Observation of wave amplification and temporal topological state in a non-synthetic photonic time crystal

Domain: physics
Published: 2025-12-17
Editorial outcome: minor_revision

Abstract
Photonictimecrystals(PTCs)arematerialswhosedielectricpermittivityis
periodicallymodulatedintime.Akintoconventionalphotoniccrystals,PTCs
giverisetomomentumbandgapaswellastotopologicalinsulatorsinthetime
domain.Here,weexperimentallydemonstratethepropertiesofakgapina
PTCwithrealdimensions,realizedinadynamicallymodulatedmicrowave
transmission-linemetamaterial.Waveamplificationisobservedwithinthek
gap,leadingtoanarrowingoftheinitialpowerspectrumanditsshifting
towardthegap.Toprobethemid-gaptopologicalstate,weintroduceatem-
poralinterfaceseparatingtwoPTCswithdistincttopologicalphases.The
phasedifferencemeasuredbetweentime-reflectedandtime-refractedwaves
attheinterface,togetherwiththetemporalconfinementofthetopological
state,provestherealizationofnontrivialtemporaltopology.Byintegratingk
gapamplificationwithtime-domaintopologicalfeatures,ourworkopensnew
avenuesforlightgenerationandmanipulationintime-varyingphotonic
materials.

Introduction
Crystals generally refer to structures with periodicity in space. In
photonics,ithasbeenknownsincethe1980sthatphotonicstructures
with dielectric permittivity periodically modulated in space, com-
monly referred to as photonic crystals1–3, can give rise to photonic
bandgaps—frequency ranges where waves cannot propagate but
only decay evanescently—similar to the energy bandgaps in
semiconductors4(Fig.1a).Inspiredbytopologicalinsulators5,photonic
bandgapshavebeenendowedwithtopologicalpropertiesinthepast
decade, leading to the development of photonic topological
insulators6–8.Asadefiningfeature,theboundarybetweentwotopo-
logically distinctphotonic topologicalinsulatorshosts topologically
protectededgestates(Fig.1b),asdeterminedbythebulk-boundary
correspondence,afundamentalprincipleintopologicalphysics.Tak-
ingthesimplestone-dimensional(1D)photonictopologicalinsulator9
as an example, the existence of an in-gap topological edge state
introducesapeakinsidethetopologicalbandgapinthewavetrans-
missionspectrumthroughsucha1Dsystem(Fig.1c).
Inaparalleldevelopment,theconceptoftimeasanadditional
dimension of crystallization has been developed in photonic time
crystals (PTCs)10–17,whicharephotonic materials whosepermittivity
variesperiodicallyintime.Ithasbeen shown thatPTCscanexhibit
bandgapsinmomentum,knownaskgaps(seeFig.1d).Incontrastto
thespatiallyevanescentmodesofconventionalphotonicbandgaps,k
gaps can host stationary, temporally amplifying modes. Here, the
vanishinggroupvelocityaccountsforthestationarity,whilethenon-
zeroimaginarypartoftheeigenfrequencydrivesamplification.Whena
kgapisexcitedthroughatemporalinterface,ratherthanbeingspa-
tiallyreflected,thepulseremainsinsidethemediumwithagrowing
temporal amplitude18. Such amplification breaks the energy con-
servationlaw,therebygreatlydiffersfromapassiveslow-lightsystem
supporting energy accumulation and field enhancement19,20. More-
over, since the spatiallyuniform modulation and conjugated eigen-
modes in k gap naturally satisfy the phase matching condition21,
temporal modulation induces non-resonant amplification over the
entirekgap,butismaximizedatthemid-gap,providingamechanism
of coherent light generation22 (e.g., lasers) by drawing energy from
timemodulationratherthangainmedia.
BandtopologycanalsobeextendedtokgapsinPTCs,givingrise
totopologicalphenomenainthetimedomain.Ithasbeenproposed18
thatakgapcarriesbandtopologycharacterizedbytheZakphase23,a
topologicalinvariantusedtodescribe1Dtopologicalinsulators.Inthis
context,anontrivialZakphaseenablesaPTCtofunctionasa“tem-
poral”topologicalinsulator.AtatemporalinterfacebetweentwoPTCs
withdistinctZakphases,amid-gaptemporaledgestatecanariseinthe
k gap (Fig. 1e), analogous to the topological edge state in spatial
topologicalinsulators7.Thistemporaltopologicaledgestateconfines
energyintimeratherthaninspaceandextendstheprincipleofbulk-
boundarycorrespondencetothetimedomain.Theexistenceofthisin-
gap topological state leads to a dip in the transmission spectrum
throughthe1Dsystem(Fig.1f).Althoughtemporallocalizationphe-
nomenawerereportedinseveralPT-symmetricplatforms24,25—where
time corresponds to the cycle number of pulses circulating in the
coupledfiberloopsandspaceisrepresentedbytherelativetemporal
positionsofthepulseswithineachcycle—aswellasinnon-photonic
systems26,27, the direct observation of a temporal topological edge
state in a genuine PTC within actual space-time has, until now,
remainedelusive.
In this work, we present an experimental study of the k-gap
propertiesinaPTCrealizedinadynamicallymodulatedtransmission-
line metamaterial and metasurfaces28–31. We employ a differential
modulationstrategy,whichenablesdirectobservationofthespatio-
temporal wave dynamics within the PTC. Moreover, the compact
constraintssofwaveinamicrostripguaranteeustofabricateasample
withasufficientlylargenumberofunitcellsandconsequentlypermits
comprehensive experimental verification of k gap characteristics.
Throughspace-timeFouriertransforms,wecapturethetemporalband
structure, revealing the k gaps. We demonstrate wave amplification
withinthekgap,whereanarbitraryinitialpowerspectrumnarrows
and shifts toward the k-gap center, confirming the mechanism of
coherentlightgenerationviatemporalmodulation.Wethenexperi-
mentallyprobethetemporaltopologyofthekgap,characterizedby
theZakphase,fromtwoperspectives.First,similartopreviousstudies
on1Dphotonictopologicalinsulators9,the time-reflectedandtime-
refracted waves through the PTC experience different phase shifts,
providing clear evidence of the quantized Zak phase. Second, we
introduceatemporalinterfaceseparatingtwoPTCswithdistinctZak
phases. A clear topological state that confines energy in time and
inducesadipinthetransmissionspectrumisobservedexperimentally.
Thismarksthefirstexperimentalobservationofatemporaltopolo-
gicaledgestateinamaterialplatform.

Results
PTCRealizationandexperimentalsetup
Inthiswork,weemployamicrostripdynamictransmissionline(DTL)
torealizeaPTC,withitsgeometrydetailedinFig.2a.Suchdiscrete
periodic loading ensures the DTL behaves as a spatially uniform
effective medium under DC bias in the long-wavelength regime
(effective-homogeneitylimit),andaccuratelymimicsaPTCwhenAC
modulationisapplied28,31,32.Theeffectivepermittivitywascomputed
using the field-circuit equivalence33 across various DC bias levels
(Fig.2b),andtheequivalentcircuitmodeloftheDTLunderthedif-
ferentialmodulationschemeispresentedinFig.2c.
In conventional DTL designs, the transmitting signal along the
DTLandtheexternalvaractormodulationremaininherentlycoupled,
preventingclearobservationofwavetemporaldynamicsintherea-
lizedPTC.Toovercomethislimitation,weadoptadifferentialmod-
ulationscheme.Toachieveprecisetemporalcontrolovertheeffective
permittivity, we use an arbitrary waveform generator (AWG) to
produceboththeprobeandmodulationsignals(Fig.2d).Thefabri-
cated DTL is presented in Fig. 2e, and the integrated measurement
platform is shown in Fig. 2f. This differential modulation scheme,
synchronizedviaanexternallygeneratedclocksignal,enablesdirect
observation of the genuine spatial-temporal evolution of electro-
magneticwaveswithinthetime-modulatedDTL.
Compared to an infinitely long PTC in theory, any practically
realizedPTCwithfinitelengthhasspecificspatialboundaries.Inthis
work,theDTLimplementedPTChastwoterminalportsatbothends,
and they affect the experiment in several aspects, as elaborated in
SupplementaryNote3.DetaileddescriptionsoftheDTLfabrication,
differentialmodulationcircuit,andexperimentalsetupareprovidedin
theMethodssection.

Dispersionandk-gapamplification
WefirstdemonstratethedispersionofastaticDTL.Theexperimen-
tallyextractedband structureofthestaticDTLispresentedinFig.3a,
withallvaractorsbiasedataDCvoltageof0.6V.Abroadbandprobe
pulse is initiallyinjected intothe DTL,and weobtain the static dis-
persion v(ω',k') by a 2D Fourier transform of the recorded spatio-
temporalvoltagev(x,t)alongthemeanderedline,wherex denotes
thepositionindex.Throughoutthiswork,frequenciesarenormalized
bythemodulationfrequencyΩ,andthewavevectorkisaccordingly
scaledbytheunitwavevectork = ε Ω=c .Afrequencybandgap
0 eff 0
emerges at ω'=1 as a consequence of the coupling between the
transmissionlineandtheperiodicallyloadedBSFs.Bymodelingthe
BSFs as effective LC resonators, we numerically calculate the
dispersion relation32,33(solidlineinFig.3a),whichshowsgoodagree-
mentwiththemeasurement.Atω'=0.5,thedispersionremainslinear,
ensuringthek-gapformationinthesubsequentmodulation.
WhentheDTLissinusoidallymodulatedintime,aPTCandits
Floquetbandsemerge,withkgapsopeninmomentumspace.Tofully
excite the Floquet eigenmode in positive and negative momentum
space,weinjectabroadbandGaussianpulsetrainfrombothportsof
theDTLsimultaneously.Byconvertingthemeasuredspace-timesignal
intothefrequency-momentumdomain,weobtaintheFloquetband
structure shown in Fig. 3b. The measured dispersion reveals two
hallmarkfeaturesofaPTC:(1)periodicreplicationofbandsalongthe
frequency axis and (2) opening of momentumbandgapsatω'=1/2.
Using the temporal Transfer Matrix Method (TMM)34 and Floquet
theory35,thetheoreticallycalculateddispersionrelation(yellowsolid
curvesinFig.3b)exhibitsexcellentagreementwiththeexperimental
data.NotethatweexcitethePTCthroughatemporalinterface.Asa
result,thek-gapmode,havingzerogroupvelocity,remainsconfined
withinthetransmissionlineandundergoesself-amplification,produ-
cingdarkspotsinthemeasureddispersionwithinthegapregion.A
horizontal stripe at ω'=1 is attributed to minor leakage of the
modulation signal, which is not fully canceled by the differential
modulation scheme. The details of the theoretical calculation are
providedinSupplementaryMethod2,andasimplifiedinvestigation
onthesystemlossimpactisprovidedinSupplementaryDiscussion3.
The k-gap modes, by virtue of their non-resonant amplification
mechanism, can spontaneously generate coherent standing waves
regardless of the input22. When a pulse with a broad momentum
spectrum is incident on a PTC, it encounters a purely temporal
boundary—unlike the spatial boundary of a conventional photonic
crystal,whichenforcestotalreflection.Undermomentumconserva-
tion, all momentum components of the incident wave lying on the
originallightlinemapontothePTCbandstructure.Themomentain
thebandregioncancoupletopropagating-waveeigenmodeswithout
experiencingamplification,whereasthemomentawithinthekgaps
induceexponentialamplification,givingrisetostandingwaves31,36and
spontaneousformationofspatiotemporalpattern22,37.Todemonstrate
these unique phenomena, we excite the DTL with a quasi-
monochromaticwavewhosecentralmomentalieswelloutsidethek
gap.DuetothefinitespatialsizeofDTL,asmallfractionofthewave’s
spatialspectrumleaksintothek-gapregion.Werecordthefullspatial-
temporalevolutionofthewaveinDTLandvisualizethedynamicsin
Fig.3c.Inaddition,wealsoplotinFig.3dfourspace-timesections,
eachwithadifferentstartingtimelabeledast0,illustratinghowthe
wave evolvesunder time modulation. Before modulation starts,the
incident wave propagates as a traveling wave (t = -10T).
Once modulationisswitchedon,amplificationoccursandastanding-wave
patterngraduallyforms(t =120T),accompaniedbyareductioninthe
effective wavelength, confirming dominant momentum conversion.
Tofurtheranalyzethedynamics,weextractthetemporalevolutionof
thenormalizedspatialpowerspectrumbyapplyingaspatialFourier
transformateachtimestep(Fig.3e).Inthisfigure,k'inc andk'gap arethe
normalizedmomentumatthecentralfrequencyoftheinputsignaland
thatatthek-gapcenter,respectively.Eventhoughtheinputmomen-
tum(k'inc=0:41)liesfarawayfromthek-gapcenter(k'gap=0:57),thek-
gapmomentadominatethepowerspectrumshortlyafterthemod-
ulation starts, overshadowing all out-of-gap components. Note that
this apparent momenta transition arises entirely from the normal-
ization strategy and does not imply a violation of momentum con-
servation in a PTC. This observation confirms the spontaneous
emissionbehaviorofPTCsaspredictedtheoretically22.Additionally,
we plot the temporal evolution of the spectral power for both the
incidentandmid-gapmomentumcomponentsinlogscale(Fig.3f)and
fitthemwithastraightline,demonstratingthepredictedexponential
amplificationofthekgap.Thesmallgainofthek'inc arisesfromthe
spatialtruncationandspectralbroadeningduetothefinitesizeofthe
DTL. A detailed discussion on thiseffectis given in Supplementary
Discussion4.Itisalsoworthnotingthatadiscrepancybetweenthe
measured result and the perfect exponential growing dashed line
becomesapparentafterasufficientlylongtime(markedinorangein
Fig.3f).Weattributesuchasaturationtopracticallimitationsinthe
experimental setup: 1. Finite modulation energy input from power
amplifiers.2.Intrinsiclossinmultiplecomponents(DCsupply,BSF,
etc.).3. Non-linearcapacitance-voltage response atlarge amplitude.
Throughout the measurement, we carefully design the modulation
schemesothatalltheexpectedphysicaleventshappenlongbefore
suchsaturationemerges.

BandtopologyandZakphase
Interestingly, PTCs also exhibitband topology in the temporal Bril-
louin zone, characterized by the Zak phase. Firstly, as theoretically
establishedinbothPTCs18andspatialphotoniccrystals9,ina1Dperi-
odicsystemwithlatticemirrorsymmetry,theZakphaseofeachband
is quantized to 0 or π. Furthermore, the sign of a wave scattering
observable, i.e., the reflection wave phase for a spatial photonic
crystal9orthephasedifferencebetweenthetime-reflectedandtime-
refracted waves for a PTC18, can be used to determine the Zak
phaseofbulkbands.InthecontextofPTCs,thisrelationisexplicitly
givenby:
sgn(φn) = (-1)n+lexp(i θZak) (1)
whereφn isthephasedifferencebetweenthetime-reflectedandtime-
refractedwavesforthenthkgap(withthelowestgapnumberindexas
1), evaluated at the last temporal interface between a PTC and the
subsequent static medium. l accounts for the number of band
crossingsunderthenthmomentumgap(l=0inourstudy),andθZak
represents the Zak phase of mth band. Note that the term “time-
reflected wave” denotes the spatially backward propagation wave
caused by a temporal interface via time-modulation18,29. We plot in
Fig. 4a the temporal profile of both PTCs with distinct Zak phase,
wherebothPTCsatisfiesthetemporalmirrorsymmetry(V(t)=V(-t))
butwithoppositesymmetrycenter.WhentwoPTCswithdistinctZak
phasesaretemporallycascadedbetweenatemporalinterface(shown
byablackdashedlineinFig.4b),amid-gaptopologicaledgestate
emergesinsidethekgap.Inourexperiment,thesinusoidalmodulation
opens only the first momentum bandgap. Therefore, we extracted
φ1(referred to simply as φ hereafter) to characterize the k-gap
topology.In the experiment, due to the finite spatial length of the
DTL,themomentumresolutioninsidethekgapislimited,makingit
difficulttoresolvetheintensity/phasespectrumwithhighprecision.
To circumvent this issue, we repeat the measurement at different
modulation frequencies and effectively increase the number of
measurable points in momentum space. Detailed methodology and
thephaseextractionprocedurearedescribedintheSupplementary
Method 3 and 4. The experimentally extracted values of φ at nine
momentapointsinthekgapareshowninFig.4cwithbluetriangles
and red diamonds corresponding to two PTC configurations with
θZak=0 and π, respectively, referred to as PTC1 and PTC2 with
opposite temporal lattice mirror symmetry center. The theoretical
relation of φ and k calculated by the Transfer Matrix Method are
plottedwithblueandredsolidlinesforcomparison.Theexperimental
resultsagreewellwiththetheory.Asexpected,PTC1andPTC2exhibit
thesamemonotonicityinφ(k)butwithoppositesigns,highlighting
theirdistinctk-gaptopologies.

Temporaltopologicaledgestate
Finally, we observe the emergence of a temporal edge state at the
interfacebetweenPTC1andPTC2.Inthecontextofspatialphotonic
crystals,ithasbeendemonstrated9thattheappearanceofanunim-
odular transmission spike at a specific frequency for two photonic
crystalswithdistinctgaptopologiesensuresatopologicaledgestate
confinedinspace,whichisalsothepointwherethesummationofthe
reflectionphasesoftherespectivespatialphotoniccrystalsapproa-
cheszero.Byanalogy,asimilarphenomenonisexpectedinthetem-
poraldomain:thezeropointofthesummationofφfortwoPTCswith
distinct Zak phase (indicated by the black line and dots in Fig. 4c)
predicts a critical momenta k, where a temporal topological edge
statearises.UnlikethatinSPCs,thetemporaltopologicaledgestatein
PTCs manifests as a dip in the total energy spectrum, while the
remaining momentum components within the k gap exhibit high
transmission.Boththetheoretical(pinksolidline)andexperimentally
extracted(reddot)totalenergyspectrumareplottedinFig.4d.The
rationaleforadoptingtotalenergy|T+R|insteadoftransmission|T|is
explainedinSupplementaryDiscussion1.Asexpected,adipappearsat
k'c =0:564inthetotalenergyspectrum,whichisclosetothetheoretical
predictionkc (markedbydashedgreenlinesinbothFig.4a,b).Note
thatowingtotheunavoidablelossinthesystem,theedgestatetotal
transmissionenergyislessthanunity.Therefore,forbettercompar-
ison,wenormalizeboththetheoreticalandexperimentalresultsby
themaximumvalueinthekgap.Asaconsequence,thenormalized
edgestatetransmissionisreducedtoaclose-to-zerovalue.Toprobe
the associated field dynamics, we excite the DTL at this critical
momentumandplotthespatiotemporalfieldevolutioninFig.4e.The
fieldshowstemporallocalizationcenteredatt=0,consistentwitha
temporallylocalizededgestate.Weextractthetemporalpowerevo-
lutionsoftheedgestatemomentaandplotitinFig.4f(bluecurve).
PriortothetemporalinterfacebetweenPTC1andPTC2,theenergy
exhibits exponential growth as a k-gap mode. After the temporal
switching,theenergyundergoesexponentialdecay,formingatem-
poral localization profile. This distinct growth-decay pattern con-
stitutes the first direct observation of a temporal topological edge
stateinagenuinePTCasamaterialplatform.Longafterthetemporal
switching,theenergyisfinallyre-amplified,whichoriginatesfromthe
simultaneous excitation of temporal growth and decaying k-gap
eigenmodes.Notethatthetemporalasymmetricalgrowth-decaypro-
file originates from intrinsic losses in the DTL, as also discussed in
Supplementary Discussion 3, which suppresses the pre-interface
amplification rate and enhances the post-interface decay rate. For
comparison,thetemporalpowerevolutionofanextendedbulkstate
atk'bulk =0:19isalsoshownastheredcurveinFig.4d,renderingafield
temporalpatternwithatemporallystablemagnitude.Fieldevolutions
of several other gap momentum components are also examined
and are provided in the Supplementary Discussion 2. Compared to
the pronounced temporal localization observed in Fig. 4f, these
components exhibit significantly shallower temporal peaks and
correspondingly higher transmission, consistent with their weaker
confinementintime.

Methods
DTLdesignandfabrication
The photonic time crystal (PTC) was realized using a microstrip
dynamictransmissionline (DTL) composed of a metallic microstrip
tracephotoetchedontoanFR4substrate(εr=2.2),periodicallyloaded
with64pairsofshuntvaractors(SkyworksSMV1249-079LF)andseries
inductors.The varactors provide non-linear,bias-voltage-dependent
capacitance,whiletheinductorsareplacedtoextendtheopticalpath
withineachunitcell.Themodulationwasappliedwithinaquasi-linear
range (indicated by the blue line in Fig. 2b) to achieve a sinusoidal
varyingpermittivitymodulation.Basedoncriticalparametersfromthe
varactor datasheet (including series resistance), typical junction
characteristics,andourmodulationparameters,thefinitetimedelay
caused by carrier transit-time and series resistance effect was esti-
matedtobeontheorderofpicoseconds,whichisnegligiblecompared
to our 7.7ns modulation period. Meanwhile, the high-purity MBE-
grownvaractorexhibitsnegligibletrap-relatedhysteresisatlowfre-
quencies. These considerations regarding varactor latency are sup-
portedbypriorstudies28,32.

Differentialmodulationscheme
Inthisscheme,eachpairofvaractorsisreverse-biasedwithopposite
polaritiesandsymmetricallyplacedoneithersideofthemicrostrip
trace line in each unit cell. In the equivalent circuit model of DTL
(Fig. 2c),L0,C0 representthe intrinsic unit-lengthseriesinductance
andshuntcapacitanceofthehostmicrostripline.Ldenotestheloaded
lumpedinductor.Notetheblueandorangecolorvaractorshavethe
opposite polarities, and the labels v(t) and -v(t) represent the time-
varyingmodulationsignalforeachvaractordiode(seealsoFig.2e).
Each varactor is shunted to the ground via a band-stop filter (BSF)
resonantatthemodulationfrequency,therebyisolatingthemodula-
tion signal. This configuration allows the modulation contributions
fromeachvaractortocanceleachotheroutandleaveacleanenvir-
onment for dynamic signal detection without diminishing the
modulationdepth.

Experimentalsetupandwaveformcontrol
Anarbitrarywaveformgenerator(AWG)wasusedtoproduceboththe
probeandmodulationsignals.Athree-stagepower-dividingnetwork
wasdesignedtoprovidecoherentmodulationsignalsforeachpairof
varactors.ThefabricatedDTL,integratedwithafieldscannerandthe
power dividing network into a compact scaffold-like measurement
platform(Fig.2f).Thissetup,synchronizedviaanexternallygenerated
clock signal, enabled direct observation of the genuine spatial-
temporal evolution of electromagnetic waves within the time-
modulated DTL. Additional details on the effective permittivity cal-
culation,experimentalsetup,andtheperformanceofthedifferential
modulation are provided in the Supplementary Method 1, Supple-
mentaryNote1and2,respectively.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "In the manuscript entitled “Observation of wave amplification and temporal topological state in a genuine photonic time crystal”, Xiong et al. present an impressive and experimentally rich demonstration of a genuine photonic time crystal (PTC), realized via a dynamically modulated transmission line platform. Through temporal periodic modulation, they succeed in observing the emergence of k-gaps and associated phenomena such as wave amplification. Even more compelling is their direct observation of a temporally localized topological edge state at a domain wall between two distinct PTC phases. The work is well-structured and highly original. I am enthusiastic about its potential impact and would recommend publication, pending the authors' response to the following points:\n- Temporally localized topological edge states have previously been demonstrated in a synthetic space (e.g., Ref. 20). The authors should explicitly discuss this earlier work and highlight how their approach differs. That said, to the best of my knowledge, this work indeed represents the first realization of temporally localized topological edge states in a genuine photonic time crystal, as opposed to previous implementations based on synthetic dimensions.\n- Line 155, the authors mention that “k gaps open in momentum space”. However, in the measured spectrum in Fig 3b, there is no clear “gap” at k = 0.5, but a dark spot occupying the expected k-gap region. Could the authors elaborate on this? Is it an experimental artifact, or does it point to a more nuanced understanding of how k-gaps manifest in this system?\n- Line 195, as the authors mentioned, the system exhibits intrinsic loss. Given that the Floquet band structure is calculated under lossless conditions, how robust is the comparison between theory and experiment? Could the inclusion of loss qualitatively or quantitatively alter the Floquet features?\n- The observation of a temporal topological edge state is one of the most exciting aspects of this work. To better understand and reproduce this result, the authors should provide the precise temporal modulation profiles corresponding to the two PTCs with Zak phases of 0 and π, respectively. Also, please include the exact modulation used to form the temporal domain wall.\n- In Fig. 1f, the temporal edge state is expected to exhibit nearly unity transmission. Yet in Fig. 4b, the measured |T + R| is close to zero. This striking difference deserves clarification—could it arise from interference effects, mode mismatch, or other loss mechanisms?\n- The asymmetry in the temporal edge state profile is intriguing. The authors suggest it stems from intrinsic losses in the transmission line (lines 258–260). Can this be backed up with simulations or theoretical modeling? If so, showing such analysis would greatly strengthen the manuscript.", "reviewer_2": "This is a very timely and interesting study of a time crystalline system experimentally realized in the microwave regime via a transmission line comprising an inductor and a varactor diode element, the latter having bias-controlled capacitance. With a time-modulated bias it hence corresponds to a time-modulated permittivity system.\nThe authors show a power gain in the system occurring within the k-gap of the time-modulated system, as well as new type of topological edge modes in the phase-space of the time-crystalline system which were observed when an interface is created between the two time crystals, each having a different modulation phase.\nThis work reports exciting new Physics that I believe will inspire further progress in the field of space-time photonics. I am therefore recommending its publication provided the authors explain more in depth the underlying Physics, while giving more expanded supportive context for the benefit of the reader.\nIn particular:\n(1) The authors state that when the incident wave, this being a broad pulse, hits the transmission line there is power gain around the k-gap coming from the modulation mechanism. They also mention that k-gap in time crystals is analogous to omega-gap in photonic crystals. When incident light within the omega-gap spectrum hits the photonic crystal interface it completely back-reflects. Here when light hits the transmission line it couples in and its intensity enhances in the k-gap spectrum. So, it appears the analogy does not quite work- something is different in their system because the EM can couples into the k-gap modes. Can the authors say a few words what allows the wave to couple into the k-gap modes, instead of getting back-reflected as one would expect from a k-gap behavior.\n(2) I think it would be helpful if the authors added a section in the supplemental material explaining the boundary conditions they are considering. In section IV they are talking about temporal boundaries but the modulation is continuous with no discontinuity so I am a bit confused where the temporal boundary (i.e. a discontinuity in permittivity in time) comes from. Also, while the wave goes through the transmission line there are clearly spatial boundaries. Elaborating further these aspects would be very helpful not only in understanding the method but also the results.\n(3) Adding to (2): I realize the transmission line has a quasi-static homogenized response, but even if I were to ignore the in between spatial boundaries there are at least two spatial boundaries at the entry and exit of the wave.\n(4) How is t0=0 defined? Is it when modulation is switched on?\n(5) In Fig. 4 is k_inc=k_0=omega_0/c with omega_0 being the central frequency of the input pulsed signal? Please elaborate.\n(6) How is wave vector k inside the time crystal defined/determined? In spatial photonic crystals the frequency is fixed by the input wave and the Floquet-Bloch wave inside the photonic crystal is a sum of various spatial harmonics, i.e., wave vector components, that relate to the spatial modulation (i.e. spatial period a). Here, the Floquet wave inside the time crystal should be a sum of different time harmonics that relate to the modulation frequency, right? In photonic crystals omega is conserved, in time crystals by analogy one would expect k to be conserved. However, Fig. 3(e) could suggest otherwise? The authors should add in the supplemental material a section describing how the wave vector inside the homogenized time-crystal is determined from the input wave.\n(7) Related to (6): is the wave vector related to the individual frequency components of the input pulse conserved but only the components around kgap end-up getting amplified at the cost of the other frequency components? I think in the caption of Fig. 3e it should be clarified that \"The momenta of the input signal ...\" is \"The momentum at the central frequency of the input pulsed signal ...”.\n(8) Where is the small gain in k_inc observed in Fig. 3(f) coming from?\n(9) Related to comment (1): I would expect at a k-gap to see high back-reflection instead of enhancement. In the band structure of the DTL seen in Fig. 3(b) in the region that the authors have terms as k-gaps, one can clearly observe flat segments in the omega versus k dispersion relation, implying domega/dk=0, i.e. frozen-light modes. Slow-light (low-group velocity) modes have been associated with amplification, by observation of enhancement of associated non-linear effects [e.g. see B. Corcoran et al, Nature Photonics volume 3, pages 206–210 (2009)], and other similar effects. Electric field intensity amplification has also been observed directly with simulations in slow-light systems; e.g. see Foteinopoulou+Vigneron, Phys. Rev. B 88, 195144 (2013)]. Could the observed power gain be due to the frozen-light mode in the region the authors term as k-gap (flat omega-k dispersion line) rather than a k-gap effect?\n(10) The authors use a model to calculate the permittivity versus the DC bias for different values of DC bias, V, of the varactor. Then they apply this model to find the modulated permittivity epsilon(t) by using each separate value V(t); i.e. the authors have implicitly assumed zero latency in the varactor response. I think it would be useful if the authors added a few words describing why the varactor latency can be considered negligible.\n(11) Eq. S.6 has a typo. In the bottom equation second term I think it should be omega_m instead of omega_a.", "reviewer_3": "In the paper, authors experimentally demonstrate both nonhermitian bandgap and temporal topological state at the temporal boundary of two bandgaps with different topologies. Although the former has been demonstrated the latter is a first-time demonstration to my knowledge. They employ a transmission line based method to mimic photonic time crystal in microwave regime. The paper is well structured and written with sufficient technical details provided in the main text and supplementary. Although similar experimental setups have been used in previous proof-of-concept demonstrations of temporally varying media, and the theoretical foundations of topology in photonic time crystals are well established, this work presents a unique and well-executed combination of both elements. So, I would recommend this paper for publication in Nature Communications, provided the authors more clearly explain the novelty and potential impact of their demonstration, more precisely for the temporal bound mode, and address the minor comments outlined below.\n1. Authors mention in the introduction the difference between a PTC bandgap and parametric amplification is the bandgap and the relaxation of phase matching. This explanation is commonly used in literature but requires further elaboration. The reason that there seems to be no phase matching requirement in PTCs is because in theory it is assumed that the whole medium is modulated simultaneously (therefore with k=0) however that is never the case if one uses a propagating pump for modulation. The phase still needs to be matched. And, as mentioned in the supplementary it is still important to control the phase between the varactors of adjacent unit cells in the experimental realization to observe the amplification. That is still part of the phase matching requirement. So, the amplification in OPA and PTCs are still fundamentally linked to each other.\n2. An addition to the first point; the modulation of varactor pairs seems to be out of phase in Fig S1 how does the device perform with different phase differences?\n3. I think it is worth mentioning in the abstract and maybe in the title that this realization is in microwave regime.\n4. I do not understand what the “genuine” refer to in the title and whether if that is necessary.\n5. I would recommend explaining the difference between PTC1 and PTC2 in the main text too, it is only explained in the figure caption that they have opposite initial phase.\n6. One other method of realizing PTCs has been done with metasurfaces[1] how does that method compare to TL based methods.\n7. [1] Wang, X., Mirmoosa, M. S., Asadchy, V. S., Rockstuhl, C., Fan, S. & Tretyakov, S. A. Metasurface-based realization of photonic time crystals. Sci. Adv. 9, eadg7541 (2023). https://doi.org/10.1126/sciadv.adg7541"}, "author_response": "Response to Reviewer #1\nDear Reviewer #1:\nWe sincerely appreciate your supportive comments and thoughtful feedback. Your insights have\ngiven us the opportunity to further clarify and emphasize the key mechanisms behind our research.\nReviewer #1 – General Comments:\n“In the manuscript entitled “Observation of wave amplification and temporal topological state in\na genuine photonic time crystal”, Xiong et al. present an impressive and experimentally rich\ndemonstration of a genuine photonic time crystal (PTC), realized via a dynamically modulated\ntransmission line platform. Through temporal periodic modulation, they succeed in observing the\nemergence of k-gaps and associated phenomena such as wave amplification. Even more\ncompelling is their direct observation of a temporally localized topological edge state at a domain\nwall between two distinct PTC phases. The work is well-structured and highly original. I am\nenthusiastic about its potential impact and would recommend publication.”\nResponse from authors:\nThank you so much for your appreciation and highly positive comment that “Xiong et al. present\nan impressive and experimentally rich demonstration of a genuine photonic time crystal (PTC)”,\n“The work is well-structured and highly original” and “I am enthusiastic about its potential\nimpact and would recommend publication”.\nIn the following, we will fully address your questions point by point.\nReviewer #1 – Comments 1:\n“Temporally localized topological edge states have previously been demonstrated in a synthetic\nspace (e.g., Ref. 20). The authors should explicitly discuss this earlier work and highlight how\ntheir approach differs. That said, to the best of my knowledge, this work indeed represents the first\nrealization of temporally localized topological edge states in a genuine photonic time crystal, as\nopposed to previous implementations based on synthetic dimensions.”\nResponse from authors:\nThank you very much for your appreciation of our work’s novelty that “this work represents the\nfirst realization of temporally localized topological edge state in a genuine photonic time crystal”.\nFor further clarification, we provide a systematic comparison of the model and methodology in\nthe context of recent progress on topological photonic time crystals, showing that our work differs\nfrom the previous studies not only in the methodology but also in the fundamental model.\nTo the best of our knowledge, only two experimental realizations have been reported in photonic\nsystems [Feis et al., Nature Photonics (2025): 1-8; Ren et al., Nature Communications 16.1 (2025):\n707], along with one preprint in acoustic wave systems [arXiv:2507.04068], and another in elastic\nwave systems [arXiv:2306.09627]. In both photonic demonstrations of temporal topological edge\nstates, as you are aware, time functions as a discrete synthetic dimension—represented by the cycle\nnumber of the pulses circulating in the coupled fiber loops—while the space is mapped to the\nrelative temporal positions of the pulses within each cycle. Therefore, the observed topological\nedge mode emerges in an effectively synthetic “space-time” domain. By contrast, in our system,\nenabled by differential modulation, we directly observe the temporal topological edge state in the\nactual space-time coordinates.\nMoreover, in all the previous works mentioned above, the “momentum bandgap” originates from\nthe spatial periodical stacking of gain and loss—namely, a non-Hermitian PT-symmetry-broken\nBloch phase. By contrast, the original theoretical prediction attributes the momentum bandgap\nsolely to temporal periodic modulation of the real part of the refractive index in a spatially\nhomogeneous Hermitian medium. Our system aligns exactly with this model (Hermitian, spatially\nhomogenous, real-index modulation), thereby constituting the first realization of a genuine\nphotonic time crystal.\nTo further highlight our novelty compared with prior works, we have added an explicit discussion\nin the revised manuscript (lines 88-95), directly contrasting our results with the existing study.\n“Although temporal localization phenomena have recently been observed in several PT-symmetric\nplatforms in synthetic space-time [24,25]—where time corresponds to the cycle number of pulses\ncirculating in the coupled fiber loops and space is represented by the relative temporal positions\nof the pulses within each cycle—as well as in non-photonic systems [26,27], the direct observation\nof a temporal topological edge state in a genuine PTC within actual space-time has, until now,\nremained elusive.”\nReviewer #1 – Comments 2:\n“Line 155, the authors mention that “k gaps open in momentum space”. However, in the measured\nspectrum in Fig 3b, there is no clear “gap” at k = 0.5, but a dark spot occupying the expected k-\ngap region. Could the authors elaborate on this? Is it an experimental artifact, or does it point to\na more nuanced understanding of how k-gaps manifest in this system?”\nResponse from authors:\nThank you for raising the concern regarding the existence of modes inside a k gap. We would like\nto clarify that this originates from the fundamental distinction between conventional photonic\nbandgaps (ω-gaps) and k gaps in photonic time crystals (PTCs).\nIn spatial photonic crystals, ω-gaps prohibit wave propagation due to their evanescent decay: when\nexcited at a frequency within the ω-gap, waves are totally reflected at a spatial interface and cannot\npenetrate the crystal. By contrast, in PTCs, k-gap eigenmodes possess both non-zero real and\nimaginary parts of the eigenfrequency. The real part enforces zero group velocity, while the\nimaginary part breaks energy conservation and drives temporal amplification. As a result, if a PTC\nis excited at a k-gap mode through a temporal interface (as in our experiment), the wave remains\nconfined within the transmission line and forms stationary amplification, making it directly\nobservable inside the PTC. This mechanism has been detailed in prior studies on PTCs [19, 22]. It\nalso explains why the experimentally extracted band structure in Fig. 3b exhibits dark spots\n(amplified fields) precisely at the theoretically predicted k-gap regions.\nTo avoid ambiguity, we have added the following explanation in the revised manuscript (lines\n192-195):\n“Note that we excite the PTC through a temporal interface. As a result, the k-gap mode, having\nzero group velocity, remains confined within the transmission line and undergoes self-\namplification, producing dark spots in the measured dispersion within the gap region.”\nReviewer #1 – Comments 3:\n“Line 195, as the authors mentioned, the system exhibits intrinsic loss. Given that the Floquet\nband structure is calculated under lossless conditions, how robust is the comparison between\ntheory and experiment? Could the inclusion of loss qualitatively or quantitatively alter the Floquet\nfeatures?”\nResponse from authors:\nThank you for raising the concern about the loss in the Floquet system, which is indeed critical.\nWe will address this issue quantitatively below.\nWe begin with a basic model in which the permittivity of a spatially homogeneous medium is\nmodulated in time:\n𝜀 (𝑡) = 𝜀̅[1+𝑚 sin(Ω𝑡)]+𝑖𝜀′′\n𝑟 𝑟 𝜀 𝑟\nwhere 𝑚 is the modulation strength, Ω modulation frequency, and 𝜀′′ the time-invariant intrinsic\n𝜀 𝑟\nloss factor. Note that 𝜀′′ captures a typical source of loss, and its effect applies similarly to other\n𝑟\npractical loss mechanisms, such as metallic loss in the microstrip line, resistance loss in the\nvaractors and BSF, and energy leakage (radiation) at the DTL ends.\nFig. R1. Floquet band structure and eigenmode of lossy PTCs. a-c. The real part (black) and\nimaginary part (red) of the eigenfrequency of PTCs with different loss. d-f. The conjugate\n𝑘\neigenmodes of the k-gap center ( = 0.5) in a-c. The increasing loss factor leads to the temporal\n𝑘0\nprofile asymmetry.\nThis model describes a PTC with uniform loss, consistent with the experiment scenario. We\ncalculate the band structure and eigenmodes under varying loss factors. Fig. R1a-c show the real\nand imaginary parts of the eigenfrequencies. As we increase the loss of the system, the real part of\neigenfrequency, particularly the location of the k-gap, remains nearly unchanged unless the loss\nbecomes exceptionally large (in this case the k gap in the Bloch band structure shifts horizontally,\nnarrows, or eventually closes). By contrast, the imaginary part of eigenfrequency is more sensitive,\nuniformly shifting into the negative half-plane. As a result, in Fig. R1d-f, the two conjugate\neigenmodes become temporally asymmetric (this also helps to clarify the origin of asymmetry in\nthe temporal edge state profile; see our reply to Comment 6). Once both eigenmodes fall into the\nnegative half-plane, the k-gap no longer supports temporal amplification and the modes decay over\ntime (see Fig. R1c and R1f).\nIn our experiment, the modulation intensity 𝑚 is carefully chosen based on real circuit parameters\n𝜀\nand estimated practical losses. This ensures that our retrieved band structure is reliable: the\ntheoretical k gaps predicted in the lossless case, which remain stable in position under the condition\nof moderate temporal amplification, are clearly identified by the dark spots in the experimental\nband structure (see Fig. 3b of the main text).\nTo illustrate the interplay between loss and modulation in our system, we have added the above\ndiscussion in Supplementary Materials as Section X: Discussion on the system loss.\nReviewer #1 – Comments 4:\n“The observation of a temporal topological edge state is one of the most exciting aspects of this\nwork. To better understand and reproduce this result, the authors should provide the precise\ntemporal modulation profiles corresponding to the two PTCs with Zak phases of 0 and π,\nrespectively. Also, please include the exact modulation used to form the temporal domain wall.”\nResponse from authors:\nThank you for highlighting one of the main findings of our work about the temporal topological\nedge state. We apologize for the absence of an exact graphic temporal modulation profile for either\nPTCs or temporal domain wall structures.\nAs predicted in [Lustig et al., Optica 5.11 (2018): 1390-1395], the quantized Zak phase is protected\nby temporal mirror symmetry, with the two symmetry centers correspond to Zak phases with 0\nand 𝜋. When two PTCs with different Zak phases are aligned in time, a domain-wall structure is\nformed. In Fig. R2 we plot the temporal profile for PTC1 (𝜃 = 0), PTC2 (𝜃 = 𝜋) and the\n𝑍𝑎𝑘 𝑍𝑎𝑘\ndomain wall structure.\nFig. R2. Temporal profiles of the effective permittivity for pure PTCs and domain wall structure.\na. The temporal profiles of PTCs with different Zak phases, 𝜃 = 0 (blue) and 𝜃 = 𝜋 (red).\n𝑍𝑎𝑘 𝑍𝑎𝑘\nb. The domain wall structure formed by aligning two PTCs with distinct Zak phases in time. At\nt=0, a temporal interface abruptly changes the effective permittivity.\nTo further improve the readability of our paper, we have included these two figures as Fig. 4a, b\nin the revised manuscript.\nReviewer #1 – Comments 5:\n“In Fig. 1f, the temporal edge state is expected to exhibit nearly unity transmission. Yet in Fig. 4b,\nthe measured |T + R| is close to zero. This striking difference deserves clarification—could it arise\nfrom interference effects, mode mismatch, or other loss mechanisms?”\nResponse from authors:\nThank you for pointing out the ambiguity regarding the scale of the transmission minimum at the\nedge state. We clarify below that this discrepancy arises from the normalization process.\nIn Fig. 1f, to highlight the distinction between spatial and temporal topological edge states, we plot\nthe exact transmission coefficient of the domain-wall structure formed by two dispersionless,\nlinear, and lossless PTCs. In this case, the in-gap mode exhibits transmission greater than unity,\nwith a dip to unity at the edge-state momenta. In contrast, in our experiment, unavoidable losses\nof various forms (see our reply to Comment 3) make the temporal edge state profile spectrally\nasymmetric, with the decay rate exceeding the exponential growing rate (this is addressed in our\nreply to your Comment 6). Therefore, when transmission is measured over equal time intervals in\nboth PTCs the edge-state transmission coefficient is smaller than unity. To connect theory and\nexperiment, we normalize the intensity |𝑇+𝑅| by the maximum value in the momentum bandgap\n|̅̅𝑇̅̅+̅̅̅̅𝑅̅̅| for both theoretically predicted and experimentally measured results. The close-to-unity\nedge state mode is then normalized to a close-to-zero transmission point.\nTo clarify this ambiguity, we add the following explanation in the revised manuscript to clarify\nthe difference (lines 292-296):\n“Note that owing to the unavoidable losses in the system, the edge state total transmission energy\nis less than unity. Therefore, for better comparison, we normalize both the theoretical and\nexperimental results by the maximum value in the k gap. As a consequence, the normalized edge\nstate transmission is reduced to a close-to-zero value.”\nReviewer #1 – Comments 6:\n“The asymmetry in the temporal edge state profile is intriguing. The authors suggest it stems from\nintrinsic losses in the transmission line (lines 258–260). Can this be backed up with simulations\nor theoretical modeling? If so, showing such analysis would greatly strengthen the manuscript.”\nResponse from authors:\nYes, as noted in our response to Comment 3, we have already addressed the physical origin of the\ntemporal asymmetry of the edge state. As illustrated in Fig. R1b, e, and R1c, f, the conjugate\neigenmodes of a lossy PTC intrinsically exhibit asymmetric temporal evolution. To further\nillustrate this effect, we calculate and present in Fig. R3a-c the field evolution in a temporal\ntopological domain-wall structure under different loss factors, where the gain-loss asymmetry\nbecomes more pronounced as the loss factor increases.\nFig. R3. Temporal topological edge mode for lossy PTC domain wall structure.\nAdditionally, as required, we have incorporated the above calculation— together with our response\nto Comment 3— as a new section in the Supplementary Materials as Section X: Discussion on the\nsystem loss.\nResponse to Reviewer #2\nDear Reviewer #2:\nWe would like to express our gratitude for your positive comments and constructive suggestions.\nFollowing your recommendations, we believe that the revisions have substantially improved the\nquality and clarity of the manuscript.\nReviewer #2 – General Comments:\n“This is a very timely and interesting study of a time crystalline system experimentally realized in\nthe microwave regime via a transmission line comprising an inductor and a varactor diode\nelement, the latter having bias-controlled capacitance. With a time-modulated bias it hence\ncorresponds to a time-modulated permittivity system.\nThe authors show a power gain in the system occurring within the k-gap of the time-modulated\nsystem, as well as new type of topological edge modes in the phase-space of the time-crystalline\nsystem which were observed when an interface is created between the two time crystals, each\nhaving a different modulation phase.\nThis work reports exciting new Physics that I believe will inspire further progress in the field of\nspace-time photonics. I am therefore recommending its publication provided the authors explain\nmore in depth the underlying Physics, while giving more expanded supportive context for the\nbenefit of the reader.”\nResponse from authors:\nThank you so much for the positive comment that “This is a very timely and interesting study”,\n“This work reports exciting new Physics that I believe will inspire further progress” and “I am\ntherefore recommending its publication”. We greatly appreciate this encouraging feedback. With\nthis supportive context, we now address your concerns point by point.\nReviewer #2 – Comments 1:\n“The authors state that when the incident wave, this being a broad pulse, hits the transmission\nline there is power gain around the k-gap coming from the modulation mechanism. They also\nmention that k-gap in time crystals is analogous to omega-gap in photonic crystals. When incident\nlight within the omega-gap spectrum hits the photonic crystal interface it completely back-reflects.\nHere when light hits the transmission line it couples in and its intensity enhances in the k-gap\nspectrum. So, it appears the analogy does not quite work- something is different in their system\nbecause the EM can couples into the k-gap modes. Can the authors say a few words what allows\nthe wave to couple into the k-gap modes, instead of getting back-reflected as one would expect\nfrom a k-gap behavior.”\nResponse from authors:\nThank you for raising the concern about the difference in wave dynamics between an ω-gap and a\nk gap. We would like to clarify that although a k gap is formally the mathematical analogue of an\nω-gap, their modal dynamics are fundamentally different.\nMathematically, the term “bandgap” describes an eigenvalue system in which the real part of the\neigenvalue (𝑘 for an ω-gap and Ω for a k gap, with subscripts B and F denoting Bloch and\n𝐵 𝐹\nFloquet modes, respectively) becomes degenerate, while the imaginary part splits into two\nopposite nonzero values. These paired imaginary components give rise to two conjugate\neigenmodes that either grow or decay spatially (in spatial photonic crystals, or SPCs) or temporally\n(in photonic time crystals, or PTCs).\nPhysically, as discussed in [Lyubarov et al. Science 377.6604 (2022): 425-428], the wave\ndynamics differ dramatically between a SPC and a PTC at their respective bandgaps. As illustrated\nin Fig R4a, when an SPC is excited at a frequency within the ω-gap, energy conservation dictates\nthat all fields decay rather than grow away from the excitation point. This evanescent spatial\nbehavior results in total reflection at a spatial interface.\nBy contrast, as shown in Fig. R4b, when a PTC is excited at a momentum within the k gap, the\nbreaking of time-translation symmetry violates energy conservation. The system simultaneously\nsupports both temporally growing and decaying modes, which overall produce a net increase in\nenergy. Importantly, temporally backscattered waves are forbidden by causality (see dashed line\nin Fig. R4b). As a result, when a PTC is excited at its k-gap momentum through a temporal\ninterface, instead of being spatially reflected, the wave remains “frozen” (due to zero group\nvelocity) and amplifies within the medium. A similar discussion can be found in [Lustig et al.,\nOptica 5.11 (2018): 1390-1395], from which we excerpt their simulated pulse dynamic of both\nbulk and k-gap modes in Fig. R4c-d.\nFig. R4. a,b: Eigenmodes in an SPC (a) and PTC (b). Excerpted from [Lyubarov et al. Science\n377.6604 (2022): 425-428]. c,d: FDTD simulated wave dynamics in PTC, when the wave\nmomentum components reside in the band and in the k gap. Excerpted from [Lustig et al.,\nOptica 5.11 (2018): 1390-1395].\nAccordingly, we have added the following clarifications in the Introduction and Section III to\ndistinguish between ω gaps and k gaps, and to explain the wave dynamics associated with\nincidence into a PTC.\n(lines 64-70) “In contrast to the spatially evanescent modes of conventional photonic bandgaps, k\ngaps can host stationary, temporally amplifying modes. Here, the vanishing group velocity\naccounts for the stationarity, while the nonzero imaginary part of the eigenfrequency drives\namplification. When a k gap is excited through a temporal interface, rather than being spatially\nreflected, the pulse remains inside the medium with a growing temporal amplitude [18].”\n(lines 203-208) “When a pulse with a broad momentum spectrum is incident on a PTC, it\nencounters a purely temporal boundary—unlike the spatial boundary of a conventional photonic\ncrystal, which enforces total reflection. Under momentum conservation, all momentum\ncomponents of the incident wave lying on the original light line map onto the PTC band structure.”\nReviewer #2 – Comments 2&3:\n“I think it would be helpful if the authors added a section in the supplemental material explaining\nthe boundary conditions they are considering. In section IV they are talking about temporal\nboundaries but the modulation is continuous with no discontinuity so I am a bit confused where\nthe temporal boundary (i.e. a discontinuity in permittivity in time) comes from. Also, while the\nwave goes through the transmission line there are clearly spatial boundaries. Elaborating further\nthese aspects would be very helpful not only in understanding the method but also the results.”\n“Adding to (2): I realize the transmission line has a quasi-static homogenized response, but even\nif I were to ignore the in between spatial boundaries there are at least two spatial boundaries at\nthe entry and exit of the wave.”\nResponse from authors:\nThank you for raising concerns about the effect of boundary conditions. Your questions are two-\nfold: (1) Where does the temporal boundary come from? and (2) How do the two DTL terminal\nports (spatial boundaries) influence the experimental results and what design considerations are\ninvolved? We address them as follows.\n1. Where does the temporal boundary come from?\nWe apologize for not sufficiently illustrating the physical meaning of the term “temporal\ninterface” in the section on the temporal topological edge state. In Fig. R5a we plot the\ntemporal profiles of two PTCs with distinct Zak phases. As predicted in [Lustig et al.,\nOptica 5.11 (2018): 1390-1395], the quantized Zak phase is protected by temporal mirror\nsymmetry, with the two symmetry centers corresponding to Zak phases of 0 and 𝜋. When\ntwo PTCs with different Zak phases are aligned in time, a domain-wall structure forms,\nand a temporal topological edge state emerges. When the two PTCs contact each other, as\nshown in Fig. R5b, a temporal discontinuity appears at t = 0, which constitutes the physical\nmeaning of the “temporal interface”.\nFig. R5: Temporal permittivity profiles for pure PTCs and the domain-wall structure. a. The\ntemporal profiles of PTCs with different Zak phases, 𝜃 = 0 (blue) and 𝜃 = 𝜋 (red).\n𝑍𝑎𝑘 𝑍𝑎𝑘\nb. The domain-wall structure formed by aligning two PTCs with distinct Zak phases in time. At\nt=0, a temporal interface abruptly changes the permittivity.\n2. How do the two DTL terminal ports (spatial boundaries) affect the experimental result and\nwhat design considerations are involved?\nYour concern regarding the spatial boundaries is indeed critical. Although we carefully\nconsidered this issue in our design, we regret not including these details in our original\nmanuscript.\nThe two spatial ports affect the experiment in several ways. Firstly, the impedance\nmatching between the DTL and transition sections at both ends needs to be carefully\nconsidered. To be concise, a trade-off is required between complete matching (ideal for\ninitial pulse injection) and complete mismatch (theoretically forming an ideal resonator\nwith no energy leakage, facilitating energy accumulation in the amplification process).\nSecondly, when exciting the k gaps (Fig. 3e), a broadband pulse is injected that contains\nboth bulk-band and k-gap momenta. Since bulk-band modes propagate while k-gap modes\nare stationary, the two components gradually separate. Escape of bulk-band modes through\nthe ports, though much weaker than the amplified k-gap modes, makes the practical DTL\ndeviate from an ideal infinitely long PTC.\nLastly, the two spatial boundaries naturally form a windowed system, which restricts the\nmomentum-space resolution and induces spectral broadening. This effect will be discussed\nexplicitly in our response to your Comment 8.\nTo address these factors, in our experiment we used an optimized impedance configuration:\n28.33Ω for the DTL Bloch impedance and 50Ω for the characteristic impedance of the\ntransition sections. This choice strikes a balance between efficient signal injection and clear\nobservation of amplification. Furthermore, we carefully tuned the circuit and modulation\nparameters so that the key physical phenomena—the k-gap topology in the transmission\nspectrum (Fig. 4b in the main text) and the temporal edge state (Fig. 4d in the main text)\n—are clearly observed within a short time window after modulation onset, thereby\nminimizing adverse effects from bulk-mode escape and spectral broadening.\nAccordingly, we have added Fig. R5 as Fig. 4a and 4b in the revised manuscript to clarify the\ntemporal boundary, and included a new section in the Supplementary Materials to elaborate on\nspatial boundary effects as Section IX: Spatial boundary influence and design considerations.\nReviewer #2 – Comments 4:\n“How is t0=0 defined? Is it when modulation is switched on?”\nResponse from authors:\nYes, t =0 denotes the moment when modulation is switched on. In Fig. 3c in the main text, t/T=0\n0\nmarks the same instant. To illustrate the wave dynamics in the DTL, we have displayed in Fig. 3d\nin the main text four different space-time sections with equal duration of 10T, each starting at a\ndifferent time labeled as t /T. This highlights the gradual wave evolution: from the initial traveling\n0\nwave before modulation begins (t = -10T) to the established standing wave long after modulation\n0\nstarts (t =120T) —a process involving both amplification and dominant momentum conversion.\ndescription (lines 214-221).\n“Fig.3| d, Four zoomed-in space-time sections in c, with their respective starting times labeled as\nt . The wave in the k-gaps exhibits a mode transition from propagating to standing-wave profile.”\n0\n“We record the full spatial-temporal evolution of the wave in DTL and visualize the dynamics in\nFig. 3c. In addition, we also plot in Fig. 3d four space-time sections, each with a different starting\ntime labeled as t , illustrating how the wave evolves under time modulation. Before modulation\n0\nstarts, the incident wave propagates as a traveling wave (𝑡 = -10T). Once modulation is switched\n0\non, amplification occurs and a standing-wave pattern gradually forms (𝑡 =120T), accompanied by\n0\na reduction in the effective wavelength, confirming dominant momentum conversion.”\nReviewer #2 – Comments 5:\n“In Fig. 4 is k_inc=k_0=omega_0/c with omega_0 being the central frequency of the input pulsed\nsignal? Please elaborate.”\nResponse from authors:\nYes, the incident momentum 𝑘 refers to the central momentum of the incident pulse. In Fig. 3e\n𝑖𝑛𝑐\nand 3f, 𝑘̅ denotes the normalized momentum. The corresponding unnormalized quantity is\n𝑖𝑛𝑐\n𝑘 = 𝜔 /𝑣, where 𝜔 is the central frequency of the input signal and v is the wave phase velocity\n𝑖𝑛𝑐 0 0\nin the unmodulated DTL.\nFollowing your suggestion, we have revised the manuscript (lines 221-228) to clarify the definition\nof both 𝑘̅ and 𝑘̅ :\n𝑖𝑛𝑐 𝑔𝑎𝑝\n“To further analyze the dynamics, we extract and show in Fig. 3e the temporal evolution of the\nnormalized spatial power spectrum by applying a spatial Fourier transform at each time step. In\nthis figure, 𝑘̅ and 𝑘̅ are the normalized momentum at the central frequency of the input\n𝑖𝑛𝑐 𝑔𝑎𝑝\nsignal and at the center of the k-gap center, respectively. Even though the input momentum (𝑘̅ =\n𝑖𝑛𝑐\n0.41) lies far away from the k-gap center (𝑘̅ = 0.57), the k-gap momenta dominate the power\n𝑔𝑎𝑝\nspectrum shortly after the modulation starts, overshadowing all out-of-gap components.”\nReviewer #2 – Comments 6&7:\n“How is wave vector k inside the time crystal defined/determined? In spatial photonic crystals the\nfrequency is fixed by the input wave and the Floquet-Bloch wave inside the photonic crystal is a\nsum of various spatial harmonics, i.e., wave vector components, that relate to the spatial\nmodulation (i.e. spatial period a). Here, the Floquet wave inside the time crystal should be a sum\nof different time harmonics that relate to the modulation frequency, right? In photonic crystals\nomega is conserved, in time crystals by analogy one would expect k to be conserved. However,\nFig. 3(e) could suggest otherwise? The authors should add in the supplemental material a section\ndescribing how the wave vector inside the homogenized time-crystal is determined from the input\nwave.”\n“Related to (6): is the wave vector related to the individual frequency components of the input\npulse conserved but only the components around kgap end-up getting amplified at the cost of the\nother frequency components? I think in the caption of Fig. 3e it should be clarified that \"The\nmomenta of the input signal ...\" is \"The momentum at the central frequency of the input pulsed\nsignal ...”.”\nResponse from authors:\nYour comments regarding spatial (temporal) harmonics in photonic spatial (temporal) crystals and\nthe corresponding ω (k) conservation are entirely correct. In particular, your speculation in\nComment 6 that “is the wave vector related to the individual frequency components of the input\npulse conserved but only the components around k end-up getting amplified at the cost of the\ngap\nother frequency components?” accurately captures the essential mechanism underlying Fig. 3e.\nWe apologize for the insufficient description of our normalization strategy in Fig. 3e which may\nhave led to confusion.\nIndeed, as the reviewer astutely noted, the power spectrum in Fig. 3e is normalized by the total\npower at each moment. This normalization causes the strongly amplified in-gap momenta to\ndominate visually, thereby obscuring other out-of-gap components and creating an illusion of\nmomentum conservation violation.\nAccordingly, we have revised the manuscript (lines 228-230) and the Fig. 3e caption to clarify the\nnormalization method and the physical meaning of the wave vectors:\n“Note that this apparent momenta transition arises entirely from the normalization strategy and\ndoes not indicate a violation of momentum conservation in a PTC.”\n“Fig.3| e, Temporal evolution of power spectrum along the DTL when an initial signal of wide\nspectrum is used for excitation. The momenta at the central frequency of the input signal and at\nthe experimentally measured k-gap center are denoted by blue and red dashed lines, respectively.\nThe power spectrum is normalized by the total power at each moment.”\nReviewer #2 – Comments 8:\n“Where is the small gain in k_inc observed in Fig. 3(f) coming from?”\nResponse from authors:\nWe thank the reviewer for raising this question regarding the anomalous gain of the bulk band\nmode. As we have mentioned in our reply to Comments 2&3, this issue is also related to the finite\nlength of our system and the presence of spatial boundaries.\nIn general, spatial truncation of a signal leads to spectral broadening. Fig.3f is a plot of the temporal\nevolution of two k components, one at the central frequency of the input signal, 𝑘 ,\n𝑖𝑛𝑐\nand the other\nat the momentum gap center, 𝑘 .\n𝑔𝑎𝑝\nBoth are obtained from a 2D Fourier transform of the recorded\nspatiotemporal voltage along the DTL. However, due to the finite length of the DTL which acts as\na spatial window, after the Fourier transformation, the spatial truncation leads to the spectral\nbroadening and leakage to their adjacent momentum component. This effect becomes pronounced\nover time as the in-gap components become sufficiently amplified and leak energy to 𝑘 .\n𝑖𝑛𝑐\nTo validate this interpretation, we performed FDTD simulations of PTCs with two different finite\nspatial lengths: one matching our experimental DTL length (approximately 5.4λ), and another\nmuch longer (50λ). The results are compared in Fig. R6. For fair comparison, we used a dielectric\nslab with time-varying permittivity, and the same impedance conditions of two spatial boundaries\nto emulate the experimental DTL environment. When excited with out-of-gap momenta, as shown\nin Fig. R6a, the time-domain wave evolution in this effective medium closely resembles that in\nFig. 3f in the main text, including the apparent increase at k .\ninc\nFig. R6c displays the k spectrum\nvia Fourier transform at a moment when in-gap components are sufficiently amplified. One sees\nclearly the significant amplitude increase at k due to spectral broadening. This behavior is indeed\ninc\npresent in Fig. 3e of the main text, though it is less apparent due to the normalization approach\nused in that plot. In contrast, when the slab length is increased to 50λ, the spectrum broadening\neffect for the in-gap momenta shown in Fig. R6d is substantially reduced, which would not raise\nthe out-of-gap momenta around the original k .\ninc\nThus, the corresponding time-domain amplitude\nof k can remain constant for much longer time until this travelling wave component starts to\ninc\nescape the system and the spectral broadening takes effect, as shown in Fig. R6b.\nTherefore, it can be concluded that the apparent increase in k observed in Fig. 3f is in fact an\ninc\nartifact of finite-size effects and Fourier processing, and it does not alter the main physical\nconclusions of this work.\nTherefore, we have provided a more detailed discussion in the Supplementary Materials as Section\nXI: Discussion of the increase of experimentally extracted 𝑘 in the time domain, and added the\n𝑖𝑛𝑐\nfollowing explanation in our revised manuscript (lines 234-235):\n“The small gain of the 𝑘 arises from spectral broadening due to the finite size of the DTL.”\n𝑖𝑛𝑐\nFig. R6 Window-induced spectral broadening. a,b Temporal power evolution of the incident\nmomenta and gap momenta for 𝐿 = 5.4𝜆 and 𝐿 = 50𝜆. c,d The power spectrum at 25T after\nmodulation for 𝐿 = 5.4𝜆 and 𝐿 = 50𝜆 where the initial spectrum is compared in red.\nReviewer #2 – Comments 9:\n“Related to comment (1): I would expect at a k-gap to see high back-reflection instead of\nenhancement. In the band structure of the DTL seen in Fig. 3(b) in the region that the authors have\nterms as k-gaps, one can clearly observe flat segments in the omega versus k dispersion relation,\nimplying domega/dk=0, i.e. frozen-light modes. Slow-light (low-group velocity) modes have been\nassociated with amplification, by observation of enhancement of associated non-linear effects\n[e.g. see B. Corcoran et al, Nature Photonics volume 3, pages 206–210 (2009)], and other similar\neffects. Electric field intensity amplification has also been observed directly with simulations in\nslow-light systems; e.g. see Foteinopoulou+Vigneron, Phys. Rev. B 88, 195144 (2013)]. Could the\nobserved power gain be due to the frozen-light mode in the region the authors term as k-gap (flat\nomega-k dispersion line) rather than a k-gap effect?”\nResponse from authors:\nThank you for raising this concern about the underlying physical principle of amplification in this\nwork. Here we wish to clarify that although slow light mode enables both field and nonlinearity\nenhancements, the phenomenon observed in this work, i.e. temporal exponential energy growth,\nstemming from the non-zero imaginary part of eigenfrequency in k-gap modes. Below we will\nexplicitly discuss the two previous works you mentioned.\nLet us start with the first paper [B. Corcoran et al, Nature Photonics volume 3, pages 206–210\n(2009)]. As described in the caption of Fig.1 from the first paper that: “The fundamental pulse at\nfrequency 𝜔 (energy ℏ𝜔) is spatially compressed in the slow-light photonic-crystal waveguide,\nincreasing the electric field intensity”, the enhancement of the local electric field originates from\nthe spatial pulse compression, instead of the real “amplification”. The total process, including the\nthird-harmonic generation reported in this paper (Fig. R7a), is still energy-conserved.\nSimilarly, the second paper [Foteinopoulou and Vigneron, Phys. Rev. B 88, 195144 (2013)]\ndiscusses the field enhancement in a plasmonic slow light biwaveguide by another mechanism. As\ndiscussed in Section VI that “EM energy, E, is fed into the biwaveguide from above via the ATR\nprism, at a faster rate than its relaxation sideways in the +x direction (M2 is a forward mode),\nbecause of the ultralow energy velocity of the M2 mode.”, the energy velocity (group velocity)\nmismatch results in energy accumulation in the slow-light system(as shown in Fig. R7b) and\ntherefore enhances the EM wave intensity. When the system reaches a steady state, the total energy\nis also time-invariant (Fig. R7c).\nFig R7: a. the schematic of slow light enhanced non-linearity, where the inset points out that total\nphoton number is conserved. Adopted from [B. Corcoran et al, Nature Photonics volume 3, pages\n206–210 (2009)]. b. the schematic of slow light assisted energy accumulation, highlighting a\nconstant energy fed instead of a finite energy pulse. c. Total energy evolution. When the system\nreaches a steady state, the energy is time-invariant. b and c are adopted from [Foteinopoulou and\nVigneron, Phys. Rev. B 88, 195144 (2013)].\nIn general, in a passive photonic system without loss, the total photon number is always conserved.\nHowever, in the k-gap mode of a PTC, although the real part of group velocity 𝑑𝜔/𝑑𝑡 is zero,\ninstead of being compressed or accumulated, the incident pulse undergoes stationary amplification\nwith its spatial line shape extended. Such amplification is the consequence of the non-zero\nimaginary part of eigenfrequency Ω, and pulse dynamic simulations have been demonstrated in\nmany literatures [Lustig et al., Optica 5.11 (2018): 1390-1395, Sharabi et al., Optica 9.6 (2022):\n585-592.]. This denotes the distinction between a k gap and a conventional photonic slow-light\nsystem.\nTo clarify and strengthen our novelty in amplification mechanisms, we have revised the main text\n(lines 66-76), where some comments are added to compare the amplification mechanisms:\n“…Here, the vanishing group velocity accounts for the stationarity, while the nonzero imaginary\npart of the eigenfrequency drives amplification. When a k gap is excited through a temporal\ninterface, rather than being spatially reflected, the pulse remains inside the medium with a growing\ntemporal amplitude [18]. Such amplification breaks energy conservation law, thereby greatly differ\nfrom passive slow-light system supporting energy accumulation and field enhancement [19,20].\nMoreover, since the spatially uniform modulation and conjugated eigenmodes in k gap naturally\nsatisfy phase matching condition [21], temporal modulation induces non-resonant amplification\nover the entire k gap, but is maximized at the mid-gap, providing a mechanism of coherent light\ngeneration [22] (e.g., lasers) by drawing energy from time modulation rather than gain media”\nReviewer #2 – Comments 10:\n“The authors use a model to calculate the permittivity versus the DC bias for different values of\nDC bias, V, of the varactor. Then they apply this model to find the modulated permittivity epsilon(t)\nby using each separate value V(t); i.e. the authors have implicitly assumed zero latency in the\nvaractor response. I think it would be useful if the authors added a few words describing why the\nvaractor latency can be considered negligible.”\nResponse from authors:\nWe thank the reviewer for the valuable comment regarding the finite response time of varactors.\nBelow we present the varactor latency mechanisms and demonstrate their negligible impact in our\nexperimental system, which employs the GaAs-based Skyworks SMV1249-079LF varactor.\n1. Carrier transit-time limitation\nThe depletion width adjustment under time-varying modulation is delayed by the finite carrier\ntransit time. With a typical carrier mobility of 8500cm2/(V⋅s), depletion region width of ~0.1-1μm,\nand built-in potential on the order of volts, the transit time is typically tens of ps. This is negligible\ncompared to our 7.7ns modulation period.\n2. Series resistance effect\nThe total series resistance (including bulk, contact, and packaging resistances) and junction\ncapacitance form a low-pass filter, and introduce an extra RC delay. Considering that the\nSMV1249 varactor has a series resistance of 0.6Ω (according to the Skyworks SMV1249\nDatasheet) and junction capacitance of 25pF (at operating point), their multiplication renders an\nRC delay of ~15ps, which is also insignificant at low frequencies (e.g. 130 MHz in our modulation).\n3. Trap-assisted response delay\nDeep-level traps (e.g., the EL2 in GaAs) could introduce longer time constants. However, the\nSkyworks SMV1249 employs high-purity MBE-grown GaAs and renders trap-related hysteresis\nnegligible at 130 MHz.\nThe above interpretation is supported by prior studies on PTCs with similar varactors ([27], [28]\nin the main text) and is further validated by the excellent agreement between our theoretical and\nexperimental results. As you suggested, we have added the following clarification in the revised\nmanuscript (lines 129-139):\n“Note in our experiment, we employ the Skyworks SMV1249-079LF varactor. Based on critical\nparameters from the datasheet (including series resistance), typical junction characteristics (carrier\nmobility, depletion region width, and built-in potential), and our modulation parameters, the finite\ntime delay caused by carrier transit-time and series resistance effect can be estimated to be on the\norder of picoseconds, which is negligible compared to our 7.7ns modulation period. Meanwhile,\nthe high-purity MBE-grown Skyworks SMV1249 also exhibits negligible trap-related hysteresis\nat low frequencies. These considerations regarding varactor latency are supported by prior studies\n[28, 32], and are further validated by the agreement between theoretical and experimental results\npresented in subsequent sections.”\nReviewer #2 – Comments 11:\n“Eq. S.6 has a typo. In the bottom equation second term I think it should be omega_m instead of\nomega_a.”\nResponse from authors:\nWe apologize for the typo, and we have corrected it in the revised Supplementary Materials.\nResponse to Reviewer #3\nDear Reviewer #3:\nWe would like to thank you for your insightful comments and feedback, which have helped us to\nrefine and strengthen our work from a different perspective. We are confident that the clarifications\nprovided below will be able to convince you of the novelty and significance of our work.\nReviewer #3 – General Comments:\n“In the paper, authors experimentally demonstrate both non-hermitian bandgap and temporal\ntopological state at the temporal boundary of two bandgaps with different topologies. Although\nthe former has been demonstrated the latter is a first-time demonstration to my knowledge. They\nemploy a transmission line based method to mimic photonic time crystal in microwave regime.\nThe paper is well structured and written with sufficient technical details provided in the main text\nand supplementary. Although similar experimental setups have been used in previous proof-of-\nconcept demonstrations of temporally varying media, and the theoretical foundations of topology\nin photonic time crystals are well established, this work presents a unique and well-executed\ncombination of both elements. So, I would recommend this paper for publication in Nature\nCommunications, provided the authors more clearly explain the novelty and potential impact of\ntheir demonstration, more precisely for the temporal bound mode, and address the minor\ncomments outlined below.”\nResponse from authors:\nThank you so much for your recommendation for publication and positive comments that “The\npaper is well structured and written with sufficient technical details provided in the main text and\nsupplementary”, “this work presents a unique and well-executed combination of both elements”,\nand “I would recommend this paper for publication in Nature Communications”.\nRegarding your comment that “provided the authors more clearly explain the novelty and potential\nimpact of their demonstration, more precisely for the temporal bound mode”, we would like to\nfirst offer a general response to both concerns.\nConcerns 1: “the novelty for the demonstration of temporal bound mode”\nTopological photonics incorporating time has attracted increasing attention in recent years. As the\nmost representative phenomenon, the temporal topological edge state has never been observed in\na classical wave system. The main challenge lies in the direct observation of a “k gap” which\nrequires both perfect separation of modulation and excitation signal and a relatively large\nmodulation region. Therefore, all the experimental demonstrations of topological PTC are\nconducted in synthetic space-time lattices and “pseudo momentum gap” arising from PT symmetry\nbroken phase [Feis et al., Nature Photonics (2025): 1-8, and Ren et al., Nature Communications\n16.1 (2025): 707], which is not a genuine PTC. In this work, we have observed the real space-time\nevolution of the mode dynamics in a k gap, extracted the quantized Zak phases with physical\nobservables (phase differences between transmission and reflection), and more importantly,\ndemonstrated the direct detection of a temporal topological edge that can focus light in time.\nConcerns 2: “potential impact of their demonstration of temporal bound mode”\nFrom an experimental perspective, by combining the existing experiment methods with our\ninnovation in the differential modulation, we establish a protocol of “in situ” observation of a\ntemporal modulated system. Such an approach has the potential to be generalized to space-time\nmodulated systems and even higher order space-time varying systems.\nFrom a physical perspective, the ability to focus light in time, or in other words, generating ultra-\nfast lasers pulses, holds major implications for electron spectroscopy and strong-field physics and\nwon the Nobel prize in 2023. The existing methods for generating ultra-fast lasers primarily rely\non enhancing the material non-linearity and exploiting higher harmonic generation. Temporal\ntopological edge mode, by contrast, provides another fundamentally different and robust pathway\ntoward realizing ultrafast lasers.\nBelow, we will address your questions point by point.\nReviewer #3 – Comments 1:\n“Authors mention in the introduction the difference between a PTC bandgap and parametric\namplification is the bandgap and the relaxation of phase matching. This explanation is commonly\nused in literature but requires further elaboration. The reason that there seems to be no phase\nmatching requirement in PTCs is because in theory it is assumed that the whole medium is\nmodulated simultaneously (therefore with k=0) however that is never the case if one uses a\npropagating pump for modulation. The phase still needs to be matched. And, as mentioned in the\nsupplementary it is still important to control the phase between the varactors of adjacent unit cells\nin the experimental realization to observe the amplification. That is still part of the phase matching\nrequirement. So, the amplification in OPA and PTCs are still fundamentally linked to each other.”\nResponse from authors:\nThank you for pointing out the underlying relation between OPA and PTCs. We agree with your\nperspective, but we wish to clarify that “to control the phase between the varactors of adjacent\nunit cells” is not for realizing a travelling wave modulation, but to ensure a uniform spatial\nmodulation.\nFirstly, we fully agree with your comment that “The reason that there seems to be no phase\nmatching requirement in PTCs is because in theory it is assumed that the whole medium is\nmodulated simultaneously (therefore with k=0)”. In other words, there are two conjugate\neigenmodes for a k-gap mode, exhibiting opposite momenta and compensating each other,\nresulting in a net zero momenta. Actually, the phase matching condition is considered as\nautomatically satisfied in the entire k gap.\nMoreover, as you mentioned, an ideal PTC theoretically requires both spatial uniformity and\nsimultaneous time modulation. In this work, discrete periodic varactor loading is first practically\nemployed to form a DTL, behaving as a spatially uniform effective medium in the long-wavelength\nregime. More crucially, to ensure simultaneous time modulation, all varactors (if without\ndifferential modulation, only one varactor for each unit cell) must receive modulation voltages\nwith identical amplitude and phase. This critical criterion is achieved by the commercially\navailable two-stage 8-way in-phase power splitters (highlighted in Fig. R1). Therefore, here the\n“control the phase” means we make sure that all the unit cells have zero phase difference.\nTherefore, we have revised Fig. S2a (see Fig. R8 below) to highlight that the two-stage 8-way\npower splitters maintain in-phase (0⁰) operation, and added the following explanations to the\nrevised Supplementary Materials in Section III:\n“Here, commercially available two-stage eight-way in-phase power dividers ensure “identical\namplitude and phase” modulation for each unit cell, a critical modulation requirement for\nmimicking a PTC. Importantly, such modulation phase is entirely independent of the transmitting\nsignal phase along the DTL; the phase difference can be indeed arbitrary. This represents the\nfundamental distinction from conventional optical parametric amplification, where strict phase\nmatching between a real pump wave and the signal wave is essential [28].”\nFig. R8. Revised experimental system diagram (replaces the original Fig. S2a). Highlighted two-\nstage 8-way splitters emphasize key modifications.\nFollowing your suggestions, we also add the modifications below to the revised manuscript to\nenhance the scientific accuracy and avoid ambiguity:\n(lines 72-76) “Moreover, since the spatially uniform modulation and conjugated eigenmodes in k\ngap naturally satisfies phase matching condition [21], temporal modulation induces non-resonant\namplification over the entire k gap, but is maximized at the mid-gap, providing a mechanism of\ncoherent light generation [22] (e.g., lasers) by drawing energy from time modulation rather than\ngain media.”\n(lines 122-125) “Such discrete periodic loading ensures the DTL behaves as a spatially uniform\neffective medium under DC bias in the long-wavelength regime (effective-homogeneity limit), and\naccurately mimics a PTC when AC modulation is applied [28,31,32]”\nReviewer #3 – Comments 2:\n“An addition to the first point; the modulation of varactor pairs seems to be out of phase in Fig\nS1 how does the device perform with different phase differences?”\nResponse from authors:\nThank you for pointing out the core mechanism of differential modulation, which represents a\nmajor technical advancement of our experiment protocol. Below we’ll explicitly explain the\nnecessity of differential modulation, which renders the out-of-phase modulation for each varactor\npair, but achieves the actual in-phase modulation across all varactor pairs along the line.\nAs you certainly know, all the previous efforts in realization of PTCs in the microwave regime,\nemployed experimental protocols with a single varactor in each unit cell (transmission line,\nmetasurface, waveguide, etc.). Such a configuration inevitably couples the modulation signal into\nthe core circuit. Moreover, to maintain the modulation strength of the varactor, the modulation\nvoltages are normally set much higher than the seed signal itself. Consequently, the transmitting\nsignal becomes obscured by the much stronger modulation signal, making it difficult to isolate and\nexclusively observe its space-time evolution in a PTC.\nTo overcome this limitation, we implemented the differential modulation scheme (Fig. 2c and Fig.\nS1a) using a pair of opposite polarized varactors in each unit cell. As shown in Fig. S1b in\nSupplementary Materials, this design achieves the desired dual functionality:\n1. The large but out-of-phase modulation voltages of each varactor pair cancel out along the trace\nline, creating a relatively “quiet” environment for transmitting signal direct detection.\n2. Each varactors pair maintains an identical time-varying phase (note the opposite polarity loading\nrelative to the microstrip trace line), thus meeting the in-phase modulation requirement for PTC\nimplementation by DTL. Therefore, in general, our DTL-realized equivalent PTC is still under\npurely temporal modulation.\nAs highlighted in our manuscript’s introduction and conclusion, this design represents a significant\nadvancement over prior time-varying platforms.\nAccordingly, to emphasize the necessity of the differential modulation scheme, and its role in\nachieving pure temporal modulation for constructing an equivalent PTC, we re-organized the\nSection II of Supplementary Materials and added the following texts/modifications to the revised\nmanuscript (lines 140-145)\n“In conventional DTL designs, the transmitting signal along the DTL and the external varactor\nmodulation remain inherently coupled, preventing clear observation of wave temporal dynamics\nin the realized PTC. To overcome this limitation, we adopt a differential modulation scheme in\nwhich each pair of varactors is reverse-biased with opposite polarities and symmetrically placed\non either side of the microstrip trace line in each unit cell.”\nReviewer #3 – Comments 3:\n“I think it is worth mentioning in the abstract and maybe in the title that this realization is in\nmicrowave regime.”\nResponse from authors:\nThank you for your suggestions, we add in the revised abstract that this experimental platform is\nin microwave regime (lines 33-35).\n“Here, we experimentally demonstrate the properties of a k gap in a genuine PTC, realized in a\ndynamically modulated microwave transmission-line metamaterial.”\nReviewer #3 – Comments 4:\n“I do not understand what the “genuine” refer to in the title and whether if that is necessary.”\nResponse from authors:\nThank you for raising concerns about the usage of term “genuine”. We stress that this usage is\nparticularly important in the context of recent works on temporal topological edge states.\nAfter Lustig et al proposed the concept of PTC and discusses the topological properties [Lustig et\nal., Optica 5.11 (2018): 1390-1395], several experiments have been reported in publication or\npreprints. The first realization of a PTC, as you mentioned in Comment 5, is based on a dynamic\nmetasurface. Although they observe the power amplification in the momentum bandgap in the\nfrequency domain, the real space-time dynamic and momentum space measurement of a PTC, are\nlacking.\nMore recently, experiments have progressed toward realizing a temporal topological edge state:\ntwo in photonic system [Feis et al. Nature Photonics (2025): 1-8, and Ren et al. Nature\nCommunications 16.1 (2025): 707], one preprint work in acoustic wave system [arXiv:2507.04068]\nand one preprint work in elastic wave system [arXiv:2306.09627]. In the two photonic realizations\nof temporal topological edge states, time serves as a synthetic space, i.e. the cycle number of the\npulse propagating in the coupled fiber loops, and the space is interpreted by the different time\npositions for the pulses in each cycle. Therefore, the observed topological edge mode resides in a\nspace-time dual-synthetic space. Moreover, due to the lack of phase measurement methods of each\npulse, the observation of k-space properties of PTC are unreachable. In our system, with the\nassistance of differential modulation, we directly observe the k gap properties and temporal\ntopological edge state in the real space-time dynamics.\nMoreover, in all the work about temporal topology mentioned above, the physical origin of the\n“momentum bandgap” is the spatial periodical stacking of gain and loss, that is, a non-Hermitian\nPT symmetry broken Bloch phase. In contrast, the original prediction of a momentum bandgap\nmerely emerges from the temporal periodic modulation of the real part of the refractive index from\na spatial homogeneous Hermitian media. Our model perfectly maps to the theoretical model\n(Hermitian, spatial homogenous, modulation on the real part), thereby our realization is the first\ngenuine 1D photonic time crystal with sufficient crystal periods.\nTo further strengthen the novelty and uniqueness of our work, in the introduction of revised\nmanuscript, we compare the previous work and highlight our innovation in experiment (lines 88-\n94).\n“Although temporal localization phenomena have recently been observed in several PT-symmetric\nplatforms in synthetic space-time [24,25]—where time corresponds to the cycle number of pulses\ncirculating in the coupled fiber loops and space is represented by the relative temporal positions\nof the pulses within each cycle—as well as in non-photonic systems [26,27], the direct observation\nof a temporal topological edge state in a genuine PTC within actual space-time has, until now,\nremained elusive.”\nReviewer #3 – Comments 5:\n“I would recommend explaining the difference between PTC1 and PTC2 in the main text too, it is\nonly explained in the figure caption that they have opposite initial phase.”\nResponse from authors:\nThank you so much for the suggestion. We highlight the temporal profile difference of PTC1 and\nPTC2 in Fig. 4a in the revised manuscript, and we add the following explanation in the revised\nmanuscript (lines 256-260):\n“In Fig. 4a we plot the temporal profile of both PTCs with distinct Zak phase, where both PTC\nsatisfies the temporal mirror symmetry (𝑉(𝑡) = 𝑉(−𝑡)) but with opposite symmetry center. When\ntwo PTCs with distinct Zak phases are temporally cascaded between a temporal interface (shown\nby black dashed line in Fig. 4b), a mid-gap topological edge state emerges inside the k gap.”\nReviewer #3 – Comments 6:\n“One other method of realizing PTCs has been done with metasurfaces[1] how does that method\ncompare to TL based methods.\n[1] Wang, X., Mirmoosa, M. S., Asadchy, V. S., Rockstuhl, C., Fan, S. & Tretyakov, S. A.\nMetasurface-based realization of photonic time crystals. Sci. Adv. 9, eadg7541 (2023).\nhttps://doi.org/10.1126/sciadv.adg7541”\nResponse from authors:\nThank you for mentioning this pioneer work on the realization of PTC. Actually, we have already\ncited this work as ref [28] in the original manuscript. Here we further compare this existing\nmetasurface-based method with our TL-based method in technical detail.\nGenerally, our TL method represents a rigorous 1-dimensional PTC. In the above reference,\nalthough they adopt a 1D metasurface to mimic a quasi-1D system where its permittivity is\nmodulated spatially uniformly, it is still a 2D system. As shown in Fig. 1B in their paper, when the\nmedium is modulated in time, all the harmonic orders beyond zero (𝜔 ,𝜔 … ) enter the light cone\n1 2\nand radiate to the surrounding vacuum. Their simulation result in Fig. 1G further validates this\npoint. This radiation greatly alters the wave dynamics in a purely 1D system (i.e. all harmonics\nshould propagate in the same direction). The leaked higher order harmonics do not experience the\nmodulation inside the circuit region, further deviating the whole system from a rigorous PTC.\nIn contrast, a transmission line, with its fully wave-guiding configuration strictly confining the\nfields, performs better in precisely representing a 1D PTC. In Fig. 3b in our manuscript, one can\nclearly find out that, apart from the zero order mode, higher harmonic modes exist concurrently\ninside the microstrip line and are under the further modulation of the transmission line.\nTherefore, we have added the following texts/modifications to provide a brief comparison between\nthis work and the reference above, highlighting our superior capability for comprehensive PTC\nfeature investigation:\n(lines 96-102) “Here, we present an experimental study of the k-gap properties in a PTC realized\nin a dynamically modulated transmission-line metamaterial and metasurfaces [28–31]. We employ\na differential modulation strategy, which enables direct observation of the spatiotemporal wave\ndynamics within the PTC. Moreover, the compact constraints of wave in a microstrip guarantee\nfabrication of a sample with a sufficiently large number of unit cells and consequently permits\ncomprehensive experimental verification of k gap characteristics.”\nResponse to Reviewer #1\nReviewer #1 – General Comments:\n“The authors have addressed all my concerns, and the presentation of this work has been\nsignificantly improved. I would like to recommend the publication as is.”\nResponse from authors:\nWe thank the reviewer for their positive feedback and for recommending our manuscript for\npublication.\nResponse to Reviewer #2\nReviewer #2 – General Comments:\n“I have read the revised manuscript and responses of the authors.\nI must commend the authors for an outstanding, in-depth and thorough effort to answer in detail\nall questions by all reviewers, thus enhancing the paper's accessibility by a wide audience!\nI especially appreciated that they have now elaborated on the nature of the k-gap in time-\nmodulated systems bringing out its unique nature, explaining that this is not a direct analogue of\nthe omega-gap in spatial modulated systems as it has been too often simplified in many\ndescriptions in the literature.\nIt is now clear how the amplification process occurs within the k-gap, where modes have a\ndRe(Omega)/dk near-zero (stationary near frozen mode) and a complex Omega, with the\nimaginary part of Omega providing the amplification.\nThe authors have also articulated well what is novel in their work, in comparison with previous\nexperimental reports that do not involve the space-time dynamics in a genuine PTC system\ncomprising a transmission line with time modulated components.\nAs mentioned in my previous report the present work has rich Physics and I believe that will inspire\nfurther exciting research in the community.\nI would therefore like to enthusiastically recommend its publication. I only have couple of minor\ncomments that I believe would be useful to the wider readership if ammended.”\nResponse from authors:\nWe are deeply grateful for the reviewer’s thorough and encouraging assessment of our revisions.\nWe are pleased that our clarifications on the k-gap nature, amplification mechanism, and\nexperimental novelty were well received, and thank the reviewer for enthusiastically supporting\npublication. Please see below our reply to your two minor comments.\nReviewer #2 – Comments 1:\n“I noticed the term “time-reflected” in abstract and other places which is a bit confusing. The\nauthors themselves state there can be no back-scattering in time due to causality. Do the authors\nmean spatial reflection caused by time-modulation? The addition of a footnote or in references of\nthe definition would make it more clear for the reader.”\nResponse from authors:\nThank you for this helpful suggestion regarding the potentially confusing term \"time-reflected\nwave.\" Your understanding is completely correct: although the reflection is induced by the\ntemporal interface introduced through time-modulation, causality requires that the reflected wave\nstill propagates in space.\nFollowing your advice, we have added the following clarifying note in the main text, along with\ntwo relevant references:\n(lines 230-232) “Note that the term “time-reflected wave” denotes the spatially backward\npropagation wave caused by a temporal interface via time-modulation [18,29]”\nReviewer #2 – Comments 2:\n“There is a bit of an abrupt transition in the manuscript between the experimental microwave\nplatform and the conceptualized system used for analysis and comparison of the experimental\nresults. After the authors present Fig. 2 it would be good to add a small schematic in another panel.\nI think it would be useful along with the magnified inset in Fig. 2(e) for the segment within the red-\ndashed rectangle to add a conceptualized schematic. In the conceptualized schematic the axis x\nshould be designated, along with the width of the transmission line along y but comprising a few\nunit cells not just one. Additionally, with a different shading (red and blue) the authors should\nidentify the areas of epsilon{r,A}(t) and epsilon{r,B}(t) which are modulated in time such that they\nhave a pi phase difference in such a conceptualized schematic. I think that would be especially\nhelpful for the wider readership.”\nResponse from authors:\nThank you for raising this problem. We believe your confusion may stem from the lack of clarity\nin the original Fig. 2 and its description. Based on your comment and suggestion, we have\nimplemented the following key revisions:\n1. We have adjusted panel e, by adding an illustration, as suggested, below the original unit-cell\nclose-up view. It can help readers clarify the DTL configuration and its logical connection to the\nschematic in panel a, and the equivalent circuit presentation in panel c. Note that in panel e, the\ntwo varactor diodes are positioned on opposite sides of the microstrip trace, consistent with the\nschematic plot in panel a (where the close-up shows only one varactor due to space constraints).\n2. To avoid potential confusion with the modulation profile used to form two consecutive PTCs in\nFig. 4b, we have removed the original v (t) and v (t) waveforms in panel a. Instead, the newly\nA B\nadded illustration in panel e now indicates the modulation voltages for each varactor pair as v(t)\nand - v(t), respectively, making clear that the signals applied to the two varactors in each pair are\nout-of-phase. Meanwhile, labels in panels c and d have been updated accordingly, and the colors\nof modulation profiles in Fig. 4b have been changed to avoid any confusion.\nFig. R1: Revised Fig. 2 in the main text.\nThe above modifications make the differential modulation mechanism more apparent: since the\ntwo varactors in each unit cell are mounted with opposite polarity, the externally applied out-of-\nphase modulation voltages cause both to behave identically. As a result, all varactor pairs\ncollectively produce, via the capacitance–voltage relation shown in panel b and field-circuit\nequivalence, a uniform effective time-varying ε(t) in the DTL-emulated PTC.\nr\nResponse to Reviewer #3\nReviewer #3 – General Comments:\n“The authors have done a good job responding to my comments about the paper. They have clearly\nanswered the questions about the experimental setup and highlighted the novelty in their method.\nTherefore, I recommend it for publishing in Nature Communications.”\nResponse from authors:\nWe thank the reviewer for acknowledging our responses and recommending publication."}]

Decision letter