What is a Quantum Shock Wave?

S. A. Simmons, F. A. Bayocboc, Jr., J. C. Pillay, D. Colas, I. P. McCulloch, and K. V. Kheruntsyan

Phys. Rev. Lett. 125, 180401 – Published 26 October 2020

Abstract

Shock waves are examples of the far-from-equilibrium behavior of matter; they are ubiquitous in nature, yet the underlying microscopic mechanisms behind their formation are not well understood. Here, we study the dynamics of dispersive quantum shock waves in a one-dimensional Bose gas, and show that the oscillatory train forming from a local density bump expanding into a uniform background is a result of quantum mechanical self-interference. The amplitude of oscillations, i.e., the interference contrast, decreases with the increase of both the temperature of the gas and the interaction strength due to the reduced phase coherence length. Furthermore, we show that vacuum and thermal fluctuations can significantly wash out the interference contrast, seen in the mean-field approaches, due to shot-to-shot fluctuations in the position of interference fringes around the mean.

Received 24 June 2020
Accepted 20 August 2020

DOI:https://doi.org/10.1103/PhysRevLett.125.180401

Physics Subject Headings (PhySH)

Bose gases Quantum fluids & solids Quantum interference effects Shock waves

Collective dynamics Superfluidity Ultracold gases

Phase space methods

Fluid DynamicsNonlinear DynamicsCondensed Matter, Materials & Applied PhysicsAtomic, Molecular & Optical

Authors & Affiliations

S. A. Simmons ¹, F. A. Bayocboc, Jr.¹, J. C. Pillay ¹, D. Colas ^1,2, I. P. McCulloch ¹, and K. V. Kheruntsyan ¹

¹School of Mathematics and Physics, University of Queensland, Brisbane, Queensland 4072, Australia
²ARC Centre of Excellence in Future Low-Energy Electronics Technologies, University of Queensland, Brisbane, Queensland 4072, Australia

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Vol. 125, Iss. 18 — 30 October 2020

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Images

Figure 1
Dispersive shock waves in an ideal and weakly interacting 1D Bose gas at $T = 0$ . In (a), we show the single-particle probability densities $ρ (x, t) = | Ψ (x, t) |^{2}$ for the ideal gas at different dimensionless times $τ$ , for $β = 1$ and $σ / L = 0.02$ . Due to the reflectional symmetry about the origin, we only show the densities for $x > 0$ . In (b) and (c), we show the density profiles in the weakly interacting regime at two instances of time, with the GPE simulations represented by solid (grey and blue) lines. We also show the results of stochastic phase-space simulations [28] using the truncated Wigner ( $W$ ) and positive- $P$ ( $+ P$ ) approaches, which incorporate the effects of vacuum fluctuations (see text). The shape of $ρ (x, 0)$ (not shown) in (b) and (c) is the same as in (a), except that it is now normalized to $N$ : (b) $N = 50$ , $γ_{bg} = 0.1$ ; (c) $N = 2000$ , $γ_{bg} = 0.01$ [30]. The dimensionless healing length $l_{h} / L = 1 / \sqrt{γ_{bg}} N_{bg}$ is $l_{h} / L ≃ 0.072$ in (b), and $l_{h} / L ≃ 0.0057$ in (c), which can be compared to $σ / L = 0.02$ .
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Figure 2
Shock waves in a 1D Bose gas at $T = 0$ for intermediate and strong interactions. In all panels, the iMPS simulation results are shown as full lines; in (d) and (e), the iMPS results are compared with exact diagonalization results (dotted lines), and show excellent agreement. In (a)–(c), the trapping potential is chosen as $\bar{V} (ξ) = - {\bar{V}}_{bg} [1 + β \exp (- ξ^{2} / 2 {\bar{σ}}^{2})]^{2}$ , with $\bar{σ} = 0.02$ in all cases, and: (a) $β = 0.98$ , ${\bar{V}}_{bg} = 1849$ (resulting in $N_{bg} ≃ 43.2$ ); (b) $β = 0.7$ , ${\bar{V}}_{bg} = 18705$ ( $N_{bg} ≃ 43.6$ ); (c) $β = 0.38$ , ${\bar{V}}_{bg} = 92450$ ( $N_{bg} ≃ 43$ ). Here, $\bar{V} (ξ) \equiv V (x) / E_{0}$ , with $E_{0} = ℏ^{2} / m L^{2}$ being the energy scale. In (d), the trapping potential is chosen according to Eq. (S33) of [32], with $N_{bg} = 44.03$ , $β = 1$ , and $\bar{σ} = 0.02$ ; in the Thomas-Fermi approximation, this would produce exactly the same initial density profile as in Fig. 1, which, however, would not display the Friedel oscillations seen here. In (e), the trapping potential has the same shape as the one used for producing the ideal Bose gas initial density profile of Fig. 1, except normalized to $N = 4$ .
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Figure 3
Shock waves in a finite-temperature quasicondensate from SPGPE simulations (thick red lines). Shown are the final-time ( $τ = 0.0007$ ) density distributions for $x > 0$ and two different initial dimensionless temperatures: (a) $\bar{T} = 0.01$ , and (b) $\bar{T} = 0.1$ [79]. Other parameters are as in Fig. 1 (c); the GPE results are shown here again as blue lines for comparison. The dimensionless thermal phase coherence length here can be expressed as $l_{T} / L = 2 / (\bar{T} N_{bg})$ , giving: (a) $l_{T} / L ≃ 0.1$ , (b) $l_{T} / L ≃ 0.01$ . The SPGPE mean densities are the averages over 100 000 stochastic trajectories, whereas the thin lines show two sample trajectories.
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