Abstract

The two modes of the Einstein-Podolsky-Rosen quadrature entangled state generated by parametric down-conversion interfere on a beam splitter of variable splitting ratio. Detection of a photon in one of the beam splitter output channels heralds preparation of a signal state in the other, which is characterized using homodyne tomography. By controlling the beam splitting ratio, the signal state can be chosen anywhere between the single-photon and squeezed state.

© 2010 Optical Society of America

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References

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    [CrossRef] [PubMed]
  4. A. Zavatta, S. Viciani, and M. Bellini, “Tomographic reconstruction of the single-photon Fock state by highfrequency homodyne detection,” Phys. Rev. A 70, 053821 (2004).
    [CrossRef]
  5. S. R. Huisman, N. Jain, S. A. Babichev, F. Vewinger, A. N. Zhang, S. H. Youn, and A. I. Lvovsky, “Instant singlephoton Fock state tomography,” Opt. Lett. 34, 2739–2741 (2009).
    [CrossRef] [PubMed]
  6. A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-Classical Transition with Single-Photon-Added Coherent States of Light,” Science 306, 660–662 (2004).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  9. E. Bimbard, N. Jain, A. MacRae, and A. I. Lvovsky, “Quantum-optical state engineering up to the two-photon level,” Nat. Photonics 4, 243–247 (2010).
    [CrossRef]
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  14. A. I. Lvovsky, “Iterative maximum-likelihood reconstruction in quantum homodyne tomography,” J. Opt. B: Quantum Semiclassical Opt. 6, S556–S559 (2004).
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    [CrossRef]
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    [CrossRef]
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2010 (2)

E. Bimbard, N. Jain, A. MacRae, and A. I. Lvovsky, “Quantum-optical state engineering up to the two-photon level,” Nat. Photonics 4, 243–247 (2010).
[CrossRef]

H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Entanglement distillation from Gaussian input states,” Nat. Photonics 4, 178–181 (2010).
[CrossRef]

2009 (3)

A. I. Lvovsky, and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81, 299–332 (2009).
[CrossRef]

S. R. Huisman, N. Jain, S. A. Babichev, F. Vewinger, A. N. Zhang, S. H. Youn, and A. I. Lvovsky, “Instant singlephoton Fock state tomography,” Opt. Lett. 34, 2739–2741 (2009).
[CrossRef] [PubMed]

G. Puentes, J. S. Lundeen, M. P. A. Branderhorst, H. B. Coldenstrodt-Ronge, B. J. Smith, and I. A. Walmsley, “Bridging Particle andWave Sensitivity in a Configurable Detector of Positive Operator-Valued Measures,” Phys. Rev. Lett. 102, 080404 (2009).
[CrossRef] [PubMed]

2007 (4)

J. Řeháček, Z. Hradil, E. Knill and A. I. Lvovsky, “Diluted maximum-likelihood algorithm for quantum tomography,” Phys. Rev. A 75, 042108 (2007).
[CrossRef]

J. Appel, D. Hoffman, E. Figueroa, and A. I. Lvovsky, “Electronic noise in optical homodyne tomography,” Phys. Rev. A 75, 035802 (2007).
[CrossRef]

A. I. Lvovsky, W. Wasilewski, and K. Banaszek, “Decomposing a pulsed optical parametric amplifier into independent squeezers,” J. Mod. Opt. 54, 721–733 (2007).
[CrossRef]

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[CrossRef]

2006 (2)

W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulsed squeezed light: simultaneous squeezing of multiple modes,” Phys. Rev. A 73, 063819 (2006).
[CrossRef]

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating Optical Schrdinger Kittens for Quantum Information Processing,” Science 312, 83–86 (2006).
[CrossRef] [PubMed]

2004 (3)

A. Zavatta, S. Viciani, and M. Bellini, “Tomographic reconstruction of the single-photon Fock state by highfrequency homodyne detection,” Phys. Rev. A 70, 053821 (2004).
[CrossRef]

A. I. Lvovsky, “Iterative maximum-likelihood reconstruction in quantum homodyne tomography,” J. Opt. B: Quantum Semiclassical Opt. 6, S556–S559 (2004).
[CrossRef]

A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-Classical Transition with Single-Photon-Added Coherent States of Light,” Science 306, 660–662 (2004).
[CrossRef] [PubMed]

2002 (1)

T. Aichele, A. I. Lvovsky, and S. Schiller, “Optical mode characterization of single photons prepared via conditional measurements on a biphoton state,” Eur. Phys. J. D 18, 237–245 (2002).
[CrossRef]

2001 (1)

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]

Aichele, T.

T. Aichele, A. I. Lvovsky, and S. Schiller, “Optical mode characterization of single photons prepared via conditional measurements on a biphoton state,” Eur. Phys. J. D 18, 237–245 (2002).
[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]

Appel, J.

J. Appel, D. Hoffman, E. Figueroa, and A. I. Lvovsky, “Electronic noise in optical homodyne tomography,” Phys. Rev. A 75, 035802 (2007).
[CrossRef]

Babichev, S. A.

Banaszek, K.

A. I. Lvovsky, W. Wasilewski, and K. Banaszek, “Decomposing a pulsed optical parametric amplifier into independent squeezers,” J. Mod. Opt. 54, 721–733 (2007).
[CrossRef]

W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulsed squeezed light: simultaneous squeezing of multiple modes,” Phys. Rev. A 73, 063819 (2006).
[CrossRef]

Bellini, M.

A. Zavatta, S. Viciani, and M. Bellini, “Tomographic reconstruction of the single-photon Fock state by highfrequency homodyne detection,” Phys. Rev. A 70, 053821 (2004).
[CrossRef]

A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-Classical Transition with Single-Photon-Added Coherent States of Light,” Science 306, 660–662 (2004).
[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]

Bimbard, E.

E. Bimbard, N. Jain, A. MacRae, and A. I. Lvovsky, “Quantum-optical state engineering up to the two-photon level,” Nat. Photonics 4, 243–247 (2010).
[CrossRef]

Branderhorst, M. P. A.

G. Puentes, J. S. Lundeen, M. P. A. Branderhorst, H. B. Coldenstrodt-Ronge, B. J. Smith, and I. A. Walmsley, “Bridging Particle andWave Sensitivity in a Configurable Detector of Positive Operator-Valued Measures,” Phys. Rev. Lett. 102, 080404 (2009).
[CrossRef] [PubMed]

Coldenstrodt-Ronge, H. B.

G. Puentes, J. S. Lundeen, M. P. A. Branderhorst, H. B. Coldenstrodt-Ronge, B. J. Smith, and I. A. Walmsley, “Bridging Particle andWave Sensitivity in a Configurable Detector of Positive Operator-Valued Measures,” Phys. Rev. Lett. 102, 080404 (2009).
[CrossRef] [PubMed]

Dowling, J. P.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[CrossRef]

Figueroa, E.

J. Appel, D. Hoffman, E. Figueroa, and A. I. Lvovsky, “Electronic noise in optical homodyne tomography,” Phys. Rev. A 75, 035802 (2007).
[CrossRef]

Furusawa, A.

H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Entanglement distillation from Gaussian input states,” Nat. Photonics 4, 178–181 (2010).
[CrossRef]

Grangier, P.

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating Optical Schrdinger Kittens for Quantum Information Processing,” Science 312, 83–86 (2006).
[CrossRef] [PubMed]

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]

Hayasaka, K.

H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Entanglement distillation from Gaussian input states,” Nat. Photonics 4, 178–181 (2010).
[CrossRef]

Hoffman, D.

J. Appel, D. Hoffman, E. Figueroa, and A. I. Lvovsky, “Electronic noise in optical homodyne tomography,” Phys. Rev. A 75, 035802 (2007).
[CrossRef]

Huisman, S. R.

Jain, N.

E. Bimbard, N. Jain, A. MacRae, and A. I. Lvovsky, “Quantum-optical state engineering up to the two-photon level,” Nat. Photonics 4, 243–247 (2010).
[CrossRef]

S. R. Huisman, N. Jain, S. A. Babichev, F. Vewinger, A. N. Zhang, S. H. Youn, and A. I. Lvovsky, “Instant singlephoton Fock state tomography,” Opt. Lett. 34, 2739–2741 (2009).
[CrossRef] [PubMed]

Kok, P.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[CrossRef]

Laurat, J.

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating Optical Schrdinger Kittens for Quantum Information Processing,” Science 312, 83–86 (2006).
[CrossRef] [PubMed]

Lundeen, J. S.

G. Puentes, J. S. Lundeen, M. P. A. Branderhorst, H. B. Coldenstrodt-Ronge, B. J. Smith, and I. A. Walmsley, “Bridging Particle andWave Sensitivity in a Configurable Detector of Positive Operator-Valued Measures,” Phys. Rev. Lett. 102, 080404 (2009).
[CrossRef] [PubMed]

Lvovsky, A. I.

E. Bimbard, N. Jain, A. MacRae, and A. I. Lvovsky, “Quantum-optical state engineering up to the two-photon level,” Nat. Photonics 4, 243–247 (2010).
[CrossRef]

A. I. Lvovsky, and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81, 299–332 (2009).
[CrossRef]

S. R. Huisman, N. Jain, S. A. Babichev, F. Vewinger, A. N. Zhang, S. H. Youn, and A. I. Lvovsky, “Instant singlephoton Fock state tomography,” Opt. Lett. 34, 2739–2741 (2009).
[CrossRef] [PubMed]

J. Appel, D. Hoffman, E. Figueroa, and A. I. Lvovsky, “Electronic noise in optical homodyne tomography,” Phys. Rev. A 75, 035802 (2007).
[CrossRef]

A. I. Lvovsky, W. Wasilewski, and K. Banaszek, “Decomposing a pulsed optical parametric amplifier into independent squeezers,” J. Mod. Opt. 54, 721–733 (2007).
[CrossRef]

W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulsed squeezed light: simultaneous squeezing of multiple modes,” Phys. Rev. A 73, 063819 (2006).
[CrossRef]

A. I. Lvovsky, “Iterative maximum-likelihood reconstruction in quantum homodyne tomography,” J. Opt. B: Quantum Semiclassical Opt. 6, S556–S559 (2004).
[CrossRef]

T. Aichele, A. I. Lvovsky, and S. Schiller, “Optical mode characterization of single photons prepared via conditional measurements on a biphoton state,” Eur. Phys. J. D 18, 237–245 (2002).
[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]

MacRae, A.

E. Bimbard, N. Jain, A. MacRae, and A. I. Lvovsky, “Quantum-optical state engineering up to the two-photon level,” Nat. Photonics 4, 243–247 (2010).
[CrossRef]

Milburn, G. J.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[CrossRef]

Mlynek, J.

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]

Munro, W. J.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[CrossRef]

Neergaard-Nielsen, J. S.

H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Entanglement distillation from Gaussian input states,” Nat. Photonics 4, 178–181 (2010).
[CrossRef]

Nemoto, K.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[CrossRef]

Ourjoumtsev, A.

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating Optical Schrdinger Kittens for Quantum Information Processing,” Science 312, 83–86 (2006).
[CrossRef] [PubMed]

Puentes, G.

G. Puentes, J. S. Lundeen, M. P. A. Branderhorst, H. B. Coldenstrodt-Ronge, B. J. Smith, and I. A. Walmsley, “Bridging Particle andWave Sensitivity in a Configurable Detector of Positive Operator-Valued Measures,” Phys. Rev. Lett. 102, 080404 (2009).
[CrossRef] [PubMed]

Radzewicz, C.

W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulsed squeezed light: simultaneous squeezing of multiple modes,” Phys. Rev. A 73, 063819 (2006).
[CrossRef]

Ralph, T. C.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[CrossRef]

Raymer, M. G.

A. I. Lvovsky, and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81, 299–332 (2009).
[CrossRef]

Sasaki, M.

H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Entanglement distillation from Gaussian input states,” Nat. Photonics 4, 178–181 (2010).
[CrossRef]

Schiller, S.

T. Aichele, A. I. Lvovsky, and S. Schiller, “Optical mode characterization of single photons prepared via conditional measurements on a biphoton state,” Eur. Phys. J. D 18, 237–245 (2002).
[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]

Smith, B. J.

G. Puentes, J. S. Lundeen, M. P. A. Branderhorst, H. B. Coldenstrodt-Ronge, B. J. Smith, and I. A. Walmsley, “Bridging Particle andWave Sensitivity in a Configurable Detector of Positive Operator-Valued Measures,” Phys. Rev. Lett. 102, 080404 (2009).
[CrossRef] [PubMed]

Takahashi, H.

H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Entanglement distillation from Gaussian input states,” Nat. Photonics 4, 178–181 (2010).
[CrossRef]

Takeoka, M.

H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Entanglement distillation from Gaussian input states,” Nat. Photonics 4, 178–181 (2010).
[CrossRef]

Takeuchi, M.

H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Entanglement distillation from Gaussian input states,” Nat. Photonics 4, 178–181 (2010).
[CrossRef]

Tualle-Brouri, R.

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating Optical Schrdinger Kittens for Quantum Information Processing,” Science 312, 83–86 (2006).
[CrossRef] [PubMed]

Vewinger, F.

Viciani, S.

A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-Classical Transition with Single-Photon-Added Coherent States of Light,” Science 306, 660–662 (2004).
[CrossRef] [PubMed]

A. Zavatta, S. Viciani, and M. Bellini, “Tomographic reconstruction of the single-photon Fock state by highfrequency homodyne detection,” Phys. Rev. A 70, 053821 (2004).
[CrossRef]

Walmsley, I. A.

G. Puentes, J. S. Lundeen, M. P. A. Branderhorst, H. B. Coldenstrodt-Ronge, B. J. Smith, and I. A. Walmsley, “Bridging Particle andWave Sensitivity in a Configurable Detector of Positive Operator-Valued Measures,” Phys. Rev. Lett. 102, 080404 (2009).
[CrossRef] [PubMed]

Wasilewski, W.

A. I. Lvovsky, W. Wasilewski, and K. Banaszek, “Decomposing a pulsed optical parametric amplifier into independent squeezers,” J. Mod. Opt. 54, 721–733 (2007).
[CrossRef]

W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulsed squeezed light: simultaneous squeezing of multiple modes,” Phys. Rev. A 73, 063819 (2006).
[CrossRef]

Youn, S. H.

Zavatta, A.

A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-Classical Transition with Single-Photon-Added Coherent States of Light,” Science 306, 660–662 (2004).
[CrossRef] [PubMed]

A. Zavatta, S. Viciani, and M. Bellini, “Tomographic reconstruction of the single-photon Fock state by highfrequency homodyne detection,” Phys. Rev. A 70, 053821 (2004).
[CrossRef]

Zhang, A. N.

Eur. Phys. J. D (1)

T. Aichele, A. I. Lvovsky, and S. Schiller, “Optical mode characterization of single photons prepared via conditional measurements on a biphoton state,” Eur. Phys. J. D 18, 237–245 (2002).
[CrossRef]

J. Mod. Opt. (1)

A. I. Lvovsky, W. Wasilewski, and K. Banaszek, “Decomposing a pulsed optical parametric amplifier into independent squeezers,” J. Mod. Opt. 54, 721–733 (2007).
[CrossRef]

J. Opt. B: Quantum Semiclassical Opt. (1)

A. I. Lvovsky, “Iterative maximum-likelihood reconstruction in quantum homodyne tomography,” J. Opt. B: Quantum Semiclassical Opt. 6, S556–S559 (2004).
[CrossRef]

Nat. Photonics (2)

E. Bimbard, N. Jain, A. MacRae, and A. I. Lvovsky, “Quantum-optical state engineering up to the two-photon level,” Nat. Photonics 4, 243–247 (2010).
[CrossRef]

H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Entanglement distillation from Gaussian input states,” Nat. Photonics 4, 178–181 (2010).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (4)

A. Zavatta, S. Viciani, and M. Bellini, “Tomographic reconstruction of the single-photon Fock state by highfrequency homodyne detection,” Phys. Rev. A 70, 053821 (2004).
[CrossRef]

J. Řeháček, Z. Hradil, E. Knill and A. I. Lvovsky, “Diluted maximum-likelihood algorithm for quantum tomography,” Phys. Rev. A 75, 042108 (2007).
[CrossRef]

J. Appel, D. Hoffman, E. Figueroa, and A. I. Lvovsky, “Electronic noise in optical homodyne tomography,” Phys. Rev. A 75, 035802 (2007).
[CrossRef]

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Figures (5)

Fig. 1.
Fig. 1.

Concept of the experimental setup. The figure is for illustration purpose only; the inset shows the actual implementation of parametric down-conversion and the variable beam splitter. HWP, half-wave plate; PBS, polarizing beam splitter; SPCM, single-photon counting module. The down-conversion is spatially and spectrally-degenerate, but polarization-nondegenerate.

Fig. 2.
Fig. 2.

Quadrature noise of the experimentally observed vacuum and signal states at different reflectivities of the variable beam splitter, for varying local oscillator phase. The angle θ of the half-wave plate and the corresponding beam splitter reflectivity (given by R = cos2 2θ) are indicated for each curve. The observed quadrature noise is influenced by preparation and detection inefficiencies.

Fig. 3.
Fig. 3.

Experimentally reconstructed Wigner functions (top row) and density matrices (absolute values, bottom row). The insets show contour diagrams associated to specificWigner functions.

Fig. 4.
Fig. 4.

Cross-section of the Wigner Function of the single-photon state (a) and the quadrature noise of the squeezed state as a function of the optical phase (b), obtained from their respective density matrices, experimentally reconstructed without correcting for detection inefficiency. The squeezing variance features a solid curve obtained from maximum-likelihood reconstruction, while the points with error-bars from 0 to π/2 are representative of the binned raw quadrature data. The error bars correspond to σ i 2 N i , σi being the width of a Gaussian distribution from Ni samples in each bin [19].

Fig. 5.
Fig. 5.

Maximum and minimum variances of the measured quadratures (a) and the Mandel parameter of the reconstructed states (b) as functions of the HWP angle. The theoretical predictions are calculated for η = 0.55, γ 2 = 0.025 in the limit of low single-photon detection efficiency. The states with the minimum quadrature variance below 1/2 or with a negative Mandel parameter are nonclassical.

Equations (2)

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Ψ = 1 γ 2 [ 0 s , 0 i + γ 1 s , 1 i + O ( γ 2 ) ] ,
Ψ out ( 1 γ 2 ) [ 0 s + γ 2 2 s ] [ 0 i γ 2 2 i ] .

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