Two-dimensional non-abelian thouless pump

ABSTRACT Non-Abelian Thouless pumps are periodically driven systems designed by the non-Abelian holonomy principle, in which quantized transport of degenerate eigenstates emerges, exhibiting

noncommutative features such that the outcome depends on the pumping sequence. The study of non-Abelian Thouless pump is currently restricted to 1D systems, while extending it to

higher-dimensional systems will not only provide effective means to probe non-Abelian physics in high-dimensional topological systems, but also expand the dimension and type of associated

non-Abelian geometric phase matrix for potential applications. Here, we propose the design and experimental realization of 2D non-Abelian Thouless pumps on a photonic chip with 2D photonic

waveguide arrays, where degenerate photonic modes are topologically pumped simultaneously along two real-space directions. We reveal the associated non-Abelian group and experimentally

demonstrate the non-Abelian feature by measuring the pumping sequence dependent output. The proposed 2D non-Abelian Thouless pump shows promising applications for robust optical

interconnections and optical computing. SIMILAR CONTENT BEING VIEWED BY OTHERS NON-ABELIAN THOULESS PUMPING IN PHOTONIC WAVEGUIDES Article 28 July 2022 THOULESS PUMPING IN DISORDERED

PHOTONIC SYSTEMS Article Open access 19 October 2020 TWO-DIMENSIONAL THOULESS PUMPING OF LIGHT IN PHOTONIC MOIRÉ LATTICES Article Open access 08 November 2022 INTRODUCTION Thouless pumps are

“quantum” pumps that enable the quantization of particle transport in slowly varying pumping cycles with the existence of periodic potentials1,2,3,4. The motion of the particle during one

pumping cycle is associated with the Chern number5,6,7,8, which is a topological invariant of the system. Protected by the topology of the pump, the particle transport is robust to

perturbations including weak disorders and interactions2. Such intriguing features have been demonstrated in a variety of Thouless pumps with 1D periodic

potentials9,10,11,12,13,14,15,16,17,18,19,20,21,22, ranging from cold atom systems9,10,11,12 to photonic systems13,14,15,16,17,18,19. These 1D Thouless pumps can be regarded as a dynamical

version of 2D integer quantum Hall systems23, where time plays the role of a second dimension. In this sense, Thouless pumps are promising platforms to directly observe the topological

consequence. To characterize the topological properties of higher-dimensional topological systems, 2D Thouless pumps are also proposed24,25,26,27,28,29. For example, 4D quantum Hall physics

has been successfully revealed in 2D Thouless pumps by employing the concept of synthetic dimensions24,25, which cannot be accessed using real-space topological systems. Recently,

non-Abelian physics30,31,32,33,34,35,36,37,38 has been introduced into the design of Thouless pumps, giving rise to non-Abelian Thouless pumps that employ the non-Abelian holonomy

principle36,37,38. The output and input of the non-Abelian Thouless pump is related via a Wilczek-Zee connection associated geometric phase matrix39, leading to non-Abelian effect that the

outcome depends on the sequence of the pumping operations, which is the key difference between Abelian and non-Abelian Thouless pumps. The intrinsic feature of non-Abelian Thouless pump,

that is, performing matrix operations which are typically not commutative, makes it especially useful for photonic disciplines, since matrix transformations lie at the heart of diverse

important photonic applications such as optical interconnections and optical computing. However, the currently proposed non-Abelian Thouless pumps are 1D pumps36,37,38, while the scheme and

experimental realization of higher-dimensional non-Abelian Thouless pumps have been elusive. Pushing non-Abelian Thouless pumps towards higher dimensions can not only offer experimental

platforms to probe non-Abelian physics in high-dimensional topological systems, but also increase the dimension of the associated non-Abelian groups and therefore enrich the generated

unitary matrices used for corresponding applications. Here, we propose the design and experimental realization of 2D non-Abelian Thouless pumps on a photonic chip with 2D photonic waveguide

arrays. We first generate a non-Abelian group associated with the 2D non-Abelian Thouless pump through the direct product of two Abelian groups: a permutation group S2 and an additive group

of quaternions. We then realize this non-Abelian group in 2D photonic waveguide arrays by introducing non-Abelian holonomy associated eigenstate degeneracy as well as manipulating the

coupling strengths within the waveguides. By investigating the generating elements of the non-Abelian group, we strictly prove that all the group elements associated with the 2D non-Abelian

Thouless pump can be realized using the proposed versatile system. As an experimental demonstration, we employ the femtosecond laser direct writing techniques to fabricate two 2D non-Abelian

Thouless pumps, which feature the motion of degenerate photonic modes both along the horizontal and vertical direction. The non-Abelian characteristics are experimentally observed by

combining the two pumps and measuring their sequence-dependent output light intensity distributions. RESULTS AND DISCUSSION CONCEPT AND GROUP REPRESENTATION OF THE 2D NON-ABELIAN THOULESS

PUMP We first study the Thouless pump from the viewpoint of group symmetry. Figure 1a illustrates the schematic of a 1D Abelian Thouless pump consisting of an infinite chain of periodic

atoms A. By enforcing appropriate potentials, the eigenfunction located at the _i_th atom can be pumped to the (_i_ + _m_)th site, where _m_ is an arbitrary integer. We use _G__m_ to

represent this pumping operation, and all the pumping operations form an additive group of integers which is an Abelian group. This is because that any two group elements _G__m_ and _G__m_′

are commutative: _G__m__G__m_′ = _G__m_′_G__m_ = _G__m+m_′. Now we add another chain of atoms B to the system, as shown in Fig. 1b. If there is no interaction between the two chains, the

system is still Abelian although all the pumping operations make up an additive group of Gaussian integers {_G__m_,_n_, …}, where _m_ and _n_ are integers associated with the location shift

of the eigenfunction in the chain A and B, respectively. The system can become non-Abelian in nature if we introduce an operation _P_ that swaps the eigenfunctions located in the chain A and

B. Such operation can be realized by enforcing these two eigenfunctions to follow a non-Abelian holonomy process in a double-degenerate subspace. In this sense, a non-Abelian group G1D = 

S2\(\otimes\){_G__m_,_n_, …} is established, where S2 = {_E_, _P_} denotes a permutation group with _E_ being the identity element. Consider two group elements _D_1 = _PG__m,n_ and _D_2 = 

\({PG}_{m{\prime},n{\prime} }\) (see the green and purple arrows respectively in Fig. 1b for example, where _m_ = 1, _n_ = 0, _m_′ = 0, and _n_′ = 1), their cascading operations yield

_D_1_D_2 = \({G}_{m^{\prime}+n,m+n^{\prime} }\) and _D_2_D_1 = \({G}_{m+n^{\prime},m^{\prime}+n}\), where for example, _D_1_D_2 indicates that the operation _D_2 and _D_1 are executed in

sequence. As long as _m’+n_ ≠ _m_ + _n’_, the two elements are noncommutative and thus the group is non-Abelian. A 2D non-Abelian Thouless pump can be realized by extending the two 1D chains

to two 2D nets, as illustrated in Fig. 1c. In this case, a non-Abelian group is constructed via G2D = S2\(\otimes\){_G__m_,_n,p,q_, …}, where the latter is an additive group of quaternions,

and the motion of the eigenfunction in the lattice A (B) along the _x_ and _y_ axis is denoted by _m_ and _n_ (_p_ and _q_), respectively. Any two group elements _D_1 = _PG__m,n,p,q_ and

_D_2 = \({PG}_{m^{\prime},n^{\prime},p^{\prime},q^{\prime} }\) are typically noncommutative since _D_1_D_2 = \({G}_{m^{\prime}+p,n^{\prime}+q,m+p^{\prime},n+q^{\prime} }\) and _D_2_D_1 = 

\({G}_{m+p^{\prime},n+q^{\prime},m^{\prime}+p,n^{\prime}+q}\). The additional dimension compared to 1D pump serves as a new degree of freedom for manipulating the switching of

eigenfunctions. DESIGN AND EXPERIMENTAL REALIZATION OF THE 2D NON-ABELIAN THOULESS PUMP We design the 2D non-Abelian Thouless pump using 2D photonic waveguide arrays, of which the schematic

diagram is illustrated in Fig. 2a. The pink box marks one unit cell of the pump that consists of six photonic waveguides namely A, B, M, N, X and Y. The waveguides A and B are working

waveguides that carry the light energy at the input and output port, playing the same role as the atoms A and B in Fig. 1c. To realize a pumping between waveguides A and B, we introduce

waveguides M, N, X and Y. Among them, the waveguides M and N are auxiliary waveguides which can store the light energy temporarily in the pumping process, while the waveguides X and Y are

coupling waveguides that are always straight. In contrast, waveguides A, B, M and N are typically curved for introducing desired coupling strengths within the waveguides. Figure 2b gives a

cross-section view of the 2D Thouless pump at the input and output port. We use Ai,j and so forth to denote the waveguides in the ith and jth unit cell. Some waveguides A in different unit

cells are also labeled to show the lattice arrangement. We use terms _g_ and _κ_ to represent the corresponding gap distance and coupling strength between adjacent waveguides, e.g., _g_AY

and _κ_AY are respectively associated with those between waveguides A and Y. In what follows, we will study two Thouless pumps namely pump I and pump II. In our experiment, we employ the

femtosecond laser direct writing technique40,41 to fabricate the photonic waveguide arrays inside the boroaluminosilicate glass (see Methods). Figure 2c shows a microscope photograph of the

fabricated sample at the input facet, where each waveguide features a cross-section size of ~6.2 μm × 7.5 μm. The refractive index contrast between the fabricated waveguide and the

background is on the order of thousandth so that each waveguide supports only one vertically polarized mode (i.e., polarized along the _y_ axis) at the working wavelength of 808 nm. We first

design a pump I which is associated with a group element _D_1 = _PG_0,0,0,-1. From the above group definition, the pump I corresponds to a process that light injected in the waveguide Ai,j

(Bi,j) is pumped to the waveguide Bi,j (Ai,j-1), which applies to all the unit cells. We employ multiple stimulated Raman adiabatic passages42 to achieve such pumping. In our specific case

with photonic waveguides, the stimulated Raman adiabatic passage refers to an adiabatic light transport process occurred in a three-waveguide system, which serves as a building block of the

whole pumping. In each building block, by exerting appropriate couplings within the three waveguides, light can be pumped from one source waveguide to a targeted waveguide, with the help of

a third coupling waveguide. In this process, light is only distributed in the source and targeted waveguides (see detailed explanation in Supplementary Note 1 and Supplementary Fig. 1). The

proposed system consists of multiple three-waveguide building blocks, in which the source/targeted waveguide could be A, B, M or N, while X and Y serve as the coupling waveguide. We briefly

show how the pump I is designed based on the above mechanism. Suppose at the initial point of the pumping, there are two modes namely mode 1 and mode 2, which are located in the waveguide

Ai,j and Bi,j, respectively. A basic principle to simultaneously pump the two modes is that their corresponding pumping paths do not interfere each other. We achieve this goal by dividing

the whole pumping into three steps. In step 1, the mode 2 is adiabatically pumped from the waveguide Bi,j to Mi,j-1 through a stimulated Raman adiabatic passage, in which Bi,j, Xi,j and

Mi,j-1 form a three-waveguide building block. Meanwhile, the mode 1 in the waveguide Ai,j keeps still. In step 2, the three waveguides Ai,j, Xi,j and Bi,j form another building block that

pumps the mode 1 from Ai,j to Bi,j, while the mode 2 is kept in Mi,j-1. A third stimulated Raman adiabatic passage is established in step 3 between waveguides Mi,j, Yi,j and Ai,j, leading to

the transfer of the mode 2 from Mi,j-1 to Ai,j-1. After the three steps, the mode 1 and mode 2 are respectively located in the waveguide Bi,j and Ai,j-1, indicating the successful

realization of the pump I. A detailed description on the pumping is also provided in Supplementary Note 1 and Supplementary Fig. 2. The above pumping requires position-dependent coupling

coefficients within the waveguides, as shown in Fig. 2d. By mapping the coupling to the gap distance between adjacent waveguides (see Supplementary Fig. 3), the architecture of the pump I

can finally be obtained. The two pumping processes in fact take place in a degenerate subspace and the working modes are photonic “zero” modes. To show this point, we calculate the

propagation constants \(\beta\) of the pump-I system (i.e., also the eigenvalues of the system) in Fig. 2e, where \({\beta }_{0}\) denotes the propagation constant of an isolated waveguide.

In the calculation, the edge modes at the boundary of the system are omitted since the pumping takes place in the bulk. We focus on the “zero” modes with \({\beta=\beta }_{0}\). We note that

at any time of the evolution, there are always four “zero” modes supported in each unit cell. Two out of them are degenerate working modes that carry out the pumping and mimic the two

degenerate eigenfunctions in the lattices A and B in Fig. 1c. The other two “zero” modes are redundant which are supported in regions beyond the pumping paths, but the corresponding

waveguides are indispensable since they would play key roles at other times. We also study the system from the viewpoint of band structure and the corresponding results are given and

discussed in Supplementary Fig. 4. We perform numerical simulations and experiments to verify the above design. Figure 3a, b shows the simulated light intensity distributions in the 2D

waveguide arrays when the Ai,j mode and Bi,j mode are excited, respectively, by using a beam propagation software RSoft (see Methods). We find that the pumping from the waveguide Ai,j to

Bi,j is realized in step 2 with the help of the waveguide Xi,j (Fig. 3a), while the other mode conversion process from the waveguide Bi,j to Ai,j-1 is achieved via two stimulated Raman

adiabatic passages in steps 1 and 3, respectively, between which the light power is temporarily stored in the waveguide Mi,j-1 (Fig. 3b). We emphasize that although theoretically the

waveguides X and Y do not carry light energy since they play the role of a coupling waveguide, there are slight but insignificant light intensity distributions in them in numerical

simulations since the system is not ideally adiabatic so that slight undesired modes can be excited. The experimentally measured light intensity distributions at the output facet are shown

in Fig. 3c, d by coupling a laser light at 808 nm into the waveguide A and B to excite the corresponding fundamental mode polarized along the vertical direction, respectively. Without loss

of generality, we excite a mode in the waveguide A1,2 (left inset of Fig. 3c) and A2,1 (right inset of Fig. 3c), and find that the light outputs via the waveguide B1,2 and B2,1,

respectively. When the laser light is coupled into the system via the waveguide B1,2 (left inset of Fig. 3d) and B2,2 (right inset of Fig. 3d), the output is found to be located in the

waveguide A1,1 and A2,1, respectively. The Thouless pump also works when we inject a laser light in other unit cells since it is a periodic system (see Supplementary Fig. 5 for additional

experimental results). We now consider a pump II that is associated with a group element _D_2 = _PG_1,1,-1,0. Its schematic diagram and the corresponding coupling coefficients are given in

Supplementary Fig. 6. As noted from the expression of _D_2, the functionality of the Thouless pump II is that a mode injected from the waveguide Ai,j and Bi,j is pumped to the waveguide

Bi+1,j+1 and Ai-1,j, respectively. This can be verified by the numerical simulations in Fig. 3e, f and the experimental measurements in Fig. 3g, h. We also study the two pumps by modeling

the system using a fitted Hamiltonian, and the corresponding numerical results are given in Supplementary Fig. 7. NON-ABELIAN FEATURE OF THE 2D NON-ABELIAN THOULESS PUMP The proposed system

is versatile as it can be used to realize all the group elements belonging to the non-Abelian group G2D. We give a simple proof. From a fundamental point of view, this non-Abelian group has

nine generating elements: _P_, _G_1,0,0,0, _G_0,1,0,0, _G_0,0,1,0, _G_0,0,0,1, _G_-1,0,0,0, _G_0,-1,0,0, _G_0,0,-1,0 and _G_0,0,0,-1. We show in Supplementary Note 2, Supplementary Figs. 8

and 9 that all these generating elements can be constructed using the proposed system. In this sense, any group element _G__m_,_n,p,q_ or _PG__m_,_n,p,q_ can be achieved by cascading the

generating elements. We demonstrate the non-Abelian feature of the 2D non-Abelian Thouless pump. We have discussed in Fig. 1c that any two group elements _PG__m,n,p,q_ and

\({PG}_{m^{\prime},n^{\prime},p^{\prime},q^{\prime} }\) are typically noncommutative. The proposed two Thouless pumps _D_1 = _PG_0,0,0,-1 and _D_2 = _PG_1,1,-1,0 in fact meet this condition.

We combine them to construct two new pumps that are represented by _D_2_D_1 and _D_1_D_2. Figure 4a illustrates a diagram of the two combined pumping operations, where the black solid and

black dashed arrows mark the pumping operation _D_1 and _D_2, respectively. We consider a mode injected via the waveguide Ai,j. When executing the pumping _D_2_D_1, we find that the mode is

first pumped to the waveguide Bi,j via _D_1, and then pumped to the waveguide Ai-1,j via _D_2. The pumping path is completely different when executing _D_1_D_2, where the input mode in the

waveguide Ai,j is first pumped to the waveguide Bi+1,j+1 (via _D_2) and then Ai+1,j (via _D_1). The experimentally measured output patterns of light intensity are given in Fig. 4b, c,

respectively, where the A1,2 mode is excited. The pumping sequence dependent outcome, i.e., the output light is mainly located in the waveguide A0,2 in the pumping _D_2_D_1 while in the

waveguide A2,2 in the pumping _D_1_D_2, is obviously due to the matrix nature of the non-Abelian Thouless pump. The non-Abelian feature can also be verified by injecting a mode via the

waveguide Bi,j. The corresponding experimental results by exciting the B1,2 mode for the pumping _D_2_D_1 and _D_1_D_2 are respectively given in Fig. 4d, e, where the two outcomes are also

different. We also provide more experimental results in Fig. 4f-i by injecting light via the waveguides A2,1 and B2,1, where the 2D non-Abelian pumping of light can also be observed. In

particular, the shift of the mode location in the 2D lattice is in accordance with the mathematical expression of the two combined pumps, i.e., _D_2_D_1 = _G_-1,0,1,0 and _D_1_D_2 = 

_G_1,0,-1,0. Their inequality is also a direct proof of the non-Abelian feature of the proposed 2D Thouless pump. A numerical study on the two combined pumps is given in Supplementary Figs. 

10-12, respectively. At last, we discuss the topological robustness of the 2D non-Abelian Thouless pump. Each pump can be assigned with a Wilczek-Zee connection associated geometric phase

matrix. Consider a 2D pump with _p_ rows along the _i_ axis and _p_ columns along the _j_ axis (see Fig. 2b for the definition of the axis), the system supports a total of 2_p_2 degenerate

working modes, since each unit cell has two degenerate working modes. A Wilczek-Zee connection associated geometric phase matrix _U_ in the size of 2_p_2 × 2_p_2 can then be constructed

which connects the input and output of the pumping process via \(|{\varphi }_{{{{\rm{OUT}}}}}\rangle=U|{\varphi }_{{{{\rm{IN}}}}}\rangle\), where \(|{\varphi }_{{{{\rm{IN}}}}}\rangle\) and

\(|{\varphi }_{{{{\rm{OUT}}}}}\rangle\) are respectively the input and output state vector in size of 2_p_2 × 1. Supplementary Note 3 gives the explicit expression of the Wilczek-Zee

connection and the geometric phase matrix. Protected by the geometric phase matrix, the performance of the 2D non-Abelian Thouless pump is robust to weak disorders and perturbations such as

those introduced to affect the propagation constant of the waveguide as well as the coupling coefficients within the waveguide arrays. We give a detailed discussion in Supplementary Note 4

and the corresponding simulation results are shown in Supplementary Figs. 13 and 14. To conclude, we have proposed the design and experimental realization of 2D non-Abelian Thouless pumps in

photonic waveguide arrays. We have revealed the non-Abelian group associated with the 2D non-Abelian Thouless pump, which exhibits more degrees of freedom than that of 1D non-Abelian

Thouless pump and therefore can generate more fruitful geometric phase matrices for on-chip photonic applications such as robust optical interconnections and optical computing. The pumping

of the photonic modes is simultaneously along two real-space directions, which makes full use of our 3D real space and thus significantly increases the on-chip information capacity. The

proposed idea that designing high-dimensional non-Abelian systems from the viewpoint of constructing high-dimensional non-Abelian groups can easily be transplanted to design other

high-dimension non-Abelian systems for studying high-dimension non-Abelian physics as well as developing associated applications. METHODS SAMPLE FABRICATION AND MEASUREMENT The proposed 2D

non-Abelian Thouless pump was fabricated by employing the femtosecond laser direct writing techniques. In the experiment, a femtosecond laser with a wavelength of 1029 nm (Light Conversion

CARBIDE 5 W) was focused at 137–241 μm below the surface of a boroaluminosilicate glass with the assistance of a 40× dry objective lens with a numerical aperture of 0.75. The pulse width,

repetition rate, and single pulse energy of the femtosecond laser are 290 fs, 1 MHz, and 230 nJ, respectively. The beam diameter of the femtosecond laser is ~1 cm before entering the

entrance pupil of the objective lens, after which the laser beam is focused for fabricating the waveguides. An Aerotech motion system was employed to control the movement of the glass at a

speed of 40 mm/s. Through the interaction between the laser and the glass material, a refractive index increase of ~2.5 × 10−3 can be induced in a region of ~6.2 μm × 7.5 μm, forming a

photonic waveguide inside the glass. By programming the moving trajectory of the glass, these waveguides with complex trajectories can be fabricated. The 2D Thouless pump contains multiple

layered waveguides along the _z_ axis. The waveguides located in the lower layers were first fabricated, while those in the upper layers were later fabricated. The length of the single pump

(i.e., pump I and II) is 50 mm (experimental results given in Fig. 3), while the device with their combination is 90 mm (experimental results given in Fig. 4). The proposed 2D Thouless pump

is consisted of curved waveguides A, B, M and N, and straight waveguides X and Y. The transmission loss for both the straight and curved waveguides is ~0.03 dB/mm. For waveguides A, B, M and

N, the curved feature is a result of the required gap distance. The gap distances for the pumps I and II are given in Supplementary Figs. 3b and 6b, which are obtained by using the

gap-coupling relation in Supplementary Fig. 3a, considering that the coupling coefficients take a linear form (see Fig. 2d and Supplementary Fig. 6c). Therefore, the gap distance and

trajectory of the curved waveguide do not have analytical forms. Their explicit data and corresponding codes for the numerical simulations have been provided on Code Ocean. To characterize

the performance of the fabricated device, vertically polarized light emitted by an 808 nm continuous laser (CNI, MDL-III-808L) with power of ~2 mW was used as the laser source. The

corresponding laser beam diameter is ~2 mm before entering the entrance pupil of the objective lens, after which it is focused and coupled into the fabricated pump. A charge-coupled device

(XG500, XWJG) was used to collect the light intensity distributions at the output side of the sample. NUMERICAL SIMULATIONS With the designed coupling strengths in Fig. 2d at hand, a

Hamiltonian \(H(z)\) modelling the 2D Thouless pump can be constructed. The propagation constants in Fig. 2e are then calculated by solving the eigenvalues of the Hamiltonian. This

Hamiltonian can also be used to simulate the wave dynamics in the Thouless pump by numerically solving the Schrödinger-like equation \(H(z)|\varphi (z)\rangle=-j{\partial }_{z}|\varphi

(z)\rangle\), where \(|\varphi (z)\rangle\) is the state vector. The corresponding results are given in Supplementary Figs. 7, 10, 13 and 14. To enhance the readability of the pumping

process, we also apply a beam propagation software RSoft to simulate the wave transmissions in the Thouless pump. Since the data are in 3D, we choose two key slices and draw the

corresponding light intensity distributions on the two slices for each pumping case. The numerical results in Fig. 3a, b, e, f are obtained using this method. DATA AVAILABILITY Part of the

source data is available on Code Ocean (https://doi.org/10.24433/CO.3068074.v1). All other data that support the plots within this paper and other findings of this study are available from

the corresponding author upon reasonable request. CODE AVAILABILITY The codes used for performing the numerical simulations are available on Code Ocean

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Science Foundation of China (Grants Nos. 623B2042, 12374350, 61827826, 62131018), the Young Top-Notch Talent for Ten Thousand Talent Program (X.L.Z. and Z.N.T.), and the Major Science and

Technology Projects in Jilin Province (20220301002GX). AUTHOR INFORMATION AUTHORS AND AFFILIATIONS * State Key Laboratory of Integrated Optoelectronics, College of Electronic Science and

Engineering, Jilin University, Changchun, China Yi-Ke Sun, Zhong-Lei Shan, Zhen-Nan Tian, Qi-Dai Chen & Xu-Lin Zhang Authors * Yi-Ke Sun View author publications You can also search for

this author inPubMed Google Scholar * Zhong-Lei Shan View author publications You can also search for this author inPubMed Google Scholar * Zhen-Nan Tian View author publications You can

also search for this author inPubMed Google Scholar * Qi-Dai Chen View author publications You can also search for this author inPubMed Google Scholar * Xu-Lin Zhang View author publications

You can also search for this author inPubMed Google Scholar CONTRIBUTIONS X.L.Z. conceived of the idea. Y.K.S., Z.L.S., and X.L.Z. performed the theoretical analysis. Y.K.S. fabricated the

samples and carried out the experimental measurements under the supervision of Z.N.T. and Q.D.C. The manuscript was written by Y.K.S. and X.L.Z. with input from all authors. The project was

supervised by Z.N.T., Q.D.C., and X.L.Z. CORRESPONDING AUTHORS Correspondence to Zhen-Nan Tian or Xu-Lin Zhang. ETHICS DECLARATIONS COMPETING INTERESTS The authors declare no competing

interests. PEER REVIEW PEER REVIEW INFORMATION _Nature Communications_ thanks Lukas Maczewsky and the other anonymous reviewer(s) for their contribution to the peer review of this work. A

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pump. _Nat Commun_ 15, 9311 (2024). https://doi.org/10.1038/s41467-024-53741-0 Download citation * Received: 27 June 2024 * Accepted: 21 October 2024 * Published: 29 October 2024 * DOI:

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