Abstract

Entangled coherent states are a fundamentally interesting class of quantum states of light, with important implications in quantum information processing, for which robust schemes to generate them are required. Here, we show that entangled coherent states emerge, with high fidelity, when mixing coherent and squeezed vacuum states of light on a beam splitter. These maximally entangled states, where photons bunch at the exit of a beam splitter, are measured experimentally by Fock-state projections. Entanglement is examined theoretically using a Bell-type nonlocality test and compared with ideal entangled coherent states. We experimentally show nearly perfect similarity with entangled coherent states for an optimal ratio of coherent and squeezed vacuum light. In our scheme, entangled coherent states are generated deterministically with small amplitudes, which could be beneficial, for example, in deterministic distribution of entanglement over long distances.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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    [Crossref]
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    [Crossref]

2016 (2)

C. Wang, Y. Y. Gao, P. Reinhold, R. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A Schrödinger cat living in two boxes,” Science 352, 1087–1091 (2016).
[Crossref]

Y. Lim, J. Joo, T. P. Spiller, and H. Jeong, “Loss-resilient photonic entanglement swapping using optical hybrid states,” Phys. Rev. A 94, 062337 (2016).
[Crossref]

2015 (1)

I. A. Walmsley, “Quantum optics: science and technology in a new light,” Science 348, 525–530 (2015).
[Crossref]

2014 (3)

P. Shadbolt, J. C. F. Mathews, A. Laing, and J. L. O’Brien, “Testing foundations of quantum mechanics with photons,” Nat. Phys. 10, 278–286 (2014).
[Crossref]

Y. Israel, S. Rosen, and Y. Silberberg, “Supersensitive polarization microscopy using NOON states of light,” Phys. Rev. Lett. 112, 103604 (2014).
[Crossref]

L. A. Rozema, J. D. Bateman, D. H. Mahler, R. Okamoto, A. Feizpour, A. Hayat, and A. M. Steinberg, “Scalable spatial superresolution using entangled photons,”Phys. Rev. Lett. 112, 223602 (2014).
[Crossref]

2013 (2)

Y. M. Zhang, X. W. Li, W. Yang, and G. R. Jin, “Quantum Fisher information of entangled coherent states in the presence of photon loss,” Phys. Rev. A 88, 043832 (2013).
[Crossref]

M. M. Weston, M. J. W. Hall, M. S. Palsson, H. M. Wiseman, and G. J. Pryde, “Experimental test of universal complementarity relations,” Phys. Rev. Lett. 110, 220402 (2013).
[Crossref]

2012 (2)

B. C. Sanders, “Review of entangled coherent states,” J. Phys. A 45, 244002 (2012).
[Crossref]

Y. Israel, I. Afek, S. Rosen, O. Ambar, and Y. Silberberg, “Experimental tomography of NOON states with large photon numbers,” Phys. Rev. A 85, 022115 (2012).
[Crossref]

2011 (2)

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
[Crossref]

J. Joo, W. J. Munro, and T. P. Spiller, “Quantum metrology with entangled coherent states,” Phys. Rev. Lett. 107, 083601 (2011).
[Crossref]

2010 (6)

T. Gerrits, S. Glancy, T. S. Clement, B. Calkins, A. E. Lita, A. J. Miller, A. L. Migdall, S. W. Nam, R. P. Mirin, and E. Knill, “Generation of optical coherent-state superpositions by number-resolved photon subtraction from the squeezed vacuum,” Phys. Rev. A 82, 031802 (2010).
[Crossref]

K. Park and H. Jeong, “Entangled coherent states versus entangled photon pairs for practical quantum-information processing,” Phys. Rev. A 82, 062325 (2010).
[Crossref]

A. Peruzzo, M. Lobino, J. C. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum walks of correlated photons,” Science 329, 1500–1503 (2010).
[Crossref]

N. Sangouard, C. Simon, N. Gisin, J. Laurat, R. Tualle-Brouri, and P. Grangier, “Quantum repeaters with entangled coherent states,” J. Opt. Soc. Am. B 27, A137–A145 (2010).
[Crossref]

J. B. Brask, I. Rigas, E. S. Polzik, U. L. Andersen, and A. S. Sørensen, “Hybrid long-distance entanglement distribution protocol,” Phys. Rev. Lett. 105, 160501 (2010).
[Crossref]

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010).
[Crossref]

2009 (2)

A. Ourjoumtsev, F. Ferreyrol, R. Tualle-Brouri, and P. Grangier, “Preparation of non-local superpositions of quasi-classical light states,” Nat. Phys. 5, 189–192 (2009).
[Crossref]

C. C. Gerry, J. Mimih, and A. Benmoussa, “Maximally entangled coherent states and strong violations of Bell-type inequalities,” Phys. Rev. A 80, 022111 (2009).
[Crossref]

2008 (4)

H. Takahashi, K. Wakui, S. Suzuki, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Generation of large-amplitude coherent-state superposition via ancilla-assisted photon subtraction,” Phys. Rev. Lett. 101, 233605 (2008).
[Crossref]

L. Pezzé and A. Smerzi, “Mach-Zehnder interferometry at the Heisenberg limit with coherent and squeezed-vacuum light,” Phys. Rev. Lett. 100, 073601 (2008).
[Crossref]

S. Glancy and H. M. de Vasconcelos, “Methods for producing optical coherent state superpositions,” J. Opt. Soc. Am. B 25, 712–733 (2008).
[Crossref]

J. Dowling, “Quantum optical metrology-the lowdown on high-NOON states,” Contemp. Phys. 49, 125–143 (2008).
[Crossref]

2007 (2)

H. F. Hofmann and T. Ono, “High-photon-number path entanglement in the interference of spontaneously down-converted photon pairs with coherent laser light,” Phys. Rev. A 76, 031806 (2007).
[Crossref]

C. F. Wildfeuer, A. P. Lund, and J. P. Dowling, “Strong violations of Bell-type inequalities for path-entangled number states,” Phys. Rev. A 76, 052101 (2007).
[Crossref]

2004 (1)

S. Janssens, B. D. Baets, and H. D. Meyer, “Bell-type inequalities for quasi-copulas,” Fuzzy Sets and Systems 148, 263–278 (2004).
[Crossref]

2003 (2)

C. C. Gerry, A. Benmoussa, and K. M. Bruno, “Single-mode squeezed vacuum states as approximate Schrödinger phase cats: relation to su (1, 1) phase operators,” J. Opt. B 5, 109–115 (2003).
[Crossref]

M. Paternostro, M. S. Kim, and B. S. Ham, “Generation of entangled coherent states via cross-phase-modulation in a double electromagnetically induced transparency regime,” Phys. Rev. A 67, 023811 (2003).
[Crossref]

2002 (1)

C. C. Gerry, A. Benmoussa, and R. A. Campos, “Nonlinear interferometer as a resource for maximally entangled photonic states: application to interferometry,” Phys. Rev. A 66, 013804 (2002).
[Crossref]

2001 (1)

A. Luis, “Equivalence between macroscopic quantum superpositions and maximally entangled states: application to phase-shift detection,” Phys. Rev. A 64, 054102 (2001).
[Crossref]

1999 (1)

C. C. Gerry, “Generation of optical macroscopic quantum superposition states via state reduction with a Mach-Zehnder interferometer containing a Kerr medium,” Phys. Rev. A 59, 4095–4098 (1999).
[Crossref]

1994 (1)

R. Jozsa, “Fidelity for mixed quantum states,” J. Mod. Opt. 41, 2315–2323 (1994).
[Crossref]

1992 (1)

B. C. Sanders, “Entangled coherent states,” Phys. Rev. A 45, 6811–6815 (1992).
[Crossref]

1982 (1)

A. Aspect, P. Grangier, and G. Roger, “Experimental realization of Einstein-Podolsky-Rosen-Bohm Gedanken experiment: a new violation of Bell’s inequalities,” Phys. Rev. Lett. 49, 91–94 (1982).
[Crossref]

Afek, I.

Y. Israel, I. Afek, S. Rosen, O. Ambar, and Y. Silberberg, “Experimental tomography of NOON states with large photon numbers,” Phys. Rev. A 85, 022115 (2012).
[Crossref]

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010).
[Crossref]

Ambar, O.

Y. Israel, I. Afek, S. Rosen, O. Ambar, and Y. Silberberg, “Experimental tomography of NOON states with large photon numbers,” Phys. Rev. A 85, 022115 (2012).
[Crossref]

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010).
[Crossref]

Andersen, U. L.

J. B. Brask, I. Rigas, E. S. Polzik, U. L. Andersen, and A. S. Sørensen, “Hybrid long-distance entanglement distribution protocol,” Phys. Rev. Lett. 105, 160501 (2010).
[Crossref]

Aspect, A.

A. Aspect, P. Grangier, and G. Roger, “Experimental realization of Einstein-Podolsky-Rosen-Bohm Gedanken experiment: a new violation of Bell’s inequalities,” Phys. Rev. Lett. 49, 91–94 (1982).
[Crossref]

Axline, C.

C. Wang, Y. Y. Gao, P. Reinhold, R. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A Schrödinger cat living in two boxes,” Science 352, 1087–1091 (2016).
[Crossref]

Baets, B. D.

S. Janssens, B. D. Baets, and H. D. Meyer, “Bell-type inequalities for quasi-copulas,” Fuzzy Sets and Systems 148, 263–278 (2004).
[Crossref]

Bateman, J. D.

L. A. Rozema, J. D. Bateman, D. H. Mahler, R. Okamoto, A. Feizpour, A. Hayat, and A. M. Steinberg, “Scalable spatial superresolution using entangled photons,”Phys. Rev. Lett. 112, 223602 (2014).
[Crossref]

Benmoussa, A.

C. C. Gerry, J. Mimih, and A. Benmoussa, “Maximally entangled coherent states and strong violations of Bell-type inequalities,” Phys. Rev. A 80, 022111 (2009).
[Crossref]

C. C. Gerry, A. Benmoussa, and K. M. Bruno, “Single-mode squeezed vacuum states as approximate Schrödinger phase cats: relation to su (1, 1) phase operators,” J. Opt. B 5, 109–115 (2003).
[Crossref]

C. C. Gerry, A. Benmoussa, and R. A. Campos, “Nonlinear interferometer as a resource for maximally entangled photonic states: application to interferometry,” Phys. Rev. A 66, 013804 (2002).
[Crossref]

Blumoff, J.

C. Wang, Y. Y. Gao, P. Reinhold, R. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A Schrödinger cat living in two boxes,” Science 352, 1087–1091 (2016).
[Crossref]

Brask, J. B.

J. B. Brask, I. Rigas, E. S. Polzik, U. L. Andersen, and A. S. Sørensen, “Hybrid long-distance entanglement distribution protocol,” Phys. Rev. Lett. 105, 160501 (2010).
[Crossref]

Bromberg, Y.

A. Peruzzo, M. Lobino, J. C. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum walks of correlated photons,” Science 329, 1500–1503 (2010).
[Crossref]

Bruno, K. M.

C. C. Gerry, A. Benmoussa, and K. M. Bruno, “Single-mode squeezed vacuum states as approximate Schrödinger phase cats: relation to su (1, 1) phase operators,” J. Opt. B 5, 109–115 (2003).
[Crossref]

Calkins, B.

T. Gerrits, S. Glancy, T. S. Clement, B. Calkins, A. E. Lita, A. J. Miller, A. L. Migdall, S. W. Nam, R. P. Mirin, and E. Knill, “Generation of optical coherent-state superpositions by number-resolved photon subtraction from the squeezed vacuum,” Phys. Rev. A 82, 031802 (2010).
[Crossref]

Campos, R. A.

C. C. Gerry, A. Benmoussa, and R. A. Campos, “Nonlinear interferometer as a resource for maximally entangled photonic states: application to interferometry,” Phys. Rev. A 66, 013804 (2002).
[Crossref]

Chekhova, M. V.

F. Töppel, M. V. Chekhova, and G. Leuchs, “Conditionally generating a mesoscopic superposition of NOON states,” arXiv:1607.01296 (2016).

Chou, K.

C. Wang, Y. Y. Gao, P. Reinhold, R. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A Schrödinger cat living in two boxes,” Science 352, 1087–1091 (2016).
[Crossref]

Clement, T. S.

T. Gerrits, S. Glancy, T. S. Clement, B. Calkins, A. E. Lita, A. J. Miller, A. L. Migdall, S. W. Nam, R. P. Mirin, and E. Knill, “Generation of optical coherent-state superpositions by number-resolved photon subtraction from the squeezed vacuum,” Phys. Rev. A 82, 031802 (2010).
[Crossref]

de Vasconcelos, H. M.

Devoret, M. H.

C. Wang, Y. Y. Gao, P. Reinhold, R. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A Schrödinger cat living in two boxes,” Science 352, 1087–1091 (2016).
[Crossref]

Dowling, J.

J. Dowling, “Quantum optical metrology-the lowdown on high-NOON states,” Contemp. Phys. 49, 125–143 (2008).
[Crossref]

Dowling, J. P.

C. F. Wildfeuer, A. P. Lund, and J. P. Dowling, “Strong violations of Bell-type inequalities for path-entangled number states,” Phys. Rev. A 76, 052101 (2007).
[Crossref]

Feizpour, A.

L. A. Rozema, J. D. Bateman, D. H. Mahler, R. Okamoto, A. Feizpour, A. Hayat, and A. M. Steinberg, “Scalable spatial superresolution using entangled photons,”Phys. Rev. Lett. 112, 223602 (2014).
[Crossref]

Ferreyrol, F.

A. Ourjoumtsev, F. Ferreyrol, R. Tualle-Brouri, and P. Grangier, “Preparation of non-local superpositions of quasi-classical light states,” Nat. Phys. 5, 189–192 (2009).
[Crossref]

Frunzio, L.

C. Wang, Y. Y. Gao, P. Reinhold, R. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A Schrödinger cat living in two boxes,” Science 352, 1087–1091 (2016).
[Crossref]

Furusawa, A.

H. Takahashi, K. Wakui, S. Suzuki, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Generation of large-amplitude coherent-state superposition via ancilla-assisted photon subtraction,” Phys. Rev. Lett. 101, 233605 (2008).
[Crossref]

Gao, Y. Y.

C. Wang, Y. Y. Gao, P. Reinhold, R. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A Schrödinger cat living in two boxes,” Science 352, 1087–1091 (2016).
[Crossref]

Gerrits, T.

T. Gerrits, S. Glancy, T. S. Clement, B. Calkins, A. E. Lita, A. J. Miller, A. L. Migdall, S. W. Nam, R. P. Mirin, and E. Knill, “Generation of optical coherent-state superpositions by number-resolved photon subtraction from the squeezed vacuum,” Phys. Rev. A 82, 031802 (2010).
[Crossref]

Gerry, C. C.

C. C. Gerry, J. Mimih, and A. Benmoussa, “Maximally entangled coherent states and strong violations of Bell-type inequalities,” Phys. Rev. A 80, 022111 (2009).
[Crossref]

C. C. Gerry, A. Benmoussa, and K. M. Bruno, “Single-mode squeezed vacuum states as approximate Schrödinger phase cats: relation to su (1, 1) phase operators,” J. Opt. B 5, 109–115 (2003).
[Crossref]

C. C. Gerry, A. Benmoussa, and R. A. Campos, “Nonlinear interferometer as a resource for maximally entangled photonic states: application to interferometry,” Phys. Rev. A 66, 013804 (2002).
[Crossref]

C. C. Gerry, “Generation of optical macroscopic quantum superposition states via state reduction with a Mach-Zehnder interferometer containing a Kerr medium,” Phys. Rev. A 59, 4095–4098 (1999).
[Crossref]

C. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge University, 2005).

Giovannetti, V.

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
[Crossref]

Girvin, S. M.

C. Wang, Y. Y. Gao, P. Reinhold, R. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A Schrödinger cat living in two boxes,” Science 352, 1087–1091 (2016).
[Crossref]

Gisin, N.

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Supplementary Material (1)

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Figures (6)

Fig. 1.
Fig. 1. Schematics for generating (a) ideal and (b) approximated entangled coherent states (ECSs). A 50/50 beam splitter combines a coherent state | β at port a and (a) coherent state superpositions (CSSs) N ˜ β ( | β + | β ) at port b to result with an exact ECS N α ( | α , 0 + | 0 , α ) in ports c , d . (b) When the squeezed vacuum state | ξ enters port b instead of CSS, the result in ports c , d approximates ECS (see text). In our experiment, a joint photon number measurement N c , N d is performed at modes c , d , respectively, using photon number-resolving detectors.
Fig. 2.
Fig. 2. Fidelity between ECS and states generated by mixing CS and SV, F = | ψ ECS α | ψ out | 2 , for the optimal SV amplitude as a function of the total photon number on average n ¯ [Eq. (4)] in solid black. The inset shows F for low values of the average photon number. The fidelity between CSS and the vacuum state ( | vac = | 0 ) is presented for comparison in dashed purple.
Fig. 3.
Fig. 3. Simulation results showing violation of the third Janssen inequality J 3 0 is shown below the gray shaded area, for the state | ψ out produced by mixing CS and SV with the optimal parameters of Eq. (9) (solid line), and for an ideal ECS (dashed line), as a function of the total average photon number n ¯ .
Fig. 4.
Fig. 4. Experimental setup for generation of entangled coherent states, detailed layout of the setup. 120-fs pulses from a Ti:sapphire oscillator operated at 80 MHz are up-converted using a lithium triborate (LBO) crystal, short-pass filtered, and then down-converted using a beta barium borate (BBO) crystal, generating a squeezed vacuum state, having correlated photon pairs at the original wavelength (808 nm). This squeezed vacuum ( H polarization) is mixed with attenuated coherent light ( V polarization) on a polarizing beam splitter (PBS). A thermally induced drift in the relative phase is corrected every few minutes with the use of a liquid crystal phase retarder, ϕ . The spatial and spectral modes are matched using a polarization-maintaining fiber (PMF) and a 3-nm (full width at half max) bandpass filter (BPF). CS and SV are mixed by a 50/50 beam splitter transformation [Fig. 1(b)] in a collinear, polarization-based inherently phase-stable design, by using a PMF fiber aligned at ± 45 ° ( D , A ) polarization axes, where ECSs are realized. Photon-number resolving detection is performed using an array of 16 single-photon counting modules (SPCM, Perkin Elmer), and 1:8 multi-mode fiber splitters (MM-FSs).
Fig. 5.
Fig. 5. Experimental Fock projection measurements of coherent and squeezed vacuum light interfered on a 50/50 beam splitter. (a)  N -photon correlation rates plotted against the photon number difference between the output ports of the beam splitter, N c N d . Error bars represent the statistical standard error of the 24 h long measurement. (b) Multiphoton correlation probabilities, normalized for every number of measured photons, P ˜ N c , N d = P N c , N d / ( k = 0 N P k , N k ) .
Fig. 6.
Fig. 6. Similarity F ˜ between the state generated by mixing of SV with CS and ECS in our setup, accounting for loss ( η = 0.1 , see text), for various amounts of SV, using experimental (circles) and simulated (solid line) photon correlation measurements. The approximate ECS is achieved for the optimal SV fraction of sinh ( 2 r ) / | α | 2 = 1 [Eq. (9)], showing maximal similarity to ECS.

Equations (13)

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| ψ ECS α = N α ( | α , 0 + | 0 , α ) ,
| ψ CSS β = N ˜ β ( | β + | β ) ,
| ψ NOON N = ( | N , 0 + | 0 , N ) / 2 ,
n ¯ = | α | 2 / ( 1 + e | α | 2 ) .
| β = e | β | 2 / 2 n = 0 β n n ! | n , β = | β | e i ϕ ,
| ξ = 1 cosh r m = 0 ( 1 ) m ( 2 m ) ! 2 m m ! ( e i θ tanh r ) m | 2 m ,
P N c , N d ( β , r , θ ) = | N c , N d | ψ out c , d | 2 .
F = | ψ ECS α | ψ out | 2 = | ψ CSS α / 2 | ξ | 2 = 1 cosh r cosh ( | α | 2 2 ) e | α | 2 2 ( cos ( θ 2 ϕ ) tanh r ) .
r = arcsinh ( | α | 2 ) / 2 , θ = 2 ϕ + π ,
Q c ( μ ) = ψ out | Q ^ c ( μ ) I ^ d | ψ out ,
Q d ( ν ) = ψ out | I ^ c Q ^ d ( ν ) | ψ out ,
Q c , d ( μ , ν ) = ψ out | Q ^ c ( μ ) Q ^ d ( ν ) | ψ out .
J 3 = Q ( α ) Q ( α , β ) Q ( α , γ ) Q ( α , δ ) + Q ( β , γ ) + Q ( β , δ ) + Q ( γ , δ ) 0 ,

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