Shock waves in a one-dimensional Bose gas: From a Bose-Einstein condensate to a Tonks gas

Bogdan Damski

Phys. Rev. A 73, 043601 – Published 3 April 2006

Abstract

We derive and analyze shock-wave solutions of hydrodynamic equations describing repulsively interacting one-dimensional Bose gas. We also use the number-conserving Bogolubov approach to verify accuracy of the Gross-Pitaevskii equation in shock wave problems. We show that quantum corrections to dynamics of shocks (dark-shock-originated solitons) in a Bose-Einstein condensate are negligible (important) for a realistic set of system parameters. We point out possible signatures of a Bose-Einstein condensate—Tonks crossover in shock dynamics. Our findings can be directly verified in different experimental setups.

Received 20 June 2005

DOI:https://doi.org/10.1103/PhysRevA.73.043601

Authors & Affiliations

Bogdan Damski

Theory Division, Los Alamos National Laboratory, MS-B213, Los Alamos, New Mexico 87545, USA

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Issue

Vol. 73, Iss. 4 — April 2006

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Images

Figure 1
Sound velocity divided by density. In the BEC limit $(γ \to 0)$ it is $\sqrt{γ}$ , while in the Tonks limit $(γ \to + \infty)$ it equals $π$ . Solid line: (3), dots: Tonks value, dashed line: $\sqrt{γ}$ . For units see Appendix .Reuse & Permissions
Figure 2
Dotted line: initial density profile (16). Solid line: solution of Eqs. (4, 5) with density at $t = 0$ given by (16) and velocity field determined from (12). Dashed line: implicit solution (18) multiplied by $ρ_{0}$ . Upper plot: subsequent density profiles correspond to $t = 0, 1.155, 2.02$ . Lower plot: $t = 2.6$ . Time of shock creation equals $t_{s} = 2.03$ , $N = 499$ , $γ_{0} = 30$ , $σ = 2$ , $l = 50$ , $η = 0.05$ . For units see Appendix .Reuse & Permissions
Figure 3
Density of a condensate, $⟨ {\hat{a}}_{0}^{†} {\hat{a}}_{0} ⟩ {∣ ϕ (x) ∣}^{2}$ , at $t = 0, 0.6$ : (a) and (c). The density of depleted atoms $(\sum_{k = 0}^{317} ∣ v_{k} (x) ∣^{2})$ at $t = 0, 0.6$ : (b) and (d). Plot (e): a comparison between condensate density (solid line) and the density of depleted atoms (dashed line) in the right-moving shock structure; condensate density is rescaled for presentational purposes. At $t = 0$ , the laser potential $V_{l}$ was $\propto - \exp (- 2 x^{2})$ and the system was in a ground state; $V_{l} (t > 0) \equiv 0$ . The number of noncondensed atoms equals $\sim 320$ during whole evolution. The total number of atoms is $1.5 \times 10^{5}$ . Other parameters: $l = 15$ , $a N = 7500$ , $T = 0$ . The time of shock creation obtained from [5] is $t_{s} = 0.36$ . For units see Appendix .Reuse & Permissions
Figure 4
Density of a condensate, $⟨ {\hat{a}}_{0}^{†} {\hat{a}}_{0} ⟩ {∣ ϕ (x) ∣}^{2}$ , at $t = 0, 0.06$ : (a) and (c). Density of depleted atoms $(\sum_{k = 0}^{349} ∣ v_{k} (x) ∣^{2})$ at $t = 0, 0.06$ : (b) and (d). Plot (e): black thick line—total density of atoms, dashed line—condensate density; both curves are for right-moving dark-shock-originated soliton train at $t = 0.75$ . Plot (f): density of depleted atoms at $t = 0.75$ . The external laser potential at $t = 0$ was $\propto \exp (- 2 x^{2})$ and the system was in a ground state; $V_{l} (t > 0) \equiv 0$ . The number of noncondensed atoms equals $\sim 320$ at $t = 0$ and $\sim 1000$ at $t = 0.75$ . The total number of atoms equals $1.5 \times 10^{5}$ . Other parameters: $l = 15$ , $a N = 7500$ , $T = 0$ . The time of shock creation obtained from [5] is $t_{s} = 0.086$ . For units see Appendix .Reuse & Permissions
Figure 5
Introducing notation: $p$ —peak density of depleted atoms in the deepest soliton moving to the right [Fig. 4e]; $d N$ —total number of depleted atoms. Dots: $\ln p (t)$ ; dotted line: a linear fit to circles, with a slope given by 2.29. Squares: $\ln [d N (t) - d N (0)]$ ; dashed line: linear fit with slope equal to 2.55. The data is plotted from $t = 0.24$ , i.e., a moment of the deepest soliton creation. For the parameters: see Fig. 4, while for units see Appendix .Reuse & Permissions

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