Symmetry Approach in the Problem of Gas Expansion into Vacuum

Abstract

A brief review of the results on the expansion of quantum and classical gases into vacuum based on the use of symmetries is presented. For quantum gases in the Gross–Pitaevskii (GP) approximation, additional symmetries arise for gases with a chemical potential μ that depends on the density n powerfully with exponent ν = 2/D, where D is the space dimension. For gas condensates of Bose atoms at temperatures T → 0, this symmetry arises for two-dimensional systems. For D = 3 and, accordingly, ν = 2/3, this situation is realized for an interacting Fermi gas at low temperatures in the so-called unitary limit (see, for example, L. P. Pitaevskii, Phys. Usp. 51, 603 (2008)). The same symmetry for classical gases in three-dimensional geometry arises for gases with the adiabatic exponent γ = 5/3. Both of these facts were discovered in 1970 independently by Talanov [V. I. Talanov, JETP Lett. 11, 199 (1970).] for a two-dimensional nonlinear Schrödinger (NLS equation, which coincides with the Gross–Pitaevskii equation), describing stationary self-focusing of light in media with Kerr nonlinearity, and for classical gases, by  Anisimov and Lysikov [S. I. Anisimov and Yu. I. Lysikov, J. Appl. Math. Mech. 34, 882 (1970)]. In the quasiclassical limit, these GP equations coincide with the equations of the hydrodynamics of an ideal gas with the adiabatic exponent γ = 1 + 2/D. Self-similar solutions in this approximation describe the angular deformations of the gas cloud against the background of an expanding gas by means of Ermakov-type equations. Such changes in the shape of an expanding cloud are observed in numerous experiments both during the expansion of gas after exposure to powerful laser radiation, for example, on metal, and during the expansion of quantum gases into vacuum.

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REFERENCES

  1. 1

    L. P. Pitaevskii, Phys. Usp. 51, 603 (2008).

    ADS  Article  Google Scholar 

  2. 2

    V. I. Talanov, JETP Lett. 11, 199 (1970).

    ADS  Google Scholar 

  3. 3

    S. I. Anisimov and Yu. I. Lysikov, J. Appl. Math. Mech. 34, 882 (1970).

    Article  Google Scholar 

  4. 4

    L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics (Fizmatlit, Moscow, 2002; Pergamon, Oxford, 1980).

  5. 5

    V. E. Zakharov and E. A. Kuznetsov, Phys. Usp. 55, 535 (2012).

    ADS  Article  Google Scholar 

  6. 6

    S. L. Cornish, S. T. Thompson, and C. E. Wieman, Phys. Rev. Lett. 96, 170401 (2006).

    ADS  Article  Google Scholar 

  7. 7

    C. Eigen, A. L. Gaunt, A. Suleymanzade, N. Navon, Z. Hadzibabic, and R. P. Smith, Phys. Rev. X 6, 041058 (2016).

    Google Scholar 

  8. 8

    H. Feshbach, Ann. Phys. 5, 357 (1958).

    ADS  MathSciNet  Article  Google Scholar 

  9. 9

    C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Rev. Mod. Phys. 82, 1225 (2010).

    ADS  Article  Google Scholar 

  10. 10

    J. Joseph, B. Clancy, L. Luo, J. Kinast, A. Turlapov, and J. E. Thomas, Phys. Rev. Lett. 98, 170401 (2007).

    ADS  Article  Google Scholar 

  11. 11

    M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Chin, J. H. Denschlag, and R. Grimm, Phys. Rev. Lett. 92, 120401 (2004).

    ADS  Article  Google Scholar 

  12. 12

    M. J. H. Ku, A. T. Sommer, L. W. Cheuk, and M. W. Zwierlein, Science (Washington, DC, U. S.) 335, 563 (2012).

    ADS  Article  Google Scholar 

  13. 13

    G. Zürn, T. Lompe, A. N. Wenz, S. Jochim, P. S. Julienne, and J. M. Hutson, Phys. Rev. Lett. 110, 135301 (2013).

    ADS  Article  Google Scholar 

  14. 14a

    E. P. Gross, Nuovo Cim. 20, 454 (1961);

    ADS  Article  Google Scholar 

  15. 14b

    L. Pitaevskii, Sov. Phys. JETP 13, 451 (1961).

    MathSciNet  Google Scholar 

  16. 15

    E. A. Kuznetsov, M. Yu. Kagan, and A. V. Turlapov, Phys. Rev. A 101, 043612 (2020).

    ADS  Article  Google Scholar 

  17. 16

    E. A. Kuznetsov and M. Yu. Kagan, Theor. Math. Phys. 202, 399 (2020).

    Article  Google Scholar 

  18. 17

    I. E. Dzyaloshinskii, private commun. (1970).

  19. 18

    E. A. Kuznetsov and S. K. Turitsyn, Phys. Lett. A 112, 273 (1985).

    ADS  Article  Google Scholar 

  20. 19

    S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, Radiophys. Quantum Electron. 14, 1062 (1971).

    ADS  Article  Google Scholar 

  21. 20

    K. Rypdal and J. J. Rasmussen, Phys. Scr. 33, 498 (1986).

    ADS  Article  Google Scholar 

  22. 21

    L. V. Ovsyannikov, Sov. Phys. Dokl. 111, 47 (1956).

    Google Scholar 

  23. 22

    S. I. Anisimov and V. A. Khokhlov, Instabilities in Laser-Matter Interaction (CRC, Boca Raton, 1995).

    Google Scholar 

  24. 23

    Yu. V. Likhanova, S. B. Medvedev, M. P. Fedoruk, and P. L. Chapovsky, JETP Lett. 103, 403 (2016).

    ADS  Article  Google Scholar 

  25. 24

    K. M. O’Hara, S. L. Hemmer, M. E. Gehm, S. R. Granade, and J. E. Thomas, Science (Washington, DC, U. S.) 298, 2179 (2002).

    ADS  Article  Google Scholar 

  26. 25

    E. Elliott, J. A. Joseph, and J. E. Thomas, Phys. Rev. Lett. 112, 040405 (2014).

    ADS  Article  Google Scholar 

  27. 26

    Yu. Kagan, E. L. Surkov, and G. V. Shlyapnikov, Phys. Rev. A 55, R18 (1997).

    ADS  Article  Google Scholar 

  28. 27

    F. J. Dyson, J. Math. Mech. 18, 91 (1968).

    Google Scholar 

  29. 28

    Ya. B. Zel’dovich, Sov. Astron. J. 8, 700 (1964).

    ADS  MathSciNet  Google Scholar 

  30. 29

    V. P. Ermakov, Differential Equations of the Second Order. Integrability Conditions in the Closed Form (Univ. Izv., Kiev, 1880) [in Russian].

    Google Scholar 

  31. 30

    F. Calogero, J. Math. Phys. 10, 2191 (1969).

    ADS  Article  Google Scholar 

  32. 31

    L. P. Pitaevskii and A. Rosch, Phys. Rev. A 55, R853 (1997).

    ADS  Article  Google Scholar 

  33. 32

    J. R. Ray and J. L. Reid, Phys. Lett. A 71, 317 (1979).

    ADS  MathSciNet  Article  Google Scholar 

  34. 33

    C. Rogers and W. K. Schief, J. Math. Anal. Appl. 198, 194 (1996).

    MathSciNet  Article  Google Scholar 

  35. 34

    A. V. Turlapov and M. Yu. Kagan, J. Phys.: Condens. Matter 29, 383004 (2017).

    ADS  Google Scholar 

  36. 35

    M. Yu. Kagan and A. V. Turlapov, Phys. Usp. 188, 225 (2019).

    Article  Google Scholar 

  37. 36

    O. I. Bogoyavlensky, in Stochastic Behavior in Classical and Quantum Hamiltonian Systems (Springer, Berlin, 1979), p. 151.

    Google Scholar 

  38. 37

    A. V. Borisov, I. S. Mamaev, and A. A. Kilin, Nonlin. Dyn. 4, 363 (2008).

    Google Scholar 

  39. 38

    B. Gaffet, J. Fluid Mech. 325, 113 (1996).

    ADS  MathSciNet  Article  Google Scholar 

  40. 39

    L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 3: Quantum Mechanics: Non-Relativistic Theory (Pergamon, Oxford, 1965).

  41. 40

    V. E. Zakharov and E. A. Kuznetsov, Sov. Phys. JETP 64, 773 (1986).

    Google Scholar 

  42. 41

    G. A. El, V. V. Geogjaev, A. V. Gurevich, and A. L. Krylov, Phys. D (Amsterdam, Neth.) 87, 186 (1995).

  43. 42

    S. K. Ivanov and A. M. Kamchatnov, Phys. Rev. A 99, 013609 (2019).

    ADS  Article  Google Scholar 

  44. 43

    V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).

    ADS  Google Scholar 

  45. 44

    E. A. Kuznetsov, M. Yu. Kagan, and A. V. Turlapov, arXiv: 1903.04245 [cond-mat.quant-gas] (2019).

  46. 45

    M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Science (Washington, DC, U. S.) 269, 198 (1995).

    ADS  Article  Google Scholar 

  47. 46

    Cl. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, Phys. Rev. Lett. 75, 1687 (1995).

    ADS  Article  Google Scholar 

  48. 47

    K. B. Davis, M.-O. Mewes, M. R. Andrew, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, Phys. Rev. Lett. 75, 3969 (1995).

    ADS  Article  Google Scholar 

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ACKNOWLEDGMENTS

The authors are grateful to A.V. Turlapov and A.M. Kamchatov for many useful discussions.

Funding

The work of one of the authors (E.A.K.) was carried out with the support of the Russian Science Foundation (Grant no. 19-72-30028). Another author (M.Yu.K.) thanks for the support of the Program of Basic Research of the National Research University “Higher School Economies” and expresses gratitude to the Russian Foundation for Basic Research (grant no. 20-02-00015). The authors made equal contributions to this work.

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Correspondence to E. A. Kuznetsov or M. Yu. Kagan.

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Contribution for the JETP special issue in honor of I.E. Dzyaloshinskii’s 90th birthday

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Kuznetsov, E.A., Kagan, M.Y. Symmetry Approach in the Problem of Gas Expansion into Vacuum. J. Exp. Theor. Phys. 132, 704–713 (2021). https://doi.org/10.1134/S1063776121040130

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