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Nat Commun
Nat Commun
Nature Communications
2041-1723
Nature Publishing Group UK London

39128899
51083
10.1038/s41467-024-51083-5
Article
Guiding charged particles in vacuum via Lagrange points
http://orcid.org/0009-0004-6523-4390
Luo Haokun 1
Wei Yunxuan 1
Pyrialakos Georgios G. 1
http://orcid.org/0000-0002-7091-1470
Khajavikhan Mercedeh khajavik@usc.edu

12
http://orcid.org/0000-0003-3630-7234
Christodoulides Demetrios N. Demetri@usc.edu

12
1 https://ror.org/03taz7m60 grid.42505.36 0000 0001 2156 6853 Ming Hsieh Department of Electrical and Computer Engineering, University of Southern California, Los Angeles, CA 90089 USA
2 https://ror.org/03taz7m60 grid.42505.36 0000 0001 2156 6853 Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089 USA
11 8 2024
11 8 2024
2024
15 688219 3 2024
29 7 2024
© The Author(s) 2024, corrected publication 2024
2024
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We propose a method for guiding charged particles such as electrons and protons, in vacuum, by employing the exotic properties of Lagrange points. This leap is made possible by the dynamics unfolding around these equilibrium points, which stably capture such particles, akin to the way Trojan asteroids are held in Jupiter’s orbit. Unlike traditional methodologies that allow for either focusing or three-dimensional storage of charged particles, the proposed scheme can guide both non-relativistic and relativistic electrons and protons in small cross-sectional areas in an invariant fashion over long distances without any appreciable loss in energy – in a manner analogous to photon transport in optical fibers. Here, particle guiding is achieved by employing twisted electrostatic potentials that in turn induce stable Lagrange points in vacuum. In principle, guidance can be realized within the fundamental mode of the resulting waveguide, thereby presenting a prospect for manipulating these particles in the quantum domain. Our findings may be useful in a wide range of applications in both scientific and technological pursuits. These applications could encompass electron microscopies and lithographies, particle accelerators, quantum and classical communication/sensing systems, as well as methods for shuttling entangled qubits between nodes within a quantum network.

Transporting charged particles in a guided manner, similar to how optical fibers carry light signals, has the potential to profoundly impact the scientific and technological landscape. Here, the authors propose a viable method to realize these waveguides by leveraging Lagrange points created by the electrostatic potential around a carefully designed twisted wire in a vacuum.

Subject terms

Particle physics
Matter waves and particle beams
https://doi.org/10.13039/100000181 United States Department of Defense | United States Air Force | AFMC | Air Force Office of Scientific Research (AF Office of Scientific Research) FA9550-20-1-0322 FA9550-21-1-0202 Khajavikhan Mercedeh https://doi.org/10.13039/100007297 United States Department of Defense | United States Navy | ONR | Office of Naval Research Global (ONR Global) N00014-20-1-2789 Khajavikhan Mercedeh https://doi.org/10.13039/100006602 United States Department of Defense | United States Air Force | AFMC | Air Force Research Laboratory (AFRL) FA8650-19-C-1692 FA86511820019 Khajavikhan Mercedeh https://doi.org/10.13039/100000888 W. M. Keck Foundation (W.M. Keck Foundation) https://doi.org/10.13039/100000183 United States Department of Defense | United States Army | U.S. Army Research, Development and Engineering Command | Army Research Office (ARO) W911NF-23-1-0312 Khajavikhan Mercedeh https://doi.org/10.13039/100006206 DOE | SC | Biological and Environmental Research (BER) DE-SC0022282 Khajavikhan Mercedeh issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

More than a hundred years have passed since the discovery of electrons by Thomson in 18971. Ever since, the use of electron transport processes within various material systems has utterly revolutionized the technological landscape, ushering in an era of electronic marvels. In this regard, of paramount importance has always been to devise effective approaches in guiding electrons and, by extension, charged particles like protons and ions in vacuum. We note that methods to store charged particles in three-dimensional settings do exist, such as, for example, Paul and Penning traps2–5. These traps rely on static electric fields when used in conjunction with an oscillating electric3–5 or a uniform magnetic field2, in a way that overcomes the limitations imposed by Earnshaw’s theorem6. On the other hand, it is also possible to focus charged particles using quadrupole magnetic7–11 or electrostatic lenses9,10,12,13. Of utmost significance is the principle of strong focusing, first proposed by Christofilos14 and Courant, Livingston, and Snyder15. Indeed, in facilities like the Large Hadron Collider (LHC), hundreds of quadrupole superconducting magnets are employed to periodically steer and focus protons throughout their journey. At this point, perhaps it is fair to say that the current use of such sequences of magnetic or electrostatic lenses for particle guiding is to a great extent reminiscent of pre-fiber era optical communication systems16,17 where light transport was envisaged to take place through a succession of optical lenses. Naturally, the question arises, as to whether it is possible to guide charged particles in vacuum in a manner akin to that used for light in optical fibers18.

In this work, we demonstrate that is possible to guide and confine charged particles in periodically twisted electrostatic fields by exploiting the counterintuitive attributes of Lagrange points. As in celestial mechanics, electrons and protons can be perpetually captured in such arrangements within induced Lagrange points via dynamic stabilization processes facilitated by Coriolis forces19,20. In principle, these electron/proton waveguiding structures can operate in single-mode formats with very small cross-sections where quantum mechanical effects can be manifested, in a manner that directly emulates the optical fiber paradigm18,21–26. Here, the problem is treated both quantum-mechanically and classically – up to the relativistic regime. By accounting for Larmor radiation, we find that these arrangements can exhibit small losses – thus opening up the possibility for transporting charged particles at high speeds over prolonged distances. As we will see, the approach proposed here introduces additional degrees of freedom in the sense that goes beyond the well-established strong focusing principle that relies on quadrupole fields. This same strategy can be deployed in other more common settings like for example, electron microscopies27–30 and lithographies31,32, particle accelerators33–36, and vacuum electronic and communication systems37,38 using high-speed charged particles. Finally, the prospect exists to physically shuttle qubits either between multiplexed traps or a quantum network’s nodes39–44 where in principle, adiabatic means are used to propel particles in the Lagrange waveguides.

Results

Lagrange waveguide setup

Figure 1a depicts a schematic of the proposed electron guiding system. In this arrangement, electrons are emitted from an electron gun after being accelerated by a static electric field. In principle, the electrons can be further accelerated using LINAC arrangements33–36 before injected in the guiding setup. As we will see, these high-speed electrons can be confined and guided around stable Lagrange points that are induced by a twisted electric field V(x,y,z). The twisted electric field configuration is produced by a helical metal wire (of pitch Λ and helical radius d) that is kept at a potential V0 with respect to a surrounding metal cylinder that is grounded (See Supplementary Note 1 and Supplementary Fig. 1). In this system, the guided electrons will then be transported along a helicoidal trajectory within a small cross-section. A similar platform can be deployed to guide heavier charged particles like protons, ions, etc. Before we proceed any further, it may be useful to briefly describe the properties of Lagrange points, as they form the basis upon which the guiding system relies.Fig. 1 Charged-particle beam dynamics in a Lagrange waveguide.

a Schematic of a possible experimental setup, to observe charged particle Trojan bound states (bright yellow beam). A stable Lagrange point is established by twisting the electrostatic potential produced by a charged helical metallic wire when kept at a potential V0. b Cross-section of the setup in (a). The electron is trapped at a stable Lagrange point and follows a helical trajectory along the z axis. Stability in trapping this charged particle is provided through the Coriolis force Fc when viewed from the co-rotating frame. R represents the distance between the center C of the grounded metallic tube and the center of the Lagrange waveguide and d is the distance between the center C and the center of the helical wire.

Lagrange points

In celestial mechanics, Lagrange points represent unique equilibrium positions where the gravitational forces from two orbiting massive bodies counteract the centrifugal force19,20. In gravitational settings (reduced three-body systems), the Lagrange points encompass five positions designated as L1,L2,..., and L5. The first three (L1, L2, and L3) are colinear with respect to the two bodies and happen to be inherently unstable, while the remaining two (L4 and L5) exhibit dynamic stability. As a result, smaller planetesimals can be indefinitely trapped around L4 and L5, for example, in the case of the Trojan asteroids in the Sun-Jupiter system. Intriguingly, what dynamically stabilizes the motion of a captured third body is the Coriolis force that acts in a Sisyphean manner in spite of the fact that in the co-rotating frame, the two-dimensional transverse potential landscape exhibits a maximum at L4 and L5. We note that quite recently, this Lagrange-induced waveguiding process has been demonstrated within the realm of optics26. In this respect, optical Trojan beams were guided and trapped even within defocusing refractive index profiles. Moreover, Lagrange points can be induced even from a single helicoidal potential without the need for a secondary source, as in reduced three-body systems.

Quantum wave mechanics of guided charged particles at a Lagrange point

In this section, we analyze the stationary quantum wavefunctions or modes associated with a charged particle, like an electron, when trapped and transported within a Lagrange waveguide (Fig. 1b). To explore this possibility, we use the Schrödinger equation under the assumption that the particles primarily move along the z axis with paraxial momenta pz=ℏkz≫px,py. To investigate this problem, we employ a moving coordinate frame z′=z,t′=t−z/vz where vz=ℏkz/me represents the dominant z component of the electron’s velocity while me denotes its corresponding mass. Moreover, kz≃2π/λdB where λdB is the electron’s de Broglie wavelength (See Supplementary Note 2). In this case, the Schrödinger equation takes the form i∂zψ=H^′ψ, where H^′=−∇x,y2/(2kz)+(me/ℏkz2)U(x,y,z) and U(x,y,z)=eV(x,y,z) indicates the twisted three-dimensional electrostatic potential in the absolute (stationary) frame (See Supplementary Note 3). At this point, it is perhaps more convenient to consider this problem within the co-rotating frame with normalized coordinates (u,v,ξ) of the helix (u,vT=R(ΩZ)X,YT) to formally decouple the twisted static electric potential U from Z, in which case U=Uu,v, (X,Y,Z) represent the normalized coordinates in the stationary frame and Ω is the normalized spatial angular velocity (See Supplementary Note 4). To do so, we introduce a vector potential A=Ω×rΩ to account for Coriolis effects where Ω=2πz0Z^/Λ with z0 being the normalization factor in z^ direction and rΩ=uu^+vv^ (See Supplementary Note 4). In the rotating frame, the normalized Schrödinger equation now takes the form1 i∂ψ∂Z=12pΩ−A2+UΩ,effu,vψ,

where pΩ=−iu^∂/∂u−iv^∂/∂v is the normalized momentum operator, and UΩ,effu,v=ex02meVX,Y,Z=0/ℏ2−Ω2rΩ2/2 is an effective potential that now also incorporates the centrifugal term −Ω2rΩ2/2 and x0 is the normalization factor in x^,y^ directions (See Supplementary Notes 3, 4). Given that the helicoidal metallic wire is kept at a repulsive voltage V0<0 within a grounded tube of radius b (Fig. 1a), the generated electrostatic potential is expected to vary in a logarithmic fashion V=V0lnr′/b/lna/b within the stationary coordinate system (x,y,z) where a is the radius of the wire and r′=x−dcosΩ0z2+y−dsinΩ0z2 where Ω0=Ω/z0=2π/Λ represents the actual spatial angular velocity (see Supplementary Note 1). Figure 2a displays the logarithmic-like potential V when spiraling around the center of the tube C with a pitch of Λ= 6 cm, a radius of a=275μm and b=2cm at V0=−2.15kV. Without loss of generality, here, we assume that the injected electrons have a kinetic energy of 30keV. The corresponding effective potential UΩ,effu,v as viewed in the co-rotating frame is depicted in Fig. 2b. For this scenario, two Lagrange points are induced (designated as LA and LB) where the electrostatic repulsion balances the centrifugal force, i.e., ∇UΩ,eff=0 (See Supplementary Note 5). In this case, it so happens that only LA is stable while LB, being a saddle point, is unstable. The positions of LA and LB are also marked in Fig. 2a for clarity. As previously noted, UΩ,eff has a maximum at LA. To obtain the electron quantum eigenfunctions, we solve Eq. (1) numerically. The wavefunction probability distribution corresponding to the ground state is shown in Fig. 2c. In this particular arrangement, the fundamental mode (Fig. 2c) has an elliptical Gaussian-like shape with a mean spot size radius of ~0.3 μm. Similarly, Fig. 2e, g display the wavefunction profiles for the next two modes. Note that, in principle, the spot size of the ground state can be further reduced to a few nanometers by either increasing V0 or by decreasing the de Broglie wavelength. To understand the nature of the trapped quantum wavefunctions, we approximately expand the effective potential around LA to second order, i.e., UΩ,eff≅−(ω12u2+ω22v2)/2, where ω12,ω22 represent the corresponding curvatures of this elliptical parabolic potential landscape. In this case, one can show that the ground state eigenfunction is analytically given by ψ0=Ne−pu2e−qv2eiηuveiβZ (See Methods). Interestingly, this mode involves a uv phase distribution that is specific to this particular arrangement—a direct byproduct of Coriolis effects. This phase term is manifested in an X-like manner and happens to persist even for high-order states (Fig. 2d, f, h). Our theoretical results are confirmed by numerically solving the Schrödinger problem in both the stationary and rotating frames. As shown in Fig. 2i, the probability profile of the ground state remains invariant (trapped) during propagation while twisting around the center C—a hallmark of the guiding behavior. Finally, we note that Lagrange waveguiding is also possible even in the case where V0 is attractive (V0>0) (See Supplementary Note 5).Fig. 2 Quantum wave dynamics of guided charged particles at a stable Lagrange point.

a An induced logarithmic, defocusing, and spiraling electrostatic potential when viewed within the stationary frame at z=0. This potential profile rotates along z at a constant spatial angular velocity Ω0 around the center C (0,0). The corresponding iso-contour potential lines are also shown. b In the co-rotating frame (u,v), the effective potential UΩ,eff now incorporates centrifugal effects and exhibits two Lagrange points, LA and LB. LB is a saddle point and hence is unstable. On the other hand, LA (ul,vl) exhibits a maximum, around which dynamics are stabilized through Coriolis effects. For comparison, the positions of LA and LB are also marked in (a). c, e, g Numerically obtained quantum wavefunction probability distributions for the fundamental (c) and next two electron Trojan modes (e, g). The quantum states in (c, e, g) at LA correspond to the potential landscapes in (a, b). In this case, the quantum modes are elliptical in the (u′,v′) system where u′=u−ul,v′=v−vl. d, f, h Respective phase structures associated with these three electron wavefunction states. In all cases, the phase profile exhibits an X-shaped pattern. i Stable propagation of the quantum Trojan ground state shown in (c), as obtained numerically by solving Eq. (1). The quantum probability function remains invariant along its helical path. The helix pitch in (a–i) is taken to be Λ=6cm while the electron energy is assumed to be 30 keV. The normalization factor x0 in (b–h) is taken to be 1 μm.

Dynamics of non-relativistic and relativistic charged particles in a Lagrange waveguide

In this section, we investigate this same charged particle guiding system under both relativistic and non-relativistic conditions. This classical treatment is imperative given that one has to understand how a Lagrange waveguide will respond under incoherent excitation of several modes. As before, we consider the periodic spiraling electric potential distribution Ux,y,z=Ux,y,z+Λ within the co-rotating frame (uΩ0,vΩ0,ξΩ0) with actual units where ξΩ0=z, [uΩ0,vΩ0]T=R(Ω0z)x,yT. In this case, the Newtonian dynamics uΩ0(z),vΩ0(z), when viewed in the transverse plane, are described by (See Supplementary Note 6)2 d2rΩ0dz2=−∇u,vUΩ0,effuΩ0,vΩ0me+2drΩ0dz×Ω0z^,

where again UΩ0,eff(uΩ0,vΩ0)=−meΩ02(uΩ02+vΩ02)/2+UΩ0(uΩ0,vΩ0) where the subscript ‘Ω0’ represents the physical parameters with actual units, and the second term in the RHS of Eq. (2) accounts for Coriolis effects. To further exemplify this situation, let us consider electron confinement around a stable Lagrange point. To do so, we analyze the same waveguide configuration considered in the previous section. In this case, 30-keV electrons are injected in a 30 cm long Lagrange waveguide after passing through an aperture of radius r0=10μm. The thermionically emitted electrons from a heated cathode (of temperature T=1000K) obey a Maxwell-Boltzmann distribution with a probability density f(vi)∝exp(−vi2/2σv2), where i=x,y, σv=kBT/me denotes the velocity variance and kB represents the Boltzmann constant. Figure 3a depicts the stable trajectory of an electron when trapped around the LA Lagrange point over a distance of 30 cm. The waveguiding action of this arrangement is obvious. Meanwhile, Fig. 3b displays both the spatial and velocity distribution of the injected electrons when the aperture is centered at LA. In the absence of a repelling potential (V0=0), the electron beam spreads or diffracts to a spot size of ~1mm after 30 cm of propagation (Fig. 3c). On the other hand, for V0=−2.15kV, a Lagrange waveguide is induced that now stably guides the electron beam with a spot size of ~65μm (Fig. 3d). Note that here, the beam spot size is considerably larger than the mean spot size of the fundamental mode (0.3 μm) since several modes are incoherently excited under these conditions. In the setup proposed here, space-charge effects can be safely ignored for electron currents below mAs (Supplementary Note 7, ref. 10).Fig. 3 Classical electron dynamics in a Lagrange waveguide.

a Stable trapping of an electron unfolding around LA for the potential landscapes depicted in Fig. 2a, b, as viewed within the co-rotating frame. The normalization factor x0 here is taken to be 1 μm. b Spatial and velocity distributions associated with the injected electrons when the aperture at the input of the Lagrange waveguide has a radius of 10 μm and is centered at LA. c Histogram of electron spatial distribution after 30 cm in free space (V0=0). In this case, the electron beam diffracts to a spot size of ~1 mm. d Histogram of electron spatial distribution after 30 cm of propagation in a Lagrange waveguide when the repelling potential is V0=−2.15kV. In this latter scenario, the electron beam is stably guided with a mean spot size of ~65 μm. e Transverse trajectory (blue curve) of a high-speed electron (ve = 0.999 c) over 100 m. The yellow dashed line depicts the Lagrange point position, located at a radius of ~112 μm from the center. The electron was positioned at a distance ~4 μm away from the Lagrange point. f Electron beam transverse distribution histogram (for ve = 0.999 c) after 100 m of propagation in a Lagrange waveguide when the repelling potential is V0 = −80kV. The electron beam is stably guided with a mean spot size of ~215 μm. The scaling bar (yellow) in (c, d, f) corresponds to 300 μm while that (green) in the inset of (d, f) to 200 μm.

This same problem is now analyzed in the relativistic regime by solving the equation of motion45 m(dve/dt)=F−(F⋅ve)(ve/c2) where F=−e∇V (Supplementary Note 8). In this case, one can show that, again, the Lagrange point is located on the same line that connects the center with the helicoidal wire source at a distance l from the center. For a stable Lagrange point LA, this distance is given by l=[d−d2+4eΘ/(m0γΩ02vz2)]/2 where γ=1/1−ve2/c2 is the Lorentz factor, m0 is the rest mass, Θ=V0/ln(a/b) and ve2=∑i=x,y,zvi2=vz2(1+Ω02l2). To confirm trapping and guidance at the relativistic Lagrange point LA, numerical simulations are carried out when ve=0.999c. The trajectory of a high-speed electron (ve=0.999c) around LA is depicted in Fig. 3e after 100 m of propagation. On the other hand, a histogram of an electron beam is also shown in Fig. 3f for this same distance. For these figures, the system parameters were taken to be Λ=6cm, a=500μm, d=1.8mm, b=2cm, and V0=−80kV. Under these conditions, the relativistic Lagrange point is located at a distance of ~112μm from the center. In obtaining the results of Fig. 3e,f, we again assume that the thermionic electrons emitted from a heated cathode (of temperature T=1000K) after acceleration, entered the relativistic Lagrange waveguide through an aperture of radius 10μm. The pertinent velocity and spatial distributions are depicted in Fig. 3b.

Transporting protons and heavier charged particles in Lagrange waveguides

This same approach can also be used to guide heavier charged particles like protons and ions. Table 1 provides possible design parameters for transporting protons, Strontium (88Sr+), and Ytterbium ions (171Yb+) after acquiring a terminal velocity vz of ~108m/s. The spot size of the quantum ground state (fundamental mode) in the Lagrange waveguide is also given for comparison purposes. The methodology used to evaluate these parameters is provided in Supplementary Note 5.Table 1 Possible design parameters for transporting other charged particles

Charged particle design parameters	Proton	88Sr+	171Yb+	
Particle kinetic energy	53.7 MeV	4.7 GeV	9.2 GeV	
Applied voltage V0	22 kV	22 kV	22 kV	
Lagrange waveguide radius R	322 μm	3 μm	1.5 μm	
Pitch Λ	60 cm	60 cm	60 cm	
Mean spot size w0 of the quantum ground state	37 nm	45 nm	45 nm	
In all the above designs, the wire radius is a=0.275 mm, the grounded outer tube radius is b=2cm, and the distance between the wire center and C is d=1.8mm.

Radiation losses

In this section, we provide an estimate for the energy loss rate associated with a charged particle when trapped in a Lagrange waveguide. Given that the particle will be transported along a helical trajectory, it will inevitably radiate energy because of transverse acceleration. In general, in this twisted arrangement, the respective dynamics can be decomposed into a uniform motion vz and a transverse velocity v⊥ (v⊥≪vz) that leads to a centripetal acceleration ae=−r^v⊥2/R which is perpendicular to the particle’s total velocity ve=vzz^+v⊥. In the latter expression, R represents the radius of the helicoidal Lagrange waveguide. In this case, the relativistic radiation loss rate can be obtained from46 (See Supplementary Note 9)3 Prad=μ0e2γ6ae26πc1−βe×a^e2,

here γ=1/1−βe2 and βe=ve/c is the velocity ratio. In this regard, the radiation loss rate experienced by a 30-keV electron in a Lagrange waveguide (Fig. 1a) with Λ=6cm and R=0.74mm will be 0.27 eV/s, which is indeed negligible, given that it translates to ~4×10−10dB/km. On the other hand, a 1-TeV proton, moving at a speed ve=(1−4.4×10−7)c in a Lagrange waveguide with Λ=60cm and R=260nm (with wire rotation radius d=1.8 mm, V0= 220 kV) is expected to encounter a radiation loss rate of 302 eV/s, which is again minute as compared to the energy of the particle. Another source for losses can be manifested when a charged particle is moving close to an imperfectly conducing surface47, in which case the drag force leads to Ohmic heating. For the 30 keV electron example considered above, if a copper outer tube has a radius of b=2cm, the drag loss rate is predicted to be only 66.3 neV/s. In addition, the thermal noise generated by the twisted wire is also considered and expected to be of no practical significance in this Lagrange waveguide arrangement (Supplementary Note 10). Finally, it should be noted that, unlike photons for which loss is equivalent to complete annihilation, in these waveguides, loss merely means a reduction in the energy of the particles. Consequently, charged particles can still preserve properties like entanglement under appropriate conditions.

The above estimates indicate that Lagrange waveguides may indeed be promising in guiding charged particles in accelerator designs or in extremely low-loss electron communication systems that operate in either the classical or quantum domain37,38.

Discussion

Here, we have demonstrated a methodology for trapping charged particles such as electrons, protons or ions by utilizing the features of Lagrange points. This approach enables charged particle guiding (up to relativistic speeds) in a manner akin to that responsible for light transport in optical fibers. At the induced Lagrange points, the electron/ion beams can be stably captured, because of Coriolis effects, in well-defined quantum states produced by long-range spiraling electrostatic forces. Unlike other charge particle transport approaches relying on quadrupole potential distributions8,9, our methodology provides additional degrees of freedom in inducing multiple parallel guiding channels that can be used to enable quantum coupling/splitting arrangements. The work presented here may potentially be employed for transporting accelerated charged particles in a vacuum where guiding has thus far remained out of reach. Of interest would be to investigate the prospect of manipulating these beams in single-mode regimes where quantum phenomena can be directly manifested. Along these lines, the possibility of shuttling entangled ion qubits over long distances among quantum charge-coupled device systems (where magnetic decoherence could be an issue39,48) using electrostatic Lagrange waveguides, can be another important direction.

Methods

Analytical solution for the fundamental Trojan mode in a twisted parabolic potential

Here we provide an analytical solution for the fundamental quantum Trojan eigenstate in a twisted parabolic elliptical potential. The stable Lagrange point is located at (ul,vl) within the (u,v) system. To do so, we first obtain this solution when the normalized potential is shifted to the center C (See Supplementary Note 4), in which case, to first order UΩ,eff=Umax−ω12u2+ω22v2/2. In this case, the quantum wavefunction envelope obeys4 i∂ψ∂Z=−12∂2ψ∂u2+∂2ψ∂v2+iΩu∂ψ∂v−v∂ψ∂u+12Ω2u2+v2ψ+Umax−ω12u2+ω22v22ψ⋅

It can be shown that the ground state of Eq. (4) is given by an elliptical Gaussian wavefunction:5 ψ=Ne−pu2e−qv2eiηuveiσZe−iUmaxZ,

where p,q>0 (p,q∈R+) and N is a normalization factor. To determine the constants involved in Eq. (5), we substitute Eq. (5) in (4) from where we find6 Hψ=−12−2p+−2pu+iηv2−2q+−2qv+iηu2+12Ω2u2+v2+iΩu−2qv+iηu−v−2pu+iηv−12ω12u2−12ω22v2ψ=−σψ⋅

By comparing terms associated with u2, v2 and uv, the following equations are obtained7 η2−4p2+Ω2−ω12−2Ωη=0,

8 η2−4q2+Ω2−ω22+2Ωη=0,

9 −ηp+q+Ωq−p=0,

and10 p+q=−σ.

From Eq. (9), we find η=Ω(q−p)/p+q<∣Ω∣. Meanwhile Eqs. (7) and (8) lead to11-1 η=Ω±ω12+4p2,

11-2 η=−Ω±ω22+4q2.

Given that η<Ω, one has to select the negative sign in Eqs. (11–1) and the positive sign in Eq. (11–2). Therefore,12 η=q−pp+qΩ=Ω−ω12+4p2=−Ω+ω22+4q2,

From Eq. (12), we obtain the following two relations13-1 2pp+qΩ=ω12+4p2,

13-2 2qp+qΩ=ω22+4q2.

By squaring Eqs. (13), one finds14 p2q2=ω12+4p2ω22+4q2,

from where we obtain15 pq=ω1ω2.

Hence, from Eq. (9), we directly establish the relation16 η=ω2−ω1ω2+ω1Ω.

Finally, from Eq. (12), p,q can now be determined:17-1 p=4Ω2−ω1+ω22122ω1+ω2ω1,

17-2 q=4Ω2−ω1+ω22122ω1+ω2ω2.

For these p,q solutions to be real and positive, one requires that26 2Ω>ω1+ω2. This ground state solution now can be translated to the actual position of the stable Lagrange point (ul,vl) where the effective potential UΩ,eff=Umax−(ω12(u−ul)2+ω22(v−vl)2)/2. In this case, the corresponding eigenmode is given by ref. 23 ψui,vi=ψu−ul,v−vleiΦ(u,v), where Φu,v=Ωulv−vlu.

The dynamics of the Trojan mode can be solved numerically under any arbitrary initial conditions using beam propagation methods that rely on fast Fourier transforms (BPM-FFT). The quantum Trojan eigenstates supported by the actual effective potential around a stable Lagrange point (like the one depicted in Fig. 2b) are obtained by numerically solving Eqs. (1) or (4). In this case, the eigenvalue problem is solved using finite-difference methods (FDM).

Supplementary information

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Supplementary information

The online version contains supplementary material available at 10.1038/s41467-024-51083-5.

Acknowledgements

This work was supported by the Air Force Office of Scientific Research (AFOSR) Multidisciplinary University Research Initiative (MURI) award on Novel light-matter interactions in topologically non-trivial Weyl semimetal structures and systems (award no. FA9550-20-1-0322)(M.K., D.N.C., H.L., Y.W., and G.G.P.), AFOSR MURI award on Programmable systems with non-Hermitian quantum dynamics(award no. FA9550-21-1-0202) (M.K., D.N.C., H.L., Y.W., and G.G.P.), ONR MURI award on the classical entanglement of light (award no. N00014-20-1-2789) (M.K., D.N.C., H.L., Y.W., and G.G.P.), the Army Research Office (W911NF-23-1-0312) (M.K., D.N.C., and G.G.P.), the Department of Energy (DE-SC0022282) (D.N.C. and H.L.), W.M. Keck Foundation (D.N.C.), MPS Simons collaboration (Simons grant no. 733682) (D.N.C.), US Air Force Research Laboratory (FA86511820019) (D.N.C.), Israel Ministry of Defense (IMOD: 4441279927) (D.N.C.) and AFRL – Applied Research Solutions (S03015) (FA8650-19-C-1692) (M.K.). All authors acknowledge the Global Research Code on the development, implementation, and communication of this research. For the purpose of transparency, we have included this statement on inclusion and ethics. This work cites a comprehensive list of research from around the world on related topics.

Author contributions

M.K. and D.N.C. conceived the idea. H.L., Y.W., G.G.P., M.K. and D.N.C. developed the theory. All the authors contributed to the writing of the original draft, review, and editing.

Peer review

Peer review information

Nature Communications thanks François Fillion-Gourdeau and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Data availability

All other data supporting the plots and findings within this paper are available from the corresponding authors upon request, subject to restrictions due to ongoing patent considerations. Source data are provided with this paper.

Code availability

The numerical codes used in this study (MATLAB and COMSOL) are available upon request from the corresponding authors, subject to restrictions due to ongoing patent considerations.

Competing interests

The authors declare the following competing interests: An invention disclosure related to the subject matter of this manuscript has been filed, and a patent application is currently underway. The details of the patent information are provided: [1] US Provisional, Serial No. 63/528,037, Applicant: Univ. of Southern California, Inventors: Mercedeh Khajavikhan, Haokun Luo, Demetrios N. Christodoulides, and Yunxuan Wei, was filed on July 20, 2023. Location and Institution: US, the US provisional. This application is now expired. [2] PCT Application, Serial No. PCT/US2024/38833, Applicant: Univ. of Southern California, Inventors: Mercedeh Khajavikhan, Haokun Luo, Demetrios N. Christodoulides, and Yunxuan Wei, was filed on July 19, 2024. Location and Institution: WIPO, the PCT. This application claims priority to the previous US provisional application and is pending.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Change history

9/11/2024

In the PDF of this article, some mathematical expressions did not display correctly. On page 3 and 4, the bolded variable Ω should not have been in italics. The original article PDF has now been corrected. The HTML was unaffected.

Change history

9/11/2024

A Correction to this paper has been published: 10.1038/s41467-024-52420-4
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References

1. Thomson, J. J. X. L. The London, Edinburgh, and Dublin of Philosophical Magazine and Journal of Science (Taylor & France, 1897).
2. Brown LS Gabrielse G Geonium theory: physics of a single electron or ion in a Penning trap Rev. Mod. Phys. 1986 58 233 311 10.1103/RevModPhys.58.233
Brown, L. S. & Gabrielse, G. Geonium theory: physics of a single electron or ion in a Penning trap. Rev. Mod. Phys. 58, 233–311 (1986).10.1103/RevModPhys.58.233
3. Paul W Electromagnetic traps for charged and neutral particles Rev. Mod. Phys. 1990 62 531 540 10.1103/RevModPhys.62.531
Paul, W. Electromagnetic traps for charged and neutral particles. Rev. Mod. Phys. 62, 531–540 (1990).10.1103/RevModPhys.62.531
4. Drewsen M Brodersen C Hornekær L Hangst JS Schifffer JP Large ion crystals in a linear Paul trap Phys. Rev. Lett. 1998 81 2878 2881 10.1103/PhysRevLett.81.2878
Drewsen, M., Brodersen, C., Hornekær, L., Hangst, J. S. & Schifffer, J. P. Large ion crystals in a linear Paul trap. Phys. Rev. Lett. 81, 2878–2881 (1998).10.1103/PhysRevLett.81.2878
5. Matthiesen C Yu Q Guo J Alonso AM Häffner H Trapping electrons in a room-temperature microwave Paul trap Phys. Rev. X 2021 11 011019
Matthiesen, C., Yu, Q., Guo, J., Alonso, A. M. & Häffner, H. Trapping electrons in a room-temperature microwave Paul trap. Phys. Rev. X 11, 011019 (2021).
6. Earnshaw S On the nature of the molecular forces which regulate the constitution of the luminiferous ether Trans. Camb. Philos. Soc. 1848 7 97
Earnshaw, S. On the nature of the molecular forces which regulate the constitution of the luminiferous ether. Trans. Camb. Philos. Soc. 7, 97 (1848).
7. Dayton IE Shoemaker FC Mozley RF The measurement of two dimensional fields. Part II: study of a quadrupole magnet Rev. Sci. Instrum. 1954 25 485 489 10.1063/1.1771107
Dayton, I. E., Shoemaker, F. C. & Mozley, R. F. The measurement of two dimensional fields. Part II: study of a quadrupole magnet. Rev. Sci. Instrum. 25, 485–489 (1954).10.1063/1.1771107
8. Enge HA Ion focusing properties of a quadrupole lens pair Rev. Sci. Instrum. 1959 30 248 251 10.1063/1.1716528
Enge, H. A. Ion focusing properties of a quadrupole lens pair. Rev. Sci. Instrum. 30, 248–251 (1959).10.1063/1.1716528
9. Pierce, J. R. Theory and Design of Electron Beams (Books on Demand, 1954).
10. Tsimring, S. E. Electron Beams and Microwave Vacuum Electronics. (John Wiley & Sons, 2006).
11. Mendel JT Quate CF Yocom WH Electron beam focusing with periodic permanent magnet fields Proc. IRE 1954 42 800 810 10.1109/JRPROC.1954.274515
Mendel, J. T., Quate, C. F. & Yocom, W. H. Electron beam focusing with periodic permanent magnet fields. Proc. IRE 42, 800–810 (1954).10.1109/JRPROC.1954.274515
12. Tien PK Focusing of a long cylindrical electron stream by means of periodic electrostatic fields J. Appl. Phys. 1954 25 1281 1288 10.1063/1.1721545
Tien, P. K. Focusing of a long cylindrical electron stream by means of periodic electrostatic fields. J. Appl. Phys. 25, 1281–1288 (1954).10.1063/1.1721545
13. Clogston AM Heffner H Focusing of an electron beam by periodic fields J. Appl. Phys. 1954 25 436 447 10.1063/1.1721659
Clogston, A. M. & Heffner, H. Focusing of an electron beam by periodic fields. J. Appl. Phys. 25, 436–447 (1954).10.1063/1.1721659
14. Christofilos, N. Focussing system for ions and electrons. US patent 2,736,799 (1956).
15. Courant ED Livingston MS Snyder HS The strong-focusing synchroton–a new high energy accelerator Phys. Rev. 1952 88 1190 1196 10.1103/PhysRev.88.1190
Courant, E. D., Livingston, M. S. & Snyder, H. S. The strong-focusing synchroton–a new high energy accelerator. Phys. Rev. 88, 1190–1196 (1952).10.1103/PhysRev.88.1190
16. Marcuse D Propagation of light rays through a lens-waveguide with curved axis Bell Syst. Tech. J. 1964 43 741 753 10.1002/j.1538-7305.1964.tb01004.x
Marcuse, D. Propagation of light rays through a lens-waveguide with curved axis. Bell Syst. Tech. J. 43, 741–753 (1964).10.1002/j.1538-7305.1964.tb01004.x
17. Miller SE Communication by laser Sci. Am. 1966 214 19 27 10.1038/scientificamerican0166-19
Miller, S. E. Communication by laser. Sci. Am. 214, 19–27 (1966).10.1038/scientificamerican0166-19
18. Snyder, A. W. & Love, J. D. Optical Waveguide Theory (Chapman and Hall, 1983).
19. Bannikova, E. & Capaccioli, M. Foundations of Celestial Mechanics (Springer Cham, 2022).
20. Pérez-Villegas A Portail M Wegg C Gerhard O Revisiting the tale of Hercules: how stars orbiting the Lagrange points visit the sun Astrophys. J. Lett. 2017 840 L2 10.3847/2041-8213/aa6c26
Pérez-Villegas, A., Portail, M., Wegg, C. & Gerhard, O. Revisiting the tale of Hercules: how stars orbiting the Lagrange points visit the sun. Astrophys. J. Lett. 840, L2 (2017).10.3847/2041-8213/aa6c26
21. Knight JC Photonic crystal fibres Nature 2003 424 847 851 10.1038/nature01940 12917699
Knight, J. C. Photonic crystal fibres. Nature 424, 847–851 (2003).12917699 10.1038/nature01940
22. Birks TA Knight JC Russell PSJ Endlessly single-mode photonic crystal fiber Opt. Lett. 1997 22 961 963 10.1364/OL.22.000961 18185719
Birks, T. A., Knight, J. C. & Russell, P. S. J. Endlessly single-mode photonic crystal fiber. Opt. Lett. 22, 961–963 (1997).18185719 10.1364/OL.22.000961
23. Ibanescu M Fink Y Fan S Thomas EL Joannopoulos JD An all-dielectric coaxial waveguide Science 2000 289 415 419 10.1126/science.289.5478.415 10903194
Ibanescu, M., Fink, Y., Fan, S., Thomas, E. L. & Joannopoulos, J. D. An all-dielectric coaxial waveguide. Science 289, 415–419 (2000).10903194 10.1126/science.289.5478.415
24. Plotnik Y Observation of unconventional edge states in ‘photonic graphene’ Nat. Mater. 2014 13 57 62 10.1038/nmat3783 24193661
Plotnik, Y. et al. Observation of unconventional edge states in ‘photonic graphene’. Nat. Mater. 13, 57–62 (2014).24193661 10.1038/nmat3783
25. Hsu CW Zhen B Stone AD Joannopoulos JD Soljačić M Bound states in the continuum Nat. Rev. Mater. 2016 1 16048 10.1038/natrevmats.2016.48
Hsu, C. W., Zhen, B., Stone, A. D., Joannopoulos, J. D. & Soljačić, M. Bound states in the continuum. Nat. Rev. Mater. 1, 16048 (2016).10.1038/natrevmats.2016.48
26. Luo H Guiding Trojan light beams via Lagrange points Nat. Phys. 2024 20 95 100 10.1038/s41567-023-02270-6
Luo, H. et al. Guiding Trojan light beams via Lagrange points. Nat. Phys. 20, 95–100 (2024).10.1038/s41567-023-02270-6
27. Muller D Structure and bonding at the atomic scale by scanning transmission electron microscopy Nat. Mater. 2009 8 263 270 10.1038/nmat2380 19308085
Muller, D. Structure and bonding at the atomic scale by scanning transmission electron microscopy. Nat. Mater. 8, 263–270 (2009).19308085 10.1038/nmat2380
28. de Boer P Hoogenboom JP Giepmans BNG Correlated light and electron microscopy: ultrastructure lights up! Nat. Methods 2015 12 503 513 10.1038/nmeth.3400 26020503
de Boer, P., Hoogenboom, J. P. & Giepmans, B. N. G. Correlated light and electron microscopy: ultrastructure lights up! Nat. Methods 12, 503–513 (2015).26020503 10.1038/nmeth.3400
29. Haider M Electron microscopy image enhanced Nature 1998 392 768 769 10.1038/33823
Haider, M. et al. Electron microscopy image enhanced. Nature 392, 768–769 (1998).10.1038/33823
30. Zewail AH Four-dimensional electron microscopy Science 2010 328 187 193 10.1126/science.1166135 20378810
Zewail, A. H. Four-dimensional electron microscopy. Science 328, 187–193 (2010).20378810 10.1126/science.1166135
31. Vieu C Electron beam lithography: resolution limits and applications Appl. Surf. Sci. 2000 164 111 117 10.1016/S0169-4332(00)00352-4
Vieu, C. et al. Electron beam lithography: resolution limits and applications. Appl. Surf. Sci. 164, 111–117 (2000).10.1016/S0169-4332(00)00352-4
32. Tu R. Direct X-ray and electron-beam lithography of halogenated zeolitic imidazolate frameworks Nat. Mater. 2021 20 93 99 10.1038/s41563-020-00827-x 33106648
Tu et al. R. Direct X-ray and electron-beam lithography of halogenated zeolitic imidazolate frameworks. Nat. Mater. 20, 93–99 (2021).33106648 10.1038/s41563-020-00827-x
33. Peralta EA Demonstration of electron acceleration in a laser-driven dielectric microstructure Nature 2013 503 91 94 10.1038/nature12664 24077116
Peralta, E. A. et al. Demonstration of electron acceleration in a laser-driven dielectric microstructure. Nature 503, 91–94 (2013).24077116 10.1038/nature12664
34. Chlouba T Coherent nanophotonic electron accelerator Nature 2023 622 476 480 10.1038/s41586-023-06602-7 37853151
Chlouba, T. et al. Coherent nanophotonic electron accelerator. Nature 622, 476–480 (2023).37853151 10.1038/s41586-023-06602-7
35. Leedle KJ Fabian Pease R Byer RL Harris JS Laser acceleration and deflection of 96.3 keV electrons with a silicon dielectric structure Optica 2015 2 158 161 10.1364/OPTICA.2.000158
Leedle, K. J., Fabian Pease, R., Byer, R. L. & Harris, J. S. Laser acceleration and deflection of 96.3 keV electrons with a silicon dielectric structure. Optica 2, 158–161 (2015).10.1364/OPTICA.2.000158
36. Black DS Net acceleration and direct measurement of attosecond electron pulses in a silicon dielectric laser accelerator Phys. Rev. Lett. 2019 123 264802 10.1103/PhysRevLett.123.264802 31951436
Black, D. S. et al. Net acceleration and direct measurement of attosecond electron pulses in a silicon dielectric laser accelerator. Phys. Rev. Lett. 123, 264802 (2019).31951436 10.1103/PhysRevLett.123.264802
37. Cozzolino D Da Lio B Bacco D Oxenløwe LK High-dimensional quantum communication: benefits, progress, and future challenges Adv. Quantum Technol. 2019 2 1900038 10.1002/qute.201900038
Cozzolino, D., Da Lio, B., Bacco, D. & Oxenløwe, L. K. High-dimensional quantum communication: benefits, progress, and future challenges. Adv. Quantum Technol. 2, 1900038 (2019).10.1002/qute.201900038
38. Paraïso TK A photonic integrated quantum secure communication system Nat. Photonics 2021 15 850 856 10.1038/s41566-021-00873-0
Paraïso, T. K. et al. A photonic integrated quantum secure communication system. Nat. Photonics 15, 850–856 (2021).10.1038/s41566-021-00873-0
39. Kielpinski D Monroe C Wineland DJ Architecture for a large-scale ion-trap quantum computer Nature 2002 417 709 711 10.1038/nature00784 12066177
Kielpinski, D., Monroe, C. & Wineland, D. J. Architecture for a large-scale ion-trap quantum computer. Nature 417, 709–711 (2002).12066177 10.1038/nature00784
40. Bruzewicz, C. D., Chiaverini, J., McConnell, R. & Sage, J. M. Trapped-ion quantum computing: progress and challenges. Appl. Phys. Rev. 6, 021314 (2019).
41. Walther A Controlling fast transport of cold trapped ions Phys. Rev. Lett. 2012 109 080501 10.1103/PhysRevLett.109.080501 23002727
Walther, A. et al. Controlling fast transport of cold trapped ions. Phys. Rev. Lett. 109, 080501 (2012).23002727 10.1103/PhysRevLett.109.080501
42. Monroe C Kim J Scaling the ion trap quantum processor Science 2013 339 1164 1169 10.1126/science.1231298 23471398
Monroe, C. & Kim, J. Scaling the ion trap quantum processor. Science 339, 1164–1169 (2013).23471398 10.1126/science.1231298
43. Brown KR Kim J Monroe C Co-designing a scalable quantum computer with trapped atomic ions npj Quantum Inf. 2016 2 16034 10.1038/npjqi.2016.34
Brown, K. R., Kim, J. & Monroe, C. Co-designing a scalable quantum computer with trapped atomic ions. npj Quantum Inf. 2, 16034 (2016).10.1038/npjqi.2016.34
44. Krutyanskiy V Entanglement of trapped-ion qubits separated by 230 meters Phys. Rev. Lett. 2023 130 050803 10.1103/PhysRevLett.130.050803 36800448
Krutyanskiy, V. et al. Entanglement of trapped-ion qubits separated by 230 meters. Phys. Rev. Lett. 130, 050803 (2023).36800448 10.1103/PhysRevLett.130.050803
45. Møller, C. The Theory of Relativity (Clarendon Press, 1952).
46. Zangwill, A. Modern Electrodynamics (Cambridge Univ. Press, 2013).
47. Boyer TH Penetration of the electric and magnetic velocity fields of a nonrelativistic point charge into a conducting plane Phys. Rev. A 1974 9 68 82 10.1103/PhysRevA.9.68
Boyer, T. H. Penetration of the electric and magnetic velocity fields of a nonrelativistic point charge into a conducting plane. Phys. Rev. A 9, 68–82 (1974).10.1103/PhysRevA.9.68
48. Wang P Single ion qubit with estimated coherence time exceeding one hour Nat. Commun. 2021 12 233 10.1038/s41467-020-20330-w 33431845
Wang, P. et al. Single ion qubit with estimated coherence time exceeding one hour. Nat. Commun. 12, 233 (2021).33431845 10.1038/s41467-020-20330-w
