Non-Abelian gauge field optics Yuntian Chen,1, 2, ∗ Ruo-Yang Zhang,3, ∗ Zhongfei Xiong,1 Jian Qi Shen,4, † and C. T. Chan3, ‡ 1 2
School of Optical and Electronic Information, Huazhong University of Science and Technology, Wuhan 430074, China
Wuhan National Laboratory of Optoelectronics, Huazhong University of Science and Technology, Wuhan 430074, China 3
Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong 4
arXiv:1802.09866v1 [physics.optics] 27 Feb 2018
Centre for Optical and Electromagnetic Research, State Key Laboratory for Modern Optical Instrumentation, Zhejiang University, Hangzhou 310058, China; The concept of gauge field is a cornerstone of modern physics and the synthetic gauge field has emerged as a new way to manipulate neutral particles in many disciplines. In optics, several schemes have been proposed to realize Abelian synthetic gauge fields. Here, we introduce a new platform for realizing synthetic SU (2) nonAbelian gauge fields acting on two-dimensional optical waves in a wide class of anisotropic materials and discover new phenomena. We show that a virtual non-Abelian Lorentz force can arise from the material anisotropy, which can induce wave packets to travel along wavy “Zitterbewegung” trajectories even in homogeneous media. We further propose an interferometry scheme to realize the non-Abelian Aharonov–Bohm effect of light, which highlights the essential non-Abelian nature of the system. We also show that the Wilson loop of an arbitrary closed optical path can be extracted from a series of gauge fixed points in the interference fringes.
Introduction Gauge fields originated from classical electromagnetism, and have become the kernel of fundamental physics after being extended to non-Abelian by Yang and Mills1 . Apart from real gauge bosons, emergent gauge fields in either real space2 or parameter spaces3,4 have recently been widely used to elucidate the complicated dynamics in a variety of physical systems, e.g. electronic5,6 , ultracold atom7–10 , and photonic11–28 systems. The geometric nature29 of gauge theory makes it a powerful tool for studying the topological phases of matter30–32 . The concept of emergent gauge fields has offered us new insights in optics and photonics, such as the manifestation of the gauge structure (Berry connection and curvature) in momentum space11–16 . Artificial gauge fields realized by breaking time reversal symmetry with magnetic effects17,18,23 or dynamic modulation20–22 have given rise to new paradigms for controlling light trajectories in real space. Even for time-reversal-invariant systems, a pair of virtual magnetic fields – each being the timereversed partner of the other – can be generated using methods such as coupled optical resonators19 , engineering lattices with strain24 , or reciprocal metamaterials25–28 . However, except for a few works revealing the non-Abelian gauge structure in momentum space13,14,16 , all of these schemes of synthetic gauge fields in real space are restricted to the Abelian type. In this work, we show that the transport of 2-dimensional (2D) optical waves in a wide class of anisotropic media can be associated with the realization of synthetic non-Abelian SU (2) gauge fields acting on light in real space. Very recently, it was demonstrated that graded metamaterials can be used to manipulate light through artificial Abelian gauge fields26–28 . The metamaterial parameters are subjected to strong restrictions
∗ † ‡
These authors contributed equally to this work.
[email protected] [email protected]
when realizing a pair of “spin-dependent” vector potentials in the wave equation. Contrary to intuition, we will show that a more exotic general SU (2) gauge framework can manifest in 2D optical dynamics, provided the restriction on the material parameters is greatly relaxed. Our non-Abelian gauge platform presents broader applicability and offers novel optical phenomena not found in Abelian systems. We will illustrate our idea with two examples. The first example is the Zitterbewegung (ZB) of light in homogeneous “non-Abelian” media, which refers to the trembling motion of wave packet33 . ZB has been realized in systems possessing Dirac dispersion34–39 , and we will see that the ZB of light can arise from a distinctly different mechanism: emergent nonAbelian Lorentz force. The other example is the design of a genuine non-Abelian Aharonov-Bohm (AB) system40 with two non-Abelian vortices. The noncommutativity of winding around the two vortices will give rise to nontrivial results of spin density interference. From the fixed points in the interference fringes under gauge transformation, we can obtain the Wilson loop of an arbitrary closed optical path. Non-Abelian gauge fields acting on light Our scheme focuses on 2D propagating optical waves in nondissipative gyrotropic media characterized by the permittivity and permeability tensors: ↔
ε /ε0 = (
↔ εT g1†
g1 ), εz
↔
µ /µ0 = (
↔ µT g2†
g2 ). µz
(1)
Here, all of the parameters depend on x, y; the block-diagonal components ↔ εT , ↔ µ T , εz , µz are real numbers, while the offblock-diagonal components gi = (gi x , gi y )⊺ = gi x ex + gi y ey (i = 1, 2) are in-plane complex vectors whose imaginary parts could be induced by the magneto-optic effect from in-plane magnetic fields. The only constraint on the medium is the “in-plane duality”, ↔ εT = α↔ µ T , where α is a constant. For simplicity, we set α = 1 in the following, and α ≠ 1 results
2 can be obtained directly by redefining ε0 → α ε0 . Under this constraint, the in-plane monochromatic wave equation can be written as
ˆ1 − A1 σ ˆ2 ). Consequently, a virtual nonand Eˆ = 2A30 (A2 σ Abelian Lorentz force emerges that affects the motion equation of wave packet
1 ↔ −1 ˆ ˆ ⋅m ˆ − Aˆ0 + V0 ] ∣ψ⟩ = 0, (2) H∣ψ⟩ = [ (ˆ p − A) ⋅ (ˆ p − A) 2 where ∣ψ⟩ = h(Ez , η0 Hz )⊺ serves as a two-component √ wave function, h can be an arbitrary function of x, y, η0 = µ0 /ε0 , ˆ = −iˆ p σ0 ∂i ei (i = 1, 2) is the canonical momentum operator, ↔ ↔ = h2↔ ε −1 σ ˆ0 is a 2D identity matrix, and m T det ( ε T )/2 can be ˆ interpreted as an effective anisotropic mass. In equation 2, H resembles the Hamiltonian of a spin-1/2 particle traveling in SU (2) non-Abelian gauge fields, where Aˆ = A1 σ ˆ1 + A2 σ ˆ2 , a Aˆ0 = A0 σ ˆa are emergent non-Abelian vector and scalar potentials respectively, σ ˆa (a = 1, 2, 3) denote Pauli matrices, and V0 is an additional Abelian scalar potential (see Methods for their expressions). The synthetic non-Abelian magnetic and electric fields acting on light can be obtained from the gauge potentials ˆ Bˆ = ∇ × Aˆ − iAˆ × A,
ˆ Eˆ = ∇Aˆ0 + i[Aˆ0 , A].
(3)
In our case, Bˆ is always along the z axis, while Eˆ is in the xy ˆ Eˆ cannot be found in the Abelian plane. The second terms of B, case since they are induced entirely by the noncommutativity of the non-Abelian gauge potentials. The two-component wave function ∣ψ⟩ behaves like a spin-1/2 spinor with the pseudo-spin s⃗ = ⟨ψ∣σ ˆ⃗ ∣ψ⟩/∣ψ∣2 where the overhead arrow indicates a vector on the Bloch sphere (pseudo-spin space). The frame orientation in the pseudo-spin space can be chosen arbitrarily. The rotation of the frame corresponds to ˆ (r)∣ψ⟩, where a gauge transformation of state vector ∣ψ ′ ⟩ = U ˆ (r) is a space-varying unitary matrix. By substiin general U tuting ∣ψ ′ ⟩ into equation 2, one can easily check that the wave equation is gauge covariant while the gauge potentials obey the ˆ Aˆµ U ˆ † + iU ˆ ∂µ U ˆ † (µ = 0, 1, 2), gauge transformation Aˆ′µ = U ˆ BˆU ˆ † , Eˆ′ = U ˆ EˆU ˆ † accordingly. The effective gauge and Bˆ′ = U potentials depend on frequency ω. Nevertheless, for quasi˜ t)e−iω0 t , the evolution of the monochromatic waves E = E(r, √ ˜ ˜ z , η0 H ˜ z )⊺ in a weak anisotropic envelope ∣ψ⟩ = k0 εz + µz (E medium simply follows the Schrödinger-like equation with the ˜ = H∣ ˜ Furtherˆ ψ⟩. same Hamiltonian as in equation 2: i∂t ∣ψ⟩ more, if the background medium is extended to bianisotropic materials, a complete construction of SU (2) × U (1) synthetic gauge fields for light can be achieved (see supplementary information).
Zitterbewegung of wave packets The most straightforward effect distinguishing emergent nonAbelian gauge fields from the Abelian type is the wavepacket dynamics in homogeneous media. The effective Abelian gauge fields vanish in homogeneous media26 whereas the nonAbelian fields persist due to the noncommutativity of Aˆµ . In our case, Bˆ = Bˆ σ3 with B = ik0 (g− × g−∗ ) along the z axis,
m
1 d2 ˆ ˆ ⟩ + ⟨E⟩ ⟨ˆr⟩ = ⟨ˆ v × Bˆ + Bˆ × v dτ 2 2 = ⟨ˆjσˆ3 ⟩ × B + E a ⟨ˆ σa ⟩,
(4)
d ˆ ˆr] = (ˆ ˆ ˆ = dτ ˆr = i[H, where v p − A)/m is the velocity operator, 1 1 ˆσ ˆ) = m p and ˆjσˆ3 = 2 (ˆ vσ ˆ3 + σ ˆ3 v ˆ3 denotes the σ ˆ3 component of pseudo-spin current. In equation 4, the right-hand side of the first line is the non-Abelian Lorentz force, and it can be further decomposed into two parts. The fσˆ3 = ⟨ˆjσˆ3 ⟩ × B term represents the force induced by non-Abelian effective magnetic field, which is perpendicular to the spin current ⟨ˆjσˆ3 ⟩ and is the counterpart of the “spin transverse force” for electrons41 . The second term represents a spin-dependent electric force arising from the non-Abelian emergent electric field. Equation 4 is valid for both monochromatic and quasi-monochromatic waves (supplementary information). In these two cases, τ represents path parameter and time respectively. As the wave packet propagates, its pseudo-spin undergoes precession as follows:
d ⃗ ˆ σ s⃗ = i⟨[H, ˆ⃗ ]⟩ = Ω(k) × s⃗, dτ
(5)
⃗ where the precession angular velocity Ω(k) = 2 (mAa0 + k ⋅ Aa ) e⃗a depends on the average wave −m vector (canonical momentum) of the wave packet k ≡ ⟨ˆ p⟩ which is conserved during propagation, and {⃗ ea } denotes the orthonormal basis in pseudo-spin space. As a result of the non-Abelian Lorentz force, the intensity centroid of a wave packet may follow a wavy trajectory: ⟨ˆr⟩ =
1 1 [k − Aa s0a + 2 Fa abc Ωb s0c ] τ m Ω
Fa sin(τ Ω) − [( cos(τ Ω) − 1)δac + abc Ωb ] s0c , 2 mΩ Ω
(6)
where Fa = E a + k × B a /m, s⃗0 represents the initial spin, and abc is the Levi-Civita symbol. This phenomenon resembles the ZB effect of Dirac particles33 . In recent years, ZB has been realized for spin-orbit coupled atoms34,35 and photons36–39 . However, unlike most schemes for the ZB of light realized in periodic systems36–38 , our result shows that light can travel along curved paths even if the background medium is homogeneous. At first glance, This counterintuitive result seems to violate momentum conservation. However, rigorous analysis shows that the conserved quantity protected by translation symmetry is Minkowski-type momentum ∫ d3 x Re(D∗ × B), but Abraham momentum – corresponding to the motion of centroid – could be unconserved in anisotropic media42 (supplementary information). The ZB effect can also be understood in terms of quasi-plane wave approximation. For a certain direction of wave vector k, there are two eigenmodes corresponding to the pair of antipodal ⃗ points along Ω(k) on the Bloch sphere. The ZB stems from
3 kz
b
ky
ky
1
σ3 s
kz θ
f
0
2 ky
-1 -1
0
1
θ
σ2
σ1
k
0
kx
s
kx
s0
-1
1
0
-1
kx
c
σ3
1
3
Ω
k
0
e
ky
a
Ω
σ1
1
s
σ2
kx
d
numerical analytical
0.2
5
g
h -0.34 5
numerical analytical
-0.36
0.1
0
y
y
y
y
-0.38 0
-0.40
-5
-0.42
-5 0 -5
0
x
5
-0.44 -5
0
5
x
-15
-10
-5
x
0
-15
-10
-5
0
x
FIG. 1. a-d, Zitterbewegung induced by a synthetic non-Abelian magnetic field in a gyrotropic medium with the parameters ↔ εT = ↔ µT = σ ˆ0 , εz = µz = 1, g1 = −g2∗ = (i 0.1, 0.04)⊺ . The synthetic gauge potential and field are, repectively, Aˆ = k0 (0.04ex σ ˆ1 − 0.1ey σ ˆ2 ) with Aˆ0 = 0, ˆ3 with Eˆ = 0. e-h, Zitterbewegung induced by a synthetic non-Abelian electric field in a biaxial crystal with the and Bˆ = −k02 0.008 ez σ parameters ↔ εT = σ ˆ0 , εz = 1.6, g1 = (0.3, 0)⊺ , and µ/µ0 = 1. The synthetic gauge potential and field are, respectively, Aˆ = −k0 0.15 ey σ ˆ1 with Aˆ0 = k02 0.255 σ ˆ3 , and Eˆ = −k03 0.0765 ey σ ˆ2 with Bˆ = 0. And k0 = 5π in the two examples. a,e, The isofrequency surfaces and their xy cross sections of both cases. The green arrows in e indicate the three principal axes of permittivity. In both cases, the beam is the equal-ratio superposition of the two eigenmodes with wave vectors along the x direction, and the average wave vectors k are marked by the red dots in the right panels of a and e. b,f, The spin precession along each beam on the Bloch sphere. c,g, Full-wave simulated intensity distributions. d,h, Numerical (black circles) and analytical (red curves) trajectories of the intensity centroid.
the interference of the two eigenmodes just as electronic ZB is induced by the superposition of positive and negative energy components (see supplementary information for details). If the ⃗ initial state is purely an eigenmode, i.e. s⃗0 is along Ω(k), the trembling term in equation 6 will vanish.
Let a light beam propagate along the x axis with incident spin along σ ˆ1 . We can obtain a trembling trajectory from the numerical result (Fig. 1g) which faithfully reproduces the analytic prediction (Fig. 1h).
In Fig. 1a-d, we show an example of spatial ZB for a monochromatic beam induced solely by a non-Abelian magnetic field. ˆ the parameters of the To realize nonzero Bˆ but vanishing E, medium should satisfy εz = µz , g1 = −g2∗ . The isofrequency surfaces of the two eigenmodes are illustrated in Fig. 1a. The wave vector k of the incident beam is along the x direction, while the pseudo-spin s⃗ initially at s⃗0 = e⃗3 precesses about ⃗ the axis Ω(k) ∝ e⃗1 . In Fig. 1c, the full-wave simulation clearly shows a transverse tremor. The trajectory of the centroid perfectly agrees with the analytical result of equation 6 as shown in Fig. 1d. The corresponding temporal ZB for quasi-monochromatic waves is discussed in supplementary information.
The non-Abelian Aharonov-Bohm effect of light
The simplest realization of ZB can be achieved in an arbitrary biaxial dielectric material with ε˜/ε0 = diag(ε1 , ε2 , ε3 ) (ε1 < ε2 < ε3 ) in the principal axis frame, and µ/µ0 = 1. If the second principal axis of permittivity is along the y axis, and the angle θ between the first principal axis and the x axis satisfies cos2 θε1 + sin2 θε3 = ε2 (Fig. 1e), the permittivity components in the xyz coordinate system reduce to ↔ ε T = ε2 σ ˆ0 , √ εz = ε1 + ε3 − ε2 , g1 = ( (ε2 − ε1 )(ε3 − ε2 ), 0)⊺ . Hence, a non-Abelian electric field Eˆ ∝ ey σ ˆ2 emerges while Bˆ = 0.
ZB discussed in the previous section can be viewed as the interference between two eigenmodes, each of which evolves with Abelian dynamics. In this sense, ZB is an apparent nonAbelian effect. Next, we will introduce a genuine non-Abelian AB effect, which cannot reduce to Abelian subsystems. The AB effect covers a group of phenomena associated with the path-dependent phase factors for particles traveling in a fieldfree region with irremovable gauge potential Aˆµ , the discovery of which confirmed the physical reality of gauge potentials and the nonlocality of gauge interactions43 . The AB effect was firstly generalized to non-Abelian by Wu and Yang29 , who showed that the scattering of nucleons (isospinors) around a non-Abelian flux tube (vortex) can generate peculiar phenomena, such as the spatial fluctuation of proton-neutron mixing ratio. However, their governing Hamiltonian can be globally diagonalized into two decoupled Abelian subsystems under a proper gauge44 , and all relevant phenomena can be interpreted from the superposition of these two subsystems. Hence, WuYang’s proposal is now viewed as an apparent non-Abelian effect10,40 . According to a rigorous definition40 , a genuine nonˆ to be Abelian AB system requires its holonomy group Hol(A)
4 e
a
f
σ3 s0
γI
s II
screen
sI
non-Abelian vortices
x0
σ2
Δ s σ1
γII
y
g
σ3 |ψ 2 s
x
b
0.5
cI γ0
γI=γ0∘cI γII=γ0∘cII
-0.5
σ1
h δθ (L0/2π)
2.0
—
○
theory simulation control
|ψ|2
1.5
c2∘c1-1
cI
c1-1∘c2 2
α
1st
π/2 0 π
j β
≃
nd
0.0 π
i
2nd
cII
2b
1.0 0.5
1st
≃ d
y —
cII c
σ2
0.0
0 -π -1.0
-0.5
0.0
0.5
1.0
y
FIG. 2. a, Sketch of the genuine non-Abelian AB system with two optical paths γI , γII interfering on the screen, where the background light blue (red) arrows denote the σ ˆ1 (ˆ σ2 ) component A1 (A2 ) of the non-Abelian vector potential. b, γI (γII ) can be divided into a closed loop cI (cII ) and a common path γ0 . c,d cI and cII can, respectively, deform continuously into a closed path that winds around the two vortices successively but in opposite sequences. e, Snapshot of the simulated field intensity for the proposed non-Abelian optical interferometry with incident spinor (1, i/5)⊺ for both beams and the vortex fluxes Φ1 = −2π/3, Φ2 = −π/3. f, Spin evolution on the Bloch sphere along two beams γI , γII , which share the same initial spin s⃗0 but achieve different final spins s⃗1 and s⃗2 . g, Spin density interference corresponding to e, where each arrow denotes the local pseudo-spin density ∣ψ∣2 s⃗ at a point on the screen. All of the local spins s⃗(y) are perpendicular to ∆⃗ s = s⃗I − s⃗II , and thus fall on the green circle in f. The corresponding intensity interference ∣ψ∣2 (y) and the two Euler angles α, β of the local spins s⃗(y) on the screen are shown in h-j, where blue circles and red curves indicate simulated and theoretical results respectively, and δθ, b are the phase shift and relative amplitude relative to the case of Aˆ = 0. The green lines correspond to the “control experiment”.
non-Abelian (see definition in Methods). As such, there should exist such loops based at a fixed point that their non-Abelian AB phase factors (holonomies) are noncommutable, i.e. if a particle travels along two such loops in opposite sequences, the obtained AB phase factors would be different. This implies that at least two vortices exist in a genuine non-Abelian system40 . Here, we use anisotropic and gyrotropic materials (see equation 1) to synthesize the vector potential Aˆ = A1 σ ˆ1 + A2 σ ˆ2 (Aˆ0 = 0) with vanishing field Bˆ = 0 in the whole space except for two small domains, taken as point singularities for simplicity. As illustrated in Fig.2a, we require Aˆ = A1 σ ˆ1 (A2 = 0) in 2 1 the upper half-space, while Aˆ = A σ ˆ2 (A = 0) in the lower half-space. We also require that A1 , A2 smoothly tend to zero in the middle region without overlap. In the vicinity of the upper (lower) singularity, A1 (A2 ) forms an irrotational vortex carrying the flux Φ1 (Φ2 ) (see supplementary information for
details). For a closed loop with a fixed base-point, its nonAbelian holonomy is invariant against continuous deformation of the path within the Bˆ = 0 region. As a consequence, the holonomies of the two kinds of loops [c1 ] and [c2 ] (where [ci ] denote the path homotopy classes; see Methods), based at x0 and winding around the upper (for [c1 ]) or lower (for ˆi = Uˆ[c ] [x0 ] = exp [iΦi σ [c2 ]) vortex once, are U ˆi ] (i = 1, 2) i respectively. As they do not commute with each other, we have thus realized a non-Abelian AB system. Consider two coherent light beams with the same initial spin s⃗0 propagating separately along the two folded paths γI and γII , and finally superposing on the screen (Fig. 2a). The final states of the two beams can be expressed as ∣ψi (y)⟩ = ˆγ eiθi (y) ∣s0 ⟩ (i = I, II), where a(y) is the envelope of a(y) U i ˆγ = P exp [i ∫ A⋅dr] ˆ both beams on the screen, U (P denotes i γi iθi (y) path-ordering) and e are, respectively, the non-Abelian AB phase factor and the dynamic phase factor of the two paths,
5
ˆγ = U ˆγ U ˆ2 U ˆ1−1 ≠ U ˆγ = U ˆγ U ˆ1−1 U ˆ2 . U 0 0 I II
(7)
Consequently, the two beams will achieve different final spins s⃗I and s⃗II on the screen (Fig. 2f), and they will interfere with each other in a non-trivial way. The term spin density interference was coined for this phenomenon and it can be calculated as follows: ⟨ψI (y) + ψII (y)∣ σ ⃗ ∣ψI (y) + ψII (y)⟩ = ∣ψ∣2 (y) s⃗(y).
(8)
Here, the angle bracket denotes the spinor inner product at a local position y on the screen, so the obtained result describes the spin density distribution on the screen. The spin density can be further decomposed into two parts: the intensity interference ∣ψ∣2 (y) and the spin orientation interference s⃗(y). For the intensity interference, ∣ψ∣2 (y) = 2a(y)2 [1 + b cos (∆θ(y) + δθ) ], where both the relative amplitude b (< 1) and the phase shift δθ, with respect to the case of Aˆ = 0, depend on the vortex fluxes Φ1 , Φ2 as well as the incident spin s⃗0 (see Methods). The interfering spin orientation s⃗(y) fluctuates around the direction ∆⃗ s = s⃗I − s⃗II , namely s⃗(y) ⋅ ∆⃗ s ≡ 0, on the screen. We have performed a full-wave simulation of this non-Abelian AB interference as shown in Fig. 2e. The spin density interference is shown in Fig. 2g, with the intensity interference ∣ψ∣2 (y) in Fig. 2h, and the spin orientation given by Euler angles in Fig. 2i,j. In Fig. 2h-j, the blue circles are the simulated results, which are consistent with the red curves obtained from equation 8. To demonstrate that the non-Abelian feature of the above design is indeed genuine, we consider a control experiment with an almost identical system except that the vector potential is ˆi = exp[iΦi σ Aˆ ∝ σ ˆ1 in the whole space. In this case, U ˆ1 ] (i = 1, 2) commute with each other, and their winding around the two vortices in opposite sequences gives the same AB phase ˆγ = U ˆγ = exp [i(Φ2 − Φ1 )ˆ factor U σ1 ]. Thus, the interfering I II spin density is uniformly orientated, and there is no phase shift (δθ ≡ 0) and amplitude contraction (b ≡ 1) compared with Aˆ = 0 case (see green lines in Fig. 2h-j). Measurement of Wilson loops In Abelian AB systems, the AB phase factor (holonomy) of a closed loop only depends on the flux inside the loop but independent of the choice of gauge. However, in non-Abelian systems, the holonomy Uˆ[c] [x0 ] of a closed path c varies as ′ ˆ (x0 )Uˆ[c] [x0 ]U ˆ † (x0 ), under a gauge transforUˆ[c] [x0 ] = U ˆ AˆU ˆ † + iU ˆ ∇U ˆ † . In spite of this, the trace of mation Aˆ′ = U
a
c = γ�-�∘γ1
γ1 γ2
b gauge fixed points
2
1
4 W (c0)
0
0
|ψ| 2
and ∣s0 ⟩ is the normalized initial spinor at x0 . The optical path of each beam can be regarded as a concatenation of a closed loop ci and a common path γ0 , i.e.,γi = γ0 ○ ci (i = I, II), as illustrated in Fig.2b. The closed loop cI can be further deformed continuously into two successive loops c2 ○ c−1 1 , which winds around the upper vortex (clockwise) first and the lower vortex (anticlockwise) second (Fig. 2c). Likewise, cII is homotopic to c−1 1 ○ c2 , namely cII winds around the lower vortex first before it does the upper vortex (Fig. 2d). Because of the noncommutativity of the sequences of winding around the two vortices, the AB phase factors of the two beams must be different:
(1/3, 1)
-6
-5
-4
-3
-2
0
-1
1
2
3
4
5
(1, i/5) (0, 1)
1
(1, 0)
2
A=0
6
Δθ/π
FIG. 3. a. For two arbitrary beams γ1 , γ2 interfering on the screen, the Wilson loop of the concatenate path c = γ2−1 ○ γ1 can be extracted from the interference fringes of the two beams. b. Four intensity interference curves corresponding to four different incident spinors (1, 0)⊺ , (0, 1)⊺ , (1, i/5)⊺ , (1/3, 1)⊺ for the non-Abelian AB system shown in Fig. 2a with vortex fluxes Φ1 = 0.22π, Φ2 = 0.33π, where circles and solid curves represent numerical and analytical results respectively. Their intersections, marked by red targets, are the gauge fixed points, which are located at the crests and troughs of the interference fringes (light blue curve) of Aˆ = 0. The maximal difference between the envelops of even and odd gauge fixed points gives the −1 Wilson loop W (c0 ) of c0 = γII ○ γI .
holonomy is an important gauge invariant observable, called the Wilson loop of the closed path c: ′ W (c) = Tr (P exp [i ∮ Aˆ ⋅ dr]) = Tr Uˆ[c] = Tr Uˆ[c] .
(9)
c
In the following, we show how to extract the Wilson loop of an arbitrary closed path via interferometry. In order to obtain the Wilson loop of a homotopy class [c] in a non-Abelian AB system, we consider the interference of two beams along any two paths γ1 and γ2 as long as γ2−1 ○ γ1 ∈ [c] forms a closed loop in the class [c] as sketched in Fig. 3a. It can be shown that, at certain positions yn (where n is an integer) in the intensity interference fringes, the intensities only depend on the Wilson loop of c and are thus fixed under gauge transformation: 2
∣ψ∣ (yn ) ≡ 2a(yn )2 [1 + (−1)n W (c) ] ,
(10)
where the two beams are supposed to share the same envelope a(y) on the screen, and the locations yn , satisfying ∆θ(yn ) = nπ, correspond to the crests and troughs in the interference fringe of Aˆ = 0 (see Methods). These particular points in the intensity fringes are called the gauge fixed points for the closed path c. Since the change of incident spin at x0 is equivalent ˆ the interference fringes to a global gauge transformation of A, for different incident spins should intersect at the gauge fixed points.
6 Using the above method, we examine the two optical paths −1 γI , γII in Fig. 2a to extract the Wilson loop of c0 = γII ○ γI ≃ −1 −1 c2 ○ c1 ○ c2 ○ c1 . Figure 3b shows the intensity interference results corresponding to four different incident spins. Indeed, these four interference curves intersect exactly at the gauge fixed points (red targets in Fig. 3b), and their locations yn coincide with the crests and troughs of the interference fringe for Aˆ = 0. By fitting the even and odd subsequences of the gauge fixed points, we obtain two curves 2a(y)2 [1 ± W (c0 )] corresponding to the two red dashed lines in Fig. 3b. Thus, the Wilson loop W (c0 ) can be identified from the difference of the two dashed curves. In the case where sin2 Φ1 sin2 Φ2 = 1/2, W (c0 ) reduces to zero, then the two dashed curves completely overlap.
also designed a genuine non-Abelian AB system with two synthetic non-Abelian vortices, and suggested a spin density interferometry to demonstrate the noncommutative feature of non-Abelian holonomies. Our scheme opens the door to the colorful non-Abelian world for light. In addition to inspiring new ideas to manipulate the flow and polarization of light, the scheme offers an optical platform to study physical effects relevant to SU (2) gauge fields, such as synthetic spin-orbit coupling45 and topological band structures in periodic nonAbelian gauge fields46,47 . Furthermore, since the SU (2) gauge field description is valid for photons down to the quantum scale, this approach might be applicable to the design of geometric gates for realizing non-Abelian holonomic quantum computation48,49 with photons. Acknowledgement
Conclusion We have shown that the dynamics of 2D optical waves in a broad class of anisotropic media can be understood through an emergent non-Abelian SU (2) gauge interaction in real space. We predicted that the Zitterbewegung effect of light can be realized even in homogeneous anisotropic media. We have
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This work was supported in part by National Natural Science Foundation of China (Grant No. 61405067, 11174250), and the Fundamental Research Funds for the Central Universities, HUST: 2017KFYXJJ027. The work in Hong Kong is supported by Research Grants Council of Hong Kong (AoE/P02/12).
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8 Methods Notations. In this paper, vectors in real space and in pseudospin space are indicated, respectively, by bold letters and letters with an overhead arrow “→”. Letters with an overhead bidirectional arrow “↔” denote 2-order tensors in real space. Symbols with an overhead hat “∧” denote operators acting on the spinor wave functions. We use Greek letters, e.g. µ, ν, to denote indices of (2+1)-dimensional spacetime. Latin letters i, j denote 2D spatial coordinate indices, and Latin letters a, b, c denote indices in pseudo-spin space. We follow the Einstein summation convention for repeated indices. The orthonormal coordinate bases in real space and pseudo-spin space are expressed as ei and e⃗a respectively. Emergent non-Abelian gauge fields. The emergent nonAbelian vector potential Aˆ = Aa σ ˆa and scalar potential Aˆ0 = Aa0 σ ˆa as well as the Abelian scalar potential V0 in wave equation 2 are determined by the material parameters with the expressions as follows A =k0 Re (g− ) × ez , 1
A = k0 Im (g− ) × ez , 2
A = 0, 3
A10 =h−2 k0 ez ⋅ [∇ × (↔ ε −1 T ⋅ Im (g+ ))] , A20 = − h−2 k0 ez ⋅ [∇ × (↔ ε −1 T ⋅ Re (g+ ))] , A30 =h−2 k02 [
εz − µz − 2 Re (g−† ⋅ ↔ ε −1 T ⋅ g+ )] , 2 ↔ εT det(↔ εT )
↔ 1 εT ⋅ (∇ h−1 ) − ∇ ⋅ ( ⋅ ∇ h−2 ) 2 det(↔ εT ) εz + µz + h−2 k02 [(g+† ⋅ ↔ ε −1 ], T ⋅ g+ ) − 2
V0 =(∇ h−1 ) ⋅
where g± = (g1 ± g2∗ )/2. For monochromatic waves, the function h(x, y) can be arbitrarily selected, and we set h = 1 in computations. For quasi-monochromatic waves, h is fixed as √ h = k0 εz + µz (see supplementary information). Meanwhile, the (2+1)-dimensional non-Abelian gauge field tensor is given by
the strict constraints on the media that (i) g1 = −g2 is real and (ii) εz = µz . In this case, the vector potential only has σ ˆ1 component Aˆ = A1 σ ˆ1 and the scalar potential Aˆ0 vanishes. As such, [Aˆi , Aˆj ] ≡ 0, and the gauge group reduces to the Abelian subgroup U (1) of SU (2). Holonomy and genuine Non-Abelian system. From a geometric viewpoint, gauge potential and field can be described as the connection and curvature in a principle fiber bundle29 . The phase factor P exp [i ∫γ Aˆµ dxµ ] ∈ G along a path γ in the physical space (base manifold) corresponds to the parallel transport in the bundle space, where G is the gauge group, and in our case G = SU (2). In particular, for a closed path c starting and ending at the same point c(0) = c(1) = x0 , the phase factor of c, ˆ = P exp [i ∮ Aˆµ dxµ ] , Uˆc (A) c
is called the holonomy of the closed path c with respect to ˆ The collection of the holonomies corresponding the gauge A. to all those closed paths based at the same point x0 forms a subgroup of the gauge group G: ˆ = {Uˆc (A) ˆ ∣ c(0) = c(1) = x0 } ⊆ G, Hol(A) ˆ In the litwhich is called the holonomy group for the gauge A. erature, a gauge system is regarded as genuinely non-Abelian if and only if the holonomy group is a non-Abelian group, namely the holonomies of some loops are noncommutable with each other10,40 . If the base manifold is simply a Euclidean space, the noncommutativity of holonomies can be traced back to noncommutable gauge fields [Fˆµν , Fˆµ′ ν ′ ] ≠ 0. However, if the base manifold possesses nontrivial topology, noncommutative holonomies can be achieved even though the gauge field vanishes everywhere (i.e. AB systems).
In contrast, the scalar potential Aˆ0 , as well as the effective ˆ is determined by both g± and εz , µz . electric field E,
For an AB system, the corresponding fiber bundle is a flat bundle, since the curvature (field) Fˆµν = 0 in the whole base manifold M (flux regions are excluded from M ). Here, the topology of the base manifold is characterized by its first fundamental group π1 (M ) = {[c] ∣ c(0) = c(1) = x0 }, which is the set of path homotopy equivalent classes [c] of closed paths based at x0 . Path homotopy is a topologically equivalent relation “≃” for paths. If two paths c1 , c2 with the same fixed base-point x0 can deform into each other continuously, they are said to be path homotopic c1 ≃ c2 and to belong to the same homotopy class [c1 ]. In flat bundles, the holonomies (AB phase factors) of all loops in the same homotopy class [c] are identical: Uˆ[c] . Based on this property, two necessary conditions for genuine non-Abelian AB systems can be obtained40 : 1. The gauge group G is non-Abelian. 2. The first fundamental group π1 (M ) is non-Abelian.
The gauge potential would not be regarded as (apparently) non-Abelian, unless some components of the potential do not commute [Aˆµ , Aˆν ] ≠ 010 . For instance, the scheme proposed by Liu and Li26 is actually a specific reduction of ours with
According to the second criterion, the Wu-Yang AB system is not genuinely non-Abelian, because the fundamental group of its base manifold (a punctured plane R2 −0) is an Abelian group π1 (R2 − 0) = Z. However, for a doubly punctured plane as
ˆµ , D ˆ ν ] = ∂µ Aˆν − ∂ν Aˆµ − i[Aˆµ , Aˆν ], Fˆµν = i[D ˆµ = σ where D ˆ0 ∂µ − i Aˆµ is the covariant derivative. The nonAbelian magnetic and electric fields in equation 3 are derived from Bˆ = 12 ij Fˆij ez and Eˆ = −Fˆ0i ei . As can be seen, the vector potential Aˆ only depends on g− . Accordingly, the nonAbelian magnetic field also only depends on g− with a rather simple expression Bˆ = i k02 (g− × g−∗ ) σ ˆ3 − k0 ∇ ⋅ [Re (g− ) σ ˆ1 + Im (g− ) σ ˆ2 ] ez .
9 shown in Fig. 2a, its fundamental group Z∗Z (where ∗ denotes a free product) is non-Abelian. Therefore, a doubly punctured plane is a qualified prototype of a genuine non-Abelian AB system. Non-Abelian AB interference. In order to realize the vector potential shown in Fig. 2a, the background media are set up as g1 = −g2∗ (i.e. g+ = 0) and εT = εz = µT = µz = const. to guarantee Aˆ0 ≡ 0 and V0 = const. Also, we use reciprocal anisotropic materials with purely real off-block-diagonal components g1 = −g2 to build the vector potential Aˆ = A1 σ ˆ1 in the upper half plane but gyrotropic materials with purely imaginary g1 = g2 to build the vector potential Aˆ = A2 σ ˆ2 in the lower half plane. In order to avoid spin flip after reflection, the mirrors shown in Fig. 2a,e are made of an impedance-matched material, namely εm /µm = 1, with a lower refractive index than the surrounding media to achieve total reflection at their surfaces. Meanwhile, the two mirrors on the right-hand side in Fig. 2e are slightly concave, so that the reflected beams with reduced widths can bypass the two singularities. The intensity interference part in equation 8 can be further derived as 2
2
∣ψ∣ (y) = ∣ψI (y) + ψII (y)∣ = 2 a(y)2 [1 + Re⟨ψII ∣ ψI ⟩(y)] = 2 a(y)2 [1 + Re (ei∆θ(y) ⟨s0 ∣ Uˆ[c0 ] ∣s0 ⟩)] = 2 a(y)2 [1 + b cos (∆θ(y) + δθ)] . ˆγ−1 U ˆγ = U ˆ −1 U ˆ1 U ˆ2 U ˆ −1 is the non-Abelian Here, Uˆ[c0 ] = U 2 1 I II −1 holonomy of the closed path c0 = γII ○ γI , ∆θ(y) = θI (y) − θII (y) is the difference between the dynamic phases of the two beams, and the initial phases of the beams have been included in θI , θII . In computations, the envelope a(y) of each beam is set to be Gaussian type with a normalized amplitude a(0) = 1. In the case of Aˆ = 0, the interference fringes only depends on the dynamic phases: ∣ψ∣2 (y) = 2a(y)[1 + cos(∆θ(y))]. ˆ the nontrivial interference However, in the presence of A, ˆ ⟨s ∣ ∣s result, indicated by 0 U[c0 ] 0 ⟩ = b eiδθ ≠ 1, relies on the ˆ1 and U ˆ2 . The phase shift δθ and the noncommutativity of U relative amplitude b are functions of the fluxes Φ1 , Φ2 and the initial spin s⃗0 with the following explicit expressions: δθ(Φ1 , Φ2 ) = arctan [
2 sin Φ1 sin Φ2 ⃗(Φ1 , Φ2 )] , s⃗0 ⋅ u 1 − 2 sin2 Φ1 sin2 Φ2
b(Φ1 , Φ2 ) =[(1 − 2 sin2 Φ1 sin2 Φ2 )
2 2
1/2
⃗(Φ1 , Φ2 )) ] + 4 sin2 Φ1 sin2 Φ2 (⃗ s0 ⋅ u
,
⃗(Φ1 , Φ2 ) (∣⃗ where u u∣ ≤ 1) is a vector defined in spin space: ⃗ = − cos Φ1 sin Φ2 e⃗1 − sin Φ1 cos Φ2 e⃗2 + cos Φ1 cos Φ2 e⃗3 . u Since b < 1 for any Φ1 , Φ2 , the relative amplitude of the interference is always contracted.
Gauge fixed points. The above derivation of the intensity interference is in fact valid for two arbitrary interfering beams γ1 , γ2 with the same initial spin and final envelop a(y): ∣ψ∣2 = 2a(y)[1 + Re(ei∆θ(y) ⟨s0 ∣ Uˆ[c] ∣s0 ⟩)], where Uˆ[c] is the holonomy of the closed path c = γ2−1 ○ γ1 . Since Uˆ[c] ∈ SU (2), it can be generically expressed as u u Uˆ[c] = ( 1∗ ∗2 ) , −u2 u1
∣u1 ∣2 + ∣u2 ∣2 = 1.
Thus, the Wilson loop reads W (c) = Tr Uˆ[c] = 2 Re u1 . For an arbitrary spinor state ∣s0 ⟩ = (cos α2 e−iβ/2 , sin α2 eiβ/2 )⊺ , we have ⟨s0 ∣ Uˆ[c] ∣s0 ⟩ = beiδθ = 2 Re u1 + i sin α Im(u2 eiβ ). Therefore, the following identity holds for any ∣s0 ⟩: W (c) = Re ⟨s0 ∣ Uˆ[c] ∣s0 ⟩ = b cos δθ. In fact, different incident spinors can interconvert through a ˆ ∣s0 ⟩. Hence, the above global gauge transformation: ∣s′0 ⟩ = U relation is straightforward ˆ −1 Uˆ[c] U ˆ ∣s0 ⟩ Re ⟨s′0 ∣ Uˆ[c] ∣s′0 ⟩ =Re ⟨s0 ∣ U ˆ −1 Uˆ[c] U ˆ ) ≡ W (c). =Tr (U As a result, at the positions such that ∆θ(yn ) = nπ, i.e. at the crests and troughs of the original interference fringes when Aˆ = 0, the intensities given in equation 10 are fixed for arbitrary incident spins, yet they are only determined by the Wilson loop W (c), provided the dynamic phases of γ1 , γ2 are unchanged.
