Manipulating topological charges via engineering zeros of wave functions

The stability of topological defects is generally guaranteed by the conservation of their topological charges MerminRMP1979 ; ChaikinCambridge1995 ; KlemanSpringer2003 . In liquid crystals, their stability is determined by the homotopy group of loops surrounding the singularities in the real space StephenRMP1974 ; BallSpringer2017 ; FumeronLCR2023 ; FumeronEPJST2023 . Meanwhile, in recent two decades, the great efforts have been devoted to studying topological defects in the Bloch bands in the momentum space using similar mathematical tools CooperRMP2019 ; ZhangAP2019 ; HasanRMP2010 ; ShenSpringer2017 ; AsbothSpringer2016 . In this way, the topological phase transition can only occur when the conduction and valence bands are closed and reopened at some critical parameters, and in the topological phase, the bulk-edge correspondence ensures the emergence of edge states HasanRMP2010 ; ShenSpringer2017 ; AsbothSpringer2016 . The search for various topological phases and their potential application in topological quantum computation remains one of the most important themes in condensed matter physics and quantum simulation HasanRMP2010 ; ShenSpringer2017 ; AsbothSpringer2016 ; NayakRMP2008 ; BulutaRPP2011 ; BlochNP2012 ; GrossSince2017 ; TerhalRMP2015 ; TokuraNRP2019 . In experiments, the topology of bands can well explain the quantum Hall effect HeNSR2014 ; KlitzingNRP2020 ; ChangRMP2023 , Thouless pumping CitroNRP2023 ; NakajimaNP2016 ; LohseNP2016 ; JurgensenNature2021 , quantized Hall drifts AidelsburgerNP2015 ; ZhuSince2023 , and other topological phenomena in terms of the Berry curvature Berry1984 ; XiaoRMP2010 .

The robustness of topological phases implies that it is hard to engineer the topological charges without destroying the geometry of the manifolds ShenSpringer2017 ; AsbothSpringer2016 ; XiaoRMP2010 ; RachelRPP2018 ; DingNRP2022 . This general conclusion also means that we can engineer the topological charges by controlling the singularities in the manifolds. These singularities correspond to the touches of conduction and valence bands of the Bloch bands or the zeros of wave functions ShenSpringer2017 ; AsbothSpringer2016 ; XiaoRMP2010 . As we have shown in Refs. ZhaoPRE2021 ; WangPS2024 , the zeros of wave functions carries the nonzero topological charges in real space, which can be transferred to the system under consideration if they are carefully designed. As a central idea of this work, we expect that engineering the zeros of wave functions could provide an alternative approach for manipulating topological charges.

This Letter reports the manipulation of topological charges in a toroidal Bose condensate by engineering the zeros of wave functions. The major idea is that the acceleration motion of dark solitons usually involves the density zeros ZhaoPRA2020 ; MengJPB2024 , which can be used to manipulate the topology of the condensate. To this end, we consider one or more dark solitons imprinted on a ring-shaped condensate and demonstrate that the topological charge, characterized by its winding number, can be manipulated by controlling the relative velocities between the solitons and their background density currents through accelerating the solitons. We propose two protocols for accelerating dark solitons: loading (I) impurity atoms or (II) bright solitons in dark solitons and applying weak forces on them. These protocols can address the challenges of directly accelerating dark solitons due to their backgrounds. Each crossing of the zero relative velocities (corresponding to the emergence of density zeros) induces a abrupt change in the winding number, with the direction of crossing determining whether winding number increases or decreases. We further show that accelerating (III) multiple dark solitons provides diverse techniques for tailoring topological charges. This idea may be used to engineer the topology of the Bloch bands, leading to the control of the associated edge states.

Physical Model: We use a binary toroidal Bose condensate as platform to demonstrate our idea. After rescaling the atomic mass and Planck’s constant to be $1$, the dimensionless mean-field energy of system in polar coordinates can be expressed as DalfovoRMP1999 ; PitaevskiiOUPress2016

where $\psi_{\rm ds}$ and $\psi_{\rm lw}$ represent the dark solitons and localized waves components, respectively. The localized waves (either impurity atoms or bright solitons) are trapped by dark solitons through the nonlinear interaction $g_{\rm cp}|\psi_{\rm ds}|^{2}|\psi_{\rm lw}|^{2}$. The dimensionless paraments $g_{\rm ds}$, $g_{\rm lw}$ and $g_{\rm cp}$ represent the corresponding interaction strengths among atoms. To avoid snaking instability CarreteroNonlinearity2008 ; FrantzeskakisJPA2010 , the trapping frequency $\omega$ must be sufficiently large along the radial direction $r$, making the condensate appearing as a quasi-one-dimensional ring along the angular direction $\theta$. The forces $F$ acting on the localized waves can accelerate the dark solitons without modifying their background density, which provides a good protocol for controlling the motions of dark solitons ZhaoPRA2020 ; MengJPB2024 . Hereafter, we set the radius of the toroidal condensate to $R=50$. In the following, we present three different strategies for manipulating the topological charges of the condensate.

(I): We consider a dark soliton coupled with some impurity atoms, where the impurity atoms are driven by a weak force $F$, as shown in Fig. 1 (a). For convenience, we replace the localized wave $\psi_{\rm lw}$ with $\psi_{\rm im}$ to account for the impurity atoms. The initial wave function of these two components can be expressed as MengNJP2023 ; MengJPB2024

In the above states, $\psi_{\rm gs}$ is the toroidal ground state of the condensate, which can be obtained through the imaginary-time evolution method YangSIAM2010 ; BaoKRM2013 ; BaoJCP2003 ; ChiofaloPRE2000 ; LehtovaaraJCP2007 by solving the two-dimensional Gross-Pitaevskii (GP) equations, assuming that two components admit the same density. The parameters $v_{\rm s}$, $v_{\rm b}$ and $c=\sqrt{g_{\rm ds}|\psi_{\rm gs}|^{2}}$ represent the soliton velocity, background density velocity, and sound speed, respectively. Therefore, $v_{\rm r}\equiv v_{\rm s}-v_{\rm b}$ signifies the relative velocity between the soliton and background density. Seeing Eq. (2), we can easily check that when $R|\theta|$ is sufficiently large, $|\psi_{\rm ds}|^{2}=|\psi_{\rm gs}|^{2}$ while $|\psi_{\rm im}|^{2}=0$.

The phase jump of dark soliton is given by $\Delta\phi=\int{(2\pi iz)^{-1}dz}$ with $z=\psi_{\rm ds}e^{-iv_{\rm b}R\theta}/\psi_{\rm gs}$, and $\Delta\phi\in[-\pi,\pi]$. Notably, when $v_{\rm r}=0$, $\psi_{\rm ds}=0$ occurs at $\theta=0$, yielding the zero of dark soliton, which is essential for its phase jump. In this way, the topological phase transition can only occur when the relative velocity $v_{\rm r}=0$. During the evolution of GP equations, the profiles of two components can still be described by Eq. (2), with $v_{\rm s}$ and $v_{\rm b}$ being time-dependent, thus $v_{\rm r}=0$ can still be employed to identify the phase transition and the associated phase jump of dark soliton. Furthermore, the single-valued nature of the order parameter requires that the phase variations along the ring should satisfy BrandJPB2001 ; MateoPRA2015 ; ShamailovPRA2019

where $2\pi Rv_{\rm b}$ represents the phase shift induced by the background density current; see Eq. (2a). The integer $Q_{\rm w}$ signifies the winding number of toroidal condensate, which can be defined as the number of times the phase winds through $2\pi$ in a closed loop counterclockwise around the ring. In recent experiments RyuPRL2007 ; RamanathanPRL2011 ; WrightPRL2013 ; RyuPRL2013 ; EckelNature2014 , the winding number can only be manipulated by a rotating weak link in the absence of dark solitons, which is by changing $v_{\rm b}$; however, an easier way to achieve this is by changing $v_{\rm r}$, see below.

We present our numerical results employing the Fourier pseudo-spectral method YangSIAM2010 ; BaoKRM2013 ; BaoJCP2003 with the Strang splitting scheme to solve the GP equations in Fig. 1. The initial parameters for this our simulation are $Q_{w}|_{t=0}=0$ and $v_{\rm r}|_{t=0}\approx 0.01$, and the weak force is designed as $F=-(-1)^{\lfloor t/500\rfloor}0.05\sin(2\pi t/500)$, where $\lfloor\ast\rfloor$ denotes the floor function. Since dark soliton admits a negative inertial mass, its acceleration is directed opposite to the direction of weak force MengNJP2023 ; FrantzeskakisJPA2010 ; MengJPB2024 ; AycockPNAS2017 ; HurstPRA2017 . The density and phase of dark soliton at various moments are shown in Fig. 1 (d), and the detailed dynamical evolutions of the dark soliton and impurity atoms are presented in Sec. SI of Supplemental Material (SM), which includes a video representation. The positions of the dips in the dark soliton are monitored to determine $v_{\rm s}$ while the phase gradient of the background density enables the determination of $v_{\rm b}$, allowing for the identification of $v_{\rm r}$. The winding number $Q_{\rm w}$ can be obtained by integrating the phase gradient of the condensate along the ring. The evolution of $v_{\rm r}$ and $Q_{\rm w}$ are depicted in Fig. 1 (b) and (c), respectively. Our results demonstrate that the density zero occurs at $v_{\rm r}=0$, precisely corresponding to the abrupt jump points of $Q_{\rm w}$. Moreover, the crossing direction of the zero relative velocity determines whether the winding number increases or decreases. As previously mentioned, the profiles of the two components can still be described by Eq. (2) during evolution, which enables us to theoretically describe the evolution of the soliton by quasiparticle theory using the Lagrangian variational method (see Sec. SII of SM for details). As depicted in Fig. 1 (b), the results given by quasiparticle theory agree with those of numerical simulation.

In Refs. ZhaoPRE2021 ; WangPS2024 , we have shown that the wave function admits a phase jump of $\pm\pi$ at the density zero, with the sign determined by the approach direction. For one dark soliton, we find that

Therefore, as the zero relative velocity is approached from different directions, the phase jump direction varies. For instance, when $v_{\rm r}$ changes from a positive value to zero, a phase jump of $-\pi$ occurs; and when $v_{\rm r}$ changes from zero to a negative value, a phase jump of $\pi$ occurs; then a total phase jump of $2\pi$ occurs, as demonstrated in Fig. 2 (a); and vice versa. In one-dimensional settings with open boundaries, this $2\pi$ phase jump occurs in accelerating dark soliton but is not observable. Linking the two ends of a one-dimensional dark soliton to form a ring configuration, we uncover a significant finding. As derived from Eq. (3), a $\pm 2\pi$ phase jump induced by the crossing of zero relative velocity brings a change of $\pm 1$ in the winding number $Q_{\rm w}$, as demonstrated in Fig. 2 (b). Explicitly, each crossing of zero relative velocity induces a change in the winding number, with the direction of the crossing determining whether the winding number increases or decreases.

The density zeros of wave functions have deep origins in physics. These singularities induce the emergence of nonintegrable phase factors in certain parameter spaces, with the integration encircling these singularities typically yielding a nonzero topological phase. This idea can even be traced back to Dirac’s work on monopoles DiracPRSLA1931 , wherein the relation between nonintegrable phases and the integer topological magnetic charges. Furthermore, Nye and Berry pointed out that the wave dislocations in wave systems correspond to the zeros of wave functions, and the phase singularities play essential roles in many striking phenomena in wave physics NyePRSLA1974 ; BerryLSA2023 . The dynamics of zeros are also closely related with the topological properties of physical systems DuanPRE1999 ; DuanPRE199960 , such as quantifying the spatiotemporal behavior of spiral wave by tracking phase singularities ZhangCPL2007 ; LiPRE2018 ; HePRE2021 . These results, together with the results presented in this work, suggest that the zeros of wave functions have the profound implications for physics.

(II): The above protocol can only realize two states, characterized by the winding number of $Q_{\rm w}|_{t=0}\pm 1$. This limitation prompts us to consider a more general question: can we manipulate the winding number to a arbitrary integer number? In following, we propose a feasible protocol to progressively increase or decrease winding number based on the oscillation of the spin soliton driven by a weak constant force $F$ ZhaoPRA2020 ; MengPRA2022 ; MengCNSNS2022 , as shown in Fig. 3 (a). Here, we replace the localized wave $\psi_{\rm lw}$ with the bright soliton $\psi_{\rm bs}$ and set $g_{\rm ds}+g_{\rm bs}=2g_{\rm cp}$. The initial wave function of these two components can be expressed as ZhaoPRA2020 ; MengPRA2022

where $c_{\rm s}=\sqrt{g_{\rm cp}-g_{\rm lw}}$, which is related to the velocity limit of spin soliton MengPRA2022 . The basic characteristic of spin soliton admits a uniform total density $|\psi_{\rm ds}|^{2}+|\psi_{\rm bs}|^{2}=|\psi_{\rm gs}|^{2}$, as shown in Fig. 3 (a). Applying a weak constant force $F=0.005$ on bright soliton and setting initial parameters as $Q_{w}|_{t=0}=0$ and $v_{\rm r}|_{t=0}\approx-0.01$, we perform numerical simulation and results are shown in Fig. 3. The density and phase of dark soliton at various moments are shown in Fig. 3 (d), and the detailed dynamical evolutions of the dark and bright solitons are presented in Sec. SI of SM, which includes a video representation. We find that that the relative velocity $v_{\rm r}$ exhibits a periodic oscillation, which corresponds to the ac oscillation induced by periodic transition between negative and positive inertial mass ZhaoPRA2020 ; MengPRA2022 ; MengCNSNS2022 . Notably, density zero only emerges when the relative velocity $v_{\rm r}$ crosses zero from positive to negative values, as pointed by the green circles in in Fig. 3 (b), where the spin soliton admits the negative inertial mass. There is no density zero for inverse crossing direction, which is due to different properties of spin solitons with different inertial masses ZhaoPRA2020 ; MengPRA2022 ; MengCNSNS2022 . Therefore, the dark soliton does admit the density zero at zero relative velocity in the same manner, which leads to a step-like increment in the winding number $Q_{\rm w}$, as presented in Fig. 3 (c). Also, as depicted in Fig. 3 (b), the results given by quasiparticle theory (see Sec. SII of SM for details) agree with those of numerical simulation. However, it should be emphasized that the system may become unstable and eventually break when the winding number $Q_{\rm w}$ reaches very high values with the increment of background density velocity $v_{\rm b}$. For the parameters used in Fig. 3, the maximum observed value for $Q_{\rm w}$ is approximately $30$.

(III): We investigate two well-separated dark solitons with coupling impurity atoms driven by two weak force $F_{1,2}$, as shown in Fig. 4 (a), which can be used as a platform to demonstrate the arbitrary tunability of winding number. The initial states for two dark solitons and coupled impurities are described by $\psi_{\rm tds}=\big{\{}iv_{\rm r}/c+\sqrt{1-v_{\rm r}^{2}/c^{2}}\tanh\big{[}\sqrt{c^{2}-v_{\rm r}^{2}}R\theta_{1}\big{]}\big{\}}\big{\{}iv_{\rm r}/c+\sqrt{1-v_{\rm r}^{2}/c^{2}}\tanh\big{[}\sqrt{c^{2}-v_{\rm r}^{2}}R\theta_{2}\big{]}\big{\}}e^{iv_{\rm b}R\theta}\psi_{\rm gs}$ and $\psi_{\rm tim}=\varepsilon\big{\{}{\rm sech}\big{[}\sqrt{c^{2}-v_{\rm r}^{2}}R\theta_{1}\big{]}e^{iv_{\rm s}R\theta_{1}}+{\rm sech}\big{[}\sqrt{c^{2}-v_{\rm r}^{2}}R\theta_{2}\big{]}e^{iv_{\rm s}R\theta_{2}}\big{\}}\psi_{\rm gs}$ with $\theta_{1,2}=\theta\pm\pi/2$. Each soliton-impurity is driven by a weak force $F_{1,2}=-(-1)^{\lfloor t/T_{1,2}\rfloor}0.03\sin(2\pi t/T_{1,2})$. We present different evolution of winding number by adjusting period ratio $T_{1}/T_{2}$ in Fig. 4 (b). When $T_{1}/T_{2}=1$, the winding number $Q_{\rm w}$ jumps two times of the case of one dark soliton (see in in Fig. 1 (c)); when $T_{1}/T_{2}\neq 1$, the winding number $Q_{\rm w}$ evolves in a near random way due to the different frequencies and directions of the density zero crossings. More manipulations of winding number and cases involving multiple spin solitons are provided in Sec. SIII of SM.

Experimental Observation: Finally, we discuss the possibilities to observe the winding number manipulations in real experiments. Let’s consider a ${}^{87}{\rm Rb}$ Bose condensate with two internal states HallPRL1998 ; MertesPRL2007 ; BeckerNP2008 to observe the results presented in Fig. 1. The related scattering lengthes are about $100a_{\rm B}$ ($a_{\rm B}$ being the Bohr radius) HallPRL1998 ; MertesPRL2007 . The condensate can be loaded in a toroidal optical dipole trap RyuPRL2007 ; RamanathanPRL2011 ; WrightPRL2013 ; RyuPRL2013 ; EckelNature2014 with radial and vertical trapping frequencies of $2\pi\times(100,200){\rm\ Hz}$ and a radius of $54{\rm\ \mu m}$. The initial states can be prepared well by imprinting a dark soliton with $2\times 10^{4}$ atoms (the background density of $1.3\times 10^{13}{\rm\ cm^{-3}}$) in state $|F=2,m_{f}=0\rangle$ and filling the density dip with $13$ atoms in state $|F=1,m_{f}=-1\rangle$ FrantzeskakisJPA2010 ; BeckerNP2008 . The effective radial length is $1.4\ \xi$ with healing length of $\xi\approx 0.76{\rm\ \mu m}$. A weak uniform gradient magnetic field can be applied along the angle coordinate as the effective force to drive atoms LuoSR2016 . By setting a sinusoidal modulating gradient magnetic field LuoSR2016 ; AnnalaPRR2024 ; RayNature2014 ; RayScience2015 with peak gradient of $0.12{\rm\ G\cdot cm}^{-1}$ and period of $0.8{\rm\ s}$, one can observe 2 times winding number jumps within the time duration of $2{\rm\ s}$. The predictions in this work could be realized through the state-of-the-art experimental techniques.

Conclusion and Discussion: To conclude, we demonstrate that the topological charge of a toroidal Bose condensate can be well manipulated by engineering its density zeros. When the relative velocities between the dark solitons and their background density currents are zeros, the density zeros occur. In our schemes, loading and driving dark solitons open up the possibility of manipulating winding number, which provides a new approach for quantum computation, quantum simulation, and other quantum-based applications NayakRMP2008 ; BulutaRPP2011 ; BlochNP2012 ; GrossSince2017 ; TerhalRMP2015 ; TokuraNRP2019 ; AmicoRMP2022 ; PoloQT2024 .

Further, vortices, as fundamental topological excitations in high-dimensional systems, also admit the presence of density zeros and their charges determine the topology of systems PitaevskiiOUPress2016 ; FetterRMP2009 ; MatthewsPRL1999 ; LeanhardtPRL2002 . The higher-order vortices exhibit distinctive properties of density zeros compared with fundamental ones ShinPRL2004 ; IsoshimaPRL2007 ; LanPRL2023 . We expect that the charges of various vortices can be well manipulated through engineering the density zeros. These results could inspire new pathways in the design of topological systems by engineering the zeros of localized waves.

This work is supported by the National Natural Science Foundation of China (Contracts No. 12375005, No. 12235007 and No. 12247103) and the Major Basic Research Program of Natural Science of Shaanxi Province (Grant No. 2018KJXX-094).