Parametric Amplification of a Terahertz Quantum Plasma Wave

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1 Paamec Amplfcaon of a Teahez Quanum Plasma Wave S. Rajasekaan E. Casanduc Y. Laplace D. Ncole G. D. Gu 4 S. R. Clak 3 D. Jaksch 5 A. Cavalle Max Planck Insue fo he Sucue and Dynamcs of Mae Luupe Chaussee Hambug Gemany Depamen of Physcs Oxfod Unvesy Claendon Laboaoy Paks Road OX 3PU Oxfod Uned Kngdom 3 Depamen of Physcs Unvesy of Bah Claveon Down BA 7AY Bah Uned Kngdom 4 Condensed Mae Physcs and Maeals Scence Depamen Bookhaven Naonal Laboaoy Upon New Yok USA 5 Cene fo Quanum Technologes Naonal Unvesy of Sngapoe 3 Scence Dve Sngapoe 7543 Sngapoe Many applcaons n phooncs eque all-opcal manpulaon of plasma waves whch can concenae elecomagnec enegy on sub-wavelengh lengh scales. Ths s dffcul n meallc plasmas because of he small opcal nonlneaes. Some layeed supeconducos suppo weakly damped plasma waves 3 nvolvng oscllaoy unnelng of he supeflud beween capacvely coupled planes. Such Josephson plasma waves (JPWs ae also hghly nonlnea 4 and exhb skng phenomena lke coopeave emsson of coheen eahez adaon 56 supeconduco-meal oscllaons 7 and solon fomaon 8. We show hee ha eahez JPWs n cupae supeconducos can be paamecally amplfed hough he cubc unnelng nonlneay. Paamec amplfcaon s sensve o he elave phase beween pump and seed waves and may be opmzed o acheve squeezng of he ode paamee phase flucuaons 9 o sngle eahez-phoon devces.

2 Cupaes ae songly ansoopc supeconducos n whch anspo s made hee-dmensonal by Josephson unnelng beween he Cu-O planes. Tunnelng educes he supeflud densy n he decon pependcula o he planes and hence he fequency of he plasmon o below he aveage pa beakng gap. Weakly damped oscllaons of he supeflud susan ansvese Josephson Plasma Waves (JPWs ha popagae along he planes. Consde a complex supeconducng ode paamee n he h Cu-O plane ( x y ( x y e θ ( x y ψ = ψ whch depends on wo n-plane spaal coodnaes x and y and on me. Fo a THz-fequency opcal feld polazed pependcula o he planes excaons above he supeconducng gap ae neglgble and he modulus of he ode paamee ψ (numbe of Coope pas s nealy consan n space and me. Hence he elecodynamcs s domnaed by he ode-paamee phase θ ( x y. Ignong a fs he spaal dependence of he phase he local unnelng sengh can fom he Josephson equaons be expessed n ems of an equvalen nducance L whch depends on he local nelaye phase dffeence θ ( + = θ ( θ+ ( as L(θ + ( ~ L / cos(θ + ( ( and + ae he ndces fo wo neghbong layes. Hee L h = s he ei c nducance a equlbum h he educed Planck s consan e he Coope pa chage and I c he ccal cuen. Denong he capacance of he Cu-O planes wh a consan C we expess he Josephson Plasma Resonance (JPR fequency as

3 = = JP cos[ θ L( θ ( C JP + + ( ] whee JP = s he equlbum value. L C Coespondngly he oscllao sengh f fo he plasma oscllaons s also a funcon of he nelaye phase and scales as f = f cos[θ + (]. The dependence of he oscllao sengh f on he cosne of he supeconducng phase coesponds o a hd ode opcal nonlneay. Accodng o he second Josephson equaon he nelaye phase dffeence θ + (advances n me wh he me negal of he nelaye volage dop as [ θ ( ] = + ev h. Fo an opcal feld made esonan wh he Josephson plasma fequency E( = E sn( JP he nelaye phase oscllaes as θ + ( =θ cos( JP ed whee E s he feld amplude and θ = E (d ~ nm s he nelaye h JP dsance. Ths mples ha he oscllao sengh f + ( = f θ + θ cos( JP cos( θ cos( JP f s modulaed a a 4 fequency JP wheneve he feld E s lage enough o make he phase excuson θ szeable. Fgue povdes a pcoal epesenaon of hs physcs. We plo a veco ha epesens boh he phase dffeence θ + ( (veco angle and he oscllao sengh f + ( (veco lengh. Ths pcue shows how fo small dvng felds only θ + ( oscllaes a he dvng fequency JP wheeas fo lage felds hese oscllaons ae accompaned by a JP modulaon of he oscllao sengh f + (.

4 Noe also ha he phenomena dscussed above can be cased n ems of a Maheu equaon (see Supplemenay Secon S. Thus a JP modulaon of he oscllao sengh can seve as a pump fo he paamec amplfcaon of a second weak plasma wave a fequency JP. In hs pape we demonsae expemenally hs effec n La.95Ba.95CuO4 (LBCO9.5 a cupae supeconduco wh he equlbum JPR a. JP 5THz. Teahez pulses geneaed wh a phooconducve anenna wee used as a weak pobe of JPWs. A ypcal THz-feld ace 3 efleced by he sample s shown n Fg. A. Two dffeen measuemens ae dsplayed: one aken below (ed lne and he ohe one above (black lne he supeconducng anson empeaue Tc = 3 K. In he supeconducng sae long-lved oscllaons wh ~ ps peod wee obseved on he alng edge of he pulse ndcave of he JPR a JP. 5THz. Fgue B (sold ed lne dsplays he coespondng eflecvy edge n fequency doman. The sold lnes n Fg. C -D ae he complex delecc pemvy ε( ε ( and he loss funcon L ( = Im =. L( peaks a JP ε ( ( ε( + ε ( whee he eal pa of he delecc pemvy ε ( cosses zeo. These opcal popees could be well epoduced by solvng he wave equaon n he supeconduco n one dmenson 8 (see Supplemenay Secon S whch yelds he space and me dependen ode paamee phase θ + (x (Fg. E and he coespondng changes (neglgble n lnea esponse egme of he oscllao sengh f = f cos[θ + (x] (Fg. F. The eflecvy complex pemvy and

5 loss funcon (dashed lnes n Fg. B C and D especvely calculaed fom hese smulaons by solvng he elecomagnec feld a he sample suface 8 ae n good ageemen wh he expemenal daa. Amplfcaon of a weak JPW lke he one above (pobe feld was acheved by mxng wh a second nense pump feld whch esonanly dove he Josephson phase o lage ampludes. Quas-sngle cycle THz pulses geneaed n LNbO3 wh he led pulse fon mehod 4 (yeldng feld senghs up o ~ kv/cm wee used o exce hese waves n nonlnea egme. The specal conen of hese pulses exended beween. and.7 THz ceneed a he JPR fequency (see Supplemenay Secon S3. Noe ha he pump feld sengh used n hs expemen exceeds he expeced heshold o access he nonlnea egme defned eed by θ = ~ h and coespondng n hs maeal o feld ampludes JP ~ kv/cm. E h JP = ed In Fg. 3 we epo he me-delay dependen specally negaed pump-pobe esponse of LBCO9.5. Changes n he efleced pobe feld wee measued a one specfc pon along he nenal delay ace of he pobe as a funcon of pumppobe me delay. Fo a sysem n whch he opcal popees ae domnaed by a sngle plasma esonance he specally negaed esponse s popoonal o he plasma oscllao sengh f. As shown n Fg. 3A-3B hs negaed esponse exhbs a educon of he eflecvy and oscllaons a a fequency ~ JP. Noe ha he oscllaon fequency dd no depend on he pump elecc feld sengh E whle he fequency

6 educed when he base empeaue of he expemen was nceased conssen wh he educon of he equlbum JP (see Supplemenay Secons S4 and S5. The effec compleely dsappeaed a T > Tc. Hence he heoecally pedced JP modulaon of he oal oscllao sengh f (see above s well epoduced by he daa n Fg. 3. Ths esponse could also be smulaed usng he space- and me-dependen sne-godon equaon (see Fg. 3C- 3D. Good ageemen beween expemen and heoy was obaned (see dashed lnes n Fg. 3A-3B. Seleced me-doman pobe aces measued befoe and afe excaon ae dsplayed n Fg. 4. Cucally a specfc me delays he pobe feld s amplfed (Fg. 4A wheeas a ohe delays s suppessed (Fg. 4B wh espec o ha measued a equlbum. In Fg. 5 we epo he me-delay and fequency dependen loss funcon L( ε ( = Im = a quany ha peaks a he zeo ε( ( ε( + ε( cossng of ε ( and s always posve fo a dsspave medum (.e. a medum wh ε ( >. The expemenal daa of Fg. 5A show ha afe excaon L acques negave values aound JP(ed egons. Ths s ndcave of a negave ε ( and hence amplfcaon. The effec s song nea zeo pump-pobe me delay hen dsappeas afe ~ ps and s obseved agan peodcally wh a epeon fequency of ~ JP (. The same effec appeas also n he smulaons (Fg. 5B yeldng peodc amplfcaon a a epeon fequency of JP.

7 The daa and heoecal analyss epoed hee demonsae ha eahez JPWs can be paamecally amplfed exhbng he expeced sensvy o he elave phase of song and weak felds mxed n hs pocess and he oscllaoy dependence a wce he fequency of he dve. Ths effec s of nees fo applcaons n phooncs o as a phase-coheen nonlnea pobe of he supeflud self 5. Moeove he ably o amplfy a plasma wave could lead o sngle-thz phoon manpulaon devces ha may opeae above K empeaues. These would explo conceps ha o dae have been developed only a mcowave fequences and n he mll-kelvn egme 6 9. Fnally he paamec phenomena dscussed hee can also poenally be used o squeeze 9 he supeflud phase and may lead o conol of flucuang supeconducvy pehaps even ove a ange of empeaues above Tc 34.

8 Mehods Lase pulses wh fs duaon and ~5 mj enegy fom a commecal T:Sa amplfe wee spl no 3 pas (9% 7% %. The mos nense beam was used o geneae song-feld THz pulses wh eneges up o ~3 µj va opcal ecfcaon n LNbO3 wh he led pulse fon echnque. These wee collmaed and hen focused a nomal ncdence ono he sample (wh polazaon pependcula o he Cu-O planes.e. along he c axs usng a Teflon lens and a paabolc mo wh focal lenghs of 5 mm and 75 mm especvely. The pump feld a he sample poson was calbaed wh eleco-opc samplng n a.-mm-hck GaP cysal yeldng a maxmum value of ~ kv/cm (see also Supplemenay Secon S3. The 7% beam was used o geneae he THz pobe pulses wh a phooconducve anenna. These had a dynamc bandwdh of.-3 THz coespondng o a me esoluon of ~5 fs. The c-axs opcal popees of he supeconduco (boh a equlbum and houghou he pump-nduced dynamcs wee pobed n eflecon geomey wh a pobe ncdence angle of 45. The efleced pobe pulses wee eleco-opcally sampled n a -mm-hck ZnTe cysal usng he emanng % of he 8 nm beam. Ths measuemen pocedue euned he quany E( τ wh beng he pump-pobe delay and τ beng he nenal eleco-opc samplng me coodnae. The complex opcal popees of he supeconduco a equlbum wee deemned by measung he complex-valued E( (pump off boh a T < Tc and T > Tc and by efeencng o he nomal-sae eflecvy measued n he same bach of samples 5. In he specally negaed pump-pobe aces of Fg. 3 E(τ was measued by scannng he pump-pobe delay a a fxed nenal delay τ. Ths was chosen o be on he alng edge of he pulse whee he JPR oscllaons ae pesen. Noe ha he obseved dynamcs and n pacula he JP oscllaons dd no depend sgnfcanly on he specfc τ value a whch he scan was pefomed. The fequency and me-delay dependen loss funcon of Fg. 5 (as well as all complex opcal popees of he peubed maeal was deemned by applyng Fesnel equaons o he pump-nduced changes n he efleced elecc feld. These wee nomalzed by ndependenly ecodng E(τ n pesence and absence of he THz pump feld. Noe ha hee was no need o ake no accoun any pumppobe peneaon deph msmach n he calculaon. In he smulaons he Josephson phase evoluon θ + ( x was deemned hough he one-dmensonal sne-godon equaon 6 : θ + ( x θ + ( x ε x γ c θ + ( x JPε = c snθ + ( x

9 beng γ a dampng consan che speed of lgh ε he delecc pemvy and JP he equlbum JPR fequency. Ths equaon was solved numecally wh he THz pump and pobe felds ovelappng a he vacuum-supeconduco neface. Fo moe deals on hs opc we efe he eade o Supplemenay Secon S.

10 Fgue. Schemac me-dependen epesenaon of JPWs n lnea and nonlnea egme n pesence of a dvng feld E( = E sn( JP. Red aows ndcae he Josephson phase whle he coespondng oscllao sengh f s epesened by he black ccle aea. A JPW n lnea egme consss of small amplude modulaons of θ + a consan oscllao sengh wheeas f s modulaed a JP. a JP f ~ JP. In nonlnea egme he Josephson phase oscllaes

11 Fgue. Lnea JPWs n LBCO9.5. (A E(τ measued n absence of pump feld boh above and below Tc = 3 K. (B Fequency-dependen c-axs eflecvy a T = 5 K (sold lne exaced fom he E(τ ace of panel (A. (C Coespondng eal and magnay pa of he complex pemvy and (D enegy loss funcon (sold lnes. Dashed lnes n (B-D wee calculaed by numecally solvng he sne-godon equaon n lnea egme. (E Phase θ + ( x and (F coespondng oscllao sengh f (no change nduced by a weak pobe THz feld also deemned fom he sne-godon equaon n lnea egme. Hozonal doed lnes ndcae he spaal coodnae x a whch he lne cus ae dsplayed (lowe panels.

12 Fgue 3. Nonlnea JPWs n LBCO 9.5. (A Specally negaed pump-pobe esponse. Expemenal daa ae dsplayed along wh calculaons based on he sne-godon equaon n nonlnea egme. Dashed lnes ndcae he backgound whch was subaced o oban he oscllaoy componen shown n (A. (B Foue ansfom of he exaced oscllaons showng a peak a ~ THz. (C Phase θ x and (D coespondng ( + oscllao sengh f (nomalzed by he equlbum value nduced by a song THz pump feld as deemned by numecally solvng he sne-godon equaon n nonlnea egme. Hozonal doed lnes ndcae he spaal coodnae x a whch he lne cus ae dsplayed (lowe panels.

13 Fgue 4. E(τ aces measued by scannng he eleco-opc samplng nenal delay τ a seleced pump-pobe delays = ps and = ps. Daa ae shown along wh he same quany measued a equlbum (pump off. Shaded egons n (A and (B ndcae amplfcaon and suppesson of he JPW amplude especvely.

14 Fgue 5: Tme-delay and fequency dependen loss funcon L( deemned (A expemenally and (B by numecally solvng he sne-godon equaon n nonlnea egme. Noe ha he equlbum L( has been mulpled by a faco of 5.

15 Refeences. Kauanen M. & Zayas A. V. Nonlnea plasmoncs. Na. Phooncs (.. Basov D. N. & Tmusk T. Elecodynamcs of hgh-tc supeconducos. Rev. Mod. Phys (5. 3. Klene R. Senmeye F. Kunkel G. & Mülle P. Innsc Josephson effecs n BSCaCuO8 sngle cysals. Phys. Rev. Le ( Savel ev S. Rakhmanov A. L. Yampol sk V. A. & No F. Analogues of nonlnea opcs usng eahez Josephson plasma waves n layeed supeconducos. Na. Phys (6. 5. Ozyuze L. e al. Emsson of Coheen THz Radaon fom Supeconducos. Scence (7. 6. Hu X. & Ln S.-Z. Phase dynamcs n a sack of nducvely coupled nnsc Josephson juncons and eahez elecomagnec adaon. Supecond. Sc. Technol (. 7. Dens A. e al. B-deconal ulafas elecc-feld gang of nelaye chage anspo n a cupae supeconduco. Na. Phooncs (. 8. Dens A. e al. Opcal excaon of Josephson plasma solons n a cupae supeconduco. Na. Mae (3. 9. Lü X.-Y. e al. Squeezed Opomechancs wh Phase-Mached Amplfcaon and Dsspaon. Phys. Rev. Le (5.. Josephson B. D. Coupled Supeconducos. Rev. Mod. Phys (964.. Dessel M. & Güne G. Elecodynamcs of Solds: Opcal Popees of Elecons n Mae (Cambdge Unvesy Pess.. Deyhaup A. Wnnel S. Dekosy T. & Helm M. Hgh-nensy eahez adaon fom a mcosucued lage-aea phooconduco. Appl. Phys. Le (5. 3. Thosmølle V. K. e al. C-axs Josephson plasma esonance obseved n TlBaCaCuO8 supeconducng hn flms by use of eahez me-doman specoscopy. Op. Le (. 4. Heblng J. Yeh K.-L. Hoffmann M. C. Baal B. & Nelson K. A. Geneaon of hgh-powe eahez pulses by led-pulse-fon excaon and he applcaon possbles. J. Op. Soc. Am. B 5 B6 (8. 5. Hohenleune M. e al. Real-me obsevaon of nefeng cysal elecons n hgh-hamonc geneaon. Naue (5. 6. Vjay R. e al. Sablzng Rab oscllaons n a supeconducng qub usng quanum feedback. Naue (. 7. Mackln C. e al. A nea quanum-lmed Josephson avelng-wave paamec amplfe. Scence (5. 8. Echle C. Salahe Y. Mlynek J. Schmd S. & Wallaff A. Quanum-Lmed Amplfcaon and Enanglemen n Coupled Nonlnea Resonaos. Phys. Rev. Le. 3 5 (4.

16 9. Casellanos-Belan M. A. & Lehne K. W. Wdely unable paamec amplfe based on a supeconducng quanum nefeence devce aay esonao. Appl. Phys. Le (7.. Almog R. Zasev S. Shempluck O. & Buks E. Nose Squeezng n a Nanomechancal Duffng Resonao. Phys. Rev. Le (7.. Zagoskn A. M. Il chev E. McCucheon M. W. Young J. F. & No F. Conolled Geneaon of Squeezed Saes of Mcowave Radaon n a Supeconducng Resonan Ccu. Phys. Rev. Le. 536 (8.. Denny S. J. Clak S. R. Laplace Y. Cavalle A. & Jaksch D. Poposed Paamec Coolng of Blaye Cupae Supeconducos by Teahez Excaon. Phys. Rev. Le (5. 3. Blbo L. S. e al. Tempoal coelaons of supeconducvy above he anson empeaue n La-xSxCuO4 pobed by eahez specoscopy. Na. Phys (. 4. Coson J. Mallozz R. Oensen J. Ecksen J. N. & Bozovc I. Vanshng of phase coheence n undedoped BSCaCuO8+δ. Naue ( Homes C. C. e al. Deemnaon of he opcal popees of La-xBaxCuO4 fo seveal dopngs ncludng he anomalous x=/8 phase. Phys. Rev. B (.

17 Paamec Amplfcaon of a Teahez Quanum Plasma Wave S. Rajasekaan E. Casanduc Y. Laplace D. Ncole G. D. Gu 4 S. R. Clak 3 D. Jaksch 5 A. Cavalle Max Planck Insue fo he Sucue and Dynamcs of Mae Luupe Chaussee Hambug Gemany Depamen of Physcs Oxfod Unvesy Claendon Laboaoy Paks Road OX 3PU Oxfod Uned Kngdom 3 Depamen of Physcs Unvesy of Bah Claveon Down BA 7AY Bah Uned Kngdom 4 Condensed Mae Physcs and Maeals Scence Depamen Bookhaven Naonal Laboaoy Upon New Yok USA 5 Cene fo Quanum Technologes Naonal Unvesy of Sngapoe 3 Scence Dve Sngapoe 7543 Sngapoe Supplemenay Maeal

18 S. Josephson equaon as Maheu equaon A Josephson juncon can be appoxmaed wh an LC ccu model. By equang he capacve cuen = V C I c o he nducve unnelng cuen ( sn ( I I L + = θ and usng hen he second Josephson equaon = + h ev ] [ ( θ we oban he empoal dependence of he Josephson phase ( ( + θ as ( sn ( x c x c p + + = θ ε θ ε ( whee ε s he delecc pemvy of he Josephson juncon c he speed of lgh e he eleconc chage I he ccal cuen C he capacance of he juncon and C e I p h = =. The equaon of moon of he Josephson phase wh dampng (γ heefoe eads ( sn ( ( x c x c x p = θ ε θ ε θ γ ( In a peubed sae n whch he oscllao sengh s modfed as + 4 cos( ( ~ ( f p θ θ (3 he me dependence of he Josephson phase s descbed by ( 4 cos( ( ( = x c x x c pobe pobe pobe θ θ θ ε θ γ θ ε (4 We noe ha Eq. (4 s a damped Maheu equaon of he fom

19 θ pobe( x θ pobe( x + β + ( a α cos( ( θ ( x = pobe (5 whee θ = 4 θ a cos( c α = and β =. 4 ε γ S. Smulaon of he nonlnea opcal popees fom he sne-godon equaon A Josephson juncon wh sem-nfne layes sacked along he decon (wh anslaonal nvaance along he y decon can be modeled wh he onedmensonal sne-godon equaon. Beng x he popagaon decon he Josephson phase evoluon s descbed by: θ + ( x θ + ( x ε θ ( x + pε = snθ x γ c c + ( x (6 The dampng facoγ s a fng paamee used o epoduce he opcal popees obseved expemenally. In hs secon we dop he subscps fo smplcy.e. we edefne θ + x = θ( x. The pump and pobe THz felds mpnge on he ( supeconduco a he bounday =. The Josephson phase evoluon s heefoe affeced by he followng bounday condons a he vacuum-sample neface 3. E +E =E x =H!! (7! H +H =H x =H λ %. (8 The subscps and c denoe he felds ncden efleced and popagang nsde he cupae especvely. Hee H =Φ /πdλ % whee Φ s he flux quanum +Φ = - and D s he dsance beween adjacen supeconducng layes. The./ equlbum Josephson Plasma Resonance (JPR s an npu paamee n he smulaons whch s chosen o be ha of La.95Ba.95CuO4.e. %3 =.5 THz.

20 Fo felds n vacuum ( < he Maxwell s equaons mply E E = 7 8 H +H =H +H. (9 By combnng Eq. (4 wh Eq. ( and (3 we oban he bounday condon. E 9 =! :! ε! /<. ( Afe solvng he Josephson phase hough Eq. (6 and Eq. ( he efleced feld s calculaed fom Eq. (7. The equlbum eflecvy of he cupae s obaned by compung he ao beween he Foue ansfoms of he efleced feld and a weak npu feld >?@AB@C =E >?@AB@C /E. ( The complex opcal popees ae hen calculaed fom >?@AB@C. In pacula he equlbum delecc pemvy and loss funcon ae compued as:. ε=d+ EFGHIHJKHGL M EFGHIHJKHGL. Loss=ImagD+ EFGHIHJKHGL M EFGHIHJKHGL Fo he pump-pobe confguaon he npu feld s he sum of he pump and pobe felds (wh a defned delay beween hem: E =E U@CU +E UVB>. ( Coespondngly he Josephson phase can be wen as

21 θ=θ +θ UVB>. (3 And he sne-godon equaon (6 decomposes no wo coupled equaons θ pump ( x θ pump ( x ε θ pump ( x pε = snθ x γ c c θ pobe( x θ pobe( x ε θ pobe( x pε = snθ x γ c c pump pobe ( x cosθ ( x cosθ pobe pump ( x ( x (4 (5 Fo a weak pobe (θ cosθ UVB> and he effec of θ UVB> on θ U@CU can be negleced n Eq. (9. The phases θ U@CU and θ UVB> ae calculaed n wo seps: ( Eqs. (4 and ( ae solved wh he dvng feld E =E U@CU o ge θ U@CU x and hen ( Eq. (5 and ( ae solved by subsung θ U@CU x wh he npu feld E =E UVB> o oban θ UVB> x and he efleced pobe feld E U>!@B. The peubed eflecvy s gven by U>!@B =E U>!@B /E. (6 The opcal esponse funcons of he peubed maeal ae exaced fom he complex opcal eflecvy U>!@B. Fo nsance he delecc pemvy and loss funcon ae calculaed as: ε=d+ ^EK_GKJ. M ^EK_GKJ Loss=ImagD+ ^EK_GKJ.! M. ^EK_GKJ! S3. Pump specum The elecc feld pofle of he THz pump pulse (geneaed wh he led pulse fon echnque n LNbO3 measued a he sample poson s dsplayed n Fg. SA

22 alongsde he coespondng fequency specum (Fg. SB. Ths s peaked a ~.5 THz beng heefoe esonan wh he JPR of LBCO9.5 (see eflecvy edge n he blue cuve of Fg. SB. The npu pump feld used n he smulaons s also dsplayed boh n me (Fg. SA and fequency doman (Fg. SB. Fgue S. (A Eleco-opc samplng ace of he THz pump pulse measued a he sample poson and (B coespondng fequency specum. The c-axs equlbum eflecvy of LBCO 9.5 a T = 5 K s also dsplayed. Dashed lnes n boh panels efe o he npu pump feld used n smulaons. The ngng obseved on he alng edge of he pulse (black lne n A s due o naow wae absopon lnes a ~.5 THz and ~. THz (see also coespondng specum n B. These can be gnoed because unlke he pump feld scan shown n hs fgue all ohe measuemens epoed n hs pape have been pefomed unde hgh vacuum condon (P = -6 mba. S4. Pump feld dependence The specally negaed D pump-pobe esponse 45 s dsplayed n Fg. S fo dffeen pump feld senghs. A mnmum feld of ~3 kv/cm was equed o nduce a esponse of suffcen amplude o be deeced n ou expemen. The oscllaoy behavo a wce he equlbum JPR fequency was found o be only weakly dependen on he pump feld sengh. Noe ha pump-feldndependen JP oscllaons ae only obseved a >.e. afe he ealy-me dynamcs ( < domnaed by peubed fee nducon decay 6 (shaded egon n Fg. S.

23 Fgue S. (A Specally-negaed D pump-pobe esponse measued fo dffeen pump feld senghs a a sample empeaue T = 5 K. The dashed lne s an example of backgound whch was subaced o exac he oscllaoy componens shown n he nse. The ealy me delay egon neesed by peubed fee nducon decay s shaded n gey. (B Nomalzed Foue ansfoms of he oscllaoy sgnals. Fgue S3. Tme-delay and fequency dependen loss funcon deemned (A expemenally and (B by numecally solvng he sne-godon equaon n nonlnea egme. The appled THz pump feld s 4 kv/cm. The me-delay and fequency dependen loss funcon measued wh a pump feld of 4 kv/cm s dsplayed n Fg. S3 along wh he coespondng heoecal calculaons. These can be compaed wh he daa of Fg. 5 n he man ex whch wee aken wh a hghe pump feld (~8 kv/cm. Remakably whle he JP oscllaoy behavo s obseved n boh daa ses peodc amplfcaon s only pesen wh songe pump feld (conssenly n boh expemen and calculaons.

24 Ths ndcaes ha phase-sensve amplfcaon of Josephson Plasma Wave can be acheved only fo THz pump feld ampludes above a heshold of ~7 kv/cm. S5. Tempeaue dependence The equlbum JPR shfs o lowe fequences wh nceasng empeaue owads Tc. Ths s clealy shown n Fg. S4A whee he measued equlbum eflecvy of LBCO9.5 s epoed a wo dffeen empeaues. The JPR exhbs a ed shf fom ~.5 THz o ~.35 THz upon nceasng he sample empeaue fom 5 K o 3 K. The empeaue dependence of he specally negaed pump-pobe esponse has also been deemned expemenally (see Fg. S4B. As expeced he measued oscllaons slow down wh nceasng T. Indeed he fequency educes fom ~ THz a 5 K o ~.75 THz a 3 K scalng as JP. Fgue S4. (A E(τ measued n absence of pump feld a dffeen empeaues above and below T c. (A Fequency-dependen eflecvy a T = 5 K and T = 3 K exaced fom he E(τ ace of panel (A. (B Oscllaoy componen of he specally-negaed D pump-pobe esponse measued a T = 5 K and T = 3 K. (B Coespondng Foue ansfoms of he oscllaoy sgnals.

25 Refeences. Hu X. & Ln S.-Z. Phase dynamcs n a sack of nducvely coupled nnsc Josephson juncons and eahez elecomagnec adaon. Supecond. Sc. Technol (.. Dens A. e al. Opcal excaon of Josephson plasma solons n a cupae supeconduco. Na. Mae (3. 3. Savel ev S. Yampol sk V. A. Rakhmanov A. L. & No F. Teahez Josephson plasma waves n layeed supeconducos: specum geneaon nonlnea and quanum phenomena. Rep. Pog. Phys (. 4. Knd J. T. & Schmuenmae C. A. Theoy fo deemnaon of he low-fequency medependen esponse funcon n lquds usng me-esolved eahez pulse specoscopy. J. Chem. Phys ( Bead M. C. Tune G. M. & Schmuenmae C. A. Tansen phooconducvy n GaAs as measued by me-esolved eahez specoscopy. Phys. Rev. B (. 6. Hamm P. Coheen effecs n femosecond nfaed specoscopy. Chem. Phys (995.

Lecture 5. Plane Wave Reflection and Transmission

Lecture 5. Plane Wave Reflection and Transmission Lecue 5 Plane Wave Reflecon and Tansmsson Incden wave: 1z E ( z) xˆ E (0) e 1 H ( z) yˆ E (0) e 1 Nomal Incdence (Revew) z 1 (,, ) E H S y (,, ) 1 1 1 Refleced wave: 1z E ( z) xˆ E E (0) e S H 1 1z H (