Collisions of Majorana Zero Modes

Introduction.– Majorana zero modes (MZMs)Majorana-1937 ; WilczekF-2009 ; BrouwerPW-2012 ; ElliottSR-2015 are exotic, neutral quasiparticles composed of the equivalent contributions of the particle and the hole. They are of fundamental scientific importance and could have profound technological applications for fault-tolerant quantum computationHasanM-2010 ; QiX-2011 ; TewariS-2007 ; NayakC-2008 , quantum memory KitaevAY-2001 ; KitaevAY-2003 ; ShorPW-1995 ; BiercukMJ-2009 ; MaurerPC-2012 and quantum random-number generationRarityJG-1994 ; DongLing-2013 . Similar to the Bardeen-Cooper-Schrieffer (BCS) pairing mechanism between electron creation and annihilationBCS-1957 , the mixing of the particle and hole results in the ground states doubly degenerate, and two MZMs then form a protected qubit that is not locally measurableKitaevAY-2001 . These non-Abelian quasiparticles are believed to merge in topological superconducting and superfluid systems, including the interfaces of s-wave superconductor and topological insulatorFuL-2008 ; LutchynRM-2010 ; OregY-2010 ; AliceaJ-2010 ; JiangL-2011 ; MourikV-2012 , intrinsic two-dimensional superconductors with p-wave pairing symmetryReadN-2000 ; MizushimaT-2008 ; BjornsonK-2015 ; MurrayJ-2015 , as well as topological ferromagnetic metal chainsPergeS-2013 ; LiJ-2014 ; PawlakR-2016 . After the realization of spin-orbit coupling (SOC) in ultracold gasesLin-2011 ; Wang-2012 ; Wu-2016 ; Meng-2016 ; Huang-2016 , atomic fermionic superfluids enter the topological state, offering a disorder-free and highly controllable platform for studying Majorana physicsDalibard-2011 ; Wu-2013 ; Devreese-2014 ; Zhai-2015 .

MZMs appear within the cores of certain soliton excitationsXu-2014 ; Liu-2015 ; Mateo-2022 , where a phase kink across the dip-like structure of the order parameter arises from the atomic phase imprinting techniquesDenschlag-2013 ; Burger-1999 ; Anderson-2000 ; Ku-2016 . These zero-energy states lead to the degeneracy of the many-body ground states, which hinges on the precise degeneracy of the MZMs in various soliton cores. Physically, soliton is a good candidate for the control and manipulation of Majorana qubits owing to its classical particle-like characterKartashov-2011 , and networks of such topological excitations have been proposed for the progress of quantum computingEl-2005 ; Tecas-2013 ; Shaukat-2017 ; Ezawa-2020 . Nevertheless, in the presence of multiple solitons, inter-soliton colliding events become possibleScottR-2012 and overlapping between Majorana states within different solitons are expected to lift the Majorana state degeneracy to some degree, even may lead to a complete breakdown. For the purposes of topological quantum computation, it is vital to figure out the stability of the MZMs in colliding processes.

In this Letter we address this question by observing soliton collisions in one-dimensional fermionic superfluids with SOC. The existence of MZMs within the solitons has important consequences for the physics of soliton collisions. Based on our numerical simulations with time-dependent Bogoliubov-de Gennes equation, we find that the soliton collision with MZMs appears repulsive and completely elastic, i.e., they do not penetrate each other but instead repel with a well distance that the soliton matters can be safely regarded as untouched. Thus the soliton matter interaction could be negligible and allow us to clearly identify the collisional nature of MZMs. We confirm that the overlapping of inter-soliton Majorana wave-functions upon collision lift the Majorana degeneracy from zero. Moreover, the energy splitting of the Majorana states creates an effective repulsive force for MZMs, which in turn protected themselves against into bulk excitation. We stress that this is distinct from other repulsion mechanisms considered previously. And, for a contrast, we also demonstrate that soliton collisions become increasingly inelastic as we tune the system into the topological trivial phase. We further reexamine the repulsive interaction in collisions between soliton-bound and defect-pinned Majorana states. Our results show that Majorana zero modes are not only topologically protected, but also self-protected. This provides new insights into the intrinsic nature of Majorana states and these unusual colliding properties may open an alternative way to detect and discriminate Majorana qubits for fault-tolerant topological quantum computations.

Theoretical model. – Our starting point is the description of dark soliton-collisions in one-dimensional spin-half fermionic superfluids with SOC. The one-dimensional setting ensures the dark solitons stable with respect to the snake instabilityBrand-2002 ; Ku-2016 and the SOC effect can be realized by two counter-propogating Raman lasersWang-2012 . Within the standard mean-field framework, the dynamics of a fermionic superfluid can be modeled by the well-known time-dependent Bogoliubov-de Gennes (TDBdG) equation, whose one-dimensional form reads

where the wave-functions is $\Phi_{\eta}\equiv[u_{\uparrow,\eta},u_{\downarrow,\eta},v_{\downarrow,\eta},-v_{\uparrow,\eta}]^{T}$ in the Nambu representation. The single particle grand-canonical Hamiltonian has the form $H_{0}=-\hbar^{2}\partial^{2}_{x}/2m+m\omega^{2}x^{2}/2-i\alpha\hbar\partial x\sigma_{y}-\mu+h_{z}\sigma_{z}$, describing the motion of fermionic atoms confined in a harmonic trapping potential with an oscillation frequency $\omega$. $\alpha$ is the SOC strength, $h_{z}$ the effective Zeeman filed, and $\mu$ the chemical potential. The order parameter has the self-consistent form $\Delta(x,t)=g_{\mathrm{1D}}\sum_{\eta}[u_{\uparrow,\eta}(x,t)v_{\downarrow,\eta}^{\ast}(x,t)f(-E_{\eta})-u_{\downarrow,\eta}(x,t)v_{\uparrow,\eta}^{\ast}(x,t)f(E_{\eta})]$ and the atom density function is given by $n_{\sigma}(x,t)=\sum_{\eta}|v_{\sigma,\eta}(x,t)|^{2}f(-E_{\eta})+|u_{\sigma,\eta}(x,t)|^{2}f(E_{\eta})$, where $g_{1D}=-2\hbar^{2}/(ma_{1D})$ is the effective interatomic coupling given by an $s$-wave scattering length $a_{1D}$. $f(E)=1/[e^{E/k_{B}T}+1]$ is the Fermi-Dirac distribution at a temperature $T$, and the summation is over the quasi-particle state $E_{n}\geqslant 0$. Eq.(1) should be calculated self-consistently with the constraints of a fixed total atomic number $N=\int dx[n_{\uparrow}(x,t)+n_{\downarrow}(x,t)]$ and the above definition of the order parameter.

The particle-hole symmetry of BdG Hamiltonian ensures that MZMs only occur at exact zero energy and typically localized in the vicinity of defects, such as phase edges or solitons. In this work, multiple dark solitons may be chosen to be real and $\pi$-phase kinks are introduced at the spots $\{x_{i},i=1,2,3,...\}$, given by

where $\Theta(x)$ is the Heaviside step function. We seek the stationary solutions self-consistently with the use of Eq. (2), after a number of iterations up to convergence. When the local superfluid becomes topologicalXu-2014 , a pair of MZMs appear inside the soliton core and it is useful to note that the interaction between MZMs inside a dark soliton vanishes due to the intrinsic property of the dark soliton: a sharp phase jump. In dynamic simulations, we use a dimensionless interaction parameter to characterize the interaction strength, $\gamma=-mg_{1D}/\hbar^{2}n$, which is basically the ration between the interaction and kinetic energy at the density $n$. We choose the Fermi vector and energy, $k_{F}=\pi n/2$ and $E_{F}=\hbar^{2}k_{F}^{2}/2m$, as the units of wave-vector and energy, respectively. In a trapped cloud with $N$ atoms, it is convenient to use the peak density of a non-interacting Fermi gas in the Thomas-Fermi approximation at the trap center, $n=(2/\pi)\sqrt{Nm\omega/\hbar}$, although the Fermi cloud itself is an interacting gas. Throughout the work, we consider only zero temperature and take the interaction parameter $\gamma=\pi$, the total atomic number $N=100$.

Collisions of Majorana zero modes.– We first prepare the system in the topological phase and then place two dark solitons with a symmetrical displace, to observe the head-on collision at the trap centre. Majorana states emerge around the nodal point of the soliton, which is in contrast to bosonic superfluids, where the Gross-Pitaevskii solitons are structureless. The initial soliton separation $x_{i}=\pm 0.5x_{F}$ is large enough that the Majorana wave-function overlapping is negligible. Before the collision, a close inspection of the stationary dark soliton at $0.5x_{F}$ is shown in Fig.(1a). Owing to the appearance of MZMs within the soliton, the inter-soliton effects should be twofold: first, the mere soliton matter interaction arises sharply at spacings of the order of the soliton core width $\xi_{S}$, and second, the soliton-induced Majorana wave-function overlap with the length scale $\xi_{M}$, which describes the MZM behavior like $\sim\cos{(\pi k_{F}x/2)\exp(-x^{2}/\xi_{M}^{2})}$. $\xi_{M}$ is larger than $\xi_{S}$ ($\xi_{M}\approx 2.5\xi_{S}$ in this case) and then we predict the Majorana soliton possesses a two-shell structure: soliton matter core surrounded by Majorana wave-functions (shown in Fig.(1b)). The collisions should be fascinating due to their multi-component nature.

In Fig.(1c), we present the evolution of the order parameter profile of the superfluid, while sequential snapshots of the lowest energy Majorana wave-functions (specifically $|u_{\uparrow}(x,t)|^{2}$) are displayed above it. Clearly, the resultant collision appears to be repulsive and elastic: the solitons slow down as they approach one another, come to a halt with a well distance of $\xi_{\mathrm{min}}\approx 0.3x_{F}$, and then reflect back to their original positions. With ($\xi_{M}>\xi_{\mathrm{min}}>\xi_{S}$), the soliton cores remain untouched and the Majorana wave-functions overlap (shown in Fig.(1c)). The soliton matter interaction could be negligible and thus we safely attribute the repulsive force just to the Majorana states, allowing us to clearly identify the nature of interactions between MZMs. To grasp the main physics, the time dependent energy expectations $\left<E_{\eta}(t)\right>=\left<\Phi_{\eta}(x,t)|H_{\mathrm{BdG}}(x,t)|\Phi_{\eta}(x,t)\right>$ are calculated to observe the energy splitting of zero modes in the collision, as shown in Fig. (1d). It’s worth noting that, far away from each other, MZMs always have zero energy irrespective of the soliton velocity. As solitons get close enough, the zero modes split into a pair of levels $(E_{0},-E_{0})$, which is proportional to the Majorana wave-function overlapping. This energy upshift creates a repulsive force for solitons that block their appropinquity and in turn drastically protects itself against scattering into bulk states. That’s why the two solitons exhibit elastic collision only.

Moreover, asymmetrical collision is shown in Fig.(2). During collision, solitons exchange energy via the Majorana wave-function overlapping, hence, solitons with different velocities (increase the soliton speed by increasing $x_{i}$), the quantity of motion is preserved, i.e., the low-speed soliton after collision propagate with high velocity, while the high-speed one runs slowly, looks like passing through one another without changing form. The observed behavior of the above collision reveal that the Majorana states are not only topological protected, but also self-incurred protected, an aspect of the Majorana state that hadn’t been explored before. This can ensure a more robust multi-Majorana quasiparticles transport.

Next, for a contrast, we tune the system into the topological trivial regime and observe the properties of the corresponding soliton collision. In Fig.(3), under the threshold $h_{z}\approx 0.7E_{F}$, the superfluid enter the topological trivial phase and finite energy Andreev-like bound states emerge near the point node of the soliton. Observably, inelastic soliton collision is presented and the lost energy is converted into small-amplitude density ripples that emanate from the point of the collision (shown in Fig.(3a)). Counter-intuitively, the dark solitons with lower energy have a higher velocity and vice versa, the soliton actually move faster after losing energy. Fig. (3b) shows the corresponding wave-function of lowest Andreev bound state (specifically $|u_{\uparrow}(x,t)|^{2}$). One can clearly see the corresponding ripples in the amplitude and, as a result, soliton-induced Andreev bound states are strongly up-shifted into the bulk quasi-particle scattering continuum, which eventually cause an emission of the sound and loss of the soliton energy (see Fig.(3c)). Furthermore, to quantify the elasticity of the collision, we evalued the soliton speed discrepancy immediately before and after the first collisions with the decreasing Zeeman field in Fig.(3d). The phase diagram for occurrence of elastic and inelastic collisions is obtained, which strongly depends on the topological properties. The speed discrepancy rises suddenly around the transition point and the collisions become increasingly inelastic in topological trivial phase. The different collision properties between Majorana and Andreev solitons may lead to an interesting technique of soliton filter for distinguishing the Majorana states.

Quasiclassical Analysis of Soliton Collision.– Dark soliton appear as wave packet that preserve its amplitude and shape during its propagation and even persist unchanged with MZMs in the colliding process, therefore being attributed as a particle-like character. In our study, the width of the Fermi cloud in the axial direction is much larger compared to the size of the soliton. Thus, under the local density approximation, the soliton can be treated as a macroscopic particle at coordinates $q_{i}$ with $p_{i}$ being generalized momentum. In our study, the overlapping between MZMs adjusts to the soliton’s motion and plays an important role in the soliton collision. According to the energy splitting from zero $\varpropto\pm\exp(-|q_{1}-q_{2}|/\xi_{M})$ induced by the Majorana wave-function overlappingCheng-2009 , their mutual repulsive potential can be assumed reasonably as $V(q_{1},q_{2})=Ae^{-|q_{1}-q_{2}|/\sqrt{\xi_{M_{1}}\xi_{M_{2}}}}$, where $A$ is the interaction intensity to be determined. The soliton matter interaction is negligible and we may finally obtain the Hamiltonian of the double solitons by evaluating the kinetic, trap and repulsive interaction energy terms in the absence of dissipation,

where $m_{\mathrm{I}}$ is the inertial mass and $m_{\mathrm{S}}=N_{\mathrm{S}}m$ bare soliton mass, which can be determined in the snaking process of single soliton (as elaborated on in Supplementary Material). With the initial state $\left\{q_{i}(t=0)=x_{i},p_{i}(t=0)=0\right\}$, following the Hamiltonian equation of motion $\dot{q}_{i}=\partial H/\partial p_{i}$, $\dot{p}_{i}=-\partial H/\partial q_{i}$, the interaction intensity is fixed to be $A=0.0137E_{F}$ compared with the soliton trajectory based on the full numerical BdG simulation in Fig.(2). The numerical results of the trajectories of the soliton centers and the time dependences of the soliton’s coordinates originating from Eq.3, are presented in Fig.(4) with different initial states. All the analytical results are in good agreement with the numerical data.

Collision with phase-edge Majorana states.– Then, in order to check the above-made statements and extend our works into the collision between MZMs within different topological defects, we perform numerical simulations of the collision between soliton-induced MZMs and one pinned at the phase edge. Due to the harmonic potential geometry, in a suitable parameter regime, a mixed phase emerges, consisting of a standard normal superfluid at the center and a topological superfluid at the two edges of the trapLiu-2012 , which is shown in Fig.(5a). A dark soliton was set in the topological region and accelerated by the trapping potential towards to the phase edge, where a MZM is prepared immovablely. As shown in Fig.(5b), the soliton slow down as it approaches the phase edge at $0.2x_{F}$, and then turn back, performing a complete oscillation (a virtual phase edge reflection) inside the topological region. Or in other words, Majorana states hosted in the soliton has been limited in the span between two fixed MZMs, reminiscent of what has been observed in the soliton collision. The soliton behavior is generated by the conjunction of the trapping effect and repulsive interaction between Majorana states, which is an reconfirmation of the repulsive interaction of MZMs.

Conclusion.– To summarize, our study shows that the zero energy splitting, induced by the overlapping of inter-soliton Majorana wave-functions upon soliton collision, generates an effective repulsive force for Majorana states, which in turn protected themselves against into bulk excitation. A complete elastic collision takes place between solitons with Majorana states, which do not penetrate each other but instead repel without any loss of energy. Remarkably, our quasi-particle analysis can fully explain the numerical findings and provide a complete description of these anomalous behaviors with Majorana states. Additionally, we perform the investigation of the collisional mechanism between Majorana states within different topological defects to confirm our conjecture. Our research provides new insights into the features of Majorana fermions, and we envision that the robustness in the collisions of Majorana states could be utilized in the topological quantum computing with a network of Majorana qubits.

We thank An-chun Ji, Qing Sun and Changan Li for useful discussions. This work was supported by the foundation of Zhejiang Province Natural Science under Grant No. LQ20A040002 and JL acknowledges support from National Natural Science Foundation of China under Project 11774317. The numerical calculations in this paper have been done on the super-computing system in the Information Technology Center of Westlake University.