Stern-Gerlach splitters for lattice quasispin
SStern-Gerlach splitters for lattice quasispin
A. S. Rosado , J. A. Franco-Villafa˜ne , C. Pineda , and E. Sadurn´ı ∗ Instituto de F´ısica, Benem´erita Universidad Aut´onoma de Puebla, Apartado Postal J-48, 72570 Puebla, M´exico Instituto de F´ısica, Universidad Nacional Aut´onoma de M´exico, 01000 M´exico D.F., Mexico (Dated: October 14, 2018)We design a Stern-Gerlach apparatus that separates quasispin components on the lattice, withoutthe use of external fields. The effect is engineered using intrinsic parameters, such as hoppingamplitudes and on-site potentials. A theoretical description of the apparatus relying on a generalizedFoldy-Wouthuysen transformation beyond Dirac points is given. Our results are verified numericallyby means of wave-packet evolution, including an analysis of
Zitterbewegung on the lattice. Thenecessary tools for microwave realizations, such as complex hopping amplitudes and chiral effects,are simulated.
PACS numbers: 03.65.Pm, 03.67.Ac, 72.80.Vp
I. INTRODUCTION
Quantum emulations have been increasingly importantfor theorists and experimentalists in areas such as ultra-cold atoms [1–5], quantum and microwave billiards [6–9], plasmonic circuits [10], and artificial solids in gen-eral [11, 12]. The concept can be used to engineer quan-tum dynamics not readily accessible in naturally occur-ring physical systems, e.g., elementary particles or chargecarriers in solids [13, 14]. For some years, the effectiveDirac theories emerging in honeycomb lattices and lin-ear chains [15–20] have led researchers to consider theuse of quasispin as an internal degree of freedom capa-ble of supporting the long-pursued realization of qubitsin solid-state physics. This interesting degree of free-dom has the property of being nonlocal, inherent to thecrystalline structure, and sufficiently robust as to pro-vide upper and lower bands around conical (Dirac) pointsin the spectrum. In the same context, there has beena recent interest in Majorana fermions [21–23], as theirtopological nature may provide robustness with respectto decoherence, hence increasing the life of qubits, andthus extending the reach of potential applications. Sev-eral theoretical developments take advantage of quasispin[24, 25] and some experiments in lattices have observedtheir effects, e.g.,
Zitterbewegung in photonic structures[26].But how does one measure quasispin on the lattice?One of the goals of this paper is to gain access to thisdegree of freedom by designing an interaction on bipar-tite lattices with the following features: (a) an adjustablecoupling with particles’ quasispin, (b) a localized regionwhere the interaction occurs, and (c) an intrinsic gen-eration of the interaction using lattice parameters. Itis worth mentioning that the electron’s true spin is noteasily accessible when immersed in a solid [27].Our tasks demand an exploration of tight-bindingmodels, oriented to an experimental setup in microwave ∗ [email protected] resonators. We establish the realization of Dirac’s equa-tion in a one-dimensional setting and solve the prob-lem of how to split the two components of the wavefunction, namely particle-antiparticle components, or, inthe language of solid-state physics, the upper and lowerbands. Under these circumstances, and using the Foldy-Wouthuysen (FW) transformation, we design and test aspatially localized Stern-Gerlach splitter represented bya banded matrix, to be used in the context of Dirac-like dynamics. In this case, the experimental restrictionsimposed by most realizations come in the form of short-range interactions. We provide a successful geometricproposal in compliance with such restrictions, using mi-crowave resonators coupled by proximity.We approach the problem in three different stages.First, in Sec. II, we study the lattice structure usingfull-band Dirac equations [20] and provide a generalizedFW transformation in Sec. II A. The explicit construc-tion of the beam splitter as a potential is achieved in Sec.II B. In Sec. III, we study wave-packet dynamics usingnumerical simulations with two important results: in Sec.III A, we show that unpolarized beams exhibit Zitterbe-wegung , while in Secs. III B and III C, we test the splitterefficiency. With the aim of ensuring the feasibility of ourmodel, in Sec. IV we establish the robustness of thesystem under random perturbations of parameters. Ourstudy is applicable to any tight-binding (TB) array withthe aforementioned structure, but, as a final step, in Sec.V we focus on plausible experiments in microwave cav-ities. Section V A describes the necessary specificationsfor the implementation and Sec. V B gives an explicitconstruction that produces negative couplings and levelinversion. We conclude in Sec. VI.
II. INTRINSIC STERN-GERLACHAPPARATUSA. Quasispin and generalized FW transformations
Let us define our periodic system, with the aim of gen-eralizing the usual FW unitary rotation [28, 29]. Con- a r X i v : . [ qu a n t - ph ] J u l x yzφ HB Aµθ FIG. 1. A visualization of the FW transformation. Usinga rotation around the z axis by an angle φ , followed by arotation around y by an angle θ , would rotate the eigenstatesof σ z to the eigenstates of the Hamiltonian given by Eq. (1).In this visualization, the angles are considered to be scalarssince they are operators that commute with the Hamiltonian. sider a one-dimensional lattice, with sites characterizedby the positions n ∈ Z , and position basis {| n (cid:105)} n ∈ Z . Wedeal with a typical TB model in this setting, with hop-ping parameter ∆ and potential V , H = ∆ T + ∆ T † + V = ∞ (cid:88) n = −∞ ∆ | n (cid:105)(cid:104) n + 1 | + h.c. + V n | n (cid:105)(cid:104) n | (1)where the translation operator is defined via T | n (cid:105) = | n +1 (cid:105) and a position-dependent potential V = (cid:80) n V n | n (cid:105)(cid:104) n | has been introduced. We have shown [20] that thisHamiltonian can be written in Dirac form without ap-proximations, with suitable definitions of Dirac matrices α in terms of projectors onto even and odd site numbers, H = ∆ α · Π + V (2)with the kinetic operatorsΠ ≡ (cid:88) n | n − (cid:105)(cid:104) n | + | n (cid:105)(cid:104) n − | = 1 + T + ( T † ) ≡ i (cid:88) n | n − (cid:105)(cid:104) n | − | n (cid:105)(cid:104) n − | = T − ( T † ) i (3)and the Dirac matrices α ≡ (cid:88) n even | n + 1 (cid:105)(cid:104) n | + | n (cid:105)(cid:104) n + 1 | α ≡ i (cid:88) n even | n + 1 (cid:105)(cid:104) n | − | n (cid:105)(cid:104) n + 1 | (4)satisfying the usual conditions, as proved in [20]. Bipartite lattices with alternating on-site potential en-ergies E , E entail the use of the potential V = E + µβ, (5)where the average energy E = ( E + E ) / µ = ( E − E ) / β , here defined as β ≡ (cid:88) n even | n (cid:105)(cid:104) n | − | n + 1 (cid:105)(cid:104) n + 1 | . (6)Our lattice operators (4) and (6) satisfy the relations { α i , β } = 0 , { α i , α j } = 2 δ ij , [ α , α ] = 2 iβ . This reorder-ing of our original TB Hamiltonian leads to an effectiveDirac Hamiltonian of the form H = ∆ α · Π + µβ + E . (7)The spectrum of H is E k, ± = E ± (cid:112) cos k + µ ≡ E ± E k and, most importantly, its eigenfunctions arewritten as spinors with up and down components rep-resented by amplitudes in the even and odd sublattices.Here we remark that this spinorial form of the eigenfunc-tions and, in general, of any wave packet on the latticeis in itself an additional discrete degree of freedom, andthus gives rise to the name: quasispin. As previouslynoted, quasispin is entirely nonlocal, given that it is a di-rect manifestation of the bipartite nature of the lattice.Returning to the discussion, we have the following com-plete set of eigenfunctions (cid:104) n | k, s (cid:105) = e ikn (cid:18) u + k,s u − k,s (cid:19) , u ± k,s = s ± / (cid:114) E k ± sµ πE k , (8)where n is an even index, k is the wave number in thereduced Brillouin zone 0 < k < π , and s = ± is theindex of upper and lower bands. For the latter use, weintroduce the parameter κ around the conical point k = π/ − κ/
2. This yields the following eigenvalues p i of Π i : p ≈ − κ , p ≈ κ (9)for momenta near the conical point. This shows that p survives, playing the role of an effective momentum of aone-dimensional (1D) Dirac equation.In order to show the role of quasispin in the solutions,one can solve the eigenvalue problem without any approx-imation by means of a rotation in the space ( α , α , β ).This is the FW transformation explained in Fig. 1, whichmaps the site model (even/odd sites) to a qubit systemof positive and negative energies [19, 20]. In terms ofPauli matrices, we write α = σ , α = σ , β = σ andwe define a vector v with components v = ∆Π , v =∆Π , v = µ . With these definitions, H becomes a purespin-orbit interaction, H − E = v · σ , [ v i , v j ] = [ v i , σ j ] = 0 . (10)This allows one to rotate the vector v independently of σ , with the aim of making it parallel to z . Equivalently,the rotation is represented by a unitary transformation U FW which block diagonalizes H , U FW = exp (cid:18) − iφ σ (cid:19) exp (cid:18) − iθ σ (cid:19) (11)In our case, this rotation allows us to guide the design ofthe polarizer. The exponential is understood in terms oftrigonometric functions, where the angles are operatorsdefined by sin θ = ∆( T + T † ) (cid:112) ∆ ( T + T † ) + µ , cos θ = µ (cid:112) ∆ ( T + T † ) + µ (12)and cos φ = 12 ( T + T † ) , sin φ = 12 i ( T − T † ) . (13)Formula (11) involves trigonometric functions of half an-gles, so we provide their expressions for completeness [wenote here that ( H − E ) is independent of Pauli matri-ces], cos (cid:18) θ (cid:19) = (cid:115) (cid:112) ( H − E ) + µ (cid:112) ( H − E ) , sin (cid:18) θ (cid:19) = (cid:115) (cid:112) ( H − E ) − µ (cid:112) ( H − E ) , (14)and cos (cid:18) φ (cid:19) = 12 ( T / + ( T † ) / ) , sin (cid:18) φ (cid:19) = 12 i ( T / − ( T † ) / ) . (15)With the unitary operator U FW , the transformationyields, in a very clean way, H FW = U † FW HU FW = (cid:18) E + (cid:112) ( H − E ) E − (cid:112) ( H − E ) (cid:19) where (cid:112) ( H − E ) = (cid:112) ∆ ( T + T † ) + µ .Adding the next-to-nearest-neighbor interaction in Eq.(1) would require a modification of the definitions (3).However, the program of the present section could alsobe carried out in a very similar fashion. The addition ofthe quartic translational terms in Eq. (3) would changeEqs. (12) and (13), and would thus make the propagationin the two bands slightly different. A splitter could thusalso be designed, but an asymmetry in the two compo-nents would indeed show up in the asymptotic evolution. FIG. 2. Lattice topologies corresponding to the polarizer,up to second neighbors (top) and third neighbors (bottom).Thick lines correspond to the (strongest) nearest-neighbor in-teraction, thin lines are the next-to-nearest-neighbor interac-tions, and, finally, dashed lines are the weakest (and in thetop model neglected) third-nearest-neighbor interactions.
B. The Stern-Gerlach apparatus as an interaction
Now that we have derived a block-diagonal Hamilto-nian, we are in the position to introduce an interactionwhich couples differently with positive- and negative-energy solutions. Moreover, we shall see that the range ofsuch interaction can be controlled at pleasure. A diagramis shown in Fig. 2.In classical relativistic dynamics, the double sign ofthe kinetic energy could be used to produce two typesof behavior in the presence of a potential well. If V ( x )interacts attractively for positive solutions (charges), theopposite case will be a potential barrier acting on nega-tive solutions (holes): E = ± (cid:112) c p + m c + V ( x ) . (16)Thus, one type of solution would be allowed to enter in acertain region while the other would be rejected; we mayregard V ( x ) as a gate keeper. We must note, however,that quantum dynamics gives rise to interference phe-nomena producing transmission and reflection in both ofthe aforementioned situations. The simplest way to sep-arate both types of waves is by introducing a potentialof the type V ± ( x ) = (cid:40) V ( x ) for particles0 for holes (17)Since the FW transformation does the job of decouplingboth types of solutions, we introduce at the level of H FW a potential V FW that separates the components as in (17),˜ H FW = H FW + + σ ⊗ V FW (18)or in matrix form, (cid:18) E + V FW + (cid:112) ( H − E ) E − (cid:112) ( H − E ) (cid:19) . In order to find the true potential V operating at the levelof lattice sites and neighbor couplings, we must returnto our original description by means of the inverse FWtransformation, V ( N ) = U FW V FW U † FW . (19)Direct computations lead to a 2 × V . Forinstance, V = e − iφ/ cos (cid:18) θ (cid:19) V FW cos (cid:18) θ (cid:19) e iφ/ (20)Here we may choose V FW at will, but using site numberkets makes it easier to provide locality: (cid:104) n | V FW | n (cid:48) (cid:105) = δ n,n (cid:48) V FW ( n ). The site dependence of V can be obtainedby inserting a complete set of Bloch waves. Let us define I s,s (cid:48) n (cid:16) µ ∆ (cid:17) ≡ (cid:90) π − π dk (cid:114) E k + sµE k e ik ( n − s (cid:48) / , (21)with s, s (cid:48) = ± and n ∈ Z . The potential blocks are then (cid:104) n | V | n (cid:48) (cid:105) = 18 π ∞ (cid:88) m = −∞ V FW ( m ) I ++ n (cid:48) − m (cid:0) I ++ n − m (cid:1) ∗ (22)for even n and n (cid:48) , (cid:104) n | V | n (cid:48) (cid:105) = 18 π ∞ (cid:88) m = −∞ V FW ( m ) I − + n (cid:48) − m (cid:0) I + − n − m (cid:1) ∗ (23)for even n and odd n (cid:48) , and finally (cid:104) n | V | n (cid:48) (cid:105) = 18 π ∞ (cid:88) m = −∞ V FW ( m ) I − + n (cid:48) − m (cid:0) I − + n − m (cid:1) ∗ (24)for odd n and n (cid:48) . It is advantageous to write our resultin the form of the series over m above: when the rangeof V FW is limited, the summation over m involves only afew terms. In the extreme case of a pointlike gate keeperin the FW picture, m = 0 is the only contribution in V .Moreover, the limits µ (cid:29) ∆ and µ (cid:28) ∆ provide usefulapproximations, I s,s (cid:48) n ≈ √ ss (cid:48) ( − ) n s (cid:48) − n + O (cid:18) ∆ µ (cid:19) (25)and in the opposite regime, I s,s (cid:48) n ≈ s (cid:48) ( − ) n s (cid:48) − n + O (cid:16) µ ∆ (cid:17) . (26)According to (22)-(24), these expansions show that theresulting potentials in space are represented by bandedmatrices, which we proceed to display as densities with-out approximations in Fig. 3. The numerical evalua-tion of matrix elements shows that a finite number ofneighbors is a reasonable approximation. For second- andthird-nearest-neighbor interactions, we depict the result-ing localized arrays in Fig. 2. FIG. 3. Top: Matrix form of the nonlocal complete polar-izer potential. The interaction zone contains different on-siteenergies indicated by the alternating pixel intensities in thediagonal. Bottom: Matrix form of a geometrical polarizerpotential with range ρ = 10 and no on-site potential. Onlycouplings to first and second order have been included. Bothpotentials are given in units of ∆. III. DYNAMICAL STUDY
In this section, we shall study two different phenom-ena. The first is a “free-particle” effect:
Zitterbewe-gung . Since its proposal by Schr¨odinger,
Zitterbewegung has been understood as a rapid oscillatory motion thatis a product of the interference between positive- andnegative-energy states present in the initial compositionof a Dirac spinor. For this oscillatory phenomena to beobserved, these positive- and negative-energy states musthave a sufficiently large overlap in position space. Thishas, in fact, been emulated in other experimental realiza-tions of the Dirac equation [26, 30, 31]. In this work, wedevelop a clean derivation that will allow us to make astationary phase approximation leading to a (cid:112) /t decayof the oscillatory part of the amplitude. In addition, weshall consider the effect of a designed potential that canspatially separate efficiently a function in its “big” and“small” contribution. The efficiency of the splitter shallbe characterized by means of reflection and transmissioncoefficients for each spin component. A. Wave packet dynamics
Zitterbewegung is the hallmark of unpolarized beams.Effective relativistic wave equations produce oscillatoryphenomena in the evolution of single-component spinorson the lattice [26]. At the heart of this effect lies theFW picture and the corresponding rotated quasispin: anobservable associated to upper and lower energy bands.The outcome of the evolution will be a superposition of”particles” and ”antiparticles” as long as the initial con-dition is a mixture of such quantum number. An obviousimplication is that
Zitterbewegung should be present inany theory with binary lattices. Noteworthy is the factthat the approximation of Bloch momenta around Diracpoints is not the essential ingredient; we may find
Zitter-bewegung in situations where the initial wave packet isa superposition of all energies in both bands, with non-negligible momentum components. We proceed to ana-lyze such physical situations.Setting (cid:126) = 1, we define the initial wave packet as | ψ (cid:105) = (cid:90) π dk (cid:88) s = ± ψ k,s | k, s (cid:105) = (cid:88) n ψ n | n (cid:105) . (27)We are interested in the average position at time t . Inorder to recover the usual definition of position x andmomentum p = − i∂/∂x in the continuous limit, wework with position operators defined over dimers (pairsof sites) and lattice constant a , X = a (cid:88) n even n [ | n (cid:105)(cid:104) n | + | n + 1 (cid:105)(cid:104) n + 1 | ] (28)with the property (cid:2) T , X (cid:3) = − aT , [ X, σ ± ] = 0 (29)(note though that it is the operator T and not T thatsatisfies this property). In the Heisenberg picture, weobtain ˙ X = aσ , ˙ Π = 0 , (30)which leads to X ( t ) = X (0) − at ∆Π H + a (cid:20) σ (0) − H (cid:21) (cid:90) t dt e − itH . (31)The first two terms describe the usual classical dynamicsfor a free particle, while the oscillations (i.e., the Zitter-bewegung ) come from the third term. The relevant partof the expectation value with respect to the state | ψ (cid:105) isthus x zitt ≡ (cid:28) (cid:20) σ (0) − H (cid:21) (cid:90) t dt e − itH (cid:29) ψ . (32) After inserting energy kets (8) and performing the timeintegral, we can write x zitt = (cid:88) s,s (cid:48) J s,s (cid:48) + (cid:88) s I s (33)where I and J are Bloch-momentum integrals of the type J s,s (cid:48) ≡ (cid:90) π dk e − iE k,s t sin( E k,s t ) E k,s ψ k,s ψ ∗ k,s (cid:48) × i (cid:104) u + k,s ( u − k,s (cid:48) ) ∗ − u − k,s ( u + k,s (cid:48) ) ∗ (cid:105) (34)and I s ≡ (cid:90) π dk e − iE k,s t sin( E k,s t ) sin k ( E k,s ) | ψ k,s | . (35)These integrals can be estimated in a long-time regimeusing the stationary phase approximation, where thestationary points are approximately determined by dE k,s /dk = 0, i.e., k = 0 , π/ , π . Since our descrip-tion involves only 0 < k < π , we see that two stationarypoints lie at the edge of the interval, and therefore theircontribution appears with a factor of 1 /
2. On the otherhand, the midpoint k = π/ µ (cid:54) = 0. From (34) and (35), we see that x zitt containsterms with a time dependence of the form e iωt (cid:112) /t , afterapplying the stationary phase approximation. Therefore,the frequencies of oscillation take the values ω = ± µ (from k = π/
2) and ω = ± (cid:112) + µ (from k = 0 , π ),while the effect vanishes with an envelope curve (cid:112) /t .We have an expression of the form x zitt ≈ (cid:114) t (cid:2) A ( µ, ∆) e − iω t + B ( µ, ∆) e − iω t + c.c. (cid:3) , (36)where A and B are coefficients related to second deriva-tives of the phase in (34) and (35). In Fig. 4, we describethe oscillations of x zitt in log scale, showing clearly an en-velope (cid:112) /t for long times. B. The potential as a beam splitter
We prepare wave packets with an adjustable widthand a proper ”thrust” or ”kick” by means of an ad-ditional plane-wave factor, imprinting an average drift.Our choice corresponds to motion from left to right.Eventually, our packets reach the gate keeper centeredat the origin described in Fig. 3, but before they do so,
Zitterbewegung is significantly observed. After the pack-ets collide with the potential, positive-energy componentsare reflected and negative-energy components are trans-mitted. This type of behavior has been verified numer-ically with specific wave packets, as we discuss now. InFig. 5, we plot the full probability density in black, upper
FIG. 4.
Zitterbewegung of the wave packet. On the left panel,we see the oscillations of x zitt without ballistic motion, aswell as the decay of amplitude predicted by stationary phaseapproximations. On the right panel, we see the same rate ofamplitude decay for three different effective masses, µ . ∆ x is the averaged maximum amplitude of the oscillations, while τ = t/T χ and T χ is the characteristic time of the simulationgiven by T χ = (cid:126) / ∆. spin component in blue, and lower spin in orange. Thedynamics is described in three steps: the first column cor-responds to times before the collision with the polarizer,the second column shows the interference produced bythe collision, and the third column finally demonstrateshow the components of the wave packet are separatedafter the collision. Upper spin is reflected and lower spinis transmitted. To make a quantitative analysis in termsof probabilities, first we define the initial wave packet as ψ n (0) = α N − P − e − an / λ e iκn + β N + P + e − an / λ e − iκn , (37)where | α | + | β | = 1, λ is the width of the discreteprobability density, and κ is the average momentum ofthe packet. P ± are the projectors onto each energy bandgiven by P s = (cid:90) π dk | k, s (cid:105)(cid:104) k, s | = U F W (cid:18) + sσ (cid:19) U † F W . (38)The matrix elements of these projectors are used afterthe scattering event takes place in the simulation, in or-der to test the sign of the spin. The results in Fig. 6 showthat after our Stern-Gerlach apparatus has done its job,only 1 .
2% of the upper spin component and 100% of thelower spin component have been transmitted. The wavepacket moving to the right still exhibits a slight hint of
Zitterbewegung as it is a mixture of components, whilethe wave packet moving to the left propagates without
Zitterbewegung , as it is only comprised by the remainder98 .
8% of the upper spin component. This quantitativeanalysis requires the reflection capacity of the upper spincomponent, denoted by R + , and transmission capacity ofthe lower spin component, T − , of the polarizer for differ-ent values of the thrust κ and the range of the polarizer ρ . These quantities are given by R + = | P + | ψ ( t ) (cid:105)| l | α | , T + = | P + | ψ ( t ) (cid:105)| r | α | ,R − = | P − | ψ ( t ) (cid:105)| l | β | , T − = | P − | ψ ( t ) (cid:105)| r | β | , (39)where subscripts l, r stand for sums over sites to the leftand right of the polarizer location, respectively. Due tocomplementarity, R + + T + = 1 and R − + T − = 1, so T + and R − are redundant. The results for R + , T − areshown in Fig. 6. When ρ is varied, both capacities re-tain near optimal values and fall to zero only for smallpolarizer sizes, as expected. Since the ”kick” is a prop-erty of the wave packet—i.e., external to the structureof the polarizer—the capacities are expected to remaininvariant for different values of κ . This is confirmed inour simulations, except for values near κ = 0 , π/ σ ) ⊗ V could be modi-fied with more refined constructions, even with transpar-ent potentials previously designed using supersymmetricmethods [32].We would like to point out that the inset in Fig. 6shows the reflection R + rising up very close to 1 for val-ues of κ > π/ π/ V , which blocks incident beams withincreasing efficiency as long as κ does not correspond toa Ramsauer resonance. C. A purely geometric beam splitter
Engineering beam splitters by means of nonlocal po-tentials include the possibility of removing all diagonalcontributions in V , in favor of the off-diagonal elementsrepresenting interactions to a certain range, as shown inFig. 3 (our approximations may include nearest neigh-bors, next-to-nearest neighbors, and so on). In the ex-perimental setup to be described in later sections, thecouplings can be determined by proximity between sites.With this technique, we can control the interaction range,as well as the zone where it operates, only using latticedeformations. Wave-packet evolution is studied numer-ically in this extreme situation and our results show asurprisingly efficient separation of components. In par-ticular, for a ρ = 10 polarizer with couplings to second-order neighbors, we see a reflection of 67 .
9% of the upperspin component and a transmission of 92 .
6% of the lowerspin component.
FIG. 5. (Color online) Evolution of a wave packet going through the lattice polarizer. The first picture shows the initialcondition of the complete wave packet, whereas the second and third pictures portray the dynamics of the upper and lowerband components of the wave packet. The collision time with the polarizer is T c = (cid:126) N κ , where N is the number of sites on thelattice.FIG. 6. (Color online) The dotted line represents the reflec-tion coefficient for the upper band component of the wavepacket as a function of the variables κ and the polarizer size ρ . The continuous line represents the transmission coefficientfor the lower band component of the wave packet as a functionof the same variables. IV. FEASIBILITY
In this section, we test the robustness of the splitterwith respect to the known experimental limitations. Inthe splitter, three parameters must be controlled: theoverall absorption, the on-site energy, and the couplingterms. This analysis will not include the overall absorp-tion because it mainly affects the width and height of theresonances without significantly disturbing the spectralpositions; therefore, it is expected that the transmissionand reflection coefficients decrease by a factor related tothe strength of the absorption.Experiments show [9] that the on-site energy can be con-trolled better than the coupling. This is the case formicrowave experiments since the variation of couplingsis at least two orders of magnitude greater than the vari-ation of the on-site energy. Thus, to estimate the robust-ness of the splitter, we will consider a Gaussian disor-der introduced randomly on the couplings. We modify∆ → (1 − δ )∆, where δ is a random variable with a stan- ρ / = κ = - - - - - - - - - - - - - - - - log ( σ δ ) FIG. 7. (Color online) Mean reflection (red) and transmis-sion (blue) coefficients as functions of the standard deviationof the coupling σ δ (see text), for a ρ = 600 splitter. The errorbars represent the fluctuations obtained from multiple real-izations. The value κ = 0 . dard deviation σ δ . Figure 7 shows that the expected co-efficients and deviations are satisfactory, even for a poorcoupling control ( σ δ ∼ . σ δ (cid:29) .
1) destroys the efficiencyof the splitter, with the latter becoming a regular wall un-able to separate the upper and lower band components.Thus, we have shown robustness and feasibility in labo-ratory implementations.
V. EXPERIMENTAL PROPOSALS
In this section, we describe a realization of the split-ter through a microwave cavity containing a set of cylin-drical resonators between parallel plates, establishing atight-binding configuration. This type of experimentalimplementation has been very useful for the emulation ofDirac equations [18], graphenelike structures [6–9], chi-ral states [33], and anomalous Anderson localization [34],among others. It is important to mention that the follow-ing experimental proposal is not unique since the split-ter can also be achieved by plasmonic circuits [10], opti-cal waveguides [35], or acoustic waves [36]. The readercan notice that these implementations rely on classicalaspects of the systems mentioned. However, the equa-tions of motion are equivalent to, say, the Schr¨odinger orDirac’s equation, depending on the regime studied. Inthis sense, we are emulating
Dirac’s equation.We show in further detail how to produce complex cou-pling constants with the aim of fabricating purely geo-metric beam splitters. The effect, important in its ownright, rests on the possibility of breaking the chiral sym-metry of polygonal geometries using dimers as individualsites. This opens the possibility of producing directedcouplings, emerging from dimeric states.
A. Experimental specifications
A set of cylindrical dielectric disks can act as the sitesof the chain, for example, Temex-Ceramics disks, E2000series, with high dielectric permittivity ( (cid:15) = 37) and lowloss (quality factor Q = 7000). Each disk has an isolatedresonance defined by the dimensions of the cylinder, e.g.,for a height of 5mm and a radius of 4mm, a resonanceclose to 6.64 GHz appears corresponding to the lowesttransverse electric mode (TE ). This resonant frequencyis equivalent to the on-site energy. For purely geometricsplitters, we have seen that on-site energies are the samethroughout the array; therefore, identical dielectric disksmust be used. On the other hand, a general type of split-ter would require disks of different dimensions and/ordielectric constants.Between two parallel metallic plates, each isolated res-onance behaves like a J -Bessel function inside of a cylin-der, and as a K -Bessel function outside of it. The func-tion K can be represented fairly well by an exponentialtail as a function of the distance with respect to the cen-ter. Therefore, any set of disks interacts by proximitythrough the overlap of their individual functions K , insuch a way that the response of the whole set is well de-scribed by a tight-binding model. The intensity of theinteraction and the main contribution of first and secondneighbors can be further manipulated by changing thedistance between the plates [19].It is possible to study the wave dynamics of the splitterby introducing two antennas into the microwave cavityconnected to different ports of a vector network analyzer(VNA). It is possible to measure both the spectrum andthe intensity of the wave functions by using only oneprobing antenna. However, for the reconstruction of wavepacket dynamics, it is necessary not only to measure theintensity but also the phase. Hence a second antennaprobing the transmission of the system is mandatory.We fix one of the antennas near to a disk wherebythe electromagnetic waves are injected, while the positionof the other antenna is varied throughout the structure,allowing one to measure the transmission spectrum on θgdf FIG. 8. Configuration of disks (indicated as circles, witha number) giving rise to the coupling structure specified byEq. (43). The six disks are organized in pairs (1 , , , d . The inner disks interact via the coupling constant f . The outer disks interact with just one of the other fourdisks (for example, 2 with 3), as the others remain screenedgeometrically. The C v symmetry is broken by tilting theouter disks with an angle θ . θ c
126 45 3 fd FIG. 9. Two particular realizations that keep the full C v symmetry are illustrated, one in which θ = θ c , and the otherfor θ = 0 in which screening sets g = 0. each disk.The evolution of the wave packet at each point of thestructure is reconstructed through a Fourier transformof the measured spectrum at that point [37]. This isallowed because we have access to the full spectrum ofthe complex transmission. B. Negative couplings and level inversion
In our purely geometric splitter, we find matrix ele-ments that are real but not positive; see, e.g., Fig. 3.Negative couplings require the control of an extra de-gree of freedom in the form of a phase factor. We showthat indeed such phases can be produced by adding morestructure in our arrays. It is worth mentioning that non-removable phase factors in hopping amplitudes are theequivalent of magnetic fields applied to charged particles[38], but our goal is to emulate these effects for a scalar frequency (GHz) (cid:1) ( d e g )
FIG. 10. Spectrum of the configuration shown in Fig. 8;dots correspond to a full 3D simulation of a microwave cav-ity using COMSOL 5.2 and continuous lines correspond totight-binding calculations. The lower band shows the desiredinversion level due to effective negative coupling. wave.First we note that any Hermitian matrix H can berewritten as a matrix with semipositive secondary diag-onals by means of a unitary transformation. We proceedto turn H into a purely positive nearest-neighbor array.Consider U sign = diag (cid:8) e − i ∆ n (cid:9) , (40)where ∆ n = (cid:80) m 00 0 d g f f g dg d . (43)The spectrum contains two degenerate doublets and twosinglets. Moreover, their eigenfrequencies are symmetri-cally disposed around E . In essence, we have producedan additional inverted copy of the spectrum due to asplitting caused by strong intradimer coupling. For di-electric disks, a numerical simulation of Maxwell equa-tions with space-dependent dielectric functions has beenrun. The results in Fig. 10 show that the inverted copycorresponds to eigenfrequencies sitting to the left of theoriginal isolated resonance at E . Moreover, this occursonly for θ > θ c ∼ 78 deg, which establishes the exis-tence of a diabolic (crossing) point in the spectrum [39].Transverse modes are shown in Fig. 11, where the panelsexhibit a change in the sign of the wave function insideat least one dimer, due to the transition at θ c .Finally, our results show that the assembled structureof alternating triangles must produce two bands open-ing around each level of a single dimer: we may chooseto work in one or the other. A similar spectral struc-ture has been achieved in other contexts: nuclear res-onances [40], flat microwave cavities [41, 42], and elec-tronic circuits [43]. VI. CONCLUSION AND OUTLOOK In this paper, we have studied a tight-binding modelthat is described by a Dirac equation. We have fo-cused on the time-dependent dynamics in the positive-and negative-energy bands. In the language of the Diracequation, this corresponds to particles and antiparticles.We have developed the theory that allows one to splitthese components by means of a localized potential; thisin turn could be a first step towards the actual measure-ment of quasispin using wave packets. We have furthershown that even though the interactions are long ranged,taking as few as next-to-nearest-neighbor interactions, ina very localized region in space, yields reasonable results.In connection with the possibility of generating pure spinwaves with our splitter, we would like to add that waveswith vanishing average momentum have been achievedand that quasispin can be indeed spatially transported.However, the mechanism relies on deformations ratherthan the application of external magnetic fields as in theusual case of spin. The local nature of the interaction ishighly desirable if an experimental emulation is pursued.We have indeed explored such scenario in the context ofa bidimensional array of dielectrics in a microwave cav-ity. In such an array, it has been necessary to consider level inversion, which we have demonstrated using a sim-ple geometric array. The next obvious step would be tocarry out the experiment. ACKNOWLEDGMENTS Financial support from CONACyT under Projects CBNo. 2012-180585 and No. 153190 and UNAM-PAPIITIN111015 is acknowledged. We are grateful to LNS-BUAP for allowing extensive use of their supercomputingfacility. Appendix: An alternative splitter A simple alternative splitter can be designed if we re-place Eq. (17) by V ± ( x ) = (cid:40) V ( x ) for particles − V ( x ) for holes . (A.1)Then, Hamiltonian (18) would be replaced by˜ H FW = H FW + σ ⊗ V FW . (A.2)In formula (20), one would need an extra term, V = e − iφ/ cos (cid:18) θ (cid:19) V FW cos (cid:18) θ (cid:19) e iφ/ + e − iφ/ sin (cid:18) θ (cid:19) V FW sin (cid:18) θ (cid:19) e iφ/ , which leads to the following changes in the matrix ele-ments: (cid:104) n | V | n (cid:48) (cid:105) = 18 π ∞ (cid:88) m = −∞ V FW ( m ) × (cid:104) I ++ n (cid:48) − m (cid:0) I ++ n − m (cid:1) ∗ + I −− n (cid:48) − m (cid:0) I −− n − m (cid:1) ∗ (cid:105) , (A.3)for even n and n (cid:48) , (cid:104) n | V | n (cid:48) (cid:105) = 18 π ∞ (cid:88) m = −∞ V FW ( m ) × (cid:104) I −− n (cid:48) − m (cid:0) I + − n − m (cid:1) ∗ − I ++ n (cid:48) − m (cid:0) I − + n − m (cid:1) ∗ (cid:105) , (A.4)for even n and odd n (cid:48) , and, finally, (cid:104) n | V | n (cid:48) (cid:105) = 18 π ∞ (cid:88) m = −∞ V FW ( m ) × (cid:104) I − + n (cid:48) − m (cid:0) I − + n − m (cid:1) ∗ + I + − n (cid:48) − m (cid:0) I + − n − m (cid:1) ∗ (cid:105) , (A.5)for odd n and n (cid:48) .1 [1] O. Morsch and M. K. Oberthaler, Rev. Mod. Phys. ,179 (2006).[2] I. Bloch, Nat. Phys. , 23 (2005).[3] M. K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmied-mayer, and A. Zeilinger, Phys. Rev. Lett. , 4980(1996).[4] T. Uehlinger, G. Jotzu, M. Messer, D. Greif, W. Hofstet-ter, U. Bissbort, and T. Esslinger, Phys. Rev. Lett. ,185307 (2013).[5] J. Struck, C. ¨Olschl¨ager, M. Weinberg, P. Hauke, J. Si-monet, A. Eckardt, M. Lewenstein, K. Sengstock, andP. Windpassinger, Phys. Rev. Lett. , 225304 (2012).[6] U. Kuhl, S. Barkhofen, T. Tudorovskiy, H.-J. St¨ockmann,T. Hosain, L. de Forges de Parny, and F. Mortessagne,Phys. Rev. B , 094308 (2010).[7] S. Barkhofen, M. Bellec, U. Kuhl, and F. Mortessagne,Phys. Rev. B , 035101 (2013).[8] S. Bittner, B. Dietz, M. Miski-Oglu, P. Oria-Iriarte,A. Richter, and F. Sch¨afer, Phys. Rev. B , 014301(2010).[9] M. Bellec, U. Kuhl, G. Montambaux, and F. Mortes-sagne, Phys. Rev. B , 115437 (2013).[10] A. J. Mart´ınez-Galera, I. Brihuega, A. Guti´errez-Rubio,T. Stauber, and J. M. G´omez-Rodr´ıguez, Sci. Rep. ,7314 (2014), arXiv:1411.5805 [cond-mat.mes-hall].[11] M. Polini, F. Guinea, M. Lewenstein, H. C. Manoharan,and V. Pellegrini, Nat. Nanotechnol. , 625 (2013).[12] K. K. Gomes, W. Mar, W. Ko, F. Guinea, and H. C.Manoharan, Nature , 306 (2012).[13] G. W. Semenoff, Phys. Rev. Lett. , 2449 (1984).[14] A. K. Geim and K. S. Novoselov, Nat. Mater. , 183(2007).[15] A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S.Novoselov, and A. K. Geim, Rev. Mod. Phys. , 109(2009).[16] G. Roati, M. Zaccanti, C. D’Errico, J. Catani, M. Mod-ugno, A. Simoni, M. Inguscio, and G. Modugno, Phys.Rev. Lett. , 010403 (2007).[17] L. Fallani, J. E. Lye, V. Guarrera, C. Fort, and M. In-guscio, Phys. Rev. Lett. , 130404 (2007).[18] J. A. Franco-Villafa˜ne, E. Sadurn´ı, S. Barkhofen,U. Kuhl, F. Mortessagne, and T. H. Seligman, Phys.Rev. Lett. , 170405 (2013).[19] E. Sadurn´ı, J. A. Franco-Villafa˜ne, U. Kuhl, F. Mortes-sagne, and T. H. Seligman, New J. Phys. , 123014(2013). [20] E. Sadurn´ı, T. H. Seligman, and F. Mortessagne, NewJ. Phys. , 053014 (2010).[21] M. Mancini, G. Pagano, G. Cappellini, L. Livi, M. Rider,J. Catani, C. Sias, P. Zoller, M. Inguscio, M. Dalmonte,and L. Fallani, Science , 1510 (2015).[22] F. Wilczek, Nat. Phys. , 614 (2009).[23] J. Alicea, Rep. Prog. Phys. , 076501 (2012).[24] V. Kagalovsky, B. Horovitz, Y. Avishai, and J. T.Chalker, Phys. Rev. Lett. , 3516 (1999).[25] D. Bernard and A. LeClair, Phys. Rev. B , 045306(2001).[26] F. Dreisow, M. Heinrich, R. Keil, A. T¨unnermann,S. Nolte, S. Longhi, and A. Szameit, Phys. Rev. Lett. (2010).[27] L. Meier, G. Salis, I. Shorubalko, E. Gini, S. Sch¨on, andK. Ensslin, Nat. Phys. , 650 (2007).[28] L. L. Foldy and S. A. Wouthuysen, Phys. Rev. , 29(1950).[29] E. D. Vries, Fortschr. Phys. , 149 (1970).[30] T. M. Rusin and W. Zawadzki, Phys. Rev. B , 195439(2007).[31] R. Gerritsma, G. Kirchmair, F. Zahringer, E. Solano,R. Blatt, and C. F. Roos, Nature , 68 (2010).[32] E. Sadurn´ı, Phys. Rev. E , 033205 (2014).[33] C. Dembowski, B. Dietz, H.-D. Gr¨af, H. L. Harney,A. Heine, W. D. Heiss, and A. Richter, Phys. Rev. Lett. (2003).[34] A. A. Fern´andez-Mar´ın, J. A. M´endez-Berm´udez, J. Car-bonell, F. Cervera, J. S´anchez-Dehesa, and V. A. Gopar,Phys. Rev. Lett. , 233901 (2014).[35] S. Longhi, Laser & Photon. Rev. , 243 (2009).[36] J. D. Maynard, Rev. Mod. Phys. , 401–417 (2001).[37] J. B¨ohm, M. Bellec, F. Mortessagne, U. Kuhl,S. Barkhofen, S. Gehler, H.-J. St¨ockmann, I. Foulger,S. Gnutzmann, and G. Tanner, Phys. Rev. Lett. (2015).[38] D. R. Hofstadter, Phys. Rev. B , 2239 (1976).[39] M. V. Berry, Proc. R. Soc. Lond. A , 45 (1984).[40] P. von Brentano and M. Philipp, Phys. Lett. B , 171(1999).[41] C. Dembowski, H.-D. Gr¨af, H. L. Harney, A. Heine,W. D. Heiss, H. Rehfeld, and A. Richter, Phys. Rev.Lett. , 787 (2001).[42] S. Bittner, B. Dietz, H. L. Harney, M. Miski-Oglu,A. Richter, and F. Sch¨afer, Phys. Rev. E (2014).[43] T. Stehmann, W. D. Heiss, and F. G. Scholtz, J. Phys.A37