## Multi-mode description of an interacting Bose-Einstein condensate

Optics Express, Vol. 8, Issue 2, pp. 92-98 (2001)

http://dx.doi.org/10.1364/OE.8.000092

Acrobat PDF (207 KB)

### Abstract

We study the equilibrium dynamics of a weakly interacting Bose-Einstein condensate trapped in a box. In our approach we use a semiclassical approximation similar to the description of a multi-mode laser. In dynamical equations derived from a full *N*-body quantum Hamiltonian we substitute all creation (and annihilation) operators (of a particle in a given box state) by appropriate c-number amplitudes. The set of nonlinear equations obtained in this way is solved numerically. We show that on the time scale of a few miliseconds the system exhibits relaxation – reaches an equilibrium with populations of different eigenstates fluctuating around their mean values.

© Optical Society of America

1. M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, and E.A. Cornell, “Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor,” Science **269**, 198–201 (1995). [CrossRef] [PubMed]

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 condensation in a gas of sodium atoms,” Phys. Rev. Lett. **75**, 3969–3972 (1995). [CrossRef] [PubMed]

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.75, 1687–1690 (1995) and Erratum 79, 1170(E) (1997). [CrossRef] [PubMed]

4. D.G. Fried, T.C. Killian, L. Willmann, D. Landhuis, S.C. Moss, D. Kleppner, and T.J. Greytak, “Bose-Einstein condensation of atomic hydrogen,” Phys. Rev. Lett. **81**, 3811–3814 (1998). [CrossRef]

5. P. Navez, D. Bitouk, M. Gajda, Z. Idziaszek, and K. Rzążewski, “Fourth statistical ensemble for the Bose-Einstein condensate,” Phys. Rev. Lett. **79**, 1789–1792 (1997). [CrossRef]

6. M. Gajda and K. Rzążewski, “Fluctuations of Bose-Einstein condensate,” Phys. Rev. Lett. **78**, 2686–2689 (1997). [CrossRef]

7. S. Grossmann and M. Holthaus, “Fluctuations of the particle number in a trapped Bose-Einstein condensate,” Phys. Rev. Lett. **79**, 3557–3560 (1997). [CrossRef]

8. S. Grossmann and M. Holthaus, “Maxwell’s Demon at work: Two types of Bose condensate fluctuations in power-law traps,” Opt. Express **1**, 262–271 (1997), http://www.opticsexpress.org/oearchive/source/2288.htm [CrossRef] [PubMed]

9. H. D. Politzer, “Condensate fluctuations of a trapped, ideal Bose gas,” Phys. Rev. A **54**, 5048–5054 (1996). [CrossRef] [PubMed]

10. M. Wilkens and C. Weiss, “Particle number fluctuations in an ideal Bose gas,” J. Mod. Opt. **44**, 1801–1814 (1997). [CrossRef]

11. M. Wilkens and C. Weiss, “Particle number counting statistics in ideal Bose gases,” Opt. Express **1**, 272–283 (1997), http://www.opticsexpress.org/oearchive/source/2372.htm [CrossRef] [PubMed]

12. S. Giorgini, L.P. Pitaevskii, and S. Stringari, “Anomalous fluctuations of the condensate in interacting Bose gases,” Phys. Rev. Lett **80**, 5040–5043 (1998). [CrossRef]

13. Z. Idziaszek, M. Gajda, P. Navez, M. Wilkens, and K. Rzążewski, “Fluctuations of the weakly interacting Bose-Einstein condensate,” Phys. Rev. Lett. **82**, 4376–4379 (1999). [CrossRef]

14. F. Meier and W. Zwerger, “Anomalous condensate fluctuations in strongly interacting superfluids,” Phys. Rev. A **60**, 5133–5135 (1999). [CrossRef]

15. V.V. Kocharovsky, V.V. Kocharovsky, and M.O. Scully, “Condensate statistics in interacting and ideal dilute Bose gases,” Phys. Rev. Lett. **84**, 2306–2309 (2000). [CrossRef] [PubMed]

16. F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose-Einstein condensation in trapped gases,” Rev. Mod. Phys. **71**, 463–512 (1999). [CrossRef]

17. R. Graham, “Condensate fluctuations in finite Bose-Einstein condensates at finite temperature,” Phys. Rev. A **62**, 023609 (2000). [CrossRef]

18. R. Graham, “Decoherence of Bose-Einstein condensates in traps at finite temperature,” Phys. Rev. Lett. **81**, 5262–5265 (1998). [CrossRef]

19. C.W. Gardiner and P. Zoller, “Quantum kinetic theory: A quantum kinetic master equation for condensation of a weakly interacting Bose gas without a trapping potential,” Phys. Rev. A **55**, 2902–2921 (1997). [CrossRef]

20. D. Jaksch, C.W. Gardiner, and P. Zoller, “Quantum kinetic theory. 2.Simulation of the quantum Boltzmann master equation,” Phys. Rev. A **56**, 575–586 (1997). [CrossRef]

21. R. Walser, J. Williams, J. Cooper, and M. Holland, “Quantum kinetic theory for a condensed bosonic gas,” Phys. Rev. A **59**, 3878–3889 (1999). [CrossRef]

22. R. Walser, J. Williams, and M. Holland, “Reversible and irreversible evolution of a condensed bosonic gas,” preprint cond-mat/0004257, http://xxx.lanl.gov/abs/cond-mat/0004257

*V*=

*L*

^{3}is a volume of the system (

*L*being a size of the box) and

*a*being the scattering length and

*m*– the mass of the atom). The field Φ is expanded in natural modes of the system – the plane waves:

*a*

_{k}are bosonic annihilation operators, and

*n*

_{i}=0,±1,±2,… (

*i*=

*x,y,z*). With this substitution the Hamiltonian assumes its final form:

*a*

_{k}=exp(-

*iξn*

^{2}

*t*)

*α*

_{k}, the Heisenberg equations of motion for the operators

*α*

_{k}acquire the following form:

*α*

_{k}by c-number complex amplitudes (we are not going to introduce a separate notation for corresponding complex fields). At very low temperatures only the lowest lying states are macroscopically occupied and quantum fluctuations in excited states become important (see [21

21. R. Walser, J. Williams, J. Cooper, and M. Holland, “Quantum kinetic theory for a condensed bosonic gas,” Phys. Rev. A **59**, 3878–3889 (1999). [CrossRef]

*N*=∑

_{k}

_{k}, as well as the total energy of the system. It therefore corresponds to a genuine microcanonical description. Moreover, the resulting equations resemble the famous Fermi-Pasta-Ulam [25] problem of a system of harmonic oscillators coupled by a nonlinear interaction. A one-dimensional version of this dynamics has been studied recently [2

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 condensation in a gas of sodium atoms,” Phys. Rev. Lett. **75**, 3969–3972 (1995). [CrossRef] [PubMed]

*T*=0) reloaded from a harmonic into a rectangular trap. Equations studied here, however, in spite of a formal analogy, describe quite a different physical situation. Our complex amplitudes are not expansion coefficients of the condensate wave function in some convenient basis. They represent a number of coupled “mean fields” – a natural extension of the condensate mean field of the Bogolubov approach. Let us notice that if we start with 100% occupation of the k=0 mode and keep only this mode in the model we simply recover the standard Gross-Pitaevskii equation for the interacting condensate.

*ξ*=71.373 Hz,

*N*=10

^{5}and

*g*=0.018 Hz (the atomic mass and the scattering length are those of

^{87}Rb and the size of the box is equal to the Thomas-Fermi radius of a condensate of

*N*atoms in a trap with frequency of

*ω*

_{0}=2π 80 Hz). We performed our calculations for the model with 729 modes (

*n*

_{i}=-4,…,4,

*i*=

*x,y,z*). Further increasing of the number of modes does not lead to a substantial change in the results for the case studied in this paper. Our calculations show that after a time of the order of a few miliseconds the system reaches a dynamical equilibrium. The mean occupation of the condensate (k=0 mode) stabilizes at some value and on larger time scales (of the order of a second) it only fluctuates around this mean value – see Figure 1.

28. J.H. Eberly, N.B. Narozhny, and J.J. Sanchez-Mondragon, “Periodic spontaneous collapse and revival in a simple quantum model,” Phys. Rev. Lett. **44**, 1323–1326 (1980). [CrossRef]

*revivals*in the system proves the quantum nature of the electromagnetic field. In our calculations the largest time scale for which we have studied the dynamics was of the order of one second. On this time scale we did not observe any revivals but this, obviously, does not exclude the possibility of revivals on much larger time scales. In fact our numerical simulations involve a finite number of modes and so the numerical implementation inevitably leads to a quasiperiodic evolution.

*N*=10

^{5}) and the energy of the system was the control parameter. Traditionally, however, the temperature, not the energy, is used as an independent thermodynamic variable. Calculation of the microcanonical temperature requires monitoring of the entropy of the system for different energies. Although, in principle, this can be done [29

29. In a recent preprint M.J. Davis, S.A. Morgan, and K. Burnett, “Simulations of Bose fields at finite temperature,” preprint cond-mat/0011431, http://xxx.lanl.gov/abs/cond-mat/0011431, using similar methods, the authors establish a link between the energy and the temperature for temperatures below the critical region.

*time-averaged*) occupation of the Bose-Einstein condensate, (ii) the spread of these distributions around the most probable value is a measure of condensate fluctuations. The figure clearly indicates that the condensate population decreases monotonically with energy while fluctuations become larger.

*E*/

*ħ*=2000 Hz.

*N*=10

^{5}atoms. Let us notice, however, that fluctuations reach the maximum value at the energy per particle close to

*E*/

*ħ*=1450

*Hz*. Moreover, this value of energy corresponds to the inflection point of the mean occupation of the condensate. Both curves distinguish the same characteristic value of the energy. Close to this energy the system undergoes rapid changes. This characteristic energy corresponds to the critical energy for the Bose-Einstein condensation.

## Acknowledgements

## References and links

1. | M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, and E.A. Cornell, “Observation 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 condensation 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.75, 1687–1690 (1995) and Erratum 79, 1170(E) (1997). [CrossRef] [PubMed] |

4. | D.G. Fried, T.C. Killian, L. Willmann, D. Landhuis, S.C. Moss, D. Kleppner, and T.J. Greytak, “Bose-Einstein condensation of atomic hydrogen,” Phys. Rev. Lett. |

5. | P. Navez, D. Bitouk, M. Gajda, Z. Idziaszek, and K. Rzążewski, “Fourth statistical ensemble for the Bose-Einstein condensate,” Phys. Rev. Lett. |

6. | M. Gajda and K. Rzążewski, “Fluctuations of Bose-Einstein condensate,” Phys. Rev. Lett. |

7. | S. Grossmann and M. Holthaus, “Fluctuations of the particle number in a trapped Bose-Einstein condensate,” Phys. Rev. Lett. |

8. | S. Grossmann and M. Holthaus, “Maxwell’s Demon at work: Two types of Bose condensate fluctuations in power-law traps,” Opt. Express |

9. | H. D. Politzer, “Condensate fluctuations of a trapped, ideal Bose gas,” Phys. Rev. A |

10. | M. Wilkens and C. Weiss, “Particle number fluctuations in an ideal Bose gas,” J. Mod. Opt. |

11. | M. Wilkens and C. Weiss, “Particle number counting statistics in ideal Bose gases,” Opt. Express |

12. | S. Giorgini, L.P. Pitaevskii, and S. Stringari, “Anomalous fluctuations of the condensate in interacting Bose gases,” Phys. Rev. Lett |

13. | Z. Idziaszek, M. Gajda, P. Navez, M. Wilkens, and K. Rzążewski, “Fluctuations of the weakly interacting Bose-Einstein condensate,” Phys. Rev. Lett. |

14. | F. Meier and W. Zwerger, “Anomalous condensate fluctuations in strongly interacting superfluids,” Phys. Rev. A |

15. | V.V. Kocharovsky, V.V. Kocharovsky, and M.O. Scully, “Condensate statistics in interacting and ideal dilute Bose gases,” Phys. Rev. Lett. |

16. | F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose-Einstein condensation in trapped gases,” Rev. Mod. Phys. |

17. | R. Graham, “Condensate fluctuations in finite Bose-Einstein condensates at finite temperature,” Phys. Rev. A |

18. | R. Graham, “Decoherence of Bose-Einstein condensates in traps at finite temperature,” Phys. Rev. Lett. |

19. | C.W. Gardiner and P. Zoller, “Quantum kinetic theory: A quantum kinetic master equation for condensation of a weakly interacting Bose gas without a trapping potential,” Phys. Rev. A |

20. | D. Jaksch, C.W. Gardiner, and P. Zoller, “Quantum kinetic theory. 2.Simulation of the quantum Boltzmann master equation,” Phys. Rev. A |

21. | R. Walser, J. Williams, J. Cooper, and M. Holland, “Quantum kinetic theory for a condensed bosonic gas,” Phys. Rev. A |

22. | R. Walser, J. Williams, and M. Holland, “Reversible and irreversible evolution of a condensed bosonic gas,” preprint cond-mat/0004257, http://xxx.lanl.gov/abs/cond-mat/0004257 |

23. | The case of boundary conditions different from the periodic ones (e.g. a rectangular trap) presents an interesting and challenging problem. In this case there are no universal eigenstates of a one-particle density matrix and therefore the definition of a condensate is unclear. |

24. | A.L. Fetter and J.D. Walecka, |

25. | E. Fermi, J. Pasta, and S. Ulam, “Studies of Nonlinear Problems. I,” in |

26. | P. Villain and M. Lewenstein, “Fermi-Pasta-Ulam problem revisited with a Bose-Einstein condensate,” Phys. Rev. A |

27. | F.M. Izrailev and B.V. Chirikov, “Statistical properties of a nonlinear string,” Dokl. Akad. Nauk SSSR166, 57–59 (1966) [Sov. Phys. Dokl.11, 30–32 (1966)]. |

28. | J.H. Eberly, N.B. Narozhny, and J.J. Sanchez-Mondragon, “Periodic spontaneous collapse and revival in a simple quantum model,” Phys. Rev. Lett. |

29. | In a recent preprint M.J. Davis, S.A. Morgan, and K. Burnett, “Simulations of Bose fields at finite temperature,” preprint cond-mat/0011431, http://xxx.lanl.gov/abs/cond-mat/0011431, using similar methods, the authors establish a link between the energy and the temperature for temperatures below the critical region. |

30. | K. Góral, M. Gajda, and K. Rzążewski, “Multi-mode dynamics of a coupled ultracold atomic-molecular system,” preprint cond-mat/0006192, http://xxx.lanl.gov/abs/cond-mat/0006192 |

**OCIS Codes**

(000.6590) General : Statistical mechanics

(000.6800) General : Theoretical physics

(270.0270) Quantum optics : Quantum optics

(270.2500) Quantum optics : Fluctuations, relaxations, and noise

**ToC Category:**

Focus Issue: Quantum control of photons and matter

**History**

Original Manuscript: November 9, 2000

Published: January 15, 2001

**Citation**

Krzysztof Goral, Mariusz Gajda, and Kazimierz Rzazewski, "Multi-mode description of an interacting Bose-Einstein condensate," Opt. Express **8**, 92-98 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-8-2-92

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### References

- M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, and E.A. Cornell, "Observation of Bose Einstein Condensation in a Dilute Atomic Vapor," Science 269, 198-201 (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 condensation in a gas of sodium atoms," Phys. Rev. Lett. 75, 3969-3972 (1995). [CrossRef] [PubMed]
- C.C. Bradle , 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, 1687-1690 (1995) and Erratum 79, 1170(E) (1997). [CrossRef] [PubMed]
- D.G. Fried, T.C. Killian, L. Willmann, D. Landhuis, S.C. Moss, D. Kleppner, and T.J. Greytak, "Bose Einstein condensation of atomic hydrogen," Phys. Rev. Lett. 81, 3811-3814 (1998). [CrossRef]
- P. Navez, D. Bitouk, M. Gajda, Z. Idziaszek, and K. Rzazewski, "Fourth statistical ensemble for the Bose Einstein condensate," Phys. Rev. Lett. 79, 1789-1792 (1997). [CrossRef]
- M. Gajda and K. Rzazewski, "Fluctuations of Bose Einstein condensate," Phys. Rev. Lett. 78, 2686-2689 (1997). [CrossRef]
- S. Grossmann and M. Holthaus, "Fluctuations of the particle number in a trapped Bose Einstein condensate," Phys. Rev. Lett. 79, 3557-3560 (1997). [CrossRef]
- S. Grossmann and M. Holthaus, "Maxwell's Demon at work: Two t pes of Bose condensate fluctuations in power law traps," Opt. Express 1, 262-271 (1997), http://www.opticsexpress.org/oearchive/source/2288.htm [CrossRef] [PubMed]
- H. D. Politzer, "Condensate fluctuations of a trapped, ideal Bose gas," Phys. Rev. A 54, 5048-5054 (1996). [CrossRef] [PubMed]
- M. Wilkens and C. Weiss, "Particle number fluctuations in an ideal Bose gas," J. Mod. Opt. 44, 1801-1814 (1997). [CrossRef]
- M. Wilkens and C. Weiss, "Particle number counting statistics in ideal Bose gases," Opt. Express 1, 272-283 (1997), http://www.opticsexpress.org/oearchive/source/2372.htm [CrossRef] [PubMed]
- S. Giorgini, L.P. Pitaevskii, and S. Stringari, "Anomalous fluctuations of the condensate in interacting Bose gases," Phys. Rev. Lett 80, 5040-5043 (1998). [CrossRef]
- Z.Idziaszek, M.Gajda, P. Navez, M. Wilkens, and K.Rzazewski, "Fluctuations of the weakly interacting Bose Einstein condensate," Phys. Rev. Lett. 82, 4376-4379 (1999). [CrossRef]
- F. Meier and W. Zwerger, "Anomalous condensate fluctuations in strongly interacting superfluids," Phys. Rev. A 60, 5133-5135 (1999). [CrossRef]
- V.V. Kocharovsk , V.V. Kocharovsk , and M.O. Scull , "Condensate statistics in interacting and ideal dilute Bose gases," Phys. Rev. Lett. 84, 2306-2309 (2000). [CrossRef] [PubMed]
- F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, "Theory of Bose Einstein condensation in trapped gases," Rev. Mod. Phys. 71, 463-512 (1999). [CrossRef]
- R. Graham, "Condensate fluctuations in finite Bose Einstein condensates at finite temperature," Phys. Rev. A 62, 023609 (2000). [CrossRef]
- R. Graham, "Decoherence of Bose Einstein condensates in traps at finite temperature," Phys. Rev. Lett. 81, 5262-5265 (1998). [CrossRef]
- C.W. Gardiner and P. Zoller, "Quantum kinetic theory: A quantum kinetic master equation for condensation of a weakly interacting Bose gas without a trapping potential," Phys. Rev. A 55, 2902-2921 (1997). [CrossRef]
- D. Jaksch, C.W. Gardiner, and P. Zoller, "Quantum kinetic theory. 2.Simulation of the quantum Boltzmann master equation," Phys. Rev. A 56, 575- 586 (1997). [CrossRef]
- R. Walser, J. Williams, J. Cooper, and M. Holland, "Quantum kinetic theory for a condensed bosonic gas," Phys. Rev. A 59, 3878-3889 (1999). [CrossRef]
- R. Walser, J. Williams, and M. Holland, "Reversible and irreversible evolution of a condensed bosonic gas," preprint cond mat/0004257, http://xxx.lanl.gov/abs/cond mat/0004257
- The case of boundary conditions different from the periodic ones (e.g. a rectangular trap) presents an interesting and challenging problem. In this case there are no universal eigenstates of a one-particle density matrix and therefore the definition of a condensate is unclear.
- A.L. Fetter and J.D.Walecka, Quantum theory of many-particle systems (McGraw Hill, New York, 1991).
- E. Fermi, J. Pasta, and S. Ulam, "Studies of Nonlinear Problems. I," in Collected Papers of Enrico Fermi (Accademia Nazionale dei Lincei and University of Chicago, Roma, 1965), Vol. II, p. 978.
- P. Villain and M. Lewenstein, "Fermi Pasta Ulam problem revisited with a Bose Einstein condensate," Phys. Rev. A 62, 043601 (2000). [CrossRef]
- F.M. Izrailev and B.V. Chirikov, "Statistical properties of a nonlinear string," Dokl. Akad. Nauk SSSR 166, 57-59 (1966) [Sov. Phys. Dokl. 11, 30-32 (1966)].
- J.H. Eberly, N.B. Narozhny, and J.J. Sanchez Mondragon, "Periodic spontaneous collapse and revival in a simple quantum model," Phys. Rev. Lett. 44, 1323-1326 (1980). [CrossRef]
- In a recent preprint M.J. Davis, S.A. Morgan, and K. Burnett, "Simulations of Bose fields at finite temperature," preprint cond-mat/0011431, http://xxx.lanl.gov/abs/cond mat/0011431, using similar methods, the authors establish a link between the energy and the temperature for temperatures below the critical region.
- K. G�ral, M.Gajda, and K. Rzazewski, "Multi-mode dynamics of a coupled ultracold atomic molecular system," preprint cond-mat/0006192, http://xxx.lanl.gov/abs/cond mat/0006192

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