Nonlinear topological pumping of solitons with time-dependent interactions

Abstract

Thouless pumping of soliton under cyclic and slow modulation of potential opens a window to understand the interplay between topology and interaction. The dynamics of a soliton change from quantized displacement per pumping cycle to its breakdown to self-trap as time-independent nonlinearity increases. Since nonlinearity can be dynamically and flexibly tuned in ultracold atomic systems, time-dependent nonlinearity can be a new degree of freedom to control behaviors of solitons in a Thouless pump. Leveraging time-dependent nonlinearity, we can not only restore quantized displacement of soliton by avoiding self-crossing structures, but also combine topological pumping and self-trap to effectively realize fractional displacement of soliton per cycle. Surprisingly, even when time translation symmetry is broken by linearly changing nonlinearity, we can still achieve the topological transport of a soliton when the initial soliton is symmetrically distributed. Our work provides a new way for dynamical and topological control of solitons.

Graphical abstract

Keywords

nonlinear Thouless pumping / solitons / ultracold atoms, optical lattice / time-dependent interactions

1 Inoduction

Thouless pumping, as quantized charge transport in a cyclically and slowly modulated potential [1], is known as a typical topological transport. If an initial state such as the Wannier state uniformly occupies and sweeps a certain band, the displacement per cycle is related to a topological invariant of the band (i.e., Chern number defined as integral of Berry curvature over a momentum-time space) [2]. More than 30 years after the first prediction, Thouless pumps have been experimentally realized in many systems involving ultracold atoms [3–6], light [7–11], and spin [12]. Hence, these experimental advances have strongly boosted the studies of Thouless pumping and motivated generalization of Thouless pumping to interacting [13–16], non-Abelian [17–19], non-adiabatic [20], non-Hermitian [21] and higher-order topological systems [22, 23].

It is appealing and challenging to study the interplay between interaction and topology in Thouless pumping [24–34], which could open a window to a largely uncharted field of correlated topological states. In the context of few-body systems, the problem can be treated within a generalized Wannier-state formalism [35]. For many-body systems with weak interactions, under mean field approximation one can treat many-body interaction as nonlinearity, so that the potential of a single particle depends on its probability amplitudes [36–38]. Thouless pumping of solitons has been theoretically studied [39–41] and experimentally observed [37, 42]. As the nonlinear strength increases, pumping dynamics of bulk solitons change from quantized displacement, fractional displacement to self-trap [37, 42]. Time-independent nonlinearity plays a crucial role in the topological pumping of bulk and edge solitons. In ultracold atomic systems, nonlinearity originates from onsite interaction, which can be flexibly and dynamically tuned by magnetic and optical Feshbach resonances [43–50]. It is interesting to study how Thouless pumping of a soliton is affected by time-dependent nonlinearity, which may provide a new degree of freedom to control solitons.

In this paper, we study topological pumping of solitons under different types of time-dependent nonlinearity, which is either periodically modulated, linearly changed, or piecewise changed. Under time-independent intermediate nonlinearity, the breakdown of quantized pumping of solitons originates from the appearance of self-crossing structures in the energy spectrum [51], which inevitably makes the evolved state fail to adiabatically follow the instantaneous soliton states.

The principle of restoring quantized transport is to avoid self-crossing structures by introducing time-dependent nonlinearity. By introducing periodic modulation of nonlinearity, overall relatively weak or strong nonlinearity does not change quantized transport or self-trap, respectively. However, by decreasing the nonlinear strength below the critical point of the self-crossing structures at the tunneling time, we can still realize quantized transport.

By linearly changing nonlinearity, even though the translation symmetry of the Hamiltonian is broken, we can still achieve topological pumping if the profile of the soliton is symmetrical and overall nonlinearity is relatively weak. Finally, we can combine quantized transport and self-trap by flexibly tuning nonlinearity to effectively achieve fractional displacement of solitons per cycle.

The rest of the paper is organized as follows. In Section 2, we introduce the nonlinear interacting Rice−Mele model and present the dynamical evolution of solitons in different regimes of time-independent nonlinearity. In Section 3, we study the pumping dynamics of solitons under time-dependent nonlinearity which is either periodically modulated or linearly changed. In Section 4, we combine quantized transport and self-trap to control solitons. In Section 5, we give a brief summary and discussion.

2 Nonlinear Rice−Mele model and Thouless pumping

We consider the motion of an ultracold atomic gas in a periodically modulated optical superlattice formed by the superposition of laser beams with short and long wavelengths, and the short wavelength is half of the long wavelength. Several experimental setups have realized such a scenario and observed Thouless pumping of ultracold atoms [3, 4], and even its breakdown in the presence of interactions [52]. Moreover, the interaction can be dynamically tuned by magnetic or optical Feshbach resonances [44, 53]. We can further assume the interaction is a function of time. Under mean field approximation, the interaction can be treated as a kind of nonlinearity, and the motion of atomic gas is governed by the nonlinear Schrödinger equation:

(1)\mathrm{i}{\displaystyle \frac{\mathrm{\partial }{\mathrm{\Psi }}_n(t)}{\mathrm{\partial }t}}= {\displaystyle \sum _m{H}_{nm}^{RM}(t){\mathrm{\Psi }}_m(t)-g(t)|{\mathrm{\Psi }}_n(t){|}^2{\mathrm{\Psi }}_n(t).}

Here, {\mathrm{\Psi }}_{n(m)}(t) is the amplitude of the wave function at site n(m) and time t, g(t)> 0 is the strength of the focusing nonlinearity at time t, and H_{nm}^{RM}(t) is the linear tight-binding Rice−Mele model [54–57], which is given by

(2)\begin{array}{rl}{H}_{nm}^{RM}(t)=& -[J+(-1{)}^{m+1}\delta \sin (\mathrm{\Omega }t+{\varphi }_0)]{\delta }_{n-1,m}\\ & -[J+(-1{)}^m\delta \sin (\mathrm{\Omega }t+{\varphi }_0)]{\delta }_{n+1,m}\\ & -\mathrm{\Delta }(-1{)}^m\cos (\mathrm{\Omega }t+{\varphi }_0){\delta }_{n,m}.\end{array}

Here, both the hopping term and the onsite energy term change with time and space. These modulations introduce circular changes of hopping differences between inter and intra-cells and energy bias between nearest neighboring sites. J, \delta, \mathrm{\Omega }, {\varphi }_0, and \mathrm{\Delta } are the averaged hopping strength, modulation strength of the hopping term, modulation frequency, modulation phase, and onsite energy, respectively. The modulation strength \mathrm{\Omega } should be small enough to make sure the adiabatic condition for Thouless pumping is satisfied in the linear limit g=0. In the calculations below, we set J=1 as a dimensionless unit, and the other parameters \delta =0.5 and \mathrm{\Delta } =1 are in the units of J. The number of cells is N=50 and periodic boundary condition is adopted.

In the linear case, there are two bands of the system with Chern numbers C_1=1, C_2=- 1 for the lower and upper ones, respectively. If an initial Wannier state uniformly occupies the lower or upper energy bands, the nonzero Chern number can support quantized displacement [58, 59]. Defining the mean position as

(3)⟨x(t) ⟩={\displaystyle \sum _{n}n|{\mathrm{\Psi }}_n(t){|}^2,}

the change of mean position in one cycle \mathrm{\Delta } x=⟨ x(T)⟩ -⟨x(0)⟩ satisfies \mathrm{\Delta }x= Cd, where T=2\pi /\mathrm{\Omega } is the time period and d=2 is the length of a unit cell.

When adding time-independent nonlinearity, solitons can bifurcate from the lower or upper bands. The soliton is quite similar to the Wannier state of the band from which it bifurcates [40, 60, 61]. By choosing a soliton bifurcating from the lower band as the initial state, we calculate the mean position as a function of time for different nonlinearity strengths; see Fig.1(a)−(c). The dynamically evolved soliton is obtained by directly solving Eq. (1) using the 4th-order Runge-Kutta method under a given initial soliton; see Appendix A for details of the numerical method. For comparison, we also calculate the mean position of instantaneous solitons by finding the static eigenstates of the nonlinear Hamiltonian at time t; see Appendix B for details of an iterative approach. For relatively weak nonlinear strength g=2, the soliton will be shifted by 2 sites in one pumping cycle [Fig.1(a)]. As nonlinear strength increases to g=4, the quantization transport of the soliton breaks down [Fig.1(b)]. As nonlinear strength further increases to g=6.5, the soliton becomes self-trapped around its initial position [Fig.1(c)]. In the first and third case, the mean positions of dynamically evolved solitons and instantaneous static solitons coincide with each other, indicating that the dynamically evolved solitons can follow the instantaneous static solitons. However, in the second case, the breakdown of quantization may come from the fact that the dynamically evolved solitons cannot follow the instantaneous static solitons.

To understand the different behaviors, we calculate the eigenvalues of instantaneous static solitons at individual time t for different nonlinearity strengths; see blue lines in Fig.1(d)−(f) with the same nonlinearity corresponding to Fig.1(a)−(c), respectively. For completeness, we also show extended eigenstates known as nonlinear Bloch states which satisfy Bloch’s theorem; see the green lines. In Appendix D, we also provide the approach for calculating the nonlinear Bloch states and demonstrate loop structures in nonlinear Bloch bands which are absent in linear systems. Since the nonlinear Bloch bands have been well studied [51], in the following, we only focus on the pumping dynamics of solitons.

In the case of quantized transport, the energy of the soliton is a continuous function of time. Even if the soliton does not occupy the energy band uniformly, its transport can still obey the Chern number of the linear energy band of the bifurcation [39]. In the case of non-quantized transport, the energy of the soliton becomes disconnected around the tunneling time when the energy difference between neighboring sites becomes zero; see Fig.1(e). Around the tunneling time [T/4,3T /4], nonlinearity becomes dominant relative to other parameters, leading to the emergence of an additional soliton. The soliton is intrinsically connected to the single-site solution observed in the limit of g\stackrel{}{\to }\mathrm{\infty } [62]. Their energies attached to the original spectrum form two X-shaped structures, which are termed as self-crossing structures. The endpoints of these self-crossing structures are termed as “dead end”. These X-shaped bands, along with looped and swallowtail structures described in previous literature [63–65] collectively form a unique feature of nonlinear systems without a linear counterpart. The existence of self-crossing structures leads to the destruction of the adiabatic path for the soliton [51]. Fig.1(b) shows that a soliton initially (at t=0) localized at the site n=51 evolves by adiabatically following the path of instantaneous soliton up to t\approx \frac{T}{4}, slightly moving towards site n=52. Meanwhile, in Fig.1(e), the energy band of the soliton reaches “dead-end”, beyond which there are no stable instantaneous solitons to adiabatically follow. The dynamically evolved soliton is then forced to perform a non-adiabatic jump to the lower part of the band. A similar non-adiabatic jump is observed again at t\approx \frac{3T}{4}. As nonlinear strength increases, the self-crossing structures will expand, and the “dead ends” of two self-crossing structures are connected to each other, leading to two continuous soliton modes. If one of the solitons is projected into the two linear energy bands, there exist Rabi oscillations between the linear energy bands [41]. Since the soliton populates the two bands, its displacement in the pumping dynamics is governed by the averaged Chern numbers of the two bands. The total zero Chern number results in the self-trap of soliton; see Fig.1(c).

We can separate the nonlinear strength into regimes of quantized transport, non-quantized transport, and self-trap, by fixing other parameters. The boundaries of the different regimes can be determined by analyzing the energy of solitons. The critical value of nonlinear strength between the regimes of quantized transport and non-quantized transport is the point at which self-crossing first appears. In the regime of quantized transport, the energy spectrum of solitons forms a smooth curve; see Fig.1(d). As the system approaches the critical value, the energy spectrum at the tunneling points starts to develop cusps, so the first derivative of energy with respect to time at the tunneling point changes from connection to disconnection. However, as soon as the nonlinear strength exceeds the critical value, the cusps evolve into self-crossing structures, around which the trajectory of the instantaneous solitons becomes discontinuous and adiabaticity is broken. Hence, the critical nonlinear strength between the regime of quantized transport and non-quantized transport is approximately equal to g_{c,1}=3.15 under the same parameters of the linear Rice−Mele model mentioned before. At the critical point between regimes of non-quantized transport and self-trap, the energy spectrum of solitons changes from the discontinuous self-crossing structures to two continuous soliton modes. We estimate the second critical value as g_{c,2}=6.2. Guided by the different pictures in different time-independent nonlinearities, in the following we will introduce different types of time-dependent nonlinearity to avoid self-crossing structures, so we can recover quantized transport and control the solitons.

3 Pumping dynamics under time-dependent interaction

Time-dependent nonlinearity provides a new degree of freedom to tailor the behaviors of solitons in pumping dynamics. In this section, we will consider three types of time-dependent nonlinearity, i.e., periodically-, linearly-, and piecewise-changing nonlinearities. For periodically-changing nonlinearity, if the nonlinear strength changes between regimes of quantized transport and self-trap, it is natural to expect that the behaviors of soliton will be quite similar to the cases of time-independent nonlinearity. It is interesting to understand how the soliton behaves when the nonlinear strength varies through different regimes of nonlinearity. For linearly-changing nonlinearity, the Hamiltonian is no longer cyclic in a pumping cycle, and time periodicity is broken. It is an open question whether the quantized transport still holds in the non-cyclic systems. For piecewise-changing nonlinearity, it is interesting to understand how to avoid self-crossing structures by connecting regimes of quantized transport and self-trap. In the following, we will address these questions one by one.

3.1 Periodically-changing interaction

Without loss of generality, we consider the time-dependent nonlinearity takes the form of the absolute value of the sine function,

(4)g(t)=g_0+{\delta }_g\left|\sin \left({\displaystyle \frac{1}{2}}\mathrm{\Omega }t\right)\right|,

where g_0, {\delta }_g, and \mathrm{\Omega } are the initial nonlinear strength, modulation strength, and frequency of nonlinearity, respectively.

We study the pumping dynamics of solitons bifurcating from the lower band in different cases, that is, g(t) is in the regimes of quantized transport and self-trap, and g(t) crosses the boundaries of different regimes. Fig.2(a)−(d) show the nonlinear strength as a function of time under different parameters, i.e., ( g_0,{\delta }_g)=(3 ,-2),(10,- 2),(7 ,-3.5),(7,- 5.5), respectively. Corresponding to the above cases, we calculate the time evolution of density distribution and mean position [Fig.2(e)−(h)] and energy of instantaneous soliton as a function of time [Fig.2(i)−(l)], where the colors in energy mark the relative position in each unit cell,

(5)r=\text{mod}(⟨x (t)⟩, d),

where r is equal to ⟨x(t) ⟩ modulo d. In the regime of quantized transport, we can still realize quantized transport of soliton, since the evolved state can smoothly follow the continuous eigenstate of soliton. The wavepacket will expand when nonlinear strength decreases and shrink when nonlinear strength increases; see Fig.2(e). This is different from the case of time-independent nonlinearity, where the profile of the soliton almost does not change. In the regime of self-trap, the soliton almost stays where it starts, similar to self-trap in the case of time-independent nonlinearity. In both cases, the basic pictures are similar to the corresponding cases of time-independent nonlinearity.

When the nonlinear strength varies through different regimes, the pumping dynamics are quite different from the time-independent case. To avoid self-crossing structures, the key tactic is to tune the nonlinear strength around the tunneling time [T/4,3T /4] below the first critical value g_{c,1}. We compare two different pumping dynamics in cases of g(T/4)>g_{c,1} and . For g(T/4)>g_{c,1}, the wavepacket of the soliton becomes diffusive after the first tunneling time \pi /(2\mathrm{\Omega }); see Fig.2(g). As a result, the mean position oscillates dramatically after time T/ 4, and the mean-position shift also deviates from the expected quantization value 2. We find that there are self-crossing structures in the energy spectrum; see Fig.2(k). Fig.2(g) shows that a soliton initially (at t=0) peaked on site n=51 evolves by adiabatically following the path formed by the green part of the energy band in Fig.2(k) up to t\approx \frac{T}{4}. Ultimately, the green band reaches a “dead-end”, beyond which there are no stable soliton solutions to adiabatically follow. The soliton is then forced to perform a non-adiabatic jump to the blue part of the energy band. In the non-adiabatic jump, most of the dynamically evolved soliton will jump to the stable nonlinear eigenstates with close energy. However, a small fraction remains in the original state. It becomes a superposition of different eigenstates and breaks the original profile. The soliton is torn apart and leads to diffusion of wavepacket. A similar “dead end” is observed again at t\approx \frac{3T}{4}. For , the soliton wavepacket maintains a well-localized profile throughout the pumping cycle; see Fig.2(h). The mean position is shifted by 2 sites in one cycle, restoring quantization transport. In this case, the eigenvalue of the soliton is a continuous function of time, and the evolved state can adiabatically follow the instantaneous static soliton; see Fig.2(l). Although the beginning and end of the pumping process are in the regime of self-trap, the quantized transport is not affected.

3.2 Linearly-changing interaction

In this subsection, we consider the nonlinear strength that takes the form of a linear function,

(6)g(t)=g_0+\nu t,

where g_0 is the initial strength, and \nu is the changing velocity. We explore how the pumping dynamics change under such a kind of noncyclic modulation. In linear Thouless pumping, the modulation of parameters should be a periodic function of both space and time. The space and time translation symmetries guarantee that the eigenstates are periodic in both quasi-momentum and time space. Hence, the Chern number governing the displacement can be defined as an integral of Berry curvature in the quasi-momentum and time space. However, because of the robustness of topological features, the quantized displacement is still preserved when adding disorder in both space and time, which breaks the spatial and time translation symmetries [14]. It is unclear whether the linearly-changing nonlinearity will break quantized transport. By preparing a soliton bifurcating from the lower band as the initial state, we show the mean position as a function of time under different modulation phases, {\varphi }_0=0 and {\varphi }_0=\pi / 3; see Fig.3(a, b), respectively. The other parameters are chosen as g_0=1.5, \nu =\frac{3}{4000}. The quantized displacement is preserved and broken in the case of {\varphi }_0=0 and {\varphi }_0=\pi / 3, respectively. To understand the difference, we show the maximally localized Wannier state (MLWS) and single-peak solitons under different nonlinear strengths; see Fig.3(c, d) for {\varphi }_0=0 and {\varphi }_0=\pi /3, respectively. For {\varphi }_0=0, we observe that if MLWS is symmetric, then the single-peak soliton will also be symmetric. This is because the system possesses inversion symmetry at the initial time,

(7)RH (-k)R=H( k),

where H(k) is the Hamiltonian in momentum space, and R is the inversion operator.

As the nonlinear strength increases, although the shape of the instantaneous soliton changes, the relative position of the single-peak soliton within a unit cell remains unchanged at the initial time due to the symmetry of the soliton. The mean position and the peak position of the soliton coincide with those of the corresponding MLWS; see Fig.3(c) and (e). When we select a symmetric single-peak soliton as the initial state, the soliton will almost follow the trajectory of the symmetric MLWS. After one pumping cycle of linear Hamiltonian, the linear Hamiltonian returns to itself and the mean position of the MLWS is shifted by 2 sites, so the mean position of the soliton is shifted by the same sites. We must emphasize that the nonlinear strength g(t) remains within the regime of quantized transport throughout the pumping cycle, which ensures the perfectly adiabatic following of the instantaneous eigenstate of the soliton. However, for {\varphi }_0=\pi / 3, both the MLWS and the single-peak soliton are not symmetric at the initial time [Fig.3(d)], due to the absence of inversion symmetry. With increasing nonlinearity, the soliton is more localized, and the relative position in each unit cell is closer to the position of the peak of the MLWS; see Fig.3(e). After one pumping cycle of the linear Hamiltonian, the linear Hamiltonian returns to itself, and the mean position of the MLWS is shifted by 2 sites. Although the peak of the soliton is dragged by the MLWS and shifted by 2 sites in one cycle, due to the asymmetric profiles, the mean position shift of the soliton departs from 2 sites.

If the Hamiltonian of a topological pump possesses inversion symmetry at the initial time, we can always realize quantized topological pumping after adding linearly changing nonlinearity. This argument is not limited to the nonlinear Rice−Mele model studied here, but it can also apply to other topological pumps. Although we cannot test the argument case by case, to increase the credibility, we also investigate the role of inversion symmetry in the nonlinear Aubry−André−Harper (AAH) model [66–68]; see Appendix C.

3.3 Piecewise-changing interaction

In the previous subsection, the nonlinear strength is in the regime of quantized transport through the pumping cycle. If we linearly vary nonlinearity from the regime of quantized transport to the regime of self-trap in one cycle, it is inevitable to cross the regime of non-quantized transport in the tunneling time [T/4,3T /4]. To avoid this obstacle, we can design a piecewise-changing nonlinearity as

(8)

Here, t_0=9T/20>T/4 and , \nu =\frac{9}{250}. In this way, the nonlinear strength in the first tunneling time T/4 is in the regime of quantized transport and in the second tunneling time 3T/4 is in the regime of self-trap. Fig.4(a)−(c) show the variations of the nonlinear strength, the mean position, and the eigenvalues of solitons in one pumping cycle, respectively. Before time t_0, the nonlinear strength g_1(t) is in the regime of quantized transport, which allows the soliton to move unidirectionally by one site around the tunneling time. After passing the first tunneling time, we linearly ramp the interaction into the regime of self-trap before the second tunneling time. This design avoids the self-crossing structure of the energy spectrum to support adiabatic evolution of the states. Because the nonlinear strength at the second tunneling time 3T/4 is in the regime of self-trap, the soliton becomes self-trapped. Finally, the mean position of the soliton changes from 51 to 52 sites, equivalent to a fractional displacement of 1/ 2. For comparison, we also calculate the variation of the mean position and eigenvalues in the case of time-independent interaction, g_2(t)= 1.6 and g_3(t)= 8.8; see the dashed lines in Fig.4(b). We find that the mean position under the piecewise-changing interaction first follows the trajectory in the case of g_2(t)= 1.6 and then smoothly changes to follow the trajectory in the case of g_3(t)= 8.8. Meanwhile, the evolved soliton under piecewise-changing interaction changes from the eigenvalue of the soliton in the case of g_2(t)= 1.6 to the one in the case of g_3(t)=8.8 around T/2. The piecewise-changing interaction combines about half of the topological pumping and half of the self-trap, which we effectively realize 1/2 fractional displacement in one pumping cycle. This feature motivates us to design periodic sequences of interaction to control the soliton in multiple pumping cycles.

4 Control of quantized transport by varying interaction

In the previous section, we have already learned the behaviors of solitons in different regimes. With the knowledge at hand, we can control the soliton by either topological pumping or self-trap, or their combination. Combining the topological pumping and self-trap, we can effectively realize fractional displacement of soliton by designing the periodic sequences of interaction. The key idea is to avoid self-crossing structures in a pumping cycle when nonlinear strength g(t) passes through the regime of non-quantized transport. Without loss of generality, we design two kinds of periodic sequences of interaction with half of the modulation frequency, g(t)=g(t+2 T). For explicitness, in two pumping cycles the nonlinearity g(t) takes the form of a modulated sine function g(t)=g_0-{\delta }_g\sin (\frac{1}{2}\mathrm{\Omega }t), and the other takes the form of a piecewise function,

Fig.5(c, d) show the time evolution of the density distribution in four pumping cycles in the two cases. In the first case, since we know the tunneling time [T /4 ,3T/4] in a linear Thouless pumping, to avoid the emergence of the self-crossing structure, we can choose the parameters as g_0=6.5 and {\delta }_g=-5.5. Such a design makes the soliton move unidirectionally by two sites in the first pumping cycle and self-trapped in the second pumping cycle. At last, the soliton is shifted by 2 unit cells in four pumping cycles, equivalent to fractional topological pumping of value 1 /2. In the second case, the parameters are chosen as g_0=2, t_0=2T/5, t_1=T/2 and \nu = \frac{3}{100}. Such design makes the soliton move unidirectionally by one site from t=0 to T/ 2, then remain self-trapped from t=T/2 to 3T/2, and move unidirectionally by one site from t=3T/2 to 2T. Equivalently, the soliton is also shifted by half of a unit cell per pumping cycle.

We can design arbitrary fractional values of topological pumping by combining quantized topological transport and self-trap. For example, we separate the pumping sequences as quantized topological pumping over N_1 cycles and self-trapping over N_2 cycles. The averaged displacement per cycle is \frac{N_1{C}_1}{N_1+N_2}d. This scheme differs from the fractional topological pumping studied in Refs. [37, 41], where the displacement in individual pumping cycles is determined by the averaged Chern number of the occupied bands. In our schemes, the effectively fractional values of displacement are just practices of the quantized topological transport and self-trap reported in Refs. [37, 41].

5 Summary and discussion

We mainly explore how to tailor the behaviors of solitons in nonlinear Thouless pumping under time-dependent nonlinearity. When the time-dependent nonlinear strength resides solely in the regimes of quantized transport or self-trap, the major features of soliton dynamics are similar to those under time-independent nonlinearity. However, when the nonlinear strength is in the regime of non-quantized transport, the appearance of self-crossing structures in the soliton spectrum can destroy the quantized transport. We propose to change the nonlinear strength around tunneling time to avoid self-crossing structures and restore the quantized transport. More interestingly, when the profile of the initial soliton is symmetric, even though linearly-changing nonlinearity breaks the time translation symmetry, we can still achieve quantized transport of solitons. By combining Thouless pumping and the self-trap of solitons, we can effectively realize fractional displacement of the soliton per pumping cycle.

It deserves further study to understand the interplay between nonlinearity and topology. In the few-body systems, we have realized that spatiotemporal modulation of interaction can induce topological pumping of multiparticle bound states out of continuum and in the continuum, where there is no topological transport at all in the single-particle case [16, 69]. In this case, the displacement per pumping cycle is related to the multiparticle Chern number of the filled multiparticle Bloch band, which turns out to be a quantized value. It is unclear how the picture changes in the case of many-body systems under the mean-field approximation. In the next steps, it is worthwhile to study fractional topological states induced by nonlinearity without a single-particle counterpart.

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