Exact density oscillations in the Tonks-Girardeau gas and their optical detection

doi:10.1364/OE.18.017569

Exact density oscillations in the Tonks-Girardeau gas and their optical detection

Mihály G. Benedict, Csaba Benedek, and Attila Czirják »View Author Affiliations

Optics Express, Vol. 18, Issue 16, pp. 17569-17575 (2010)
http://dx.doi.org/10.1364/OE.18.017569

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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.

OCIS Codes
(120.4290) Instrumentation, measurement, and metrology : Nondestructive testing
(140.7010) Lasers and laser optics : Laser trapping
(020.1475) Atomic and molecular physics : Bose-Einstein condensates
(260.2710) Physical optics : Inhomogeneous optical media

ToC Category:
Atomic and Molecular Physics

History
Original Manuscript: June 22, 2010
Manuscript Accepted: July 21, 2010
Published: July 30, 2010

Citation
Mihály G. Benedict, Csaba Benedek, and Attila Czirják, "Exact density oscillations in the Tonks-Girardeau gas and their optical detection," Opt. Express 18, 17569-17575 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-16-17569

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References

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).
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]
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]
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, 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]
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]
L. Tonks, “The Complete Equation of State of One, Two and Three-Dimensional Gases of Hard Elastic Spheres,” Phys. Rev. 50, 955 (1936). [CrossRef]
M. D. Girardeau, “Relationship between Systems of Impenetrable Bosons and Fermions in One Dimension,” J. Math. Phys. 1, 516 (1960). [CrossRef]
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).
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. 80, 885 (2008). [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]
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]
I. S. Gradshtein, and I. M. Ryzhik, Table of Integrals, Series, and products, 7th ed. (Academic Press, 2007).
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]
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]
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]
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]
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).

J. Math. Phys.

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

Nat. Phys.

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

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.

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

Opt. Express

Phys. Rev.

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

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.

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.

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

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

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

State Phys. Rev.

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

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).

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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).

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Exact density oscillations in the Tonks-Girardeau gas and their optical detection

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References

J. Math. Phys.

Nat. Phys.

Nature

Nuovo Cim.

Opt. Express

Phys. Rev.

Phys. Rev. A

Phys. Rev. Lett.

Rev. Mod. Phys.

Science

Sov. Phys. JETP

State Phys. Rev.

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