Abstract

We construct the exact time dependent density profile for a superposition of the ground and singly excited states of a harmonically trapped one dimensional Bose gas in the limit of strongly interacting particles, the Tonks-Girardeau gas. We obtain analytic results that allows one to determine the number of particles in the gas, as well as the quantum amplitudes in the superposition, from measurement results in an off-resonant light scattering experiment.

© 2010 Optical Society of America

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  1. E. P. Gross, “Structure of a quantized vortex in boson systems,” Nuovo Cim. 20, 454 (1961).
    [CrossRef]
  2. L. P. Pitaevskii, “Vortex lines in an imperfect Bose gas,” Sov. Phys. JETP 13, 451–454 (1961).
  3. F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose-Einstein condensation in trapped gases,” Rev. Mod. Phys. 71, 463 (1999).
    [CrossRef]
  4. E. B. Kolomeisky, T. J. Newman, J. P. Straley, and X. Qi, “Low-dimensional Bose liquids: Beyond the Gross-Pitaevskii approximation,” Phys. Rev. Lett. 85, 1146 (2000).
    [CrossRef] [PubMed]
  5. H. Moritz, T. Stöferle, M. Köhl, and T. Esslinger, “Exciting collective oscillations in a trapped 1D gas,” Phys. Rev. Lett. 91, 250402 (2003).
    [CrossRef]
  6. K. Bongs, S. Burger, S. Dettmer, D. Hellweg, J. Arlt, W. Ertmer, and K. Sengstock, “Waveguide for Bose-Einstein condensates,” Phys. Rev. A 63, 031602 (2001).
    [CrossRef]
  7. A. Minguzzi, P. Vignolo, M. L. Chiofalo, and M. P. Tosi, “Sum rule for the dynamical response of a confined Bose-Einstein condensed gas,” Phys. Rev. A 64, 033605 (2001).
    [CrossRef]
  8. C. Menotti, and S. Stringari, “Collective oscillations of a one-dimensional trapped Bose-Einstein gas,” Phys. Rev. A 66, 043610 (2002).
    [CrossRef]
  9. M. D. Girardeau, and E. M. Wright, “Breakdown of time-dependent mean-field theory for a one-dimensional condensate of impenetrable bosons,” Phys. Rev. Lett. 84, 5239 (2000).
    [CrossRef] [PubMed]
  10. B. Paredes, A. Widera, V. Murg, O. Mandel, S. Folling, I. Cirac, G. V. Shlyapnikov, T. W. Hansch, and I. Bloch, “Tonks-Girardeau gas of ultracold atoms in an optical lattice,” Nature 429, 277 (2004).
    [CrossRef] [PubMed]
  11. T. Kinoshita, T. Wenger, and D. S. Weiss, “Observation of a one-dimensional Tonks-Girardeau Gas,” Science 305, 1125 (2004).
    [CrossRef] [PubMed]
  12. L. Tonks, “The Complete Equation of State of One, Two and Three-Dimensional Gases of Hard Elastic Spheres,” Phys. Rev. 50, 955 (1936).
    [CrossRef]
  13. M. D. Girardeau, “Relationship between Systems of Impenetrable Bosons and Fermions in One Dimension,” J. Math. Phys. 1, 516 (1960).
    [CrossRef]
  14. E. Lieb, and W. Liniger, “Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground,” State Phys. Rev. 130, 1605 (1963).
  15. C. J. Pethick, and H. Smith, Bose-Einstein condensation in dilute gases, (Cambridge, 2008).
    [CrossRef]
  16. I. Bloch, J. Dalibard, and W. Zwerger, “Many-body physics with ultracold gases,” Rev. Mod. Phys. 80, 885 (2008).
    [CrossRef]
  17. D. S. Petrov, G. V. Shlyapnikov, and J. T. M. Walraven, “Regimes of quantum degeneracy in trapped 1D gases,” Phys. Rev. Lett. 85, 3745 (2000).
    [CrossRef] [PubMed]
  18. V. Dunjko, V. Lorent, and M. Olshanii, “Bosons in cigar-shaped traps: Thomas-Fermi Tonks-Girardean regime, and in between,” Phys. Rev. Lett. 86, 5413 (2001).
    [CrossRef] [PubMed]
  19. M. Olshanii, and V. Dunjko, “Short-distance correlation properties of the Lieb-Liniger system and momentum distributions of trapped one-dimensional atomic gases,” Phys. Rev. Lett. 91, 090401 (2003).
    [CrossRef] [PubMed]
  20. P. Pedri, and L. Santos, “Three-dimensional quasi-Tonks gas in a harmonic trap,” Phys. Rev. Lett. 91, 110401 (2003).
    [CrossRef] [PubMed]
  21. M. D. Girardeau, E. M. Wright, and J. M. Triscari, “Ground-state properties of a one-dimensional system of hard-core bosons in a harmonic trap,” Phys. Rev. A 63, 033601 (2001).
    [CrossRef]
  22. I. S. Gradshtein, and I. M. Ryzhik, Table of Integrals, Series, and products, 7th ed. (Academic Press, 2007).
  23. O. Morice, Y. Castin, and J. Dalibard, “Refractive-index of a dilute Bose-gas,” Phys. Rev. A 51, 3896 (1995).
    [CrossRef] [PubMed]
  24. J. Javanainen, and J. Ruostekoski, “Off-resonance light-scattering from low-temperature Bose and Fermi gases,” Phys. Rev. A 52, 3033 (1995).
    [CrossRef] [PubMed]
  25. A. Csordás, R. Graham, and P. Szépfalusy, “Off-resonance light scattering from Bose condensates in traps,” Phys. Rev. A 54, R2543 (1996).
    [CrossRef] [PubMed]
  26. E. A. Ostrovskaya, and Y. S. Kivshar, “Photonic crystals for matter waves: Bose-Einstein condensates in optical lattices,” Opt. Express 12(1), 19 (2004).
    [CrossRef] [PubMed]
  27. I. Mekhov, C. Maschler, and H. Ritsch, “Probing quantum phases of ultracold atoms in optical lattices by transmission spectra in cavity quantum electrodynamics,” Nat. Phys. 3, 319 (2007).
    [CrossRef]
  28. H.-W. Cho, Y.-C. He, T. Peters, Y.-H. Chen, H.-C. Chen, S.-C. Lin, Y.-C. Lee, and I. A. Yu, “Direct Measurement of the Atom Number in a Bose Condensate,” Opt. Express 15(19), 12114 (2007).
    [CrossRef] [PubMed]
  29. Z. Dutton, M. Budde, C. Slowe, and L. Vestergaard Hau, “Observation of quantum shock waves created with ultra-compressed slow light pulses in a Bose-Einstein condensate,” Science 293, 663 (2001).
    [CrossRef] [PubMed]
  30. Handbook of Mathematical Functions, edited by M. Abramowitz and I. Stegun (Dover, New York, 1965).
  31. D. A. Steck, “Rubidium 87 D Line Data,” available at http://steck.us/alkalidata (rev. 2.1.2, 12 August 2009).

2008 (1)

I. Bloch, J. Dalibard, and W. Zwerger, “Many-body physics with ultracold gases,” Rev. Mod. Phys. 80, 885 (2008).
[CrossRef]

2007 (2)

I. Mekhov, C. Maschler, and H. Ritsch, “Probing quantum phases of ultracold atoms in optical lattices by transmission spectra in cavity quantum electrodynamics,” Nat. Phys. 3, 319 (2007).
[CrossRef]

H.-W. Cho, Y.-C. He, T. Peters, Y.-H. Chen, H.-C. Chen, S.-C. Lin, Y.-C. Lee, and I. A. Yu, “Direct Measurement of the Atom Number in a Bose Condensate,” Opt. Express 15(19), 12114 (2007).
[CrossRef] [PubMed]

2004 (3)

E. A. Ostrovskaya, and Y. S. Kivshar, “Photonic crystals for matter waves: Bose-Einstein condensates in optical lattices,” Opt. Express 12(1), 19 (2004).
[CrossRef] [PubMed]

B. Paredes, A. Widera, V. Murg, O. Mandel, S. Folling, I. Cirac, G. V. Shlyapnikov, T. W. Hansch, and I. Bloch, “Tonks-Girardeau gas of ultracold atoms in an optical lattice,” Nature 429, 277 (2004).
[CrossRef] [PubMed]

T. Kinoshita, T. Wenger, and D. S. Weiss, “Observation of a one-dimensional Tonks-Girardeau Gas,” Science 305, 1125 (2004).
[CrossRef] [PubMed]

2003 (3)

M. Olshanii, and V. Dunjko, “Short-distance correlation properties of the Lieb-Liniger system and momentum distributions of trapped one-dimensional atomic gases,” Phys. Rev. Lett. 91, 090401 (2003).
[CrossRef] [PubMed]

P. Pedri, and L. Santos, “Three-dimensional quasi-Tonks gas in a harmonic trap,” Phys. Rev. Lett. 91, 110401 (2003).
[CrossRef] [PubMed]

H. Moritz, T. Stöferle, M. Köhl, and T. Esslinger, “Exciting collective oscillations in a trapped 1D gas,” Phys. Rev. Lett. 91, 250402 (2003).
[CrossRef]

2002 (1)

C. Menotti, and S. Stringari, “Collective oscillations of a one-dimensional trapped Bose-Einstein gas,” Phys. Rev. A 66, 043610 (2002).
[CrossRef]

2001 (5)

K. Bongs, S. Burger, S. Dettmer, D. Hellweg, J. Arlt, W. Ertmer, and K. Sengstock, “Waveguide for Bose-Einstein condensates,” Phys. Rev. A 63, 031602 (2001).
[CrossRef]

A. Minguzzi, P. Vignolo, M. L. Chiofalo, and M. P. Tosi, “Sum rule for the dynamical response of a confined Bose-Einstein condensed gas,” Phys. Rev. A 64, 033605 (2001).
[CrossRef]

M. D. Girardeau, E. M. Wright, and J. M. Triscari, “Ground-state properties of a one-dimensional system of hard-core bosons in a harmonic trap,” Phys. Rev. A 63, 033601 (2001).
[CrossRef]

Z. Dutton, M. Budde, C. Slowe, and L. Vestergaard Hau, “Observation of quantum shock waves created with ultra-compressed slow light pulses in a Bose-Einstein condensate,” Science 293, 663 (2001).
[CrossRef] [PubMed]

V. Dunjko, V. Lorent, and M. Olshanii, “Bosons in cigar-shaped traps: Thomas-Fermi Tonks-Girardean regime, and in between,” Phys. Rev. Lett. 86, 5413 (2001).
[CrossRef] [PubMed]

2000 (3)

D. S. Petrov, G. V. Shlyapnikov, and J. T. M. Walraven, “Regimes of quantum degeneracy in trapped 1D gases,” Phys. Rev. Lett. 85, 3745 (2000).
[CrossRef] [PubMed]

M. D. Girardeau, and E. M. Wright, “Breakdown of time-dependent mean-field theory for a one-dimensional condensate of impenetrable bosons,” Phys. Rev. Lett. 84, 5239 (2000).
[CrossRef] [PubMed]

E. B. Kolomeisky, T. J. Newman, J. P. Straley, and X. Qi, “Low-dimensional Bose liquids: Beyond the Gross-Pitaevskii approximation,” Phys. Rev. Lett. 85, 1146 (2000).
[CrossRef] [PubMed]

1999 (1)

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose-Einstein condensation in trapped gases,” Rev. Mod. Phys. 71, 463 (1999).
[CrossRef]

1996 (1)

A. Csordás, R. Graham, and P. Szépfalusy, “Off-resonance light scattering from Bose condensates in traps,” Phys. Rev. A 54, R2543 (1996).
[CrossRef] [PubMed]

1995 (2)

O. Morice, Y. Castin, and J. Dalibard, “Refractive-index of a dilute Bose-gas,” Phys. Rev. A 51, 3896 (1995).
[CrossRef] [PubMed]

J. Javanainen, and J. Ruostekoski, “Off-resonance light-scattering from low-temperature Bose and Fermi gases,” Phys. Rev. A 52, 3033 (1995).
[CrossRef] [PubMed]

1963 (1)

E. Lieb, and W. Liniger, “Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground,” State Phys. Rev. 130, 1605 (1963).

1961 (2)

E. P. Gross, “Structure of a quantized vortex in boson systems,” Nuovo Cim. 20, 454 (1961).
[CrossRef]

L. P. Pitaevskii, “Vortex lines in an imperfect Bose gas,” Sov. Phys. JETP 13, 451–454 (1961).

1960 (1)

M. D. Girardeau, “Relationship between Systems of Impenetrable Bosons and Fermions in One Dimension,” J. Math. Phys. 1, 516 (1960).
[CrossRef]

1936 (1)

L. Tonks, “The Complete Equation of State of One, Two and Three-Dimensional Gases of Hard Elastic Spheres,” Phys. Rev. 50, 955 (1936).
[CrossRef]

Arlt, J.

K. Bongs, S. Burger, S. Dettmer, D. Hellweg, J. Arlt, W. Ertmer, and K. Sengstock, “Waveguide for Bose-Einstein condensates,” Phys. Rev. A 63, 031602 (2001).
[CrossRef]

Bloch, I.

I. Bloch, J. Dalibard, and W. Zwerger, “Many-body physics with ultracold gases,” Rev. Mod. Phys. 80, 885 (2008).
[CrossRef]

B. Paredes, A. Widera, V. Murg, O. Mandel, S. Folling, I. Cirac, G. V. Shlyapnikov, T. W. Hansch, and I. Bloch, “Tonks-Girardeau gas of ultracold atoms in an optical lattice,” Nature 429, 277 (2004).
[CrossRef] [PubMed]

Bongs, K.

K. Bongs, S. Burger, S. Dettmer, D. Hellweg, J. Arlt, W. Ertmer, and K. Sengstock, “Waveguide for Bose-Einstein condensates,” Phys. Rev. A 63, 031602 (2001).
[CrossRef]

Budde, M.

Z. Dutton, M. Budde, C. Slowe, and L. Vestergaard Hau, “Observation of quantum shock waves created with ultra-compressed slow light pulses in a Bose-Einstein condensate,” Science 293, 663 (2001).
[CrossRef] [PubMed]

Burger, S.

K. Bongs, S. Burger, S. Dettmer, D. Hellweg, J. Arlt, W. Ertmer, and K. Sengstock, “Waveguide for Bose-Einstein condensates,” Phys. Rev. A 63, 031602 (2001).
[CrossRef]

Castin, Y.

O. Morice, Y. Castin, and J. Dalibard, “Refractive-index of a dilute Bose-gas,” Phys. Rev. A 51, 3896 (1995).
[CrossRef] [PubMed]

Chen, H.-C.

Chen, Y.-H.

Chiofalo, M. L.

A. Minguzzi, P. Vignolo, M. L. Chiofalo, and M. P. Tosi, “Sum rule for the dynamical response of a confined Bose-Einstein condensed gas,” Phys. Rev. A 64, 033605 (2001).
[CrossRef]

Cho, H.-W.

Cirac, I.

B. Paredes, A. Widera, V. Murg, O. Mandel, S. Folling, I. Cirac, G. V. Shlyapnikov, T. W. Hansch, and I. Bloch, “Tonks-Girardeau gas of ultracold atoms in an optical lattice,” Nature 429, 277 (2004).
[CrossRef] [PubMed]

Csordás, A.

A. Csordás, R. Graham, and P. Szépfalusy, “Off-resonance light scattering from Bose condensates in traps,” Phys. Rev. A 54, R2543 (1996).
[CrossRef] [PubMed]

Dalfovo, F.

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose-Einstein condensation in trapped gases,” Rev. Mod. Phys. 71, 463 (1999).
[CrossRef]

Dalibard, J.

I. Bloch, J. Dalibard, and W. Zwerger, “Many-body physics with ultracold gases,” Rev. Mod. Phys. 80, 885 (2008).
[CrossRef]

O. Morice, Y. Castin, and J. Dalibard, “Refractive-index of a dilute Bose-gas,” Phys. Rev. A 51, 3896 (1995).
[CrossRef] [PubMed]

Dettmer, S.

K. Bongs, S. Burger, S. Dettmer, D. Hellweg, J. Arlt, W. Ertmer, and K. Sengstock, “Waveguide for Bose-Einstein condensates,” Phys. Rev. A 63, 031602 (2001).
[CrossRef]

Dunjko, V.

M. Olshanii, and V. Dunjko, “Short-distance correlation properties of the Lieb-Liniger system and momentum distributions of trapped one-dimensional atomic gases,” Phys. Rev. Lett. 91, 090401 (2003).
[CrossRef] [PubMed]

V. Dunjko, V. Lorent, and M. Olshanii, “Bosons in cigar-shaped traps: Thomas-Fermi Tonks-Girardean regime, and in between,” Phys. Rev. Lett. 86, 5413 (2001).
[CrossRef] [PubMed]

Dutton, Z.

Z. Dutton, M. Budde, C. Slowe, and L. Vestergaard Hau, “Observation of quantum shock waves created with ultra-compressed slow light pulses in a Bose-Einstein condensate,” Science 293, 663 (2001).
[CrossRef] [PubMed]

Ertmer, W.

K. Bongs, S. Burger, S. Dettmer, D. Hellweg, J. Arlt, W. Ertmer, and K. Sengstock, “Waveguide for Bose-Einstein condensates,” Phys. Rev. A 63, 031602 (2001).
[CrossRef]

Esslinger, T.

H. Moritz, T. Stöferle, M. Köhl, and T. Esslinger, “Exciting collective oscillations in a trapped 1D gas,” Phys. Rev. Lett. 91, 250402 (2003).
[CrossRef]

Folling, S.

B. Paredes, A. Widera, V. Murg, O. Mandel, S. Folling, I. Cirac, G. V. Shlyapnikov, T. W. Hansch, and I. Bloch, “Tonks-Girardeau gas of ultracold atoms in an optical lattice,” Nature 429, 277 (2004).
[CrossRef] [PubMed]

Giorgini, S.

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose-Einstein condensation in trapped gases,” Rev. Mod. Phys. 71, 463 (1999).
[CrossRef]

Girardeau, M. D.

M. D. Girardeau, E. M. Wright, and J. M. Triscari, “Ground-state properties of a one-dimensional system of hard-core bosons in a harmonic trap,” Phys. Rev. A 63, 033601 (2001).
[CrossRef]

M. D. Girardeau, and E. M. Wright, “Breakdown of time-dependent mean-field theory for a one-dimensional condensate of impenetrable bosons,” Phys. Rev. Lett. 84, 5239 (2000).
[CrossRef] [PubMed]

M. D. Girardeau, “Relationship between Systems of Impenetrable Bosons and Fermions in One Dimension,” J. Math. Phys. 1, 516 (1960).
[CrossRef]

Graham, R.

A. Csordás, R. Graham, and P. Szépfalusy, “Off-resonance light scattering from Bose condensates in traps,” Phys. Rev. A 54, R2543 (1996).
[CrossRef] [PubMed]

Gross, E. P.

E. P. Gross, “Structure of a quantized vortex in boson systems,” Nuovo Cim. 20, 454 (1961).
[CrossRef]

Hansch, T. W.

B. Paredes, A. Widera, V. Murg, O. Mandel, S. Folling, I. Cirac, G. V. Shlyapnikov, T. W. Hansch, and I. Bloch, “Tonks-Girardeau gas of ultracold atoms in an optical lattice,” Nature 429, 277 (2004).
[CrossRef] [PubMed]

He, Y.-C.

Hellweg, D.

K. Bongs, S. Burger, S. Dettmer, D. Hellweg, J. Arlt, W. Ertmer, and K. Sengstock, “Waveguide for Bose-Einstein condensates,” Phys. Rev. A 63, 031602 (2001).
[CrossRef]

Javanainen, J.

J. Javanainen, and J. Ruostekoski, “Off-resonance light-scattering from low-temperature Bose and Fermi gases,” Phys. Rev. A 52, 3033 (1995).
[CrossRef] [PubMed]

Kinoshita, T.

T. Kinoshita, T. Wenger, and D. S. Weiss, “Observation of a one-dimensional Tonks-Girardeau Gas,” Science 305, 1125 (2004).
[CrossRef] [PubMed]

Kivshar, Y. S.

Köhl, M.

H. Moritz, T. Stöferle, M. Köhl, and T. Esslinger, “Exciting collective oscillations in a trapped 1D gas,” Phys. Rev. Lett. 91, 250402 (2003).
[CrossRef]

Kolomeisky, E. B.

E. B. Kolomeisky, T. J. Newman, J. P. Straley, and X. Qi, “Low-dimensional Bose liquids: Beyond the Gross-Pitaevskii approximation,” Phys. Rev. Lett. 85, 1146 (2000).
[CrossRef] [PubMed]

Lee, Y.-C.

Lieb, E.

E. Lieb, and W. Liniger, “Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground,” State Phys. Rev. 130, 1605 (1963).

Lin, S.-C.

Liniger, W.

E. Lieb, and W. Liniger, “Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground,” State Phys. Rev. 130, 1605 (1963).

Lorent, V.

V. Dunjko, V. Lorent, and M. Olshanii, “Bosons in cigar-shaped traps: Thomas-Fermi Tonks-Girardean regime, and in between,” Phys. Rev. Lett. 86, 5413 (2001).
[CrossRef] [PubMed]

Mandel, O.

B. Paredes, A. Widera, V. Murg, O. Mandel, S. Folling, I. Cirac, G. V. Shlyapnikov, T. W. Hansch, and I. Bloch, “Tonks-Girardeau gas of ultracold atoms in an optical lattice,” Nature 429, 277 (2004).
[CrossRef] [PubMed]

Maschler, C.

I. Mekhov, C. Maschler, and H. Ritsch, “Probing quantum phases of ultracold atoms in optical lattices by transmission spectra in cavity quantum electrodynamics,” Nat. Phys. 3, 319 (2007).
[CrossRef]

Mekhov, I.

I. Mekhov, C. Maschler, and H. Ritsch, “Probing quantum phases of ultracold atoms in optical lattices by transmission spectra in cavity quantum electrodynamics,” Nat. Phys. 3, 319 (2007).
[CrossRef]

Menotti, C.

C. Menotti, and S. Stringari, “Collective oscillations of a one-dimensional trapped Bose-Einstein gas,” Phys. Rev. A 66, 043610 (2002).
[CrossRef]

Minguzzi, A.

A. Minguzzi, P. Vignolo, M. L. Chiofalo, and M. P. Tosi, “Sum rule for the dynamical response of a confined Bose-Einstein condensed gas,” Phys. Rev. A 64, 033605 (2001).
[CrossRef]

Morice, O.

O. Morice, Y. Castin, and J. Dalibard, “Refractive-index of a dilute Bose-gas,” Phys. Rev. A 51, 3896 (1995).
[CrossRef] [PubMed]

Moritz, H.

H. Moritz, T. Stöferle, M. Köhl, and T. Esslinger, “Exciting collective oscillations in a trapped 1D gas,” Phys. Rev. Lett. 91, 250402 (2003).
[CrossRef]

Murg, V.

B. Paredes, A. Widera, V. Murg, O. Mandel, S. Folling, I. Cirac, G. V. Shlyapnikov, T. W. Hansch, and I. Bloch, “Tonks-Girardeau gas of ultracold atoms in an optical lattice,” Nature 429, 277 (2004).
[CrossRef] [PubMed]

Newman, T. J.

E. B. Kolomeisky, T. J. Newman, J. P. Straley, and X. Qi, “Low-dimensional Bose liquids: Beyond the Gross-Pitaevskii approximation,” Phys. Rev. Lett. 85, 1146 (2000).
[CrossRef] [PubMed]

Olshanii, M.

M. Olshanii, and V. Dunjko, “Short-distance correlation properties of the Lieb-Liniger system and momentum distributions of trapped one-dimensional atomic gases,” Phys. Rev. Lett. 91, 090401 (2003).
[CrossRef] [PubMed]

V. Dunjko, V. Lorent, and M. Olshanii, “Bosons in cigar-shaped traps: Thomas-Fermi Tonks-Girardean regime, and in between,” Phys. Rev. Lett. 86, 5413 (2001).
[CrossRef] [PubMed]

Ostrovskaya, E. A.

Paredes, B.

B. Paredes, A. Widera, V. Murg, O. Mandel, S. Folling, I. Cirac, G. V. Shlyapnikov, T. W. Hansch, and I. Bloch, “Tonks-Girardeau gas of ultracold atoms in an optical lattice,” Nature 429, 277 (2004).
[CrossRef] [PubMed]

Pedri, P.

P. Pedri, and L. Santos, “Three-dimensional quasi-Tonks gas in a harmonic trap,” Phys. Rev. Lett. 91, 110401 (2003).
[CrossRef] [PubMed]

Peters, T.

Petrov, D. S.

D. S. Petrov, G. V. Shlyapnikov, and J. T. M. Walraven, “Regimes of quantum degeneracy in trapped 1D gases,” Phys. Rev. Lett. 85, 3745 (2000).
[CrossRef] [PubMed]

Pitaevskii, L. P.

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose-Einstein condensation in trapped gases,” Rev. Mod. Phys. 71, 463 (1999).
[CrossRef]

L. P. Pitaevskii, “Vortex lines in an imperfect Bose gas,” Sov. Phys. JETP 13, 451–454 (1961).

Qi, X.

E. B. Kolomeisky, T. J. Newman, J. P. Straley, and X. Qi, “Low-dimensional Bose liquids: Beyond the Gross-Pitaevskii approximation,” Phys. Rev. Lett. 85, 1146 (2000).
[CrossRef] [PubMed]

Ritsch, H.

I. Mekhov, C. Maschler, and H. Ritsch, “Probing quantum phases of ultracold atoms in optical lattices by transmission spectra in cavity quantum electrodynamics,” Nat. Phys. 3, 319 (2007).
[CrossRef]

Ruostekoski, J.

J. Javanainen, and J. Ruostekoski, “Off-resonance light-scattering from low-temperature Bose and Fermi gases,” Phys. Rev. A 52, 3033 (1995).
[CrossRef] [PubMed]

Santos, L.

P. Pedri, and L. Santos, “Three-dimensional quasi-Tonks gas in a harmonic trap,” Phys. Rev. Lett. 91, 110401 (2003).
[CrossRef] [PubMed]

Sengstock, K.

K. Bongs, S. Burger, S. Dettmer, D. Hellweg, J. Arlt, W. Ertmer, and K. Sengstock, “Waveguide for Bose-Einstein condensates,” Phys. Rev. A 63, 031602 (2001).
[CrossRef]

Shlyapnikov, G. V.

B. Paredes, A. Widera, V. Murg, O. Mandel, S. Folling, I. Cirac, G. V. Shlyapnikov, T. W. Hansch, and I. Bloch, “Tonks-Girardeau gas of ultracold atoms in an optical lattice,” Nature 429, 277 (2004).
[CrossRef] [PubMed]

D. S. Petrov, G. V. Shlyapnikov, and J. T. M. Walraven, “Regimes of quantum degeneracy in trapped 1D gases,” Phys. Rev. Lett. 85, 3745 (2000).
[CrossRef] [PubMed]

Slowe, C.

Z. Dutton, M. Budde, C. Slowe, and L. Vestergaard Hau, “Observation of quantum shock waves created with ultra-compressed slow light pulses in a Bose-Einstein condensate,” Science 293, 663 (2001).
[CrossRef] [PubMed]

Stöferle, T.

H. Moritz, T. Stöferle, M. Köhl, and T. Esslinger, “Exciting collective oscillations in a trapped 1D gas,” Phys. Rev. Lett. 91, 250402 (2003).
[CrossRef]

Straley, J. P.

E. B. Kolomeisky, T. J. Newman, J. P. Straley, and X. Qi, “Low-dimensional Bose liquids: Beyond the Gross-Pitaevskii approximation,” Phys. Rev. Lett. 85, 1146 (2000).
[CrossRef] [PubMed]

Stringari, S.

C. Menotti, and S. Stringari, “Collective oscillations of a one-dimensional trapped Bose-Einstein gas,” Phys. Rev. A 66, 043610 (2002).
[CrossRef]

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose-Einstein condensation in trapped gases,” Rev. Mod. Phys. 71, 463 (1999).
[CrossRef]

Szépfalusy, P.

A. Csordás, R. Graham, and P. Szépfalusy, “Off-resonance light scattering from Bose condensates in traps,” Phys. Rev. A 54, R2543 (1996).
[CrossRef] [PubMed]

Tonks, L.

L. Tonks, “The Complete Equation of State of One, Two and Three-Dimensional Gases of Hard Elastic Spheres,” Phys. Rev. 50, 955 (1936).
[CrossRef]

Tosi, M. P.

A. Minguzzi, P. Vignolo, M. L. Chiofalo, and M. P. Tosi, “Sum rule for the dynamical response of a confined Bose-Einstein condensed gas,” Phys. Rev. A 64, 033605 (2001).
[CrossRef]

Triscari, J. M.

M. D. Girardeau, E. M. Wright, and J. M. Triscari, “Ground-state properties of a one-dimensional system of hard-core bosons in a harmonic trap,” Phys. Rev. A 63, 033601 (2001).
[CrossRef]

Vestergaard Hau, L.

Z. Dutton, M. Budde, C. Slowe, and L. Vestergaard Hau, “Observation of quantum shock waves created with ultra-compressed slow light pulses in a Bose-Einstein condensate,” Science 293, 663 (2001).
[CrossRef] [PubMed]

Vignolo, P.

A. Minguzzi, P. Vignolo, M. L. Chiofalo, and M. P. Tosi, “Sum rule for the dynamical response of a confined Bose-Einstein condensed gas,” Phys. Rev. A 64, 033605 (2001).
[CrossRef]

Walraven, J. T. M.

D. S. Petrov, G. V. Shlyapnikov, and J. T. M. Walraven, “Regimes of quantum degeneracy in trapped 1D gases,” Phys. Rev. Lett. 85, 3745 (2000).
[CrossRef] [PubMed]

Weiss, D. S.

T. Kinoshita, T. Wenger, and D. S. Weiss, “Observation of a one-dimensional Tonks-Girardeau Gas,” Science 305, 1125 (2004).
[CrossRef] [PubMed]

Wenger, T.

T. Kinoshita, T. Wenger, and D. S. Weiss, “Observation of a one-dimensional Tonks-Girardeau Gas,” Science 305, 1125 (2004).
[CrossRef] [PubMed]

Widera, A.

B. Paredes, A. Widera, V. Murg, O. Mandel, S. Folling, I. Cirac, G. V. Shlyapnikov, T. W. Hansch, and I. Bloch, “Tonks-Girardeau gas of ultracold atoms in an optical lattice,” Nature 429, 277 (2004).
[CrossRef] [PubMed]

Wright, E. M.

M. D. Girardeau, E. M. Wright, and J. M. Triscari, “Ground-state properties of a one-dimensional system of hard-core bosons in a harmonic trap,” Phys. Rev. A 63, 033601 (2001).
[CrossRef]

M. D. Girardeau, and E. M. Wright, “Breakdown of time-dependent mean-field theory for a one-dimensional condensate of impenetrable bosons,” Phys. Rev. Lett. 84, 5239 (2000).
[CrossRef] [PubMed]

Yu, I. A.

Zwerger, W.

I. Bloch, J. Dalibard, and W. Zwerger, “Many-body physics with ultracold gases,” Rev. Mod. Phys. 80, 885 (2008).
[CrossRef]

J. Math. Phys. (1)

M. D. Girardeau, “Relationship between Systems of Impenetrable Bosons and Fermions in One Dimension,” J. Math. Phys. 1, 516 (1960).
[CrossRef]

Nat. Phys. (1)

I. Mekhov, C. Maschler, and H. Ritsch, “Probing quantum phases of ultracold atoms in optical lattices by transmission spectra in cavity quantum electrodynamics,” Nat. Phys. 3, 319 (2007).
[CrossRef]

Nature (1)

B. Paredes, A. Widera, V. Murg, O. Mandel, S. Folling, I. Cirac, G. V. Shlyapnikov, T. W. Hansch, and I. Bloch, “Tonks-Girardeau gas of ultracold atoms in an optical lattice,” Nature 429, 277 (2004).
[CrossRef] [PubMed]

Nuovo Cim. (1)

E. P. Gross, “Structure of a quantized vortex in boson systems,” Nuovo Cim. 20, 454 (1961).
[CrossRef]

Opt. Express (2)

Phys. Rev. (1)

L. Tonks, “The Complete Equation of State of One, Two and Three-Dimensional Gases of Hard Elastic Spheres,” Phys. Rev. 50, 955 (1936).
[CrossRef]

Phys. Rev. A (7)

K. Bongs, S. Burger, S. Dettmer, D. Hellweg, J. Arlt, W. Ertmer, and K. Sengstock, “Waveguide for Bose-Einstein condensates,” Phys. Rev. A 63, 031602 (2001).
[CrossRef]

A. Minguzzi, P. Vignolo, M. L. Chiofalo, and M. P. Tosi, “Sum rule for the dynamical response of a confined Bose-Einstein condensed gas,” Phys. Rev. A 64, 033605 (2001).
[CrossRef]

C. Menotti, and S. Stringari, “Collective oscillations of a one-dimensional trapped Bose-Einstein gas,” Phys. Rev. A 66, 043610 (2002).
[CrossRef]

O. Morice, Y. Castin, and J. Dalibard, “Refractive-index of a dilute Bose-gas,” Phys. Rev. A 51, 3896 (1995).
[CrossRef] [PubMed]

J. Javanainen, and J. Ruostekoski, “Off-resonance light-scattering from low-temperature Bose and Fermi gases,” Phys. Rev. A 52, 3033 (1995).
[CrossRef] [PubMed]

A. Csordás, R. Graham, and P. Szépfalusy, “Off-resonance light scattering from Bose condensates in traps,” Phys. Rev. A 54, R2543 (1996).
[CrossRef] [PubMed]

M. D. Girardeau, E. M. Wright, and J. M. Triscari, “Ground-state properties of a one-dimensional system of hard-core bosons in a harmonic trap,” Phys. Rev. A 63, 033601 (2001).
[CrossRef]

Phys. Rev. Lett. (7)

M. D. Girardeau, and E. M. Wright, “Breakdown of time-dependent mean-field theory for a one-dimensional condensate of impenetrable bosons,” Phys. Rev. Lett. 84, 5239 (2000).
[CrossRef] [PubMed]

E. B. Kolomeisky, T. J. Newman, J. P. Straley, and X. Qi, “Low-dimensional Bose liquids: Beyond the Gross-Pitaevskii approximation,” Phys. Rev. Lett. 85, 1146 (2000).
[CrossRef] [PubMed]

H. Moritz, T. Stöferle, M. Köhl, and T. Esslinger, “Exciting collective oscillations in a trapped 1D gas,” Phys. Rev. Lett. 91, 250402 (2003).
[CrossRef]

D. S. Petrov, G. V. Shlyapnikov, and J. T. M. Walraven, “Regimes of quantum degeneracy in trapped 1D gases,” Phys. Rev. Lett. 85, 3745 (2000).
[CrossRef] [PubMed]

V. Dunjko, V. Lorent, and M. Olshanii, “Bosons in cigar-shaped traps: Thomas-Fermi Tonks-Girardean regime, and in between,” Phys. Rev. Lett. 86, 5413 (2001).
[CrossRef] [PubMed]

M. Olshanii, and V. Dunjko, “Short-distance correlation properties of the Lieb-Liniger system and momentum distributions of trapped one-dimensional atomic gases,” Phys. Rev. Lett. 91, 090401 (2003).
[CrossRef] [PubMed]

P. Pedri, and L. Santos, “Three-dimensional quasi-Tonks gas in a harmonic trap,” Phys. Rev. Lett. 91, 110401 (2003).
[CrossRef] [PubMed]

Rev. Mod. Phys. (2)

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose-Einstein condensation in trapped gases,” Rev. Mod. Phys. 71, 463 (1999).
[CrossRef]

I. Bloch, J. Dalibard, and W. Zwerger, “Many-body physics with ultracold gases,” Rev. Mod. Phys. 80, 885 (2008).
[CrossRef]

Science (2)

Z. Dutton, M. Budde, C. Slowe, and L. Vestergaard Hau, “Observation of quantum shock waves created with ultra-compressed slow light pulses in a Bose-Einstein condensate,” Science 293, 663 (2001).
[CrossRef] [PubMed]

T. Kinoshita, T. Wenger, and D. S. Weiss, “Observation of a one-dimensional Tonks-Girardeau Gas,” Science 305, 1125 (2004).
[CrossRef] [PubMed]

Sov. Phys. JETP (1)

L. P. Pitaevskii, “Vortex lines in an imperfect Bose gas,” Sov. Phys. JETP 13, 451–454 (1961).

State Phys. Rev. (1)

E. Lieb, and W. Liniger, “Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground,” State Phys. Rev. 130, 1605 (1963).

Other (4)

C. J. Pethick, and H. Smith, Bose-Einstein condensation in dilute gases, (Cambridge, 2008).
[CrossRef]

I. S. Gradshtein, and I. M. Ryzhik, Table of Integrals, Series, and products, 7th ed. (Academic Press, 2007).

Handbook of Mathematical Functions, edited by M. Abramowitz and I. Stegun (Dover, New York, 1965).

D. A. Steck, “Rubidium 87 D Line Data,” available at http://steck.us/alkalidata (rev. 2.1.2, 12 August 2009).

Supplementary Material (1)

» Media 1: MOV (208 KB)     

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Figures (1)

Fig. 1.
Fig. 1.

Plot of the time dependent particle density ρ(x, t) for 10 atoms as function of x measured in units of the radius xT . The thin solid line (red in color) is at t = 0, the thick solid line (blue in color) is at t = π/2ω and the thin dotted line (orange in color) is at t = π/ω. We plot the average density ρ ¯ (x) for comparison as a black dashed line. The attached video file (Media 1) shows an animation of the density oscillations vs. time.

Equations (22)

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Ψ S ( x 1 , , x N , t ) = c 0 Ψ 0 exp [ i E 0 h ¯ t ] + c 1 Ψ 1 exp [ i ( E 0 h ¯ + ω ) t ]
ρ ( x , t ) = N Ψ S ( x , x 2 , x N , t ) 2 d x 2 d x N
ρ ( x , t ) = c 0 2 ρ 0 ( x ) + c 1 2 ρ 1 ( x ) + 2 Re c 1 * c 0 ρ 01 ( x ) exp ( i ω t )
ρ 01 ( x ) = N Ψ 0 * ( x , x 2 , x N ) Ψ 1 ( x , x 2 , x N ) d x 2 d x N
Ψ 0 = 1 N ! det ( n , j ) = 0 , 1 ( N 1 , N ) φ n ( x j ) Π 1 j < k N sign ( x k x j )
Ψ 1 = 1 N ! Σ j = 1 N ( 1 ) N + j 1 φ N ( x j ) D j
ρ 0 ( x ) = n = 0 N 1 φ n ( x ) 2 , ρ 1 ( x ) = n = 0 N 2 φ n ( x ) 2 + φ N ( x ) 2 .
Ψ 0 Ψ 1 = 1 N ! j = 1 N ( 1 ) j 1 φ N 1 ( x j ) D j × j = 1 N ( 1 ) j 1 φ N ( x j ) D j
ρ 01 ( x ) = φ N 1 ( x ) φ N ( x )
n ( x , t ) = ( 1 + β ρ ( x , t ) 1 β ρ ( x , t ) 3 ) 1 2
2 x 2 E ( x , t ) + n 2 ( x , t ) k 2 E ( x , t ) = 0
E f ( x , t ) = 1 n ( x , t ) E 0 exp ( i k x 0 x n ( x , t ) d x i n ( x ) 4 k n 2 ( x ) i x 0 x ( n ( x ) ) 2 8 k n 3 ( x ) d x ) ,
T ( t ) = E f ( x , t ) E 0 2 = exp ( 2 k x 0 x Im n ( x , t ) d x + 1 4 k x 0 x Im ( n ( x ) ) 2 n 3 ( x ) d x )
T ( t ) = exp ( N k Im β + k Im β 2 12 ρ 2 ( x , t ) d x + Im β 2 16 k ( ρ ( x , t ) ) 2 d x )
T ( t ) = T N exp [ ζ cos ( 2 ( ω t + α ) ) ] ,
T N = exp ( N k Im β k 12 Im ( β 2 ) ( R 0 + R 1 2 ) + Im β 2 16 k ( R 01 + R 11 2 ) )
ζ = Im ( β 2 ) ( k R 1 24 + R 11 32 k ) ,
R 0 = r 0 2 ( x ) d x , R 1 = r 1 2 ( x ) d x , R 01 = ( r 0 ( x ) ) 2 d x , R 11 = ( r 1 ( x ) ) 2 d x
T ( t ) = T N [ I 0 ( ζ ) + 2 S = 1 I s ( ζ ) cos ( 2 s ( ω t + α ) ) ]
T ˜ ( 2 s ω ) = T N I s ( ζ ) exp ( 2 s i α ) , s = 0 , 1 , 2 , . . . ,
N = ( 2 k Im β ) 1 ln [ T ˜ Δ ( 0 ) T ˜ Δ ( 0 ) ] .
ζ = 2 I 1 ( ζ ) I 0 ( ζ ) I 2 ( ζ ) = 2 T ˜ ( 2 ω ) exp ( 2 i α ) T ˜ ( 0 ) T ˜ ( 4 ω )

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