Formation of shock waves in a Bose-Einstein condensate

Bogdan Damski

Phys. Rev. A 69, 043610 – Published 16 April 2004

Abstract

We consider propagation of density wave packets in a Bose-Einstein condensate. We show that the shape of initially broad, laser-induced, density perturbation changes in the course of free time evolution so that a shock wave front finally forms. Our results are well beyond predictions of commonly used zero-amplitude approach, so they can be useful in extraction of a speed of sound from experimental data. We discuss a simple experimental setup for shock propagation and point out possible limitations of the mean-field approach for description of shock phenomena in a Bose-Einstein condensate.

Received 29 September 2003

DOI:https://doi.org/10.1103/PhysRevA.69.043610

Authors & Affiliations

Bogdan Damski

Instytut Fizyki, Uniwersytet Jagielloński, Ulica Reymonta 4, PL-30-059 Kraków, Poland and Institut für Theoretische Physik, Universität Hannover, D-30167 Hannover, Germany

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Vol. 69, Iss. 4 — April 2004

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Images

Figure 1
Density of atoms during free time evolution. Dotted line—an initial density profile, dashed line—the prediction (4), solid line—a numerical solution of Eq. (1) with the QP term. Plot (a) [(b)]: $η = 0.0175$ and $t = 21$ [ $η = 0.0175$ and $t = 42$ ]. Plot (c) [(d)]: the right-moving wave packet for $η = 0.0175$ $[η = - 0.0175]$ at $t = 63$ . Other parameters: $g = 7.5 \times 10^{3}$ , $σ = 6.5$ , $l = 350$ .Reuse & Permissions
Figure 2
Schematic plot of functions: $- \ln γ$ (dashed line) and $(1 + η γ) ∕ (2 + η γ)$ (solid line) vs $γ$ . Dotted lines define bounds leading to inequalities (13).Reuse & Permissions
Figure 3
Density of atoms during free time evolution. Dotted line—an initial density of atoms, dashed line—the analytical solution (6), solid line—a numerical solution of Eq. (1) with the QP term. Plot (a) [(b)] is for $t = 24$ $[t = 51]$ . Other parameters: $η = 0.06$ , $t_{s} = 60.4$ , $x_{s} = 253.8$ , $g = 7.5 \times 10^{3}$ , $σ = 12.5$ , $l = 250$ .Reuse & Permissions
Figure 4
Plots (a) and (b): density profile of the right-moving wave packet. Dashed line—the analytical solution (6), solid line—a solution of Eq. (1) with the QP term, thick solid line in (b) shows multivalued part of the solution (6). Plot (a) [(b)] is for $t = 18.9$ $[t = 27]$ . Plot (c): relative slope (16) of the right-moving wave packet. Dotted line indicates time of shock formation (11, 12). Parameters: $η = 0.2$ , $σ = 12.5$ , $t_{s} = 19$ , $x_{s} = 93.5$ , $g = 7.5 \times 10^{3}$ , $l = 250$ , $ξ (2 \times 10^{- 3}) = 0.18$ .Reuse & Permissions
Figure 5
Plot (a): density of atoms at different time moments. Plot (b). the same as in the plot (a) but with the $x^{2} ∕ 2 g$ term added to the density profile in order to remove a trivial position dependence of atomic density due to external harmonic trapping potential—see Eq. (17). Dotted line, $t = 0$ , dashed line, $t = 0.3$ ; dashed-dotted, $t = 0.6$ ; solid line, $t = 0.8$ . Both $t$ and $x$ are expressed in harmonic oscillator units [17]. Other parameters: $u_{0} = - 90$ , $η = 0.184$ , $g = 7.5 \times 10^{3}$ , $σ = 1.4$ , $R_{T F} \approx 22$ .Reuse & Permissions
Figure 6
Plots (a)–(c): a position of impulse’s maximum vs time. Solid line—the semianalytical prediction (20), dashed line—numerics on the basis of Eq. (1) with the QP term, dotted line—time of shock formation extracted from numerics (see text). Dashed-dotted line on the plot (b): $x_{m} (t, 0)$ . Parameters $(u_{0}, η)$ for plots (a)–(c): $(- 30, 0.06)$ , $(- 90, 0.184)$ , $(- 180, 0.39)$ , respectively. Plot (d): time of shock formation vs relative amplitude of an impulse. Solid line comes from numerics, while dotted line is an approximation (21). Both $t$ and $x$ are expressed in harmonic oscillator units [17]. Other parameters: $g = 7.5 \times 10^{3}$ , $σ = 1.4$ , $R_{T F} \approx 22$ .Reuse & Permissions

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