Unconventional level attraction in cavity axion polariton of antiferromagnetic topological insulator
Yang Xiao, Huaiqiang Wang, Dinghui Wang, Ruifeng Lu, Xiaohong Yan, Hong Guo, C. -M. Hu, Ke Xia, Haijun Zhang, Dingyu Xing
aa r X i v : . [ c ond - m a t . m e s - h a ll ] N ov Unconventional level attraction in cavity axion polariton of antiferromagnetictopological insulator
Yang Xiao , ∗ , Huaiqiang Wang , , ∗ , Dinghui Wang , Ruifeng Lu , XiaohongYan , Hong Guo , C. -M. Hu , Ke Xia , Haijun Zhang , , † and Dingyu Xing , College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China National Laboratory of Solid State Microstructures and Physics School, Nanjing University, Nanjing 210093, China Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China Department of Applied Physics, Nanjing University of Science and Technology, Nanjing 210094, China School of Material Science and Engineering, Jiangsu University, Zhenjiang, 212013, China Department of Physics, McGill University, Montreal, Quebec H3A 2T8, Canada Department of Physics and Astronomy, University of Manitoba, Winnipeg R3T 2N2, Canada and Beijing Computational Science Research Center, Beijing 100193, China ∗ Strong coupling between cavity photons and various excitations in condensed matters boosts thefield of light-matter interaction and generates several exciting sub-fields, such as cavity optome-chanics and cavity magnon polariton. Axion quasiparticles, emerging in topological insulators, werepredicted to strongly couple with the light and generate the so-called axion polariton. Here, wedemonstrate that there arises a gapless level attraction in cavity axion polariton of antiferromag-netic topological insulators, which originates from a nonlinear interaction between axion and theodd-order resonance of cavity. Such a novel level attraction is essentially different from conventionallevel attractions with the mechanism of either a linear coupling or a dissipation-mediated interaction,and also different from the level repulsion induced by the strong coupling in common polaritons.Our results reveal a new mechanism of level attractions, and open up new roads for exploring theaxion polariton with cavity technologies. They have potential applications for quantum informationand dark matter research.
Introduction.
Axion was first postulated as an el-ementary particle to solve the charge-parity puzzle inthe strong interaction between quarks in the particlephysics[1]. Now, axion is also considered as a dark mattercandidate of the universe[2]. But its existence in natureis another puzzle. Interestingly, axion emerges as a quasi-particle in three-dimensional (3D) topological insulatorsthrough the axion action S θ = ( θ/ π )( e /hc ) R d xdt E · B from the topological field theory[3, 4], in which E and B are the electromagnetic fields inside the insulators, e isthe charge of an electron, h is Plank’s constant, θ (mod-ulo 2 π ) is the dimensionless pseudoscalar parameter asthe axion field. The axion field θ can only be 0 for trivialinsulators and π for topological insulators because it isodd under the time reversal symmetry T or the inver-sion symmetry P . In antiferromagnetic (AFM) insula-tors, once both T and P are broken, θ can deviate fromthe quantized values (0 or π ) and becomes dynamicaldue to the spin-wave excitation, denoted as a dynamicalaxion field (DAF) θ ( r , t ) = θ + δθ ( r , t )[5–7].In order to describe the electromagnetic response of3D topological insulators, the axion action should beincluded to the conventional Maxwell’s action S =(8 /π ) R d xdt ( ǫ E − (1 /µ ) B ) to obtain a set of mod-ified Maxwell’s equations[8], which indicates DAF natu-rally couples with the light. Many exotic electromagneticphenomena were predicted for DAF systems, such as, ax-ion polariton[5], the chiral magnetic effect[9–11], and soon[12–15]. Interestingly, it was also proposed to detectaxion dark matter in the universe[16]. But most of themare theoretical studies due to the lack of realistic DAF materials. Recently, several topological dynamical axioninsulators (DAIs) were proposed to host large DAF pro-tected by nonzero spin Chern number[17] and stronglycouple with the light, for example, van der Waals(vdW)layered material Mn Bi Te [7] and MnBi Te /Bi Te superlattice[17]. Especially, some of us predicted tun-able DAF in MnBi Te films[18] which has been success-fully synthesized in experiments[19–25]. Detailed discus-sions on axion materials can also be found in a recentreview[26]. Therefore, the cavity with embedded DAIcan provide an important platform to study cavity axionpolariton induced by the strong coupling between DAFand cavity photon and develop potential applications.Before turning to cavity axion polariton, we first in-troduce generic concepts of the level repulsion (LR) andlevel attraction (LA). As two harmonic oscillators are co-herently coupled with strength g , the eigenfrequencies ofhybridized modes are repelled with the gap 2 g at reso-nance, which is known as the LR[27–31], schematicallyshown in Fig. 1a. When the coupling is of dissipativenature[32–36], a LA may occur, schematically shown inFig. 1b. The LA spectrum can be generally written as, ω ± = ( ω a + ω b ) ± p ( ω a − ω b ) − g ( ω ) , (1)where ω a and ω b are frequencies of two oscillators a and b , and g ( ω ) denotes the coupling strength. Once | g ( ω ) | > | ω a − ω b | , the square root gives a pure imaginaryvalue, causing that the real parts of complex frequenciesin Eq. (1) are degenerate, i.e. the attraction of mode fre-quencies. g ( ω ) is frequency-independent in most cases, Even modeOdd mode DAI
L l s S Trivial Non-trivialTopological phase transition (M) A x i on - pho t on c oup li ng ( m e V ) FIG. 1:
Schematic of level repulsion (LR), level attraction (LA), and photon cavity. a-c , Conventional LR ( a ),conventional LA ( b ) and unconventional LA ( c ) depicted with solid red lines. The dashed blue lines generally present twouncoupled modes. d , Schematic of photon cavity geometry with a dynamical axion insulator (DAI) film. The photon propagatesfrom the left port to the right port through the DAI film. Depending on the mode number, the cavity resonance mode canbe the even mode or the odd mode. As for the odd mode, the electric field at the DAI is negligibly small, indicating that thephoton-axion coupling would be dominated by the nonlinear interaction. But for the even mode, a large electric field can lead toa linear photon-axion coupling. The static magnetic field is applied with an angle ϕ with respect to the z axis and induces thecoupling between the electric field E of photon and dynamical axion field (DAF) δθ . e , The schematic of the coupling strengthbetween axion mode and the photon mode as a function of the mass parameter M , distinguishing between topologically trivial( M <
0) and nontrivial (
M >
0) DAI. but it may be frequency dependent, for example, in oursystem discussed below. In principle, a LA is attributedto various mechanisms, and it has been observed in afew quantum physics, e.g., cavity optomechanics[33] andcavity magnon polariton[34, 35, 37], as well as in classicphysics[38], e.g. coupled piano strings. In cavity optome-chanics, the Stokes process of photon-phonon interactionresults in mechanical amplification and a LA. In cavitymagnon polariton, the dissipation mediates the photon-magnon coupling and explicitly induces a LA. Moreover,the phase manipulation of microwave photons can inducea LA in photon-magnon systems as well[39, 40].In this article, we studied the cavity axion polaritonof DAIs and proposed an unconventional LA with a newmechanism. It is achieved by a nonlinear-type strongcoupling between DAF and cavity photon. As the cou-pling strength is greatly enhanced near the topologicalphase transition, a giant nonlinear axion-photon interac-tion is induced, producing this unconventional LA andhigh-order modes. Though the dissipation effect is notthe key to producing the LA, it induces a gap of the LA.
Cavity axion polariton.
The coupling between pho-ton and DAF in AFM topological insulators can be de- scribed by the effective action S tot = S Maxwell + S topo + S axion [5] (see Supplementary Materials for the details).The dynamics of the coupled DAF and electromagneticfield can be obtained as, ∂ δθ∂t − v ▽ δθ + m δθ + Γ ∂δθ∂t − α B π g J E = 0 , (2) ∂ E ∂t − c ▽ E + α B πǫ ∂ δθ∂t = 0 , (3)where c = c / √ ǫµ with c the speed of light in the vac-uum, ǫ and µ represent dielectric constant and magneticpermeability, α = e / ~ c is the fine-structure constant,and δθ represents the massive DAF originating from spin-wave mode with material-dependent stiffness J , velocity v i , mass m , and coefficient g defined in Ref. [5]. v i = v is assumed to be isotropic, and Γ presents the decay rateof the axion mode.Furthermore, in the presence of an external uniformmagnetic field B , the coupling strength between the ax-ion mode and the electric field E of the photon can bederived as b = p α B x / π ǫg J , depending on the x component of B . As an illustration, we picked repre-sentative parameters B within (0 − ǫ ∼
20 for (a) (b) (c)(d) (e) (f) | S || S | GG nn B (T) B (T) FIG. 2: | S | transmission spectra. a , | S | transmission spectra of the even mode as a function of the static magnetic fieldand the photon frequency, which is calculated based on Eq. (4) for the DAI film thickness l s = 0 . µm . The strong couplingmanifests the anticrossing gap. b , Spectrum at resonant magnetic field represented by the white dashed line in ( a ). c , Absolutevalue of the nth − order terms in G. Here, | S | is formally written as F/G, and G is Taylor-expanded into nth − order terms. Asfor the even mode, we can see that the linear term ( n =1) dominates. d-f , The same as ( a - c ) but for the odd mode. In orderto avoid divergence, a small imaginary part is added into the frequency ω in Eq. (5). As for the odd mode, the linear termof type-I interaction (represented by ∆ ω = 0) is vanishing. This results in the fact that the high-order terms dominate type-Iinteraction and generate high-order modes. Moreover, the linear term and second-order term of type-II interaction (representedby ∆ ω = 1) dominate and give rise to Eq. (5). AFM insulators and the coupling strength b as a func-tion of the mass term M is schematically shown in Fig.1e. It is distinguished between topologically nontrivial( M >
0) and trivial (
M <
0) DAIs. The magnitude ofthe coupling strength b in the nontrivial regime is signif-icantly larger than that in the trivial regime. The max-imum b can be obtained near the topological transition( M ∼ δM z , we consider parallel pumping of antiferro-magnet inside a cavity[41]. As shown in Fig. 1d, a DAI isplaced in the middle of one-dimensional cylindrical cavitywith input and output ports at the left and right ends.The electromagnetic wave propagates along the y axis,the magnetization is aligned along the z axis but thestatic magnetic field is oriented with the angle ϕ withrespect to the z axis. In this setup, the z magnetic com-ponent B z of the electromagnetic wave induces a periodic oscillation of magnetization δM z and thereby establishesa DAF. The x component of external magnetic field B couples with the electric field E x and thus contributes tothe coupling between the electric field and DAF. Suchlongitudinal excitation of magnetization is usually re-ferred to parallel pumping. Since parallel pumping is aprocess that one photon splits into two magnon with thesame frequency, we take m = 2 ω m with ω m = ω ± γB z as the AFM resonance frequency, γ is the the gyromag-netic ratio and B z is the z component of B .Combining Eqs. (2) and (3) gives rise to the propaga-tion state inside the DAI. With the wave vector k , onecan directly obtain the transfer matrix that connects theelectromagnetic field at input port and output port. Asfor the finite-velocity DAF, the boundary condition needsto be further considered. The scattering matrix and con-tinuity condition at input and output ports are usedto calculate transmission coefficient S . The detailedmethods can been seen in Supplementary Material. Forour calculations, the axion frequency is m = 2 .
07 meV (a) (b) (c)
R=0.95R=0.97R=0.99 F2F1S | S | | S | B (T) FIG. 3:
Dissipation effect. a , | S | spectra of the odd mode calculated with full-wave numeric method (without usingTaylor expansion) for the DAI film thickness l s = 0 . µm and the reflectivity R = 0 .
99 of the cavity. The reflectivity R ofelectromagnetic wave at two ports determines the decay rate of intracavity field. b , | S | spectra for the reflectivity R =0.95(black), 0.97 (blue), 0.99 (red) at the resonant magnetic field. As the reflectivity R increases, the dissipation is graduallysuppressed, and the gap (dip) width is enhanced little by little. But we notice that the position of the gap remains unchanged. c , The schematic mechanism of gaped LA. The approximated formula of | S | spectra is written as ( ω − m ′ ) / [( ω − ω ′ c )( ω − ω ′ )](S ,black), which can be considered as the combination of two functions 1 / [( ω − ω ′ c )( ω − ω ′ )] (F1, brown) and ( ω − m ′ ) (F2,green). Here, ‘ m ′ , ω ′ c and ω ′ ’ are renormalized by the dissipation. We can see that the function F2 (the axion mode) is thekey to opening the LA gap. as B =0, thus the cavity resonance frequency is about0.5 THz and the cavity length is taken to be L =5 mm .Depending on the mode number n , the resonance modesof cavity can be either the even mode or the odd mode atthe center of the cavity. Strong interaction between theaxion mode and the electric field E , as demonstrated inEqs. (2) and (3), generates LR between the axion modeand the even mode (see Fig. S1 in Supplementary Ma-terial). For the odd mode, our calculations surprisinglypresent an unconventional LA, seen in Fig. 2.In order to reveal the origin of such unexpected LA,we derive an analytic expression of transmission spec-trum S , written in the form of S = F/G (see Supple-mentary Material for details). The expression of S iscomposed of Taylor’s expansions around cavity resonancefrequency ω c and the dimension l s =0. The expansion re-sults in two types of interaction, i.e. pure axion interac-tion (called type-I) and mixed axion-photon interaction(type-II). For the even mode, the type-I interaction dom-inates and S can be written as S e = C e ( ω − m )( ω − ω c )( ω − m ) − η , (4)here, C e denotes a constant and η = ǫ r l s b / L − l s ). Itwell reproduces the characteristic LR (Fig. 2a and 2b),implying the linear coupling between the even mode andthe axion mode. This can be further confirmed in Fig.2c where the linear term is the largest among all termsof G in S .But for the odd mode, the the first- and second-orderterms of type-I interaction are vanishing (Fig. 2f). Asthe film thickness l s is small, the nonlinear type-II inter-action mainly contributes to S . Through simplifying the function, we reach a compact formula for the oddmode S o = C o ω − m ( ω − ω c )( ω − ω ) , (5)where C o also denotes a constant, ω = (4 m + λω c ) / (4 + λ ) and λ = ( L − l s ) l s ǫ s b / c . From Eq. (5), one can seethat there appears a new high-order mode with frequency ω . Since the high-order mode has a smaller slope thanthe axion mode, it bends to the cavity resonance with anattractive characteristics, shown in Fig. 2d. Moreover,the high-order mode ω reduces to the axion mode atresonance ( m = ω c ), reproducing S = C o / ( ω − ω c )shown in Fig. 2e. Therefore, the LA under this study isof both gapless and nonlinear nature. Dissipation effects.
The reflectivity R indicates thedecay rate (dissipation) of photons inside a cavity. Asmaller R represents a larger photon leakage (dissipa-tion). In Eq. (5) and Fig. 2, we assume the reflectivity R =1, implying no leakage of intracavity photons, and weobtained a gapless LA. Here, we further consider the dis-sipation ( R < S spectrum of theodd mode with R = 0 .
99 and a gap at resonance can beclearly seen, different from that gapless LA in Fig. 2d.Furthermore, a larger dissipation (a smaller R ) results ina wider gap, shown in Fig. 3b. To understand the behav-ior in Fig. 3, we deduce an analytic expression of S inthe presence of the dissipation. Interestingly, the new for-mula is of the same form as Eq. (5). The only differenceis that the dissipation of each mode is renormalized, i.e. ω ′ c = ω c − iη (1 − R ), ω ′ = ω − iζ (1 − R ) and m ′ = m − i Γ ω .Here, η and ζ are parameters the detailed forms of whichare given in Supplementary Materials. The new formula B (T) B (T) B (T) FIG. 4: | S | transmission spectra dependent on coupling strength and DAI film thickness. a,b,c , | S | spectra forthe DAI film thickness l s =0.1 µm with different coupling strength b = 0( a ), 1( b ) and 2( c ) meV. Controllable coupling strength b can be obtained through applying the external magnetic field in topologically nontrivial DAI. We can see that an enhancedgap in LA is induced by a strong coupling. d-f , The same as ( a - c ), but for a thicker DAI film l s =0.5 µm . Note that high-ordermodes arise in the spectrum for which the nonlinear interaction dominates. For all data here, the reflectivity R is fixed to be0.99. satisfactorily reproduces the gapped LA (see Supplemen-tary Materials), indicating that the gap originates fromthe dissipation effect. Further analysis (Fig. 3c) showsthat the R -dependent line-width of cavity resonance andhigh-order mode result in a gap at m , in contrast to thepeak (Fig. 2d) in the absence of the dissipation. There-fore, the dissipation is not the key to generating the non-linear LA, but induces a gap in the LA. Discussions.
The LA found in this work is essentiallydifferent from the conventional LA. First of all, the LAhere is attributed to a nonlinear interaction, but the con-ventional LA originates from the linear coupling of twomodes[32–35, 37, 39, 40]. Second, the dissipation is notthe essence for the LA in this work and it merely tunesthe LA gap of the axion polariton. Third, the axion modeunder this study arises from longitudinal magnetizationfluctuation, instead of transverse magnetization fluctua-tion in the case of cavity magnon polariton[42–48].To demonstrate the LA here is controllable, we varythe coupling strength b and the thickness l s of the DAIfilm. As shown in Fig. 1e, the topologically trivial DAIpossesses negligible axion-photon coupling strength andthus does not couple with cavity resonance (Fig. 4a).Around the topological transition, the coupling strengthquickly increases, resulting in a large gap of LA (Fig. 4b and 4c). This trend of increasing gap can also be obtainedthrough increasing the thickness l s . But, extra high-ordermodes may appear. As for a large l s case, the type-II in-teraction does not overwhelm the type-I interaction thatis absent for a small l s case. The third-order type-I termwhich reproduces LR, together with other terms, start tocontribute to S . The high-order terms imply more newmodes as shown in Fig. 4e and 4f. The competition be-tween type-I and type-II nonlinear interaction results inthe coexistence of LR and LA. Therefore, it is expectedto observe LA, LR and nonlinearity-induced high-ordermodes by tuning the topological phases ( M ) and the DAIdimension l s .To make possible experimental measurement, high-frequency cavity and high-quality DAI materials are im-portant. Recently, Scalari et al. reported the fabricationof µm -size electronic meta-material cavity with resonancefrequency up to 4 THz[49]. The ultra-strong couplingwith a low dissipation rate have been achieved, implyingthe potential application in cavity quantum electrody-namics. On the other hand, thanks to the fast-growingfield of MnBi Te -based magnetic topological materialsand state-of-art experimental techniques[19–25], it seemsquite promising that candidate materials hosting largetopological DAF, such as MnBi Te films, Mn Bi Te and MnBi Te /Bi Te superlattice proposed by some ofthe authors[18, 20], can be realized in the short futureand pave the way for the development of cavity axionploarition and potential applications in quantum infor-mation and discovery of dark matter in the universe. Acknowledgements.
This work is supported by theFundamental Research Funds for the Central Universities(Grant No. 020414380149), the Natural Science Founda-tion of China (Grants No. 61974067, No. 12074181, No.11674165 and No. 11834006), Natural Science Founda-tion of Jiangsu Province (No. BK20200007) and the FokYing-Tong Education Foundation of China (Grant No.161006). ∗ Electronic address: [email protected][1] R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. , 1440(1977).[2] J. Preskill, M. B. Wise, and F. Wilczek, Physics LettersB , 127 (1983), ISSN 0370-2693.[3] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. , 1057(2011).[4] X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B , 195424 (2008).[5] R. Li, J. Wang, X. L. Qi, and S. C. Zhang, Nat. Phys. ,284 (2010).[6] J. Wang, R. Li, S.-C. Zhang, and X.-L. Qi, Phys. Rev.Lett. , 126403 (2011).[7] J. Zhang, D. Wang, M. Shi, T. Zhu, H. Zhang, andJ. Wang, Chin. Phys. Lett. , 077304 (2020).[8] F. Wilczek, Phys. Rev. Lett. , 1799 (1987).[9] A. Sekine and K. 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