Abstract

The possibilities of applying tomographic techniques to a Bose–Einstein condensate to reconstruct its ground state are investigated by means of numerical simulations. Two situations for which the density-matrix elements can be retrieved from atom counting probabilities are considered. The methods presented here allow one to distinguish among various possible quantum states.

© 2000 Optical Society of America

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  1. D. F. Walls, G. J. Milburn, Quantum Optics (Springer-Verlag, Berlin, 1994).
  2. M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, E. A. Cornell, “Observation of Bose–Einstein condensation in a dilute atomic vapor,” Science 269, 198–201 (1995); K. B. Davies, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, W. Ketterle, “Bose–Einstein condensation in a gas of sodium atoms,” Phys. Rev. Lett. 75, 3969–3973 (1995); C. C. Bradley, C. A. Sackett, J. J. Tollett, R. G. Hulet, “Evidence of Bose–Einstein condensation in an atomic gas with attractive interactions,” Phys. Rev. Lett. 75, 1687–1690 (1995).
    [CrossRef] [PubMed]
  3. M. R. Andrews, C. G. Townsend, H. J. Miesner, D. S. Durfee, D. M. Kurn, W. Ketterle, “Observation of interference between two Bose–Einstein condensates,” Science 275, 637–641 (1997).
    [CrossRef] [PubMed]
  4. W. Ketterle, H. J. Miesner, “Coherence properties of Bose–Einstein condensates and atom lasers,” Phys. Rev. A 56, 3291–3293 (1997).
    [CrossRef]
  5. M. Lewenstein, L. You, “Quantum phase diffusion of a Bose–Einstein condensate,” Phys. Rev. Lett. 77, 3489–3493 (1996); M. J. Steel, M. J. Collett, “Quantum state of two trapped Bose–Einstein condensates with a Josephson coupling,” Phys. Rev. A 57, 2920–2930 (1998).
    [CrossRef] [PubMed]
  6. H. Zeng, W. Zhang, F. Lin, “Nonclassical Bose–Einstein condensate,” Phys. Rev. A 52, 2155–2160 (1995).
    [CrossRef] [PubMed]
  7. J. Ruostekoski, J. Javanainen, “Quantum field theory of cooperative atom response: low light intensity,” Phys. Rev. A 55, 513–526 (1997); E. A. Burt, R. W. Ghrist, C. J. Myatt, M. J. Holland, E. A. Cornell, C. E. Wieman, “Coherence, correlations and collisions: what one learns about Bose–Einstein condensates from their decay,” Phys. Rev. Lett. 79, 337–340 (1997).
    [CrossRef]
  8. See, e.g., U. Leonhardt, Measuring the Quantum State of Light (Cambridge U. Press, Cambridge, UK, 1997); J. Mod. Opt. 44, 11/12, special issue on quantum state preparation and measurement.
  9. S. Mancini, P. Tombesi, “Quantum state reconstruction of a Bose–Einstein condensate,” Europhys. Lett. 40, 351–355 (1997).
    [CrossRef]
  10. E. L. Bolda, S. M. Tan, D. F. Walls, “Reconstruction of the joint state of a two-mode Bose–Einstein condensate,” Phys. Rev. Lett. 79, 4719–4723 (1997); R. Walser, “Measuring the state of a bosonic two-mode quantum field,” Phys. Rev. Lett. 79, 4724–4727 (1997).
    [CrossRef]
  11. E. L. Bolda, S. M. Tan, D. F. Walls, “Measuring the quantum state of a Bose–Einstein condensate,” Phys. Rev. A 57, 4686–4694 (1998).
    [CrossRef]
  12. D. T. Smithey, M. Beck, M. G. Raymer, 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] [PubMed]
  13. R. Spreeuw, T. Pfau, U. Janicke, M. Wilkens, “Laser-like scheme for atomic matter waves,” Europhys. Lett. 32, 469–474 (1995); M. Holland, K. Burnett, C. Gardiner, J. I. Cirac, P. Zoller, “Theory of an atom laser,” Phys. Rev. A 54, 1757–1760 (1996); H. Wiseman, A. M. Martins, D. F. Walls, “An atom laser based on evaporative cooling,” Quantum Semiclassic. Opt. 8, 737–753 (1996); H. Steck, M. Naraschewski, H. Wallis, “Output of a pulsed atom laser,” Phys. Rev. Lett. 80, 1–5 (1998); J. Schneider, A. Schenzle, “Output from an atom laser: theory vs. experiment,” Appl. Phys. B 69, 353–356 (1999); I. Bloch, T. W. Hänsch, T. Esslinger, “Atom laser with a cw output coupler,” Phys. Rev. Lett. 82, 3008–3011 (1999).
    [CrossRef]
  14. J. A. Dunningham, K. Burnett, “Phase standard for Bose–Einstein condensates,” Phys. Rev. Lett. 82, 3729–3733 (1999).
    [CrossRef]
  15. S. Mancini, V. I. Man’ko, P. Tombesi, “Different realizations of the tomography principle in quantum state measurement,” J. Mod. Opt. 44, 2281–2292 (1997).
    [CrossRef]
  16. G. M. D’Ariano, “Group theoretical quantum tomography,” Acta Phys. Slovaca 49, 513–522 (1999).
  17. E. P. Wigner, Perspectives in Quantum Theory, W. Yourgrau, A. van der Merwe, eds. (Dover, New York, 1979).
  18. P. Berman, ed., Atom Interferometry (Academic, New York, 1997); M.-O. Mewes, M. R. Andrews, D. M. Kurn, D. S. Durfee, C. G. Townsend, W. Ketterle, “Output coupler for Bose–Einstein condensed atoms,” Phys. Rev. Lett. 78, 582–585 (1997); H. Hinderthur, A. Pautz, F. Ruschewitz, K. Sengstock, W. Ertmer, “Atomic interferometry with polarizing beam splitters,” Phys. Rev. A 57, 4730–4735 (1998).
    [CrossRef]
  19. L. C. Biedenharn, H. Van Dam, eds., Quantum Theory of Angular Momentum (Academic, New York, 1965).
  20. V. V. Dodonov, V. I. Man’ko, “Positive distribution description for spin states,” Phys. Lett. A 229, 335–339 (1997).
    [CrossRef]
  21. V. I. Man’ko, O. V. Man’ko, “Spin state tomography,” Zh. Eksp. Teor. Fiz. 112, 796–800 (1997). [JETP 85, 430–434 (1997)].
    [CrossRef]
  22. S. Wallentowitz, W. Vogel, “Unbalanced homodyning for quantum state measurements,” Phys. Rev. A 53, 4528–4533 (1996); K. Banaszek, K. Wodkiewicz, “Direct probing of quantum phase space by photon counting,” Phys. Rev. Lett. 76, 4344–4347 (1996); S. Mancini, P. Tombesi, V. I. Man’ko, “Density matrix from photon number tomography,” Europhys. Lett. 37, 79–83 (1997).
    [CrossRef] [PubMed]
  23. T. Opatrny, D. G. Welsch, “Density matrix reconstruction by unbalanced homodyning,” Phys. Rev. A 55, 1462–1465 (1997).
    [CrossRef]
  24. See, e.g., D. J. S. Robinson, A Course in Linear Algebra with Applications (World Scientific, Singapore, 1991); W. T. Vetterling, S. A. Teukolsky, W. H. Press, B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1985).
  25. M. O. Scully, W. E. Lamb, “Quantum theory of optical maser. III. Theory of photoelectron counting statistics,” Phys. Rev. 179, 368–374 (1969).
    [CrossRef]
  26. K. Vogel, W. Schleich, “More on interference in phase space,” in Fundamental Systems in Quantum Optics, J. Dalibard, J.-M. Raimond, J. Zinn-Justin, eds. (North-Holland, Amsterdam, 1992), pp. 713–765.
  27. M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1986). The Wigner function is a more refined picture of the quantum state in phase space and is particularly suited to display quantum characters of the state. In simple words, the Q function is a smoothed version of the Wigner function.
    [CrossRef]
  28. F. T. Arecchi, E. Courtens, R. Gilmore, H. Thomas, “Atomic coherent states in quantum optics,” Phys. Rev. A 6, 2211–2237 (1972); G. S. Agarwal, “Relation between atomic coherent representation, state multipoles and generalized phase space distributions,” Phys. Rev. A 24, 2889–2896 (1981).
    [CrossRef]
  29. D. Forster, Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions (Benjamin, New York, 1975); K. Huang, Statistical Mechanics, 2nd ed. (Wiley, New York, 1987).
  30. G. M. D’Ariano, U. Leonhardt, H. Paul, “Homodyne detection of the density matrix of the radiation field,” Phys. Rev. A 52, R1801–R1804 (1995).
    [CrossRef] [PubMed]
  31. P. Braun-Munzinger, H. J. Specht, R. Stock, H. Stocker, eds., Quark Matter, Nucl. Phys. A610, 1c–565c (1996); T. Csorgo, P. Levai, J. Zimanyi, eds., Strangeness in Hadronic Matter, Heavy Ion Phys.4, 1–464 (1996); S. Pratt, “Pion lasers from high-energy collisions,” Phys. Lett. B 301, 159–164 (1993).
    [CrossRef]

1999 (2)

J. A. Dunningham, K. Burnett, “Phase standard for Bose–Einstein condensates,” Phys. Rev. Lett. 82, 3729–3733 (1999).
[CrossRef]

G. M. D’Ariano, “Group theoretical quantum tomography,” Acta Phys. Slovaca 49, 513–522 (1999).

1998 (1)

E. L. Bolda, S. M. Tan, D. F. Walls, “Measuring the quantum state of a Bose–Einstein condensate,” Phys. Rev. A 57, 4686–4694 (1998).
[CrossRef]

1997 (9)

J. Ruostekoski, J. Javanainen, “Quantum field theory of cooperative atom response: low light intensity,” Phys. Rev. A 55, 513–526 (1997); E. A. Burt, R. W. Ghrist, C. J. Myatt, M. J. Holland, E. A. Cornell, C. E. Wieman, “Coherence, correlations and collisions: what one learns about Bose–Einstein condensates from their decay,” Phys. Rev. Lett. 79, 337–340 (1997).
[CrossRef]

S. Mancini, P. Tombesi, “Quantum state reconstruction of a Bose–Einstein condensate,” Europhys. Lett. 40, 351–355 (1997).
[CrossRef]

E. L. Bolda, S. M. Tan, D. F. Walls, “Reconstruction of the joint state of a two-mode Bose–Einstein condensate,” Phys. Rev. Lett. 79, 4719–4723 (1997); R. Walser, “Measuring the state of a bosonic two-mode quantum field,” Phys. Rev. Lett. 79, 4724–4727 (1997).
[CrossRef]

M. R. Andrews, C. G. Townsend, H. J. Miesner, D. S. Durfee, D. M. Kurn, W. Ketterle, “Observation of interference between two Bose–Einstein condensates,” Science 275, 637–641 (1997).
[CrossRef] [PubMed]

W. Ketterle, H. J. Miesner, “Coherence properties of Bose–Einstein condensates and atom lasers,” Phys. Rev. A 56, 3291–3293 (1997).
[CrossRef]

V. V. Dodonov, V. I. Man’ko, “Positive distribution description for spin states,” Phys. Lett. A 229, 335–339 (1997).
[CrossRef]

V. I. Man’ko, O. V. Man’ko, “Spin state tomography,” Zh. Eksp. Teor. Fiz. 112, 796–800 (1997). [JETP 85, 430–434 (1997)].
[CrossRef]

S. Mancini, V. I. Man’ko, P. Tombesi, “Different realizations of the tomography principle in quantum state measurement,” J. Mod. Opt. 44, 2281–2292 (1997).
[CrossRef]

T. Opatrny, D. G. Welsch, “Density matrix reconstruction by unbalanced homodyning,” Phys. Rev. A 55, 1462–1465 (1997).
[CrossRef]

1996 (2)

S. Wallentowitz, W. Vogel, “Unbalanced homodyning for quantum state measurements,” Phys. Rev. A 53, 4528–4533 (1996); K. Banaszek, K. Wodkiewicz, “Direct probing of quantum phase space by photon counting,” Phys. Rev. Lett. 76, 4344–4347 (1996); S. Mancini, P. Tombesi, V. I. Man’ko, “Density matrix from photon number tomography,” Europhys. Lett. 37, 79–83 (1997).
[CrossRef] [PubMed]

M. Lewenstein, L. You, “Quantum phase diffusion of a Bose–Einstein condensate,” Phys. Rev. Lett. 77, 3489–3493 (1996); M. J. Steel, M. J. Collett, “Quantum state of two trapped Bose–Einstein condensates with a Josephson coupling,” Phys. Rev. A 57, 2920–2930 (1998).
[CrossRef] [PubMed]

1995 (4)

H. Zeng, W. Zhang, F. Lin, “Nonclassical Bose–Einstein condensate,” Phys. Rev. A 52, 2155–2160 (1995).
[CrossRef] [PubMed]

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, E. A. Cornell, “Observation of Bose–Einstein condensation in a dilute atomic vapor,” Science 269, 198–201 (1995); K. B. Davies, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, W. Ketterle, “Bose–Einstein condensation in a gas of sodium atoms,” Phys. Rev. Lett. 75, 3969–3973 (1995); C. C. Bradley, C. A. Sackett, J. J. Tollett, R. G. Hulet, “Evidence of Bose–Einstein condensation in an atomic gas with attractive interactions,” Phys. Rev. Lett. 75, 1687–1690 (1995).
[CrossRef] [PubMed]

R. Spreeuw, T. Pfau, U. Janicke, M. Wilkens, “Laser-like scheme for atomic matter waves,” Europhys. Lett. 32, 469–474 (1995); M. Holland, K. Burnett, C. Gardiner, J. I. Cirac, P. Zoller, “Theory of an atom laser,” Phys. Rev. A 54, 1757–1760 (1996); H. Wiseman, A. M. Martins, D. F. Walls, “An atom laser based on evaporative cooling,” Quantum Semiclassic. Opt. 8, 737–753 (1996); H. Steck, M. Naraschewski, H. Wallis, “Output of a pulsed atom laser,” Phys. Rev. Lett. 80, 1–5 (1998); J. Schneider, A. Schenzle, “Output from an atom laser: theory vs. experiment,” Appl. Phys. B 69, 353–356 (1999); I. Bloch, T. W. Hänsch, T. Esslinger, “Atom laser with a cw output coupler,” Phys. Rev. Lett. 82, 3008–3011 (1999).
[CrossRef]

G. M. D’Ariano, U. Leonhardt, H. Paul, “Homodyne detection of the density matrix of the radiation field,” Phys. Rev. A 52, R1801–R1804 (1995).
[CrossRef] [PubMed]

1993 (1)

D. T. Smithey, M. Beck, M. G. Raymer, 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] [PubMed]

1986 (1)

M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1986). The Wigner function is a more refined picture of the quantum state in phase space and is particularly suited to display quantum characters of the state. In simple words, the Q function is a smoothed version of the Wigner function.
[CrossRef]

1972 (1)

F. T. Arecchi, E. Courtens, R. Gilmore, H. Thomas, “Atomic coherent states in quantum optics,” Phys. Rev. A 6, 2211–2237 (1972); G. S. Agarwal, “Relation between atomic coherent representation, state multipoles and generalized phase space distributions,” Phys. Rev. A 24, 2889–2896 (1981).
[CrossRef]

1969 (1)

M. O. Scully, W. E. Lamb, “Quantum theory of optical maser. III. Theory of photoelectron counting statistics,” Phys. Rev. 179, 368–374 (1969).
[CrossRef]

Anderson, M. H.

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, E. A. Cornell, “Observation of Bose–Einstein condensation in a dilute atomic vapor,” Science 269, 198–201 (1995); K. B. Davies, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, W. Ketterle, “Bose–Einstein condensation in a gas of sodium atoms,” Phys. Rev. Lett. 75, 3969–3973 (1995); C. C. Bradley, C. A. Sackett, J. J. Tollett, R. G. Hulet, “Evidence of Bose–Einstein condensation in an atomic gas with attractive interactions,” Phys. Rev. Lett. 75, 1687–1690 (1995).
[CrossRef] [PubMed]

Andrews, M. R.

M. R. Andrews, C. G. Townsend, H. J. Miesner, D. S. Durfee, D. M. Kurn, W. Ketterle, “Observation of interference between two Bose–Einstein condensates,” Science 275, 637–641 (1997).
[CrossRef] [PubMed]

Arecchi, F. T.

F. T. Arecchi, E. Courtens, R. Gilmore, H. Thomas, “Atomic coherent states in quantum optics,” Phys. Rev. A 6, 2211–2237 (1972); G. S. Agarwal, “Relation between atomic coherent representation, state multipoles and generalized phase space distributions,” Phys. Rev. A 24, 2889–2896 (1981).
[CrossRef]

Beck, M.

D. T. Smithey, M. Beck, M. G. Raymer, 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] [PubMed]

Bolda, E. L.

E. L. Bolda, S. M. Tan, D. F. Walls, “Measuring the quantum state of a Bose–Einstein condensate,” Phys. Rev. A 57, 4686–4694 (1998).
[CrossRef]

E. L. Bolda, S. M. Tan, D. F. Walls, “Reconstruction of the joint state of a two-mode Bose–Einstein condensate,” Phys. Rev. Lett. 79, 4719–4723 (1997); R. Walser, “Measuring the state of a bosonic two-mode quantum field,” Phys. Rev. Lett. 79, 4724–4727 (1997).
[CrossRef]

Burnett, K.

J. A. Dunningham, K. Burnett, “Phase standard for Bose–Einstein condensates,” Phys. Rev. Lett. 82, 3729–3733 (1999).
[CrossRef]

Cornell, E. A.

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, E. A. Cornell, “Observation of Bose–Einstein condensation in a dilute atomic vapor,” Science 269, 198–201 (1995); K. B. Davies, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, W. Ketterle, “Bose–Einstein condensation in a gas of sodium atoms,” Phys. Rev. Lett. 75, 3969–3973 (1995); C. C. Bradley, C. A. Sackett, J. J. Tollett, R. G. Hulet, “Evidence of Bose–Einstein condensation in an atomic gas with attractive interactions,” Phys. Rev. Lett. 75, 1687–1690 (1995).
[CrossRef] [PubMed]

Courtens, E.

F. T. Arecchi, E. Courtens, R. Gilmore, H. Thomas, “Atomic coherent states in quantum optics,” Phys. Rev. A 6, 2211–2237 (1972); G. S. Agarwal, “Relation between atomic coherent representation, state multipoles and generalized phase space distributions,” Phys. Rev. A 24, 2889–2896 (1981).
[CrossRef]

D’Ariano, G. M.

G. M. D’Ariano, “Group theoretical quantum tomography,” Acta Phys. Slovaca 49, 513–522 (1999).

G. M. D’Ariano, U. Leonhardt, H. Paul, “Homodyne detection of the density matrix of the radiation field,” Phys. Rev. A 52, R1801–R1804 (1995).
[CrossRef] [PubMed]

Dodonov, V. V.

V. V. Dodonov, V. I. Man’ko, “Positive distribution description for spin states,” Phys. Lett. A 229, 335–339 (1997).
[CrossRef]

Dunningham, J. A.

J. A. Dunningham, K. Burnett, “Phase standard for Bose–Einstein condensates,” Phys. Rev. Lett. 82, 3729–3733 (1999).
[CrossRef]

Durfee, D. S.

M. R. Andrews, C. G. Townsend, H. J. Miesner, D. S. Durfee, D. M. Kurn, W. Ketterle, “Observation of interference between two Bose–Einstein condensates,” Science 275, 637–641 (1997).
[CrossRef] [PubMed]

Ensher, J. R.

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, E. A. Cornell, “Observation of Bose–Einstein condensation in a dilute atomic vapor,” Science 269, 198–201 (1995); K. B. Davies, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, W. Ketterle, “Bose–Einstein condensation in a gas of sodium atoms,” Phys. Rev. Lett. 75, 3969–3973 (1995); C. C. Bradley, C. A. Sackett, J. J. Tollett, R. G. Hulet, “Evidence of Bose–Einstein condensation in an atomic gas with attractive interactions,” Phys. Rev. Lett. 75, 1687–1690 (1995).
[CrossRef] [PubMed]

Faridani, A.

D. T. Smithey, M. Beck, M. G. Raymer, 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] [PubMed]

Forster, D.

D. Forster, Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions (Benjamin, New York, 1975); K. Huang, Statistical Mechanics, 2nd ed. (Wiley, New York, 1987).

Gilmore, R.

F. T. Arecchi, E. Courtens, R. Gilmore, H. Thomas, “Atomic coherent states in quantum optics,” Phys. Rev. A 6, 2211–2237 (1972); G. S. Agarwal, “Relation between atomic coherent representation, state multipoles and generalized phase space distributions,” Phys. Rev. A 24, 2889–2896 (1981).
[CrossRef]

Hillery, M.

M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1986). The Wigner function is a more refined picture of the quantum state in phase space and is particularly suited to display quantum characters of the state. In simple words, the Q function is a smoothed version of the Wigner function.
[CrossRef]

Janicke, U.

R. Spreeuw, T. Pfau, U. Janicke, M. Wilkens, “Laser-like scheme for atomic matter waves,” Europhys. Lett. 32, 469–474 (1995); M. Holland, K. Burnett, C. Gardiner, J. I. Cirac, P. Zoller, “Theory of an atom laser,” Phys. Rev. A 54, 1757–1760 (1996); H. Wiseman, A. M. Martins, D. F. Walls, “An atom laser based on evaporative cooling,” Quantum Semiclassic. Opt. 8, 737–753 (1996); H. Steck, M. Naraschewski, H. Wallis, “Output of a pulsed atom laser,” Phys. Rev. Lett. 80, 1–5 (1998); J. Schneider, A. Schenzle, “Output from an atom laser: theory vs. experiment,” Appl. Phys. B 69, 353–356 (1999); I. Bloch, T. W. Hänsch, T. Esslinger, “Atom laser with a cw output coupler,” Phys. Rev. Lett. 82, 3008–3011 (1999).
[CrossRef]

Javanainen, J.

J. Ruostekoski, J. Javanainen, “Quantum field theory of cooperative atom response: low light intensity,” Phys. Rev. A 55, 513–526 (1997); E. A. Burt, R. W. Ghrist, C. J. Myatt, M. J. Holland, E. A. Cornell, C. E. Wieman, “Coherence, correlations and collisions: what one learns about Bose–Einstein condensates from their decay,” Phys. Rev. Lett. 79, 337–340 (1997).
[CrossRef]

Ketterle, W.

M. R. Andrews, C. G. Townsend, H. J. Miesner, D. S. Durfee, D. M. Kurn, W. Ketterle, “Observation of interference between two Bose–Einstein condensates,” Science 275, 637–641 (1997).
[CrossRef] [PubMed]

W. Ketterle, H. J. Miesner, “Coherence properties of Bose–Einstein condensates and atom lasers,” Phys. Rev. A 56, 3291–3293 (1997).
[CrossRef]

Kurn, D. M.

M. R. Andrews, C. G. Townsend, H. J. Miesner, D. S. Durfee, D. M. Kurn, W. Ketterle, “Observation of interference between two Bose–Einstein condensates,” Science 275, 637–641 (1997).
[CrossRef] [PubMed]

Lamb, W. E.

M. O. Scully, W. E. Lamb, “Quantum theory of optical maser. III. Theory of photoelectron counting statistics,” Phys. Rev. 179, 368–374 (1969).
[CrossRef]

Leonhardt, U.

G. M. D’Ariano, U. Leonhardt, H. Paul, “Homodyne detection of the density matrix of the radiation field,” Phys. Rev. A 52, R1801–R1804 (1995).
[CrossRef] [PubMed]

See, e.g., U. Leonhardt, Measuring the Quantum State of Light (Cambridge U. Press, Cambridge, UK, 1997); J. Mod. Opt. 44, 11/12, special issue on quantum state preparation and measurement.

Lewenstein, M.

M. Lewenstein, L. You, “Quantum phase diffusion of a Bose–Einstein condensate,” Phys. Rev. Lett. 77, 3489–3493 (1996); M. J. Steel, M. J. Collett, “Quantum state of two trapped Bose–Einstein condensates with a Josephson coupling,” Phys. Rev. A 57, 2920–2930 (1998).
[CrossRef] [PubMed]

Lin, F.

H. Zeng, W. Zhang, F. Lin, “Nonclassical Bose–Einstein condensate,” Phys. Rev. A 52, 2155–2160 (1995).
[CrossRef] [PubMed]

Man’ko, O. V.

V. I. Man’ko, O. V. Man’ko, “Spin state tomography,” Zh. Eksp. Teor. Fiz. 112, 796–800 (1997). [JETP 85, 430–434 (1997)].
[CrossRef]

Man’ko, V. I.

V. I. Man’ko, O. V. Man’ko, “Spin state tomography,” Zh. Eksp. Teor. Fiz. 112, 796–800 (1997). [JETP 85, 430–434 (1997)].
[CrossRef]

S. Mancini, V. I. Man’ko, P. Tombesi, “Different realizations of the tomography principle in quantum state measurement,” J. Mod. Opt. 44, 2281–2292 (1997).
[CrossRef]

V. V. Dodonov, V. I. Man’ko, “Positive distribution description for spin states,” Phys. Lett. A 229, 335–339 (1997).
[CrossRef]

Mancini, S.

S. Mancini, P. Tombesi, “Quantum state reconstruction of a Bose–Einstein condensate,” Europhys. Lett. 40, 351–355 (1997).
[CrossRef]

S. Mancini, V. I. Man’ko, P. Tombesi, “Different realizations of the tomography principle in quantum state measurement,” J. Mod. Opt. 44, 2281–2292 (1997).
[CrossRef]

Matthews, M. R.

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, E. A. Cornell, “Observation of Bose–Einstein condensation in a dilute atomic vapor,” Science 269, 198–201 (1995); K. B. Davies, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, W. Ketterle, “Bose–Einstein condensation in a gas of sodium atoms,” Phys. Rev. Lett. 75, 3969–3973 (1995); C. C. Bradley, C. A. Sackett, J. J. Tollett, R. G. Hulet, “Evidence of Bose–Einstein condensation in an atomic gas with attractive interactions,” Phys. Rev. Lett. 75, 1687–1690 (1995).
[CrossRef] [PubMed]

Miesner, H. J.

W. Ketterle, H. J. Miesner, “Coherence properties of Bose–Einstein condensates and atom lasers,” Phys. Rev. A 56, 3291–3293 (1997).
[CrossRef]

M. R. Andrews, C. G. Townsend, H. J. Miesner, D. S. Durfee, D. M. Kurn, W. Ketterle, “Observation of interference between two Bose–Einstein condensates,” Science 275, 637–641 (1997).
[CrossRef] [PubMed]

Milburn, G. J.

D. F. Walls, G. J. Milburn, Quantum Optics (Springer-Verlag, Berlin, 1994).

O’Connell, R. F.

M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1986). The Wigner function is a more refined picture of the quantum state in phase space and is particularly suited to display quantum characters of the state. In simple words, the Q function is a smoothed version of the Wigner function.
[CrossRef]

Opatrny, T.

T. Opatrny, D. G. Welsch, “Density matrix reconstruction by unbalanced homodyning,” Phys. Rev. A 55, 1462–1465 (1997).
[CrossRef]

Paul, H.

G. M. D’Ariano, U. Leonhardt, H. Paul, “Homodyne detection of the density matrix of the radiation field,” Phys. Rev. A 52, R1801–R1804 (1995).
[CrossRef] [PubMed]

Pfau, T.

R. Spreeuw, T. Pfau, U. Janicke, M. Wilkens, “Laser-like scheme for atomic matter waves,” Europhys. Lett. 32, 469–474 (1995); M. Holland, K. Burnett, C. Gardiner, J. I. Cirac, P. Zoller, “Theory of an atom laser,” Phys. Rev. A 54, 1757–1760 (1996); H. Wiseman, A. M. Martins, D. F. Walls, “An atom laser based on evaporative cooling,” Quantum Semiclassic. Opt. 8, 737–753 (1996); H. Steck, M. Naraschewski, H. Wallis, “Output of a pulsed atom laser,” Phys. Rev. Lett. 80, 1–5 (1998); J. Schneider, A. Schenzle, “Output from an atom laser: theory vs. experiment,” Appl. Phys. B 69, 353–356 (1999); I. Bloch, T. W. Hänsch, T. Esslinger, “Atom laser with a cw output coupler,” Phys. Rev. Lett. 82, 3008–3011 (1999).
[CrossRef]

Raymer, M. G.

D. T. Smithey, M. Beck, M. G. Raymer, 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] [PubMed]

Robinson, D. J. S.

See, e.g., D. J. S. Robinson, A Course in Linear Algebra with Applications (World Scientific, Singapore, 1991); W. T. Vetterling, S. A. Teukolsky, W. H. Press, B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1985).

Ruostekoski, J.

J. Ruostekoski, J. Javanainen, “Quantum field theory of cooperative atom response: low light intensity,” Phys. Rev. A 55, 513–526 (1997); E. A. Burt, R. W. Ghrist, C. J. Myatt, M. J. Holland, E. A. Cornell, C. E. Wieman, “Coherence, correlations and collisions: what one learns about Bose–Einstein condensates from their decay,” Phys. Rev. Lett. 79, 337–340 (1997).
[CrossRef]

Schleich, W.

K. Vogel, W. Schleich, “More on interference in phase space,” in Fundamental Systems in Quantum Optics, J. Dalibard, J.-M. Raimond, J. Zinn-Justin, eds. (North-Holland, Amsterdam, 1992), pp. 713–765.

Scully, M. O.

M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1986). The Wigner function is a more refined picture of the quantum state in phase space and is particularly suited to display quantum characters of the state. In simple words, the Q function is a smoothed version of the Wigner function.
[CrossRef]

M. O. Scully, W. E. Lamb, “Quantum theory of optical maser. III. Theory of photoelectron counting statistics,” Phys. Rev. 179, 368–374 (1969).
[CrossRef]

Smithey, D. T.

D. T. Smithey, M. Beck, M. G. Raymer, 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] [PubMed]

Spreeuw, R.

R. Spreeuw, T. Pfau, U. Janicke, M. Wilkens, “Laser-like scheme for atomic matter waves,” Europhys. Lett. 32, 469–474 (1995); M. Holland, K. Burnett, C. Gardiner, J. I. Cirac, P. Zoller, “Theory of an atom laser,” Phys. Rev. A 54, 1757–1760 (1996); H. Wiseman, A. M. Martins, D. F. Walls, “An atom laser based on evaporative cooling,” Quantum Semiclassic. Opt. 8, 737–753 (1996); H. Steck, M. Naraschewski, H. Wallis, “Output of a pulsed atom laser,” Phys. Rev. Lett. 80, 1–5 (1998); J. Schneider, A. Schenzle, “Output from an atom laser: theory vs. experiment,” Appl. Phys. B 69, 353–356 (1999); I. Bloch, T. W. Hänsch, T. Esslinger, “Atom laser with a cw output coupler,” Phys. Rev. Lett. 82, 3008–3011 (1999).
[CrossRef]

Tan, S. M.

E. L. Bolda, S. M. Tan, D. F. Walls, “Measuring the quantum state of a Bose–Einstein condensate,” Phys. Rev. A 57, 4686–4694 (1998).
[CrossRef]

E. L. Bolda, S. M. Tan, D. F. Walls, “Reconstruction of the joint state of a two-mode Bose–Einstein condensate,” Phys. Rev. Lett. 79, 4719–4723 (1997); R. Walser, “Measuring the state of a bosonic two-mode quantum field,” Phys. Rev. Lett. 79, 4724–4727 (1997).
[CrossRef]

Thomas, H.

F. T. Arecchi, E. Courtens, R. Gilmore, H. Thomas, “Atomic coherent states in quantum optics,” Phys. Rev. A 6, 2211–2237 (1972); G. S. Agarwal, “Relation between atomic coherent representation, state multipoles and generalized phase space distributions,” Phys. Rev. A 24, 2889–2896 (1981).
[CrossRef]

Tombesi, P.

S. Mancini, P. Tombesi, “Quantum state reconstruction of a Bose–Einstein condensate,” Europhys. Lett. 40, 351–355 (1997).
[CrossRef]

S. Mancini, V. I. Man’ko, P. Tombesi, “Different realizations of the tomography principle in quantum state measurement,” J. Mod. Opt. 44, 2281–2292 (1997).
[CrossRef]

Townsend, C. G.

M. R. Andrews, C. G. Townsend, H. J. Miesner, D. S. Durfee, D. M. Kurn, W. Ketterle, “Observation of interference between two Bose–Einstein condensates,” Science 275, 637–641 (1997).
[CrossRef] [PubMed]

Vogel, K.

K. Vogel, W. Schleich, “More on interference in phase space,” in Fundamental Systems in Quantum Optics, J. Dalibard, J.-M. Raimond, J. Zinn-Justin, eds. (North-Holland, Amsterdam, 1992), pp. 713–765.

Vogel, W.

S. Wallentowitz, W. Vogel, “Unbalanced homodyning for quantum state measurements,” Phys. Rev. A 53, 4528–4533 (1996); K. Banaszek, K. Wodkiewicz, “Direct probing of quantum phase space by photon counting,” Phys. Rev. Lett. 76, 4344–4347 (1996); S. Mancini, P. Tombesi, V. I. Man’ko, “Density matrix from photon number tomography,” Europhys. Lett. 37, 79–83 (1997).
[CrossRef] [PubMed]

Wallentowitz, S.

S. Wallentowitz, W. Vogel, “Unbalanced homodyning for quantum state measurements,” Phys. Rev. A 53, 4528–4533 (1996); K. Banaszek, K. Wodkiewicz, “Direct probing of quantum phase space by photon counting,” Phys. Rev. Lett. 76, 4344–4347 (1996); S. Mancini, P. Tombesi, V. I. Man’ko, “Density matrix from photon number tomography,” Europhys. Lett. 37, 79–83 (1997).
[CrossRef] [PubMed]

Walls, D. F.

E. L. Bolda, S. M. Tan, D. F. Walls, “Measuring the quantum state of a Bose–Einstein condensate,” Phys. Rev. A 57, 4686–4694 (1998).
[CrossRef]

E. L. Bolda, S. M. Tan, D. F. Walls, “Reconstruction of the joint state of a two-mode Bose–Einstein condensate,” Phys. Rev. Lett. 79, 4719–4723 (1997); R. Walser, “Measuring the state of a bosonic two-mode quantum field,” Phys. Rev. Lett. 79, 4724–4727 (1997).
[CrossRef]

D. F. Walls, G. J. Milburn, Quantum Optics (Springer-Verlag, Berlin, 1994).

Welsch, D. G.

T. Opatrny, D. G. Welsch, “Density matrix reconstruction by unbalanced homodyning,” Phys. Rev. A 55, 1462–1465 (1997).
[CrossRef]

Wieman, C. E.

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, E. A. Cornell, “Observation of Bose–Einstein condensation in a dilute atomic vapor,” Science 269, 198–201 (1995); K. B. Davies, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, W. Ketterle, “Bose–Einstein condensation in a gas of sodium atoms,” Phys. Rev. Lett. 75, 3969–3973 (1995); C. C. Bradley, C. A. Sackett, J. J. Tollett, R. G. Hulet, “Evidence of Bose–Einstein condensation in an atomic gas with attractive interactions,” Phys. Rev. Lett. 75, 1687–1690 (1995).
[CrossRef] [PubMed]

Wigner, E. P.

M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1986). The Wigner function is a more refined picture of the quantum state in phase space and is particularly suited to display quantum characters of the state. In simple words, the Q function is a smoothed version of the Wigner function.
[CrossRef]

E. P. Wigner, Perspectives in Quantum Theory, W. Yourgrau, A. van der Merwe, eds. (Dover, New York, 1979).

Wilkens, M.

R. Spreeuw, T. Pfau, U. Janicke, M. Wilkens, “Laser-like scheme for atomic matter waves,” Europhys. Lett. 32, 469–474 (1995); M. Holland, K. Burnett, C. Gardiner, J. I. Cirac, P. Zoller, “Theory of an atom laser,” Phys. Rev. A 54, 1757–1760 (1996); H. Wiseman, A. M. Martins, D. F. Walls, “An atom laser based on evaporative cooling,” Quantum Semiclassic. Opt. 8, 737–753 (1996); H. Steck, M. Naraschewski, H. Wallis, “Output of a pulsed atom laser,” Phys. Rev. Lett. 80, 1–5 (1998); J. Schneider, A. Schenzle, “Output from an atom laser: theory vs. experiment,” Appl. Phys. B 69, 353–356 (1999); I. Bloch, T. W. Hänsch, T. Esslinger, “Atom laser with a cw output coupler,” Phys. Rev. Lett. 82, 3008–3011 (1999).
[CrossRef]

You, L.

M. Lewenstein, L. You, “Quantum phase diffusion of a Bose–Einstein condensate,” Phys. Rev. Lett. 77, 3489–3493 (1996); M. J. Steel, M. J. Collett, “Quantum state of two trapped Bose–Einstein condensates with a Josephson coupling,” Phys. Rev. A 57, 2920–2930 (1998).
[CrossRef] [PubMed]

Zeng, H.

H. Zeng, W. Zhang, F. Lin, “Nonclassical Bose–Einstein condensate,” Phys. Rev. A 52, 2155–2160 (1995).
[CrossRef] [PubMed]

Zhang, W.

H. Zeng, W. Zhang, F. Lin, “Nonclassical Bose–Einstein condensate,” Phys. Rev. A 52, 2155–2160 (1995).
[CrossRef] [PubMed]

Acta Phys. Slovaca (1)

G. M. D’Ariano, “Group theoretical quantum tomography,” Acta Phys. Slovaca 49, 513–522 (1999).

Europhys. Lett. (2)

R. Spreeuw, T. Pfau, U. Janicke, M. Wilkens, “Laser-like scheme for atomic matter waves,” Europhys. Lett. 32, 469–474 (1995); M. Holland, K. Burnett, C. Gardiner, J. I. Cirac, P. Zoller, “Theory of an atom laser,” Phys. Rev. A 54, 1757–1760 (1996); H. Wiseman, A. M. Martins, D. F. Walls, “An atom laser based on evaporative cooling,” Quantum Semiclassic. Opt. 8, 737–753 (1996); H. Steck, M. Naraschewski, H. Wallis, “Output of a pulsed atom laser,” Phys. Rev. Lett. 80, 1–5 (1998); J. Schneider, A. Schenzle, “Output from an atom laser: theory vs. experiment,” Appl. Phys. B 69, 353–356 (1999); I. Bloch, T. W. Hänsch, T. Esslinger, “Atom laser with a cw output coupler,” Phys. Rev. Lett. 82, 3008–3011 (1999).
[CrossRef]

S. Mancini, P. Tombesi, “Quantum state reconstruction of a Bose–Einstein condensate,” Europhys. Lett. 40, 351–355 (1997).
[CrossRef]

J. Mod. Opt. (1)

S. Mancini, V. I. Man’ko, P. Tombesi, “Different realizations of the tomography principle in quantum state measurement,” J. Mod. Opt. 44, 2281–2292 (1997).
[CrossRef]

Phys. Lett. A (1)

V. V. Dodonov, V. I. Man’ko, “Positive distribution description for spin states,” Phys. Lett. A 229, 335–339 (1997).
[CrossRef]

Phys. Rep. (1)

M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1986). The Wigner function is a more refined picture of the quantum state in phase space and is particularly suited to display quantum characters of the state. In simple words, the Q function is a smoothed version of the Wigner function.
[CrossRef]

Phys. Rev. (1)

M. O. Scully, W. E. Lamb, “Quantum theory of optical maser. III. Theory of photoelectron counting statistics,” Phys. Rev. 179, 368–374 (1969).
[CrossRef]

Phys. Rev. A (8)

S. Wallentowitz, W. Vogel, “Unbalanced homodyning for quantum state measurements,” Phys. Rev. A 53, 4528–4533 (1996); K. Banaszek, K. Wodkiewicz, “Direct probing of quantum phase space by photon counting,” Phys. Rev. Lett. 76, 4344–4347 (1996); S. Mancini, P. Tombesi, V. I. Man’ko, “Density matrix from photon number tomography,” Europhys. Lett. 37, 79–83 (1997).
[CrossRef] [PubMed]

T. Opatrny, D. G. Welsch, “Density matrix reconstruction by unbalanced homodyning,” Phys. Rev. A 55, 1462–1465 (1997).
[CrossRef]

F. T. Arecchi, E. Courtens, R. Gilmore, H. Thomas, “Atomic coherent states in quantum optics,” Phys. Rev. A 6, 2211–2237 (1972); G. S. Agarwal, “Relation between atomic coherent representation, state multipoles and generalized phase space distributions,” Phys. Rev. A 24, 2889–2896 (1981).
[CrossRef]

G. M. D’Ariano, U. Leonhardt, H. Paul, “Homodyne detection of the density matrix of the radiation field,” Phys. Rev. A 52, R1801–R1804 (1995).
[CrossRef] [PubMed]

H. Zeng, W. Zhang, F. Lin, “Nonclassical Bose–Einstein condensate,” Phys. Rev. A 52, 2155–2160 (1995).
[CrossRef] [PubMed]

J. Ruostekoski, J. Javanainen, “Quantum field theory of cooperative atom response: low light intensity,” Phys. Rev. A 55, 513–526 (1997); E. A. Burt, R. W. Ghrist, C. J. Myatt, M. J. Holland, E. A. Cornell, C. E. Wieman, “Coherence, correlations and collisions: what one learns about Bose–Einstein condensates from their decay,” Phys. Rev. Lett. 79, 337–340 (1997).
[CrossRef]

W. Ketterle, H. J. Miesner, “Coherence properties of Bose–Einstein condensates and atom lasers,” Phys. Rev. A 56, 3291–3293 (1997).
[CrossRef]

E. L. Bolda, S. M. Tan, D. F. Walls, “Measuring the quantum state of a Bose–Einstein condensate,” Phys. Rev. A 57, 4686–4694 (1998).
[CrossRef]

Phys. Rev. Lett. (4)

D. T. Smithey, M. Beck, M. G. Raymer, 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] [PubMed]

J. A. Dunningham, K. Burnett, “Phase standard for Bose–Einstein condensates,” Phys. Rev. Lett. 82, 3729–3733 (1999).
[CrossRef]

M. Lewenstein, L. You, “Quantum phase diffusion of a Bose–Einstein condensate,” Phys. Rev. Lett. 77, 3489–3493 (1996); M. J. Steel, M. J. Collett, “Quantum state of two trapped Bose–Einstein condensates with a Josephson coupling,” Phys. Rev. A 57, 2920–2930 (1998).
[CrossRef] [PubMed]

E. L. Bolda, S. M. Tan, D. F. Walls, “Reconstruction of the joint state of a two-mode Bose–Einstein condensate,” Phys. Rev. Lett. 79, 4719–4723 (1997); R. Walser, “Measuring the state of a bosonic two-mode quantum field,” Phys. Rev. Lett. 79, 4724–4727 (1997).
[CrossRef]

Science (2)

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, E. A. Cornell, “Observation of Bose–Einstein condensation in a dilute atomic vapor,” Science 269, 198–201 (1995); K. B. Davies, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, W. Ketterle, “Bose–Einstein condensation in a gas of sodium atoms,” Phys. Rev. Lett. 75, 3969–3973 (1995); C. C. Bradley, C. A. Sackett, J. J. Tollett, R. G. Hulet, “Evidence of Bose–Einstein condensation in an atomic gas with attractive interactions,” Phys. Rev. Lett. 75, 1687–1690 (1995).
[CrossRef] [PubMed]

M. R. Andrews, C. G. Townsend, H. J. Miesner, D. S. Durfee, D. M. Kurn, W. Ketterle, “Observation of interference between two Bose–Einstein condensates,” Science 275, 637–641 (1997).
[CrossRef] [PubMed]

Zh. Eksp. Teor. Fiz. (1)

V. I. Man’ko, O. V. Man’ko, “Spin state tomography,” Zh. Eksp. Teor. Fiz. 112, 796–800 (1997). [JETP 85, 430–434 (1997)].
[CrossRef]

Other (9)

D. F. Walls, G. J. Milburn, Quantum Optics (Springer-Verlag, Berlin, 1994).

See, e.g., D. J. S. Robinson, A Course in Linear Algebra with Applications (World Scientific, Singapore, 1991); W. T. Vetterling, S. A. Teukolsky, W. H. Press, B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1985).

K. Vogel, W. Schleich, “More on interference in phase space,” in Fundamental Systems in Quantum Optics, J. Dalibard, J.-M. Raimond, J. Zinn-Justin, eds. (North-Holland, Amsterdam, 1992), pp. 713–765.

P. Braun-Munzinger, H. J. Specht, R. Stock, H. Stocker, eds., Quark Matter, Nucl. Phys. A610, 1c–565c (1996); T. Csorgo, P. Levai, J. Zimanyi, eds., Strangeness in Hadronic Matter, Heavy Ion Phys.4, 1–464 (1996); S. Pratt, “Pion lasers from high-energy collisions,” Phys. Lett. B 301, 159–164 (1993).
[CrossRef]

D. Forster, Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions (Benjamin, New York, 1975); K. Huang, Statistical Mechanics, 2nd ed. (Wiley, New York, 1987).

See, e.g., U. Leonhardt, Measuring the Quantum State of Light (Cambridge U. Press, Cambridge, UK, 1997); J. Mod. Opt. 44, 11/12, special issue on quantum state preparation and measurement.

E. P. Wigner, Perspectives in Quantum Theory, W. Yourgrau, A. van der Merwe, eds. (Dover, New York, 1979).

P. Berman, ed., Atom Interferometry (Academic, New York, 1997); M.-O. Mewes, M. R. Andrews, D. M. Kurn, D. S. Durfee, C. G. Townsend, W. Ketterle, “Output coupler for Bose–Einstein condensed atoms,” Phys. Rev. Lett. 78, 582–585 (1997); H. Hinderthur, A. Pautz, F. Ruschewitz, K. Sengstock, W. Ertmer, “Atomic interferometry with polarizing beam splitters,” Phys. Rev. A 57, 4730–4735 (1998).
[CrossRef]

L. C. Biedenharn, H. Van Dam, eds., Quantum Theory of Angular Momentum (Academic, New York, 1965).

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Figures (4)

Fig. 1
Fig. 1

Squeezed two-mode state: (a) ideal Q function when the displacement parameter is x0=5 and the squeezing parameter is r=e, (b) corresponding Q function reconstructed by the method given in Section 3. To obtain this figure we simulated experimental data by adding to each probability w a noise term with a Gaussian distribution whose width was proportional to the ratio between the probability itself and the number of runs for the given parameters.

Fig. 2
Fig. 2

Squeezed state for the single mode: (a) ideal Wigner function when the displacement parameter is x0=3 and the squeezing parameter is r=e, (b) corresponding Wigner function reconstructed by the method given in Section 4. The reconstruction parameters are |β|=1.1 and η=0.9, and 3×105 simulated experimental data per each phase have been used (see text).

Fig. 3
Fig. 3

Number state for the single mode: (a) ideal Wigner function for the Fock state |n=|5, (b) corresponding Wigner function reconstructed by the method given in Section 4. The reconstruction parameters are |β|=0.3 and η=0.9, and 3×105 simulated experimental data per each phase have been used (see text).

Fig. 4
Fig. 4

Squeezed state with a random phase between 0 and 2π. Here a value of the displacement parameter x0=5 has been used.

Equations (36)

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

ρˆ=ρˆ,Trρˆ=1,
v|ρˆ|v=ρv,v0.
Πˆv=|vv|
ρv,v=Tr(Πˆvρˆ).
ρˆσ=Uˆ-1(σ)ρˆUˆ(σ).
v|ρˆσ|vw(v, σ)0.
dvw(v, σ)=1.
Πˆv(σ)=Uˆ(σ)ΠˆvUˆ-1(σ),
w(v, σ)=Tr[ρˆΠˆv(σ)].
ρˆUˆ(θ, ϕ)ρˆUˆ(θ, ϕ),
Uˆ(θ, ϕ)=exp{-i(θ/2)[b1b2 exp(-iϕ)+b1b2exp(iϕ)]}.
|Ψ=n=0Ncn|N-n1|n2.
Jˆ+=Jˆ-=bˆ1bˆ2,Jˆz=(1/2)(bˆ1bˆ1-bˆ2bˆ2),
Jˆ2=Jˆz2+(1/2)(Jˆ+Jˆ-+Jˆ-Jˆ+).
Uˆ(θ, ϕ)=exp(-iθJˆ·uϕ).
m|Uˆ(θ, ϕ)|mDmm(j)(ψ=0, θ, ϕ),
w(m, θ, ϕ)=m1=-jjm2=-jjDmm1(j)(ψ, θ, ϕ)ρm1m2(j)×Dmm2(j)*(ψ, θ, ϕ).
ρm1m2(j)=(-1)m2j=02jm=-jj(2j+1)2×m=-jj(-1)mw(m, θ, ϕ)×D0m(j)(ψ, θ, ϕ)Wm-m0jjjWm1-m2mjjjdΩ8π2,
dΩ=02πdψ0π sin θ d θ02πdϕ.
w(n, β)=Tr[Uˆ-1(θ=π/2, ϕ)ρˆ|β¯22×β¯|Uˆ(θ=π/2, ϕ)|n11n|],
w(n, β)=exp(-|β|2)n!k,m=0N1k|ρ|m×1k!m!|β|m+k-2n exp[i(m-k)φ]×Ln(m-n)(|β|2)Ln(k-n)(|β|2),
w(s)(n, |β|)=12π02πdφw(n, β)exp(isφ)
w(s)(n, |β|)=m=0N1-sAn,m(s)(|β|)m+s|ρ|m,
An,m(s)(|β|)=exp(-|β|2)n!1(m+s)!m!×|β|2(m-n)+sLn(m-n)(|β|2)Ln(m+s-n)(|β|2).
m+s|ρ|m=n=0NMm,n(s)(|β|)w(s)(n, |β|),
n=0NMm,n(s)(|β|)An,m(s)(|β|)=δm,m,
m+s|ρ|m=12πn=0NdˆφMm,n(s)(|β|)exp(isφ)w(n, β),
wη(k, β)=n=knkηk(1-η)n-kw(n, β),
|Ψ=n=0cn|n,
cn=2r+11/2r1/4r-1r+1n/2(2nn!)-1/2Hn×2r2r2-1x0exp-rr+1x02,
Q(α)=α|ρˆ|α,
Q(α)=exp(-|α|2)n=0(α*)nn!cn2.
|Ψ=Nm=-jjcj+m|m,
|θ, ϕ=m=-jjDm,-j(j)(ψ=0, θ, ϕ)|m=m=-jj2jm+j1/2sinθ2j+mcosθ2j-m×exp(-imϕ)|m,
Q(θ, ϕ)=θ, ϕ|ρˆ|θ, ϕ.
Q(θ, ϕ)=m=-jjDm,-j(j)*(ψ=0, θ, ϕ)cj+m2.

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