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 OSA

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    [Crossref]
  2. L. P. Pitaevskii, “Vortex lines in an imperfect Bose gas” Soviet Physics 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]
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  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]
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    [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).
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  15. C. J. Pethick and H. Smith, Bose-Einstein condensation in dilute gases, (Cambridge, 2008).
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  16. I. Bloch, J. Dalibard, and W. Zwerger, “Many-body physics with ultracold gases,” Rev. Mod. Phys., 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).
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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  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]
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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]

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]

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]

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,” Il Nuovo Cimento 20, 454 (1961).
[Crossref]

L. P. Pitaevskii, “Vortex lines in an imperfect Bose gas” Soviet Physics 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.

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]

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

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.

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

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

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]

Gradshtein, I. S.

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

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,” Il Nuovo Cimento 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]

Hau, L. Vestergaard

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]

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.

Pethick, C. J.

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

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” Soviet Physics 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]

Ryzhik, I. M.

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

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

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]

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]

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]

Smith, H.

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

Steck, D. A.

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

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

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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]

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).
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Yu, I. A.

Zwerger, W.

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

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[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]

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]

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]

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[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]

Phys. Rev. Lett. (7)

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

Rev. Mod. Phys. (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]

Science (2)

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

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]

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Other (5)

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

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

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).

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

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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