Chuu, X.-M. Jin., J. Schmiedmayer, Z.-S. Yuan, S. Chen, and J.-W. Pan, “A millisecond quantum memory for scalable quantum networks,” Nature Phys. 5, 95–99 (2009).
[Crossref]

Chen, Y.-A.

B. Zhao, Y.-A. Chen, X.-H. Bao, T. Strassel, Ch.-S. Chuu, X.-M. Jin., J. Schmiedmayer, Z.-S. Yuan, S. Chen, and J.-W. Pan, “A millisecond quantum memory for scalable quantum networks,” Nature Phys. 5, 95–99 (2009).
[Crossref]

Cho, Y. W.

Chrapkiewicz, R.

M. Jachura, R. Chrapkiewicz, R. Demkowicz-Dobrzański, W. Wasilewski, and K. Banaszek, “Mode engineering for realistic quantum-enhanced interferometry,” Nature Comm. 7, 11411 (2016).
[Crossref]

Chuu, Ch.-S.

B. Zhao, Y.-A. Chen, X.-H. Bao, T. Strassel, Ch.-S. Chuu, X.-M. Jin., J. Schmiedmayer, Z.-S. Yuan, S. Chen, and J.-W. Pan, “A millisecond quantum memory for scalable quantum networks,” Nature Phys. 5, 95–99 (2009).
[Crossref]

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 Phys. 10, 321–326 (2014).
[Crossref]

Cleland, A.N.

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

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M. Napolitano, M. Koschorreck, B. Dubost, N. Behbood, R. J. Sewell, and M. W. Mitchell, “Interaction-based quantum metrology showing scaling beyond the Heisenberg limit,” Nature 471, 486–489 (2011).
[Crossref] [PubMed]

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G. Kurizki, P. Bertetb, Y. Kubob, K. Molmer, D Petrosyan, P. Rabl, and J. Schmiedmayer, “Quantum technologies with hybrid systems,” PNAS 112, 3866–3873 (2015).
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Palomaki, T.A.

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X.-H. Bao, A. Reingruber, P. Dietrich, J. Rui, A. Dück, T. Strassel, L. Li, N. Liu, B. Zhao, and J.-W. Pan, “Efficient and long-lived quantum memory with cold atoms inside a ring cavity,” Nature Phys. 8, 517–521 (2012).
[Crossref]

B. Zhao, Y.-A. Chen, X.-H. Bao, T. Strassel, Ch.-S. Chuu, X.-M. Jin., J. Schmiedmayer, Z.-S. Yuan, S. Chen, and J.-W. Pan, “A millisecond quantum memory for scalable quantum networks,” Nature Phys. 5, 95–99 (2009).
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R. Gaiba and M. G. Paris, “Squeezed vacuum as a universal quantum probe,” Phys. Lett. A 373, 934–939 (2009).
[Crossref]

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[Crossref]

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G. Kurizki, P. Bertetb, Y. Kubob, K. Molmer, D Petrosyan, P. Rabl, and J. Schmiedmayer, “Quantum technologies with hybrid systems,” PNAS 112, 3866–3873 (2015).
[Crossref] [PubMed]

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O. Pinel, P. Jian, N. Treps, C. Fabre, and D. Braun, “Quantum parameter estimation using general single-mode Gaussian states,” Phys. Rev. A 88, 040102(2013).
[Crossref]

O. Pinel, J. Fade, D. Braun, P. Jian, N. Treps, and C. Fabre, “Ultimate sensitivity of precision measurements with intense Gaussian quantum light: A multimodal approach,” Phys. Rev. A 85, 010101 (2012).
[Crossref]

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K. Hammerer, A. S. Sørensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. 82, 1041 (2010).
[Crossref]

W. Wasilewski, K. Jensen, H. Krauter, J. J. Renema, M. V. Balabas, and E. S. Polzik, “Quantum noise limited and entanglement-assisted magnetometry,” Phys. Rev. Lett. 104, 133601 (2010).
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Predojevic, A.

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A. D. Cronin, J. Schmiedmayer, and D. E. Pritchard, “Optics and interferometry with atoms and molecules,” Rev. Mod. Phys. 81, 1051 (2009).
[Crossref]

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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 Phys. 10, 321–326 (2014).
[Crossref]

Qi, M.

Rabl, P.

G. Kurizki, P. Bertetb, Y. Kubob, K. Molmer, D Petrosyan, P. Rabl, and J. Schmiedmayer, “Quantum technologies with hybrid systems,” PNAS 112, 3866–3873 (2015).
[Crossref] [PubMed]

M. Wallquist, K. Hammerer, P. Rabl, M. Lukin, and P. Zoller, “Hybrid quantum devices and quantum engineering,” Phys. Scripta 2009(T137), 014001 (2009).
[Crossref]

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

Regal, C. A.

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 Phys. 10, 321–326 (2014).
[Crossref]

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X.-H. Bao, A. Reingruber, P. Dietrich, J. Rui, A. Dück, T. Strassel, L. Li, N. Liu, B. Zhao, and J.-W. Pan, “Efficient and long-lived quantum memory with cold atoms inside a ring cavity,” Nature Phys. 8, 517–521 (2012).
[Crossref]

Renema, J. J.

W. Wasilewski, K. Jensen, H. Krauter, J. J. Renema, M. V. Balabas, and E. S. Polzik, “Quantum noise limited and entanglement-assisted magnetometry,” Phys. Rev. Lett. 104, 133601 (2010).
[Crossref] [PubMed]

Riedinger, R.

R. Riedinger, S. Hong, R. A. Norte, J. A. Slater, J. Shang, A. G. Krause, V. Anant, M. Aspelmeyer, and S. Gröblacher, “Non-classical correlations between single photons and phonons from a mechanical oscillator,” Nature 530, 313–316 (2016).
[Crossref] [PubMed]

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[Crossref] [PubMed]

Rui, J.

X.-H. Bao, A. Reingruber, P. Dietrich, J. Rui, A. Dück, T. Strassel, L. Li, N. Liu, B. Zhao, and J.-W. Pan, “Efficient and long-lived quantum memory with cold atoms inside a ring cavity,” Nature Phys. 8, 517–521 (2012).
[Crossref]

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L. Ruppert and R. Filip, “Estimation of nonclassical independent Gaussian processes by classical interferometry,” Sci. Rep. 7, 39641 (2017).
[Crossref] [PubMed]

Šafránek, D.

D. Šafránek, A. R. Lee, and I. Fuentes, “Quantum parameter estimation using multi-mode Gaussian states,” New J. Phys. 17, 073016 (2015).
[Crossref]

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E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature 482, 63–67 (2012).
[Crossref] [PubMed]

Schmiedmayer, J.

G. Kurizki, P. Bertetb, Y. Kubob, K. Molmer, D Petrosyan, P. Rabl, and J. Schmiedmayer, “Quantum technologies with hybrid systems,” PNAS 112, 3866–3873 (2015).
[Crossref] [PubMed]

A. D. Cronin, J. Schmiedmayer, and D. E. Pritchard, “Optics and interferometry with atoms and molecules,” Rev. Mod. Phys. 81, 1051 (2009).
[Crossref]

B. Zhao, Y.-A. Chen, X.-H. Bao, T. Strassel, Ch.-S. Chuu, X.-M. Jin., J. Schmiedmayer, Z.-S. Yuan, S. Chen, and J.-W. Pan, “A millisecond quantum memory for scalable quantum networks,” Nature Phys. 5, 95–99 (2009).
[Crossref]

Sellars, M.J.

M.P. Hedges, J.J. Longdell, Y. Li, and M.J. Sellars, “Efficient quantum memory for light,” Nature 465, 1052–1056 (2010).
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Seshadreesan, K.P.

J.P. Dowling and K.P. Seshadreesan, “Quantum Optical Technologies for Metrology, Sensing, and Imaging,” J. Lightwave Tech. 33, 2359–2370 (2015).
[Crossref]

Sewell, R. J.

M. Napolitano, M. Koschorreck, B. Dubost, N. Behbood, R. J. Sewell, and M. W. Mitchell, “Interaction-based quantum metrology showing scaling beyond the Heisenberg limit,” Nature 471, 486–489 (2011).
[Crossref] [PubMed]

Shang, J.

R. Riedinger, S. Hong, R. A. Norte, J. A. Slater, J. Shang, A. G. Krause, V. Anant, M. Aspelmeyer, and S. Gröblacher, “Non-classical correlations between single photons and phonons from a mechanical oscillator,” Nature 530, 313–316 (2016).
[Crossref] [PubMed]

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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 Phys. 10, 321–326 (2014).
[Crossref]

Simmonds, R.W.

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]

Simon, J.

J. Simon, H. Tanji, J.K. Thompson, and V. Vuletic, “Interfacing collective atomic excitations and single photons,” Phys. Rev. Lett. 98, 183601 (2007).
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R. Riedinger, S. Hong, R. A. Norte, J. A. Slater, J. Shang, A. G. Krause, V. Anant, M. Aspelmeyer, and S. Gröblacher, “Non-classical correlations between single photons and phonons from a mechanical oscillator,” Nature 530, 313–316 (2016).
[Crossref] [PubMed]

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K. Hammerer, A. S. Sørensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. 82, 1041 (2010).
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M. Hosseini, B. M. Sparkes, G. Campbell, P.K. Lam, and B.C. Buchler, “High efficiency coherent optical memory with warm rubidium vapour,” Nature Commun. 2, 174 (2011).
[Crossref]

Sparkes, B.M.

M. Hosseini, B.M. Sparkes, G. Hétet, J.J. Longdel, P.K. Lam, and B.C. Buchler, “Coherent optical pulse sequencer for quantum applications,” Nature 461, 241–245 (2009).
[Crossref] [PubMed]

Strassel, T.

X.-H. Bao, A. Reingruber, P. Dietrich, J. Rui, A. Dück, T. Strassel, L. Li, N. Liu, B. Zhao, and J.-W. Pan, “Efficient and long-lived quantum memory with cold atoms inside a ring cavity,” Nature Phys. 8, 517–521 (2012).
[Crossref]

B. Zhao, Y.-A. Chen, X.-H. Bao, T. Strassel, Ch.-S. Chuu, X.-M. Jin., J. Schmiedmayer, Z.-S. Yuan, S. Chen, and J.-W. Pan, “A millisecond quantum memory for scalable quantum networks,” Nature Phys. 5, 95–99 (2009).
[Crossref]

Tanji, H.

J. Simon, H. Tanji, J.K. Thompson, and V. Vuletic, “Interfacing collective atomic excitations and single photons,” Phys. Rev. Lett. 98, 183601 (2007).
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Teng, M.

Teufel, J.D.

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]

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J. Simon, H. Tanji, J.K. Thompson, and V. Vuletic, “Interfacing collective atomic excitations and single photons,” Phys. Rev. Lett. 98, 183601 (2007).
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X. B. Wang, T. Hiroshima, A. Tomita, and M. Hayashi, “Quantum information with Gaussian states,” Phys. Rep. 448(1), 1–111 (2007).
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O. Pinel, P. Jian, N. Treps, C. Fabre, and D. Braun, “Quantum parameter estimation using general single-mode Gaussian states,” Phys. Rev. A 88, 040102(2013).
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O. Pinel, J. Fade, D. Braun, P. Jian, N. Treps, and C. Fabre, “Ultimate sensitivity of precision measurements with intense Gaussian quantum light: A multimodal approach,” Phys. Rev. A 85, 010101 (2012).
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J. Bochmann, A. Vainsencher, D.D. Awschalom, and A.N. Cleland, “Nanomechanical coupling between microwave and optical photons,” Nature Phys. 9, 712–716 (2013).
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Verhagen, E.

E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature 482, 63–67 (2012).
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Vostrosablin, N.

A. A. Rakhubovsky, N. Vostrosablin, and R. Filip, “Squeezer-based pulsed optomechanical interface,” Phys. Rev. A 93, 033813 (2016).
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J. Simon, H. Tanji, J.K. Thompson, and V. Vuletic, “Interfacing collective atomic excitations and single photons,” Phys. Rev. Lett. 98, 183601 (2007).
[Crossref] [PubMed]

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M. Wallquist, K. Hammerer, P. Rabl, M. Lukin, and P. Zoller, “Hybrid quantum devices and quantum engineering,” Phys. Scripta 2009(T137), 014001 (2009).
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Wang, J.

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X. B. Wang, T. Hiroshima, A. Tomita, and M. Hayashi, “Quantum information with Gaussian states,” Phys. Rep. 448(1), 1–111 (2007).
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M. Jachura, R. Chrapkiewicz, R. Demkowicz-Dobrzański, W. Wasilewski, and K. Banaszek, “Mode engineering for realistic quantum-enhanced interferometry,” Nature Comm. 7, 11411 (2016).
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W. Wasilewski, K. Jensen, H. Krauter, J. J. Renema, M. V. Balabas, and E. S. Polzik, “Quantum noise limited and entanglement-assisted magnetometry,” Phys. Rev. Lett. 104, 133601 (2010).
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Weis, S.

E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature 482, 63–67 (2012).
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W. Wieczorek, S. G. Hofer, J. Hoelscher-Obermaier, R. Riedinger, K. Hammerer, and M. Aspelmeyer, “Optimal State Estimation for Cavity Optomechanical Systems,” Phys. Rev. Lett. 114, 223601 (2015).
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Z.-L. Xiang, S. Ashhab, J. Q. You, and F. Nori, “Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems,” Rev. Mod. Phys. 85, 623 (2013).
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Z.-L. Xiang, S. Ashhab, J. Q. You, and F. Nori, “Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems,” Rev. Mod. Phys. 85, 623 (2013).
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B. Zhao, Y.-A. Chen, X.-H. Bao, T. Strassel, Ch.-S. Chuu, X.-M. Jin., J. Schmiedmayer, Z.-S. Yuan, S. Chen, and J.-W. Pan, “A millisecond quantum memory for scalable quantum networks,” Nature Phys. 5, 95–99 (2009).
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Zhao, B.

X.-H. Bao, A. Reingruber, P. Dietrich, J. Rui, A. Dück, T. Strassel, L. Li, N. Liu, B. Zhao, and J.-W. Pan, “Efficient and long-lived quantum memory with cold atoms inside a ring cavity,” Nature Phys. 8, 517–521 (2012).
[Crossref]

B. Zhao, Y.-A. Chen, X.-H. Bao, T. Strassel, Ch.-S. Chuu, X.-M. Jin., J. Schmiedmayer, Z.-S. Yuan, S. Chen, and J.-W. Pan, “A millisecond quantum memory for scalable quantum networks,” Nature Phys. 5, 95–99 (2009).
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Zhao, R.

R. Zhao, Y. O. Dudin, S. D. Jenkins, C. J. Campbell, D. N. Matsukevich, T. A. B. Kennedy, and A. Kuzmich, “Long-lived quantum memory,” Nature Phys. 5, 100–104 (2009).
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Zoller, P.

M. Wallquist, K. Hammerer, P. Rabl, M. Lukin, and P. Zoller, “Hybrid quantum devices and quantum engineering,” Phys. Scripta 2009(T137), 014001 (2009).
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J. Lightwave Tech. (1)

J.P. Dowling and K.P. Seshadreesan, “Quantum Optical Technologies for Metrology, Sensing, and Imaging,” J. Lightwave Tech. 33, 2359–2370 (2015).
[Crossref]

Nature (5)

R. Riedinger, S. Hong, R. A. Norte, J. A. Slater, J. Shang, A. G. Krause, V. Anant, M. Aspelmeyer, and S. Gröblacher, “Non-classical correlations between single photons and phonons from a mechanical oscillator,” Nature 530, 313–316 (2016).
[Crossref] [PubMed]

M. Napolitano, M. Koschorreck, B. Dubost, N. Behbood, R. J. Sewell, and M. W. Mitchell, “Interaction-based quantum metrology showing scaling beyond the Heisenberg limit,” Nature 471, 486–489 (2011).
[Crossref] [PubMed]

M. Hosseini, B.M. Sparkes, G. Hétet, J.J. Longdel, P.K. Lam, and B.C. Buchler, “Coherent optical pulse sequencer for quantum applications,” Nature 461, 241–245 (2009).
[Crossref] [PubMed]

M.P. Hedges, J.J. Longdell, Y. Li, and M.J. Sellars, “Efficient quantum memory for light,” Nature 465, 1052–1056 (2010).
[Crossref] [PubMed]

E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature 482, 63–67 (2012).
[Crossref] [PubMed]

Nature Comm. (1)

M. Jachura, R. Chrapkiewicz, R. Demkowicz-Dobrzański, W. Wasilewski, and K. Banaszek, “Mode engineering for realistic quantum-enhanced interferometry,” Nature Comm. 7, 11411 (2016).
[Crossref]

Nature Commun. (1)

M. Hosseini, B. M. Sparkes, G. Campbell, P.K. Lam, and B.C. Buchler, “High efficiency coherent optical memory with warm rubidium vapour,” Nature Commun. 2, 174 (2011).
[Crossref]

Nature Photon. (1)

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nature Photon. 5, 222 (2011).
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Nature Phys. (5)

X.-H. Bao, A. Reingruber, P. Dietrich, J. Rui, A. Dück, T. Strassel, L. Li, N. Liu, B. Zhao, and J.-W. Pan, “Efficient and long-lived quantum memory with cold atoms inside a ring cavity,” Nature Phys. 8, 517–521 (2012).
[Crossref]

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

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 Phys. 10, 321–326 (2014).
[Crossref]

B. Zhao, Y.-A. Chen, X.-H. Bao, T. Strassel, Ch.-S. Chuu, X.-M. Jin., J. Schmiedmayer, Z.-S. Yuan, S. Chen, and J.-W. Pan, “A millisecond quantum memory for scalable quantum networks,” Nature Phys. 5, 95–99 (2009).
[Crossref]

R. Zhao, Y. O. Dudin, S. D. Jenkins, C. J. Campbell, D. N. Matsukevich, T. A. B. Kennedy, and A. Kuzmich, “Long-lived quantum memory,” Nature Phys. 5, 100–104 (2009).
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New J. Phys. (1)

D. Šafránek, A. R. Lee, and I. Fuentes, “Quantum parameter estimation using multi-mode Gaussian states,” New J. Phys. 17, 073016 (2015).
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Opt. Express (1)

Opt. Lett. (1)

Optica (1)

Phys. Lett. A (1)

R. Gaiba and M. G. Paris, “Squeezed vacuum as a universal quantum probe,” Phys. Lett. A 373, 934–939 (2009).
[Crossref]

Phys. Rep. (1)

X. B. Wang, T. Hiroshima, A. Tomita, and M. Hayashi, “Quantum information with Gaussian states,” Phys. Rep. 448(1), 1–111 (2007).
[Crossref]

Phys. Rev. A (3)

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

O. Pinel, J. Fade, D. Braun, P. Jian, N. Treps, and C. Fabre, “Ultimate sensitivity of precision measurements with intense Gaussian quantum light: A multimodal approach,” Phys. Rev. A 85, 010101 (2012).
[Crossref]

O. Pinel, P. Jian, N. Treps, C. Fabre, and D. Braun, “Quantum parameter estimation using general single-mode Gaussian states,” Phys. Rev. A 88, 040102(2013).
[Crossref]

Phys. Rev. Lett. (6)

S. A. Haine, “Information-Recycling Beam Splitters for Quantum Enhanced Atom Interferometry,” Phys. Rev. Lett. 110, 053002 (2013).
[Crossref] [PubMed]

W. Wasilewski, K. Jensen, H. Krauter, J. J. Renema, M. V. Balabas, and E. S. Polzik, “Quantum noise limited and entanglement-assisted magnetometry,” Phys. Rev. Lett. 104, 133601 (2010).
[Crossref] [PubMed]

F. Wolfgramm, A. Cere, F.A. Beduini, A. Predojević, M. Koschorreck, and M. W. Mitchell, “Squeezed-light optical magnetometry,” Phys. Rev. Lett. 105, 053601 (2010).
[Crossref] [PubMed]

J. Simon, H. Tanji, J.K. Thompson, and V. Vuletic, “Interfacing collective atomic excitations and single photons,” Phys. Rev. Lett. 98, 183601 (2007).
[Crossref] [PubMed]

J. Appel, E. Figueroa, D. Korystov, M. Lobino, and A. I. Lvovsky, “Quantum memory for squeezed light,” Phys. Rev. Lett. 100, 093602 (2008).
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W. Wieczorek, S. G. Hofer, J. Hoelscher-Obermaier, R. Riedinger, K. Hammerer, and M. Aspelmeyer, “Optimal State Estimation for Cavity Optomechanical Systems,” Phys. Rev. Lett. 114, 223601 (2015).
[Crossref] [PubMed]

Phys. Scripta (1)

M. Wallquist, K. Hammerer, P. Rabl, M. Lukin, and P. Zoller, “Hybrid quantum devices and quantum engineering,” Phys. Scripta 2009(T137), 014001 (2009).
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PNAS (1)

G. Kurizki, P. Bertetb, Y. Kubob, K. Molmer, D Petrosyan, P. Rabl, and J. Schmiedmayer, “Quantum technologies with hybrid systems,” PNAS 112, 3866–3873 (2015).
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Progress in Optics (1)

R. Demkowicz-Dobrzański, M. Jarzyna, and J. Kołodyński, “Quantum Limits in Optical Interferometry,” Progress in Optics 60, 345–435 (2015).
[Crossref]

Rev. Mod. Phys. (4)

K. Hammerer, A. S. Sørensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. 82, 1041 (2010).
[Crossref]

Z.-L. Xiang, S. Ashhab, J. Q. You, and F. Nori, “Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems,” Rev. Mod. Phys. 85, 623 (2013).
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A. D. Cronin, J. Schmiedmayer, and D. E. Pritchard, “Optics and interferometry with atoms and molecules,” Rev. Mod. Phys. 81, 1051 (2009).
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M. Asplemeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391 (2014).
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Sci. Rep. (1)

L. Ruppert and R. Filip, “Estimation of nonclassical independent Gaussian processes by classical interferometry,” Sci. Rep. 7, 39641 (2017).
[Crossref] [PubMed]

Science (1)

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]

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H. Cramér, Mathematical Methods of Statistics (Princeton University, 1946).

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Figures (5)

Fig. 1
Fig. 1 Schematic representation of light-matter interferometry with quantum interfaces. A material sample (mode M) in thermal state with variance V interfaces with a coherent light (mode L) with amplitude r; their interaction is approximated well by a beam-splitter (BS) with transmittance T1. Further on, the matter mode is modified by an unknown Gaussian process (G.P.) stimulated by the environment. The two modes then interact again on a second BS-type interface with transmittance T2. Finally, the quadratures of the light are measured with heterodyne (or homodyne) detection (HOM./HET.) to estimate the parameters of the Gaussian process. The other, matter mode is not measured.
Fig. 2
Fig. 2 Variations of non-interferometric setups: (left) the most simplistic setup, where the the interface is applied only after the measured unknown Gaussian process, (right) light interfaces with the matter at the first BS interaction, but is blocked after that, so the coherent light is only used for the preparation of the state, not for read-out by the second interface. (We used the notations of Fig. 1.)
Fig. 3
Fig. 3 (left) The MSE of the estimators of the displacement (d) if only displacement is present. The solid line corresponds to interferometric setup with T1 = T2, the dashed line corresponds to simplistic setup (T1 = 0), the dotted line corresponds to the case of blocked optical beam with T1 = T2. We have used the following parameters: V = 100, N = 105, d = 4, β = 0.5. (right) The MSE of the estimators of the phase-shift (Φ) if only phase-shift is present. The solid line comes from the estimators using only the covariance matrix (second moments), the dashed line comes from the estimators using only the means (first moments), the dotted line comes from the maximum likelihood estimator, the red (lighter) solid line is the theoretical limit coming from Cramér-Rao inequality (note the last two lines basically coincide). We have applied the following parameters for the scheme: T1 = T2 = 0.1, V = 100, N = 105, Φ = 0.7.
Fig. 4
Fig. 4 The MSE of the estimators of the Gaussian process as a function of (left) the strength of the coherent light (r), (right) the variance of the matter (V). The purple (dark) lines correspond to the phase shift, the orange (medium) lines correspond to the magnitude of the displacement and cyan (light) lines correspond to the magnitude of the squeezing. The solid lines come from the estimators using the covariance matrix, the dashed lines come from the estimators using only the means. We have applied the following parameters for the scheme: T1 = T2 = 0.1, N = 105, (left) V = 100, (right) r = 100, and for the unknown process: Φ = 0.7, d = 4, β = 0.5, q = 2, α = −0.3.
Fig. 5
Fig. 5 The MSE of the estimators of the Gaussian process as a function of (left) T = T1 = T2 and (right) loss. The purple (dark) lines correspond to the phase shift, the orange (medium) lines correspond to the magnitude of the displacement and cyan (light) lines correspond to the magnitude of the squeezing. The solid lines come from the estimators using the covariance matrix, the dashed lines come from the estimators using only the means, the dotted lines come from the naive estimators (assuming that the channel is ideal, i.e. not using a calibration phase) using only the means (note that the purple dashed and the purple dotted lines coincide in the right subfigure). We have applied the following parameters for the scheme: r = 100, V = 100, N = 105 (right) T = 0.1, VC = 1.2 and for the unknown process: Φ = 0.7, d = 4, β = 0.5, q = 2, α = −0.3.

Equations (13)

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ρ * = D ( γ ) R ( Φ ) S ( ξ ) ρ S ( ξ ) R ( Φ ) D ( γ ) ,
I ( θ ) = E x _ ( θ log f ( x _ ; θ ) ) 2 = E x _ ( 2 θ 2 log f ( x _ ; θ ) ) .
Var ( θ ^ ) I 1 ( θ ) .
MSE M ( θ ^ ) : = 1 M k = 1 M ( θ ^ k θ ) 2 ,
x o = T 2 x 1 * + 1 T 2 x 2
d x ^ = x o 1 T 2 r T 2 .
d p ^ = p o T 2 .
I ( d ) = T 2 1 + T 2 ( V 1 ) .
I ( d ) = T 2 1 T 2 + T 2 ( ( 1 T 1 ) V + T 1 ) .
I ( d ) = T ,
Var ( x o ) = Var ( p o ) = u + v cos ( Φ ) ,
Φ ^ 1 = arccos ( v 1 ( Var ( x o ) + Var ( p o ) 2 u ) )
Φ ^ 2 = arctan ( p o x o r ( 1 T 1 ) ( 1 T 2 ) ) .

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