PT-symmetric spectral singularity and negative frequency resonance
aa r X i v : . [ phy s i c s . op ti c s ] D ec PT -symmetric spectral singularity and negative frequency resonance Sarang Pendharker, Yu Guo, Farhad Khosravi, and Zubin Jacob
1, 2, ∗ Department of Electrical and Computer Engineering,University of Alberta, Edmonton, Alberta T6G 1H9, Canada Birck Nanotechnology Center, School of Electrical and Computer Engineering,Purdue University, West Lafayette, IN 47906, USA ∗ (Dated: Thursday 8 th December, 2016)Vacuum consists of a bath of balanced and symmetric positive and negative frequency fluctuations.Media in relative motion or accelerated observers can break this symmetry and preferentially amplifynegative frequency modes as in Quantum Cherenkov radiation and Unruh radiation. Here, weshow the existence of a universal negative frequency-momentum mirror symmetry in the relativisticLorentzian transformation for electromagnetic waves. We show the connection of our discoveredsymmetry to parity-time ( PT ) symmetry in moving media and the resulting spectral singularityin vacuum fluctuation related effects. We prove that this spectral singularity can occur in thecase of two metallic plates in relative motion interacting through positive and negative frequencyplasmonic fluctuations (negative frequency resonance). Our work paves the way for understandingthe role of PT -symmetric spectral singularities in amplifying fluctuations and motivates the searchfor PT -symmetry in novel photonic systems. I. INTRODUCTION
Systems with PT -symmetric Hamiltonians have in-voked interest in recent years, primarily because they en-able the extension of quantum mechanical formulation tosystems with complex non-Hermitian Hamiltonians [1].Bender et. al. discovered that [2, 3] that for an en-ergy eigen-spectrum to be real, the stringent conditionof Hermiticity of a Hamiltonian can be replaced by aweaker PT -symmetry condition. A major consequence ofthis extension of quantum mechanical framework to non-Hermitian systems, is a new class of optical structures[4] with spatially distributed loss and gain profiles [5–7].Such PT -symmetric non-Hermitian optical systems withcomplex dielectric profiles find promising applications inoptical components ranging from couplers [8] and waveg-uides [9] to microresonators [10, 11] and lasers [12–18].An important characteristic of the PT -symmetric sys-tems is that they exhibit spectral singularities (zero-width resonance) [19, 20]. The PT -symmetric spectralsingularities have been observed in a variety of systemssuch as, periodic finite gap systems [21], confined opti-cal potential [22], and unidirectional singularities in Fanocoupled disk resonators [23]. Recently, the PT -symmetricsingularity in a graphene metasurface has been employedfor enhanced sensing applications [24]. The stabilitiesand instabilities in a complex potential system are alsorelated to PT -symmetry [25]. Therefore characterizationof the PT -symmetry in a complex Hamiltonian systemis important not just to enable consistent quantum me-chanical formulation, but also to identify the stable andunstable regimes in photonic systems, and to predict sin-gularities.A moving lossy medium such as a plasma in motionis known to exhibit electromagnetic instabilities [26]. ∗ [email protected] These instabilities in a moving medium are caused bythe Cherenkov amplification of negative energy waves[27] and have been recently linked to the noncontact vac-uum friction [28–31] between media at relative motion.Vacuum friction arises from quantum-fluctuation inducednear-field photonic interactions [32], and has also beenstudied in particles moving near surfaces [33–36] and inrotating bodies [37, 38]. The nature of vacuum friction isto oppose the relative motion, and therefore the energyspent in maintaining the relative velocities is utilized inthe amplification of vacuum fluctuations, which resultsin the instabilities. Recently, Silveirinha [26, 39, 40] re-ported that these instabilities in moving media occur be-cause of PT -symmetry breaking. Recently, Guo et. al.[41] have shown that moving media can support singularresonances, which are manifested in giant vacuum frictionand enhanced non-equilibrium heat transfer between twomoving slabs [42]. However, the origin of these singularresonances in view of the symmetries present in movingmedia remains unexplained.In this paper, we reveal a PT -symmetric spectral sin-gularity (zero width resonance) which occurs for bodiesin relative motion. We show that this PT -symmetry is aconsequence of a universal frequency-momentum mirrorsymmetry observed under the relativistic Lorentz trans-formations, which was surprisingly overlooked so far. Weanalyze the case of metallic media in relative motion andshow that the spectral singularity occurs because of theperfect coupling between positive and negative frequencysurface plasmon polaritons. This is fundamentally differ-ent from the case of the balance between spatially dis-tributed gain and loss profiles known in conventional PT -symmetric systems. These PT -symmetric spectral singu-larities are manifested at the transition between stableregions (loss dominant regime) and region of instabilities(gain dominant regime) in the dispersion of a movingsystem. Our work explains the underlying cause of agiant enhancement in all phenomena related to vacuumand thermal fluctuations in moving media eg.: vacuumforces and radiative heat transfer. We show that the gi-ant enhancement is caused by the universal phenomenaof coupling between negative and positive frequencies inthe near-field and therefore can be used to explain sim-ilar effects in acoustic systems [43], hydrodynamic flows[44] and experiments on Coulomb drag [45]. II. FREQUENCY-MOMENTUM MIRRORSYMMETRY
In this section, we show the existence of a frequency-momentum mirror symmetry in the Lorentz transforma-tion laws. The time-dependent electromagnetic field so-lutions to Maxwell’s equations are real variables. Thusthe spectral decomposition of modes necessarily consistof positive and negative frequencies which are complexconjugates of each other. For the dispersion relationin the ω − k plane, this implies that positive frequencybranches are necessarily accompanied by a symmetricnegative frequency branch. Under stationary conditions,the positive frequency components alone contain all thephysics in the system, and therefore it generally sufficesto restrict our analysis to the positive frequencies. How-ever, both the positive and negative frequency solutionsbecome relevant when there is a relative translatory mo-tion in the system. This is because the Doppler shiftsare velocity dependent causing the symmetry betweenforward/backward traveling waves and positive/negativefrequencies to be broken.The transformation of frequency ( ω ) and momentum( k x ) from a stationary frame of reference S to an inertialframe of reference S ′ , under the relativistic Doppler shiftis governed by [46], k ′ x = γ (cid:0) k x − β x ωc (cid:1) (1) ω ′ = γ ( ω − k x β x c ) (2)where ω ′ and k ′ x is the frequency and the propaga-tion constant (respectively), as seen in the transformedframe of reference S ′ moving with a velocity v motion ; β x = v motion /c is the normalized velocity of transla-tion, and γ = 1 / (1 − β x ) / . c is the velocity of lightin vacuum. For simplicity, we have considered transla-tory motion along the x -axis only. The central theme ofthis paper is the unique relativistic transformation frompositive frequencies to an equal and opposite frequencygiven by ω ′ = − ω (3)Note that the momentum of waves is invariant to thistransformation and is conserved, k ′ x = k x (4)This unique transformation is satisfied by the equationof a line k x = γβ x c ( γ − ω (5) − . − k x / k − − − − (cid:2) x = 0.3 k x m irror sym m etry lines (cid:1) (cid:1) k x / k (cid:1) (cid:1) (cid:1) (cid:1) (cid:0) x = . m irror sym m etry lines (b)(a) Light lines L i g h t li n e s L i gh t li ne s S t a t i ona r y f r a m e M o v i ng f r a m e kxk x ' = - kx S t a t i ona r y f r a m e M o v i ng f r a m e FIG. 1. (a) Shows the frequency-mirror-symmetry conditionfor β x = 0 .
3. The equation of the line satisfying frequency-mirror symmetry condition (equation(5)) in frame S ′ is shownby the solid red line, while the corresponding Lorentz trans-formed line in the frame of reference S ′ is shown by thedashed red line. It can be seen that the frequency ( ω ) flipsits sign while the momentum k x is conserved. (b) showsthe momentum-symmetry-condition (equation(7)), where thesolid red line is the momentum-mirror-symmetry line in the S frame of reference which transforms to the dashed red linethe moving frame of reference S ′ . It can be seen that the mo-mentum ( k x ) changes its sign while frequency ω is conserved. We call this as the frequency-mirror-symmetry condition,on which the frequency flips its sign in a transformedframe of reference while maintaining its momentum.On similar lines, it can be shown that another specialrelativistic transformation exists which maps the wavemomentum in the stationary to an equal and oppositemomentum in the moving frame i.e. k ′ x = − k x (6)This is satisfied on the line ω = γβ x cγ − k x (7)Note, that in this case, the frequency is invariant to theLorentz transformation ω ′ = ω (8)We call this as the momentum-mirror-symmetry con-dition.Thus, the fundamental Lorentz transformation equa-tions (1),(2) have universal symmetry properties in the ω − k x plane such that for a given velocity v motion , thereexists,1. A line given by equation (5), on which the momen-tum is conserved ( k ′ x = k x ) while the frequency flipsits sign (frequency-mirror-symmetry ω ′ = − ω ).2. A line given by equation (7), on which the fre-quency is conserved ( ω ′ = ω ) while the momen-tum flips its sign (momentum-mirror symmetry, k ′ x = − k x ).These two symmetry conditions in the ω − k x planeare shown in Fig. 1, panel (a) and (b) respectively. Thedashed black lines in the figure represent the light lines.In Fig. 1(a), the solid red line represents the equation (5)for β x = 0 .
3, in the stationary frame of reference denotedby S . The line then transforms to the dashed line in themoving frame of reference denoted by S ′ . It can be seenthat a positive frequency mode satisfying equation (5)transforms to its negative frequency counterpart whilethe momentum is invariant, i.e. it exhibits frequency-mirror-symmetry. Similarly, Fig. 1(b) shows the transfor-mation of a momentum-mirror symmetry line. It shouldbe noted that while the momentum-mirror symmetry linelies inside the light line, the frequency-mirror-symmetrycondition can be satisfied only outside the light line, i.e.when the phase velocity is lower than the velocity of light.The frequency-mirror-symmetry condition with itsflipped frequency and invariant momentum (shown inFig. 1(a)) is of particular interest, because it enables theobservation of the negative frequency electromagnetic re-sponse of a medium at positive frequencies. Thus, an EMmode at ( ω, k x ) on the frequency-mirror-symmetry line,will be transformed to ( − ω, k x ) in a moving medium,and consequently its negative frequency response will beobserved in the stationary frame of reference. III. NEGATIVE FREQUENCY RESPONSE ATPOSITIVE FREQUENCIES
Our mirror-symmetry arguments are completely gen-eral and apply in the relativistic case. Here, we showa practical scenario where this symmetry is manifested.First,we provide a physical interpretation of negative fre-quency modes followed by the role of the universal sym-metry described above. The electromagnetic properties of a metal slab in relative motion with a velocity v motion ,as observed in the stationary lab frame of reference aregoverned by the Lorentz transformation of constitutiverelations. However at low velocities (( v x /c ) ≪
1) underthe First Order Lorentz transformation (FOLT) limit,the dielectric response of the slab as seen from the sta-tionary lab frame is ǫ r ≈ ǫ ′ r ( ω ′ ) [46], where ǫ ′ r is thedielectric response of the metal and ω ′ is the frequencyas seen in its proper frame of reference. ω ′ is obtained bytransforming ω as per equation (2), which simplifies to ω ′ = ω − k x v motion under FOLT limit. The dielectric re-sponse as seen in the lab frame of reference is then givenby ǫ r = ǫ r ( ω − k x v motion ) (9)It can be seen that the dielectric response of a movingmetal slab is not just dependent on frequency, but alsoon the propagation constant and the velocity of motion.As a consequence, an incident wave of frequency ω andpropagation constant k x will observe a negative frequencydielectric response of the metal when, ω ′ = ω − k x v motion < v motion > v p (11)This is the Cherenkov condition at which the velocity ofmotion is greater than the phase velocity ( v p = ω/k x ) ofthe wave [47, 48].The condition of v motion > v p can be physically satis-fied only when v p ≪ c . This requires the momentum tobe larger than the free space wavevector k x > k = ω/c causing the waves to decay in vacuum. Thus the nega-tive frequency transformation for metallic slab in motioncan occur for near-field evanescent waves. We now ana-lyze the reflection properties of such evanescent waves atthe vacuum-metal interface. The origin of such evanes-cent waves could be quantum emitters or another sta-tionary slab in the near-field of the moving slab. Thenormal component of Poynting vector ( S z ) of an evanes-cent wave absorbed at the metal interface is proportionalto the imaginary component of the reflection coefficient( r ′′ p ) [41]. S z ∝ r ′′ p (12)Therefore the reflection coefficient of a semi-infiniteDrude metal ( ǫ r = 1 − ω p / ( ω + iω Γ)) slab sheds lighton the absorption and amplification characteristics of themedium. Note that the tangential boundary conditionsand hence the Snell’s reflection law at moving media in-terface is valid when the motion is in the plane of theinterface [46] [see Appendix A].The sign of the imaginary component of the reflectioncoefficient ( r ′′ p ) is representative of the loss in the metalslab and it’s peak follows the dispersion curves of a sur-face plasmon polariton (SPP). For any lossy metal, r ′′ p ispositive. However, this is strictly true only at positive fre-quencies. At negative frequencies, the dielectric responseand the reflection coefficient is the complex conjugate ofits respective positive frequency values ( ǫ ( − ω ) = ǫ ∗ ( ω )). FIG. 2. Imaginary part of reflection coefficient for (a) mov-ing with velocity v x = 0 . c and (b) stationary metallic slabin the four quadrants of ω − k x plane. p-polarized plas-monic mode is considered such that r p = ǫ k z − ǫ k z ǫ k z + ǫ k z . Here, k z = p ǫ k − k x , k z = p ǫ k − k x and dielectric responseis governed by Drude model ǫ ( ω ) = 1 − ω p ω + i Γ ω . It can be seenthat peak in r ′′ p follows the dispersion curve of surface plasmonpolaritons. The r ′′ p is proportional to the normal componentof the Poynting vector in the evanescent plasmonic wave, andis representative of the loss in the slab. For a lossy medium, r ′′ p is positive for positive frequencies and negative for nega-tive frequencies as depicted in panel (a). For a moving slab,in the region where k x v motion > ω , the negative frequencymode (represented by the blue peak in r ′′ p ) is dragged into thepositive frequency quadrant, which exhibits gain at positivefrequencies. Therefore in a moving slab there are two forwardpropagating modes in the positive frequency region; one is thepositive frequency lossy mode represented by positive peaksin r ′′ p , while the other is the gain mode with negative r ′′ p inthe positive frequency region. Fig. 2(a) shows the dispersion of the p-polarized plas-monic mode in the ω − k x plane. It can be seen that theSPPs have positive peaks in r ′′ p for positive frequencies( ω > ω < k x and ω have the same sign and when they have opposite signsthe mode is backward propagating. Therefore the com-plete representation of a forward propagating mode in-cludes the first and third quadrant solution, while that ofa backward propagating mode includes second and fourthquadrant. It should also be noted that in the stationary case, the dispersion characteristics are symmetric for theforward ( k x >
0) and the backward propagation ( k x < r ′′ p , implying gain character-istics in the positive frequency domain. A slab movingabove the Cherenkov limit has two forward propagatingmodes and no backward propagating mode. One of theforward propagating modes is an ordinary lossy mode(shown by red peak in r ′′ p ), while the other mode is am-plified (growing mode shown by blue peak in r ′′ p ). Themotion of the dielectric slab results in the violation oftime reversal symmetry. ω ( k x ) = ω ( − k x ) (13)We would like to emphasize that the above argument isvalid even at relativistic velocities, even without FOLTapproximations.Note the existence of a special frequency-mirror-symmetry when the observed response of the moving slabis the exact complex conjugate of its stationary value.This is shown by the starred point in the ω − k x plane.While the dielectric response at this frequency-mirror-symmetry point is ǫ r ( ω, k x ) for a stationary slab, it is ǫ r ( − ω, k x ) = ǫ ∗ r ( ω, k x ) for the moving slab. If we nowconsider two identical parallel slabs (see Fig. 3), one inrelative motion to the other, our analysis shows the exis-tence of a unique velocity at which the dielectric responseof the moving slab is the complex conjugate of the sta-tionary slab. This occurs for a specific frequency andmomentum wavevector dictated by two conditions - thedispersion relation of surface waves on the slab and thefrequency-momentum mirror symmetry condition. IV. PT -SYMMETRIC RESONANCE IN AMOVING MIM WAVEGUIDE Here, we show how the relativistic negative frequency-mirror-symmetry is connected to the achievement ofparity-time symmetry in a moving system. Note ourwork in this section is connected to the mirror-symmetrycondition and not the instabilities or spontaneous PT -symmetry breaking in moving media [40]. If we placea stationary and a moving metal slab close enough toallow evanescent wave interactions, they form a metal-insulator-metal (MIM) waveguide structure. In this Negative Frequency Resonance d k x k x ' = k x d PT -symmetric Spectral Singularity v motion = 2 k x xz FIG. 3. Metal slabs separated by a distance d at relativemotion interact via near field evanescent waves. When thevelocity of motion is greater than the Cherenkov velocity( v motion > ω/k x ), the interaction is between positive frequen-cies of stationary slab and negative frequencies of the movingslab. This can result in a perfect coupling between positiveand negative frequency modes which we call as a negative fre-quency resonance. When the velocity of motion is twice theCherenkov limit ( v motion = 2 ω/k x ), the dielectric responseof the two slabs become complex conjugate pairs, resultingin PT -symmetric spectral singularity. The PT -symmetry isachieved as a consequence of the negative frequency-mirror-symmetry. structure, the stationary slab will have a dielectric re-sponse ǫ ( ω ) and the moving slab will exhibit dielectricresponse of ǫ ( ω ) = ǫ ( ω ′ ). This is shown in Fig. 3. Theseparation between the slabs is d . The overall dielectricdistribution of the waveguide as a function of z is thenwritten as, ǫ ( z, ω ) = ǫ ( ω ) ; z < − d/ ǫ = 1 ; − d/ < z < d/ ǫ ( ω ′ ) ; d/ < z (14)The dielectric function of the waveguide is complex inthe region z < − d/ z > d/
2. To investigate the PT -symmetry properties of this complex dielectric system,we write the Hamiltonian formulation for a plane wavepropagation ( e ik x x ) along ˆ x direction [9, 46]ˆ H em Ψ k x ( z ) = k x Ψ k x ( z ) (15)where eigenfunctions Ψ k x can be either ~E ( y, z ) or ~H ( y, z )and k x is the eigenvalue. Assuming non-magnetic mediaand in the FOLT limit, ˆ H em can be written asˆ H em ( z, ω ) = ∇ t + ω ǫ µ ǫ ( z, ω ) , (16)in which ∇ t = ˆ x × ∇ . Equation (15) will be PT -symmetric with real eigenvalues and unitary time-evolution if Ψ k x is eigenfunction of PT operator and,[ ˆ H em , PT ] = 0 (17)From the properties of P and T operators, it is straight-forward to show that [see Appendix B] P . ˆ H em ( z, ω ) = ˆ H em ( − z, ω ) . P (18a) T . ˆ H em ( z, ω ) = ˆ H ∗ em ( z, ω ) . T (18b) Combining Eq. (18a) and (18b) together, we concludethat the Hamiltonian is PT -symmetric if ˆ H em ( z, ω ) =ˆ H ∗ em ( − z, ω ). This means that the condition of PT -symmetry on dielectric function, as obtained from equa-tion (16), is the well known condition [8] ǫ ( − z, ω ) = ǫ ∗ ( z, ω ) . (19)Using equation (14) and the fact that the imaginarypart of dielectric response is an odd function of frequency[49], the condition for PT -symmetry in the moving sys-tem translates to, ǫ ( ω ) = ǫ ( − ω ′ ) (20)This condition is only met when ω ′ = − ω for the samevalue of k x in stationary as well as moving frame of refer-ence, i.e. on the frequency-mirror-symmetry line of equa-tion (5). This is a unique case where the system responseis PT -symmetric only for a specific electromagnetic mode.Our moving slab systems does not possess time-reversal-symmetry or parity-symmetry individually for any mode.In the FOLT limit, the frequency-mirror-symmetry linesimplifies to [see Appendix C], k x = 2 ωv motion (21)We will henceforth refer to this line as the PT -symmetry line along which the Hamiltonian (equa-tion (16)) is PT -symmetric or [ PT , ˆ H em ] = 0. Note thatthis condition is independent of the separation d but thespectral singularity depends on the gap distance. A modeof the system on the PT -symmetry line will not undergoattenuation or amplification, because at this conditionthe loss in the stationary slab is perfectly balanced bythe gain in the moving slab. V. CHERENKOV AMPLIFICATION
We emphasize that the parametric amplification ofvacuum fluctuations is well-known for the phenomenonof vacuum friction which occurs for bodies in relativemotion [28]. The parametric nature arises since theevanescent wave on reflection is amplified without changein frequency or momentum. Similarly, growing elec-tromagnetic waves and instabilities in moving plasmasand their connection to negative energy waves have beenwell-studied [27]. However, the role of the frequency-momentum mirror symmetry condition in Lorentz trans-formations, the perfect coupling of positive and negativefrequencies in the near-field and PT -symmetric spectralsingularity has never been pointed out till date.We note that gain in the moving system arises fromCherenkov amplification also known as the anomalousDoppler effect fundamentally different from conventional PT -symmetric systems. These growing waves in mov-ing media can be seeded by vacuum fluctuations. Wenote that unlike the classical Cherenkov radiation wherecharged particles are necessary, this effect only requires k x ( ω p (a) v motion =0 (b) v motion =0.2 c P T - s y m m e t r y li n e k x ( ω p − − − − FIG. 4. Dispersion curves for plasmonic MIM waveguide in four quadrants of ω − k x plane for (a) stationary slabs (b) slabswith relative velocity of 0 . c . Solid colored lines represent the real part of propagation constant k ′ x while the dash-dot linesrepresent the imaginary part k ′′ x of the respective mode. Black dotted lines represent line cone. In (b), the dispersion curvelying below the PT -symmetry line is the negative frequency mode which is dragged into the first quadrant. This the is thegain mode. The gain and loss modes merge at the intersection of PT -symmetry line with imaginary component of propagationconstant equal to zero at the intersection point. At this point, loss in stationary metal slab is exactly compensated by gain inthe moving metal slab. All points on the dispersion curve below the PT -symmetry line have gain while all the points abovehave loss. Inset shows H z mode profile at loss (b1) , gain (b2) and PT -symmetric (b3) points. The magnetic field profile in thelossy region of the dispersion curve (b1) attenuates as it propagates along x direction, and amplifies in the gain region (b2).On the PT -symmetry point of the dispersion curve, the field profiles propagates without any attenuation or gain. Drude metalwith plasma frequency ω p = 10 Hz , collision frequency Γ = 0 . ω p is considered in the simulation. The separation betweenthe slabs is d = 25 nm. the motion of neutral polarizable particles or harmonicoscillators with internal degrees of freedom. The classicaldispersion relation of modes enters the classical thermalfluctuations and quantum vacuum fluctuations throughthe fluctuation-dissipation theorem (FDT) [50]. h E j ( ~r, ω ) E ∗ k ( ~r ′ , ω ) i = ω πc ǫ Θ( ω, T )Im (cid:8) G Ejk ( ~r, ~r ′ ; ω ) (cid:9) (22)where ˆ j and ˆ k represent the three spatial orthogonal co-ordinates (ˆ j, ˆ k ∈ { ˆ x, ˆ y, ˆ z } ). Θ( ω, T ) is the energy of aquantum oscillator at equilibrium, given by Θ( ω, T ) = ~ ω/ ~ ω/ ( e ~ ω/ ( K B T ) − G jk is an element ofthe electric field Green’s tensor. We note that the FDTformalism ensures that classical mode structure affectsthe noise properties and effects such as Casimir forces[51, 52], near-field heat transfer [53] and vacuum friction[28]. VI. PT -SYMMETRIC SPECTRALSINGULARITY In this final section, we show the spectral singularityassociated with PT -symmetric systems is manifested inthe perfect coupling of positive and negative frequencybranches in moving plasmonic media. This perfect cou- pling occurs at a critical velocity and gap distance whennegative frequency mirror symmetry is achieved for sur-face wave solutions. We call this a negative frequencyresonance. Note that, even though PT -symmetry is satis-fied on all points of the PT -symmetry line, not all ( ω, k x )on the line are valid waveguide modes. This is becausea mode has to satisfy additional boundary conditions atthe interfaces, which also makes the solution gap-size ( d )dependent. To get the precise location of a mode on the PT -symmetry line, we compute the full dispersion curvefor an MIM waveguide with one moving slab, by solvingthe dispersion relation, e − ik z d = k z /ǫ + k z /ǫ k z /ǫ − k z /ǫ k z /ǫ + k z /ǫ k z /ǫ − k z /ǫ (23)in the full ω − k x plane. Here, k zj = p ( ǫ j k − k x ), j ∈{ , , } ; ǫ is the Drude dielectric response, ǫ = 1 (ora constant) and ǫ = ǫ ( ω − k x v motion ). We consider p-polarized ( H y = 0) wave propagation as it alone supportsplasmonic modes.Fig. 4 contrasts the computed dispersion curves of theMIM waveguide with (a) stationary slabs and (b) one slabmoving with v motion = 0 . c . Background color in the plotindicates the sign of the imaginary part of the dielectricresponse of the two slabs ( ǫ and ǫ ). In Fig. 4a, darkyellow background represents the region where ǫ ′′ and ǫ ′′ are positive, while dark green represents the region where
90 0.92 0.94 0.96 0.98 1.00 1.02 / v p Q - f a c t o r (b) v p ( c ) / p P T - sy mm e t r y li ne Gain regionLoss region (a) k x ( (cid:3) p /c) (cid:4) / (cid:4) p Dispersion curve P T - sy mm e t r y li ne v motion /2 v motion PT -symmetric Spectral Singularity v motion v p FIG. 5. (a) Phase velocity along the dispersion curve asfunction of frequency when v motion = 0 . c . The gain andloss regions of the dispersion curve are separated by the PT -symmetry line with v motion /
2. The mode intersects the PT -symmetry line when v motion = 2 v p . (b) Q -factor of the res-onance as a function of the ratio v motion / v p . The modeexhibits spectral singularity as v motion → v p , at the inter-section of the PT -symmetry line and dispersion curve. both the constants are negative. For a lossy medium, ǫ ′′ ispositive for positive frequencies and negative for negativefrequencies. Thus both slabs ( ǫ and ǫ ) are lossy in thestationary case.When one of the slabs starts moving, its dielectric re-sponse transforms according to ǫ ( ω − k x v motion ). Abovethe Cherenkov limit of v motion > ω/k x , negative fre-quency characteristics are dragged into the positive fre-quency region as shown by the overlap of light green andlight yellow region in Fig. 4b. In this overlap regionmedium-1 is lossy, while medium-3 exhibits amplifyingcharacteristics. The PT -symmetry line lies in this over-lap region and forms the diagonal to the rhombus formedbetween the region ǫ < − ǫ and ǫ < − ǫ . The condi-tion ǫ = ǫ ∗ is satisfied at all points on PT -symmetry linewhich implies the moving slab dielectric response is thecomplex conjugate of the stationary slab.In Fig. 4, the real component of propagation constant( k ′ x ) is represented by solid lines and the imaginary com-ponent ( k ′′ x ) is shown by the dashed line of correspondingcolor. For the stationary MIM waveguide (Fig. 4a), bothpositive as well as negative momentum waves attenuateas they propagate in either direction, as indicated by thesame sign of k ′ x and k ′′ x for all positive frequencies ( Thesigns are opposite in the complex conjugate region of neg-ative frequencies). However when one slab is moving, wenotice two unique modes in first quadrant, one is a lossymode ( k ′ x and k ′′ x have same sign) and another is a mode exhibiting gain ( k ′ x and k ′′ x have opposite signs). Thesetwo modes are shown by red and blue colors in Fig. 4b,respectively. The gain mode in first quadrant arises formthe negative frequency component of the backward prop-agating mode which is dragged to the positive frequencyregion from the fourth quadrant. The lossy and amplifiedmodes converge and meet on a point on PT -symmetricline (shown by magenta colored line with star marker).We emphasize that the propagation constant at this pointof intersection is purely real, k ′′ x = 0. At this point onthe dispersion curve, the PT -symmetry is achieved andthe wave propagates without any attenuation. All pointson the dispersion curve above the PT -symmetry line arelossy, while those below exhibit gain as shown by the H y mode profile in inset. It can also be seen that, in contrastto the stationary case, the dispersion diagrams becomesunsymmetrical ω ( k x ) = ω ( − k x ).The dispersion curve and the PT -symmetry line inter-sect when the phase velocity ( v p ) of the mode is equal tohalf the slab velocity. Fig. 5(a) shows the phase velocityof points along the dispersion curve when v motion = 0 . c .The corresponding dispersion curve in the first quad-rant of the ω − k x plane is shown in the inset. Allthe points on dispersion curve with phase velocity higherthan v motion / v motion > v p (24)The more stringent condition for amplification arisesfrom the fact that the gain in the moving slab has tocompensate for the loss in stationary slab, and thereforeto achieve net gain the slab velocity has to be twice theconventional Cherenkov limit. The PT -symmetry con-dition lies at the boundary of stability (loss dominantregime) and instability (gain dominant regime).A mode at the PT -symmetry condition ( v motion = 2 v p )exhibits zero-width resonance, as depicted by the Q -factor of resonance in Fig. 5(b). The Q -factor is definedas a ratio k ′ x / k ′′ x [see Appendix D]. It can be seen thatat the PT -symmetry condition, the Q -factor tends to in-finity, indicating zero-width of resonance or spectral sin-gularity. Note that non-equilibrium phenomena such asradiative heat transfer and vacuum friction will exhibita giant enhancement when the velocity and gap size istuned to achieve this resonance [42] . Our future workwill focus on theoretical work beyond linear response the-ory to regularize the fluctuations near this spectral sin-gularity. VII. CONCLUSION
In this paper, we have shown the existence of a uni-versal frequency and momentum mirror symmetry con-ditions in relativistic Lorentz transformations. We haveshown that frequency-mirror-symmetry is the fundamen-tal origin of the PT -symmetry condition in the case ofmetallic slabs in relative motion. We show that the PT -symmetry condition is achieved only on a line which sat-isfy frequency-mirror-symmetry. Our work provides aclear connection between negative frequency resonancesand PT -symmetric spectral singularity in moving media.We have considered two metallic slabs in motion to showhow the spectral singularity results from perfect coupling of positive and negative frequency surface plasmon po-lariton branches. Our work on the coupling of negativeand positive frequencies in the near-field is a universalphenomenon and can lead to similar effects being discov-ered in acoustic systems [43], hydrodynamic flows [44]and experiments on Coulomb drag [45]. [1] C. M. Bender and S. 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The generalized boundary conditions at moving me-dia interface are derived in A. J. Kong’s book [46]. Forcompleteness, we restate these conditions and show thatthey simplify to stationary boundary conditions in ourconfiguration.At moving interface the electric and magnetic bound-ary conditions are not decoupled [46]ˆ n × (cid:16) ~E − ~E (cid:17) − (ˆ n · ~v motion ) ( ~B − ~B ) = 0 (A1)ˆ n × (cid:16) ~H − ~H (cid:17) + (ˆ n · ~v motion ) ( ~D − ~D ) = ~J s (A2)where ˆ n is the normal to the interface. In our case the v motion is along the interface with ˆ n ⊥ ~v . Therefore thecoupled terms with ˆ n · ~v have zero contribution and theboundary conditions simplify to, (cid:16) ~E tangential − ~E tangential (cid:17) = 0 (A3) (cid:16) ~H tangential − ~H tangential (cid:17) = ~J s (A4)The boundary conditions on ~D and ~B derived from thedivergence equations remain unaltered for moving me-dia. Therefore the Snell’s law is valid when the motionis in the plane of the interface. The reflection propertiescan then be computed by Lorentz transforming the ma-terial properties of moving medium to stationary frameof reference and applying the boundary conditions forstationary interface. Alternatively, the same reflectionproperties can be obtained by solving the boundary con-ditions in the proper frame of reference of the movingmedium followed by Lorentz transformation of the solu-tion to stationary frame of reference. Appendix B: Parity and Time reversal operation
Parity reversal operation P performs r → r ′ on thesystem. Therefore P acting on the Hamiltonian of equa-tion (16) gives, P . ˆ H em ( z, ω ) = P . (cid:2) ∇ t + ω ǫ µ ǫ ( z, ω ) (cid:3) (B1)= ⇒ P . ˆ H em ( z, ω ) = (cid:2) ( −∇ t ) + ω ǫ µ ǫ ( − z, ω ) (cid:3) . P (B2)= ⇒ P . ˆ H em ( z, ω ) = ˆ H em ( − z, ω ) . P (B3)The time reversal operator T is defined to reverse thedirection of time, by performing t → − t and i → i ∗ on the system. Therefore T acting on the Hamiltonian ofequation (16) gives, T . ˆ H em ( z, ω ) = T . (cid:2) ∇ t + ω ǫ µ ǫ ( z, ω ) (cid:3) (B4)= ⇒ T . ˆ H em ( z, ω ) = (cid:2) ∇ t + ω ǫ µ ǫ ∗ ( − z, ω ) (cid:3) . T (B5)= ⇒ T . ˆ H em ( z, ω ) = ˆ H ∗ em ( z, ω ) . T (B6) Appendix C: Frequency-mirror-symmetry in FirstOrder Lorentz Transform (FOLT) limit
The frequency-mirror-symmetry line for a given β x is, k x = γβ x c ( γ − ω (C1)Substituting the value γ = 1 / p − β x , we get k x = ωc β x − p − β x (C2)Using the binomial expansion, (cid:0) − β x (cid:1) / = 1 − β x + 12 12 (cid:18) − (cid:19) β x + . . . (C3)In the first order Lorentz transform limit, β x ≪
1, andtherefore we neglect the higher powers of β x . Substitut-ing equation C3) in equation (C2) and simplifying, weget k x = 2 ωv x (C4) Appendix D: Q-factor of a propagating waveresonance
The line width of resonance in a system is characterizedby its Q -factor, defined as, Q = ω Average stored Energy( W )Average power dissipated( P ) (D1)The line width is inversely proportional to Q value andthere is singularity in the spectrum when Q → ∞ .The average stored energy in the system for time har-monic fields propagating in x direction is, W = Z vol ~E ( x, t ) · ~E ∗ ( x, t ) d ( vol )+ Z vol ~H ( x, t ) · ~H ∗ ( x, t ) d ( vol )(D2)The electric and magnetic fields of a mode in one-dimensional propagation can be written as, ~E ( x, t ) = ~Ee i ( k ′ x x − ωt ) e − k ′′ x x (D3) ~H ( x, t ) = ~He i ( k ′ x x − ωt ) e − kx ′′ x (D4)0where the complex propagation constant k x = k ′ x + ik ′′ x .From (D2), (D3) and (D4), the average stored energy inthe mode is, W = Z vol ~E · ~E ∗ e − k ′′ x x d ( vol ) + Z vol ~H · ~H ∗ e − k ′′ x x d ( vol )(D5)Average power dissipated by the mode as it propagatesa distance ∂x in time ∂t is, P = − ∂∂t ( W ) (D6)Substituting (D5) in (D6), P = Z vol ~E · ~E ∗ e − k ′′ x x (cid:18) k ′′ x ∂x∂t (cid:19) d ( vol )+ Z vol ~H · ~H ∗ e − k ′′ x x (cid:18) k ′′ x ∂x∂t (cid:19) d ( vol ) (D7) Since a phase front ( k ′ x x − ωt = const ) travels ∂x distancein time ∂t , we have ∂x∂t = ωk ′ x (D8)From (D5), (D7) and (D8), the average power dissi-pated is given by, P = 2 k ′′ x k ′ x ωW (D9)From (D9) and (D1) we get the Q -factor of the propa-gating mode as, Q = k ′ x k ′′ xx