Entanglement, violation of Kramers-Kronig relation and curvature in spacetime
EEntanglement, violation of Kramers-Kronig relation and curvature in spacetime
Mehmet Emre Tasgin
Institute of Nuclear Sciences, Hacettepe University, 06800, Ankara, Turkey [email protected] and [email protected] (Dated: December 11, 2019)Recent studies show that a maximally entangled Schwinger pair creates a nontraversable Einstein-Rosen (ER) bridge (wormhole) in the gravitational theory, which bridges causally not connectedparts of the AdS spacetime [PRL 111, 211603 and 211602]. Same authors raise the possibility of ageneral correspondence between a weaker entanglement and curvature. Here, we provide some cluesfor such a generalized relation. First, we show that (i) entanglement and (ii) violation of Kramers-Kronig (KK) relations appear at the same critical parameters for a standard two-mode (squeezing)entanglement interaction. We also include the dampings. Second, we bring a study into attention:Presence of a spacetime-curvature in vacuum polarization, electron-positron Schwinger pairs, makesQED also violate the KK relations without violating the causality. Then, we discuss that thesefindings are possible to provide new clues for a generalized relation between entanglement andspacetime-curvature. Such an interpretation may also save violation of KK relations from implyingthe violation of the causality.
Entanglement poses a spooky action between a pairof well-separated particles. Although entanglement can-not be used for instant communication [1], choice of themeasurement on one of the particles affect the state ofthe other instantaneously . Similarly, a nontraverablewormhole, a phenomenon emerging in general relativity(gravitational theory), can connect (shortcut) two dis-tant particles at a space-like separation. Although a non-traversable wormhole cannot be used for instant commu-nication, as in entanglement, action on one of the parti-cles can affect the state of the other one [4].Recent influential works [5, 6],[4] demonstrate a con-crete relation between the two phenomena; entanglementin quantum theories and Einstein-Rosen (ER) bridge (awormhole) in the gravitational theory. In simple words,Refs. [5, 6] use equivalence (holography) principle [7, 8]between the solutions of conformal (quantum) field the-ory (CFT) and super-symmetric gravitational theory (asa low energy string theory), i.e. AdS [9]. On the CFTside, they provide analytical solutions [10] to a quark-antiquark pair, in a maximally entangled state, createdvia Schwinger effect [11] in vacuum. The two particlesare not in causal contact such that the light emitted fromone particle cannot reach the other one [6]. On the AdSside, they explicitly show that this maximally entangledstate, of CFT, produces a nontraversable wormhole (ERbridge) in AdS [5, 6, 11], via the mathematical corre-spondence (holography) [10] between 4-dimensional CFTand AdS . A “tunneling instanton” solution accompa-nies the Scwinger effect [11], which corresponds to an ERbridge [14–16] between the horizons of the particles outof causal contact [6, 11]. In a many-particle (ensemble) entangled state, measurement ona single particle can affect the state of all remaining particles.This appears in systems like single-photon superradiance or in-teracting Bose-Einstein condensates [2, 3]. A solution who can change (tunnel), e.g., between two minimumenergies almost instantaneously [12, 13].
Such an explicit demonstration of the correspondencebetween a maximally entangled
Schwinger pair and emer-gence of a nontraversable ER bridge (wormhole) led Son-ner [5] raise the following intriguing question. What ifthe system is not in a maximally entangled state but ina weaker inseparable state?In this paper, we provide new clues on theentanglement-ER bridge correspondence. Our resultsshow a possible extension of the connection between en-tanglement and ER bridge (very special solutions of Ein-stein Field equations, EFE, in curved spacetime) into akind of correspondence between (i) a particular, small,value of entanglement and (ii) presence of curvature inspacetime.We handle the correspondence in a completely differ-ent point of view. Instead of CFT, we use the standardsecond-quantized theory of quantum optics and QED inthe curved spacetime [17, 18]. Similar to Refs. [5, 6, 11],QED in curved spacetime deals with electron-positronSchwinger pairs, i.e. vacuum polarization.First, we show that the most standard, exactly solv-able, two-mode squeezing ˆ H = ¯ h ( g ˆ c † ˆ a † + g ∗ ˆ a ˆ c ), interac-tion [19] in a cavity manifests the violation of Kramers-Kronig (KK) relations in the input/output (transfer func-tions) of the cavity. More interestingly, we show thatin such a damped cavity, nonclassicality [entanglementor single-mode nonclassicality (SMNc), e.g. squeezing]shows up at exactly the same critical (e.g. cavity-mirror) coupling, g , parameter. Moreover, this coin-cidence appears for any values of the system parameters!(see Fig. 1) We note that SMNc of a (collective) quasi-excitation is the collective entanglement of the back-ground (particles) generating the. e.g. squeezed, exci-tation [2, 20], see Fig. 2 for an illustration.Second, we examine (compare with) the quantum elec-trodynamics (QED) solutions of vacuum-polarization incurved spacetime. Hollowood and Shore calculate the re-fractive index n ( ω ) for vacuum-polarization (Schwingerproduction of electron-positron pairs) of light in the pres- a r X i v : . [ phy s i c s . g e n - ph ] J u l ence of background curvature in spacetime [17, 18] us-ing an unperturbed (sigma wordline [21–24]) formalism.They provide worldline instanton solutions describingSchwinger pair creation which can be interpreted as tun-neling in this formalism [17, 18]. They show that re-fractive index violates the KK relations due to theinteraction of electron-positron Schwinger pairs with aweak background curvature . We kindly remind that inRefs. [5, 6, 11], maximally entangled Schwinger pairs areresponsible for the ER bridge (tunneling instanton).Fortunately, Hollowood and Shore show that in thepresence of background curvature in spacetime, violationof KK relations does not imply the violation of causal-ity [17, 18, 25]. Shortly, they argue that interaction ofthe generated electron-positron pair with the background“curvature” violates only the strong equivalence princi-ple (SEP) [25, 26] in general relativity . Validity of SEPwould imply the existence of global reference frames andthis would indicate a superluminal propagation whichcould violate the causality [25]. Weak equivalence prin-ciple, however, implies only the existence of local iner-tial reference frames which is not sufficient to establish alink between superluminal propagation and violation ofcausality [25] .Both (i) a weakly-entangled system and (ii) a sys-tem, where a background spacetime curvature is present,violate the Kramers-Kronig relations. Hence, consider-ing a possibility of a relation between entanglement andspacetime curvature could allow us to (a) circumventthe appearance of violation of KK relation to imply thecausality violation in entangled devices and (b) extendthe entanglement-ER bridge correspondence, bearing inRefs. [5, 6, 11], into an entanglement-curvature duality.In advance, we state that our findings do not providea proof for the entanglement-curvature (spacetime) rela-tion. The reason we raise the common appearance of vio-lation of KK relations, showing up both in the onset of (i)weak entanglement and (ii) weak spacetime curvature, asa clue is the following. Actually there are several reasonsfor this. First, KK violation is obviously not a commonphenomenon which otherwise would not be so contro-versial to the audience. Second, Refs. [5, 6, 11] alreadyprovide a direct derivation between maximally entangledSchwinger pair and ER bridge. So, such a relation be-tween weak entanglement and spacetime curvature is al-ready something expected [5]. Third, violation of KKand onset of entanglement appear at exactly the samecritical coupling g > g gcr , see Fig. 1. This obviously pro-vides a stronger clue , e.g. compared to Ref. [29] where(fourth) small fluctuations in entanglement are shown to of the light which creates the vacuum polarization Here, light interacts with the background curvature through thecreated electron-positron pairs. SEP is not violated if the interaction depends only on theChristoffel symbols but not directly on the curvature [25, 27, 28]. An extended discussion can be found in Refs. [25] introduce curvature (weight) in a non-dissipative system.That is, there is no critical point in the valuable work [29],but the two critical values coincide in our case. Fifth, theobservation on the violation of KK relations via entan-glement or SMNc (especially in a continuous regime ofvariables, i.e. not at some resonances), “forces” us to be-lieve in (check the existence of) such a correspondence.Induction of curvature with inseparability can avoid theviolation of causality [17, 18, 25] appearing via entangle-ment [5].So this work aims to gather all clues in a single boxbesides providing new ones. We propose a way for cir-cumventing acceptance of the presence of the violationof causality under the light of Refs. [5, 6, 17, 18]. Weunderline that the work presented here has its roots atRef. [30].
Tunneling and KK relations — Before demonstratingour results in the next paragraphs, it is appropriate todiscuss an important point. We note that violation of KKrelations are observed also in “causal” interferometry de-vices [31–33] where multiple interferences take place.It is well-known that interference can avoid/limit elec-tromagnetic field to occupy particular spatial domains.These domains are tunneled [34]. In these systems, themajor problem is to define the “tunneling times” for pho-tons [35] which are calculated to be superluminal [36–38].Although this is discussed to appear due to the instanta-neous spreading of the wavefunctions [39, 40], also rela-tivistic equations demonstrate the superluminal tunnel-ing times [41, 42]. It is also shown that, still open tofurther questions, weak Einstein causality (which statesthat expectations or ensemble averages may not be su-perluminal) is not violated in tunneling process [36]. InRef. [43], we discuss the interference phenomenon in de-tails. In Ref. [43], we also show that the faster-than-light“peak” velocities observed in various experiments [44] aresuperluminal since the group-index (mathematically canbe shown to govern the peak velocity [45]) violates theKK relations. This provides a support for superluminalpropagation needs to be accompanied by a violation ofKK relations which Refs. [17, 18, 25] show: does not nec-essarily imply the violation of causality.Actually the appearance of nonanalyticity in theupper-half of the complex-frequency-plane (CFP), in oursystem, is different than the ones discussed in “causal”interferometry devices [32, 46, 47]. The nonanalyticitywe observe does not depend on the cavity length, un-like interferometry devices, hence possibly do not origi-nate from interference . Our system is merely absorptivewhere violation of KK relations do not appear [46, 48]. M. Suhail Zubbairy –private communication (group meeting). Actually it is not so important whether it appears via in-terference or not. Statement “entanglement induces (or in-duced with) superluminal tunneling” also lines with our dis-cussions, which strongly accompanies the tunneling instanton inRefs. [5, 6, 17, 18] wherein tunneling time can be more superlu-minal (faster) compared to Refs. [36–38].
Entanglement & violation of Kramers-Kronigrelations — First, we consider an optomechanical sys-tem [49, 50] in which a cavity mode ˆ c interacts with thevibrating mirror ˆ a placed inside the cavity. Then, wetune the cavity to favor ˆ H int = ¯ h ( g ˆ c † ˆ a † + g ∗ ˆ a ˆ c ) two-mode squeezing [49] type interaction. We show that en-tanglement and violation of KK relations appear at thesame critical coupling g = g crt , in Fig. 1.Entanglement features of an optomechanical systemhave already been studied extensively [51]. Hamiltoniancan be written [30, 49] asˆ H = ¯ h ∆ c ˆ c † ˆ c + ¯ hω m ˆ a † ˆ a + ¯ hg ˆ c † ˆ c ˆ q + i ¯ hε L (ˆ c † − ˆ c ) (1)in the frame rotating with the laser (pump) frequency ω L ,i.e. ∆ c = ω c − ω L . ω c is the frequency of the optical cav-ity mode. Nonclassicalities, e.g entanglement and single-mode nonclassicality (SMNc) such as squeezing, are de-termined by noise operators [52, 53], i.e. δ ˆ c = ˆ c − (cid:104) ˆ c (cid:105) and δ ˆ q = ˆ q −(cid:104) ˆ q (cid:105) , with ˆ q = (ˆ a † +ˆ a ) / √
2. After the linerization,Langevin equations δ ˙ˆ q = ω m δ ˆ p (2) δ ˙ˆ p = − γ m δ ˆ p − ω m δ ˆ q + g ( α ∗ c δ ˆ c + α c δ ˆ c † ) + g m ˆ (cid:15) in ( t ) (3) δ ˙ˆ c = − ( γ c + i ∆) δ ˆ c + igα c δ ˆ q + g c ˆ a in ( t ) (4)become analytically solvable, where ˆ a in ( t ) and ˆ (cid:15) in ( t ) arethe optical and mechanical noises, leaking in, from thetwo vacua [30]. The laser pump is used for increasing theeffective coupling between the mirror and cavity mode. γ c,m = πD ( ω c,m ) g c,m are the damping rates and ∆ =∆ c − g | q s | with q s and α c are the steady-state values for (cid:104) ˆ q (cid:105) and (cid:104) ˆ c (cid:105) [54, 55].In this work, in difference to Refs. [49, 50], weare particularly interested in single-mode nonclassical-ity (SMNc), e.g. squeezing, of the ˆ c -mode. The reasonbecomes apparent in the following section. We quantifythe SMNc of the cavity mode —stays Gaussian due tolinearization [49]— using a beam-splitter (BS) approachdescribed in Ref. [56] in full details. Shortly, a strongernonclassical state generates stronger entanglement at aBS output [57].The linearized version of the hamiltonian (1) con-tains two terms. One of them is the entangler partˆ H ent ∝ ˆ c † ˆ a † +ˆ a ˆ c . The second one is ˆ H BS ∝ ˆ a † ˆ c +ˆ c † ˆ a . Thetwo-mode squeezing hamiltonian [58] ˆ H ent generates pureentanglement and ˆ H BS interaction just distributes it be-tween different amounts of two-mode entanglement andsingle-mode nonclassicality, where a conservation holdsbetween the two [57, 59–61]. That is, ˆ H ent alone, with-out the presence of ˆ H BS , cannot produce SMNc in thecavity-mode ˆ c .We tune ∆ = − ω m . This brings the interaction ˆ H ent into resonance [49, 50] and favors the generation of ˆ c -ˆ a entanglement. When ∆ = − ω m , however, instabilitysets up at a small g st = (cid:112) γ c γ m / S M N c (a) g/g crt I m { p ( ) } -7 (b) FIG. 1. For ∆ = − ω m , ¯ hg (ˆ c † ˆ a † + ˆ a ˆ c ) entangler interaction isfavored. In this case, (a) single-mode nonclassicality (SMNc)and (b) violation of KK relations appear at the same criticalcoupling g . In graphics (a), plot is ended at g = 1 .
2. Afterthat value system becomes unstable. tuning, action of the ˆ H BS is too small due to the off-resonance. We can circumvent this problem by introduc-ing a variable interactionˆ H int = ¯ hg (ˆ c † ˆ a † + ˆ a ˆ c ) + ¯ hg (ˆ c † ˆ a + ˆ a † ˆ c ) (5)hamiltonian. Here, we aim to control ˆ H ent and ˆ H BS terms independently, via g , . For ∆ = − ω m , we increase g accordingly merely to stabilize the system. That is,smaller values of g does not stabilize the system for usto see the increase in the SMNc.In Fig. 1a, we plot the degree of the SMNc of the cavitymode for ∆ = − ω m and g = 10 g . We observe thatthere is a dramatic increase after g ≥ g crt . Here g crt ∼ (cid:112) γ c γ m /
2. In Fig. 1a, system becomes unstable wherethe plot ends. We note that g is chosen larger onlyto demonstrate the audience that when the system isstabilized, SMNc and violation of KK relations appearat the same parameters. Although we use g =10 g inFig. 1, one can appreciate that choice of ∆ = − ω m , off-resonant, already weakens its act substantially [49].Next, we leave the second-quantized picture and workwith the expectation values of the operators [54, 55], e.g., c = c + c + α p e − i ∆ p t + c − α ∗ p e i ∆ p t , (6)similarly for q and p . We assume an extremely smallprobe field α p , | α p | is the number of probe photons, offrequency ∆ p in the rotating frame. Since the probe field α p is extremely small, system can be safely describedwith the linear response, i.e. α p ∼
0. In both treatmentssecond order terms, e.g. ( δ ˆ q ) and α p , are neglected. Thenonanalyticities of the transfer function can be obtainedfrom the roots of c + = 0 [30], i.e. c + ∝ [ γ c − i (∆ + ∆ p )](∆ p − ω m + iγ m ∆ p ) − iω m | G | = 0 . (7)We also checked if the zeros from the denominator of c + cancels the ones from the nominator and we observedthat they are completely different. Denominator does nothave a zero in the upper half of the complex frequencyplace (CFP). In our cavity system, there appears somenonanalyticities also due to interference [36, 46, 47], e.g. − (2 γ c ˜ c + − + (2 γ c ˜ c + ) e i k p L + ... = 0 with k p = ω p /c ,which we do not consider here. c + = 0 has 3 complexroots ∆ (1 , , p , one of them relies in the upper half of theCFP for g ≥ g crt , see Fig. 1b.In Fig. 1, we clearly observe that for an entanglerhamiltonian ˆ H ent = ¯ hg (ˆ c † ˆ a † + ˆ a ˆ c ), violation of KK andSMNc onset at the same critical point. Moreover, thisis independent of γ c,m . In Fig. 1a, system becomes un-stable where the plot ends. We recall that for SMNc ofthe cavity mode to emerge, ˆ H ent interaction is not suf-ficient. ˆ H ent only generates the entanglement, but ˆ H BS distributes the entanglement into SMNc.Hence, for the simplest entanglement generator(namely the two-mode squeezing) hamiltonian, SMNcand violation of KK relations appear together . Wenote that, the final hamiltonianˆ H = ¯ h ∆ˆ c † ˆ c + ¯ hω m ˆ a † ˆ a + ¯ hg (ˆ c † ˆ a † + ˆ a ˆ c ) + ¯ hg (ˆ c † ˆ a + ˆ a † ˆ c )(8)does not include any gain . We introduce only the lossesinto Langevin equations. There is no physical phe-nomenon (restriction) which avoids the achievement of g ≥ g crt in principle. Our work bases on the solutionsof the simplest (standard) exactly solvable nonclassical-ity (entanglement and SMNc) generator hamiltonian (8).We put the optomechanics hamiltonian as an examplephysical system where one can approximately obtain theinteraction (5). Other systems could also result similarinteraction. For instance, in Ref. [62] it is discussed that aradiation pressure like hamiltonian is responsible for theStokes and anti-Stokes shifts in surface enhanced Ramanscattering (SERS).Besides its fundamental implications, such a phe-nomenon can also be used to “guess” the onset regimefor measurements below the standard quantum limit(squeezing) by measuring the transfer functions . Why SMNc?
Now we are in the right position tostate the physical reasoning for considering the degree ofSMNc of the cavity field, instead of ˆ c − ˆ a entanglement.As explicitly demonstrated in Ref. [2], the quasiparticleexcitations of an ensemble become nonclassical (SMNc)when the particles, in the ensemble, are entangled “col-lectively”. For a better visualization, in Fig. 2 we plot asqueezed “phonon” wavepacket. The findings of Ref. [2] Even though probe field and frequency ∆ p appear explicitly inthe calculation of the transfer functions, violation of KK relationsis related with the complete frequency response of the transferfunctions. The credit for this idea belongs to Peter Zoller at IQOQI ofInnsbruck. collectively entangledsqueezed wavepacket
FIG. 2. If, for instance, a phonon is squeezed the the motionaldegree of freedom of the vibrating atoms are collectively en-tangled within the extent of the phonon wavepacket. A simi-lar visualization can be made for a squeezed photon, e.g. thecavity mode. state that if the phonon is squeezed [63], vibrational (mo-tional) degree of freedom of atoms are entangled witheach other. This happens within the extent of the phononwavepacket [64]. Hence, the following question would beintriguing. Analogously, can one imagine that, when thelight inside the cavity is SMNc (e.g. squeezed), the differ-ent positions in the background spacetime of the cavity(maybe Schwinger pairs at different positions) are entan-gled [20] ? This is the explicit reason we calculate theSMNc of the cavity field instead of the ˆ c − ˆ a entangle-ment. Interestingly, only the SMNc of ˆ c displays such anincrease, not the ˆ c -ˆ a entanglement. Hence, in this sense,the type of entanglement we treat here can be differentthan the bipartite entanglement in Refs. [5, 6].We repeat ourselves: we do not base a theory relying onthese coincidences. We only raise them to the attentionof the society, which we think they are important issues. Conclusion — The relation between a “maximally en-tangled” Schwinger pair and a wormhole connecting thetwo particles has already been demonstrated explicitly.Here, we question if there could exist a more generalconnection between entanglement of quantum optics andcurvature in general relativity [5]. Unlike Refs. [5, 6, 11],we do not use String Theory, however, nor we can providea direct derivation like the one stated in Refs. [5, 6].We consider the possibility of such a relation in atotally different point of view. We provide reasonable“clues”. On one side we show that (a.i) entanglementand (a.ii) violation of Kramers-Kronig relations appearmutually in the simplest exactly solvable hamiltonian.On a second side, Refs. [17, 18, 25] show that a small(b.i) curvature in the background spacetime makes the(b.ii) refractive index violate the KK relations in QED.Keeping in mind that ( ) violation of KK relationsis not a common (is an unresolved) phenomenon, ( )superluminal propagation is accompanied by violationof KK relations [43], and ( ) presence of superluminalpropagation and violation of KK relations do not imply This can be checked via coupled QED-EFE equations. the violation of causality in a curved spacetime back-ground [17, 18, 25]; makes us consider the possibility of ageneralization of the (into weak) entanglement-curvaturerelation. Actually, this is better than accepting the vio-lation of causality with entanglement in Fig. 1 or in other systems.We believe that the conjectures we raise and the di-rection we point out in this work will stimulate new andfundamental works on QED in curved spacetime. [1] Phillippe H Eberhard and Ronald R Ross, “Quantumfield theory cannot provide faster-than-light communica-tion,” Foundations of Physics Letters , 127–149 (1989).[2] Mehmet Emre Tasgin, “Many-particle entanglement cri-terion for superradiantlike states,” Physical review letters , 033601 (2017).[3] A Sørensen, L-M Duan, JI Cirac, and Peter Zoller,“Many-particle entanglement with bose–einstein conden-sates,” Nature , 63–66 (2001).[4] Juan Maldacena and Leonard Susskind, “Cool horizonsfor entangled black holes,” Fortschritte der Physik ,781–811 (2013).[5] Julian Sonner, “Holographic schwinger effect and the ge-ometry of entanglement,” Physical Review Letters ,211603 (2013).[6] Kristan Jensen and Andreas Karch, “Holographic dual ofan einstein-podolsky-rosen pair has a wormhole,” Physi-cal Review Letters , 211602 (2013).[7] Juan Maldacena, “The large-n limit of superconformalfield theories and supergravity,” International journal oftheoretical physics , 1113–1133 (1999).[8] Edward Witten, “Anti de sitter space and holography,”arXiv preprint hep-th/9802150 (1998).[9] Mitsutoshi Fujita, Tadashi Takayanagi, and Erik Tonni,“Aspects of ads/bcft,” Journal of High Energy Physics , 43 (2011).[10] Bo-Wen Xiao, “On the exact solution of the acceleratingstring in ads5 space,” Physics Letters B , 173–177(2008).[11] Gordon W Semenoff and Konstantin Zarembo, “Holo-graphic schwinger effect,” Physical review letters ,171601 (2011).[12] Gerald V Dunne and Christian Schubert, “Worldline in-stantons and pair production in inhomogenous fields,”Physical Review D , 105004 (2005).[13] Gerald V Dunne, Qing-hai Wang, Holger Gies, andChristian Schubert, “Worldline instantons and the fluc-tuation prefactor,” Physical Review D , 065028 (2006).[14] Jin Young Kim, HW Lee, and YS Myung, “Classical in-stanton and wormhole solutions of type iib string theory,”Physics Letters B , 32–36 (1997).[15] Michael Gutperle and Wafic Sabra, “Instantons andwormholes in minkowski and (a) ds spaces,” NuclearPhysics B , 344–356 (2002).[16] AS Gorsky, KA Saraikin, and KG Selivanov, “Schwingertype processes via branes and their gravity duals,” Nu-clear Physics B , 270–294 (2002).[17] Timothy J Hollowood and Graham M Shore, “The re-fractive index of curved spacetime: the fate of causalityin qed,” Nuclear physics B , 138–171 (2008).[18] Timothy J Hollowood and Graham M Shore, “Causalityand micro-causality in curved spacetime,” Physics Let-ters B , 67–74 (2007).[19] Vincent Josse, Aur´elien Dantan, Alberto Bramati, and Elisabeth Giacobino, “Entanglement and squeezing in atwo-mode system: theory and experiment,” Journal ofOptics B: Quantum and Semiclassical Optics , S532(2004).[20] Stefano Liberati and Luca Maccione, “Astrophysical con-straints on planck scale dissipative phenomena,” PhysicalReview Letters , 151301 (2014).[21] Richard Phillips Feynman, “Mathematical formulationof the quantum theory of electromagnetic interaction,”Physical Review , 440 (1950).[22] Julian Schwinger, “The theory of quantized fields. i,”Physical Review , 914 (1951).[23] Fiorenzo Bastianelli and Andrea Zirotti, “Worldline for-malism in a gravitational background,” Nuclear PhysicsB , 372–388 (2002).[24] Christian Schubert, “Perturbative quantum field theoryin the string-inspired formalism,” Physics Reports ,73–234 (2001).[25] Graham M Shore, “Quantum gravitational optics,” Con-temporary Physics , 503–521 (2003).[26] Ian T Drummond and SJ Hathrell, “Qed vacuum polar-ization in a background gravitational field and its effecton the velocity of photons,” Physical Review D , 343(1980).[27] B Bertotti and LP Grishchuk, “The strong equiva-lence principle,” Classical and Quantum Gravity , 1733(1990).[28] Graham M Shore, “Faster than light photons in gravi-tational fields ii.: Dispersion and vacuum polarisation,”Nuclear Physics B , 271–294 (2002).[29] David Edward Bruschi, “On the weight of entanglement,”Physics Letters B , 182–186 (2016).[30] Devrim Tarhan and Mehmet Emre Tasgin, “Mutualemergence of noncausal optical response and nonclas-sicality in an optomechanical system,” arXiv preprintarXiv:1502.01294 (2015).[31] M Beck, IA Walmsley, and JD Kafka, “Group delaymeasurements of optical components near 800 nm,” IEEEjournal of quantum electronics , 2074–2081 (1991).[32] LJ Wang, “Causal all-pass filters and kramers–kronig re-lations,” Optics communications , 27–32 (2002).[33] Liron Stern and Uriel Levy, “Transmission and time delayproperties of an integrated system consisting of atomicvapor cladding on top of a micro ring resonator,” Opticsexpress , 28082–28093 (2012).[34] Raymond Y Chiao, “Tunneling times and superluminal-ity: A tutorial,” in AIP Conference Proceedings , Vol. 461(AIP, 1999) pp. 3–13.[35] Paul Charles William Davies, “Quantum tunnelingtime,” American journal of physics , 23–27 (2005).[36] Zhi-Yong Wang and Cai-Dong Xiong, “Theoretical ev-idence for the superluminality of evanescent modes,”Physical Review A , 042105 (2007).[37] Herbert G Winful, “Optics (communication arising): Mechanism for’superluminal’tunnelling,” Nature ,638 (2003).[38] Herbert G Winful, “Nature of superluminal” barrier tun-neling,” Physical review letters , 023901 (2003).[39] Gerhard C Hegerfeldt, “Instantaneous spreading and ein-stein causality in quantum theory,” Annalen der Physik , 716–725 (1998).[40] Gerhard C Hegerfeldt, “Remark on causality and particlelocalization,” Physical Review D , 3320 (1974).[41] J Fernando Perez and Ivan F Wilde, “Localization andcausality in relativistic quantum mechanics,” PhysicalReview D , 315 (1977).[42] Gerhard C Hegerfeldt and Simon NM Ruijsenaars, “Re-marks on causality, localization, and spreading of wavepackets,” Physical Review D , 377 (1980).[43] M. E. Tasgin, “A Lorentzian group-index violatesKramers-Kronig relations,” (2019), see on Researchgateor arXiv with the title.[44] S Chu and S Wong, “Linear pulse propagation in anabsorbing medium,” Physical Review Letters , 738(1982).[45] John David Jackson, Classical electrodynamics , 3rd ed.(Wiley, New York, NY, 1999).[46] Li-Gang Wang, Lin Wang, M Al-Amri, Shi-Yao Zhu, andM Suhail Zubairy, “Counterintuitive dispersion violatingkramers-kronig relations in gain slabs,” Physical reviewletters , 233601 (2014).[47] Lin Wang, Li-Gang Wang, Lin-Hua Ye, M Al-Amri, Shi-Yao Zhu, and M Suhail Zubairy, “Counterintuitive dis-persion effect near surface plasmon resonances in ottostructures,” Physical Review A , 013806 (2016).[48] Li-Gang Wang and Shi-Yao Zhu, “Superluminal pulse re-flection from a weakly absorbing dielectric slab,” Opticsletters , 2223–2225 (2006).[49] C Genes, A Mari, P Tombesi, and D Vitali, “Robust en-tanglement of a micromechanical resonator with outputoptical fields,” Physical Review A , 032316 (2008).[50] David Vitali, Sylvain Gigan, Anderson Ferreira,HR B¨ohm, Paolo Tombesi, Ariel Guerreiro, Vlatko Ve-dral, Anton Zeilinger, and Markus Aspelmeyer, “Op-tomechanical entanglement between a movable mirrorand a cavity field,” Physical Review Letters , 030405(2007).[51] Florian Marquardt and Steven M Girvin, “Optomechan-ics (a brief review),” arXiv preprint arXiv:0905.0566(2009).[52] R Simon, N Mukunda, and Biswadeb Dutta, “Quantum-noise matrix for multimode systems: U (n) invariance,squeezing, and normal forms,” Physical Review A ,1567 (1994).[53] R. Simon, “Peres-horodecki separability criterion for con-tinuous variable systems,” Phys. Rev. Lett. , 2726–2729 (2000).[54] GS Agarwal and Sumei Huang, “Electromagnetically in-duced transparency in mechanical effects of light,” Phys-ical Review A , 041803 (2010).[55] Devrim Tarhan, Sumei Huang, and ¨Ozg¨ur EM¨ustecaplıo˘glu, “Superluminal and ultraslow light prop-agation in optomechanical systems,” Physical Review A , 013824 (2013).[56] Mehmet Emre Tasgin, “Single-mode nonclassicalitymeasure from simon-peres-horodecki criterion,” arXivpreprint arXiv:1502.00992 (2015). [57] Wenchao Ge, Mehmet Emre Tasgin, and M SuhailZubairy, “Conservation relation of nonclassicality and en-tanglement for gaussian states in a beam splitter,” Phys-ical Review A , 052328 (2015).[58] M. O. Scully and M. S. Zubairy, Quantum Optics (Cam-bridge University Press, New York, 1997).[59] Ievgen I Arkhipov, Jan Peˇrina Jr, Jiˇr´ı Svozil´ık, andAdam Miranowicz, “Nonclassicality invariant of generaltwo-mode gaussian states,” Scientific reports , 26523(2016).[60] Ievgen I Arkhipov, Jan Peˇrina Jr, Jan Peˇrina, and AdamMiranowicz, “Interplay of nonclassicality and entangle-ment of two-mode gaussian fields generated in opticalparametric processes,” Physical Review A , 013807(2016).[61] Anton´ın ˇCernoch, Karol Bartkiewicz, Karel Lemr, andJan Soubusta, “Experimental tests of coherence andentanglement conservation under unitary evolutions,”Physical Review A , 042305 (2018).[62] Philippe Roelli, Christophe Galland, Nicolas Piro, andTobias J Kippenberg, “Molecular cavity optomechanicsas a theory of plasmon-enhanced raman scattering,” Na-ture nanotechnology , 164 (2016).[63] GA Garrett, AG Rojo, AK Sood, JF Whitaker, andR Merlin, “Vacuum squeezing of solids: macroscopicquantum states driven by light pulses,” Science ,1638–1640 (1997).[64] Mehmet Emre Tasgin, Mehmet Gunay, and M SuhailZubairy, “Nonclassicality and entanglement forwavepackets,” arXiv preprint arXiv:1904.13149 (2019). single-mode nonclassicality (SMNc) Entanglement of the background(ensemble generating the excitation)
Violates KK relations presence of 𝑔 𝜇, 𝜈 = small curvaturein QEDnot a common phenomenon vacuum polarization ( 𝑒 + − 𝑒 − pair s ) a maximally entangled Schwinger pair (QFT) ER-bridge (wormhole) general relativity + string
SMNc
Violates KK relations presence of 𝑔 𝜇, 𝜈 = small curvature in QED avoids violation of causality does not imply violation of causality (only local inertial frame I f c o rr e s p on d s t o c u r v a t u r e tunneling instanton solutionsuperluminal propagation a cc o m pa n i e ss