Effective spontaneous -symmetry breaking in hybridized metamaterials
We show that metamaterials can be used as a testing ground to investigate spontaneous symmetry breaking associated with non-Hermitian quantum systems. By exploring the interplay between near-field dipolar coupling and material absorption or gain, we demonstrate various spontaneous breaking processes of the -symmetry for a series of effective Hamiltonians associated to the scattering matrix. By tuning the coupling parameter, coherent perfect absorption, laser action and gain-induced complete reflection ( reflector) by using an ultra-thin metamaterial can be obtained. Moreover, an ideal -symmetry can be established effectively in a passive system by exploring the balance between scattering and absorption loss using metamaterials.
Metamaterials, periodic arrangements of artificial subwavelength atoms, have paved the way for realizing different novel concepts and applications in optics, such as negative refraction, sub-diffraction imaging and invisibility cloakingR01 (); R02 (); R03 (); R04 (). Instead of using the macroscopic material parameters for description, many of the optical properties, including metamaterial-based chiralityR05 (); R06 () and electromagnetically induced transparency (EIT)R07 (); R08 (); R09 (); R10 (); R11 (); R12 (); R13 (); R14 (), can be understood by studying the coupling between different magnetic and electric artificial atoms in the so called hybridization picture R07 (); R12 (); R15 (); R16 (). Similar to optical lattices with controllable couplingR17 (); R18 (); R19 (), the design flexibility in the coupling between artificial atoms opens up a new perspective to realize quantum simulators to achieve effects which are difficult to obtain in quantum systemsR17 (); R18 (). Here, we theoretically investigate the possibility in constructing a metamaterial system with an effective -symmetric Hamiltonian which can still show real eigenenergies. One of the most interesting properties of a -symmetric Hamilton is the existence of a phase transition arising from spontaneous symmetry breaking by tuning an external parameter, defined as the coalescence of two eigenvalues and their corresponding eigenvectorsR20 (). Unlike the previously considered symmetry breaking effects of metamaterialsR21 (), such a spontaneous symmetry breaking is implicit in nature. It means that the phase transition occurs without structurally destroying a specific symmetry (e.g. mirror, rotational symmetry). The external parameter being tuned still respect the -symmetry.
During the past few years, -symmetric Hamiltonians have in fact been extensively investigated within the framework of optics. The most direct analog is represented by a pair of optical waveguides with a balanced gain and loss profileR19 (); R22 (); R23 (); R24 (); R25 (); R26 (); R27 (); R28 (); R29 (); R30 (); R31 (). Spontaneous -symmetry breaking has also been theoretically and experimentally investigatedR19 (); R22 (); R23 (); R24 (); R25 (); R26 (); R27 (); R28 (); R29 (); R30 (); R31 (). Intriguing phenomena, such as asymmetric transmissionR19 (); R24 (), power oscillation R24 (), unidirectional invisibilityR19 (); R29 (), coherent perfect absorption and laserR30 () have been predicted. Moreover, extending from an ideal -symmetric system with balanced gain and loss, non-ideal -phase transition of a passive system has also been proposed and experimentally demonstrated R25 (); R27 (). Unconventional loss-induced optical transparency is demonstrated around the phase transition pointR25 (). Actually, -symmetry has not been discussed within the context of metamaterials. By employing metamaterials, the larger flexibility in designing the coupling between different artificial atoms gives us additional possibilities in realizing non-trivial -symmetry. Here, we show that by mapping a metamaterial consisting of a bright electric and a dark magnetic atom to an effective Hamiltonian, we can establish an ideal spontaneous symmetry breaking transiting from little absorption to complete absorption even in a passive system. It relies on the inverse relationship that scattering is much larger than loss for a bright electric atom while the reverse is true for a dark atom. Such a system has been previously considered to achieve EIT. However, we are interested in the phase transition to obtain complete absorptionR32 (); R33 () across a threshold in the coupling parameter, in contrast to the high transmission for EIT emerging from the strong coupling regimeR07 (). Moreover, by incorporating a small material gain into the dark atom of the ultrathin metamaterial, we can obtain laser actionR34 (); R35 (); R36 () and gain-assisted complete reflection by considering the spontaneous symmetry breaking effect of a class of Hamiltonians associated with the scattering matrix.
The discussion of -symmetric Hamiltonians originates from Bender and colleague’s work, showing that a Hamiltonian respecting a parity-time symmetry, i.e. , can have completely real eigenenergiesR20 (). To find the analog of -symmetry in a metamaterial system, the simplest way is to look for a pair of coupled artificial atoms, since their response can be understood within a hybridization picture in quantum mechanics as a two state problem, and the -symmetry in two-state quantum system is already well studiedR20 (). Here, we consider a metamaterial constituting a planar array of atoms. Each unit cell consists of an electric atom coupled to a magnetic atom through the near fields. For simplicity, we assume normal incidence and there is no polarization conversion for the incoming plane waves. The optical response of such a metamaterial can be expressed using the scattering matrix as
where and are the E-field amplitudes of the incoming and outgoing waves in the positive and negative directions. The same scattering process has a microscopic representation using the atomic responses and their associated far-field radiations. By further assuming each atom is dominated by only one dipolar resonating mode, the atomic responses can be summarized as
while the outgoing fields and the atomic responses are related to each other by
In the above equations, () represents the total resonating linewidth for each atom where , a positive number, represents the part of scattering and is the part due to absorption loss when it is positive. We also normalize the electric dipole and the magnetic dipole by the oscillator strengths (with details in establishing the above model given in Appendix) and assume the two resonant frequencies at the same frequency . In fact, Eq. (4) defines a series of Hamiltonian with meaning in the following. By taking an appropriate arithmetic average of Eq.(2) and Eq.(3), one can prove that yields an eigenvalue of matrix . We can also obtain
For example, leads to the pole of , which corresponds to laser actionR30 (); R34 (); R35 (); R36 (). implies a zero of to completely absorb the incoming EM waves, namely coherent perfect absorption (CPA)R30 (); R32 (); R33 (). For a metamaterial with mirror symmetry in the propagation direction, the two eigenvectors of are with eigenvalue and with eigenvalue by defining as the complex transmission/reflection coefficient for electric field with respect to the mirror plane. If we further confine our discussion to the coupling between a bright electric and a dark magnetic atom, we have the additional requirement . Then, implies a complete reflection and . In general, a particular /eigenvalue of denotes a particular optical phenomenon and it corresponds to solving the eigenvalue problem of . If the eigenvalue of is real, the phenomenon can occur at a real frequency with a pole/zero on the corresponding optical spectrum. Therefore, by considering the spontaneous symmetry breaking process of a -symmetric , one can observe the phase transition when the real eigenfrequencies are turned into complex conjugates or vice versa. For our Hamiltonian, the P-operation switches rows and columns of the matrix while the T-operation takes complex conjugate on the elements. A -symmetric is defined by . We have to emphasize that -symmetry being discussed here is an effective one through the establishment of an effective -symmetric Hamiltonian, which is different to the approach employing complex potentialsR20 ().
As an example, a -symmetric corresponds to the condition , which can be satisfied by a representative hybridized metamaterial with a bright electric atom (scatters most) and a dark magnetic atom(absorbs most). We should emphasize this kind of balance between scattering loss and absorption loss to obtain effective -symmetric Hamiltonian is new and different to previous approaches with balanced gain and lossR19 (); R22 (); R23 (); R24 (); R25 (); R26 (); R27 (); R28 (); R29 (); R30 (); R31 (). For a fixed set of parameters, e.g. THz, THz ( THz), , this condition is represented by the vertical red line in Fig. 1 at THz to indicate a -symmetric . Then, the spontaneous symmetry breaking process can be observed by changing the coupling coefficient (still respecting the -symmetry) across a threshold value. The eigenfrequencies of -symmetric are real(solid red line) when , and becomes a complex conjugate pair(dotted red line) when . The splitting of the real/imaginary part of the eigenfrequencies with varying is showed in the inset. This behavior reveals itself through the absorption coefficient when the metamaterial is excited by the incident waves of , giving CPA for at the two real frequencies. We note that such a configuration of coupling between a bright and a dark atom (without the -symmetric condition satisfied in general) has been used to give EIT in the strong coupling regimeR07 (). We endow the same system to visualize effective -symmetry breaking and focus on its optical properties in the regime near the critical point.
In the following, we demonstrate the interesting optical properties associated with -symmetry breaking with metamaterial structures. In particular, undergoing the phase transition, the metamaterial becomes perfectly absorptive with real frequencies. The bright atom (inset of Fig. 2(a)) is a silver strip of length nm, width nm and thickness nm. Fig. 2 (a) shows the electric field near one end of the strip with numerical simulation (symbols) and theoretical model (lines). The numerical simulation is carried out using finite element analysis (Comsol MultiPhysics) with Drude model of silver being , where Hz, Hz, and . The theoretical result is fitted using a Lorentz model , where THz, THz, and THz, is the scattering(incident) field. can be estimated by the probing the response of the bright atom with and without metal loss. The dark atom is a magnetic resonator tuned to the same resonating frequency with two parallel silver strips and a dielectric gap material ( with introduced to represent a slight absorption or gain) in between. The geometric parameters are nm, nm, nm, as showed in the inset of Fig. 2(b). Due to the large quality factor of the dark mode resonance, a small of the gap material can effectively alter the half-linewidth . As shown in Fig. 2 (b), numerically calculated field (symbols) near one of the metal strip ends is again fitted well with theoretical Lorentz model (solid line) for (hollow, solid symbols). Comparing between numerical and analytical results with different , a simple approximated formula between and can be established as . We need to emphasize that, dark atom at magnetic resonance can provide flexibility to realize , bright atom provide flexibility to realize , and hybridized system with these two atoms can provide a opportunity to satisfy the requirement of -symmetric condition for . Now, we hybridize these two atoms within a metamaterial with a square lattice of periodicity nm and a distance in separating the two atoms, as shown in Fig. 2(c). The change of distance between the two meta-atoms can tune the coupling between them, but there exists no simple relationship between them, as shown in Fig. 2(d). The can be calculated from the bright meta-atom probe response for by changing the distance between them. The fitted formula can represent main variation character of , but exhibits slight deviation in large region.
Now, we illustrate the spontaneous symmetry breaking with fixed geometry of the meta-atoms with constant , , and . The variable parameters are the coupling coefficient (or equivalently ) between the two atoms and (through the tuning of ). We can then choose a particular to satisfy -symmetry requirement of . For a -symmetric , we need THz, i.e. , which is a passive configuration to display the effective symmetry. In the light of the above theoretical discussion, the eigenvalue problem for a -symmetric is equivalent to the zeros of the , which can be easily observed in the coherent outgoing spectra when we change the parameter . The -phase transition point in this case occurs at THz ( nm). Below the critical point, THz ( nm), the eigenvalues of is a complex conjugate pair, there are no crossing in real frequency axis, so we cannot observe CPA in this region, shown in Fig. 3(a). As we increase (decrease) , at THz( nm), the two eigenvalues of collapse to one point in the real frequency axis, i.e. CPA() emerges in the spectra. When passes over the , there are two real frequencies satisfies the eigenvalue solution of , CPA branching phenomenon can be clearly shown in Fig. 3(a) from the analytical results obtained from Eq. (2) and (3) with the atomic model established from the previous simulations shown in Fig. 2. The same phenomenon can be observed from the numerical simulations (with microstructures included) shown in Fig. 3(b). Real parts (white dashed line in Fig. 3(a),(b)) of the eigenvalues of agree well with the CPA positions in the spectra. In both Fig. 3(a) and Fig. 3(b), the absorption loss ( for symmetric incoming waves) of the hybridized system increases dramatically with increasing coupling (decreasing separation) below . At , the presence of interferes with to the right size and gives and . This is the onset of CPA phase. Going beyond the transition point (larger coupling), the CPA peak splits into two. On the other hand, this splitting makes both atoms cannot be efficiently excited at and loss decreases at that frequency. In fact, the non-monotonic function of loss against coupling coefficient is a direct manifestation of spontaneous symmetry breaking, which is similar to the loss-induced optical transparency in a passive -symmetric system deviated from the ideal -symmetryR25 (). Here, using metamaterials, we can actually establish an ideal -symmetry in a completely passive system. CPA becomes possible besides the counterintuitive trend of loss function.
Besides in revealing -symmetry breaking process, is also a potential candidate to display the same process. As predicted in Eq. (5) and Fig. 1, by taking into account the eigenvalue problem of , we need to fulfill the case THz (the black vertical line in Fig. 1), i.e. , where potential amplifying media can be available in natureR34 (); R35 (); R36 (). The emergence of effective -symmetry breaking process can be observed though transmission spectra, and in spectra implies real frequency solution for eigenvalue of . It indicates the system without absorption loss, , and transmission, , with perfect reflection with and a -phase shift, . The phase transition point in this configuration is at THz ( nm). Illustrated in Fig. 3(c), and numerically verified in Fig. 3(d), the phase transition point is at nm, () in the transmission spectra agrees well with the real parts of the eigenvalues of . Below the phase transition point (), the eigenvalue of implies no cross point in the real frequency axis, a complex conjugate pair, so transmission is dominated by the bright atom dipole oscillation, although , the eigenvalue of is not (Absorption loss is not zero). Above the phase transition point , the eigenvalue of implies two real eigenfrequencies, so () can be observed at two different frequencies in the transmission spectra. Real parts (white dashed line in Fig. 3(c),(d)) of the eigenvalues of match the positions well in spectra. The effective spontaneous -symmetry breaking for can thus be observed in the spectra. It is worth to note that the transmission exhibits maximum value at the position THz ( nm for analytical prediction and nm for numerical results) at , because gain behavior of the system appears, when pass over the phase transition point . As we increase , we can encounter a real eigenvalue of , indicating pole of , which implies the system shows laser action at this parameter positionR30 (); R34 (); R35 (); R36 (), as indicted in the cross point between the black vertical line indicating -symmetric and the blue curve displaying real eigenvalue of in Fig. 1. The laser action results from non-ideal -symmetry of whose origin will be discussed in the following.
For , ideal -symmetry condition implies that huge gain is required to compensate metal losses. This is difficult to be satisfied using natural materials. However, in a practical (non-ideal) system, even though ideal condition is not satisfied, the eigenvalues still split in the same way as an ideal -symmetric Hamiltonian. Only an additional constant bias in the imaginary part of the eigenvalues occurs. Similar strategies have been employed in studying non-indeal -symmetry in passive systemsR25 (); R27 (). Then, phase transition (branching) point is not on the real frequency axis, and is shifted into the lower-half of the complex -plane. One of the eigenvalues (the one with increasing imaginary part) will cross the real frequency axis, corresponding to the original optical phenomenon, e.g. laser action for the case of as a pole of at R30 (); R34 (); R35 (); R36 (). It is the solid blue curve in Fig. 1 and is lying below branching condition (branching occurs when ) where (dashed green line). In contrary, ideal -symmetric has the poles of on the real frequency axis beyond branching condition (vertical solid blue line) at two different frequencies. As an example, THz (), Fig. 4 gives the outgoing intensity of the system with symmetric incoming light as a function of . Laser action collapses to one position at , nm for theoretical prediction and nm for numerical results. The branching phenomenon still occurs (without real poles of ) and agrees well with the real parts of the eigenvalues of (black dashed lines), as displayed in Fig. 4. Laser action and branch phenomenon is the direct demonstration of non-ideal -symmetry for . Besides , non-ideal -symmetry can also occur for and . In Fig. 1, the real eigenvalue of (zero of ), i.e. CPA, is at (solid red curved line) while the branching condition is at (dashed purple line). For , perfect reflection, (one eigenvalue is ) is at (solid black curved line) while the branching condition is at (dashed black line). So, for a non-ideal -symmetric system, the optical phenomenon (e.g. laser, CPA or -reflector) associated to collapses to a single point (with a a bit less than the one triggers branching) instead of the two branches.
In the above discussion, is the external parameter in demonstrating -symmetry. If we change with a fixed instead (travelling in a horizontal direction in Fig.1), the metamaterial can function as a laser, the cross point for , or a CPA, the cross point for at . As an example for demonstration, nm, besides appearance of laser action and CPA in spectra, schematically shown in Fig. 5, above CPA position, an increase of loss (increasing ) leads to more output of the system, which is similar to loss induced optical transparency in a non-ideal passive systemR25 (). Similar counter-intuitive trend also occurs beyond the laser position, an increase in gain (decreasing negative ) causes less output observed in the spectra.
In conclusion, we investigated the effective -symmetric Hamiltonians using a metamaterial system to reveal spontaneous symmetry breaking processes. Theoretical and numerical results clearly confirm the analog of spontaneous breaking of -symmetry using metamaterials. Our results not only open up a new perspective towards testing the intriguing properties of spontaneous -symmetry breaking in non-Hermitian quantum systems, but also provide feasibility for future applications in plasmonic and metamaterial systems, such as laser actionR30 (); R34 (); R35 (); R36 (),CPA with sub-wavelength thicknessR32 (),and asymmetric transmissionR37 (). Moreover, by using metamaterials with both electric and magnetic responses, it becomes possible to establish an ideal -symmetry by exploring the balance between scattering and absorption loss using a passive system.
Appendix A Appendix
The metamaterial in this work is modeled as a planar array of electric atoms coupled with an array of magnetic atoms on the same plane with surface normal vector and a unit cell of area . For simplicity, we assume the plane is excited by normally incident plane waves with electric field in the x-direction () and magnetic field in the y-direction (). Suppose each electric atom only generates electric dipole moment () along the -direction dominated by a single resonance while each magnetic atom only generates magnetic dipole moment () along its principal direction also dominated by a single resonance (at a general angle with being the special case of a completely dark magnetic atom). Then, the response of these atoms can be described using a simple dipolar model as
where and with being the wavenumber in vacuum and the positive numbers , , being the resonance frequencies, oscillator strength and total resonance linewidth of the -th atom ( or for electric or magnetic atom). is the coupling parameter (controlled by the displacement between the electric and the magnetic atom in a unit cell), is assumed as a real number in the long wavelength limit in this work. We have employed the Heaviside-Lorentz unit system and the normalization ( and ) is chosen so that , , and are in the same unit of frequency.
If we use to indicate the E-field amplitudes of the incoming plane waves in the positive and negative -direction, the incident fields ( and ) applying on the electric and magnetic atoms can be expressed as
Substituting A.7 into A.6 gives the atomic response to the incoming plane-wave amplitudes as
where is a postive number defined as .
On the other hand, the scattering plane waves (with being the E-field amplitudes in the positive and negative -direction) are the radiation plane waves generated by the dipole moments:
If we use to indicate the E-field amplitudes of the outgoing plane waves in the positive and negative -direction, substituting A.9 and A.8 into it expresses the outgoing plane waves in terms of the dipolar moments as
In fact, we can also express the atomic response using the incoming and scattering plane waves by substituting A.9 into A.8 as
where the matrix is defined by
If the system is without gain, we have the passivity condition requiring . By substituting A.11 into it for arbitrary excitaiton, after some algebra, the passivity condition means being a negative-definite matrix. Equivalently, it means and directly obtained from A.12. Therefore, for the general system, we can decompose the total resonance linewidth by where the part due to scattering is , and is the part due to absorption loss when it is positive. With such decompositions of the linewidths, A.8 becomes Eq. 2 and A.10 becomes Eq. 3 in text immediately.
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