Measurement of a one-dimensional matter-wave quantum breather

Piotr Staroń, Andrzej Syrwid, and Krzysztof Sacha

Phys. Rev. A 102, 063308 – Published 7 December 2020

Abstract

Employing the Bethe ansatz approach and numerical simulations of measurements of particles' positions we investigate a post-quench many-body dynamics of attractively interacting bosons on a ring, which in the mean-field approach corresponds to the so-called breather solution. Despite the fact that the initial many-body ground state is translationally invariant, the measurements reveal breather dynamics if quantum fluctuations of the center of mass of the system are extracted. Moreover, the analysis of the many-body evolution shows signatures of dissociation of the solitons that form the breather.

Received 23 July 2020
Revised 6 November 2020
Accepted 9 November 2020

DOI:https://doi.org/10.1103/PhysRevA.102.063308

Physics Subject Headings (PhySH)

Bose-Einstein condensates Ultracold collisions

Breathers Clusters Collective dynamics Solitons Ultracold gases

Bethe ansatz Mean field theory

Atomic, Molecular & OpticalGeneral PhysicsNonlinear Dynamics

Authors & Affiliations

Piotr Staroń ¹, Andrzej Syrwid ¹, and Krzysztof Sacha ^1,2

¹Instytut Fizyki Teoretycznej, Uniwersytet Jagielloński, ulica Profesora Stanisława Łojasiewicza 11, PL-30-348 Kraków, Poland
²Mark Kac Complex Systems Research Center, Uniwersytet Jagielloński, ulica Profesora Stanisława Łojasiewicza 11, PL-30-348 Kraków, Poland

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Vol. 102, Iss. 6 — December 2020

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Images

Figure 1
Periodic evolution of the probability density corresponding to the mean-field breather solution, Eq. (4), for $g = 11.85$ and $x_{c . m .} = 0$ . Note that at $t = m T_{B}, m \in Z$ (solid blue line) the mean-field breather solution coincides with the fundamental bright soliton, Eq. (3). After half of the period, i.e., at $t = (m + \frac{1}{2}) T_{B}$ , we can observe a very high and narrow central peak accompanied by two small side maxima located symmetrically on the left and on the right (dashed red line).
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Figure 2
(a) Illustrates how the center of mass of particles on a ring is determined in the three-particle case. First, the barycenter in the 2D space is calculated and next the line starting from the center of the ring and passing through the barycenter is drawn, and its crossing with the ring determines the center-of-mass position of particles. Note that the barycenter is associated with the angle $θ_{c . m .}$ , which measures the angular distance from the reference position $x = 0$ . (b) The normalized histogram (solid blue line) presents the single-particle probability density $ρ_{rel} (x)$ [Eq. (7)], where $Ψ = Ψ_{0}$ is the many-body ground state of $N = 4$ particles which attract each other with the strength $g_{0} (N - 1) = - 11.85$ . The histogram consists of $10^{6}$ realizations of the measurements of the positions of $N = 4$ particles (see Appendix pp2). A similar result but obtained for $N = 10$ is illustrated by a gray shading. For comparison, the dashed red line shows the probability density of the corresponding mean-field ground state which is the fundamental bright soliton [Eq. (3)].
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Figure 3
Time evolution of the four-particle system. The system is initially prepared in the ground state for $g_{0} = - 3.95$ and then the interactions are quenched by a factor of 4, i.e., $g_{0} \to 4 g_{0}$ . Left column shows $ρ_{rel} (x; t)$ ]Eq. (7)] (solid blue line) and $G^{(2)} (x; t)$ [Eq. (9)] (dashed red line) for different moments of time as indicated above the panels. Right column: color-coded plots of $P (x, y; t)$ [Eq. (10)] at the same time moments as in the corresponding left panels. By monitoring the time evolution of the considered system we estimate the period of the quantum breather oscillations $T_{Q B} \approx 0.0168$ . At $t_{d} \approx 0.073$ the oscillations nearly die out (cf. Fig. 4 where the time moments presented in the current figure are indicated by vertical black lines).
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Figure 4
Time evolution of $ρ_{rel} (0; t)$ (dashed red line) and $G^{(2)} (0; t)$ (solid blue line) which corresponds to the results presented in Fig. 3 and illustrates decay and revival of the amplitudes of the central peak oscillations in the plots of $ρ_{rel} (x; t)$ and $G^{(2)} (x; t)$ . While initially the oscillations are significant and indicate the breatherlike dynamics known from the mean-field description, for longer times (of the order of $t_{d} \approx 0.073$ ) they nearly die out. The decay of the central peak oscillations is accompanied by the increasing distance between the central peak and side maxima (cf. Fig. 3).
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