## Higher energy collective excitations in trapped Bose condensates

Optics Express, Vol. 1, Issue 10, pp. 293-299 (1997)

http://dx.doi.org/10.1364/OE.1.000293

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### Abstract

We investigate theoretically the collective excitations of trapped Bose condensates with energies of the order of the chemical potential of the system. For the MIT sodium Bose condensate in a Cloverleaf trap, we find interesting level crossing behavior for high energy excitations and calculate the spatial magnetic dipole moments for selective creation of condensate excitations.

© Optical Society of America

^{1–31. M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell,“Observations of Bose-Einstein condensation in a dilute atomic vapor”, Science 269, 198 (1995). [CrossRef] [PubMed] }has generated great interest in studying the properties of these confined weakly interacting quantum gases. Tremendous progress has been made in the past two years: many single particle properties of trapped condensates have been measured; direct nondestructive optical imaging techniques have been developed

^{4∆54. M. R. Andrews, M. -O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Direct, nondestructive observation of a Bose condensate”, Science 273, 84 (1996). [CrossRef] [PubMed] }; and several of the low energy collective excitation modes have been detected

^{6∆86. D. S. Jin, J. R. Ensher, M. R. Matthews, C. Wieman, and E. A. Cornell, “Collective excitations of a Bose-Einstein condensate in a dilute gas”, Phys. Rev. Lett. 77, 420 (1996). [CrossRef] [PubMed] }. Recently, the macroscopic coherence properties of the condensate were displayed in a spectacular fashion by the demonstration of the interference of two condensates

^{99. M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Observation of interference between two Bose condensates”, Science 275, 637 (1997). [CrossRef] [PubMed] }, and by the suppression of condensate collisional losses from inelastic collisions

^{1010. E. A. Burt, R. W. Ghrist, C. J. Myatt, M. J. Holland, E. A. Cornell, and C. E. Wieman, “Coherence, correlations, and collisions: What one learns about Bose-Einstein condensates from their decay”, Phys. Rev. Lett. 79, 337 (1997). [CrossRef] }due to multi-particle correlations. Rudimentary atom lasers with pulsed output couplers have also recently been deveoped

^{99. M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Observation of interference between two Bose condensates”, Science 275, 637 (1997). [CrossRef] [PubMed] ,1111. M. -O. Mewes, M. R. Andrews, D. M. Kurn, D. S. Durfee, C. G. Townsend, and W. Ketterle, “Output coupler for Bose-Einstein condensed atoms”, Phys. Rev. Lett. 78, 582 (1997); [CrossRef] }, and the zero-sound velocity of the condensate has been measured

^{1212. M. R. Andrews, D. M. Kurn, H. J. Miesner, D. S. Durfee, C. G. Townsend, S. Inouye, and W. Ketterle, “Propagation of sound in a Bose-Einstein condensate”, Phys. Rev. Lett. 79, 553 (1997). [CrossRef] }.

^{1313. M. Edwards, P. A. Ruprecht, K. Burnett, R. J. Dodd, and C. W. Clark, “Collective excitations of atomic Bose-Einstein condensates”, Phys. Rev. Lett. 77, 1671 (1996). [CrossRef] [PubMed] ,1414. K. G. Singh and D. S. Rokhsar, “Collective excitations of a confined Bose condensate”, Phys. Rev. Lett. 77, 1667 (1996). [CrossRef] [PubMed] }. An (analytic) asymptotic expression for excitation eigenfrequencies, valid in the Thomas-Fermi/hydrodynamic limit, was first presented by S. Stringari

^{1515. S. Stringari, “Collective excitations of a trapped Bose condensed gas”, Phys. Rev. Lett. 77, 2360 (1996). [CrossRef] [PubMed] }. A variational approach was developed by A. Fetter

^{1616. A. L. Fetter, “Ground state and excited states of a confined condensed Bose gas”, Phys. Rev. A 53, 4245 (1996). [CrossRef] [PubMed] }, and a dynamical variational approach was applied by V. M. Pírez-García,

*et al*

^{1717. V. M. Pérez-García, H. Michinel, J. I. Cirac, M. Lewenstein, and P. Zoller, “Low energy excitations of a Bose-Einstein condensate: a variational analysis”, Phys. Rev. Lett. 77, 5230 (1996). [CrossRef] }. More recent studies in this area are listed in Ref. 18 ∆ 23

18. Yu. Kagan, E. L. Surkov, and G. V. Shlyapnikov, “Evolution of a Bose-condensed gas under variations of the confining potential”, Phys. Rev. A **54**, R1753 (1996). [CrossRef] [PubMed]

^{88. D. S. Jin, M. R. Matthews, J. R. Ensher, C. E. Wieman, and E. A. Cornell, “Temperature-dependent damping and frequency shifts in collective excitations of a dilute Bose-Einstein condensate”, Phys. Rev. Lett. 78, 764 (1997). [CrossRef] }and theory

^{24∆2624. D. A. W. Hutchinson, E. Zaremba, and A. Griffin, “Finite temperature excitations of a trapped Bose gas”, Phys. Rev. Lett. 78, 1842 (1997). [CrossRef] }may be difficult to achieve at finite temperatures and for high energy excitations.

^{2626. L. You, W. Hoston, M. Lewenstein, and M. Marinescu, “Low energy excitation spectra of trapped Bose condensates”, Acta Phys. Pol. A,(to appear).,2727. L. You, W. Hoston, and M. Lewenstein, “Low energy excitations of trapped Bose condensates”, Phys. Rev. A 55, R1581 (1997); [CrossRef] }by two of us and co-workers, indicate that quasi-particles (i.e. collective excitations) display interesting dynamics. A careful numerical study of the dependence of the excitation frequencies on the total number of condensate atoms at zero temperature

^{2727. L. You, W. Hoston, and M. Lewenstein, “Low energy excitations of trapped Bose condensates”, Phys. Rev. A 55, R1581 (1997); [CrossRef] }(i.e. the interaction strength of the condensate), found complicated level crossings and avoided crossings

^{2626. L. You, W. Hoston, M. Lewenstein, and M. Marinescu, “Low energy excitation spectra of trapped Bose condensates”, Acta Phys. Pol. A,(to appear).}. Similar behavior for high energy excitations was also recently predicted by Fliesser, et. al.

^{2828. M. Fliesser, A. Csordás, R. Graham, and P. Szépfalusy, “Classical quasi-particle dynamics in trapped Bose condensates”, (preprint, cond-mat/9707122).,2929. M. Fliesser, A. Csordás, P. Szépfalusy, and R. Graham, “Hydrodynamic excitations of Bose condensates in anisotropic traps”, (preprint, cond-mat/9706002).}, where they studied the classical dynamics of the condensate quasi-particles using a semiclassical approach. They found that for non-isotropic harmonic traps, quasi-particles can be chaotic when their excitation energy is of the order of the system’s chemical potential. It will, of course, be important to study quasi-particle dynamics quantum mechanically to confirm the semiclassical studies. In this paper, we further such investigation of high energy collective excitations. We present new calculations, for the MIT sodium condensate in a Cloverleaf trap

^{77. M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. M. Kurn, D. S. Durfee, C. G. Townsend, and W. Ketterle, “Collective excitations of a Bose-Einstein condensate in a magnetic trap”, Phys. Rev. Lett. 77, 988 (1996). [CrossRef] [PubMed] }, of the symmetry properties of the quasi-particle level structure, and of selective quasi-particle excitation using spatial magnetic resonance

^{30}.

*N*spinless bosonic atoms trapped in a potential

*V*

_{t}(

*r*⃗) is given by

*r*⃗) and Ψ

^{†}(

*r*⃗) are atomic (bosonic) annihilation and creation fields,

*u*

_{0}= 4

*πħ*

^{2}

*a*

_{sc}/

*M*, with

*M*being the atomic mass, and

*a*

_{sc}is the scattering length of the atom-atom interaction. The chemical potential

*μ*is introduced to guarantee conservation of the average of the total number of atoms

*N̂*= ∫

*dr*⃗Ψ̂†(

*r*⃗)Ψ̂(

*r*⃗). The mean field BH theory describes the single particle excitations of the system in terms of non-interacting quasi-particles, thus casting the Hamiltonian (1) into the form

*T*= 0, the the Bogoliubov approximation starts with the assumption

*c*-number condensate wave function

*ψ*

_{0}(

*r*⃗) is assumed to be real without loss of generality, and is normalized according to ∫

*dr*⃗|

*ψ*

_{0}(

*r*⃗)|

^{2}= 1.

*δ*Ψ̂(

*r*⃗) represents the contributions of quantum fluctuations, and fulfills the same standard bosonic commutation relations as Ψ̂(

*r*⃗). We substitute Eq. (3) into Eq. (1) and neglect both 3-rd and 4-th order fluctuation terms. Linear fluctuation terms vanish provided

*ψ*

_{0}(

*r*⃗) satisfies the nonlinear Schrödinger equation (NLSE), i.e.

*ρ*

_{0}(

*r*⃗) =

*N*|

*ψ*

_{0}

^{2}. The resulting linearized Hamiltonian is then quadratic in

*δΨ̂*and

*δΨ̂*

^{†}, and can be diagonalized by the quasi-particle annihilation operator

*U*

_{k}(

*r*⃗) and

*V*

_{k}(

*r*⃗) are the mode functions of the quasi-particles, and must be calculated for particular traps and condensates. To determine the functions

*U*

_{k}(

*r*⃗) and

*V*

_{k}(

*r*⃗) (for

*k*≠ 0) we solve [

*g͂*

*k*,

*Η*] =

*ħῶ*

_{k}

*g͂*

_{k}, which gives the following coupled Bogoliubov-de Gennes equations:

_{0}(

*r*⃗) =

*N*

*r*⃗). The mode functions obey the standard

*orthonormality*condition

^{2121. A. Griffin, W.-C. Wu, and S. Stringari, “Hydrodynamic modes in a trapped Bose gas above the Bose-Einstein transition”, Phys. Rev. Lett. 78, 1838 (1997). [CrossRef] }. The above Eqs. (4) and (6) may be solved for the condensate wave function

*ψ*

_{0}(

*r*⃗) and the properties of quasi-particle excitations

^{2626. L. You, W. Hoston, M. Lewenstein, and M. Marinescu, “Low energy excitation spectra of trapped Bose condensates”, Acta Phys. Pol. A,(to appear).,2727. L. You, W. Hoston, and M. Lewenstein, “Low energy excitations of trapped Bose condensates”, Phys. Rev. A 55, R1581 (1997); [CrossRef] }. The details of our numerical approach are described in Ref. 26. Here we present new calculations for the MIT Cloverleaf trap

^{77. M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. M. Kurn, D. S. Durfee, C. G. Townsend, and W. Ketterle, “Collective excitations of a Bose-Einstein condensate in a magnetic trap”, Phys. Rev. Lett. 77, 988 (1996). [CrossRef] [PubMed] }with (

*ω*

_{x}:

*ω*

_{y}:

*ω*

_{z}) = (13 : 13 : 1)(2

*π*)19.28(Hz), the scattering length

*a*

_{sc}= 2.7 (nm) for the sodium atom (the actual hyperfine state of the condensed atoms).

*N*increases, i.e. as the mean field interactions grow. In Fig. 2 the

*N*-dependence of the quasi-particle level structure is plotted for

*L*

_{z}= 0 excitations only, i.e. for excitations with no axial angular momentum. This figure shows the correspondence between two shape oscillations recently observed in the MIT BEC experiment

^{77. M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. M. Kurn, D. S. Durfee, C. G. Townsend, and W. Ketterle, “Collective excitations of a Bose-Einstein condensate in a magnetic trap”, Phys. Rev. Lett. 77, 988 (1996). [CrossRef] [PubMed] }, which operates in the high atom number limit (

*N*≈ 10

^{6}), and the

*ῶ*

_{k}= 2 and

*ῶ*

_{k}= 26 quasi-particle states in the low density limit. Note that level crossings become common for higher energy

*L*

_{z}=0 quasi-particles when the excitation energy is of the order of the system’s chemical potential. It is worth emphasizing that the Eq. (6) is in fact a set of linear equations, and the observed complicated level repulsion behavior is not due to a simple nonlinear coupling between different quasiparticle modes.

*spatial magnetic resonance*(SMR) recently analyzed by two of us for the selective creation and manipulation of quasi-particles

^{3030. R. Walsworth and L. You, “Selective creation of quasi-particles in trapped Bose condensates”, Phys. Rev. A 55, 555 (1997). [CrossRef] }. The physics of SMR is straightforward and may be thought of as an optimized variation of the trap potential (or other applied magnetic fields). The variation is optimized to excite the condensate into higher quasi-particle states with a desired spatial symmetry while minimizing the perturbation of the trap potential, thus reducing the excitation of unwanted modes. In contrast, simple resonant variations of the trap potential will not be generally efficient at exciting high energy quasi-particles; in addition, such non-optimal trap potential variations will significantly perturb the condensate during application.

*i*(e.g. the condensate) is assumed to be spatially symmetric (an even function of spatial coordinate

*x*in a one-dimensional symmetric trap), and the higher energy quasi-particle state

*f*is assumed to be spatially antisymmetric (an odd function of

*x*). The two trap state wave functions

*ψ*

_{i}(

*r*⃗) and

*ψ*

_{f}(

*r*⃗) describe the spatial extent of the respective quasi-particles; while the internal spin state |IN〉, which describes the ensemble magnetic dipole moment of the atomic cloud, remains the same for both trapped states (in order for atoms to remain trapped ). We introduce a time and space dependent magnetic field

*B*⃗

_{SMR}(

*r*⃗,

*t*) =

*ê*

_{B}(

*r*⃗)

*B*

_{SMR}(

*r*⃗)

*F*(

*t*), where

*F*(

*t*) =

*F*(

*t*)(

*e*

^{-iωBt}+

*e*

^{iωBt}),

*F*(

*t*) is a slowly varying envelope function, and

*ω*

_{B}is close to resonance with the transition of the two quasi-particle states

*ω*

_{f}-

*ω*

_{i}. The unit vector

*ê*

_{B}(

*r*⃗) denotes the direction of the time-dependent magnetic field. Then the ensemble magnetic dipole transition matrix element between the two quasi-particle states is

*μ*⃗

_{m}=

*μ*

_{m}

*ê*

_{z}(

*r*⃗), and we have assumed that the strong trapping field completely spin polarizes the atoms and is pointing along the z-axis

^{20}. Therefore the transition matrix element is a product of two parts: an internal part (a constant) due to the permanent magnetic moment

*μ*⃗

_{m}of the atoms; and a spatial part, which is the magnetic dipole transition analog of the Franck-Condon factor in electric dipole transitions in diatomic molecules. Now if

*B*

_{SMR}(

*r*⃗) is spatially symmetric, as in a simple trap potential variation, then the transition matrix element is zero when the initial and final quasi-particle states have opposite spatial parity. As long as

*B*

_{SMR}(

*r*⃗) has some spatial asymmetry, however, the matrix element is non-zero. For optimal SMR in this simple example,

*B*

_{SMR}(

*r*⃗) should be antisymmetric.

^{20}by the term

*ħu*

_{kk′}= -

*μ*

_{m}∫

*d*

*r*⃗

*U*

_{k}(

*r*⃗)

*r*⃗)

*B*

_{SMR}(

*r*⃗) (and similarly for

*ħv*

_{kk′}with

*V*

_{k}replacing

*U*

_{k}, etc.) and

*ħγ*

_{kk′}= -

*μ*

_{m}∫

*d*

*r*⃗

*U*

_{k}(

*r*⃗)

*V*

_{k′}(

*r*⃗)

*B*

_{SMR}(

*r*⃗).

*U*

_{0}(

*r*) and

*V*

_{0}(

*r*). The excited state wavefunctions factor into radial parts,

*U*(

_{kLLz}*r*) and

*V*(

_{kLLz}*r*); and into angular parts, the spherical harmonics

*Y*(

_{LLz}*θ*,

*ϕ*) that are eigenstates of the angular momentum

*L̂*

^{2}and its projection on the z-axis

*L̂*

_{z}. (Here

*k*is the quantum number for the radial direction.) If the applied SMR field is chosen to have

*B*

_{SMR}(

*r*⃗) proportional to a particular spherical harmonic

*Y*(

_{LLz}*δ*,

*ϕ*), then only quasi-particle states with this particular spatial symmetry have non-zero SMR coupling matrix elements from the condensate. A similar argument applies for the radial dependence of

*B*

_{SMR}(

*r*⃗). So by selecting the spatial symmetries of the applied magnetic field we can control the induced dipole coupling given in Eq. (9), and selectively create either specific quasi-particles or a coherent superposition of quasi-particles. (Note that when the initial state is not spherically symmetric, the spatial symmetries of the optimal SMR field will not in general be the same as the final quasi-particle state.) For the selective creation of a particular quasi-particle state

*k*

_{s}, one wants non-zero coupling constants with

*k*→ condensate and

*k*′ →

*k*

_{s}, and also small coupling constants linking the condensate to other quasi-particles. Determining the appropriate SMR field is an iterative process. One can set the initial symmetry of

*B*

_{SMR}(

*r*⃗) according to the above discussion, and adjust both

*B*

_{SMR}(

*r*) and

*F*(

*t*) during the SMR excitation pulse (spatial and spectral chirping) to stay on resonance with the difference in quasi-particle energies and the quasi-particle mode structure (both of which change with depletion of the condensate and accordant reductions in the BH mean field). If the evolution dynamics (i.e. SMR Rabi frequency) is sufficiently slow, then the rotating wave approximation allows us to drop all other states not resonantly coupled

^{3030. R. Walsworth and L. You, “Selective creation of quasi-particles in trapped Bose condensates”, Phys. Rev. A 55, 555 (1997). [CrossRef] }.

*ῶ*

*k*= 26 when

*N*= 1. Note that the coupling to the higher energy mode grows significantly as

*N*is increased; this is consistent with the relative ease with which this shape oscillation has been excited experimentally using a simple trap potential variation

*N*considered. We are currently investigating the use of SMR fields employing tailored spatial symmetries and chirping techniques to excite specific quasi-particles not well coupled to simple, resonant trap potential modulations (e.g. the lower energy mode shown in Fig. 3).

*ω*

_{k}=

*ω*

_{r}= 13(

*ω*), presumably because the radial (

*ω*

_{r}=

*ω*

_{x}=

*ω*

_{y}) and axial (

*ω*

_{z}) trap frequencies are significantly different, such that the families of quasi-particles studied here are adiabatically well connected with levels of different radial excitation levels. Interesting level crossing behavior becomes evident when the chemical potential

*μ*becomes of the order of the higher energy shape oscillation mode [i.e.

*ω*

_{k}= 2

*ω*

_{r}= 26(

*ω*

_{z}=

*ω*)]. The detailed crossing/anticrossing character of the quasi-particle level structure will be studied in future work. We also discussed spatial magnetic resonance, or SMR, a technique for the selective creation of quasi-particles and coherent superpositions of quasi-particles. We presented new calculations for the excitation of high energy quasi-particles in the MIT trap using SMR, outlined additional SMR calculations we will pursue in future work, and discussed possible experimental ramifications.

## Footnotes

* | Also with Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 |

## References

1. | M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell,“Observations of Bose-Einstein condensation in a dilute atomic vapor”, Science |

2. | K.B. Davis, M. -O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Bose-Einstein condensations in a gas of sodium atoms”, Phys. Rev. Lett. |

3. | C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, “Evidence of Bose-Einstein condensation in an atomic gas with attractive interactions”, Phys. Rev. Lett. |

4. | M. R. Andrews, M. -O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Direct, nondestructive observation of a Bose condensate”, Science |

5. | C. C. Bradley, C. A. Sackett, and R. G. Hulet, “Analysis of in situ images of Bose-Einstein condensates of lithium”, Phys. Rev. A |

6. | D. S. Jin, J. R. Ensher, M. R. Matthews, C. Wieman, and E. A. Cornell, “Collective excitations of a Bose-Einstein condensate in a dilute gas”, Phys. Rev. Lett. |

7. | M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. M. Kurn, D. S. Durfee, C. G. Townsend, and W. Ketterle, “Collective excitations of a Bose-Einstein condensate in a magnetic trap”, Phys. Rev. Lett. |

8. | D. S. Jin, M. R. Matthews, J. R. Ensher, C. E. Wieman, and E. A. Cornell, “Temperature-dependent damping and frequency shifts in collective excitations of a dilute Bose-Einstein condensate”, Phys. Rev. Lett. |

9. | M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Observation of interference between two Bose condensates”, Science |

10. | E. A. Burt, R. W. Ghrist, C. J. Myatt, M. J. Holland, E. A. Cornell, and C. E. Wieman, “Coherence, correlations, and collisions: What one learns about Bose-Einstein condensates from their decay”, Phys. Rev. Lett. |

11. | M. -O. Mewes, M. R. Andrews, D. M. Kurn, D. S. Durfee, C. G. Townsend, and W. Ketterle, “Output coupler for Bose-Einstein condensed atoms”, Phys. Rev. Lett. |

12. | M. R. Andrews, D. M. Kurn, H. J. Miesner, D. S. Durfee, C. G. Townsend, S. Inouye, and W. Ketterle, “Propagation of sound in a Bose-Einstein condensate”, Phys. Rev. Lett. |

13. | M. Edwards, P. A. Ruprecht, K. Burnett, R. J. Dodd, and C. W. Clark, “Collective excitations of atomic Bose-Einstein condensates”, Phys. Rev. Lett. |

14. | K. G. Singh and D. S. Rokhsar, “Collective excitations of a confined Bose condensate”, Phys. Rev. Lett. |

15. | S. Stringari, “Collective excitations of a trapped Bose condensed gas”, Phys. Rev. Lett. |

16. | A. L. Fetter, “Ground state and excited states of a confined condensed Bose gas”, Phys. Rev. A |

17. | V. M. Pérez-García, H. Michinel, J. I. Cirac, M. Lewenstein, and P. Zoller, “Low energy excitations of a Bose-Einstein condensate: a variational analysis”, Phys. Rev. Lett. |

18. | Yu. Kagan, E. L. Surkov, and G. V. Shlyapnikov, “Evolution of a Bose-condensed gas under variations of the confining potential”, Phys. Rev. A |

19. | B. D. Esry, “Hartree-Fock theory for Bose-Einstein condensates and the inclusion of correlation effects”, Phys. Rev. A |

20. | Y. Castin and R. Dum, “Bose-Einstein condensation in time dependent traps”, Phys. Rev. Lett. |

21. | A. Griffin, W.-C. Wu, and S. Stringari, “Hydrodynamic modes in a trapped Bose gas above the Bose-Einstein transition”, Phys. Rev. Lett. |

22. | P. Öhberg, E.L. Surkov, I. Tittonen, S. Stenholm, M. Wilkens, and G. V. Shlyapnikov, “Low-energy elementary excitations of a trapped Bose-condensed gas”, (preprint, 5/97). |

23. | A. L. Fetter and D. Rokhsar, “Excited states of a dilute Bose-Einstein condensate in a harmonic trap”, (preprint, 4/97). |

24. | D. A. W. Hutchinson, E. Zaremba, and A. Griffin, “Finite temperature excitations of a trapped Bose gas”, Phys. Rev. Lett. |

25. | M. Edwards, (private communications). |

26. | L. You, W. Hoston, M. Lewenstein, and M. Marinescu, “Low energy excitation spectra of trapped Bose condensates”, Acta Phys. Pol. A,(to appear). |

27. | L. You, W. Hoston, and M. Lewenstein, “Low energy excitations of trapped Bose condensates”, Phys. Rev. A |

28. | M. Fliesser, A. Csordás, R. Graham, and P. Szépfalusy, “Classical quasi-particle dynamics in trapped Bose condensates”, (preprint, cond-mat/9707122). |

29. | M. Fliesser, A. Csordás, P. Szépfalusy, and R. Graham, “Hydrodynamic excitations of Bose condensates in anisotropic traps”, (preprint, cond-mat/9706002). |

30. | R. Walsworth and L. You, “Selective creation of quasi-particles in trapped Bose condensates”, Phys. Rev. A |

31. | M. Lewenstein and L. You, “Quantum phase diffusion of the Bose-Einstein condensate”, Phys. Rev. Lett. |

**OCIS Codes**

(020.0020) Atomic and molecular physics : Atomic and molecular physics

(270.0270) Quantum optics : Quantum optics

**ToC Category:**

Focus Issue: Fluctuations and oscillations of Bose-Einstein

**History**

Original Manuscript: September 11, 1997

Published: November 10, 1997

**Citation**

Li You, Ron Walsworth, and W. Hoston, "Higher energy collective excitations in
trapped Bose condensates," Opt. Express **1**, 293-299 (1997)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-1-10-293

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### References

- M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell,"Observations of Bose-Einstein condensation in a dilute atomic vapor", Science 269, 198 (1995). [CrossRef] [PubMed]
- K.B. Davis, M. -O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, "Bose-Einstein condensations in a gas of sodium atoms", Phys. Rev. Lett. 75, 3969 (1995). [CrossRef] [PubMed]
- C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, "Evidence of Bose-Einstein condensation in an atomic gas with attractive interactions", Phys. Rev. Lett. 75, 1687 (1995). [CrossRef] [PubMed]
- M. R. Andrews, M. -O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, "Direct, nondestructive observation of a Bose condensate", Science 273, 84 (1996). [CrossRef] [PubMed]
- C. C. Bradley, C. A. Sackett, and R. G. Hulet, "Analysis of in situ images of Bose-Einstein condensates of lithium", Phys. Rev. A 55, 3951 (1997). [CrossRef]
- D. S. Jin, J. R. Ensher, M. R. Matthews, C. Wieman, and E. A. Cornell, "Collective excitations of a Bose-Einstein condensate in a dilute gas", Phys. Rev. Lett. 77, 420 (1996). [CrossRef] [PubMed]
- M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. M. Kurn, D. S. Durfee, C. G. Townsend, and W. Ketterle, "Collective excitations of a Bose-Einstein condensate in a magnetic trap", Phys. Rev. Lett. 77, 988 (1996). [CrossRef] [PubMed]
- D. S. Jin, M. R. Matthews, J. R. Ensher, C. E. Wieman, and E. A. Cornell, "Temperature-dependent damping and frequency shifts in collective excitations of a dilute Bose-Einstein condensate", Phys. Rev. Lett. 78, 764 (1997). [CrossRef]
- M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn, and W. Ketterle, "Observation of interference between two Bose condensates", Science 275, 637 (1997). [CrossRef] [PubMed]
- E. A. Burt, R. W. Ghrist, C. J. Myatt, M. J. Holland, E. A. Cornell, and C. E. Wieman, "Coherence, correlations, and collisions: What one learns about Bose-Einstein condensates from their decay", Phys. Rev. Lett. 79, 337 (1997). [CrossRef]
- M.-O. Mewes, M. R. Andrews,D.M.Kurn,D. S. Durfee,C.G.Townsend,andW. Ketterle, "Output coupler for Bose-Einstein condensed atoms", Phys. Rev. Lett. 78, 582 (1997); [CrossRef]
- M. R. Andrews, D. M. Kurn, H. J. Miesner, D. S. Durfee, C. G. Townsend, S. Inouye, and W. Ketterle, "Propagation of sound in a Bose-Einstein condensate", Phys. Rev. Lett. 79, 553 (1997). [CrossRef]
- M. Edwards, P. A. Ruprecht, K. Burnett, R. J. Dodd, and C. W. Clark, "Collective excitations of atomic Bose-Einstein condensates", Phys. Rev. Lett. 77, 1671 (1996). [CrossRef] [PubMed]
- K. G. Singh and D. S. Rokhsar, "Collective excitations of a confined Bose condensate", Phys. Rev. Lett. 77, 1667 (1996). [CrossRef] [PubMed]
- S. Stringari, "Collective excitations of a trapped Bose condensed gas", Phys. Rev. Lett. 77, 2360 (1996). [CrossRef] [PubMed]
- A. L. Fetter, "Ground state and excited states of a confined condensed Bose gas", Phys. Rev. A 53, 4245 (1996). [CrossRef] [PubMed]
- V. M. Perez-Garcia, H. Michinel, J. I. Cirac, M. Lewenstein, and P. Zoller, "Low energy excitations of a Bose-Einstein condensate: a variational analysis", Phys. Rev. Lett. 77, 5230 (1996). [CrossRef]
- Yu. Kagan, E. L. Surkov, and G. V. Shlyapnikov, "Evolution of a Bose-condensed gas under variations of the confining potential", Phys. Rev. A 54, R1753 (1996). [CrossRef] [PubMed]
- B. D. Esry, "Hartree-Fock theory for Bose-Einstein condensates and the inclusion of correlation effects", Phys. Rev. A 55, 1147 (1997). [CrossRef]
- Y. Castin and R. Dum, "Bose-Einstein condensation in time dependent traps", Phys. Rev. Lett. 77, 5315 (1996). [CrossRef] [PubMed]
- A. Grin, W.-C. Wu, and S. Stringari, "Hydrodynamic modes in a trapped Bose gas above the Bose-Einstein transition", Phys. Rev. Lett. 78, 1838 (1997). [CrossRef]
- P. Ohberg, E.L. Surkov, I. Tittonen, S. Stenholm, M. Wilkens, G. V. Shlyapnikov, "Low-energy elementary excitations of a trapped Bose-condensed gas", (preprint, 5/97).
- A. L. Fetter and D. Rokhsar, "Excited states of a dilute Bose-Einstein condensate in a harmonic trap", (preprint, 4/97).
- D. A. W. Hutchinson, E. Zaremba, and A. Grin, "Finite temperature excitations of a trapped Bose gas", Phys. Rev. Lett. 78, 1842 (1997). [CrossRef]
- M. Edwards, (private communications).
- L. You, W. Hoston, M. Lewenstein, and M Marinescu, "Low energy excitation spectra of trapped Bose condensates", Acta Phys. Pol. A,(to appear).
- L. You, W. Hoston, and M. Lewenstein, "Low energy excitations of trapped Bose condensates", Phys. Rev. A 55, R1581 (1997); [CrossRef]
- M. Fliesser, A. Csordas, R. Graham, P. Szepfalusy, "Classical quasi-particle dynamics in trapped Bose condensates", (preprint, cond-mat/9707122).
- M. Fliesser, A. Csordas, P. Szepfalusy, R. Graham, "Hydrodynamic excitations of Bose condensates in anisotropic traps", (preprint, cond-mat/9706002).
- R. Walsworth and L. You, "Selective creation of quasi-particles in trapped Bose condensates", Phys. Rev. A 55, 555 (1997). [CrossRef]
- M. Lewenstein and L. You, "Quantum phase diffusion of the Bose-Einstein condensate", Phys. Rev. Lett. 77, 3489 (1996). [CrossRef] [PubMed]

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