## On coherence of Bose field

Optics Express, Vol. 8, Issue 2, pp. 106-111 (2001)

http://dx.doi.org/10.1364/OE.8.000106

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### Abstract

We study a system of interacting bosons at zero temperature in an atomic trap. Using wave function that models the ground state of interacting bosons we examine the concepts of the order parameter, off-diagonal order and coherence of the system. We suggest that the coherence length becomes much smaller than the size of the system if the number of trapped particles exceeds a certain limit. This behavior is related to the unavoidable existence of two different length scales – one determined by the external potential and the second one depending on the two-body forces.

© Optical Society of America

1. D. Kleppner, “A Beginner’s Guide to the Atom Laser,” Phys. Today , **8**, 11–13 (1997). [CrossRef]

2. M. R. Andrews*et. al*. “Observation of interference between two Bose condensates,” Science **275**, 637–641 (1997). [CrossRef] [PubMed]

3. J. Stenger*et al*. “Bragg spectroscopy of a Bose-Einstein condensate,” Phys. Rev. Lett. **82**, 4569–4573 (1998). [CrossRef]

4. E. W. Hagley*et al*. “Measurement of coherence of a Bose-Einstein condensate,” Phys. Rev. Lett. **83**, 312–315 (1999). [CrossRef]

5. I. Bloch, T. W. Hänsch, and T. Esslinger, “Measurement of the spatial coherence of a trapped Bose gas at the phase transition,” Nature **403**, 166–170 (2000). [CrossRef] [PubMed]

7. O. Penrose and L. Onsager, “Bose-Einstein condensation and liquid helium,” Phys. Rev. **104**, 576–584 (1956). [CrossRef]

9. C.N. Yang, “Concept of off-diagonal long-range order and quantum phases of liquid He and of superconductors,” Rev. Mod. Phys. **34**, 694–704 (1962) [CrossRef]

10. J. Javanainen and S. M. Yoo, “Quantum Phase of a Bose-Einstein Condensate with an Arbitrary Number of Atoms,” Phys. Rev. Lett. **76**, 161–164 (1996). [CrossRef] [PubMed]

11. S. M. Barnett, K. Burnett, and J.A. Vaccaro, “Why a condensate can be thought of as having a definite phase,” J. Res. Natl. Inst. Stan. **101**593–600 (1996). [CrossRef]

*N*-body state with a fixed energy and particle number.

*N*-particle wave function which corresponds to the ground state of bosonic system:

*x*

_{i}is the position of the

*i*–

*th*particle and

*x*

_{CM}=

_{i}/√

*N*is the collective variable proportional to the center-off-mass coordinate. Let us list some important properties of this wave function:

*N*-fold product (unless

*ω*=1) of one-particle states. This feature signifies an interacting system.

*N*– 1 independent internal degrees of freedom only (the second term). The relative-coordinates wave function is transitionally invariant. These features are generic for any two-body central forces.

*ω*=1 signifies the ideal gas case and the function given in Eq.(1) becomes an

*N*-fold product of identical gaussian functions. For attractive forces

*ω*>1 and the center-off-mass extension is larger than the spatial extension of the internal coordinates. Repulsive forces are characterized by 0<

*ω*<1. In such a case the center-off-mass length scale is smaller than the length scale of the relative motion.

*N*– 1 normal modes has its own characteristic length scale. In addition, it is easy to check that this wave function is the ground state of the Hamiltonian with quadratic two-body interactions, i.e. it corresponds to a long distance forces. The solution to this many body problem as well as analysis of some properties of the Bose condensate interacting by two-body harmonic forces are presented in [12

12. M. ZaFluska-Kotur, M. Gajda, A. OrFlowski, and J. Mostowski, “Soluble model of many interacting quantum particles in a trap,” Phys. Rev. A **61**, 033613–8 (2000). [CrossRef]

13. M. Gajda, M. ZaFluska-Kotur, and J. Mostowski, “Destruction of a Bose-Einstein condensate by strong interactions,” J.Phys.B: At. Mol. Opt. Phys. **33**4003–4016 (2000). [CrossRef]

7. O. Penrose and L. Onsager, “Bose-Einstein condensation and liquid helium,” Phys. Rev. **104**, 576–584 (1956). [CrossRef]

*by Bose condensate we mean a bosonic many-body system under such a special situation in which quantum mechanics will produce its own characteristic effects on a large or “macroscopic” scale*- the scale of a spatial extension of the system. Being more specific we may say that coherence is the feature which, similarly as in the case of laser light, proves (or disproves) quantum character of a Bose system.

*ρ*1(x; y)=∫d

**r**

_{2},…, r

_{N}Ψ*(x,

**r**

_{2},…,

**r**

_{N})Ψ(y,

**r**

_{2},…

**r**

_{N}). Similarly, one can define s-particle reduced matrices for s larger than one. Although direct expressions for these matrices can be easily obtained [13

13. M. Gajda, M. ZaFluska-Kotur, and J. Mostowski, “Destruction of a Bose-Einstein condensate by strong interactions,” J.Phys.B: At. Mol. Opt. Phys. **33**4003–4016 (2000). [CrossRef]

*ρ*1(

**r; r**) is equal to a one-particle density. Similarly, the

*ρ*2(

**r**

_{1},

**r**

_{2};

**r**

_{1},

**r**

_{2}) is the probability density of a joint detection of one particle at

**r**

_{1}and the other at

**r**

_{2}.

_{1},…x

_{s}) and eigenvalues

*s*-particle “orbitals” and their mean populations. The

*s*-particle subsystem is an open system because it interacts with

*N*–

*s*particle “reservoir”. Therefore it is typically in a mixed state. However, there are some particular situations when the s-particle subsystem is in a pure state, i.e

*ρ*1(x; y)≈[

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

*ρ*

_{1}(x; y)≠0

*ρ*1(x; x), (ii) and

*ρ*

_{1}(x; y). The first scale is approximately equal to larger of the two:

*ℓ*

_{CM}and

*ℓ*

_{int}. It determines the region (around the trap center) where an individual particle can be effectively detected. The second scale is associated with an extension of region of a “one-particle phase correlation”. The off-diagonal long-range order or, in other words, the long range coherence is present in the system if the spatial extension of the off-diagonal elements is at least of the size of the spatial extension of the particle density, i.e. if

*N*≫1) the one-particle state is macroscopically occupied if |

*κ*|=|log

*ω*|/log

*N*<1. The latter inequality defines the weak interaction limit of our model. Interestingly, in the opposite case of strong interactions the one-particle density matrix has a form characteristic for a high temperature thermal state. Note however, that the total system is in its ground state given by a wave function, see Eq.(1). Let us add that in the case of trapped atomic gases a relative occupation of the condensate at zero temperature is estimated to be about

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

^{4}He the relative occupation of the condensate is only of the order of a few percent [16].

*κ*|<1. For the coherent state of electromagnetic field similar equalities hold for all orders of coherence functions. This cannot be true for Bose systems of particles with mass. Such systems are in a Fock state (state with a fixed number of particles). Superselection rules forbid to superpose states with different particle numbers. In the case of noninteracting system equality of the type of the Eq.(5) holds up to the

*N*-th order of coherence[20

20. R.M. Ziff, G.E. Uhlenbeck, and M. Kac, “The ideal Bose-Einstein gas, revisited”, *Phys. Rep*.32C, 169–248 (1977). [CrossRef]

*g*2 introduces, similarly as

*g*1, at least two characteristic length scales. The first one

**r**). The spatial extension

*R*

_{N}of this one-particle wave function grows with

*N*to infinity and in the Thomas-Fermi limit it can be approximated by

*R*

_{N}=(15

*Na*

_{s})

^{1/5}[15

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

*a*

_{s}is the scattering length measured in units of the external harmonic trap extension

*ℓ*

_{CM}. Then the

*N*-particle system can be described as the

*N*-fold product of the order parameter and the central limit theorem leads to the following probability distribution of the center-of-mass variable

*x*

_{CM}=1/√

*N*∑

*N*

_{i=1}

**r**

_{i}:

*N*-body solution gives

*N*-independent size of the center-of-mass probability distribution

*N*) cannot exhibit high order coherences over its total spatial extension. High order coherence may exists on a smaller scale only. Such system cannot be easily distinguished from the filtered (i.e. monochromatic) beam of atoms produced by a chaotic source and the notion of the Bose-Einstein condensate can be ambiguous.

## References and links

1. | D. Kleppner, “A Beginner’s Guide to the Atom Laser,” Phys. Today , |

2. | M. R. Andrews |

3. | J. Stenger |

4. | E. W. Hagley |

5. | I. Bloch, T. W. Hänsch, and T. Esslinger, “Measurement of the spatial coherence of a trapped Bose gas at the phase transition,” Nature |

6. | O. Penrose, “On the quantum mechanics of helium II,” Phil. Mag. |

7. | O. Penrose and L. Onsager, “Bose-Einstein condensation and liquid helium,” Phys. Rev. |

8. | S. T. Beliaev, “Application of the method of quantum field theory to a system of bosons,” J. Exp. Theor. Phys. (USSR) |

9. | C.N. Yang, “Concept of off-diagonal long-range order and quantum phases of liquid He and of superconductors,” Rev. Mod. Phys. |

10. | J. Javanainen and S. M. Yoo, “Quantum Phase of a Bose-Einstein Condensate with an Arbitrary Number of Atoms,” Phys. Rev. Lett. |

11. | S. M. Barnett, K. Burnett, and J.A. Vaccaro, “Why a condensate can be thought of as having a definite phase,” J. Res. Natl. Inst. Stan. |

12. | M. ZaFluska-Kotur, M. Gajda, A. OrFlowski, and J. Mostowski, “Soluble model of many interacting quantum particles in a trap,” Phys. Rev. A |

13. | M. Gajda, M. ZaFluska-Kotur, and J. Mostowski, “Destruction of a Bose-Einstein condensate by strong interactions,” J.Phys.B: At. Mol. Opt. Phys. |

14. | R. P. Feynman, “The Feynman lectures on physics” vol. III, (Addison-Wesley, 1965). |

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

16. | K. Huang, “Statistical Mechanics,” (Wiley, New York, 1987). |

17. | D. F. Walls, “Evidence for the quantum nature of light,” Nature |

18. | R.J. Dodd, C.W. Clark, M. Edwards, and K. Burnett, “Characterizing the coherence of Bose-Einstein condensates and atom lasers,” Optics Express |

19. | R. J. Glauber, “Quantum Optic and photon statistics” in Quantum Optics and Electronics, C. De-Witt, A. Blandin, and C. Cohen-Tannoudji, eds. (Gordon and Breach, New York, 1965). |

20. | R.M. Ziff, G.E. Uhlenbeck, and M. Kac, “The ideal Bose-Einstein gas, revisited”, |

21. | C. J. Pethick and L. P. Pitaevskii, “On the criterion for Bose-Einstein condensaton for particles in trap,” preprint cond-mat/0004187. http://xxx.lanl.gov/abs/cond-mat/0004187 |

**OCIS Codes**

(000.6590) General : Statistical mechanics

(000.6800) General : Theoretical physics

(020.7010) Atomic and molecular physics : Laser trapping

(030.1640) Coherence and statistical optics : Coherence

(270.0270) Quantum optics : Quantum optics

**ToC Category:**

Focus Issue: Quantum control of photons and matter

**History**

Original Manuscript: November 9, 2000

Published: January 15, 2001

**Citation**

Mariusz Gajda, Magdalena Zaluska-Kotur, and Jan Mostowski, "On coherence of Bose field," Opt. Express **8**, 106-111 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-8-2-106

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### References

- D. Kleppner, "A Beginner's Guide to the Atom Laser," Phys. Today 8, 11-13 (1997). [CrossRef]
- M. R. Andrews et. al. "Observation of interference between two Bose condensates," Science 275, 637-641 (1997). [CrossRef] [PubMed]
- J. Stenger et al. "Bragg spectroscopy of a Bose Einstein condensate," Phys. Rev. Lett. 82, 4569-4573 (1998). [CrossRef]
- E. W. Hagle et al. "Measurement of coherence of a Bose Einstein condensate," Phys. Rev. Lett. 83, 312-315 (1999). [CrossRef]
- I. Bloch, T. W. H�nsch, and T. Esslinger, "Measurement of the spatial coherence of a trapped Bose gas at the phase transition," Nature 403, 166-170 (2000). [CrossRef] [PubMed]
- O. Penrose, "On the quantum mechanics of helium II," Phil. Mag. 42, 1373-1377 (1951).
- O. Penrose, L. Onsager, "Bose Einstein condensation and liquid helium," Phys. Rev. 104, 576-584 (1956). [CrossRef]
- S. T. Beliaev, "Application of the method of quantum field theory to a system of bosons," J. Exp. Theor. Phys. (USSR) 34, 417-432 (1958).
- C.N. Yang, "Concept of off diagonal long range order and quantum phases of liquid He and of superconductors," Rev. Mod. Phys. 34, 694-704 (1962). [CrossRef]
- J. Javanainen and S. M. Yoo, "Quantum Phase of a Bose Einstein Condensate with an Arbitrary Number of Atoms," Phys. Rev. Lett. 76, 161-164 (1996). [CrossRef] [PubMed]
- S. M .Barnett , K. Burnett, J.A. Vaccaro, "Why a condensate can be thought of as having a definite phase," J. Res. Natl. Inst. Stan. 101 593-600 (1996). [CrossRef]
- M. Zaluska-Kotur, M. Gajda, A. Orlowski, and J. Mostowski, "Soluble model of many interacting quantum particles in a trap," Phys. Rev. A 61, 033613-8 (2000). [CrossRef]
- M.Gajda, M. Zaluska-Kotur, and J. Mostowski, "Destruction of a Bose Einstein condensate by strong interactions," J. Phys. B: At. Mol. Opt. Phys. 33 4003-4016 (2000). [CrossRef]
- R. P. Feynman, "The Feynman lectures on physics" vol. III, (Addison Wesley, 1965).
- F. Dalfovo, S. Giorgini, L. Pitaevskii, S. Stringari, "Theory of Bose Einstein condensation in trapped gases," Rev. Mod. Phys. 71, 463-512 (1999). [CrossRef]
- K.Huang, "Statistical Mechanics," (Wiley, New York, 1987).
- D. F. Walls, "Evidence for the quantum nature of light," Nature 280, 451 (1979). [CrossRef]
- R. J. Dodd, C. W. Clark, M. Edwards, and K. Burnett, "Characterizing the coherence of Bose Einstein condensates and atom lasers," Opt. Express 1, 284-292 (1997). http://www.opticsexpress.org/oearchive/source/2369.htm [CrossRef] [PubMed]
- R. J. Glauber, "Quantum Optic and photon statistics" in Quantum Optics and Electronics, C. De Witt, A. Blandin, and C. Cohen Tannoudji, eds. (Gordon and Breach, New York, 1965).
- R. M. Ziff, G. E. Uhlenbeck, M. Kac, "The ideal Bose Einstein gas, revisited", Phys. Rep. 32C, 169-248 (1977). [CrossRef]
- C. J. Pethick and L. P. Pitaevskii, "On the criterion for Bose Einstein condensaton for particles in trap," preprint cond mat/0004187. http://xxx.lanl.gov/abs/cond mat/0004187

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