KVANT TOMCHISIDAGI ICHKI QO‘ZG‘ALISHLAR DIAGRAMMASI

Diagram of internal excitations in a quantum droplet

S.M.Usanov ¹,  SH.M. Usanov ²

¹Kimyo International University in Tashkent, 156 Usman Nasyr Str., 100121, Tashkent, Uzbekistan

²Tashkent Institute of Irrigation and Agricultural Mechanization Enginees, National Research University

Annatatsiya: Small droplets collide quasi-elastically, featuring the soliton-like behavior. On the other hand, large colliding droplets may merge or suffer fragmentation, depending on their relative velocity. The frequency of a breathing excited state of droplets, as predicted by the dynamical variational approximation based on the Gaussian ansatz, is found to be in good agreement with numerical results.

Keywords: Gaussian, soliton, surface tension

The stability diagram for the excited droplet in the plane of (N,k) is displayed in Fig. 1, 2, [2-5]

      (1)

We demonstrate below that

    (2)

determines a critical number of particles separating two different physical regimes.

Thus, rescaling

t = t₀t^', x = x₀x^', ψ = ψ₀ψ^',     (3)

casts Eq. (1) in an equation without free coefficients (where the primes are omitted):

        (4)

in which symbols correspond to values k_c extracted from systematic simulations of Eq. (4), according to the procedure outlined in subsubsection IIIB2 [6-10]. The droplet remains undivided

Figure 1: (Color online) The frequency, ω (the left axis), and damping ratio, ζ (right axis) of oscillations following the application of the density modulation to the droplet, as per Eq. (2), for N = 0.1. The critical value of wavenumber, k_c, above which the perturbed droplet splits, is obtained by fitting the frequency in the stable region to . The fit is shown by the solid black line. The dashed line shows the frequency of the breathing mode, , from Fig.

at k < k_c. It is seen that the strongest stability corresponds to N ≈ 1. The stability-threshold line may be interpreted in terms of energy considerations, by comparing the collisional kinetic energy [12] associated with the imposed wavenumber, E_kin = Nħ²k²/(2m), and the surface energy, E_s, see Eq. (5). 

        (5)

for N⁻¹ → 0. The coefficients in Eq. define the volume, surface and curvature tension, respectively. The surface tension τ is related to  as , where the unit-volume radius r₀ is defined by condition .

In the 1D system, the expansion parameter is N⁻¹, instead of N^−1/3 in Eq. (5), and the corresponding coefficients can be obtained analytically. The bulk energy density is 

E_v = −2/9, and the surface-energy coefficient is E_s = 16/(27e²). In one dimension, the “surface” is reduced to two points, hence its size is independent of the size of the droplet. The respective surface tension is

       (6)

ψ(x,t = 0) = ψ_e(x)cos(kx) ,    (7)

The ratio of the two energies is known as the (modified) Weber number [13, 14]. Curved lines in Fig. 2 correspond to We = 1, 2, and 3. We find that, for N ≳ 4, the classical prediction based on a fixed value of the Weber number explains the stability diagram reasonably well.

                 (8)

Figure 2: (Color online) The stability diagram for a single droplet in the plane of norm N and wavenumber k of the initially applied density modulation (2). Symbols show the stability border k_c obtained by fitting the oscillation frequency, cf. Fig. 1. Lines correspond to different values of the Weber number, defined as per

Eq. (3): We = 1, 2, and 3 (dashed, dashed-dotted, and dashed-dotted-dotted lines, respectively).

On the other hand, the stability for N ≤ 1 is quite different. In this regime the perturbation with wavenumber k may efficiently create an excitation in the droplet only at , where W is the width of the droplet.

F. Cinti, A. Cappellaro, L. Salasnich, and T. Macr`ı, Phys. Rev. Lett. 119, 215302 (2017).

T. D. Lee and C. N. Yang, Phys. Rev. 105, 1119 (1957).

G. Kirchhoff, Monatsb. Deutsch. Akad. Wiss. Berlin 144, 48 (1877).

M. Gaudin, Phys. Rev.A4, 386 (1971).

Y. S. Kivshar and B. A. Malomed, Rev. Mod. Phys. 61, 763 (1989).

A. E. Astrakharchik and A. I. Maimistov, JETP 81, 275 (1995).

E. L. F. ao Filho, C. B. de Arau´jo, and J. J. J. Rodrigues, J. Opt. Soc. Am. B24, 2948 (2007).

D. Anderson, Phys. Rev.A27, 3135 (1983).

L. Salasnich, International Journal of Modern Physics B 14, 1 (2000).

B. A. Malomed, Progress in Optics(Elsevier, 2002) pp. 71–193.

L. Khaykovich and B. A. Malomed, Phys. Rev.A74, 023607 (2006).

M. Orme, Progress in Energy and Combustion Science 23, 65 (1997).

S. Chandra and C. T. Avedisian, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 432, 13 (1991).

A. Frohn and N. Roth, Dynamics of Droplets(Springer Berlin Heidelberg, 2000).

J. Nespolo, G. E. Astrakharchik, and A. Recati, New Journal of Physics 19, 125005 (2017).

S.M. Usanov, M.E. Akramov, Sh.M. Usanov, “Tarmoqlangan to'lqin o'tkazgichlarda ikkinchi garmonik avlod”

https://scholar.google.com/citations?view_op=view_citation&hl=ru&user=nXwUCG8AAAAJ&citation_for_view=nXwUCG8AAAAJ:u-x6o8ySG0sC

17. M. Akramov, K. Sabirov, D. Matrasulov, H. Susanto, S. Usanov, and O. Karpova, Phys. Rev. E 105, 054205 (2022).