Acoustic vortex beams in synthetic magnetic fields
AAcoustic vortex beams in synthetic magnetic fields
Irving Rond´on and Daniel Leykam Center for Theoretical Physics of Complex Systems,Institute for Basic Science (IBS), Daejeon 34126, Republic of Korea (Dated: August 23, 2019)We analyze propagation of acoustic vortex beams in longitudinal synthetic magnetic fields. Weshow how to generate two field configurations using a fluid contained in circulating cylinders: auniform synthetic magnetic field hosting Laguerre-Gauss modes, and an Aharonov-Bohm flux tubehosting Bessel beams. For non-paraxial beams we find qualitative differences from the well-studiedcase of electron vortex beams in magnetic fields, arising due to the vectorial nature of the acousticwave’s velocity field. In particular, the pressure and velocity components of the acoustic wave can beindividually sensitive to the relative sign of the beam orbital angular momentum and the magneticfield. Our findings illustrate how analogies between optical, electron, and acoustic vortex beams canbreak down in the presence of external vector potentials.
I. INTRODUCTION
Analogies between wave phenomena in different physi-cal systems are a powerful tool to understand and controlwave propagation. For example, the idea of characteris-ing electronic Bloch waves using topological invariants,responsible for the discovery of a wealth of new topo-logical insulator materials [1], has been fruitfully trans-lated to photonics [2] and acoustics [3–6]. Similarly, waveorbital angular momentum, long-studied in the contextof optical vortex beams [7], can also be created for freeelectron beams in transmission electron microscopes [8].Going beyond the simplest scalar waves, spin and spin-orbit-interactions provide extra degrees of freedom to de-sign and control structured vectorial wave fields [9].Acoustic waves are longitudinal and conventionallyviewed as spinless, purely scalar waves. Recently, how-ever, several studies have found that acoustic waves canexhibit nontrivial vectorial properties in the form of spintextures and spin-momentum locking [10–14]. Thesefeatures are hidden in the commonly-used second or-der Schr¨odinger-like wave equation governing the acous-tic velocity potential φ and require analysis of the morefundamental first order equations describing the coupledacoustic pressure P and velocity v fields. While theacoustic spin necessarily vanishes for the simplest planewaves, nontrivial spin-related phenomena can emerge forlocalized fields such as surface waves and non-paraxialbeams [11–13]. Spin therefore provides an additional de-gree of freedom to control acoustic waves, with potentialapplications including particle manipulation [15], selec-tive excitation of surface waves [11], and design of noveltopological materials. For example, Ref. [11] recently ob-served the spin-controlled directional excitation of acous-tic waves at the surface of a metamaterial waveguide.Since spin is a form of angular momentum, it is nat-ural to ask how acoustic spin interacts with the morewidely-studied orbital angular momentum degree of free-dom, exemplified by vortex beams [16–19]. Nontrivialspin in acoustic vortex beams was already hinted at byZhang and Marston in 2011 [18], who showed that the full acoustic angular momentum flux tensor contains non-paraxial corrections. Bliokh and Nori recently employeda quantum-like formalism analogous to that used for elec-tromagnetic waves to show that the angular momentumof acoustic beams naturally divides into spin and orbitalparts, obtaining nonzero spin densities of acoustic Besselbeams in free space [13]. Nevertheless, the total spin al-ways vanishes, as required for longitudinal waves. Dueto the time-reversal symmetry of the underlying acousticwave equations, reversing the orbital angular momentumof an acoustic vortex beam must flip its spin density.In this manuscript we aim to complement these recentstudies by analyzing the effect of broken time-reversalsymmetry on the spin, momentum, and energy densitiesof acoustic beams. To break time-reversal symmetry weconsider acoustic waves propagating through a slowly cir-culating fluid, which generates synthetic vector potentialsand magnetic fields for sound. Previous studies focusedon the quasi-2D limit of acoustic waves propagating inthe plane of circulation, studying scattering off a mag-netic flux tube [20], acoustic isolation and nonreciprocaltransmission [21], and topological phononic crystals [3].In these examples, the synthetic magnetic field is per-pendicular to the wavevector. Here we instead studypropagation parallel to the synthetic magnetic field, inthe waveguide geometry illustrated in Fig. 1 consistingof fluid confined between concentric rotating cylinders.By controlling the cylinder radii and rotation speeds,one can induce either an isolated flux tube or a uniformmagnetic field, resulting in effective Schr¨odinger equa-tions previously analyzed in the context of electron vor-tex beams [22]. We demonstrate some interesting fea-tures and qualitative differences compared to this pre-vious study due to the vectorial nature of the acousticwaves and their differing boundary conditions.We find that a uniform synthetic magnetic field canhost localized guided modes, which is a nontrivial re-sult due to the constraint that the field must be weakto avoid Taylor instabilities of the background Couetteflow [3, 23]. While the modal energy density is indepen-dent of the relative sign of the field and beam orbitalangular momentum, the individual pressure and velocity a r X i v : . [ phy s i c s . c l a ss - ph ] A ug xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxxxxxxx xxxxx z Ω Ω l (a) (b)(c) Uniform magnetic fieldAharonov-Bohm flux
FIG. 1: Schematic of the studied system. (a) Propagation ofa charge l acoustic vortex beam with frequency ω through awaveguide formed by concentric cylinders with radii R , ro-tating with angular frequencies Ω , . (b) Inner rotating cylin-der creates an Aharonov-Bohm tube of flux α = ω Ω R /c ,where c is the speed of sound. (c) The outer (hollow) cylindercreates a uniform synthetic magnetic field of strength Ω . field components are sensitive to this relative sign. As lo-calized beams carrying orbital angular momentum, theirlongitudinal spin density is also nonzero near the vor-tex core. A synthetic flux tube does not host localizedmodes but can nevertheless strongly affect the profiles ofnon-diffracting Bessel beams. By tuning the strength ofthe enclosed flux one can tune both the magnitude andsign of the spin density, as well as the beam radius. Inboth cases, the beam energy density remains finite at ze-ros of the corresponding Schr¨odinger wavefunction. Nearthese points the beam’s canonical and spin momentumhave opposite directions and similar magnitudes, mean-ing that a weak vector potential can significantly affectthe direction of the local energy flow.The structure of this article is as follows: Sec. IIpresents a self-contained overview of the quantum-likeformalism for acoustic waves, extending the free spaceanalysis of Ref. [13] to a background vector potential.Sec. III analyzes two interesting limits of the acousticwaveguide shown in Fig. 1: Laguerre-Gauss beams sup-ported by a uniform synthetic magnetic flux (Sec. III A)and Bessel beams threaded by an Aharonov-Bohm fluxtube (Sec. III B). Sec. IV concludes with discussion anda summary of the main results. II. VECTOR POTENTIALS FOR ACOUSTICWAVES
We study the propagation of small amplitude acousticwaves in a fluid with uniform density and a steady stateposition-dependent velocity profile u ( r ). We assume thefluid speed is much smaller than the speed of sound c ,i.e. u /c (cid:28)
1, and that u has a slow spatial variationcompared to the acoustic wavelength. Under these con-ditions, the acoustic velocity potential φ obeys the waveequation [3, 24] ∇ φ − c D t φ = 0 , D t = ∂ t + u · ∇ , (1)where c = 1 / ( ρβ ) is the speed of sound, determined bythe mass density ρ and compressibility β . The local pres-sure P ( r , t ) and particle velocity v ( r , t ) fields induced bythe acoustic wave are given by P = ρD t φ and v = −∇ φ .Eliminating φ yields coupled first order equations [24], βD t P = −∇ · v , ρD t v = −∇ P. (2)Using Eqs. (2) and the fact that ∇ · u = 0 for a timeindependent background flow, one can show that P and v obey the continuity equation D t (cid:18) β P + ρ v (cid:19) + ∇ · ( P v ) = 0 , (3)which we will see corresponds to an energy density in-dependent of the background flow, and an energy fluxdensity modified by the background flow.Now we specialise to monochromatic waves with timedependence e − iωt described by complex φ , P , and v fields, such that D t = − iω + u · ∇ . Physical observ-ables are then computed from the real parts of the fields.Assuming u /c (cid:28)
1, Eq. (1) can be recast as [3]( ∇ − i A ) φ + ω c φ = 0 , A = − ω u c , (4)such that the background fluid flow u resembles a vectorpotential in the time independent Schr¨odinger equation.The time-averaged energy W and energy flux Π densitiesof Eq. (3) are W = 14 (cid:0) β | P | + ρ | v | (cid:1) , (5) Π = 12 Re[ P ∗ v ] + W u . (6)Following Ref. [13], one can introduce a quantum-like for-malism by defining the four-component “wavefunction” | Ψ (cid:105) = ( P, v ) T and inner product (cid:104) Ψ | Ψ (cid:105) = 14 ω ( β | P | + ρ | v | ) , (7)such that W = (cid:104) Ψ | ω | Ψ (cid:105) becomes the local expec-tation value of the energy operator ω and Π is the ki-netic momentum density. One can similarly introducethe canonical momentum density p = (cid:104) Ψ | ( − i ∇ ) | Ψ (cid:105) characterizing the local phase gradient [13], p = 14 ω Im[ βP ∗ ∇ P + ρ v ∗ · ( ∇ ) v ] , (8)where v ∗ · ( ∇ ) v ≡ (cid:80) j v ∗ j ∇ v j . We stress that these quan-tities generally differ from the energy, canonical momen-tum, and kinetic momentum densities of the velocity po-tential, which are | φ | , Im[ φ ∗ ∇ φ ], and Im[ φ ∗ ∇ φ ] − A | φ | respectively.Expanding Eq. (6) in terms of the canonical momen-tum density p yields Π c = p + 14 ∇ × S − Wω A , (9)where the spin angular momentum density S = ρ ω Im ( v ∗ × v ) , (10)also contributes to Π [13]. The last term in Eq. (9) can bewritten as the expectation value of the vector potential, Wω A = (cid:104) Ψ | A | Ψ (cid:105) . (11)which resembles the vector potential contribution to thekinetic momentum in the Schr¨odinger equation.For completeness, we also give the angular momentumdensities: the canonical angular momentum density, L = r × p , (12)the total canonical angular momentum density J = L + S , (13)and the kinetic angular momentum density M = r × Π /c . (14)In particular, it was shown in Ref. [13] that for local-ized acoustic fields in the absence of a vector potential, (cid:104) S (cid:105) = 0 and (cid:104) M (cid:105) = (cid:104) J (cid:105) = (cid:104) L (cid:105) . In the presence of anacoustic vector potential L and S are unchanged, but M is modified by the background fluid flow, M = 12 c Re[ P ∗ r × v ] − Wω ( r × A ) . (15)Interestingly this Schr¨odinger-like formalism for acousticvector potentials differs from both the scalar Schr¨odingerequation (since | Ψ (cid:105) is a vector field) and the spinorSchr¨odinger-Pauli equation (since no Stern-Gerlach term S · ( ∇ × A ) appears in the equations of motion).As a simple example illustrating these differences con-sider the inhomogeneous transverse background flow u =(0 , B z x, z axis. There exists acousticwave solutions propagating along the z axis and evanes-cent along the x axis, described by the velocity potential φ = e ik z z − κx , where k z and κ are the propagation con-stant and localization length respectively. The pressureand velocity fields are P = − iωρe ik z z − κx , v = κ − ik z e ik z z − κx , (16)and the dispersion relation is ω = c ( k z − κ ). Thissolution is independent of the synthetic magnetic fieldstrength B z , which would couple v x and v y if there werea Stern-Gerlach term similar to the Pauli-Schr¨odingerequation. Only the kinetic momentum density is affectedby the vector potential, Π W = (cid:32) , B z x, c (cid:115) − κ k z (cid:33) , (17)acquiring a deflection parallel to the background flow.Since the derivation of the acoustic wave equations as-sumes a weak background flow ( B z x/c ) (cid:28)
1, this de-flection is small in the plane wave limit κ = 0. However,as the field becomes more strongly evanescent the lon-gitudinal kinetic momentum Π z becomes smaller, suchthat Π tilts more strongly in the direction of the back-ground flow. There are two ways this stronger tilt canbe interpreted: (i) the evanescent part of the wavevectorcontributes to the energy density W , but not the first( u -independent) term of Eq. (6), or (ii) the evanescentwave has a transverse spin S y which suppresses the longi-tudinal component of the kinetic momentum in Eq. (9).The origin of both these effects is ultimately the vectorialnature of the acoustic wavefunction | Ψ (cid:105) . By contrast,for scalar (Schr¨odinger) waves with fixed k z the directionof Π is independent of κ . III. ACOUSTIC BEAMS IN CYLINDRICALWAVEGUIDES
Now we will study the effect of the acoustic vectorpotential on the propagation of acoustic vortex beams.We shall consider the simplest case of cylindrical symme-try, corresponding to a purely azimuthal background flow u = u θ ( r )ˆ θ , where ( r, θ, z ) are cylindrical coordinates.The velocity potential for a cylindrically-symmetric beamwith canonical orbital angular momentum l is φ ( r ) = ψ ( r ) e i ( k z z + lθ ) . (18)Substituting ψ ( r ) into Eq. (4) yields an equation for theradial profile, (cid:20) r ∂ r ( r∂ r ) − r (cid:16) l + rωu θ c (cid:17) + ω c (cid:21) ψ = k z ψ. (19)The pressure P and velocity v fields are P = − iρ (cid:18) ω − u θ lr (cid:19) ψe i ( k z z + lθ ) , (20)( v r , v θ , v z ) = (cid:18) ∂ r ψ, ilr ψ, ik z ψ (cid:19) e i ( k z z + lθ ) . (21)Only P depends explicitly on the vector potential.To proceed we need to specify the background flow u .We will consider Couette flow in an incompressible fluidconfined between circulating concentric cylinders withradii R , rotating at angular frequencies Ω , , see Fig. 1.The small amplitude acoustic waves obey the hard wallboundary conditions ˆ r · v ( R , ) = 0. Assuming the cylin-ders are sufficiently long that boundary effects at theirtop and bottom are negligible, the cylinders establish theazimuthal background flow [23] u ( r ) = Ω − Ω R R − R /R r + R (Ω − Ω )1 − R /R r ˆ θ. (22)In the following we will focus on two simple limits: a thinmagnetic flux line at r = 0 ( u ( r ) ∝ r ˆ θ ) and a uniformmagnetic field ( u ( r ) ∝ r ˆ θ ).At first glance this might seem identical to the elec-tron vortex beam problem studied in Ref. [22]. How-ever, we stress that there are a few important differenceseven without taking the vectorial nature of the acousticwaves into account: Acoustic beams obey the Neumannboundary conditions ∂ r ψ ( R , ) = 0, whereas an electronbeam in a hard wall cylindrical waveguide would have theDirichlet boundary conditions ψ ( R , ) = 0. Addition-ally, the acoustic vector potential describes a real back-ground fluid flow which much have a finite speed muchslower than the fluid speed of sound c : the Schr¨odinger-like Eq. (4) neglects terms of order ( u/c ) , and more im-portantly, the azimuthal flow Eq. (22) becomes unstableabove a critical fluid speed [3, 23]. A. Uniform synthetic magnetic field
We obtain a uniform synthetic magnetic field fromEq. (22) by assuming a stationary inner cylinder Ω = 0and taking the limit R →
0, which yields u ( r ) = r Ω ˆ θ and the radial equation (cid:34) r ∂ r ( r∂ r ) − r (cid:18) l + r ω Ω c (cid:19) + ω c (cid:35) ψ = k z ψ. (23)The well-known solutions are Laguerre-Gauss modes [22], ψ ( r ) = (cid:16) rw (cid:17) | l | L | l | n (cid:18) r w (cid:19) e − r /w , (24)with beam waist w = (cid:112) c / ( ω | Ω | ) and dispersion (cid:18) ck z ω (cid:19) = 1 + 2 l Ω ω − | Ω | ω (2 n + | l | + 1) , (25) Ω / ω R ω /c R Ω = u max ~ c R > w R < w FIG. 2: Phase diagram of acoustic vortex beams supportedby uniform magnetic field induced by rotation Ω of the outercylinder of radius R . ω and c are the acoustic wave frequencyand speed, respectively. Unshaded region u max > . c : Thelarge peak fluid velocity u max renders the effective Schr¨odingerinvalid. Red region R > w : Vortex beams are localized bythe synthetic magnetic field. Blue region R < w : Vortexbeams are localized by the cylinder wall. Black circle denotesthe parameters used in Figs. 3 and 4. where n is the radial mode number. The mode localiza-tion (determined by the beam waist w ) and energy bothdepend on the synthetic magnetic field strength, withthe dispersion relation including a Zeeman-like interac-tion between orbital angular momentum l and syntheticmagnetic field. This results in an l -dependent modal cut-off frequency when l Ω <
0. In contrast, the modal pro-file ψ ( r ) is independent of the sign of l , even though thevector potential breaks time-reversal symmetry. Thesefeatures resemble the electron beam case [8, 22].The Laguerre-Gauss solution is obtained by assum-ing an unbounded fluid, so it is only valid if the beamwidth ∼ w is much smaller than the radius of the outercylinder R . However, R cannot be too large becausethe effective Schr¨odinger equation is only valid if thefluid speed is slow compared to the speed of sound, i.e.( u max /c ) (cid:28)
1. Moreover, R must be remain somewhatlarger than the acoustic wavelength, meaning Ω /ω mustbe small. Fig. 2 illustrates these various constraints.The dispersion relation Eq. (25) implies the beamswill be close to paraxial, i.e. k ≈ k z ˆ z . There isa “Goldilocks” zone of sufficiently large (but not toolarge) R supporting acoustic vortex beams localized bythe synthetic magnetic field. In the following we use(Ω , R ) = (0 . ω, c/ω ), considering vortex beamswithin this zone. We note that vortex beams solutionsstill exist for R (cid:46) w , just they will not take the simpleLaguerre-Gauss form Eq. (24).Evaluating the field profiles, there is a l Ω -dependentredistribution of energy between the pressure P and lon- r ω /c
10 10 10 202020 30 30 30 | P(r) | | v(r) | W(r)
FIG. 3: Pressure P , velocity | v | = | v x | + | v y | + | v z | , andenergy density W profiles of Laguerre-Gauss vortex beams ofcharge l = 0 , ± , ± . Vortex beams exhibit a field direction-dependent redis-tribution of energy between their P and v components, while W is independent of sgn( l Ω ). gitudinal velocity v z components, | P | = ρ ω (1 − l Ω ω ) ψ , (26) | v z | = ω c ψ (cid:18) l Ω ω − | Ω | ω (2 n + | l | + 1) (cid:19) . (27)However, the total energy density W remains insensitiveto sgn( l Ω ). To illustrate these features, Fig. 3 plotsthe energy densities for vortex beams of various charges.Since Ω /ω is small there is only a slight (a few percent)redistribution of energy between the P and v fields. Nev-ertheless, this shows that a detector sensitive to only oneof the field components can measure shifts due to the in-teraction between the beam’s orbital angular momentumand the synthetic magnetic field, unobservable for elec-tron vortex beams localized by a magnetic field [8, 22].Since we are close to the paraxial limit, other featuresof the Laguerre-Gauss beam do not differ significantlyfrom the Schr¨odinger limit. The only exceptions are closeto zeros of the potential φ , where non-paraxial correctionscan still be significant. Fig. 4 shows how both the trans-verse and longitudinal spin are nonzero (comparable tothe energy density W ) close to the core of vortex beams.Consequently the longitudinal kinetic momentum Π z issuppressed by the ∇ × S term in Eq. (9). Since thebackground flow is small close to the beam core, thereare no obvious synthetic magnetic field-induced changesto the spin densities. On the other hand, the kineticangular momentum density M z shows a clear crossoverfor counter-rotating vortex beams with sgn( l Ω ) < r the canonical AM (determined by the vor-tex charge l ) dominates, whereas at large r the syntheticmagnetic field provides the dominates. B. Aharonov-Bohm flux
The second example we consider is a synthetic mag-netic flux tube centred at r = 0. This is obtained bysetting Ω = 0 and R → ∞ in Eq. (22), such that r ω /c
10 20 30 S θ /W
01 1 02 S z /WM z /W r ω /c r ω /c
10 20 300 40042-26 Π z /Wr ω /c
10 20 300 4010.50.2500.75 0 -2-1+1+2
FIG. 4: Normalised transverse spin S θ , longitudinal spin S z ,longitudinal kinetic momentum Π z , and kinetic angular mo-mentum M z densities of the Laguerre-Gauss vortex beams.The synthetic magnetic field does not significantly affect thespin or momentum densities, but induces a change in the signof M z for counter-rotating vortex beams with sgn( l Ω ) < u = Ω R r ˆ θ . Substituting u θ into the radial equation (19), (cid:34) r ∂ r ( r∂ r ) − r (cid:18) l + ω Ω R c (cid:19) + ω c (cid:35) ψ = k z ψ. (28)The general solution decaying to zero as r → ∞ is [25] ψ ( r ) = aJ l + α ( k ⊥ r ) + bJ − ( l + α ) ( k ⊥ r ) , (29)where J m ( x ) are Bessel functions, α ≡ ω Ω R /c is thesynthetic magnetic flux, and k ⊥ = ω /c − k z . The hardwall boundary condition ∂ r ψ ( R ) = 0 requires b = − a J l + α − ( k ⊥ R ) − J l + α +1 ( k ⊥ R ) J − l − α − ( k ⊥ R ) − J − l − α +1 ( k ⊥ R ) . (30)For the special case of integer values of the flux α , theBessel functions J l + α and J − ( l + α ) become linearly de-pendent, such that the correct solution is ψ ( r ) = aJ l + α ( k ⊥ r ) + bY l + α ( k ⊥ r ) , (31)where Y m ( x ) is a Bessel function of the second kind, and b = a J l + α +1 ( k ⊥ R ) − J l + α − ( k ⊥ R ) Y l + α − ( k ⊥ R ) − Y l + α +1 ( k ⊥ R ) . (32) r ω /c
10 10 10 1515155 5 5 | P(r) | | v(r) | W(r)
FIG. 5: Pressure P , velocity | v | , and energy density W pro-files of Bessel beams with charges l = 0 , ± , ± α . There is a flux-dependent shiftof the minima of P , v , and W , with all sensitive to sgn( αl ). The synthetic magnetic flux α is constrained by( u max /c ) (cid:28)
1, which ensures the background flowis slow compared to the acoustic wave speed. TheAharonov-Bohm background flow has its maximum ve-locity u max = Ω R at the edge of the inner cylinder, r = R . Therefore to maximize the flux subject to theseconstraints a large, slowly rotating inner cylinder is re-quired. In the following we take Ω = 0 .
02 and R = 5,corresponding to a flux α = 1 / k ⊥ ) is anadditional free parameter of the Bessel beam solutions.In the paraxial limit k ⊥ behaviour similar to Ref. [22]is expected, since the acoustic wave becomes effectivelyscalar. On the other hand, in the strongly non-paraxiallimit the fluid flow described by the synthetic vector po-tential is small compared to the large in-plane momentum k ⊥ . Therefore to maximize the effect of the acoustic vec-tor potential, we take a moderately non-paraxial beamangle ϕ = π/
8, where k z = k cos ϕ and k ⊥ = k sin ϕ .Fig. 5 plots P , v , and W profiles of the acoustic Besselbeams. The synthetic flux α strongly breaks the symme-try between beams with orbital angular momentum ± l ;the beam radius is larger when sgn( αl ) >
0. While beamswith the same | l + α | have identical velocity potentials ψ ,they can be still distinguished through their P , v , and W profiles. P vanishes at zeros (corresponding to nodesof ψ ), while due to non-paraxiality v and W are alwaysnonzero. Finally, the inner cylinder boundary r = R can be either a global maximum or local minimum of P , v , and W , depending on α , k ⊥ , and l .Fig. 6 plot some other observables. The transverse S θ and longitudinal S z spin densities become large closeto the minima of W . In this non-paraxial regime, thepeak values of S z /W are sensitive to the enclosed flux α .The longitudinal momentum Π z is determined purely by | l + α | , with no visible difference between the l = 0 , − l = +1 , − W . Away from the minima of W , the kinetic angularmomentum M z ≈ ( l + α ) W ; at the minima the canonicalangular momentum contribution vanishes, leaving M z ≈ αW controlled purely by the enclosed flux. r ω /c
10 15 20 S θ /W S z /WM z /W r ω /c r ω /c Π z /Wr ω /c FIG. 6: Normalised transverse spin S θ , longitudinal spin S z ,longitudinal kinetic momentum Π z , and kinetic angular mo-mentum M z densities of the Bessel beams. Shifts in the max-ima of S θ and minima of Π z follow the shifts in the minimaof W . The peak values of S z and M z are sensitive to α . IV. CONCLUSION
In conclusion, we have studied acoustic vortex beamsin the presence of synthetic magnetic fields. We analyzedtwo simple limits; a uniform field and a flux tube, wherethe vortex beams can be obtained exactly from solutionsof the radial Schr¨odinger equation. This enables a directcomparison between previously-studied optical [17] andelectron [22] vortex beams. The main important differ-ences are: • The vector potential of the electronic Schr¨odingerequation is a gauge field, whereas the syntheticvector potential in acoustics is the gauge-invariantbackground fluid speed which must be slow com-pared to the acoustic wave speed. This lim-its acoustics to weak uniform synthetic mag-netic fields, which can host weakly-localized (near-paraxial) Laguerre-Gauss beams. • Beam symmetries present in the effectiveSchr¨odinger equation and total energy density W can be broken when considering the microscopicacoustic pressure P and velocity v fields. This isbecause the synthetic vector potential induces aredistribution of energy between P and v . • The synthetic magnetic fields can be used to fine-tune the beam profiles, and therefore control thedistribution of the acoustic spin density in non-paraxial beams.In the future it would be interesting to extend this anal-ysis to acoustic surface waves, where analogies with sur-face plasmon polaritons have recently been explored [12–14]. In particular, the breaking of time-reversal symme-try due to the synthetic magnetic field is expected to in-duce unidirectional surface wave propagation [26]. Otherinteresting directions are the analysis of forces on smallparticles induced by structured acoustic beams [27, 28] and the characterization of random acoustic wave fields,including their distribution of phase and polarization sin-gularities and emergence of superoscillations [29]. Herewe anticipate further non-trivial differences between thescalar effective Schr¨odinger equation governing the acous-tic velocity potential and the real measurable pressureand velocity fields.We thank Konstantin Bliokh for useful discussions.This work was supported by the Institute for Basic Sci-ence in Korea (IBS-R024-Y1). [1] M. Z. Hasan and C. L. Kane,
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