Abstract

Complete and accurate quantum state characterization is a key requirement of quantum information science and technology. The Wigner quasi-probability distribution function provides such a characterization. We reconstructed the Wigner function of a narrowband single-photon state from photon-number-resolving measurements with transition-edge sensors (TESs) at system efficiency 58(2)%. This method makes no assumption on the nature of the measured state, although a limitation on continuous-wave photon flux was imposed by the TES. The negativity of the Wigner function was observed in the raw data without any inference or correction for decoherence.

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

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References

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    [Crossref]
  31. See supplemental document.
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    [Crossref]
  33. S. Feng and O. Pfister, “Stable nondegenerate optical parametric oscillation at degenerate frequencies in Na:KTP,” J. Opt. B 5, 262 (2003).
    [Crossref]
  34. S. Feng and O. Pfister, “Quantum interference of ultrastable twin optical beams,” Phys. Rev. Lett. 92, 203601 (2004).
    [Crossref]
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    [Crossref]
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    [Crossref]

2014 (2)

2013 (1)

B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L. Frunzio, S. M. Girvin, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “Deterministically encoding quantum information using 100-photon Schrödinger cat states,” Science 342, 607–610 (2013).
[Crossref]

2012 (1)

2010 (1)

K. Laiho, K. N. Cassemiro, D. Gross, and C. Silberhorn, “Probing the negative Wigner function of a pulsed single photon point by point,” Phys. Rev. Lett. 105, 253603 (2010).
[Crossref]

2009 (3)

M. Bondani, A. Allevi, and A. Andreoni, “Wigner function of pulsed fields by direct detection,” Opt. Lett. 34, 1444–1446 (2009).
[Crossref]

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

J. Niset, J. Fiurášek, and N. J. Cerf, “No-go theorem for Gaussian quantum error correction,” Phys. Rev. Lett. 102, 120501 (2009).
[Crossref]

2008 (1)

2007 (1)

S. Ghose and B. C. Sanders, “Non-Gaussian ancilla states for continuous variable quantum computation via Gaussian maps,” J. Mod. Opt. 54, 855–869 (2007).
[Crossref]

2006 (1)

N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. Nielsen, “Universal quantum computation with continuous-variable cluster states,” Phys. Rev. Lett. 97, 110501 (2006).
[Crossref]

2004 (2)

S. Feng and O. Pfister, “Quantum interference of ultrastable twin optical beams,” Phys. Rev. Lett. 92, 203601 (2004).
[Crossref]

S. Feng and O. Pfister, “Realization of an ultrastable twin-beam source for continuous-variable entanglement of bright beams,” Proc. SPIE 5161, 109–115 (2004).
[Crossref]

2003 (1)

S. Feng and O. Pfister, “Stable nondegenerate optical parametric oscillation at degenerate frequencies in Na:KTP,” J. Opt. B 5, 262 (2003).
[Crossref]

2002 (3)

P. Bertet, A. Auffeves, P. Maioli, S. Osnaghi, T. Meunier, M. Brune, J. M. Raimond, and S. Haroche, “Direct measurement of the Wigner function of a one-photon Fock state in a cavity,” Phys. Rev. Lett. 89, 200402 (2002).
[Crossref]

S. D. Bartlett, B. C. Sanders, S. L. Braunstein, and K. Nemoto, “Efficient classical simulation of continuous variable quantum information processes,” Phys. Rev. Lett. 88, 097904 (2002).
[Crossref]

J. Eisert, S. Scheel, and M. B. Plenio, “Distilling Gaussian states with Gaussian operations is impossible,” Phys. Rev. Lett. 89, 137903 (2002).
[Crossref]

2001 (2)

D. Gottesman, A. Kitaev, and J. Preskill, “Encoding a qubit in an oscillator,” Phys. Rev. A 64, 012310 (2001).
[Crossref]

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

1999 (2)

S. Lloyd and S. L. Braunstein, “Quantum computation over continuous variables,” Phys. Rev. Lett. 82, 1784–1787 (1999).
[Crossref]

K. Banaszek, C. Radzewicz, K. Wódkiewicz, and J. S. Krasiński, “Direct measurement of the Wigner function by photon counting,” Phys. Rev. A 60, 674–677 (1999).
[Crossref]

1998 (1)

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[Crossref]

1996 (4)

D. Leibfried, D. M. Meekhof, B. E. King, C. Monroe, W. M. Itano, and D. J. Wineland, “Experimental determination of the motional quantum state of a trapped atom,” Phys. Rev. Lett. 77, 4281–4285 (1996).
[Crossref]

M. G. A. Paris, “Displacement operator by beam splitter,” Phys. Lett. A 217, 78–80 (1996).
[Crossref]

S. Wallentowitz and W. Vogel, “Unbalanced homodyning for quantum state measurements,” Phys. Rev. A 53, 4528–4533 (1996).
[Crossref]

K. Banaszek and K. Wódkiewicz, “Direct probing of quantum phase space by photon counting,” Phys. Rev. Lett. 76, 4344 (1996).
[Crossref]

1993 (1)

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[Crossref]

1983 (1)

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[Crossref]

1977 (1)

A. Royer, “Wigner function as the expectation value of a parity operator,” Phys. Rev. A 15, 449–450 (1977).
[Crossref]

1974 (1)

R. L. Hudson, “When is the Wigner quasi-probability density non-negative?” Rep. Math. Phys. 6, 249–252 (1974).
[Crossref]

1969 (1)

K. E. Cahill and R. J. Glauber, “Density operators and quasiprobability distributions,” Phys. Rev. 177, 1882–1902 (1969).
[Crossref]

1932 (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[Crossref]

Aichele, T.

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

Allevi, A.

Andreoni, A.

Auffeves, A.

P. Bertet, A. Auffeves, P. Maioli, S. Osnaghi, T. Meunier, M. Brune, J. M. Raimond, and S. Haroche, “Direct measurement of the Wigner function of a one-photon Fock state in a cavity,” Phys. Rev. Lett. 89, 200402 (2002).
[Crossref]

Banaszek, K.

K. Banaszek, C. Radzewicz, K. Wódkiewicz, and J. S. Krasiński, “Direct measurement of the Wigner function by photon counting,” Phys. Rev. A 60, 674–677 (1999).
[Crossref]

K. Banaszek and K. Wódkiewicz, “Direct probing of quantum phase space by photon counting,” Phys. Rev. Lett. 76, 4344 (1996).
[Crossref]

Bartlett, S. D.

S. D. Bartlett, B. C. Sanders, S. L. Braunstein, and K. Nemoto, “Efficient classical simulation of continuous variable quantum information processes,” Phys. Rev. Lett. 88, 097904 (2002).
[Crossref]

Beck, M.

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[Crossref]

Bell, J. S.

J. S. Bell, “EPR correlations and EPW distributions,” in Speakable and Unspeakable in Quantum Mechanics (Cambridge University, 1987), Chap. 21, pp. 196–200.

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]

Bertet, P.

P. Bertet, A. Auffeves, P. Maioli, S. Osnaghi, T. Meunier, M. Brune, J. M. Raimond, and S. Haroche, “Direct measurement of the Wigner function of a one-photon Fock state in a cavity,” Phys. Rev. Lett. 89, 200402 (2002).
[Crossref]

Bondani, M.

Braunstein, S. L.

S. D. Bartlett, B. C. Sanders, S. L. Braunstein, and K. Nemoto, “Efficient classical simulation of continuous variable quantum information processes,” Phys. Rev. Lett. 88, 097904 (2002).
[Crossref]

S. Lloyd and S. L. Braunstein, “Quantum computation over continuous variables,” Phys. Rev. Lett. 82, 1784–1787 (1999).
[Crossref]

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[Crossref]

Brune, M.

P. Bertet, A. Auffeves, P. Maioli, S. Osnaghi, T. Meunier, M. Brune, J. M. Raimond, and S. Haroche, “Direct measurement of the Wigner function of a one-photon Fock state in a cavity,” Phys. Rev. Lett. 89, 200402 (2002).
[Crossref]

Cahill, K. E.

K. E. Cahill and R. J. Glauber, “Density operators and quasiprobability distributions,” Phys. Rev. 177, 1882–1902 (1969).
[Crossref]

Calkins, B.

Cassemiro, K. N.

K. Laiho, K. N. Cassemiro, D. Gross, and C. Silberhorn, “Probing the negative Wigner function of a pulsed single photon point by point,” Phys. Rev. Lett. 105, 253603 (2010).
[Crossref]

Cerf, N. J.

J. Niset, J. Fiurášek, and N. J. Cerf, “No-go theorem for Gaussian quantum error correction,” Phys. Rev. Lett. 102, 120501 (2009).
[Crossref]

D’Auria, V.

Devoret, M. H.

B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L. Frunzio, S. M. Girvin, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “Deterministically encoding quantum information using 100-photon Schrödinger cat states,” Science 342, 607–610 (2013).
[Crossref]

Drever, R. W. P.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[Crossref]

Eisert, J.

J. Eisert, S. Scheel, and M. B. Plenio, “Distilling Gaussian states with Gaussian operations is impossible,” Phys. Rev. Lett. 89, 137903 (2002).
[Crossref]

Fabre, C.

Faridani, A.

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[Crossref]

Feng, S.

S. Feng and O. Pfister, “Realization of an ultrastable twin-beam source for continuous-variable entanglement of bright beams,” Proc. SPIE 5161, 109–115 (2004).
[Crossref]

S. Feng and O. Pfister, “Quantum interference of ultrastable twin optical beams,” Phys. Rev. Lett. 92, 203601 (2004).
[Crossref]

S. Feng and O. Pfister, “Stable nondegenerate optical parametric oscillation at degenerate frequencies in Na:KTP,” J. Opt. B 5, 262 (2003).
[Crossref]

Fiurášek, J.

J. Niset, J. Fiurášek, and N. J. Cerf, “No-go theorem for Gaussian quantum error correction,” Phys. Rev. Lett. 102, 120501 (2009).
[Crossref]

Ford, G. M.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[Crossref]

Frunzio, L.

B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L. Frunzio, S. M. Girvin, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “Deterministically encoding quantum information using 100-photon Schrödinger cat states,” Science 342, 607–610 (2013).
[Crossref]

Fuchs, C. A.

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[Crossref]

Furusawa, A.

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[Crossref]

Gerrits, T.

Ghose, S.

S. Ghose and B. C. Sanders, “Non-Gaussian ancilla states for continuous variable quantum computation via Gaussian maps,” J. Mod. Opt. 54, 855–869 (2007).
[Crossref]

Girvin, S. M.

B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L. Frunzio, S. M. Girvin, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “Deterministically encoding quantum information using 100-photon Schrödinger cat states,” Science 342, 607–610 (2013).
[Crossref]

Glauber, R. J.

K. E. Cahill and R. J. Glauber, “Density operators and quasiprobability distributions,” Phys. Rev. 177, 1882–1902 (1969).
[Crossref]

Gottesman, D.

D. Gottesman, A. Kitaev, and J. Preskill, “Encoding a qubit in an oscillator,” Phys. Rev. A 64, 012310 (2001).
[Crossref]

Gross, D.

K. Laiho, K. N. Cassemiro, D. Gross, and C. Silberhorn, “Probing the negative Wigner function of a pulsed single photon point by point,” Phys. Rev. Lett. 105, 253603 (2010).
[Crossref]

Gu, M.

N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. Nielsen, “Universal quantum computation with continuous-variable cluster states,” Phys. Rev. Lett. 97, 110501 (2006).
[Crossref]

Hall, J. L.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[Crossref]

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]

Haroche, S.

P. Bertet, A. Auffeves, P. Maioli, S. Osnaghi, T. Meunier, M. Brune, J. M. Raimond, and S. Haroche, “Direct measurement of the Wigner function of a one-photon Fock state in a cavity,” Phys. Rev. Lett. 89, 200402 (2002).
[Crossref]

Hough, J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[Crossref]

Hudson, R. L.

R. L. Hudson, “When is the Wigner quasi-probability density non-negative?” Rep. Math. Phys. 6, 249–252 (1974).
[Crossref]

Itano, W. M.

D. Leibfried, D. M. Meekhof, B. E. King, C. Monroe, W. M. Itano, and D. J. Wineland, “Experimental determination of the motional quantum state of a trapped atom,” Phys. Rev. Lett. 77, 4281–4285 (1996).
[Crossref]

Kimble, H. J.

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[Crossref]

King, B. E.

D. Leibfried, D. M. Meekhof, B. E. King, C. Monroe, W. M. Itano, and D. J. Wineland, “Experimental determination of the motional quantum state of a trapped atom,” Phys. Rev. Lett. 77, 4281–4285 (1996).
[Crossref]

Kirchmair, G.

B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L. Frunzio, S. M. Girvin, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “Deterministically encoding quantum information using 100-photon Schrödinger cat states,” Science 342, 607–610 (2013).
[Crossref]

Kitaev, A.

D. Gottesman, A. Kitaev, and J. Preskill, “Encoding a qubit in an oscillator,” Phys. Rev. A 64, 012310 (2001).
[Crossref]

Kowalski, F. V.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[Crossref]

Krasinski, J. S.

K. Banaszek, C. Radzewicz, K. Wódkiewicz, and J. S. Krasiński, “Direct measurement of the Wigner function by photon counting,” Phys. Rev. A 60, 674–677 (1999).
[Crossref]

Laiho, K.

K. Laiho, K. N. Cassemiro, D. Gross, and C. Silberhorn, “Probing the negative Wigner function of a pulsed single photon point by point,” Phys. Rev. Lett. 105, 253603 (2010).
[Crossref]

Laurat, J.

Leghtas, Z.

B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L. Frunzio, S. M. Girvin, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “Deterministically encoding quantum information using 100-photon Schrödinger cat states,” Science 342, 607–610 (2013).
[Crossref]

Leibfried, D.

D. Leibfried, D. M. Meekhof, B. E. King, C. Monroe, W. M. Itano, and D. J. Wineland, “Experimental determination of the motional quantum state of a trapped atom,” Phys. Rev. Lett. 77, 4281–4285 (1996).
[Crossref]

Leonhardt, U.

U. Leonhardt, Measuring the Quantum State of Light (Cambridge University, 1997).

Lita, A.

Lita, A. E.

Lloyd, S.

S. Lloyd and S. L. Braunstein, “Quantum computation over continuous variables,” Phys. Rev. Lett. 82, 1784–1787 (1999).
[Crossref]

Lvovsky, A. I.

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

Maioli, P.

P. Bertet, A. Auffeves, P. Maioli, S. Osnaghi, T. Meunier, M. Brune, J. M. Raimond, and S. Haroche, “Direct measurement of the Wigner function of a one-photon Fock state in a cavity,” Phys. Rev. Lett. 89, 200402 (2002).
[Crossref]

Meekhof, D. M.

D. Leibfried, D. M. Meekhof, B. E. King, C. Monroe, W. M. Itano, and D. J. Wineland, “Experimental determination of the motional quantum state of a trapped atom,” Phys. Rev. Lett. 77, 4281–4285 (1996).
[Crossref]

Menicucci, N. C.

N. C. Menicucci, “Fault-tolerant measurement-based quantum computing with continuous-variable cluster states,” Phys. Rev. Lett. 112, 120504 (2014).
[Crossref]

N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. Nielsen, “Universal quantum computation with continuous-variable cluster states,” Phys. Rev. Lett. 97, 110501 (2006).
[Crossref]

Meunier, T.

P. Bertet, A. Auffeves, P. Maioli, S. Osnaghi, T. Meunier, M. Brune, J. M. Raimond, and S. Haroche, “Direct measurement of the Wigner function of a one-photon Fock state in a cavity,” Phys. Rev. Lett. 89, 200402 (2002).
[Crossref]

Milburn, G. J.

D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 1994).

Miller, A. J.

Mirrahimi, M.

B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L. Frunzio, S. M. Girvin, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “Deterministically encoding quantum information using 100-photon Schrödinger cat states,” Science 342, 607–610 (2013).
[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]

Monroe, C.

D. Leibfried, D. M. Meekhof, B. E. King, C. Monroe, W. M. Itano, and D. J. Wineland, “Experimental determination of the motional quantum state of a trapped atom,” Phys. Rev. Lett. 77, 4281–4285 (1996).
[Crossref]

Morin, O.

Munley, A. J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[Crossref]

Nam, S. W.

Nemoto, K.

S. D. Bartlett, B. C. Sanders, S. L. Braunstein, and K. Nemoto, “Efficient classical simulation of continuous variable quantum information processes,” Phys. Rev. Lett. 88, 097904 (2002).
[Crossref]

Nielsen, M. A.

N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. Nielsen, “Universal quantum computation with continuous-variable cluster states,” Phys. Rev. Lett. 97, 110501 (2006).
[Crossref]

Nigg, S. E.

B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L. Frunzio, S. M. Girvin, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “Deterministically encoding quantum information using 100-photon Schrödinger cat states,” Science 342, 607–610 (2013).
[Crossref]

Niset, J.

J. Niset, J. Fiurášek, and N. J. Cerf, “No-go theorem for Gaussian quantum error correction,” Phys. Rev. Lett. 102, 120501 (2009).
[Crossref]

Osnaghi, S.

P. Bertet, A. Auffeves, P. Maioli, S. Osnaghi, T. Meunier, M. Brune, J. M. Raimond, and S. Haroche, “Direct measurement of the Wigner function of a one-photon Fock state in a cavity,” Phys. Rev. Lett. 89, 200402 (2002).
[Crossref]

Paris, M. G. A.

M. G. A. Paris, “Displacement operator by beam splitter,” Phys. Lett. A 217, 78–80 (1996).
[Crossref]

Pfister, O.

N. Sridhar, R. Shahrokhshahi, A. J. Miller, B. Calkins, T. Gerrits, A. Lita, S. W. Nam, and O. Pfister, “Direct measurement of the Wigner function by photon-number-resolving detection,” J. Opt. Soc. Am. B 31, B34–B40 (2014).
[Crossref]

S. Feng and O. Pfister, “Realization of an ultrastable twin-beam source for continuous-variable entanglement of bright beams,” Proc. SPIE 5161, 109–115 (2004).
[Crossref]

S. Feng and O. Pfister, “Quantum interference of ultrastable twin optical beams,” Phys. Rev. Lett. 92, 203601 (2004).
[Crossref]

S. Feng and O. Pfister, “Stable nondegenerate optical parametric oscillation at degenerate frequencies in Na:KTP,” J. Opt. B 5, 262 (2003).
[Crossref]

Plenio, M. B.

J. Eisert, S. Scheel, and M. B. Plenio, “Distilling Gaussian states with Gaussian operations is impossible,” Phys. Rev. Lett. 89, 137903 (2002).
[Crossref]

Polzik, E. S.

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[Crossref]

Preskill, J.

D. Gottesman, A. Kitaev, and J. Preskill, “Encoding a qubit in an oscillator,” Phys. Rev. A 64, 012310 (2001).
[Crossref]

Radzewicz, C.

K. Banaszek, C. Radzewicz, K. Wódkiewicz, and J. S. Krasiński, “Direct measurement of the Wigner function by photon counting,” Phys. Rev. A 60, 674–677 (1999).
[Crossref]

Raimond, J. M.

P. Bertet, A. Auffeves, P. Maioli, S. Osnaghi, T. Meunier, M. Brune, J. M. Raimond, and S. Haroche, “Direct measurement of the Wigner function of a one-photon Fock state in a cavity,” Phys. Rev. Lett. 89, 200402 (2002).
[Crossref]

Ralph, T. C.

N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. Nielsen, “Universal quantum computation with continuous-variable cluster states,” Phys. Rev. Lett. 97, 110501 (2006).
[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]

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[Crossref]

Royer, A.

A. Royer, “Wigner function as the expectation value of a parity operator,” Phys. Rev. A 15, 449–450 (1977).
[Crossref]

Sanders, B. C.

S. Ghose and B. C. Sanders, “Non-Gaussian ancilla states for continuous variable quantum computation via Gaussian maps,” J. Mod. Opt. 54, 855–869 (2007).
[Crossref]

S. D. Bartlett, B. C. Sanders, S. L. Braunstein, and K. Nemoto, “Efficient classical simulation of continuous variable quantum information processes,” Phys. Rev. Lett. 88, 097904 (2002).
[Crossref]

Scheel, S.

J. Eisert, S. Scheel, and M. B. Plenio, “Distilling Gaussian states with Gaussian operations is impossible,” Phys. Rev. Lett. 89, 137903 (2002).
[Crossref]

Schiller, S.

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]

Schoelkopf, R. J.

B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L. Frunzio, S. M. Girvin, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “Deterministically encoding quantum information using 100-photon Schrödinger cat states,” Science 342, 607–610 (2013).
[Crossref]

Shahrokhshahi, R.

Silberhorn, C.

K. Laiho, K. N. Cassemiro, D. Gross, and C. Silberhorn, “Probing the negative Wigner function of a pulsed single photon point by point,” Phys. Rev. Lett. 105, 253603 (2010).
[Crossref]

Smithey, D. T.

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[Crossref]

Sørensen, J. L.

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[Crossref]

Sridhar, N.

van Loock, P.

N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. Nielsen, “Universal quantum computation with continuous-variable cluster states,” Phys. Rev. Lett. 97, 110501 (2006).
[Crossref]

Vlastakis, B.

B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L. Frunzio, S. M. Girvin, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “Deterministically encoding quantum information using 100-photon Schrödinger cat states,” Science 342, 607–610 (2013).
[Crossref]

Vogel, W.

S. Wallentowitz and W. Vogel, “Unbalanced homodyning for quantum state measurements,” Phys. Rev. A 53, 4528–4533 (1996).
[Crossref]

Wallentowitz, S.

S. Wallentowitz and W. Vogel, “Unbalanced homodyning for quantum state measurements,” Phys. Rev. A 53, 4528–4533 (1996).
[Crossref]

Walls, D. F.

D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 1994).

Ward, H.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[Crossref]

Weedbrook, C.

N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. Nielsen, “Universal quantum computation with continuous-variable cluster states,” Phys. Rev. Lett. 97, 110501 (2006).
[Crossref]

Wigner, E.

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[Crossref]

Wineland, D. J.

D. Leibfried, D. M. Meekhof, B. E. King, C. Monroe, W. M. Itano, and D. J. Wineland, “Experimental determination of the motional quantum state of a trapped atom,” Phys. Rev. Lett. 77, 4281–4285 (1996).
[Crossref]

Wódkiewicz, K.

K. Banaszek, C. Radzewicz, K. Wódkiewicz, and J. S. Krasiński, “Direct measurement of the Wigner function by photon counting,” Phys. Rev. A 60, 674–677 (1999).
[Crossref]

K. Banaszek and K. Wódkiewicz, “Direct probing of quantum phase space by photon counting,” Phys. Rev. Lett. 76, 4344 (1996).
[Crossref]

Appl. Phys. B (1)

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[Crossref]

J. Mod. Opt. (1)

S. Ghose and B. C. Sanders, “Non-Gaussian ancilla states for continuous variable quantum computation via Gaussian maps,” J. Mod. Opt. 54, 855–869 (2007).
[Crossref]

J. Opt. B (1)

S. Feng and O. Pfister, “Stable nondegenerate optical parametric oscillation at degenerate frequencies in Na:KTP,” J. Opt. B 5, 262 (2003).
[Crossref]

J. Opt. Soc. Am. B (1)

Opt. Express (1)

Opt. Lett. (2)

Phys. Lett. A (1)

M. G. A. Paris, “Displacement operator by beam splitter,” Phys. Lett. A 217, 78–80 (1996).
[Crossref]

Phys. Rev. (2)

K. E. Cahill and R. J. Glauber, “Density operators and quasiprobability distributions,” Phys. Rev. 177, 1882–1902 (1969).
[Crossref]

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[Crossref]

Phys. Rev. A (4)

S. Wallentowitz and W. Vogel, “Unbalanced homodyning for quantum state measurements,” Phys. Rev. A 53, 4528–4533 (1996).
[Crossref]

D. Gottesman, A. Kitaev, and J. Preskill, “Encoding a qubit in an oscillator,” Phys. Rev. A 64, 012310 (2001).
[Crossref]

A. Royer, “Wigner function as the expectation value of a parity operator,” Phys. Rev. A 15, 449–450 (1977).
[Crossref]

K. Banaszek, C. Radzewicz, K. Wódkiewicz, and J. S. Krasiński, “Direct measurement of the Wigner function by photon counting,” Phys. Rev. A 60, 674–677 (1999).
[Crossref]

Phys. Rev. Lett. (13)

D. Leibfried, D. M. Meekhof, B. E. King, C. Monroe, W. M. Itano, and D. J. Wineland, “Experimental determination of the motional quantum state of a trapped atom,” Phys. Rev. Lett. 77, 4281–4285 (1996).
[Crossref]

P. Bertet, A. Auffeves, P. Maioli, S. Osnaghi, T. Meunier, M. Brune, J. M. Raimond, and S. Haroche, “Direct measurement of the Wigner function of a one-photon Fock state in a cavity,” Phys. Rev. Lett. 89, 200402 (2002).
[Crossref]

K. Laiho, K. N. Cassemiro, D. Gross, and C. Silberhorn, “Probing the negative Wigner function of a pulsed single photon point by point,” Phys. Rev. Lett. 105, 253603 (2010).
[Crossref]

S. Feng and O. Pfister, “Quantum interference of ultrastable twin optical beams,” Phys. Rev. Lett. 92, 203601 (2004).
[Crossref]

N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. Nielsen, “Universal quantum computation with continuous-variable cluster states,” Phys. Rev. Lett. 97, 110501 (2006).
[Crossref]

N. C. Menicucci, “Fault-tolerant measurement-based quantum computing with continuous-variable cluster states,” Phys. Rev. Lett. 112, 120504 (2014).
[Crossref]

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[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]

K. Banaszek and K. Wódkiewicz, “Direct probing of quantum phase space by photon counting,” Phys. Rev. Lett. 76, 4344 (1996).
[Crossref]

S. Lloyd and S. L. Braunstein, “Quantum computation over continuous variables,” Phys. Rev. Lett. 82, 1784–1787 (1999).
[Crossref]

S. D. Bartlett, B. C. Sanders, S. L. Braunstein, and K. Nemoto, “Efficient classical simulation of continuous variable quantum information processes,” Phys. Rev. Lett. 88, 097904 (2002).
[Crossref]

J. Eisert, S. Scheel, and M. B. Plenio, “Distilling Gaussian states with Gaussian operations is impossible,” Phys. Rev. Lett. 89, 137903 (2002).
[Crossref]

J. Niset, J. Fiurášek, and N. J. Cerf, “No-go theorem for Gaussian quantum error correction,” Phys. Rev. Lett. 102, 120501 (2009).
[Crossref]

Proc. SPIE (1)

S. Feng and O. Pfister, “Realization of an ultrastable twin-beam source for continuous-variable entanglement of bright beams,” Proc. SPIE 5161, 109–115 (2004).
[Crossref]

Rep. Math. Phys. (1)

R. L. Hudson, “When is the Wigner quasi-probability density non-negative?” Rep. Math. Phys. 6, 249–252 (1974).
[Crossref]

Rev. Mod. Phys. (1)

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

Science (2)

B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L. Frunzio, S. M. Girvin, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “Deterministically encoding quantum information using 100-photon Schrödinger cat states,” Science 342, 607–610 (2013).
[Crossref]

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[Crossref]

Other (4)

See supplemental document.

U. Leonhardt, Measuring the Quantum State of Light (Cambridge University, 1997).

D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 1994).

J. S. Bell, “EPR correlations and EPW distributions,” in Speakable and Unspeakable in Quantum Mechanics (Cambridge University, 1987), Chap. 21, pp. 196–200.

Supplementary Material (1)

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Figures (5)

Fig. 1.
Fig. 1. Implementation of a displacement by a beam splitter. The initial coherent state amplitude is β with |β|1, so that we can have t1 in order to preserve the purity of the quantum signal ρ^, while still retaining a large enough value of |α|=t|β|, as needed for the raster scan of the Wigner function in phase space.
Fig. 2.
Fig. 2. Experimental setup. The red dotted lines denote the locking beam paths for the on/off PDH servo loops of the OPO and the FC. The displacement operation is contained in the black dashed-dotted box at the top. BP, Brewster prism; DM, dichroic mirror; EOM, electro-optic modulator; FI, Faraday isolator; FR, Faraday rotator; HWP, half-wave plate; IF, interference filter; LO, local oscillator input to the displacement field; PBS, polarizing beam splitter; PD, photodiode; POL, polarizer; PZT, piezoelectric transducer.
Fig. 3.
Fig. 3. On/off cycles of alternated active locking and data acquisition. Data collection begins for a period of 800 ms while the auxiliary locking beam is blocked, followed by a period of 200 ms where the auxiliary locking (broken red line in the experiment schematic) is enabled for active locking, and the signal channel is blocked. This process occurs cyclically during data collection to prevent excess photon flux from damaging the TES while ensuring a stable OPO cavity mode.
Fig. 4.
Fig. 4. Top, reconstructed Wigner function; black points, reconstructed values from raw data; solid surface, least-square Wigner-function fit, Eq. (9). Bottom, fit residuals.
Fig. 5.
Fig. 5. Phase-averaged Wigner function. The Wigner function fit yielded η=0.57(3), which is consistent with the heralding efficiency 0.58(2). Error bars are discussed in the text.

Tables (1)

Tables Icon

Table 1. Experimentally Measured Number of Single-Photon Counts on Both Channelsa

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

W(q,p)=1πe2ipyqy|ρ^|q+ydy,
dpW(q,p)=|ψ(q)|2,
dqW(q,p)=|ψ(p)|2.
Wρ^(α)=1πTr[ρ^D^(α)(1)N^D^(α)],
W(0)=1πn(1)nρnn.
ωs=ωi,
msL+ns(T)=miL+ni(T),
ηh=NcNi=0.58±0.02,
W(q,p)=ηW|11|(q,p)+(1η)W|00|(q,p),
W(0,0)=0.035±0.005.

Metrics