⇦Chinese Physics Letters, 2019, Vol. 36, No. 12, Article code 120301 Solitonic Diffusion of Wavepackets in One-Dimensional Bose–Einstein Condensates * Yu Mo (莫宇), Cong Zhang (张聪), Shiping Feng (冯世平), Shi-Jie Yang (杨师杰)** Affiliations Department of Physics, Beijing Normal University, Beijing 100875 Received 28 July 2019, online 25 November 2019 *Supported by the National Natural Science Foundation of China under Grant Nos 11774034 and 11734002.
**Corresponding author. Email: yangshijie@tsinghua.org.cn Citation Text: Mo Y, Zhang C, Feng S P and Yang S J 2019 Chin. Phys. Lett. 36 120301    Abstract Solitonic characteristics are revealed in the diffusion process of a hump or a notch wave packet in a one-dimensional Bose–Einstein condensate. By numerically solving the time-dependent Gross–Pitaevskii equation, we find completely different spreading behavior for attractive or repulsive condensates. For the attractive condensate, a series of bright solitons are continuously generated one after another at the wave front and they nearly stay at the positions where they are generated in the whole diffusion process. In contrast, for the repulsive condensate, the initial wave packet splits at the beginning into a series of grey solitons that travel at different velocities. The moving velocity of the grey soliton depends on nonlinear interaction strength, as well as the shape of a particular grey soliton. DOI:10.1088/0256-307X/36/12/120301 PACS:03.75.Mn, 03.75.Lm, 05.30.Jp © 2019 Chinese Physics Society Article Text A soliton is a localized wave of nonlinear dispersive media, which has drawn extensive interest in numerous physical branches, such as water waves, fiber optics,[1] polyacetylene,[2] magnets,[3] ultracold atom systems,[4,5] and so on. A stable soliton can travel in a medium without changing their shape and velocity, which is a result of a nonlinear interaction that compensates for wave packet dispersion. The moving velocity depends on the strength of the nonlinear interaction. The realization of Bose–Einstein condensates (BECs) of dilute atom gases provides an ideal playground for investigation of solitons. In the condensates, solitons can be engineered by phase or density imprinting,[6] or by quantum quenching processes.[7] For example, dark (grey) solitons and bright solitons have been observed in repulsive[8,9] and attractive[10–15] condensates. Recently, non-equilibrium processes such as quenching processes have played an increasingly important role in revealing dynamical properties of the systems.[5] The high tunability of the nonlinear interaction strength[16] and easy observation by time-of-flight imaging of the ultracold atomic gas provide us a particularly convenient way to study various processes. Experimentalists have observed the collapse and revival of macroscopic matter waves in an optical lattice,[17] and self-trapping in a double well,[18] coherent quenching dynamics of Fermi sea in a Fermi-Bose mixture,[19] non-thermal behavior in nearly integrable experimental regimes, and equilibration in Bose–Hubbard-like systems, and so on. Prediction of Bose gas shows a connection of dark solitons to quantum many-body eigenstates of the Bethe-ansatz solvable Lieb-Liniger model.[20–22] In this Letter, we focus on the dynamical evolution of a dump or a notch in attractive and repulsive condensates. Two completely different solitonic modes of diffusion of the wave packet are found. In the attractive condensate, a series of bright solitons are continuously generated one after another at the wave front. The generated solitons do not move but exhibit breathing feature. In contrast, the initial wave packet in the repulsive condensate splits at the beginning into a series of grey solitons that travel at different velocities; namely, the diffusion process is solitonic. The moving velocity of the generated soliton depends on nonlinear interaction strength, as well as the depth of a particular grey soliton. The wave packet splits into a sequence of solitons in the process of diffusion. This indicates that a wave packet is unstable and degenerates into a series of stable solitons that move at different speeds. These solitons act as stationary eigenstates and the initial wavepacket seems to be a superposition of them. This nonlinear system has equivalent eigenstates as linear system but, regretfully, we cannot prove it mathematically. This Letter is arranged as follows. First, we describe our model and the basic characteristics of solitons in a BEC. Second, we explore the solitonic behavior in the diffusion processes of an attractive BEC and study the spreading speed of the wave packet. Finally, we study the diffusion of a repulsive BEC with a uniform background, which exhibits completely different behavior in comparison to the attractive case. The dynamics of dilute atom BEC is well described by the mean-field Gross–Pitaevskii (GP) equation.[23] Theoretical predictions based on this equation, such as solitons and vortices, and many other nonlinear phenomena, have been successfully tested experimentally. The three-dimensional (3D) GP equation could reduce into a quasi-one-dimensional (1D) nonlinear Schrödinger equation (NLSE) by confining the transverse dimensions of the condensate on the order of its healing length and its longitudinal dimension is much longer than its transverse ones. The 3D time-dependent NLSE in a uniform potential reads $$ i\hbar\frac{\partial}{\partial{t}}\psi({\boldsymbol r},t)=\Big[-\frac{\hbar^2}{2M}\nabla^2+g|\psi({\boldsymbol r},t)|^2\Big]\psi({\boldsymbol r},t),~~ \tag {1} $$ where $|\psi({\boldsymbol r},t)|^2$ is the particle density. The nonlinear coupling constant $g\equiv4\pi\hbar^2 aN/M$ comes from the interatomic interaction, with $a$ being the s-wave scattering length. The sign of $a$ corresponds to attractive or repulsive interaction respectively and the coupling strength $g$ is proportional to the total number of atoms in the system. Defined with the characteristic length scale of variations of the condensate wave function, the healing length[24] is $\xi\equiv(8\pi\bar{\rho}|a|)^{(-1/2)}$, where $\bar{\rho} \equiv N/(LA_t)$ is the mean particle density, $L$ is the longitudinal length of the confining potential, and $A_t=L_yL_z$ is the transverse area with the transverse lengths $L_y$ and $L_z$. The quasi-1D condition requires $L_y$ and $L_z$ satisfying $L_y,L_z\sim\xi$ and $L_y,L_z\ll L$. Under these conditions, Eq. (1) reduces to $$ i\partial_{\tilde{t}}f(x,t)=[-{(\xi_{\rm eff})}^2{\partial_{x}}^2+\gamma|f(x,t)|^2]f(x,t),~~ \tag {2} $$ where all terms are dimensionless: $f(x,t)$ is the longitudinal part of the wave function, $\tilde{t}\equiv(\hbar/2M\xi^2)t$ the natural time; $\xi_{\rm eff}$ an effective healing length; and $\gamma$ the effective interaction, respectively. For our purpose, Eq. (2) can be simply rewritten as $$ i\frac{\partial}{\partial{t}}\psi(x,t)= \Big[-\frac{1}{2}\frac{\partial^2}{\partial{x^2}}+\gamma|\psi(x,t)|^2\Big]\psi(x,t).~~ \tag {3} $$ Equation (1) is deduced to Eq. (3) in detail in Ref. [25]. Some analytical travelling solutions to Eq. (3) have been found for either attractive or repulsive situations. For example, in an attractive condensate ($\gamma < 0$) one has the solitonic solution[25] $$ \psi_\textrm{bright}(x,t)=\sqrt{-\frac{2}{\gamma}} k \exp\Big[i\frac{c}{2}x-i\mu t\Big]\textrm{sech}[k(x-ct)],~~ \tag {4} $$ where $c=2\sqrt{\mu+k^2}$ is the velocity of the bright soliton with $\mu$ the chemical potential of the condensate. Meanwhile, in a repulsive condensate ($\gamma>0$), one has the dark or grey solitonic solution,[26] $$ \psi_\textrm{grey}(x,t)\!=\!\exp(\!-i\mu t)\Big\{i\frac{c}{\sqrt {2\gamma}}+\!\sqrt{\frac{2}{\gamma}}k\tanh[k(x-ct)]\Big\},~~ \tag {5} $$ where the soliton velocity $c=2\mu-4k^2$. The parameter $k$ specifies the width of the soliton. In a practical process, one cannot create the exact form of solitons as in Eq. (4) or Eq. (5). Actually, we may imprint the condensate density by a hump or a notch. Hence we are interested in the diffusion of an arbitrary wave packet in the condensate. We consider an initial hump or notch in a constant condensate background. In the present study we take a Gaussian function as an example. We have tried various shapes of the wave packet in our simulations and we find that the conclusions for them are the same. The initial wave functions reads $$\begin{align} \psi(x)=&1+\lambda\delta\psi(x),\\ \delta\psi(x)=&A_0\exp[-(x-x_0)^2/2\sigma^2],~~ \tag {6} \end{align} $$ where $\delta\psi(x)$ is an Gaussian function with $A_0$, $\sigma$, $x_0$ referring to the height, width and position of the initial wave packet, respectively; $\lambda=+1$ corresponds to a hump and $\lambda=-1$ a notch. The real-time evolution of Eq. (3) is carried out by the Fourier domain method. In the calculations, we take $\gamma=\pm 1$ with normalization of the wave function in each step to the total number of atom $N$, which is taken from $500$ to $5000$ according to Ref. [27]. Variation of $N$ is equivalent to adjust the nonlinear coupling strength $\gamma$.
cpl-36-12-120301-fig1.png
Fig. 1. Diffusion of a Gaussian hump on a uniform attractive condensate background. The initial hump height $A_0=0.1$ and width $\sigma=0.025$. A series of bright solitons are generated one-by-one at the wavefront in the whole diffusing process. The solitons exhibit breathing oscillations and nearly do not travel. The total numbers of atoms are (a) $N=500$, (b) $N=2000$, (c) $N=3500$, (d) $N=5000$, which are equivalent to different nonlinear coupling strengths. More solitons are generated for stronger nonlinear interactions or more atom numbers.
Theoretically, in an attractive condensate the bright soliton is stable. We first take a Gaussian hump ($\lambda=+1$) on a uniform condensate background. The parameters are $A_0=0.1$ and $\sigma=0.025$. Figure 1 shows the real-time evolution of the initial wave packet for total atom numbers $N=500$, 2000, 3500, 5000, respectively. We observe that in the process of diffusion the hump decays by generating a series of bright solitons one after another at the wave front. These newly created solitons do not travel and exhibit breathing modes. A stationary bright soliton with velocity $c=0$ implies $\mu=-k^2$ in Eq. (4) for a single soliton solution. The breathing mode means that the generated solitons are not stationary completely. The stronger the nonlinear interaction or the more the atom number is, the denser the solitons are, with a lower height. All of the solitons have the same breathing frequency in a given process of diffusion, while the breathing frequency increases as the nonlinear interaction becomes stronger. In the Gaussian notch case ($\lambda=-1$), the diffusion process is shown in Fig. 2. We note that the wave packet also decays as in the hump case. However, there is a slight difference: in the hump case (as shown in Fig. 1), a remnant bright soliton remains at the original position of the initial hump; while there is no soliton in the notch case (as shown in Fig. 2). In both situations, pairs of stationary bright solitons are centered on the initial position. The solitonic diffusing characteristic is universal. We have taken various other shapes of wave packet including sawtooth, square and sine functions, in either symmetrical or asymmetrical form, as the initial wave packet to verify our conclusions. The parameters such as depth or width do not alter the results qualitatively. The solitonic feature increases with the total atom number $N$. If we take the linear limit (i.e., $N=1$), then the solitonic feature disappears.
cpl-36-12-120301-fig2.png
Fig. 2. The same as Fig. 1 but for an initial Gaussian notch. The initial wave packet depth $A_0=-0.1$ and width $\sigma=0.025$. Again, bright solitons are continuously generated in the diffusing process but no soliton appears at the initial notch position, in contrast to the Gaussian hump case.
Next, we examine the spreading speed $v_w$ of the wave packet. In the simulations we take $A_0=\pm0.1$ and $\sigma=0.025$. $N$ is taken from $5\times10^2$ to $1\times10^4$, such that $\gamma$ changes from $1.3\times10^3$ to $2.5\times10^4$. Figure 3 plots the spreading speed versus the relative coupling strength. The relative coupling strength is referenced to the case of $N=500$. We find that the wave spreading speed largely depends on $\gamma$ or $N$ and reveals a nearly linear relation. It is demonstrated that the solitonic diffusion is a result of nonlinear effect. Dark or grey solitons can be stable in repulsive condensates. We presumably consider that the diffusion of a wave packet will exhibit the same behavior in repulsive condensates as in attractive condensate. Unexpectedly, numerical simulations reveal completely different characteristics.
cpl-36-12-120301-fig3.png
Fig. 3. Spreading speed of a hump (solid line) and a notch (dashed line) in an attractive condensate versus the relative nonlinear coupling strength. The height/depth of the hump/notch are $A_0=\pm0.1$, respectively. The relative coupling strength is referenced to the case of $N = 500$. It shows that the spreading speeds for both the cases are almost the same.
cpl-36-12-120301-fig4.png
Fig. 4. Diffusion of a Gaussian hump (upper row with $\lambda=+1$) or a Gaussian notch (lower row with $\lambda=-1$) on a uniform repulsive condensate background. The parameters $A_0=0.5$ and $\sigma=0.05$. From left to right: the total numbers of atoms are $N=500$, 2000, 3500, 5000, respectively. In contrast to the attractive case, the initial packet splits at the beginning into several grey solitons, which travel at different velocities.
We consider a Gaussian hump ($\lambda=+1$) and notch ($\lambda=-1$), respectively, on the constant condensate background. The upper and the lower panels of Fig. 4 are, respectively, the diffusion of a Gaussian hump and notch for the total number of atom $N=500$, 2000, 3500, 5000. The parameters are $A_0=0.1$ and $\sigma=0.025$. In contrast to the attractive case, the initial packet splits at the beginning into several grey solitons, which travel separately leftward and rightward at different velocities. The soliton travels faster when the notch is shallower. The split always takes place, yet the number of solitons depends on the shape of the initial wave packet and the nonlinear coupling strength. This behavior is somehow similar to the standard D'Alembert's solution to the linear wave equation. Other shapes are chosen and the basic conclusion remains the same.
cpl-36-12-120301-fig5.png
Fig. 5. Log-log plot of the traveling velocity of grey solitons split from a Gaussian notch versus the relative nonlinear interaction strength. The initial packet depth $A_0=-0.5$ and width $\sigma=0.05$. The relative coupling strength is referenced to the case of $N = 500$. The approximate linearity implies a power law relation of the soliton velocity with the nonlinear coupling strength.
Our simulations show that the velocity of the grey soliton $v_{\rm s}$ depends on nonlinear coupling strength or total number of atoms, as well as the depth and the width of the initial wave packet. Figure 5 shows the diffusion of a Gaussian notch on a uniform condensate. The log-log plot of the soliton velocity versus relative coupling strength reveals a good linear relation, implying a power law that is dependent on the soliton velocity and the nonlinear interaction. This relation somehow differs from the analytical solution to Eq. (5), in which the velocity $c$ of the grey soliton is independent of the coupling $\gamma$. The largest soliton velocity $v_{\rm s}$ can be taken as the spreading speed of the wave packet, which is also quite different from the attractive case. In summary, we have investigated the dynamical diffusion of a hump or notch packet in the attractive or repulsive condensates. We report numerically two distinguished characteristics of solitonic behavior in these two situations. In the attractive condensate, a series of bright solitons are continuously generated at the wave front in the whole diffusing process. The solitons exhibit breathing oscillations and nearly do not travel. In the repulsive condensate, the initial packet splits immediately at the beginning into several grey solitons which travel at different velocities. The velocity of the soliton depends on the nonlinear interaction. Unfortunately, the underlying physics are still unclear. However, it seems that the wavepacket decays until a stable soliton forms and remains. The residues also decay until a new soliton forms. This process happens continuously until all wavepackets degenerate into solitons. References Solitons in PolyacetyleneSoliton relaxation in magnetsMagnetic Solitons in a Binary Bose-Einstein CondensateMany-body physics with ultracold gasesControlled Generation of Dark Solitons with Phase ImprintingColloquium : Nonequilibrium dynamics of closed interacting quantum systemsDark Solitons in Bose-Einstein CondensatesGenerating Solitons by Phase Engineering of a Bose-Einstein CondensateHybrid-order Poincaré sphereFormation and propagation of matter-wave soliton trainsBright Bose-Einstein Gap Solitons of Atoms with Repulsive InteractionCondensate bright solitons under transverse confinementModulational Instability and Complex Dynamics of Confined Matter-Wave SolitonsBright Solitonic Matter-Wave InterferometerFormation and transformation of vector solitons in two-species Bose-Einstein condensates with a tunable interactionTime-resolved observation of coherent multi-body interactions in quantum phase revivalsDirect Observation of Tunneling and Nonlinear Self-Trapping in a Single Bosonic Josephson JunctionCoherent quench dynamics in the one-dimensional Fermi-Hubbard modelRotating Bose-Einstein condensates with a finite number of atoms confined in a ring potential: Spontaneous symmetry breaking beyond the mean-field approximationQuantum states of dark solitons in the 1D Bose gasLieb-Liniger model: Emergence of dark solitons in the course of measurements of particle positionsStructure of a quantized vortex in boson systemsGround-State Properties of Magnetically Trapped Bose-Condensed Rubidium GasStationary solutions of the one-dimensional nonlinear Schrödinger equation. I. Case of repulsive nonlinearityStationary solutions of the one-dimensional nonlinear Schrödinger equation. II. Case of attractive nonlinearityNumerical solution of the Gross–Pitaevskii equation for Bose–Einstein condensation
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