Geodesic
Quantum Integrable and Nonintegrable Nonlinear Schrödinger Models for Realizable Bose–Einstein Condensation in $d+1$ Dimensions $(d=1,2,3)$
Teoretičeskaâ i matematičeskaâ fizika, Tome 134 (2003) no. 1, pp. 55-73 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We evaluate finite-temperature equilibrium correlators корреляторы $\langle T_\tau \hat{\psi}({\bold r}_1) \hat{\psi}^\dagger({\bold r}_2)\rangle$ for thermal time $\tau$ ordered Bose fields полей $\hat{\psi}$, $\hat{\psi}^\dagger$ to good approximations by new methods of functional integration in $d=1,2,3$ dimensions and with the trap potentials $V({\bold r})\not\equiv0$. As in the translationally invariant cases, asymptotic behaviors fall as $R^{-1}\equiv|{\bold r}_1-{\bold r}_2|^{-1}$ to longer-range condensate values for and only for $d=3$ in agreement with experimental observations; but there are generally significant corrections also depending on ${\bold S}\equiv({\bold r}_1+{\bold r}_2)/2$ due to the presence of the traps. For $d=1$, we regain the exact translationally invariant results as the trap frequencies $\Omega\rightarrow0$. In analyzing the attractive cases, we investigate the time-dependent $c$-number Gross–Pitaevskii (GP) equation with the trap potential for a generalized nonlinearity $-2c\psi|\psi|^{2n}$ and $c<0$. For $n=1$, the stationary form of the GP equation appears in the steepest-descent approximation of the functional integrals. We show that collapse in the sense of Zakharov can occur for $c<0$ and $nd\geq2$ and a functional $E_{\textup{NLS}}[\psi]\leq0$ even when $V({\bold r})\not\equiv0$. The singularities typically arise as $\delta$-functions centered on the trap origin ${\bold r}={\bold 0}$.
Keywords: Bose–Einstein condensation, functional integral method, quantum model of nonlinear Schrödinger equation, finite-temperature theory, magnetic traps, two-point correlations, coherence functions. 
@article{TMF_2003_134_1_a5,
     author = {R. K. Bullough and N. M. Bogolyubov and V. S. Kapitonov and K. L. Malyshev and I. Timonen and A. V. Rybin and G. G. Varzugin and M. Lindberg},
     title = {Quantum {Integrable} and {Nonintegrable} {Nonlinear} {Schr\"odinger} {Models} for {Realizable} {Bose{\textendash}Einstein} {Condensation} in $d+1$ {Dimensions} $(d=1,2,3)$},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {55--73},
     year = {2003},
     volume = {134},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2003_134_1_a5/}
}
TY  - JOUR
AU  - R. K. Bullough
AU  - N. M. Bogolyubov
AU  - V. S. Kapitonov
AU  - K. L. Malyshev
AU  - I. Timonen
AU  - A. V. Rybin
AU  - G. G. Varzugin
AU  - M. Lindberg
TI  - Quantum Integrable and Nonintegrable Nonlinear Schrödinger Models for Realizable Bose–Einstein Condensation in $d+1$ Dimensions $(d=1,2,3)$
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2003
SP  - 55
EP  - 73
VL  - 134
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TMF_2003_134_1_a5/
LA  - ru
ID  - TMF_2003_134_1_a5
ER  - 
%0 Journal Article
%A R. K. Bullough
%A N. M. Bogolyubov
%A V. S. Kapitonov
%A K. L. Malyshev
%A I. Timonen
%A A. V. Rybin
%A G. G. Varzugin
%A M. Lindberg
%T Quantum Integrable and Nonintegrable Nonlinear Schrödinger Models for Realizable Bose–Einstein Condensation in $d+1$ Dimensions $(d=1,2,3)$
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2003
%P 55-73
%V 134
%N 1
%U http://geodesic.mathdoc.fr/item/TMF_2003_134_1_a5/
%G ru
%F TMF_2003_134_1_a5
R. K. Bullough; N. M. Bogolyubov; V. S. Kapitonov; K. L. Malyshev; I. Timonen; A. V. Rybin; G. G. Varzugin; M. Lindberg. Quantum Integrable and Nonintegrable Nonlinear Schrödinger Models for Realizable Bose–Einstein Condensation in $d+1$ Dimensions $(d=1,2,3)$. Teoretičeskaâ i matematičeskaâ fizika, Tome 134 (2003) no. 1, pp. 55-73. http://geodesic.mathdoc.fr/item/TMF_2003_134_1_a5/

[1] N. M. Bogoliubov, R. K. Bullough, V. S. Kapitonov, C. Malyshev, J. Timonen, Europhys. Lett., 55:6 (2001), 755 | DOI

[2] N. M. Bogolyubov, A. G. Izergin, V. E. Korepin, Korrelyatsionnye funktsii integriruemykh sistem i kvantovyi metod obratnoi zadachi, Nauka, M., 1992 | MR | Zbl

[3] R. K. Bullough, J. T. Timonen, “Quantum and classical integrable models and statistical mechanics”, Statistical Mechanics and Field Theory, eds. V. V. Bazhanov and C. J. Burden, World Scientific, Singapore, 1995, 336

[4] V. I. Arnold, Matematicheskie metody klassicheskoi mekhaniki, Nauka, M., 1974 | MR

[5] I. Bloch, T. W. Hänsch, T. Esslinger, Nature, 403 (2000), 166 | DOI

[6] V. E. Zakharov, ZhETF, 62 (1972), 1745; Л. М. Дегтярев, В. Е. Захаров, Л. И. Рудаков, ЖЭТФ, 68 (1975), 115

[7] V. E. Zakharov, V. S. Synakh, ZhETF, 68 (1975), 940

[8] A. V. Rybin, G. G. Varzugin, M. Lindberg, J. Timonen, R. K. Bullough, Phys. Rev. E, 62:5 (2000), 6224 | DOI | MR

[9] M. H. Anderson, J. R. Ensher, M. R. Andrews, C. E. Wieman, E. A. Cornell, Science, 269 (1995), 198 | DOI | MR

[10] K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, W. Ketterle, Phys. Rev. Lett., 75 (1995), 3969 ; D. G. Fried et al., Phys. Rev. Lett., 81 (1999), 3969 | DOI

[11] C. C. Bradley et al., Phys. Rev. Lett., 75 (1995), 1687 | DOI

[12] S. L. Cornish et al., Phys. Rev. Lett., 85:9 (2000), 1795 ; Physics Today, August 2000, 17 | DOI | Zbl

[13] J. L. Roberts et al., Phys. Rev. Lett., 86:19 (2001), 4211 ; P. A. Ruprecht, M. J. Holland, K. Burnett, M. Edwards, Phys. Rev. A, 51 (1995), 4704 | DOI | DOI

[14] R. K. Bullough, Yu-zhong Chen, J. Timonen, “Soliton statistical mechanics – thermodynamic limits for quantum and classical integrable models”, Nonlinear World, V. 2, eds. V. G. Bary'akhtar, V. M. Chernousenko, N. S. Erokhin, A. G. Sitenko and V. E. Zakharov, World Scientific, River Edge, NJ, 1990, 1377 | MR

[15] N. M. Bogoliubov, R. K. Bullough, J. Timonen, Phys. Rev. Lett., 72 (1994), 3993 | DOI | MR

[16] N. A. Slavnov, TMF, 121:1 (1999), 117 | DOI | MR | Zbl

[17] T. Kojima, V. E. Korepin, N. A. Slavnov, Commun. Math. Phys., 188 (1997), 657 ; V. E. Korepin, N. A. Slavnov, J. Phys. A, 30 (1997), 8241 ; А. Р. Итс, Н. А. Славнов, ТМФ, 119:2 (1999), 179 ; V. Korepin, N. Slavnov, J. Phys. A, 30 (1997), 8623 | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl

[18] P. A. Deift, A. R. Its, Xin Zhou, Ann. Math., 146 (1997), 149 | DOI | MR | Zbl

[19] R. K. Bullough, N. M. Bogoliubov, G. D. Pang, J. Timonen, Chaos Solitons Fractals, 5 (1995), 2639 | DOI | MR | Zbl

[20] N. N. Bogoliubov, J. Phys., 9 (1947), 23 | MR

[21] N. N. Bogoliubov, V. V. Tolmachev, D. V. Shirkov, A New Method in the Theory of Superconductivity, Chapman Hall, London, 1959 | MR

[22] F. Dalfovo, S. Giorgini, L. Pitaevskii, S. Stringari, Rev. Mod. Phys., 71 (1999), 463 | DOI

[23] V. N. Popov, Funktsionalnye integraly v kvantovoi teorii polya i statisticheskoi fizike, Atomizdat, M., 1976 | MR

[24] N. M. Bogolyubov, Zap. nauchn. semin. POMI, 269, 2000, 125 | Zbl

[25] N. N. Bogoliubov, Quasi-expectation values in problems of statistical mechanics, JINR, Dubna, 1963

[26] D. S. Petrov, G. V. Shlyapnikov, J. T. M. Walraven, Phys. Rev. Lett., 85 (2000), 3745 | DOI

[27] R. K. Bullough, J. Mod. Opt., 47:11 (2000), 2029 ; Erratum, J. Mod. Opt., 48:4 (2001), 747 | DOI | MR | Zbl | DOI

[28] N. M. Bogoliubov, R. K. Bullough, V. S. Kapitonov, C. Malyshev, J. Timonen, “Finite-temperature correlations in the one-dimensional trapped and untrapped Bose gas”, Phys. Rev. A, 2003, submitted

[29] T. Tsurumi, M. Wadati, J. Phys. Soc. Japan, 66 (1997), 3031 ; 3035 ; M. Wadati, T. Tsurumi, Phys. Lett. A, 247 (1998), 287 ; L. Bergé, T. J. Alexander, Y. S. Kivshar, Phys. Rev. A, 62 (2000), 023607 | DOI | Zbl | Zbl | DOI | MR | DOI | MR

[30] R. K. Bullough, M. Wadati, “Quantum solitons as possible qubits”, Coherence and Quantum Optics 8 (CQO8), eds. N. Bigelow, J. H. Eberly et al., Plenum, New York, 2001

[31] Y. Lai, H. A. Haus, Phys. Rev. A, 40:2 (1989), 844 ; 854 | DOI | MR | Zbl

[32] R. K. Bullough, N. M. Bogoliubov, A. V. Rybin, G. G. Varzugin, J. Timonen, J. Phys. A, 34 (2001), 1 | DOI | MR | Zbl

[33] A. V. Rybin, G. G. Varzugin, J. Timonen, R. K. Bullough, J. Phys. A, 34 (2001), 157 | DOI | MR | Zbl