Relativistic Anyon Beam: Construction and Properties

TL;DR

This paper proposes a method to construct and analyze a relativistic anyon beam, deriving explicit solutions and currents, and discusses potential laboratory realization, advancing the understanding of anyon dynamics.

Contribution

It introduces a novel construction of a relativistic anyon beam using superpositions of plane wave solutions, with explicit current expressions and property analysis.

Findings

Derived explicit relativistic plane wave solutions for anyons.

Constructed a planar anyon beam from superposed solutions.

Analyzed the properties and currents of the proposed anyon beam.

Abstract

Motivated by recent interest in photon and electron vortex beams, we propose the construction of a relativistic anyon beam. Following Jackiw and Nair [Phys. Rev. D 43, 1933 (1991)] we derive explicit form of relativistic plane wave solution of a single anyon. Subsequently we construct the planar anyon beam by superposing these solutions. Explicit expressions for the conserved anyon current are derived. Finally, we provide expressions for the anyon beam current using the superposed waves and discuss its properties. We also comment on the possibility of laboratory construction of anyon beam.

Equations

$${\partial }_a{ϵ}^{abc}{F}_c\pm m{F}^b=0;(p\cdot j{)}_b^a{F}^b+ms{F}^a=0.$$

$$F^{a}(p)=\frac{1}{\sqrt{2}}\Bigg{[}\begin{bmatrix}0\\ 1\\ i\\ \end{bmatrix}+\frac{p^{x}+ip^{y}}{m(E+m)}\begin{bmatrix}E+m\\ p^{x}\\ p^{y}\\ \end{bmatrix}\Bigg{]}\sqrt{\frac{m}{E}}.$$

$$F^{a}(p)=B^{a}_{b}(p)N^{b}_{c}F^{c}_{0}(p),~{}~{}N^{b}_{c}=\frac{1}{\sqrt{2}}\left[\begin{tabular}[]{ccc}$\sqrt{2}$&0&0\\ 0&1&$i$\\ 0&-$i$&$i$\end{tabular}\right]$$

$$B(p)=e^{i{\Omega }_a(p)j^a};[B(p){]}_b^a=[B^{-1}(p){]}_b^a$$

$$={\delta }_b^a-\frac{(p^a+{\eta }^{a}m)(p_b+{\eta }_{b}m)}{m(p\cdot \eta +m)}+\frac{2p^a{\eta }_b}{m}$$

$$P\cdot (K+j{)}_{an{a}^{\prime }{n}^{\prime }}{f}_{n^{\prime }}^{a^{\prime }}+ms{f}_{an}=0,(D_a{f}^a{)}_n=0,$$

$$P\cdot K_{an{a}^{\prime }{n}^{\prime }}=P\cdot K_{n{n}^{\prime }}{\delta }_{a{a}^{\prime }},P\cdot j_{an{a}^{\prime }{n}^{\prime }}=P\cdot j_{a{a}^{\prime }}{\delta }_{n{n}^{\prime }};$$

$$K_{n{n}^{\prime }}^a=<\lambda ,n\mid K^a\mid \lambda ,n^{\prime }>,(j^a{)}_{a^{\prime }{a}^{\prime \prime }}=i{ϵ}_{a^{\prime }{a}^{\prime \prime }}^a$$

$$f_n^{a(\pm )}(p)=B_{n0}(p)B_b^a(p)N_c^b{f}^{c(\pm )}(p),$$

$$f_n^{a+}={\left(\frac{2m}{E+m}\right)}^{\lambda }\sqrt{\frac{\Gamma (2\lambda +n)}{n!\Gamma (2\lambda )}}{\left(\frac{p^x+i{p}^y}{E+m}\right)}^n$$

$$\times F^a(p)e^{-ip\cdot x}$$

$${\partial }^{\mu }{j}_{\mu }^{(s=1)}={\partial }^0\left[F^{0†}{F}^0+F^{x†}{F}^x+F^{y†}{F}^y\right]$$

$$-{\partial }^x\left[F^{0†}{F}^x+F^{x†}{F}^0\right]-{\partial }^y\left[F^{0†}{F}^y+F^{y†}{F}^0\right]=0$$

$${\partial }^0\sum _{n=0}^{\infty }\left[(f_n^{0†}{f}_n^0+f_n^{x†}{f}_n^x+f_n^{y†}{f}_n^y)-i(f_n^{y†}{K}_{n{n}^{\prime }}^0{f}_{n^{\prime }}^x-f_n^{x†}{K}_{n{n}^{\prime }}^0{f}_{n^{\prime }}^y)\right]$$

$$-{\partial }^x\sum _{n=0}^{\infty }\left[(f_n^{0†}{f}_n^x+f_n^{x†}{f}_n^0)-i(f_n^{y†}{K}_{n{n}^{\prime }}^x{f}_{n^{\prime }}^x-f_n^{x†}{K}_{n{n}^{\prime }}^x{f}_{n^{\prime }}^y)\right]$$

$$-{\partial }^y\sum _{n=0}^{\infty }\left[(f_n^{0†}{f}_n^y+f_n^{y†}{f}_n^0)-i(f_n^{y†}{K}_{n{n}^{\prime }}^y{f}_{n^{\prime }}^x-f_n^{x†}{K}_{n{n}^{\prime }}^y{f}_{n^{\prime }}^y)\right]=0$$

$$j^0=(1-\lambda )E/m;j^x=(1-\lambda )p^x/m;$$

$$j^y=(1-\lambda )p^y/m\to j^{\mu }=(1-\lambda )\frac{p^{\mu }}{m}=s\frac{p^{\mu }}{m}.$$

$$F_n^a(p,x)={\mathcal{A}}_n\underset{-\pi /2}{\overset{\pi /2}{\int }}\left[\begin{array}{c}\frac{p}{m}{e}^{in\alpha }{e}^{i\alpha }\\ \left(1+\frac{p^2}{M}\cos \alpha e^{i\alpha }\right)e^{in\alpha }\\ \left(i+\frac{p^2}{M}\sin \alpha e^{i\alpha }\right)e^{in\alpha }\end{array}\right]$$

$$\left(e^{i(x\cos \alpha +y\sin \alpha )}+e^{i(x\cos \alpha -y\sin \alpha )}\right)d\alpha$$

$$J^0(\rho ,\theta )$$

$$M=e\int dA({ϵ}_{ik}{r}^i{j}_s^k)/\int dA{j}_s^0$$

$$d\mathcal{A}=-i\lambda \left(\frac{\omega d\bar{\omega }-\bar{\omega }d\omega }{1-\omega \bar{\omega }}\right).$$

$$g=\left(\begin{array}{cc}1& \omega \\ \bar{\omega }& 1\end{array}\right)\frac{1}{\sqrt{1-\bar{\omega }\omega }}\times \left(\begin{array}{cc}{e}^{i\varphi /2}& 0\\ 0& e^{-i\varphi /2}\end{array}\right)$$

$$g=\left(\begin{array}{cc}1& \omega \\ \bar{\omega }& 1\end{array}\right)\frac{1}{\sqrt{1-\bar{\omega }\omega }}$$

$$[K_0,K_{\pm }]=\pm K_{\pm },[K_{+},K_{-}]=-2K_0$$

$$K_0=\frac{1}{2}{\sigma }_3,K_1=\frac{i}{2}{\sigma }_1,K_2=-\frac{i}{2}{\sigma }_2.$$

$$e^{-i\omega K_{+}}{e}^{\log (1-\bar{\omega }\omega )K_0}{e}^{-i\bar{\omega }{K}_{-}}$$

$$g=e^{-i\omega K_{+}}{e}^{\log (1-\bar{\omega }\omega )K_0}{e}^{-i\bar{\omega }{K}_{-}}.$$

$$K^0\mid \lambda ,n>=(\lambda +n)\mid \lambda ,n>;$$

$$K^{+}\mid \lambda ,n>=\sqrt{(2\lambda +n)(n+1)}\mid \lambda ,n+1>;$$

$$K^{-}\mid \lambda ,n>=\sqrt{(2\lambda +n-1)n}\mid \lambda ,n-1>.$$

$$K_{-}\mid \lambda ,0⟩=0,K_0\mid \lambda ,0⟩=\lambda \mid \lambda ,0⟩$$

$$f(n)$$

$$f(n)=n!(2\lambda +n-1)(2\lambda +n-2)\cdot \cdot \cdot (2\lambda )=\frac{\Gamma (n+1)\Gamma (2\lambda +n)}{\Gamma (2\lambda )}$$

Full text

We propose construction of a relativistic beam of anyons in a plane. Anyons are planar excitations with arbitrary spin and statistics. The procedure is similar to 3D optical (spin $1$) and electron (spin $1/2$) vortex beam.

Vortex beams, of recent interest, carrying intrinsic orbital angular momentum are non-diffracting wave packets in motion. In 3D, vortex beams for photons and electrons were proposed by Durnin et al. dur and by Bliokh et al. bl1 respectively (see bl3 ). Properties and experimental signatures of twisted optical and electron vortex beam were studied respectively in allen (for spinless electron bl1 ) and silenko (spinor electron bl1 ). Wave packets were constructed out of free plane wave, orbital angular momentum solutions having vorticity. In 3D, non-trivial Lie algebra of rotation generators along with finite dimensional representations of angular momentum eigenstates restricts the angular momentum to quantize in units of $\hbar/2$. Absence of such an algebra in 2D allows excitations (anyons wil1 ; lein ) with arbitrary spin and statistics with infinite dimensional representations. Here, we construct directed transversely localized anyon wave packet or wave beam in 2D.

Geometry imposes a qualitative difference between monoenergetic wave packet and wave beam both in 3D and 2D. In 3D the wave packet is described by three discrete quantum numbers as it is localized in three directions whereas ideally an energy dispersionless wave packet where wave components having identical momentum along the direction of propagation will be completely delocalized along that direction and will constitute a beam dur ; bl1 . But, in 2D a monoenergetic wave beam will necessary have dispersion in momentum along propagation direction and hence cannot be completely delocalized. The delocalization along propagation direction will be more for narrower angle of superposition.

Apart from earlier interest wil1 , graphene also involves anyons graph . Non-Abelian anyons nonab gr are touted as theoretical building blocks for topological fault-tolerant quantum computers kit . An exact chiral spin liquid with non-Abelian anyons has been reported spin . The direct observational status of anyons has shown promising development expt .

The minimal field theoretic and relativistic model of a single anyon was constructed by Jackiw and Nair (JN) jn . JN anyon is the most suitable one that serves our purpose since, being first order in spacetime derivatives, it closely resembles the Dirac equation used by bl1 . A major difference is that unitary representation for arbitrary spin JN anyon requires an infinite component wave function bin , (to maintain covariance), along with subsidiary constraints that restrict the number of independent components to a single one jn .

We propose: (i) an explicit form of the single anyon solution of the JN anyon equation jn (which is a new result), (ii) anyon conserved current (again a new result), (iii) anyon wave packet in 2D, (iv) anyon wave packet in the conserved current that constitutes the anyon beam, (iv) anyon charge and current densities for the anyon beam numerically since closed analytic expressions could not be obtained, (v) possible laboratory setup for observing anyon beams, and (vi) finally, future prospects. The new results (i, ii) lead us to our principal outputs (iii, iv) and hopefully (v).

(i) Jackiw-Nair anyon equation and its solution: We start with a familiar system, free spin one particle in $2+1$-dimensions jn . The dynamical equation in co-ordinate and momentum space ($i\partial_{a}=p_{a}$) is given by

[TABLE]

The solution of the three-vector $F^{a},~{}a=0,1,2$ (we use Minkowski metric $\eta_{\mu\nu}=diag(1,-1,-1)$) is given by

[TABLE]

The same $F^{a}$ can also be expressed as a Lorentz boosted form

[TABLE]

where the boost is expressed by the generators $j^{a}$

[TABLE]

with $\eta_{a}=(1,0,0)$. It is straightforward to check that $F^{a}$ describes a spin one particle ($s=1$) of mass $m$.

This construction has been extended in an elegant way to the JN anyon equation jn to describe an anyon of arbitrary spin $s=1-\lambda$, whose dynamics in momentum space is given by,

[TABLE]

($D^{a}_{nn^{\prime}}=\epsilon^{a}_{~{}bc}P^{b}K^{c}_{nn^{\prime}}$) where the second equation is the subsidiary (constraint) relation. For $\lambda=0$ the anyon reduces to the spin one model discussed earlier jn . The actions of $j^{a},K^{a}$ are given by

[TABLE]

Generalizing the spin-$1$ case (3), the free anyon solution is formally given by jn

[TABLE]

where $B_{nn^{\prime}}(p),~{}B^{a}_{b}(p)$ are the spin $\lambda$ and spin $1$ representations of the boost transformation respectively and $N^{b}_{c}$ is the same numerical matrix as in (3).

Exploiting coherent state formalism for $SU(1,1)$ per ; vpn along with the $SU(1,1)\sim SO(2,1)$ connection, we have constructed explicit form of $B_{n0}(p)$ (in a matrix representation of $K^{a}$ jn (see e.g. wy ). Computational details are in Suppl. material A. The free anyon solution is,

[TABLE]

where $F^{a}(p)$ is same as the spin-$1$ case defined in (2). This is our primary result that we exploit to construct the wave packet and subsequent anyon beam.

(ii) Conserved current for single anyon: Next we derive the conserved probability current for anyon $\partial^{\mu}j^{(s=1-\lambda)}_{\mu}=0$ where $j_{0}^{s}$ is the probability density. Since the anyon model of jn is an extension of the spin-$1$ case we can take a cue from the latter where the conservation law $\partial^{\mu}j^{(s=1)}_{\mu}=0$ reads

[TABLE]

Considering the Fourier transform of the anyon equation of motion (5) in position space, a long calculation yields the conserved free (single) anyon current $j^{(1-\lambda)}_{\mu}$,

[TABLE]

where we have explicitly shown the summation over $n$, the anyonic index. For $\lambda=0$ the current $j_{\mu}^{(1-\lambda)}$ reduces to the spin $1$ current $j_{\mu}^{(1)}$ of (8). A nontrivial check of the consistency of the expressions for anyon current (9) is to substitute $f^{a}_{n}$ from (7) to yield

[TABLE]

For computational details see Suppl. material B.

(iii) Anyon wave packet: Our aim is to construct the anyon current, not for a single anyon as done above, but for an anyon wave packet which can be amenable to experimental verification. Let us now construct the anyon wave packet that we want to move towards, say, $+x$-direction. Since we have superposed plane waves, later figures will reveal that the current density has a sharply peaked profile with the $y$-component of current density having a comparatively reduced value. Note an important difference in geometry between our construction and that of the three-dimensional vortex beam bl1 . In the latter case the free monoenergetic plane wave solutions (to be superposed) are distributed over the surface of a right circular cone with identical momentum amplitude in the propagation direction. However, for our anyon wave packet, in a planar geometry the above is not possible. Instead we use the superposition scheme where the azimuthal angle $\phi$ of momenta of the plane wave is integrated symmetrically from $\phi_{0}=-\pi/2$ to $\phi_{0}=+\pi/2$. In Fig. 1 we have shown profiles for charge density of anyon beam, $J_{0}^{\lambda}$, for $\lambda=0.6\rightarrow s=1-\lambda=0.4$ for $\phi_{0}=\pm\pi/2$.

Hence, considering superpositions of anyon plane waves with fixed spin $s=1-\lambda$, fixed energy $E$ and fixed kinetic energy $p={\sqrt{(p^{x})^{2}+(p^{y})^{2}}}$, for the special case of integration limits $\phi_{0}=\pm\pi/2$, the superposition $F^{a}_{n}$ appears as

[TABLE]

where ${\mathcal{A}}_{n}=\frac{1}{2}\left(\frac{2m}{E+m}\right)^{\lambda}\sqrt{\frac{\Gamma(2\lambda+n)}{n!\Gamma(2\lambda)}}\sqrt{\frac{m}{E}}\left(\frac{p}{E+m}\right)^{n}$, $x\equiv px$, $y\equiv py$ and $M=m(E+m)$. The expression is symmetric separately under $x\rightarrow-x$ and $y\rightarrow-y$.

Conserved current for anyon wave packet: The final analytical task is to substitute the anyon wave packet (11) in the expression of the anyon current (9). Since the current components are quadratic in the packet wavefunctions $F^{a}_{n}$, the final expressions are quite long and involved. We have shown only the expression for probability density $J^{0}$ and have relegated $J^{x}$ and $J^{y}$ to Suppl. material C together with a few computational steps. The cherished form of anyon beam probability density, in polar coordinates $x=\rho\cos\theta$, $y=\rho\sin\theta$, is

[TABLE]

where, $\sigma=\left(\frac{p}{E+m}\right)$.

(iv) Visualizing the anyon beam: Unfortunately, closed form expressions for the anyon beam current components $J^{0}$, $J^{x}$, and $J^{x}$ are not possible to obtain. For different choices of $\lambda$ and $\phi_{0}$ the profiles of $J^{0}$ are given in Suppl. material D. Subsequently, in Fig. 2 and Fig. 3, for the above values of $s,\phi_{0}$, we have plotted the profiles of $J^{a},~{}~{}a\equiv x,y$ respectively, in three ways: a two-dimensional plot of $J^{a}$ against the polar angle $\theta$ for a few

values of the (planar) radial distance $\rho$, panel (a) in each group of figures. Another two-dimensional graph of the same data as panel (a) with magnitude of $J^{a}$ against $\theta$ for the same values of $\rho$ is shown in panel (b). A density plot in co-ordinate plane $x$-$y$ is given in panel (c). Finally, a three-dimensional plot of $J^{a}$ in co-ordinate plane $x$-$y$ is provided in panel (d). Note that in panel (a), each continuous curve represents a fixed polar distance in coordinate plane $x$-$y$ with the height being a measure of $J^{a}$ whereas in panel (b), the radial distance is a measure of the intensity of $J^{a}$. Hence the curves that are further away from the centre in panel (b) represent points that are closer to the coordinate plane $x$-$y$.

As expected, all the wave packet profiles are symmetric about the abscissa

since the packets are superposition of plane waves, that are symmetrically placed about the $x$-axis. Contrasting with the three-dimensional wave packets bl1 it is clear that the planar anyon beams do not possess a vortex nature since the axial symmetry is manifestly broken while constructing a propagating anyon beam. This is also corroborated in the figures that do not have any destructive interference at the origin, a characteristic feature of vortex beams bl1 . Hence the anyon beams are characterized by the spin value $s$ of the wave packet, which is same as that of individual plane wave single anyon component.

An important observation is that in the cases we have considered, $J^{0}$ is always positive, which has to be the case since it is the probability density. But $J^{x}$ is also positive throughout whereas

$J^{y}$ has positive and negative values in equal amount. Furthermore, maximum value of $J^{y}$ is far lower than each of $J^{0}$ and $J^{x}$. These reflect the nature of our construction of the anyon beam where all the plane waves have positive velocity along $x$-direction but have pairwise opposite (both $+$ve and $-$ve ) velocities along $y$-direction. Hence, the anyon beam will predominantly move in the positive $x$-direction with the $y$-component effectively canceled out.

(v) Experimental possibilities: Anyons were detected in quantum antidot experiments anyonobs and in Laughlin quasiparticle interferometer obsan . In relation to simulation of high $T_{c}$ superconductivity by charged anyon fluid hos their Josephson frequency has been observed ch an in 2D electrons in high magnetic field.

External electromagnetic field affects anyon with charge $e$ and magnetic moment (see e.g. bl1 )

[TABLE]

for the single anyon current $j_{a}^{s}$ (8) using (7). We replace $j^{a}$ by $J^{a}$ using the wave packet (11). The nonstationary quantum superposition state (wave packet) can be created by exciting matter coherently with an ultrafast laser pulse, which is composed of eigenstates spanned by the frequency bandwidth of the laser wpacket .

Possibility of fault tolerant quantum computation by (non-abelian) anyons has generated interest in controlled production of anyonic expt : collective anyon excitations from electrons in the Fractional Quantum Hall (FQH) systems or from atoms in 1D optical lattices, creation of FQH effect for photons (using 1D or 2D

cavity array), using a nonlinear resonator lattice subject to dynamic modulation for creating anyons from photons, among others.

Topologically ordered many-body states (with quenched kinetic energy) are generated with strongly interacting particles in magnetic field. The system minimizes interaction energy forming intricate patterns of long-range entanglement as observed in FQH (semiconductor heterojunction, graphene 12 ; 13 and van der Waals bilayers 14 ). Formation of Laughlin states in synthetic quantum systems (ultracold atoms 7 ; 8 , photons 9 ; 10 ; 11 ) have been developed. Recently in anyon optics, Laughlin states are made out of photon pairs (in a synthetic magnetic field for light induced from twisted optical cavity 19 , strong photonic interactions via Rydberg atoms). These are modeled exploiting anyonic Hubbard Hamiltonian in ultracold-atom 1D lattices pra . Anyon imaging with STM has also been achieved pra . Theoretical 16 ; 17 and experimental 18 studies for simulating anyonic NOON states with photons in waveguide lattices have appeared.

Our experimental proposal: Following the work in 18 , a two-photon NOON state with arbitrary anyonic symmetry is first prepared in a detuned directional coupler, and subsequently evolved in a Bloch oscillator emulated by a curved array. Then we use planar analogue of a spiral phase plate where phase shift proportional to the path length of the waves passing through the plate will occur. Upon superposition this will generate the anyon beam. This is schematically depicted in Fig. 4. The left panel of Fig. 4 shows a convex lens in three-dimensions $x,y,z$ with parallel rays shown only along $x$-$y$ plane that converge on $x$-axis. We consider an extremely thin slice of the lens in the $x$-$y$ plane which can be thought as the shaded area in the left panel. The same slice is drawn separately in the right panel that shows the planar superposition in our work.

(vi) Summary and future prospects: We have suggested the construction of relativistic anyon beam, a symmetrical superposition of Jackiw-Nair single anyon solutions. Explicit forms of wave packets, their significant features and numerically plotted profiles of anyon beam current are given. A laboratory model of anyon beam construction is provided.

We thank Prof. V. Parameswaran Nair for actively helping us in this project.