Abstract

Considering matter wave bright solitons from weakly coupled Bose-Einstein condensates trapped in a double-well potential, we study the formation of macroscopic non-classical states, including Schrödinger-cat superposition state and maximally path entangled N00N-state. We examine these macroscopic states by Mach-Zehnder interferometer in the context of parity measurements, which has been done to obtain Heisenberg limit accuracy for linear phase shift measurement. We reveal that the ratio of two-body scattering length to intra-well hopping parameter can be measured with the scaling beyond this limit by using nonlinear phase shift with interacting quantum solitons.

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

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2017 (2)

C. Schneider, K. Winkler, M. D. Fraser, M. Kamp, Y. Yamamoto, E. A. Ostrovskaya, and S. Hofling, “Exciton-polariton trapping and potential landscape engineering,” Rep. Prog. Phys. 80, 016503 (2017).
[Crossref]

Y. Sun, P. Wen, Y. Yoon, G. Liu, M. Steger, L. N. Pfeiffer, K. West, D. W. Snoke, and K. A. Nelson, “Bose-Einstein Condensation of Long-Lifetime Polaritons in Thermal Equilibrium,” Phys. Rev. Lett. 118, 016602 (2017).
[Crossref] [PubMed]

2016 (2)

H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, “Detection of 15 dB Squeezed States of Light and their Application for the Absolute Calibration of Photoelectric Quantum Efficiency,” Phys. Rev. Lett. 117, 110801 (2016).
[Crossref]

Ch. Weiss, S.L. Cornish, and S. A. Gardiner, “Superballistic center-of-mass motion in one-dimensional attractive Bose gases:Decoherence-induced Gaussian random walks in velocity space,” Phys. Rev. A 93, 013605 (2016).
[Crossref]

2015 (2)

R. Demkowicz-Dobrzanski, M. Jarzyna, and J. Kolodynski, “Quantum limits in optical interferometry,” Progress in Optics,  60, 345–435 (2015).
[Crossref]

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

2014 (5)

Qi-Ping Su and Chui-Ping Yang and Shi-Biao Zheng, “Fast and simple scheme for generating NOON states of photons in circuit QED,” Scientific Reports 4, 3898 (2014).
[Crossref] [PubMed]

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] [PubMed]

C. Sturm, D. Tanese, H.S. Nguyen, H. Flayac, E. Galopin, A. Lemaitre, I. Sagnes, D. Solnyshkov, A. Amo, G. Malpuech, and J. Bloch, “All-optical phase modulation in a cavity-polariton Mach-Zehnder interferometer,” Nature Comm. 5, 3278 (2014).
[Crossref]

L. Cohen, D. Istrati, L. Dovrat, and H. S. Eisenberg, “Super-resolved phase measurements at the shot noise limit by parity measurement,” Optics Express 22, 11945–11953 (2014).
[Crossref] [PubMed]

N. Takemura, S. Trebaol, M. Wouters, M. T. Portella-Oberli, and B. Deveaud, “Polaritonic Feshbach resonance”, Nat. Phys.  10, 500 (2014).
[Crossref]

2013 (2)

J. Kolodyski and R. Demkowicz-Dobrzaski, “Efficient tools for quantum metrology with uncorrelated noise,” New J. Phys.  15073043 (2013).
[Crossref]

I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 88, 299 (2013).
[Crossref]

2012 (3)

M. Sich, D. N. Krizhanovskii, M. S. Skolnick, A. V. Gorbach, R. Hartley, D. V. Skryabin, E. A. Cerda-Méndez, K. Biermann, R. Hey, and P. V. Santos, “Observation of bright polariton solitons in a semiconductor microcavity,” Nat. Photon. 6, 50–55 (2012).
[Crossref]

T. Byrnes, Kai Wen, and Y. Yamamoto, “Macroscopic quantum computation using Bose-Einstein condensates,” Phys. Rev. A 85, 040306 (2012).
[Crossref]

Q. Y. He, P. D. Drummond, M. K. Olsen, and M. D. Reid, “Einstein-Podolsky-Rosen entanglement and steering in two-well Bose-Einstein-condensate ground states,” Phys. Rev. A 86, 023626 (2012).
[Crossref]

2011 (4)

G. Mazzarella, L. Salasnich, A. Parola, and F. Toigo, “Coherence and entanglement in the ground state of a bosonic Josephson junction: From macroscopic Schrödinger cat states to separable Fock states,” Phys. Rev. A 83, 053607 (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]

Jaewoo Joo, W.J. Munro, and T. P. Spiller, “Quantum Metrology with Entangled Coherent States,” Phys. Rev. Lett. 107, 083601 (2011).
[Crossref] [PubMed]

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

2010 (5)

C. Gross, T. Zibold, E. Nickolas, J. Estève, and M.K. Oberthaler, “Nonlinear atom interferometer surpasses classical precision limit,” Nature 464, 1165–1169 (2010).
[Crossref] [PubMed]

S. T Merkel and F. K Wilhelm, “Generation and detection of NOON states in superconducting circuits,” New Journal of Physics 12, 093036 (2010).
[Crossref]

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON States by Mixing Quantum and Classical Light,” Science 328, 879–881 (2010).
[Crossref] [PubMed]

Hui Deng, H. Haug, and Yoshihisa Yamamoto, “Exciton-polariton Bose-Einstein condensation,” Rev. Mod. Phys.  82, 1489 (2010).
[Crossref]

Yu-Ao Chen, Bao Xiao-Hui, Yuan Zhen-Sheng, Chen Shuai, Zhao Bo, and Pan Jian-Wei, “Heralded Generation of an Atomic NOON State,” Phys. Rev. Letts 104, 043601 (2010).
[Crossref]

2009 (4)

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal Quantum Phase Estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[Crossref]

K. Banaszek, R. Demkowicz-Dobrzaski, and I. A. Walmsley, “Quantum states made to measure,” Nature Photonics 3, 673 (2009).
[Crossref]

Kim Heonoh, Park Hee Su, and Choi Sang-Kyung, “Three-photon N00N states generated by photon subtraction from double photon pairs,” Optics Express 17, 19720–19726 (2009).
[Crossref]

D. Maldonado-Mundo and A. Luis, “Metrological resolution and minimum uncertainty states in linear and nonlinear signal detection schemes,” Phys. Rev. A 80, 063811 (2009).
[Crossref]

2008 (5)

J. Esteve, C. Gross, A. Weller, S. Giovanazzi, and M. K. Oberthaler, “Squeezing and entanglement in a BoseâĂŞEinstein condensate,” Nature 455, 1216 (2008).
[Crossref]

J. P. Dowling, “Quantum optical metrology - the lowdown on high-N00N states,” Contem. Phys. 49, 125–143 (2008).
[Crossref]

L. Pezze and A. Smerzi, “Mach-Zehnder Interferometry at the Heisenberg Limit with Coherent and Squeezed-Vacuum Light,” Phys. Rev. Lett. 100, 073601 (2008).
[Crossref] [PubMed]

S. M. Roy and S. L. Braunstein, “Exponentially Enhanced Quantum Metrology,” Phys. Rev. Lett. 100, 220501 (2008).
[Crossref] [PubMed]

S. Boixo, A. Datta, M. J. Davis, S. T. Flammia, A. Shaji, and C. M. Caves, “Quantum Metrology: Dynamics versus Entanglement,” Phys. Rev. Lett. 101, 040403 (2008).
[Crossref] [PubMed]

2007 (3)

S. Boixo, Steven T. Flammia, C. M. Caves, and J.M Geremia, “Generalized Limits for Single-Parameter Quantum Estimation,” Phys. Rev. Lett. 98, 090401 (2007).
[Crossref] [PubMed]

A. M. Dudarev and M. G. Raizen, and Qian Niu, “Quantum Many-Body Culling: Production of a Definite Number of Ground-State Atoms in a Bose-Einstein Condensate,” Phys. Rev. Lett. 98, 063001 (2007).
[Crossref]

C. C. Gerry, A. Benmoussa, and R. A. Campos, “Parity measurements, Heisenberg-limited phase estimation, and beyond,” J. Mod. Opt. 54, 2177 (2007).
[Crossref]

2006 (2)

O. Morsch and M. Oberthaler, “Dynamics of Bose-Einstein condensates in optical lattices,” Rev. Mod. Phys. 78, 179 (2006).
[Crossref]

L. Fu and J. Liu, “Quantum entanglement manifestation of transition to nonlinear self-trapping for Bose-Einstein condensates in a symmetric double well,” Phys. Rev. A 74, 063614 (2006).
[Crossref]

2005 (1)

J. Beltran and A. Luis, “Breaking the Heisenberg limit with inefficient detectors,” Phys. Rev. A 72, 045801 (2005).
[Crossref]

2004 (2)

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-Enhanced Measurements: Beating the Standard Quantum Limit,” Science 306, 1330–1336 (2004).
[Crossref] [PubMed]

P. Kok, S.L. Braunstein, and J. P. Dowling, “Quantum lithography, entanglement and Heisenberg-limited parameter estimation,” J. Opt. B: Quantum Semiclass. Opt 6, S811 (2004).
[Crossref]

2002 (2)

K. E. Strecker, G. B. Partridge, A. G. Truscott, and R. G. Hulet, “Formation and propagation of matter-wave soliton trains,” Nature 417, 150–153 (2002).
[Crossref] [PubMed]

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2001 (3)

A. Sorensen, L.-M. Duan, J. I. Cirac, and P. Zoller, “Many-particle entanglement with Bose-Einstein condensates,” Nature 409, 63–66 (2001).
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2000 (2)

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum Interferometric Optical Lithography: Exploiting Entanglement to Beat the Diffraction Limit,” Phys. Rev. Lett. 85, 2733 (2000).
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S. Raghavan and G. P. Agrawal, “Switching and self-trapping dynamics of Bose-Einstein solitons,” J. Mod. Opt. 47, 1155–1169 (2000).
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1999 (1)

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1998 (5)

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1997 (4)

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1996 (2)

J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, “Optimal frequency measurements with maximally correlated states,” Phys. Rev. A 54, R4649 (1996).
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1995 (1)

A. P. Alodjants and S. M. Arakelian, “Quantum chaos and its observation in coupled optical solitons,” Zh. Eksp. i Teor. Fiz.  107, 1792 (1995).

1992 (1)

D. J. Wineland, J. J. Bollinger, W. M. Itano, F. L. Moore, and D. J. Heinzen, “Spin squeezing and reduced quantum noise in spectroscopy,” Phys. Rev. A 46, R6797 (1992).
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1990 (1)

1989 (2)

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1987 (3)

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1986 (1)

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I. Afek, O. Ambar, and Y. Silberberg, “High-NOON States by Mixing Quantum and Classical Light,” Science 328, 879–881 (2010).
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A. P. Alodjants and S. M. Arakelian, “Quantum chaos and its observation in coupled optical solitons,” Zh. Eksp. i Teor. Fiz.  107, 1792 (1995).

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I. Afek, O. Ambar, and Y. Silberberg, “High-NOON States by Mixing Quantum and Classical Light,” Science 328, 879–881 (2010).
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C. Sturm, D. Tanese, H.S. Nguyen, H. Flayac, E. Galopin, A. Lemaitre, I. Sagnes, D. Solnyshkov, A. Amo, G. Malpuech, and J. Bloch, “All-optical phase modulation in a cavity-polariton Mach-Zehnder interferometer,” Nature Comm. 5, 3278 (2014).
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B. P. Anderson and M. A. Kasevich, “Macroscopic quantum interference from atomic tunnel arrays,” Science 282, 1686–1689 (1998).
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Andrews, M.R.

M.R. Andrews, C.G. Townsend, H. Miesner, D.S. Durfee, D.M. Kurn, and W. Ketterle, “Observation of Interference Between Two Bose Condensates,” Science 275, 637–641 (1997).
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Banaszek, K.

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J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, “Optimal frequency measurements with maximally correlated states,” Phys. Rev. A 54, R4649 (1996).
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A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum Interferometric Optical Lithography: Exploiting Entanglement to Beat the Diffraction Limit,” Phys. Rev. Lett. 85, 2733 (2000).
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P. Kok, S.L. Braunstein, and J. P. Dowling, “Quantum lithography, entanglement and Heisenberg-limited parameter estimation,” J. Opt. B: Quantum Semiclass. Opt 6, S811 (2004).
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S. Boixo, Steven T. Flammia, C. M. Caves, and J.M Geremia, “Generalized Limits for Single-Parameter Quantum Estimation,” Phys. Rev. Lett. 98, 090401 (2007).
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H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, “Detection of 15 dB Squeezed States of Light and their Application for the Absolute Calibration of Photoelectric Quantum Efficiency,” Phys. Rev. Lett. 117, 110801 (2016).
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S. Boixo, A. Datta, M. J. Davis, S. T. Flammia, A. Shaji, and C. M. Caves, “Quantum Metrology: Dynamics versus Entanglement,” Phys. Rev. Lett. 101, 040403 (2008).
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Figures (2)

Fig. 1
Fig. 1 (a) The dependence of the “cat size” 1/ϵ (14) on the population imbalance p0 for different numbers of particles N. One can see that “cat size” tends to infinity when |p0| tends to 1. Also 1/ϵ ≈ 0 when |p0| ≈ 0. Infinite “cat size” corresponds to macroscopic SC-state and can be approximately taken as a N00N-state. Zero “cat size” corresponds to microscopic SC-sate which means almost no entanglement. (b) Illustration of the precision measurement of the phase shift, based on a Mach-Zehnder interferometer (MZI). Here, QSPD denotes a quantum state preparation device, ϕ1 and ϕ2 are two resulting phases accumulated at the arms of interferometer, BS is a beam splitter, and D is a parity detector that runs in the particle counting regime.
Fig. 2
Fig. 2 (a) Reduced phase uncertainty N σ ϕ against total particle number N, for an initial SC-state used in the measurement procedure. The value N σ ϕ = 1 corresponds to SQL limit. (b) The dependence of σΘ on Θ demonstrating a second-order like phase transition from the state possessing non-zero σΘ beyond the linear Heisenberg limit (gray area) to the state nonapplicable for such measurements. The number of particles N = Nc = 6000 is taken for Lithium atomic condensates with negative scattering length, as example.

Equations (43)

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σ ϕ C 0 N 1 ,
H ^ = H ^ 1 + H ^ 2 + H ^ i n t ,
H ^ j = d z a ^ j ( z ) ( 1 2 M 2 z 2 + U 2 a ^ j ( z ) a ^ j ( z ) ) a ^ j ( z ) H ^ i n t = κ d z a ^ 2 ( z ) a ^ 1 ( z ) + H . C .
[ a ^ i ( z ) , a ^ j ( z ) ] = δ ( z z ) δ i j ; i , j = 1 , 2 .
| Ψ N = 1 N ! [ d z ( Ψ 1 a ^ 1 + Ψ 2 a ^ 2 ) ] N | 0 ,
L = j = 1 2 ( i 2 [ Ψ j * Ψ ˙ j Ψ ˙ j * Ψ j ] + 1 2 M Ψ j * 2 Ψ j z 2 U 2 | Ψ j | 4 ) κ ( Ψ 1 * Ψ 2 + Ψ 1 Ψ 2 * ) .
Ψ j = N j 2 | U | s e c h ( N j | U | 2 z ) e i M θ j .
L = L d z = M ( N 1 θ ˙ 1 + N 2 θ ˙ 2 ) + U 2 24 M ( N 1 3 + N 2 3 ) 4 κ N 1 N 2 N I ( p ) cos [ θ ] .
I ( p ) = 0 d z cosh 2 ( z ) + sinh 2 ( z p ) .
I ( p ) 1 α p 2
p ˙ = 1 M ( 1 p 2 ) ( 1 α p 2 ) sin [ θ ] ,
θ ˙ = Λ p + 2 p M cos [ θ ] [ 1 + α 2 α p 2 ] .
p 0 2 = 1 2 α [ 1 + α Λ 2 ] ,
cos ( θ 0 ) = M ;
p 0 2 = 1 ,
cos θ 0 = M Λ 2 ( 1 α ) .
| Ψ ( ± ) = 1 N ! [ d z ( Ψ a ^ 1 Ψ a ^ 2 ) ] N | 0 ,
Ψ ± = N U 4 ( 1 ± | p 0 | ) s e c h ( N U 4 ( 1 ± | p 0 | ) z ) ,
| Ψ = C ( | Ψ ( + ) + | Ψ ( ) ) .
ϵ = | Ψ ( ± ) | Ψ ( ) = X N .
| Φ ( ± ) = 1 N ! [ d z ( Φ a ^ 2 , 1 ) ] N | 0 ,
Φ = N U 2 s e c h ( N U 2 z ) .
| Φ = 1 2 ( | Φ ( + ) + e i θ N | Φ ( ) ) ,
( Δ ϕ ) 2 = ( Δ P ^ ) 2 | P ^ ϕ | 2 ,
S ^ 0 = 1 2 ( a ^ 1 a ^ 1 + a ^ 2 a ^ 2 ) d z ,
S ^ 1 = 1 2 ( a ^ 1 a ^ 1 a ^ 2 a ^ 2 ) d z ,
S ^ 2 = 1 2 ( a ^ 1 a ^ 2 + a ^ 2 a ^ 1 ) d z ,
S ^ 3 = i 2 ( a ^ 2 a ^ 1 a ^ 1 a ^ 2 ) d z .
P ^ a ^ 2 exp [ i π ( S ^ 0 S ^ 1 ) ] .
P ^ a ^ 2 = U ^ M Z I P ^ a ^ 2 U ^ M Z I = e i π S ^ 0 e i ϕ S ^ 1 e i π S ^ 3 e i ϕ S ^ 1 .
| Ψ = C ( | j , m + | j , m ) ,
| Φ = 1 2 ( | j , j + e i θ N | j , j ) .
Ψ | P ^ a 2 | Ψ = ( 1 ) N cos [ ( ϕ π 2 ) N | p 0 | ] ,
Φ | P ^ a ^ 2 | Φ = { ( 1 ) N 2 cos [ ϕ N + θ N ] ; N is even ( 1 ) N + 1 2 sin [ ϕ N + θ N ] ; N is odd
Ψ | ( Δ P ^ a ^ 2 ) 2 | Ψ sin 2 [ ( ϕ π 2 ) N | p 0 | ] ,
Φ | Δ P ^ a ^ 2 | Φ = { sin 2 [ ϕ N + θ N ] ; N is even cos 2 [ ϕ N + θ N ] ; N is odd
Ψ | ( Δ ϕ ) 2 | Ψ = 1 N 2 | p 0 | 2 ,
Φ | ( Δ ϕ ) 2 | Φ = 1 N 2 .
θ N = N arccos ( Θ N 2 2 ( 1 α ) ) ,
θ N = π 2 N + N 3 2 ( 1 α ) Θ + O ( Θ 3 ) ,
Φ | P ^ a ^ 2 | Φ = { ( 1 ) N 2 cos [ θ N ] ; N is even ( 1 ) N + 1 2 sin [ θ N ] ; N is odd
Φ | ( Δ P ^ a ^ 2 ) 2 | Φ = { sin 2 [ θ N ] ; N is even cos 2 [ θ N ] ; N is odd
( Δ Θ ) 2 = 4 ( 1 α ) 2 N 6 .

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