PHYSICAL REVIEW RESEARCH 2, 013281 (2020)

Analogues of gravity-induced instabilities in anisotropic metamaterials

Caio C. Holanda Ribeiro * and Daniel A. Turolla Vanzella † Instituto de Física de São Carlos, Universidade de São Paulo, Caixa Postal 369, 13560-970, São Carlos, São Paulo, Brazil

(Received 27 November 2019; accepted 11 February 2020; published 9 March 2020)

In the context of field theory in curved space-times, it is known that suitable background space-time geometries can trigger instabilities of fields, leading to exponential growth of their (quantum and classical) fluctuations, a phenomenon called vacuum awakening in the quantum context, which in some classical scenarios seeds spontaneous scalarization or vectorization. Despite its conceptual interest, an actual observation in nature of this effect is uncertain since it depends on the existence of fields with appropriate masses and couplings in strong-gravity regimes. Here, we propose analogues for this gravity-induced instability based on nonlinear optics of metamaterials which could, in principle, be observed in laboratory.

DOI: 10.1103/PhysRevResearch.2.013281

I. INTRODUCTION spin-1 fields through appropriate nonminimal couplings [18] and, in analogy with the scalar case, the stabilization pro- The influence of a background material medium on the cess was termed “spontaneous vectorization.” To the best of propagation of mechanic and electromagnetic waves is well our knowledge, condensed-matter and optical analogues of known to be formally analogous to that of an effective curved these gravity-induced instabilities have not been proposed space-time geometry. This idea was first presented, in the to this date. In this work, we propose and explore possible electromagnetic context, by Gordon in 1923 [1] and it has analogues of gravity-induced instabilities in the context of since been developed in a number of different scenarios, par- electromagnetism in polarizable and magnetizable anisotropic ticularly after Unruh’s [2] and Visser’s [3] works on acoustic (meta)materials. analogues of black holes and their associated Hawking-type Electromagnetic instabilities in flat space-time are ex- radiation. More recent applications of this formal analogy pected to occur in some materials. One celebrated example include mimicking in material media quantum light-cone appeared in the context of plasma physics in the late 1950s fluctuations [4] and anisotropy in cosmological space-times and became known as Weibel instability [19]. The system, a [5]. The most appealing feature of these condensed-matter neutral plasma whose components have anisotropic velocity analogues of gravitational backgrounds is the possibility of distribution, possesses growing electromagnetic transverse observing in laboratory subtle but conceptually interesting waves. Related effects have been studied since then, with effects which can be virtually unobservable in their original recent applications to solar plasma instability [20] and solid- contexts, Hawking radiation being certainly the most emblem- state devices [21]. Moreover, causal aspects of classical prop- atic among them, with claims of having already been observed agation in active materials were discussed in Ref. [22], where in laboratory [6–8]. properties of the refractive index were established. Neverthe- An interesting effect in the context of (quantum) fields less, aside from the fairly recurrence in the literature, usually in curved space-times is the triggering of field instabili- quantization in such scenarios is not considered [23–25]orit ties due to the background space-time geometry, a phe- is regarded as inconsistent [26,27]. nomenon called vacuum awakening in the quantum context It is noteworthy that instability of the electromagnetic [9–12]. These gravity-induced instabilities exponentially am- field is always accompanied by evolution of the background, plify vacuum fluctuations to the point they decohere and seed ending with the stabilization of the system as a whole. In classical perturbations [13], which, depending on field pa- the case of gravity-induced instability, the gravitational field rameters, eventually evolve to a nonzero classical field con- changes with time, whereas electromagnetic instability in the figuration (“spontaneous scalarization” in the case of scalar presence of plasmas involves growing plasmons. In the case fields [14–17]), stabilizing the whole system. More recently, of electromagnetic fields in the presence of matter, for what- this mechanism was also predicted to occur for massless ever form of the interaction with the background, the field’s evolution is ruled by Maxwell’s equations in the presence of polarizable and magnetizable media, and the interaction with *[email protected] the background is encapsulated in the functional dependence †[email protected] of the electric displacement (magnetic) vector field D (H) with the true (microscopic) fields E and B. If the magnitudes Published by the American Physical Society under the terms of the involved are small (e.g., in the beginning of the instability Creative Commons Attribution 4.0 International license. Further action), these functional relations become linear and one may distribution of this work must maintain attribution to the author(s) find the form of the coefficients for such systems. For the case and the published article’s title, journal citation, and DOI. of Weibel instability, for instance, if the velocity anisotropy is

2643-1564/2020/2(1)/013281(20) 013281-1 Published by the American Physical Society RIBEIRO AND VANZELLA PHYSICAL REVIEW RESEARCH 2, 013281 (2020) taken in the z direction, the instability is modeled by a negative where ∂a is the derivative operator compatible with the flat squared refractive index in the direction perpendicular to z. metric ηab (but in arbitrary coordinates) and the square brack- We apply Gordon’s method to propose a family of optical- ets denote antisymmetrization over the indices enclosed by based analogue models for electromagnetic fields presenting them. These equations must be supplemented by medium- ab instabilities in curved space-times. We show how anisotropies dependent constitutive relations between Fab and G ,aswell of the background enter the effective equations in the form of as initial and boundary conditions, in order to provide a well- nonminimal couplings, and in the case of strong anisotropy posed problem. These constitutive relations are usually set at a a (just like for the Weibel instability), this coupling results in the level of (observer-dependent) fields Ea, B , D , and Ha, ab unstable solutions. We also discuss that for these systems the related to Fab and G through stabilization process occurs through the nonlinear nature of = b, the background, which may seed spontaneous vectorization Ea Fabu (3) a ab in analogy to the Einstein’s field equations in the gravitational D = G ub, (4) scenario. Ba =−1 abcd F u , (5) The paper is organized as follows. In Sec. II, we present 2 bc d the covariant formalism of electromagnetism in anisotropic =−1  bc d , Ha 2 abcd G u (6) polarizable and magnetizable materials, establishing the for- a mal analogy with nonminimally coupled electromagnetism where u is the four-velocity of the observer measuring these   = in curved space-times. In Sec. II A, we consider a particular fields√ and abcd is the Levi-Civita pseudotensor [with 0123 type of nonminimal coupling inspired by one-loop quantum −η, η := det(ημν )]. Moreover, the constitutive relations electrodynamics (QED) corrections to electromagnetism in usually take a simpler form in the reference frame in which curved space-times. In Sec. III, we apply the formalism the medium is (locally and instantaneously) at rest. presented in the previous section to the scenario of a plane- Here, we consider a polarizable and magnetizable medium symmetric anisotropic medium at rest in an inertial frame. whose constitutive relations in its instantaneous rest frame Although plane-symmetric curved space-times (in four di- take the form mensions) are not really (physically) appealing, we consider Da = εabE , (7) this scenario for its simplicity and for its possible implications b b for the physics of the material medium. We construct the Ha = μabB , (8) ˆ electromagnetic quantum-field operator A (in the generalized εab μ Coulomb gauge) in the standard-vacuum representation, dis- where the tensors and ab may depend on space-time cuss the conditions for appearance of instabilities and their coordinates, and the system is assumed dispersionless. We types (Sec. III A), and present a concrete example (homoge- return to this point later. The fact that Eqs. (7) and (8) are valid in the medium’s instantaneous rest frame means that the fields neous medium; Sec. III B) where calculations can be carried a a ab over to the end. In Sec. IV, we repeat the treatment of the Ea, B , D , and Ha appearing in them are related to Fab and G a = a previous section, but now for a more appealing scenario through Eqs. (3)–(6) with u v , the medium’s four-velocity field. We proceed by splitting the “spatial” [30] tensors εab and on the gravitational side: spherically symmetric, stationary μ anisotropic media. Conditions for triggering instabilities and ab into isotropic and traceless anisotropic parts, their types are shown to be very similar to those in the ab ab ab ε = ε h + χ ε , (9) plane-symmetric case (Sec. IV A). As a concrete applica- ( ) μ = μ−1 + χ (μ), tion, in Sec. IV B we show how to mimic QED-inspired ab hab ab (10) nonminimally coupled electromagnetism in the background a = δa + a space-time of a Schwarzschild black hole. Then, Sec. V is where h b : b v vb is the projection operator orthogonal a a = a dedicated to discuss possible stabilization mechanisms which to v . Inverting Eqs. (4) and (6) (with u v ), might bear analogy to some curved-space-time phenomena, ab [a b] abcd G = 2v D −  Hcvd , (11) such as spontaneous vectorization [18] and particle bursts due to tachyonic instability [28]. Finally, in Sec. VI we present and substituting Eqs. (7)–(10) and (3) and (5), we obtain some final remarks. We leave for an Appendix tedious calcu- Gab = (gacgbd + χ abcd )F , (12) lations related to the orthonormalization of modes of Sec. IV. cd We adopt the abstract-index notation to represent tensorial where we have defined the tensors quantities (see, e.g., Ref. [29]) and, unless stated otherwise, 1 we use natural units (in whichh ¯ = c = 1). gab := √ [ηab − (n2 − 1)vavb], (13) n II. COVARIANT ELECTROMAGNETISM IN n | χ abcd := − 1 ga[cgd]b − 2χ [a [c vd]v|b] ANISOTROPIC MATERIAL MEDIA μ (ε) Electromagnetism in material media, in flat space-time and 1 + abe f cdghχ (μ) , in the absence of free charges, is described by two antisym- eg v f vh (14) ab 2 metric (observer-independent) tensors, Fab and G , satisfying the macroscopic covariant Maxwell’s equations and the squared refractive index n2 = με. The idea, then, is to bc = ∂ ab = , consider the symmetric tensor gab, defined through gabg aG 0 (1) δc a,asaneffective metric of a curved background space-time ∂[aFbc] = 0, (2) perceived by the electromagnetic field Fab. Note that the

013281-2 ANALOGUES OF GRAVITY-INDUCED INSTABILITIES IN … PHYSICAL REVIEW RESEARCH 2, 013281 (2020)

ab ηab ¯   components of g and satisfy Ab (respectively, Ab) with Eq. (19) applied to Fcd (respectively, F¯cd ) and subtract one from the other, arriving at det(gαβ ) = det(ηαβ ) (15) ∇ ac bd + χ abcd ¯  −  ¯ = . √ √ a[(g g )(AbFcd AbFcd )] 0 (20) and, thus, −η = −g, where g := det(gαβ ). One can easily check that gab is explicitly given by This continuitylike equation ensures that the quantity √ 2 (n − 1) ,  = ac bd + χ abcd ¯  −  ¯ g = n η + v v . (16) (A A ): i d Na(g g )(AbFcd AbFcd ) ab ab n2 a b (21) Therefore, in an arbitrary coordinate system, Eq. (1) reads as is independent of the spacelike hypersurface where the 1 √ αβ 1 √ αβ 0 = √ ∂α ( −η G ) = √ ∂α ( −gG ). (17) integration is performed, provided we restrict attention to −η −g solutions satisfying “appropriate” boundary condition, where b Up to this point, it was understood that the physical d is the physical volume element on and Na = ηabN , ab a background metric ηab and its inverse η were responsible with N being a unit, future-pointing vector orthogonal to for lowering and raising tensorial indices. Now, with the (according to ηab). More specifically, considering that the M ∼ introduction of an effective metric gab, we should be careful system of interest is contained in the space-time region = when performing these isomorphisms. In order to minimize T × , where T ⊆ R is a real open interval, then the appro- chances of confusion, we shall avoid lowering and raising priate boundary condition amounts to imposing that the flux tensorial indices using the effective metric, making explicit of the (sesquilinear) current appearing in Eq. (20) vanishes ab × ˙ S˙ most appearances of gab and g in the equations below, with through T (where denotes the boundary of the space few exceptions which will be clearly stated. One obvious S). In particular, in stationary situations which we shall treat exception is the definition of g as the inverse of gab. Another here, this condition translates to ab such exception is the use of ∇a to denote covariant derivative ac bd + χ abcd ¯  −  ¯ = , compatible with gab. With this in mind, from Eqs. (2) and (17), dSsa(g g )(AbFcd AbFcd ) 0 (22) ˙ the electromagnetic tensor Fab satisfies where dS is the physical area element on ˙ and sa is the 0 = ∇[aFbc], (18) unit vector field normal to T × ˙ (according to ηab). Thus, ac bd abcd 0 = ∇a[(g g + χ )Fcd ]. (19) these conditions being satisfied, Eq. (21) provides a legiti- mate sesquilinear form on the space SC of complex-valued Notice that Eqs. (18) and (19) applied to homogeneous solutions of Eqs. (18) and (19). Notice that for pure-gauge ∇ ε = ∇ μ = χ ab = = χ (μ) ( a a 0), isotropic ( (ε) 0 ab ) materials, solutions, i.e., A = ∇ ψ, for some scalar function ψ —, a a a with arbitrary four-velocity field v , lead to the same equa- (A, A) = 0. (The converse, however, is not true.) tions which rule minimally coupled vacuum√ electromag- The relevance of this sesquilinear form is that it provides μ/ netism in a curved space-time with metric ngab. Optical a legitimate inner product on a (nonunique choice of) sub- analogue models in these configurations with μ = 1were S+  S space C C of “positive-norm solutions,” which, together studied in [31,32]. Here, we shall focus on electromagnetism S−  S with its complex conjugate C C, generates all solutions in anisotropic materials, more specifically, materials with + − + S : S ⊕ S = S . Loosely speaking, upon completion, S χ [ab] = = χ (μ) C C C C C only “shearlike” anisotropies: (ε) 0 [ab]. In this case, yields a Hilbert space H from which the (symmetrized) Fock χ abcd the tensor defined in Eq. (14) has the same algebraic space Fs(H) is canonically constructed to represent states of symmetries as the Riemann curvature tensor, namely, χ abcd = S+ the electromagnetic field. In particular, choosing C to be χ cdab and χ a[bcd] = 0, in addition to χ abcd = χ [ab][cd], which generated by positive-frequency solutions (those proportional is always true. to e−iωt , with ω>0), the vacuum state of this Fock represen- Equations (18) and (19) can be seen as analogous to tation corresponds to the usual physical vacuum state of the some nonminimally coupled electromagnetic field equations field. in curved space-time. Although in general χ abcd is inde- pendent of the Riemann tensor associated with the effective A. QED-inspired nonminimal couplings metric gab, one can construct cases where they are related. This is interesting because some one-loop QED corrections As mentioned earlier, Eqs. (18) and (19) can be interpreted to Maxwell’s field equations in curved space-time [33,34] can as ruling electromagnetism in curved space-times with some be emulated by such nonminimal coupling, as we shall discuss QED-inspired nonminimal coupling χ abcd with the back- below, in Sec. II A. ground geometry. In fact, in the one-loop QED approximation Before considering particular applications of the equations [33,34], above, let us define a sesquilinear form on the space of χ abcd = α Rabcd + α R[a|[cgd]|b] + α Rga[cgd]b, (23) complexified solutions, which will be relevant when apply- 1 2 3 α =−α / = α =−α/ π 2 α ing the canonical quantization procedure. As usual, let us with 1 2 13 2 3 (90 me ), where is the abcd solve Eq. (18) by introducing the four-potential Aa such that fine-structure constant, me is the electron’s mass, and R ,  ab Fab = ∇aAb − ∇bAa. Then, let Fab and F be two complex R , and R are, respectively, the Riemann, Ricci, and Ricci- ab  solutions of Eq. (19), associated to Aa and Aa, respectively. scalar curvature tensors associated with the (effective) metric With overbars representing complex conjugation, we contract gab. By leaving α1,α2,α3 unconstrained, Eq. (23) represents

013281-3 RIBEIRO AND VANZELLA PHYSICAL REVIEW RESEARCH 2, 013281 (2020) a three-parameter family of couplings of the electromagnetic tensor field with the background effective geometry (see Ref. [35] 3 for some interesting particular cases). T ab = ρuaub + p eaeb, (31) For a generic medium, χ abcd is not related to the geometry j j j j=1 associated with gab. However, we can simulate couplings given by Eq. (23) by conveniently relating n and va (which with {ua, ea, ea, ea} being a tetrad and ua timelike, then μ χ ab 1 2 3 determine gab) with and the anisotropic tensors (ε) and χ (μ) χ abcd 3 ab (which appear in ). From Eqs. (14) and (23), and 1 ρ + p j = 0 (32) their contractions with gab, 3 j=1 ac 3/2 3 n χ ε n μ g χ abcd = − 1 gac + ( ) + gabgcd χ ( ) and bd 2 μ 2n3/2 2 bd V ea (ρ + p ) = 0, j = 1, 2, 3. (33) = (α + α /2)Rac + (α /4 + 3α /2)Rgac, (24) a j j 1 2 2 3 n In particular, if V a = ua, then Eq. (32) is the only additional g g χ abcd = − = α + α / + α R, ac bd 6 μ 1 ( 1 3 2 2 6 3) (25) constraint to be enforced. Returning attention to the background-effective geometry we can solve for μ and the anisotropic tensors, obtaining and recalling that all the geometric tensors are obtained from gab given in Eq. (16), we see that Eq. (29) actually comprises a system of four differential equations which n and va must sat- μ = n , (26) isfy. Electromagnetism with nonminimal coupling described 1 + (α1/6 + α2/4 + α3)R by Eq. (23) can only be simulated in these anisotropic media if −3/2 ab acbd R ab the background space-time geometry is associated to solutions n χ ε =−2α1 R VcVd + H ( ) 12 of this system [via Eq. (16)]. We shall treat a particular α R solution to these differential equations later. + 2 Rab − gab , (27) 2 4 III. PLANE-SYMMETRIC ANISOTROPIC μ R n3/2χ ( ) = 2α R V cV d + H MEDIUM AT REST ab 1 acbd 12 ab In this section, we consider the simplest case of an (4α + α ) R + 1 2 Rab − gab , (28) anisotropic medium: a plane-symmetric medium at rest in the 2 4 inertial laboratory frame. The purpose of this section is not yet to establish an analogy with some interesting gravitational / where V a = n3 4va is the four-velocity of the medium normal- system, but to present the analysis in a simple context. In ized according to the effective metric gab and Hab := gab + Sec. IV we apply the analysis to a more appealing scenario. VaVb.InEqs.(27) and (28), indices are lowered and raised by Let us consider a medium at rest in an inertial laboratory, 3 the effective metric and its inverse. Notice that, unless α1 = { , , , }= × μ such that in inertial Cartesian coordinates (t x y z) R α = χ ab = = χ ( ) 4 μ 2 0, which implies (ε) 0 ab , as a consequence I ⊆ R we have v = (1, 0, 0, 0), μ = μ(z), ε = ε(z), μ of χ ab v = 0 = χ ( )vb, only geometries associated with g (ε) b ab ab χ αβ = (ε)/ δαδβ − δαδβ − δαδβ , which can be put in the form given by Eq. (16) and satisfying (ε) ( 3) 2 z z x x y y (34) and R a b = a, R bv v (29) (μ) (μ) z z x x y y 4 χαβ = ( /3) 2δαδβ − δαδβ − δαδβ , (35)

a for some timelike four-vector v , can be emulated by these with (ε) = (ε)(z), (μ) = (μ)(z), z ∈ I.(I is an open real a anisotropic media, with v then set as the medium’s four- interval.) This simply means that velocity. Using Einstein’s equations to map this constraint to ab j the stress-energy-momentum tensor T of the corresponding D = ε⊥E j, j = x, y (36) gravitational source, we have that z D = ε Ez, (37) T = μ−1 j, = , a b = a, Hj ⊥ B j x y (38) T bv v (30) 4 −1 z Hz = μ B , (39) where, again, the effective metric and its inverse are used to ε − ε ≡ (ε) ε + ε / ≡ ε μ−1 − μ−1 ≡ lower and raise indices (and T := T a ). One can easily check where ⊥ ,(2⊥ ) 3 , ⊥ a (μ) μ−1 + μ−1 / ≡ μ−1 that in case of perfect fluids, characterized by a proper energy , and (2 ⊥ ) 3 √. −2 density ρ and (isotropic) pressure p,Eq.(30) is only satisfied In these coordinates, gμν = n diag(−n , 1, 1, 1). For for p =−ρ, i.e., for a cosmological-constant-type “fluid.” convenience, we shall work in the generalized Coulomb gauge However, if one allows for sources with anisotropic pres- [36] in which Aμ = (0, A) and ∂⊥ · (ε⊥A⊥) + ∂z(ε Az ) = 0, sures (p1, p2, p3), described by the stress-energy-momentum where we have defined A⊥ := (Ax, Ay ), ∂⊥ := (∂x,∂y ). In this

013281-4 ANALOGUES OF GRAVITY-INDUCED INSTABILITIES IN … PHYSICAL REVIEW RESEARCH 2, 013281 (2020) gauge, the t component of Eq. (19) is automatically satisfied, z direction (transverse magnetic modes A(TM), for short (TM) = ε−1∂2 φ while the spatial components lead to [37]) is obtained by conveniently setting Az ⊥ , 2 where φ is an auxiliary function. Our gauge condition then ε⊥ 1 1 (TM) − − ∂2 + ∂2 + ∂ leads to A =−ε 1∂ ∂ φ, j = x, y. Using, again, staticity t ⊥ z A⊥ j ⊥ j z μ⊥ μ⊥μ μ⊥ and planar symmetry, we find solutions of the form φ = −i(ωt−k⊥·x⊥ ) (TM) (TM) e fωk⊥ (z), where fωk⊥ (z) satisfies 1 −1 −1 = [μ ∂z(μ⊥ ∂⊥) − ε⊥ ∂z(ε ∂⊥)]Az, (40) μ⊥μ 2 2 μ ω2 d k⊥ ⊥ (TM) − + − f = 0, (49) μ 2 ξ 2 ε ε ε ωk⊥ ⊥ 2 1 2 1 d ⊥ ⊥ − ∂ + ∂ + ∂ ε A = . ε t ε ε ⊥ ε z ( z ) 0 (41) ⊥ ⊥ ⊥ with ξ being a spatial coordinate such that dξ = ε⊥dz.The = boundary condition imposed by Eq. (22) now leads to First, let us consider solutions A such that Az 0, which describe electric fields which are perpendicular to the z direc- 2 (TM) d (TM) 2 (TM) d (TM) (TE) ω f f  − ω f  f = 0. (50) tion (transverse electric modes A , for short [37]). In this ωk⊥ ω k⊥ ω k⊥ ωk⊥ dξ dξ I˙ case, our gauge condition ensures that there exists a scalar (TE) (TE) (TM) field ψ such that A = ∂ ψ and A =−∂ ψ. Moreover, E ⊥ ω x y y x Let k⊥ be the (k -dependent) set of values for which making use of the staticity and planar symmetry of the present Eqs. (49) and (50) are satisfied for f (TM) ≡ 0. For ω, ω ∈ −i(ωt−k⊥·x⊥ ) (TE) ωk⊥ scenario, we can write ψ = e f (z), where x⊥ = ∗ ωk⊥ E (TM) = E (TM) ∩ (TE) k⊥+ : k⊥ R+, we can normalize these modes accord- (x, y, 0), k⊥ = (k , k , 0), k⊥ = k⊥ , and f (z) satisfies x y ωk⊥ ing to 2 2 ε ω2 d k⊥ ⊥ (TE) (TM) (TM) (TM) (TM) − + − = , Aω , Aω  =−Aω , Aω  fωk⊥ 0 (42) k⊥ k⊥ k⊥ k⊥ dζ 2 μ⊥μ μ⊥  = δ  δ − , ζ ζ = μ ωω (k⊥ k⊥) (51) with being a spatial coordinate such that d ⊥dz. Equa- tion (42) must be supplemented by boundary conditions for (TM) (TM) Aω , A   = 0 (52) (TE) k⊥ ω k⊥ fωk⊥ . Imposing Eq. (22) to these modes leads to δ  ( ωω now being the appropriate Dirac-delta distribution on (TE) d (TE) (TE) d (TE) E (TM)  −  = , k⊥ ), obtaining (up to a global phase) fωk⊥ fω k⊥ fω k⊥ fωk⊥ 0 (43) dζ dζ I˙ −i(ωt−k⊥·x⊥ ) 2 (TM) e k⊥ k⊥ d (TM) where [...]|I˙ denotes the flux of the quantity in square brack- A = √ n + i f (z), (53) ωk⊥ ε z ε ωk⊥ ets through I˙. This condition restricts the possible values of 2πk⊥ 2ω3 ⊥ dz 2 (TE) ω E ⊥ ω .Let ⊥ be the (k -dependent) set of values for which with k Eqs. (42) and (43) are satisfied for f (TE) ≡ 0. For ω, ω ∈ ωk⊥ (TM) (TM) ∗ dzμ⊥(z) f (z) f  (z) = δωω . (54) E (TE) = E (TE) ∩ ωk⊥ ω k⊥ k⊥+ : k⊥ R+, we can orthonormalize these modes ac- I cording to (TM) (TM) Moreover, modes Aω and Aω are orthogonal to modes (TE) (TE) (TE) (TE) k⊥ k⊥ Aω , A   =−Aω , A   k⊥ ω k⊥ k⊥ ω k⊥ (TE) (TE) Aωk⊥ and Aωk⊥ .  = δωω δ(k⊥ − k⊥), (44) The solutions expressed in Eqs. (47) and (53), dubbed positive-frequency normal modes, play a central role in the (TE) (TE) Aω , A   = 0 (45) k⊥ ω k⊥ construction of the Fock (Hilbert) space of the quantized theory, as described at the end of the previous section. With (TE) δωω E ( being the appropriate Dirac-delta distribution on k⊥ ), these solutions, the quantum-field operator Aˆ is represented where the sesquilinear form given in Eq. (21), applied to the by current scenario, takes the form 2 (J) (J) Aˆ = d k⊥ dω aˆ A + H.c. , (55)  3   ωk⊥ ωk⊥ A, A := i d x [ε⊥(A¯ ⊥ · ∂ A − A · ∂ A¯ ⊥) R2 E (J) t ⊥ ⊥ t J∈{TE,TM} k⊥+

  + ε (A¯ ∂ A − A ∂ A¯ )]. (46) where “H.c.” stands for “Hermitian conjugate” of the pre- z t z z t z (J) (J)† ceding term anda ˆωk⊥ (respectively,a ˆωk⊥ ) is the annihilation We obtain (up to a global phase) (J) (respectively, creation) operator associated with mode Aωk⊥ , k⊥ × n (TE) = √ z −i(ωt−k⊥·x⊥ ) (TE) , satisfying the canonical commutation relations: Aω ⊥ e fω ⊥ (z) (47) k k  2πk⊥ 2ω (J) (J )† JJ  aˆω , aˆ   = δ δωω δ(k⊥ − k⊥), (56) k⊥ ω k⊥ with (J) (J ) aˆω , aˆ   = 0. (57) (TE) (TE) k⊥ ω k⊥ ε⊥  = δωω dz (z) fωk⊥ (z) fω k⊥ (z) (48) I As an application of our quantization scheme, one can and nz := (0, 0, 1). use the above formulas to obtain, for instance, the Carniglia- The second set of solutions of Eqs. (40) and (41), which Mandel quantization [38] in a straightforward way. The sys- describe magnetic fields which are perpendicular to the tem in this case is composed by a dielectric-vacuum interface

013281-5 RIBEIRO AND VANZELLA PHYSICAL REVIEW RESEARCH 2, 013281 (2020) at z = 0 and a nonmagnetizable (μ = μ⊥ = 1) homogeneous ω2 < 0 in what follows. The possibility of having this type of 2 isotropic nondispersive dielectric (ε = ε⊥ = ε ≡ n ) filling material will be discussed later. the half-space z < 0. These data enter Eqs. (42) and (49), thus Let ω2 =−2 (with >0) be such value for which at describing the background in terms of effective potentials of least one of the constitutive functions is negative for z ∈ I. one-dimensional Schrödinger-type problems. Thus, both the effective potentials of Eqs. (42) and (49)take the general form

A. Instability analysis 2 2 Veff = C1k⊥ + C2 , (58) In the analysis presented above, it was implicitly assumed that all constitutive functions ε⊥, ε , μ⊥, and μ are positive with C1 and C2 being functions of z. Two interesting possibil- functions of z ∈ I. This condition ensures that the field modes ities arise: presented in Eqs. (47) and (53), together with their complex (i) C1 < 0: In this case, the larger the value of k⊥,the conjugates, constitute a complete set of (complexified) solu- more negative the effective potential gets. Therefore, it is quite tions of Maxwell equations in R3 × I; in other words, the reasonable to expect that, for a given size of the interval I, one boundary-value problems defined by Eqs. (42) and (43) and can always find “large enough” values of k⊥ (certainly satisfy- 2 2 2 (49) and (50) admit solutions only for (a subset of) ω > 0. ing k⊥ > C2 /|C1|) such that the Schrödinger-type equation This is easily seen by interpreting them as null-eigenvalue with effective potential Veff admits null-eigenvalue solutions. problems for the linear operators defined in the square brack- We shall refer to this situation as large-k⊥ instability. etsofEqs.(42) and (49). Experience with Schrödinger-type (ii) C1 > 0 and C2 < 0: Under these conditions, the effec- equations teaches us that these equations have solutions pro- tive potential Veff, as a function of k⊥, is bounded from below: 2 vided the associated effective potentials (terms in parentheses) Veff  −|C2| . Therefore, a Schrödinger-type equation with become sufficiently negative in a given region, which implies effective potential Veff only admits null-eigenvalue solutions 2 2 2 2 ω > 0 and, typically, the larger the k⊥, the larger the ω . provided k⊥ is “sufficiently small’ (certainly satisfying k⊥ < 2 Here, however, we shall consider a more interesting situa- |C2| /C1) and the size of the interval where Veff is negative tion. It has been known for almost two decades that materials is “sufficiently large.” We shall refer to this situation as can be engineered so that some of their constitutive functions minimum-width instability. (J) can assume negative values [39–43]. These exotic materials Let us call gk⊥ the null-eigenvalue solutions mentioned in have been termed metamaterials. In this case, the effective either case above, with J ∈{TE, TM} depending on whether potentials appearing in Eqs. (42) and (49) may become suf- it refers to Eq. (42)or(49) with ω2 =−2 (without loss ficiently negative, granting solutions to these boundary-value of generality, >0). These solutions are associated with 2 problems, without demanding ω > 0. For instance, if μ < 0 unstable electromagnetic modes whose temporal behavior is ±t (with μ⊥,ε⊥ > 0), then the larger the value of k⊥,themore proportional to e . Although it might be tempting not to negatively it contributes to the effective potential of Eq. (42), consider these “runaway” solutions [25,26], they are essential, favoring the appearance of solutions with smaller (possibly if they exist, to expand an arbitrary initial field configura- 2 negative) values of ω . The same is true for Eq. (49)ifε < 0 tion satisfying the boundary-value problems set by Eqs. (42) and similar analysis can be done if any other constitutive and (43) and (49) and (50); in other words, the stationary function becomes negative. modes alone do not constitute a complete set of solutions At this point, we must introduce an element of reality of Maxwell’s equations with the given boundary conditions. concerning the constitutive functions. We have been treating And even if, on the classical level, one might want to restrict these quantities as given functions of z alone, neglecting attention to initial field configurations which have no contri- dispersion effects, since we are, here, interested in gravity bution coming from these unstable modes, which is certainly analogues. However, these material properties generally de- unnatural, for causality forbids the system to constrain its pend on characteristics of the electromagnetic field itself, initial configuration based on its future behavior, inevitable particularly on its time variation (i.e., on ω), in which case quantum fluctuations of these modes would grow, making Eqs. (7) and (8) would be valid mode by mode, with the them dominant some time e-foldings (t ∼ N−1, N  1) ab constitutive tensors ε and μab possibly being different for after the proper material conditions having been engineered. different modes. When translated to space-time-dependent Therefore, these modes are as physical as the oscillatory quantities, Eqs. (7) and (8) would be substituted by sums over ones. In fact, artificial inconsistencies have been reported in the set of allowed field modes [44]. the literature, regarding field quantization in active media Therefore, the precise key assumption about our meta- [25,26], which are completely cured when unstable modes are material media is that some of their anisotropic constitutive included in the analysis [44]. functions ε⊥, ε , μ⊥, μ can become negative for some ω It is interesting to note that depending on which consti- on the positive imaginary axis ω2 < 0. Notwithstanding, the tutive function is negative, Eqs. (42) and (49) may incur less restrictive condition Im(ω) > 0 would suffice for our different types of instabilities. For instance, if μ⊥ < 0for purposes. However, dealing with the case Im(ω)Re(ω) = 0 agivenω2 =−2 < 0, with all other constitutive functions would involve quantization in active media, which we shall being positive, then Eq. (42) exhibits case (i) instability, treat elsewhere [44]. Moreover, our focus here is to show that while Eq. (49) incurs in case (ii) instability.√ This means the electromagnetic field itself can exhibit interesting behavior that unstable TE modes, with some k⊥ > μ ε⊥ , would without need to exchange energy with the medium (which certainly be present, while unstable TM modes, with some occurs in dispersive/active media). This justifies our focus on k⊥ < |μ⊥|ε , would only appear if the width of the

013281-6 ANALOGUES OF GRAVITY-INDUCED INSTABILITIES IN … PHYSICAL REVIEW RESEARCH 2, 013281 (2020) material (size of the interval I) is larger than some critical (and orthogonal to all other modes) read as (up to a time value. We shall illustrate these facts in a simple example translation) below. But first, let us analyze some features of these unstable ik⊥·x⊥ 2 e k k⊥ d modes. In order not to rely on particular initial field configu- A(uTM) = √ ⊥ n + i g(TM)(z) k⊥ ε z ε k⊥ rations, let us focus on the inevitable quantum fluctuations of 4πk⊥ 3 sin κ ⊥ dz  + ⊥κ/ − − ⊥κ/ these modes. × (e t isμ 2 + e t isμ 2 ), (67) <κ<π (TM) where, again, 0 , gk⊥ is normalized according to 1. Unstable TE modes (TM) (TM) Repeating the procedure which led us from Eq. (42)to μ⊥  = δ , dz (z) gk⊥ (z) g k⊥ (z) (68) (uTE) I Eq. (47) for the stable modes, unstable TE modes Ak⊥ ⊥ properly orthonormalized according to and sμ is the sign of the integral above. Calculating the electric (uTM) (uTM) Ek⊥ and magnetic Bk⊥ fields associated to these modes, (uTE) (uTE) (uTE) (uTE) A , A  =−A , A  we have k⊥ k⊥ k⊥ k⊥ ik⊥·x⊥ 2  (uTM) e k⊥ k⊥ d (TM) = δ δ(k⊥ − k⊥), (59) =− √ + Ek⊥ nz i gk⊥ (z) 4πk⊥  sin κ ε ε⊥ dz (uTE), (uTE) = A A  0 (60) t+is⊥κ/ −t−is⊥κ/ k⊥ k⊥ × (e μ 2 − e μ 2 ), (69) √ μ⊥  (uTM) i ik⊥·x⊥ (TM) (and orthogonal to all other modes) read as (up to a time = √ × ⊥ Bk⊥ (nz k ) e gk⊥ (z) translation) 4πk⊥ sin κ  + ⊥κ/ − − ⊥κ/ × (e t isμ 2 + e t isμ 2 ). (70) k⊥ × n A(uTE) = √ z eik⊥·x⊥ g(TE)(z) k⊥ k⊥ The modes given by Eqs. (61) and (67), if present, must 4πk⊥  sin κ be added to the expansion of the field operator Aˆ given  − ⊥κ/ − + ⊥κ/ ×(e t isε 2 + e t isε 2 ), (61) in Eq. (55), along with their complex conjugates, with cor- (uJ) (uJ)† responding annihilationa ˆk⊥ and creationa ˆk⊥ operators, <κ<π (TE) J ∈{TE, TM}. The resulting operator expansion can then be with 0 , gk⊥ normalized according to used to calculate electromagnetic-field fluctuations and corre- lations. In the presence of unstable modes, it is easy to see (TE) (TE)  −1 ε⊥  = δ , that the field’s vacuum fluctuations are eventually (t ) dz (z) gk⊥ (z) g k⊥ (z) (62) I dominated by these exponentially growing modes. Obviously, this instability cannot persist indefinitely as these wild fluctu- ⊥ and sε being the sign of the integral above. Calculating the ations will affect the medium’s properties, supposedly leading (uTE) (uTE) the whole system to a final stable state. In some gravitational electric Ek⊥ and magnetic Bk⊥ fields associated to these modes, we have contexts, stabilization occurs by decoherence of these growing vacuum fluctuations [13], giving rise to a nonzero classical √  field configuration, a phenomenon called spontaneous scalar- (uTE) ik⊥·x⊥ (TE) E = √ (n × k⊥) e g (z) ization (for spin-0) [14–17] or vectorization (for spin-1 fields) k⊥ π κ z k⊥ 4 k⊥ sin [18]. It is possible that something similar might occur in the  − ⊥κ/ − + ⊥κ/ × (e t isε 2 − e t isε 2 ), (63) analogous system. We shall discuss this point further in Sec. V. ik⊥·x⊥ (uTE) e 2 d (TE) = √ − + ⊥ Bk⊥ ik⊥nz k gk⊥ (z) 4πk⊥  sin κ dz B. Example  − ⊥κ/ − + ⊥κ/ × (e t isε 2 + e t isε 2 ). (64) Let us consider a very simple system just to illustrate the results above in a concrete scenario: a slab of width L (in the region −L/2 < z < L/2), made of a homogeneous material 2. Unstable TM modes 2 2 with, say, μ⊥ < 0 for a given ω =− (>0) and all Now, turning to the TM modes, we repeat the procedure other constitutive functions positive. For concreteness sake, which led us from Eq. (49)toEq.(53) for the stable modes. here we assume that this value ω2 =−2 is isolated and (uTM) ω2 μ < Unstable TM modes Ak⊥ properly orthonormalized accord- that it is the most negative value of for which ⊥ 0. ing to This latter assumption is merely a matter of choice, while the former only affects the measure on the set of quantum 2  ⊥ ⊥ → θ π ⊥/ ⊥ δ ⊥ − → (uTM) (uTM) (uTM) (uTM) numbers k : d k d k⊥ 2 k L , (k k⊥) A , A   =−A , A    k⊥  k⊥ k⊥  k⊥ δ  δ θ − θ / π L⊥ k⊥k⊥ ( ) (2 k⊥), where L⊥ is the length scale asso-   = δ δ(k⊥ − k⊥), (65) ciated with the area of the “infinite” slab (L⊥ L). According to the discussion presented earlier, in this sce- (uTM) (uTM) A , A   = 0 (66) nario, TE modes incur in case (i) (large-k⊥) instability, while k⊥  k⊥ TM modes undergo case (ii) (minimum-width) instability. The

013281-7 RIBEIRO AND VANZELLA PHYSICAL REVIEW RESEARCH 2, 013281 (2020)

FIG. 1. Effective potential well which represents the homoge- FIG. 2. Graphic representation of solutions of Eqs. (73)and neous slab with negative μ⊥ for the unstable electromagnetic modes. (74). The solid black curves in the upper (respectively, lower) half plane represent the left-hand side (lhs) of Eq. (73) [respectively, minus the lhs of Eq. (74)], with am replaced by the variable a,for (J) ω2 =−2  solutions gk⊥ of Eqs. (42) and (49) with are different values of L. The dashed blue lines (respectively, dotted given by the normalizable [according to Eqs. (62) and (68)] red lines) represent the function − tan a (respectively, cot a). The solutions of the null-eigenvalue, Schrödinger-type equation values am appearing in Eq. (72) are determined by the crossing of the corresponding solid black curve with the dashed blue lines (for m d2 − + V g(J) = 0, even) and the dotted red lines (for m odd). dz2 eff k⊥

   | |/ with Veff being the well potential represented in Fig. 1.The while for the TM modes, 0 am L n 2 and depth of the potential is given by  √ 2L2 2 (TM) −2 2 |μ⊥| 2 ε⊥ε |n |[1 +|n⊥ | ] − 1 |μ⊥|ε⊥ + k⊥, J = TE 2 = μ 4am V0 2 ε⊥ 2 (71)  |μ⊥|ε⊥ − k⊥, J = TM. ε , = tan am m even − , . (74) Although here we focus only on unstable modes, associated cot am m odd (J) with g , note that in this example there would also appear k⊥ The transverse momentum k⊥ is given in terms of am by (TE) μ⊥ < ⎧  stationary bound solutions associated with fω ⊥ (if 0 0k μ 2 2 (TE) ⎨ 2 2 −| 2| L , ω ∈ 2 > {ω2, ω 2} |μ | am n TE modes for some 0 R), for some k⊥ max 0 (n⊥ 0 ) , where (m) L  ⊥ 4 √ k⊥ = k⊥ := (75) (TE) ⎩ ε 2 2 n⊥ := μ ε⊥ is the transverse refractive index for the TE 2 | 2| L − 2 , . ε n am TM modes modes. For such a hypothetical mode, the slab would act as L ⊥ 4 N (J) a waveguide, keeping the mode confined due to total internal The explicit form of m is not particularly important, so reflections at its boundaries. The only peculiar feature here we only present its asymptotic behavior for k⊥ →∞for the is that k⊥ would assume arbitrarily large values (in practice, TE modes ⎧ limited only by the inverse length scale below which the +|μ |μ ω ⎨ 2(1 ⊥ ) , m even continuous-medium idealization breaks down) for a given 0. (TE) Lε⊥ N ≈  , k⊥   (76) Back to the unstable modes, a straightforward calculation m ⎩ +|μ⊥|μ 2(1 ) , m odd leads to the familiar even and odd solutions to the square-well Lε⊥|μ⊥|μ potential, with g(J) (z) exponentially suppressed for |z| > L/2 k⊥ and for k⊥ → 0 for both TE and TM modes and ⎧ ⎧ ⎨ 2(ε⊥+|μ⊥|) N (J) , m even ⎨ m Lε2 cos(2amz/L), 0  m even N (TE) ≈  ⊥ ,   (J) cos am m k⊥ (77) g (z) = (72) ⎩ 2(ε⊥+|μ⊥|) k⊥ N (J) , m odd ⎩ m Lε⊥|μ⊥| sin(2amz/L), 1  m odd  sin am ⎧ ⎨ 2(ε⊥+|μ⊥|) , |μ |2 m even (−L/2  z  L/2), where N (J) are normalization constants N (TM) ≈  L ⊥ ,  . m m k⊥ (78) 2 ⎩ 2(ε⊥+|μ⊥|) and, for the TE modes, a  L|n |/2 (with n := μ⊥ε⊥)are , m odd m Lε⊥|μ⊥| solutions of the transcendental equations  In Fig. 2, we plot, for different values of L and given 2 2 μ μ⊥ ε ε⊥ L (TE) − values of , , , and , the left-hand side of Eq. (73) |μ⊥|μ 1 − |n2|[1 − (n ) 2] 2 ⊥ (solid black curves in the upper half plane) minus the left-hand 4am  side of Eq. (74) (solid black curves in the lower half plane), − tan a , m even substituting, in both a by the variable a and the functions = m (73) m cot am, m odd − tan a and cot a (blue dashed lines and red dotted lines,

013281-8 ANALOGUES OF GRAVITY-INDUCED INSTABILITIES IN … PHYSICAL REVIEW RESEARCH 2, 013281 (2020) respectively). Crossing of the blue dashed lines (respectively, than some minimum width L0, given by red dotted lines) with a fixed solid black curve determines val-  (J) −1 = 2 ε⊥ ues a am for even (respectively, odd) solutions gk⊥ ,forthe −1 L0 = tan . (79) corresponding value of L. The figure clearly corroborates |n | |μ⊥| our preliminary analysis, showing that unstable TE modes appear with arbitrarily large values of am (and, therefore, of The unstable TE and TM modes inside the slab can then be k⊥) and that unstable TM modes only appear if L is larger put in the form

(TE) (m) N (k⊥ × nz ) cos (2amz/L + mπ/2) (m)·  − κ / − + κ / A(uTE) = m √ eik⊥ x⊥ (e t i m 2 + e t i m 2 ), m  m(TE) (80) k(m) (m) + π/ ⊥ 4πk⊥  sin κm cos (am m 2)

(m) (TM) ik ·x⊥ N e ⊥  − κ / − + κ / (uTM) = m t i m 2 + t i m 2 A (m) (e e ) k⊥ 4π 3 sin κ m k(m)n cos (2a z/L + mπ/2) k(m) 2a sin (2a z/L + mπ/2) × ⊥ z m − i ⊥ m m , 0  m  m(TM) (81) ε + π/ (m) + π/ cos (am m 2) ε⊥k⊥ L cos (am m 2) with    L 2 ε⊥ m(TE) := 1 + − 1 tan−1 , (82) L0 π |μ⊥|    L 2 ε⊥ m(TM) := − 1 tan−1 (83) L0 π |μ⊥| (x represents the smallest integer larger than, or equal to, x, while x represents the largest integer smaller than, or equal to, x). The corresponding electric and magnetic field modes are √ (TE) (m) N (nz × k⊥ )  cos (2amz/L + mπ/2) (m)·  − κ / − + κ / E(uTE) = m √ eik⊥ x⊥ (e t i m 2 − e t i m 2 ), m  m(TE) (84) k(m) (m) + π/ ⊥ 4πk⊥ sin κm cos (am m 2)

(TE) ik⊥·x⊥ N e  − κ / − + κ / (uTE) = m√ t i m 2 + t i m 2 B (m) (e e ) k⊥ 4π  sin κ m cos (2a z/L + mπ/2) k(m) 2a sin (2a z/L + mπ/2) × −ik(m)n m − ⊥ m m , m  m(TE) (85) ⊥ z + π/ (m) + π/ cos (am m 2) k⊥ L cos (am m 2)

(m) (TM) ik ·x⊥ −N e ⊥  − κ / − + κ / (uTM) = m√ t i m 2 − t i m 2 E (m) (e e ) k⊥ 4π  sin κ m k(m)n cos (2a z/L + mπ/2) k(m) 2a sin (2a z/L + mπ/2) × ⊥ z m − i ⊥ m m , 0  m  m(TM) (86) ε + π/ (m) + π/ cos (am m 2) ε⊥k⊥ L cos (am m 2) √ (TM) (m) iN μ⊥ (nz × k⊥ ) (m)· cos (2amz/L + mπ/2)  − κ / − + κ / B(uTM) = m √ eik⊥ x⊥ (e t i m 2 + e t i m 2 ), 0  m  m(TM). (87) k(m) (m) + π/ ⊥ 4πk⊥ sin κm cos (am m 2)

Let us recall that these modes give information about netic field in flat space-time in the presence of arbitrary plane- fluctuations and correlations of the electromagnetic field; as symmetric anisotropic polarizable and magnetizable media at long as decoherence does not come into play, the expectation linear order. The vacuum of such system was then identified values of the field are null, Aˆ =Eˆ =Bˆ =0. We shall with the vacuum of some nonminimally coupled spin-1 field use these modes later, when discussing possible consequences in a true√ curved space-time described by the effective metric −2 of these analogue instabilities. But first, let us explore more gαβ = n diag(−n , 1, 1, 1). The analysis had the advan- interesting analogies. tage of generalizing in a unified language the quantization of various interesting models coming from quantum optics in terms of simple equations (e.g., the Carniglia-Mandel modes IV. SPHERICALLY SYMMETRIC, STATIONARY [38]). However, the analogue space-time for these configura- ANISOTROPIC MEDIUM tions is of mathematical interest only and does not capture In the previous section, we presented with great amount of the symmetry of physical space-times. In order to study more detail the canonical quantization scheme for the electromag- appealing analogues, in this section we turn to spherically

013281-9 RIBEIRO AND VANZELLA PHYSICAL REVIEW RESEARCH 2, 013281 (2020) symmetric configurations, presenting them in a more concise above can be made to represent Schwarzschild space-time by way, for the nuances of the quantization were already ex- tuning v so that F ≡ (1 − rs/r), where rs is some positive plained previously. In this context, we may obtain interesting constant. This is achieved by a velocity field satisfying v2 = 2 −1 analogues by also assuming that the medium is able to flow. If [1 + (n − 1)r/rs] (n = 1). the refractive index in a flowing material is high enough, such Despite this apparent simplification, the coordinate τ = that the velocity of light becomes smaller than the medium’s t − p(r) with p satisfying Eq. (91) is not convenient to express velocity, then it is clear that a sort of event horizon will form Maxwell’s equations in anisotropic media. This is due to the (restricted only to some frequency band which may contain kinematic polarization (respectively, magnetization) caused unstable modes). This kind of phenomenon enables us to by the magnetic (respectively, electric) field. In the case of study analogues of unstable black holes, for instance. small velocities and isotropic materials, this effect is modeled We start working in standard spherical coordinates by Minkowski’s equations [45]. The coordinates (τ,r,θ,ϕ) 2 2 2 (t, r,θ,ϕ), such that ημν = diag(−1, 1, r , r sin θ ). Let the defined using Eq. (91) “diagonalizes” only the isotropic part μ medium’s four-velocity field be v = γ (1, v, 0, 0), where v = of the theory and do not take into account the anisotropies. − / v(r) and γ = (1 − v2 ) 1 2. The effective-metric components It turns out that a much better choice is obtained by setting then take the form τ = − ⎛ ⎞ : t p(r) and replacing condition given in Eq. (91)by 2 −2 2 −2 2 −γ (n − v ) −(1 − n )γ v 00 2 dp (n − 1)v √ ⎜− − −2 γ 2 γ 2 − −2 2 ⎟ =− , ⎜ (1 n ) v (1 n v )0 0 ⎟ 2 (93) gαβ = n⎜ ⎟, dr 1 − n v2 ⎝ 00r2 0 ⎠ 2 2 2 000r sin θ where, again, n := μ⊥ε⊥. This choice fully decouples the electromagnetic field modes in the anisotropic, moving ma- (88) terial medium, as we shall see below. where the isotropic parts of the constitutive tensors (in the Introducing again the four-potential Aμ via Fμν = ∂μAν − local, instantaneous rest frame of the medium) are functions ∂ν Aμ, in these new coordinates (τ,r,θ,ϕ), the convenient of r [ε = ε(r),μ= μ(r)] and, as usual, n2 = με.Asforthe (generalized Coulomb) gauge conditions read as Aτ = 0 and ab (μ) traceless anisotropic tensors χ ε and χ , their components 2 ( ) ab ∂ (ε r A ) + ∂⊥ · A⊥ = 0, (94) read as r αβ β α  /  ≡ χ = ((ε)/3) 2γ 2v2δαδ + 4γ 2vδ( δβ) + 2γ 2δαδβ where is merely an auxiliary variable such that dr d (ε) t t t r r r γ 2 − 2 2 /ε = , ∂ (1 n v ) ⊥, A⊥ (Aθ Aϕ ), ⊥ is the derivative oper- α β −2 α β −2 −2 − δθ δθ r − δϕ δϕ r sin θ (89) ator on the unit sphere compatible with its metric, and it is understood that r is a function of the auxiliary variable .In and this gauge, Maxwell’s equations lead to (μ) μ χ = ( )/ γ 2v2δt δt − γ 2vδt δr + γ 2δr δr αβ ( 3) 2 α β 4 (α β) 2 α β μ γ 2 − 2 2 ⊥ (1 n v ) (0) ϕ − ∂2 + ∂2 +  ε 2 = , − δθ δθ 2 − δϕδ 2 2 θ . τ  S ( r Ar ) 0 (95) α β r α β r sin (90) ε⊥ ε⊥ε r2 Similarly to the plane-symmetric case, these anisotropic ten- ε γ 2(1 − n2v2 ) − ⊥ ∂2 + ∂2 + (1) − sors simply mean that in the instantaneous local rest frame τ ρ S 1 A⊥ μ⊥ μ⊥μ r2 of the medium, its electric permittivity and magnetic per- meability in the radial direction (ε and μ ) and in the dr μ⊥ dr = ∂ ∂ − ∂ ε 2 , ε⊥ μ⊥ ⊥ ρ Ar ρ ( r Ar ) (96) angular directions ( and ) satisfy the same relations dρ r2μ ε⊥ dρ (ε) given below Eqs. (37)–(39): ε − ε⊥ ≡  ,(2ε⊥ + ε )/3 ≡ −1 −1 (μ) −1 −1 −1 ρ ε, μ − μ⊥ ≡  , and (2μ⊥ + μ )/3 ≡ μ . where appearing in Eq. (96) is another auxiliary variable de- / ρ ≡ γ 2 − 2 2 /μ (0) (1) Not surprisingly, the laboratory coordinates (t, r,θ,ϕ)are fined through dr d (1 n v ) ⊥ and S and S not the most convenient ones to express Eqs. (18) and (19)in are the Laplacian operators defined on the unit sphere, acting the case of a moving medium. One might initially think that on scalar and covector fields, respectively. coordinates (τ,r,θ,ϕ) which diagonalize the components of In order to solve these equations, we proceed in close the effective metric, obtained by defining τ := t − p(r), with analogy to the plane-symmetric case. First, let us find solu- (TE) p(r) satisfying tions with Ar = 0, the transverse electric modes A .The 2 gauge conditions imply that these solutions can be written as dp (n − 1)v (TE) = ,∂ ψ/ θ,− θ∂ ψ ψ =− , (91) A (0 ϕ sin sin θ ), where is an auxiliary dr 1 − n2v2 function to be determined. Making use of the stationarity and would lead to the simplest form of the field equations. In these spherical symmetry of the present scenario, we can look for 2 −iωτ (TE) coordinates, the effective line element ds becomes field modes of the form ψ = e Ym(θ,ϕ) fω (r), where √ eff − − Ym are the scalar spherical harmonics. Substituting this into ds2 = n[−n 2Fdτ 2 + F 1dr2 + r2(dθ 2 + sin2 θ dϕ2 )], (TE) eff Eq. (96), fω must satisfy (92) $ % 2 γ 2(1 − n2v2 )( + 1) ε ω2 − d + − ⊥ (TE) = , where F = γ 2(1 − n2v2 ). It is noteworthy that for n = fω 0 dρ2 r2μ⊥μ μ⊥ > 2 constant 0 (such that the factors of n in dseff can be ab- sorbed via τ → n3/4τ and r → n−1/4r), then the line element (97)

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(TM) where it is understood that r is a function of the auxiliary with fω satisfying Eqs. (102) and (103), and normalized variable ρ. Notice the similarity between this equation and according to Eq. (42). In fact, the boundary condition given by Eq. (22) assumes the same form here as it does in the plane-symmetric (TM) (TM) dρ f f  = δωω . (105) case: ω ω  Iρ (TE) d (TE) (TE) d (TE) f f  − f  f = . ω ρ ω  ω  ρ ω 0 (98) d d I˙ Similarly to the TE case, note that the integration variable is This boundary condition ensures that these modes can be not the same which appears in the differential equation (102). orthonormalized according to the sesquilinear form given (Iρ stands for the domain of integration in the variable ρ in Eq. (21), which in this spherically symmetric scenario corresponding to I in coordinate r.) assumes the form The electromagnetic field operator can be represented in  terms of the TE and TM modes (and their complex conjugates)   ε⊥A¯ ⊥ · ∂τ A⊥ , = ε ¯ ∂τ + as A A i d Ar Ar 2 γ 2(1 − n v2 ) t & 2 2 ˆ = ω (J) (J) + . . , γ (n − 1)v A d aˆωmAωm H c (106) + ¯ · ∂  − ¯ · ∂  E (J) [A⊥ rA⊥ (A⊥ ⊥)Ar] J∈{TE,TM} m + μ⊥ − ¯ ↔  , (J) (J) ∗ (J) (A A ) (99) where E+ := E ∩ R+, with E being the set of ω values for which Eqs. (97) and (98), for J = TE, and Eqs. (102) and with being a spacelike surface t = constant. After some t (103), for J = TM, have nontrivial solutions. The orthonor- tedious but straightforward manipulations (presented in the mality of TE and TM modes, Appendix), we obtain the final form of normalized, positive- frequency TE modes: (J) , (J ) =− (J) , (J ) (0, im/ sin θ,− sin θ∂θ ) Aω Aω  Aω Aω  (TE) = √ −iωτ θ,ϕ (TE) , m m m m Aωm e Ym( ) fω (r) 2ω( + 1) = δJJ δωω δ δmm , (107) (100) (J) , (J ) = , Aωm Aωm 0 (108) (TE) with fω satisfying Eqs. (97) and (98), and normalized according to requires that the canonical commutation relations

(TE) (TE)   d f f  = δωω . (101) (J) (J )† ω ω  , = δJJ δ  δ  δ  , I aˆωm aˆωm ωω  mm (109)   ρ (J) , (J ) = Note that the integration variable is [instead of appearing aˆωm aˆωm 0 (110) in Eq. (97)] and I stands for the domain of integration in this variable corresponding to I in coordinate r. hold. Now, let us look for solutions with Ar ≡ 0, the trans- (TM) φ (0)φ = verse magnetic modes A .Let be such that S 2 (TM) −r ε Ar. Thus, the gauge conditions lead to A = A. Instability analysis − −2ε−1(0),∂ ∂ ,∂ ∂ φ ( r S θ  ϕ  ) . Using again stationarity and The close similarity between Eqs. (42) and (97) and be- −iωτ (TM) tween Eqs. (49) and (102) makes the instability analysis in this spherical symmetry φ = e Ym(θ,ϕ) fω (r), we obtain (TM) spherically symmetric scenario essentially identical to the one that f (r) satisfies ω $ % performed in the plane-symmetric case, with ( + 1) playing 2 γ 2(1 − n2v2 )( + 1) μ ω2 2 − d + − ⊥ (TM) = . the role k⊥ didinEq.(58). So, putting the effective potentials fω 0 ω2 =−2 d2 r2ε⊥ε ε⊥ of Eqs. (97) and (102), with ,intheform

(102) 2 Veff = C1( + 1) + C2 , (111) Notice, again, the similarity between this equation and Eq. (49). And, again, the boundary condition imposed by we again have two types of instabilities: (i) large- instability, Eq. (22) to these modes takes the same form as in the plane- when C1 < 0 somewhere, and (ii) minimum-thickness insta- bility, when C > 0butC < 0 in a sufficiently thick spherical symmetric case: 1 2 shell [see discussion below Eq. (58)]. The only additional 2 (TM) d (TM) 2 (TM) d (TM) ω fω fω − ω fω fω = 0. (103) feature is that, by allowing the medium to flow, type-(i) (large- d d I˙ ) instability for both TE and TM modes can arise when the −1 Properly orthonormalizing these modes using Eq. (A1) (see medium’s velocity v(r) exceeds the radial light velocity n . Appendix) leads to the positive-frequency TM normal modes (J) Let g represent the solutions of Eqs. (97)(forJ= TE) −2 −1 (r ε ( + 1),∂θ ∂, im∂ ) and (102) (for J = TM), subject to the boundary conditions (TM) = Aωm given by Eqs. (98) and (103), respectively, with ω2 =−2 2ω3( + 1) (>0, without loss of generality). The normalized, unstable −iωτ (TM) × e Ym(θ,ϕ) fω (r), (104) modes are presented below (see Appendix for details).

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1. Unstable TE modes (TE) with κ being a constant (0 <κ<π), g normalized ac- Unstable TE modes orthonormalized according to the anal- cording to ogou of Eqs. (107) and (108) read as (up to global phase and ε ⊥ (TE) (TE) time translation) dr g (r) g  (r) = δ , (113) 2 2 2    I γ (1 − n v ) τ− ⊥κ/ −τ+ ⊥κ/ isε 2 + isε 2 (uTE) (e e ) (TE) ⊥ A = √ g (r) and sε being the sign of the integral above. Calculating the m 2 ( + 1)sin κ (uTE) (uTE) electric Em and magnetic Bm vector fields associated to × (0, im/ sin θ,− sin θ∂θ )Ym(θ,ϕ), (112) these modes in the laboratory frame,wehave

√ (−im eθ / sin θ + eϕ ∂θ ) ⊥ ⊥ (uTE) (TE) τ−isε κ/2 −τ+isε κ/2 E = √ g (r)Ym(θ,ϕ)(e − e ), (114) m 2r ( + 1)sin κ [( + 1)e + im eϕ/ sin θ + eθ ∂θ r∂ ] ⊥ ⊥ (uTE) r r (TE) τ−isε κ/2 −τ+isε κ/2 B = √ g (r)Ym(θ,ϕ)(e + e ). (115) m 2r2 ( + 1)sin κ

2. Unstable TM modes (TM) with, again, κ being a constant (0 <κ<π), g normalized Finally, the unstable TM modes orthonormalized according according to to the analogues of Eqs. (107) and (108) read as (up to global μ ⊥ (TM) (TM) phase and time translation) dr g (r) g  (r) = δ , (117) 2 2 2    I γ (1 − n v ) −2 −1 (r ε ( + 1),∂θ ∂, im∂ ) (uTM) = (TM) ⊥ A g (r) and sμ being the sign of the integral above. Calculating the m 2 3( + 1)sin κ electric E(uTM) and magnetic B(uTM) vector fields associated ⊥ ⊥ m m τ+isμ κ/2 −τ−isμ κ/2 ×Ym(θ,ϕ)(e + e ), (116) to these modes in the laboratory frame,wehave

[( + 1)e /ε + (im eϕ/ sin θ + eθ ∂θ )r∂] ⊥ ⊥ (uTM) r (TM) τ+isμ κ/2 −τ−isμ κ/2 E =− √ g (r)Ym(θ,ϕ)(e − e ), (118) m 2r2 ( + 1)sin κ √ μ  − / θ + ∂ (uTM) ⊥ ( im eθ sin eϕ θ ) (TM) τ+is⊥κ/2 −τ−is⊥κ/2 B = √ g (r)Y (θ,ϕ)(e μ + e μ ). (119) m 2 2 2  m 2rγ (1 − n v ) ( + 1)sin κ

As argued in the previous case, when instability is triggered which lead to the material properties (uJ) ' α ( and modes A appear, they must be included in the field 1rs m ε⊥ = n 1 − , (123) expansion given by Eq. (106), along with their complex n1/2r3 − conjugates. Eventually (t   1), these modes dominate the 2α1rs field fluctuations. ε = n 1 + , (124) n1/2r3 ' (− α1rs 1 μ⊥ = n 1 − , (125) B. Example n1/2r3 −1 Now, let us consider a concrete scenario where electromag- 2α1rs μ = n 1 + . (126) netism in a gravitationally interesting system, nonminimally n1/2r3 coupled to the background geometry via χ abcd given by √ We promptly see that n := μ⊥ε⊥ = n, which shows α ,α ,α Eq. (23) (but with arbitrary 1 2 3), can be mimicked by that the analogue horizon for these nonminimally coupled an anisotropic, stationary moving medium. We have already modes, located where v2 = n−2, coincides with the analogue = 2 = + 2 − / −1 seen that setting n constant and v [1 (n 1)r rs] Schwarzschild radius r . [Note, however, that this system leads to an effective line element which describes the vacuum s is analogous to a physical black hole√ with Schwarzschild Schwarzschild space-time. In this case, Eq. (29) is trivially 1/4 radius Rs = n rs, due to absorption of n in Eq. (92).] As satisfied and Eqs. (26)–(28)give (TE) √ (TM) for the other refractive indices n := μ ε⊥ and n := √ ⊥ ⊥ 2 (TE) μ⊥ε (= n /n ), Fig. 3 shows their squared values (in μ = , ⊥ n (120) black and red, respectively) for positive (solid lines) and r α (ε) = 3α n1/2 s , (121) negative (dashed lines) values of 1. Note that, depending 1 3 α / 1/2 2 r on the values of 1 (n rs ), some kind of metamaterial 3α r (possibly with some negative squared refractive indices) may (μ) = 1 s , (122) n3/2 r3 be needed in order to mimic this nonminimal coupling of the

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electromagnetic field with the Riemann curvature tensor in the exterior region of a Schwarzschild black hole. Conversely, regardless how difficult it may be to set up such an experimen- tal configuration in the laboratory, it is interesting in its own that QED-inspired nonminimally coupled electromagnetism in the background of a black hole behaves as in such an exotic metamaterial in flat space-time. Turning to the question of possible instabilities, in Fig. 4 we show the behavior of the terms C1 and C2 appearing in Eq. (111) for the TE (in blue) and TM (in red) modes extracted, respectively, from Eqs. (97) and (102): ⎧ α α ⎨ −2 −9 3 √1rs 3 2√1rs n r (r − rs ) r − r + , = n n C1 −2 − 3 (127) ⎩ n√ (r rs )r √ , 3 3 (r −α1rs/ n)(r +2α1rs/ n) (TE) FIG. 3. Squared values of the refractive indices n (in black) α 2 ⊥ 1 − 1√rs , and n(TM) (in red) for positive (solid lines) and negative (dashed lines) = r3 n ⊥ C2 α −2 (128) α (TE) − 1√rs , values of 1. The black and red dotted lines mark where n⊥ (for 1 r3 n (TM) negative α1)andn⊥ (for positive α1) are singular, respectively. where the first and second lines in the expressions above refer to the TE and TM modes, respectively. Figure 4(a) is

FIG. 4. Plot of the coefficients C1 [(a) and (b)] and C2 [(c) and (d)] appearing in Eq. (111) for electromagnetic modes TE (blue curves) and TM (red curves), nonminimally coupled√ to the background√ geometry of a Schwarzschild black hole via Eq. (23). (a), (c) Illustrate√ the − 2 / <α < 2 α < − 2 / general behavior√ of C1 and C2 for rs n 2 1 rs n, while (b), (d) are representative of the behavior of C1 and C2 for 1 rs n 2 α > 2   or 1 rs√ n. According to√ the instability discussion, only large- instability can appear in this case since C2 0 everywhere. Moreover, for α < − 2 / α > 2 > 1 rs n 2or 1 rs n, the unstable modes can be mostly supported outside the analogous event horizon r rs.

013281-13 RIBEIRO AND VANZELLA PHYSICAL REVIEW RESEARCH 2, 013281 (2020) √ − 2 / <α < representative√ of the behavior of C1 for rs n 2 1 the analogous systems), the qualitative features of the whole 2 rs n, while Fig. 4(b)√ gives the correct√ qualitative behavior process, described above, seem quite reasonable to occur in α < − 2 / α > 2 of C1 for 1 rs n 2or 1 rs n. Figures 4(c) and 4(d) generic field stabilization processes. show the behavior of C2 for the same values of α1 used in It is important to mention that the timescale set by the − Figs. 4(a) and 4(b). instability  1 is typically of the order of the time light It is clear, from the expressions above, that C2 is every- takes to travel the typical size of the system, L. Therefore, in where non-negative, while C1 assumes negative√ values in the the analogous laboratory scenarios, the stabilization process α / 1/3 ∼ / × −10 region with radial coordinate r between√ ( 1rs n) and rs would occur almost instantaneously [ L (1 cm) 10 s] 1/3 (if α1 > 0) or between [|α1|rs/(2 n)] and rs (if α1 < 0). once the instability conditions are met, which, for a given sys- Therefore, according to the discussion of Sec. IV A, this non- tem, may depend on external parameters such as temperature, minimally coupled electromagnetic theory in Schwarzschild external fields, etc., through their influence on the constitutive  α > ε ε μ μ space-time√ exhibits√ large- instability. In particular, if 1 functions ⊥, , ⊥, . The whole process would most likely 2 α < − 2 rs n or 1 2rs n, then the unstable modes influence the be interpreted as a kind of phase transition, where the “long- exterior region of the back hole. range” emergent correlations in the material would come from interaction of its constituents with a common (initially V. STABILIZATION: SPONTANEOUS VECTORIZATION, unstable vacuum) fluctuating mode and/or the stabilized field PHOTO PRODUCTION, AND LONG-RANGE configuration. INDUCED CORRELATIONS For concreteness sake, let us consider the explicit form of the unstable modes found in the example of Sec. III, where We now turn our attention to discussing what can possibly instability occurs due to a negative value of μ⊥,forsome happen to the analogous system when the vacuum instability (isolated) ω2 =−2 < 0, in a homogeneous slab of width L. is triggered. In the gravitational scenario, it has been shown Although this system is not analogous to vacuum nonmini- that in some cases (for instance, depending on the field- mally coupled electromagnetism in any realistic space-time, background coupling), stabilization occurs due to the appear- it serves to illustrate general features of the mechanism itself, ance of a nonzero value for the field (spontaneous scalariza- in addition to being much simpler to set up in the laboratory. tion or vectorization) [14–18], seeded by decoherence of the This is no different than looking for fingerprints of analogue growing initial-vacuum fluctuations [13]. In this process, field Hawking radiation in systems whose only similarity with particles and waves are produced [14,28] and carry away the realistic black holes is the presence of an effective event energy excess of the initial vacuum state in comparison to the horizon, which is the common approach in condensed-matter stabilized configuration. and optical experimental analogues. mutatis mutandis If we transpose these conclusions, ,to As argued before, once instability sets in, the unstable our analogous systems, then an electromagnetic field should modes must be added to the expansion of the field operator spontaneously appear in the material, bringing the whole Aˆ , along with their complex conjugates, with corresponding system to a new equilibrium configuration through nonlinear annihilationa ˆ (uJ) and creationa ˆ (uJ)† operators. It is easy to effects brought in by field-dependent constitutive tensors εab k⊥ k⊥ μ see that the field’s vacuum fluctuations and correlations are and ab [see Eqs. (7) and (8)], with photons being emitted, eventually (t, t   −1) dominated by these unstable modes, carrying away the energy excess. Although the detailed dy- at least as long as decoherence does not come into play. namics of the stabilization processes in the gravitational and The dominant contribution to the vacuum correlations in the in the analogous systems are quite different (ruled by Einstein example of Sec. III B reads as (the reader should refer to equations in the gravitational case and by the macroscopic εab μ Sec. III B for the definition of all quantities appearing in these Maxwell’s equations with field-dependent and ab in expressions) ⎧ ⎫ (TM) π 2π ⎨ m ∞ ⎬   ∼2 ϕ (m) (uTM) (uTM)  + (m) (uTE) (uTE)  A j (x)Al (x ) d k A (m) (x) A (m) (x ) k A (m) (x) A (m) (x ) ⎩ ⊥ k j k l ⊥ k j k l ⎭ L⊥ 0 ⊥ ⊥ ⊥ ⊥ m=0 m=m(TE) ⎧ ⎫ (TM) (t+t  ) ⎨ m (m) ∞ (m) ⎬ e k⊥ (TM) (TM) (TM)  k⊥ (TE) (TE) (TE)  ∼ D (d⊥)g (z)g (z ) + D (d⊥)g (z)g (z ) , (129) ⎩ jl(m) k(m) k(m) jl(m) k(m) k(m) ⎭ 4L⊥ sin κm ⊥ ⊥ sin κm ⊥ ⊥ m=0 m=m(TE) ϕ −  = −  D(J) (J) (J)  where is the angle between k⊥ and (x⊥ x⊥), d⊥ : x⊥ x⊥ , and the operators jl(m)(d⊥) acting on g (m) (z)g (m) (z )are k⊥ k⊥ defined by (m) 2 (m) 2 (m) (m) (m) (TM) 1  (m)   J1(k⊥ d⊥) ϕ ϕ d (k⊥ ) J0(k⊥ d⊥) z z k⊥ J1(k⊥ d⊥)  z d z  d D (d⊥):= J (k d⊥)δ δ + δ δ + δ δ − δ δ − δ δ , jl(m) 2ε2 1 ⊥ j l (m) j l  2ε2 j l 2ε ε j l j l  ⊥ k⊥ d⊥ dzdz ⊥ dz dz (130)

(m) (TE) J1 k⊥ d⊥    (m) ϕ ϕ D ⊥ = δ δ + ⊥ δ δ , jl(m)(d ): (m) j l J1(k⊥ d ) j l (131) k⊥ d⊥

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 −1  FIG. 5. Equal-time (t = t   ), two-point correlation function A(x)A(x ) of the component of the quantum field Aˆ along the vector  separation x⊥ − x⊥, for points in the z = 0 plane, for different values of L, with same values of constitutive functions given in Fig. 2.The dotted (blue) lines represent the contribution coming from the TE modes, while the dashed (red) lines depict the contribution coming from the  TM modes. The solid (black) lines give the sum of both contributions. Notice that long-range ( x⊥ − x⊥  L) correlations are mainly due to the TM modes, which undergo minimum-width instability. with indices  and ϕ standing for vector components VI. FINAL REMARKS   along (x⊥ − x⊥) and nz × (x⊥ − x⊥), respectively; Jn and  We have shown that gravity-induced instabilities, related Jn stand for the Bessel functions of first kind and their  to the vacuum-awakening effect in the quantum context first derivatives, respectively. Field correlations E j (x)El (x )  [9–11,28] and spontaneous scalarization or vectorization in and B j (x)Bl (x ) can be similarly obtained, in particular,  2  the classical one [14–18], can be mimicked by electromag- E j (x)El (x )∼ A j (x)Al (x ). As an illustration, in Fig. 5  − netism in anisotropic metamaterials with appropriate constitu- we plot the equal-time (t = t   1), longitudinal correla-   tive functions. This follows from the formal analogy between tion function A(x)A(x ) for points x, x in the plane z = 0, electromagnetism in anisotropic media and nonminimally for the same values of constitutive functions used in Fig. 2 and coupled electromagnetism in curved space-times, presented four different values of L. The vertical-axis scale is arbitrary, in Sec. II. We explored two concrete scenarios: (i) a plane- but the same in all plots, since the correlations grow exponen- symmetric, static slab, whose main interest is its simplicity re- tially in time, from their typical (stable-vacuum) values of or- 2 −8 3 2 garding experimental setup (see Sec. III), and (ii) a spherically derh ¯/(cLd⊥) ∼ [1 cm /(Ld⊥)] × 10 eV/(cm GHz ), un- symmetric, moving media, whose main feature is its analogy til decoherence and vectorization take over. Notice that once with QED-inspired nonminimally coupled electromagnetism minimum-width (TM) instability sets in, macroscopic (∼L) in Schwarzschild space-time [33,34] for given velocity and field correlations are enhanced. It is an interesting question constitutive-functions profiles (see Secs. II A and IV). whether any such “long-range” correlation would survive or Once instability is triggered in the analogous systems, leave an imprint in the final stable configuration. Although some stabilization process must take place, leading the system not directly relevant for the analogy with gravity-induced to a new stable configuration. The details of this stabilization instability itself, such correlations might lead to interesting process and of the final configuration will most likely depend material behavior. on specific nonlinear properties of the metamaterial, but it

013281-15 RIBEIRO AND VANZELLA PHYSICAL REVIEW RESEARCH 2, 013281 (2020) seems reasonable that they might involve the appearance of instability. For instance, they can be used to study light nonzero electromagnetic fields in the material (analogous to ray propagation in the corresponding space-times and one spontaneous vectorization in curved space-times) and photo possible application is the QED-induced birefringence in the production which carries away the energy excess with re- Schwarzschild space-time [33]. For this particular experiment, spect to the stable configuration. As discussed earlier, the one can work far from the effective horizon, where the consti- timescale involved in the stabilization process can be very tutive coefficients (123)–(126) are positive. short (∼10−10 s), which would make it very difficult to Our main purpose here was to lay down an unexplored even identify the unstable phase. This is similar to what class of analogue models of curved-space-time phenomena, might occur with negative conductivity, which has never been with main interest on the gravitational side of the analogy. directly measured but which is predicted to lead to zero- Notwithstanding, the consequences of the analogue gravity- dc-resistance states [46] which were observed in laboratory induced instability to the metamaterial side may be inter- [47,48], although an alternative explanation has been pro- esting on its own. The electromagnetic field instability may posed [49]. mark, lead, or mediate some kind of phase transition in the Clearly, the feasibility of such analogues is bound to the metamaterial, where the spontaneously created field and/or existence of material configurations with the required consti- its amplified “long-range” correlations may play some im- tutive functions. As briefly pointed out in the Introduction, this portant role (see discussion in Sec. V). Investigations along can be achieved at least for anisotropic neutral plasmas, and these lines are currently in course and will be presented the recent advances in metamaterial science offer a plethora of elsewhere. possible candidates, especially the hyperbolic metamaterials [42,43], that possess precisely the form given in Eqs. (9) and (10) with the required “negativeness.” In particular, we call ACKNOWLEDGMENT attention to the increase in the spontaneous light emission in such configurations, which may be related to the process of C.C.H.R. was mainly supported by São Paulo Research stabilization in active scenarios. Foundation (FAPESP), through Grant No. 2015/26438-8, and It is also important to mention that the QED-inspired in part by the Coordenação de Aperfeiçoamento de Pessoal de analogues (Sec. II A) are not restricted to the study of vacuum Nível Superior-Brasil (CAPES)-Finance Code 001.

APPENDIX: NORMALIZATION OF STABLE AND UNSTABLE MODES IN THE SPHERICALLY SYMMETRIC CASE Here, we present in detail the calculations involved in normalizing the electromagnetic modes in the spherically symmetric case. Since we are dealing with analogues to which there is a natural physical notion of time, the laboratory-frame time t,itis convenient to use t = constant surfaces ( t ) to normalize the modes. Obviously, this choice bears no physical consequence on our results. The sesquilinear form given in Eq. (21), applied to the scenario described in Sec. IV, takes the form (notice that the integrand is a scalar and, as such, can be evaluated in any coordinate system) &  γ 2 2 −   ε⊥A¯ ⊥ · ∂τ A⊥ (n 1)v    (A, A ) = i d ε A¯ ∂τ A + + A¯ ⊥ · ∂ A − (A¯ ⊥ · ∂⊥)A − (A¯ ↔ A ). (A1) r r γ 2 − 2 2 μ r ⊥ r t (1 n v ) ⊥

Below, we evaluate this expression for each type of mode.

1. TE modes a. Stable (TE) −iωτ (TE) Substituting A = (0,∂ϕψ/sin θ,− sin θ∂θ ψ) into Eq. (A1), with ψ = e Ym(θ,ϕ) fω (r), one gets  ε (TE) (TE) mm i(ω−ω )τ  ⊥ (TE) (TE) , = ∂θ  ∂θ   +    ρ ω + ω   (A A ) dS ( Y m )( Y m ) 2 Y mY m d e ( ) fω fω  S2 sin θ I μ⊥ γ 2 2 − i (n 1)v (TE) d (TE) (TE) d (TE) + fω fω − fω fω , (A2) μ⊥ dρ dρ

2 2 2 2 where S is the unit sphere, recall that dr/dρ = γ (1 − n v )/μ⊥, and it is understood that this last integral must be evaluated at τ + p(r) = t = constant [recall definition of τ right above Eq. (93)]. It is straightforward to show that the first integral evaluates   + δ  δ    = δ  δ  to ( 1)  mm provided we normalize Ym according to S2 dSYmY m  mm . As for the second integral, let us first consider the quantity () 1 (TE) d (TE) (TE) d (TE) W  := f f  − f  f . (A3) ωω (ω − ω) ω dρ ω  ω  dρ ω

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() Making use of Eq. (97), Wωω clearly satisfies ε d () ⊥  (TE) (TE) Wωω = (ω + ω ) fω fω . (A4) dρ μ⊥ Therefore, (TE) (TE) i(ω−ω )τ d ()  dp () A , A = ( + 1)δ δmm dρ e Wωω − i(ω − ω ) Wωω I dρ dρ i(ω−ω )[τ+p(r)] d −i(ω−ω )p(r) () = ( + 1)δ δmm dρ e e Wωω I dρ i(ω−ω )t −i(ω−ω )p(r) () = ( + 1)δ δmm e e Wωω , (A5) I˙ where we made use that t = τ + p(r) is kept constant along integration in r (or ρ) and [...]|I˙ indicates that we must calculate the flux of the quantity in square brackets at the boundaries of I. We see that in order to guarantee orthogonality between modes with different ω, without worrying about the specific form of p(r), we must impose boundary conditions at I˙ such that, in () | = ω = ω Eq. (A3), Wωω I˙ 0for . Then, referring back to Eq. (A4) and writing ρ ε ()   ⊥ (TE) (TE) W  ρ = ω + ω dρ f f  , ωω ( ) ( ) μ ω ω  (A6) ρ− ⊥ we finally obtain (TE) (TE) (TE) (TE) (A , A ) = 2ω( + 1)δ δmm d fω fω , (A7) I which justifies the normalization of the TE modes in Sec. IV. [Notice that the integration variable is , defined through dr/d = 2 2 2 γ (1 − n v )/ε⊥.]

b. Unstable TE modes (uTE) Generic unstable TE modes are given by A = (0,∂ϕψ/sin θ,− sin θ∂θ ψ) with

τ −τ (TE) ψ = (αe + βe )Ym(θ,ϕ)g (r), (A8)

(TE) 2 2 where α and β are complex constants and g (r) is a solution of Eq. (97) with ω =− (>0, without loss of generality) and proper boundary conditions (see below). Sesquilinearity of Eq. (A1) makes it easy to calculate (A(uTE), A(uTE) ) from Eq. (A5) with the appropriate substitution ω →∓i and ω →±i: (uTE) (uTE) (+ )τ (u) −(+ )τ (u) (− )τ (u) (A , A ) = ( + 1)δ δmm ααe W + ββe W− − + αβe W − − − τ (u) + β α  ( ) ,   e W− I˙ (A9) where (u) i (TE) d (TE) (TE) d (TE) W  := g g  − g  g . (A10) ± ± (± ± )  dρ     dρ 

(TE) As with the stable case, we must impose boundary conditions on g (r) such that (TE) d (TE) (TE) d (TE) g g − g g = 0, dρ dρ I˙ (u)| = (u) | =  =  (u) | = (u) | = which implies W I˙ W− − I˙ 0 and, for , W − I˙ W− I˙ 0. Therefore, using the analogue of Eq. (A6), ρ ε (u)   ⊥ (TE) (TE) W  ρ =−i ± ∓  dρ g g  , ± ± ( ) ( ) μ    ρ− ⊥ into Eq. (A9), we finally obtain (uTE) (uTE) (TE) (TE) (A , A ) = 4( + 1)δ δmm Im(αβ) d g g , (A11) I

013281-17 RIBEIRO AND VANZELLA PHYSICAL REVIEW RESEARCH 2, 013281 (2020) where Im(z) stands for the coefficient of the imaginary part of the complex number z. Thus, imposing orthonormality of these (TE) 2 modes, for orthonomalized g [in the L (I, d) inner product], the general expression for α and β (up to rephasing, iδ iδ t −t α → e α, β → e β, and time resetting, α → e 0 α, β → e 0 β) read as e−iκ/2 α = √ , (A12) 2 ( + 1)sin κ eiκ/2 β = √ , (A13) 2 ( + 1)sin κ with 0 <κ<πbeing an arbitrary constant.

2. TM modes a. Stable (TM) = −2ε−1(0),∂ ∂ ,∂ ∂ φ φ = −iωτ θ,ϕ (TM) Now, substituting A (r S θ  ϕ  ) into Eq. (A1), with e Ym( ) fω (r), and evaluating the angular integrals (similarly to the previous TE case), we obtain 2 2 2  γ (1 − n v )( + 1) (TM) (TM) i(ω−ω )τ  d (TM) d (TM) (TM) (TM) (A , A ) = ( + 1)δ δmm d e (ω + ω ) fω fω + fω fω I d d ε ε⊥r2 & iγ 2(n2 − 1)v 2 (TM) d (TM) 2 (TM) d (TM) + ω fω fω − ω fω fω , (A14) ε⊥ d d

2 2 2 where recall that dr/d = γ (1 − n v )/ε⊥. The strategy to simplify the expression above is the same applied in the TE case. Define () 1 2 (TM) d (TM) 2 (TM) d (TM) W  := ω f f  − ω f  f . (A15) ωω (ω − ω) ω d ω  ω  d ω One can easily check, using Eq. (102), that γ 2 − 2 2   + d ()  d (TM) d (TM) (1 n v ) ( 1) (TM) (TM) Wωω = (ω + ω ) fω fω + fω fω . (A16) d d d ε ε⊥r2

Therefore, we can put Eq. (A14) in the same form as Eq. (A5), with W → W and ρ → . Now, orthogonality of the modes (TM) demand that fω satisfy either Dirichlet or Neumann boundary conditions at I˙, which leads to  () (TM), (TM) =   + δ  δ  W . (A A ) ( 1)  mm ωω I˙ (A17) In order to simplify even further the expression above, note that using again Eq. (102)inEq.(A16), we can write ω + ω 2 μ d () ( ) d ⊥ 2 2 (TM) (TM) Wωω = + (ω + ω ) fω fω , (A18) d 2 d2 ε⊥ I W () | whose integration on gives us [ ωω ] I˙ , which substituted into Eq. (A17) finally leads to (TM) (TM) 3 (TM) (TM) (A , A ) = 2ω ( + 1)δ δmm dρ fω fω . (A19) I (Notice that the integration variable is ρ.)

b. Unstable (uTM) = −2ε−1(0),∂ ∂ ,∂ ∂ φ Generic unstable TM modes are given by A (r S θ  ϕ  ) with

τ −τ (TM) φ = (αe + βe )Ym(θ,ϕ)g (r), (A20)

(TM) 2 2 where α and β are complex constants and g (r) is a solution of Eq. (102) with ω =− (>0, without loss of generality) satisfying Dirichlet or Neumann boundary conditions. Once more, sesquilinearity of Eq. (A1) makes it easy to calculate (A(uTM), A(uTM) )fromEq.(A17) with the substitution ω →∓i and ω →±i:  (u) (u) (uTM), (uTM) =   + δ  δ  α β  W + β α  W , (A A ) ( 1)  mm     −    − I˙ (A21)

013281-18 ANALOGUES OF GRAVITY-INDUCED INSTABILITIES IN … PHYSICAL REVIEW RESEARCH 2, 013281 (2020) where − (u) i 2 (TM) d (TM) 2 (TM) d (TM) W  :=  g g  −  g  g ± ± (± ± )  dρ     dρ  ± ∓  ' (  μ i( ) d (TM) (TM) 2 2  ⊥ (TM) (TM) = − g g  +  +  d g g  .     ( ) ε    (A22) 2 d − ⊥ In the last passage of the expression above we used the analogue of Eq. (A18): (uTM) (uTM) 3 (TM) (TM) (A , A ) =−4 ( + 1)δ δmm Im(αβ) dρ g g . (A23) I

(TM) 2 Imposing orthonormality of these modes, for orthonormalized g [in the L (I, dρ) inner product], the general expression for α and β (again, up to rephasing and time resetting) can be expressed as eiκ/2 α = , (A24) 2 3( + 1)sin κ e−iκ/2 β = , (A25) 2 3( + 1)sin κ with 0 <κ<π.

[1] W. Gordon, Zur Lichtfortpflanzung nach der Relativitätstheorie, [13] A. G. S. Landulfo, W. C. C. Lima, G. E. A. Matsas, and D. A. T. Ann. Phys. (Leipzig) 72, 421 (1923). Vanzella, From quantum to classical instability in relativistic [2] W. G. Unruh, Experimental Black-Hole Evaporation? Phys. stars, Phys. Rev. D 91, 024011 (2015). Rev. Lett. 46, 1351 (1981). [14] P. Pani, V. Cardoso, E. Berti, J. Read, and M. Salgado, Vacuum [3] M. Visser, Acoustic black holes: Horizons, ergospheres and revealed: The final state of vacuum instabilities in compact Hawking radiation, Classical Quantum Gravity 15, 1767 stars, Phys. Rev. D 83, 081501(R) (2011). (1998). [15] T. Damour and G. Esposito-Farèse, Nonperturbative Strong- [4]C.H.G.Bessa,V.A.DeLorenci,L.H.Ford,andC.C.H. Field Effects in Tensor-Scalar Theories of Gravitation, Phys. Ribeiro, Model for lightcone fluctuations due to stress tensor Rev. Lett. 70, 2220 (1993). fluctuations, Phys. Rev. D 93, 064067 (2016). [16] T. Damour and G. Esposito-Farèse, Tensor-scalar gravity and [5] P. Jain, S. Weinfurtner, M. Visser, and C. W. Gardiner, Analog binary-pulsar experiments, Phys. Rev. D 54, 1474 (1996). model of a Friedmann-Robertson-Walker universe in Bose- [17] E. Berti, Astrophysical black holes as natural laboratories for Einstein condensates: Application of the classical field method, fundamental physics and strong-field gravity, Braz. J. Phys. 43, Phys. Rev. A 76, 033616 (2007). 341 (2013). [6] S. Weinfurtner, E. W. Tedford, M. C. J. Penrice, W. G. Unruh, [18] L. Annulli, V. Cardoso, and L. Gualtieri, Electromagnetism and and G. A. Lawrence, Measurement of Stimulated Hawking hidden vector fields in modified gravity theories: Spontaneous Emission in an Analogue System, Phys. Rev. Lett. 106, 021302 and induced vectorization, Phys.Rev.D99, 044038 (2019). (2011). [19] E. S. Weibel, Spontaneously Growing Transverse Waves in a [7] W. G. Unruh, Has Hawking radiation been measured? Found. Plasma Due to an Anisotropic Velocity Distribution, Phys. Rev. Phys. 44, 532 (2014). Lett. 2, 83 (1959). [8] J. R. M. de Nova, K. Golubkov, V. I. Kolobov, and J. Steinhauer, [20] N. Rubab, A. C.-L. Chian, and V. Jatenco-Pereira, On the ordi- Observation of thermal Hawking radiation and its temperature nary mode Weibel instability in space plasmas: A comparison of in an analog black hole, Nature (London) 569, 688 (2019). three–particle distributions, J. Geophys. Res. Space Phys. 121, [9] W. C. C. Lima and D. A. T. Vanzella, Gravity-Induced Vacuum 1874 (2016). Dominance, Phys. Rev. Lett. 104, 161102 (2010). [21] T. A. Morgado and M. G. Silveirinha, Negative Landau [10]W.C.C.Lima,G.E.A.Matsas,andD.A.T.Vanzella,Awaking Damping in Bilayer Graphene, Phys.Rev.Lett.119, 133901 the Vacuum in Relativistic Stars, Phys. Rev. Lett. 105, 151102 (2017). (2010). [22] B. Nistad and J. Skaar, Causality and electromagnetic properties [11] W. C. C. Lima, R. F. P. Mendes, G. E. A. Matsas, and D. A. T. of active media, Phys. Rev. E 78, 036603 (2008). Vanzella, Awaking the vacuum with spheroidal shells, Phys. [23] R. Matloob, R. Loudon, S. M. Barnett, and J. Jeffers, Electro- Rev. D 87, 104039 (2013). magnetic field quantization in absorbing dielectrics, Phys. Rev. [12] R. F. P. Mendes, G. E. A. Matsas, and D. A. T. Vanzella, In- A 52, 4823 (1995). stability of nonminimally coupled scalar fields in the spacetime [24] R. Matloob, R. Loudon, M. Artoni, S. M. Barnett, and J. Jeffers, of slowly rotating compact objects, Phys.Rev.D90, 044053 Electromagnetic field quantization in amplifying dielectrics, (2014). Phys. Rev. A 55, 1623 (1997).

013281-19 RIBEIRO AND VANZELLA PHYSICAL REVIEW RESEARCH 2, 013281 (2020)

[25] C. Raabe, S. Scheel, and D.-G. Welsch, Unified approach to [38] C. K. Carniglia and L. Mandel, Quantization of evanescent QED in arbitrary linear media, Phys. Rev. A 75, 053813 (2007). electromagnetic waves, Phys. Rev. D 3, 280 (1971). [26] B. Huttner and S. M. Barnett, Quantization of the electromag- [39] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and netic field in dielectrics, Phys.Rev.A46, 4306 (1992). S. Schultz, Composite Medium with Simultaneously Negative [27] C. Raabe and D.-G. Welsch, QED in arbitrary linear media: Permeability and Permittivity, Phys. Rev. Lett. 84, 4184 (2000). Amplifying media, Eur. Phys. J. Special Topics 160, 371 [40] R. A. Shelby, D. R. Smith, and S. Schultz, Experimental verifi- (2008). cation of a negative index of refraction, Science 292, 77 (2001). [28] A. G. S. Landulfo, W. C. C. Lima, G. E. A. Matsas, and [41] W. J. Padilla, D. N. Basov, and D. R. Smith, Negative refractive D. A. T. Vanzella, Particle creation due to tachyonic instability index metamaterials, Mater. Today 9, 28 (2006). in relativistic stars, Phys. Rev. D 86, 104025 (2012). [42] A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, Hyperbolic [29] R. M. Wald, General Relativity (University of Chicago Press, metamaterials, Nat. Photonics 7, 948 (2013). Chicago, 1984). [43] V. Caligiuri, R. Dhama, K. Sreekanth, G. Strangi, and A. De [30] By “spatial” here we mean that the contraction of any index of Luca, Dielectric singularity in hyperbolic metamaterials: the these tensors with the medium’s four-velocity va vanishes. inversion point of coexisting anisotropies, Sci. Rep. 6, 20002 [31] U. Leonhardt and P. Piwnicki, Optics of nonuniformly moving (2016). media, Phys. Rev. A 60, 4301 (1999). [44] C. C. H. Ribeiro and D. A. T. Vanzella (unpublished). [32] U. Leonhardt and P. Piwnicki, Relativistic Effects of Light in [45] H. Minkowski, Die Grundgleichungen für die elektromagnetis- Moving Media with Extremely Low Group Velocity, Phys. Rev. chen Vorgänge in bewegten Körpern, Math. Ann. 68, 472 Lett. 84, 822 (2000). (1910). [33] I. T. Drummond and S. J. Hathrell, QED vacuum polarization [46] A. V. Andreev, I. L. Aleiner, and A. J. Millis, Dynamical in a background gravitational field and its effect on the velocity Symmetry Breaking as the Origin of the Zero-dc-Resistance of photons, Phys.Rev.D22, 343 (1980). State in an ac-Driven System, Phys. Rev. Lett. 91, 056803 [34] A. B. Balakin, V. V. Bochkarev, and J. P. S. Lemos, Nonmin- (2003). imal coupling for the gravitational and electromagnetic fields: [47] R. Mani, J. H. Smet, K. von Klitzing, V. Narayanamurti, W. B. Black hole solutions and solitons, Phys.Rev.D77, 084013 Johnson, and V. Umansky, Zero-resistance states induced by (2008). electromagnetic-wave excitation in GaAs/AlGaAs heterostruc- [35] A. B. Balakin and J. P. S. Lemos, Non-minimal coupling for tures, Nature (London) 420, 646 (2002). the gravitational and electromagnetic fields: a general system [48] M. A. Zudov, R. R. Du, L. N. Pfeiffer, and K. W. West, of equations, Classical Quantum Gravity 22, 1867 (2005). Evidence for a New Dissipationless Effect in 2D Electronic [36] R. J. Glauber and M. Lewenstein, Quantum optics of dielectric Transport, Phys.Rev.Lett.90, 046807 (2003). media, Phys. Rev. A 43, 467 (1991). [49] J. Inarrea and G. Platero, Theoretical Approach to Microwave- [37] A. M. Contreras Reyes and C. Eberlein, Completeness of Radiation-Induced Zero-Resistance States in 2D Electron Sys- evanescent modes in layered dielectrics, Phys.Rev.A79, tems, Phys. Rev. Lett. 94, 016806 (2005); 111, 229903(E) 043834 (2009). (2013).

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