Photothermal fluctuations as a fundamental limit to low-frequency squeezing in a degenerate optical parametric oscillator

Transcription

1 PHYSICAL REVIEW A Phototherml fluctutions s fundmentl limit to low-frequency squeezing in degenerte opticl prmetric oscilltor Keisuke God Kirk McKenzie Eugeniy E. Mikhilov Ping Koy Lm 3 Dvid E. McClellnd nd Nergis Mvlvl LIGO Lortory Msschusetts Institute of Technology Cmridge Msschusetts 039 USA Center for Grvittionl Physics Deprtment of Physics Fculty of Science The Austrlin Ntionl University ACT 000 Austrli 3 Quntum Optics Group Deprtment of Physics Fculty of Science The Austrlin Ntionl University ACT 000 Austrli Received 8 June 005; pulished 4 Octoer 005 We study the effect of phototherml fluctutions on squeezed sttes of light through the photo-refrctive effect nd therml expnsion in degenerte opticl prmetric oscilltor OPO. We lso discuss the effect of the phototherml noise in vrious cses nd how to minimize its undesirle consequences. We find tht the phototherml noise in the OPO introduces significnt mount of noise on phse squeezed ems mking them less thn idel for low-frequency pplictions such s grvittionl wve GW interferometers wheres mplitude squeezed ems re reltively immune to the phototherml noise nd my represent the est choice for ppliction in GW interferometers. DOI: 0.03/PhysRevA PACS numer s : 4.50.Dv Nn T Ym I. INTRODUCTION Opticl squeezed sttes re used in mny res of quntum optics to improve the sensitivity of mesurements to eyond the shot-noise limit SNL. For exmple squeezed sttes cn e used in interferometers sptil nd spectroscopic mesurements 3 nd potentilly to improve the quntum noise limit of grvittionl wve GW interferometers 4 6. Opticl prmetric oscilltors OPOs re often the systems of choice to produce squeezed sttes since in theory they cn produce sttes with very high levels of squeezing. The level of squeezing tht cn e produced in these systems is limited y the introduction of noise from vriety of sources. The noise sources tht hve een reported to limit squeezing in these systems re pump noise 7 9 nd seed noise 0. In experiments reported to dte the mximum mount of squeezing inferred efore detection is round 7dB 0. This result nd in most results the mximum squeezing is mesured t sidend frequencies ove MHz rther thn t lower frequencies where idelly greter squeezing is predicted. There hs een recent interest in producing squeezed sttes t lower frequencies primrily for use in grvittionl wve detectors. For such sttes the squeezing ndwidth should cover the GW detection nd 0 Hz 0 khz. Severl results hve een pulished elow 300 khz 4 with the lowest result 80 Hz. These frequencies represent different regime experimentlly to the mjority of squeezed stte production nd s such other potentil limiting lowfrequency noise sources need to e considered. One such effect which is lrge t low frequencies in opticl cvity systems is the phototherml-effect-induced noise 5 7. By nture phototherml effects re importnt t low frequencies nd my e significnt for limiting the genertion of squeezed light in the GW detection nd. This effect is investigted theoreticlly in n OPO cvity system in this pper. The phototherml effect cn e descried s the sorption of opticl power in medium cusing temperture chnge to the medium. This effect my e significnt in most nonliner crystls since mny hve reltively high sorption rtes. For exmple the crystl MgO:LiNO 3 hs the liner sorption rte of 4% cm t 53 nm 8. High sorption rtes coupled with the high circulting power required for strong nonlinerity result in lrge mount of opticl power sored into the crystl which my cuse significnt temperture chnge. The verge temperture chnge due to the power sored in the crystl cn e compensted for y using temperture controller nd does not pose significnt prolems for most experimentl systems. Insted we focus our investigtion on the effect of photothermlly induced temperture fluctutions cused y fluctutions in the circulting power in the OPO cvity. The circulting power fluctutions could hve clssicl nd quntum origins or in the cse of shot-noise-limited system only quntum-mechnicl origin. The phototherml noise cused y therml-expnsive noise nd therml-refrctive noise hs two degrding effects on the production of squeezed light in the OPO. The first effect is vi fluctutions in the nonlinerity. This rises s the nonliner strength is temperture dependent due to the phse-mtching condition. The second effect is vi opticl pth length fluctutions. The temperture fluctutions cuse the opticl pth length to chnge potentilly cusing detuning of the opticl cvity. These effects pper s / + T in vrince where T is the therml relxtion cutoff frequency of the nonliner medium nd re therefore primrily significnt t low frequencies. This pper is orgnized s follows: In Sec. II we write down the equtions of motion for the fundmentl nd second-hrmonic modes in n OPO with extr terms required to tke into ccount the phototherml effect. In Sec. III we /005/7 4 / /$ The Americn Physicl Society

2 KEISUKE GODA et l. quntify the fluctuting phototherml effect. In Sec. III A the reltion etween fluctutions in the power sored into the crystl nd in its temperture is descried. In Secs. III B nd III C the coupling of the temperture fluctutions to the fluctutions in the nonliner coupling strength nd cvity resonnce frequencies through the photorefrctive effect nd therml expnsion of the crystl is descried. In Sec. IV the equtions of motion with these phototherml contriutions re solved nd the qudrture field mplitudes in oth mplitude nd phse qudrtures re studied. In Sec. V the squeezed nd ntisqueezed qudrture vrinces with the inclusion of the phototherml noise re derived nd plotted. In Sec. V A we discuss the results for stndrd experimentl prmeters nd for the shot-noise-limited cse. In Sec. V B we consider the effect of squeezing with the phototherml noise on grvittionl wve interferometers t low frequencies. The conclusions of the pper re summrized in Sec. VI. II. FIELD EVOLUTION EQUATIONS IN THE DEGENERATE OPTICAL PARAMETRIC AMPLIFIER In this section the model of the OPO sed on the Heisenerg equtions of motion is introduced then these equtions re linerized nd dditionl terms for the phototherml fluctutions re introduced. This sets up the formlism to include the phototherml fluctutions which re descried in terms of the input fields in the following section. The modes cn then e -coupled nd the vrinces clculted. Strting from the quntum Lngevin eqution the equtions of motion for the intrcvity fields t the fundmentl PHYSICAL REVIEW A frequency nd t the second-hrmonic frequency re given y 9 ȧ = i c + tot + * + in A in e i t + v + sc v sc + s v s ḃ = i c + tot + in B in e i t + v + sc v sc + s v s. The fields nd coupling rtes here re shown schemticlly in Fig.. A in nd B in re the fundmentl nd secondhrmonic input fields to the cvity t frequencies nd respectively =. is the nonliner coupling constnt. c nd c re the cvity resonnce frequencies of the fundmentl nd second-hrmonic fields. in sc s nd re the cvity dmping constnts ssocited with the inputcoupling put-coupling intrcvity scttering nd intrcvity sorption t oth frequencies. v v sc s nd v re the ssocited vcuum fields tht couple in. The following commuttion reltions re stisfied: ss =0 ss = 3 for s=a in B in v v v sc v sc v s nd v s nd ll others vnish. Trnsforming to the rotting frme of ech field with e i t e i t nd similrly for the input fields the equtions of motion ecome ȧ = i det + tot + * + in A in + v + sc v sc + s v s 4 ḃ = i det + tot + in B in + v + sc v sc + s v s 5 FIG.. Color online A schemtic of the OPO cvity. A in nd B in re the input fields to the OPO cvity A nd B re the put fields nd nd re the intrcvity fields t the fundmentl nd second-hrmonic frequencies respectively. in sc nd s re the cvity dmping constnts ssocited with the inputcoupling put-coupling intrcvity scttering nd intrcvity sorption respectively. The index refers to the fundmentl nd second-hrmonic frequencies respectively. v v sc nd s re the ssocited vcuum fields tht couple in. v where the cvity detunings det = c nd det = c. The most common method of generting the nlytic form of squeezed qudrture vrinces is to expnd the opertors their stedy-stte vlues nd then linerize the resulting expressions to first order in the fluctution terms 0. To linerize the equtions of motion the following sustitution for the nnihiltion nd cretion opertors re used: = ā + = ā *

3 PHOTOTHERMAL FLUCTUATIONS AS A FUNDAMENTAL PHYSICAL REVIEW A = + = * + 7 nd similrly for the input fields A in B in. x for x= is the complex expecttion vlue x nd x is the opertor for the fluctutions of x so tht x =0. To otin the equtions of motion for the fluctuting components we consider the following fluctutions due to the phototherml effect in the crystl: i fluctutions in the resonnce frequencies of the cvity due to fluctutions in the length of the cvity ssuming tht the lser frequency is perfectly stle nd ii fluctutions in the nonliner coupling constnt due to fluctutions in the temperture of the crystl nd in the length of the nonliner region. We then mke the following sustitutions: det = det + det det = det + det 8 det nd det re rel = + * = * +. 9 The fluctution terms re otined y tking the fluctution components in Eqs. 4 nd 5 in B tot in 5 i det where nd re the phses of the fundmentl nd second-hrmonic input fields such tht Ā in = Ā in e i nd B in = B in e i. The reltive phse of the fundmentl nd second-hrmonic fields determines whether the fundmentl field is prmetriclly mplified or de-mplified. Equtions 0 nd nd their correlted fluctution opertors cn e rewritten in compct form Ẋ c = M c X c + M in X in + M V + M sc V sc + M s V s + X pt 6 where the intrcvity nd input field vectors re defined y X c A Xin Ain B in B in 7 the vcuum field vectors ssocited with the put coupling sorption loss nd scttering loss re respectively defined y ȧ = iā det + i det tot + * + *ā * + ā * + in A in + v + sc v sc + s v s 0 v V v v v s s v Vs v v v s ḃ = i det + i det tot ā ā + in B in + v + sc v sc + s v s. The coherent components of the equtions of motion re otined similrly y tking the coherent terms in Eqs. 4 nd 5 in the stedy stte ssuming tht the pump field is undepleted ā in B in sc sc v V sc v v v sc 8 nd the field vector due to phototherml fluctutions X pt is split into the fluctuting nonliner coupling constnt component nd the fluctuting cvity detuning component where X pt = X + X w 9 0= i det tot ā + *ā * + in Ā in 0 i det tot + in B in 3 = ā* X ā * ā ā* Xw = det iā w iā * det w det i w i * w det. 0 from which we find the coherent intrcvity field mplitudes in Ā in i det + tot + * e i ā = 4 tot + det det nd their djoints will e derived in the following section. The coupling mtrices ssocited with the intrcvity field input coupling put coupling sorption nd scttering re respectively defined y

4 KEISUKE GODA et l. PHYSICAL REVIEW A M c i det tot * *ā * in * i det tot 0 ā 0 in 0 0 ā 0 i det Min tot 0 0 *ā * 0 i tot tot M s s 0 0 Ms in in 0 0 s s sc sc 0 0 M sc 0 0 sc sc. In terms of frequency components defined y 9 Eq. 6 ecomes Q Q t e = i t dt i X c = M c X c + M in X in + M Ṽ + M sc Ṽ sc + M s Ṽ s + X pt 3 where is the sidend frequency reltive to. The lst field vector X pt will e derived in the next section. The commuttion reltions in Eq. 3 imply s s =0 s s = 4 for s= A in B in v v v sc v sc v s nd v s nd the commuttion reltions etween ny two different sttes is zero. Since the nonliner coupling strength is function of the refrctive index long the ordinry nd extrordinry xes nd the crystl length fluctutions in the temperture of the crystl cuse fluctutions in the nonliner coupling strength. In ddition since the cvity resonnce frequencies re functions of the crystl length t oth the fundmentl nd second-hrmonic frequencies fluctutions in the crystl s temperture cuse fluctutions in the resonnce frequencies. In generl nonliner crystls re sorptive nd therefore the fluctutions in the crystl s temperture re directly coupled with the fluctutions in the mplitudes of the input fields. Here we do not consider the three-dimensionl expnsion of the crystl. A. Power sorption in the crystl The fluctutions in the temperture of the crystl re due to the fluctutions in the opticl power sored into the crystl which is directly relted to the intrcvity field fluctutions. Tking into considertion tht the sorption occurs over the entire length of the crystl the totl sored power is given y P s = P s + P s = A s A s + B s B s III. PHOTOTHERMAL NOISE Through this pper we consider only type-i phse mtching which is simple nd hs een shown to generte squeezing t low frequencies. The phototherml noise is descried s follows: squeezing is degrded y fluctutions in i the nonliner coupling strength nd ii the cvity resonnce frequencies cused y temperture fluctutions due to fluctutions in the photon power sored in the crystl. Ā s + Ā s A s + A s + B s + B s B s + B s 5 where the fluctution terms re A s = s v s A s = s v s

5 PHOTOTHERMAL FLUCTUATIONS AS A FUNDAMENTAL nd the coherent terms re B s = s v s B s = s v s 7 Ā s = s ā * Ā s = s ā * 8 B s = s * B s = s *. 9 Hence we find the power fluctution term P s P s = Ā s A s + A s + B s B s + B s = s ā s + v s + v s + s s + v s + v s. 30 Assuming tht the power sorption in the crystl is uniform over the crystl length it is directly coupled with chnge in the crystl s temperture through the eqution C V Ṫ + T T = P s 3 where is the crystl density V is the mode volume C is the specific het nd T is the therml relxtion time of the crystl. T sets the criticl frequency ditic limit for the response to the fluctutions in the opticl power nd is therefore given y 7 T = C r 0 T 3 where is the therml conductivity of the crystl nd r 0 is the rdius of the nonliner interction etween the seed nd pump fields ssuming they hve Gussin trnsverse profile nd the interction distnce is within the Ryleigh rnge of the fields. Here we hve ssumed tht the rdius of the ems is much smller thn the length of the crystl nd the cross section of the ems is much smller thn the cross section of the crystl. We then find the ssocited temperture fluctutions T in the frequency domin T = P s i + T C V. 33 B. Fluctutions in the nonliner coupling strength In this section the fluctutions in the nonliner coupling strength re clculted for given temperture fluctutions. This result will then e used in Eq. 3. The nonliner coupling constnt is function of the phse mismtch prmeter k defined y k=k k = 0 ze i kz/ sinc kz 34 where 0 is constnt. The refrctive index of nonliner crystl such s mgnesium-oxide doped lithium-niote MgO:LiNO 3 is dependent on the temperture through the photorefrctive effect which is used for chieving type-i phse mtching. The temperture nd wvelength dependence of the phse-mtching condition for MgO:LiNO 3 is descried y the Sellmeier eqution 3 which cn e pproximted round the optimum temperture T 0 t the fundmentl frequency k = T T 0 35 where is constnt whose vlue depends on the crystl s properties nd T is the crystl s temperture. The fluctutions in the crystl s temperture cuse the fluctutions in the nonliner coupling strength through the photorefrctive effect nd therml expnsion = k + k z z = d k k dt + dz z dt T = + k z z T. 36 Here we hve used the Selmeier eqution nd the thermlexpnsion eqution d k dt = 37 dz dt = z 38 where is the liner therml-expnsion coefficient. From Eq. 34 we otin k = iz k + z cot kz 39 z = i z + cot kz. 40 Sustituting Eq. 33 into Eq. 36 we express the effect of the fluctuting nonliner coupling constnt in terms of X c nd V s where PHYSICAL REVIEW A X = M c X c + M s Ṽ s

6 KEISUKE GODA et l. PHYSICAL REVIEW A T * ā* C s ā * C s M c i + ā C s ā* C * s ā * * C s ā * C s ā C s ā* * C s ā * * C s ā * C s ā C s ā* * C s ā * * C s ā * C s ā C s ā* * C s 4 * ā* C ā M s i + T * * C ā * * C ā * * C ā * C ā * C ā * C ā * C ā C ā C ā C ā C ā* * C ā* * C ā* * C ā* C * nd C = s ā + C V k z z C = s + C V k z z. C. Fluctutions in the cvity detunings In this section the fluctutions in the opticl pth length re clculted for given temperture fluctutions. This result will then e used in Eq. 3. The phototherml fluctutions couple to the fluctutions in the opticl pth length from the following two mechnisms: the photorefrctive effect nd therml expnsion. The fluctutions in the opticl pth length cn e converted into cvity resonnce frequency fluctutions using 4 c = c dn n dt + T c = c dn n dt + T nd therefore ssuming tht the lser frequencies re stle the cvity detuning fluctutions re det = c det nd = c. Sustituting Eq. 33 into Eq. 46 we similrly write the effect of the fluctuting cvity detuning component in terms of X c nd V s where X = M c X c + M s Ṽ s 48 M c T s iāk iā * K s i + i K s i * K s iāk s iā * K s i K s i * K s iāk s iā * K s i K s i * K s iāk s iā * K s i K s i * K s 49 iāk iāk iāk iāk M s iā i + T * K iā * K iā * K iā * K 50 i K i K i K i K i * K i * K i * K i * K

7 PHOTOTHERMAL FLUCTUATIONS AS A FUNDAMENTAL PHYSICAL REVIEW A where = s ā 5 C V = s 5 C V K = c dn n dt + 53 It is importnt to note tht s the phototherml effect is turned off y setting s 0 nd s 0 the phototherml coupling mtrices M c M s M c nd M s s well s M s ll ecome zero nd then Eq. 58 reduces to the solutions of the field evolution equtions with the phototherml effect X = M i I M c M in X in + M i I M c M I Ṽ 60 K = c dn n dt +. IV. QUADRATURE FIELD AMPLITUDES WITH THE PHOTOTHERMAL NOISE 54 Now tht we hve descried the equtions of motion the sored power fluctutions nd the ssocited fluctutions in the nonliner coupling strength nd cvity detunings we re in position to put these equtions together nd solve the equtions of motion with the phototherml effect. We lso discuss limiting cse in which the qudrture vrinces cn e pproximted to simple nlytic forms under relistic ssumptions. Sustituting Eqs. 4 nd 48 into Eq. 9 yields X pt = M c + M c X c + M s + M s Ṽ s nd sustituting this into Eq. 3 gives i I M c M c M c X c = M in X in + M Ṽ + M sc Ṽ sc 55 + M s + M s + M s Ṽ s 56 where I is the identity mtrix. We thus find the intrcvity field fluctutions X c = i I M c M c M c M in X in + M Ṽ + M sc Ṽ sc + M i I M c M sc Ṽ sc. 6 We define the mplitude nd phse qudrture field fluctution mplitudes in the frequency domin reltive to the fundmentl frequency respectively X s s + + s 6 X s i s + s 63 for s=a in A B in B v v v sc v sc v s nd v s. The commuttion reltions 4 imply the following vlues for the commuttors of the qudrture field mplitudes nd their djoints: X s X s = X s X s = i 64 for s=a in A B in B v v v sc v sc v s nd v s nd ll others vnish. It is convenient to express Eq. 7 in terms of the qudrture field mplitudes X = X X in = X in Ṽ sc = Ṽ sc Ṽ = Ṽ Ṽ s = Ṽ s 65 + M s + M s + M s Ṽ s. 57 Defining the extrcvity field vector y 9 we find à X à 58 B B X = M X c Ṽ = M i I M c M c M c M in X in + M i I M c M c M c M I Ṽ + M i I M c M c M c M sc Ṽ sc + M i I M c M c M c M s + M s + M s Ṽ s. 59 where nd X 0 0 i i i i X A X A X B X B X in X A in X A in X B in X B in

8 KEISUKE GODA et l. PHYSICAL REVIEW A Ṽ X v Ṽ s Ṽsc X X v s s s s Eqution 58 cn e rewritten s X = in X in + Ṽ v sc sc sc sc sc Ṽ sc + s Ṽ s 68 where the qudrture field coupling mtrices re defined y in M i I M c M c M c M in M i I M c M c M c M I sc M i I M c M c M c M sc s M i I M c M c M c M s + M s + M s. 69 Normlized mplitude nd phse qudrture vrinces re given y 9 Ṽ s = X s Ṽ s = X s 70 for s=a B A in B in v v v sc v sc v s nd v s respectively. The normlized qudrture vrinces of the fundmentl put field cn e written s liner comintion of Ṽ Ain Ṽ Bin Ṽ v Ṽ v Ṽ v sc Ṽ v sc Ṽ v s nd Ṽ v s 5. Since the vcuum fields tht couple in t the opticl losses re in the minimum uncertinty stte Ṽ v = Ṽ v = Ṽ v sc = Ṽ v sc = Ṽ v s = Ṽ v s =. 7 Therefore we find the normlized mplitude nd phse qudrture vrinces of the fundmentl put field respectively Ṽ A = in Ṽ Ain + in Ṽ Ain + 3 in Ṽ Bin + 4 in Ṽ Bin 4 + j= j + j sc + j s Ṽ A = in Ṽ Ain + in Ṽ Ain + 3 in Ṽ Bin + 4 in Ṽ Bin 4 + j= j + j sc + j s 7 73 where the superscripts ij of s denote the mtrix elements. in nd 3 in re the mplitude noise coupling constnts of the seed or pump fields 4 in nd in re the phse noise coupling constnts of the seed nd pump fields respectively. The rest of the s re the mplitude nd phse noise coupling constnts of the vcuum fields t the fundmentl nd second-hrmonic frequencies. Note tht the normlized qudrture vrinces re completely chrcterized y the normlized qudrture vrinces of the two input nd vcuum fields nd the coupling constnts. If the seed nd pump fields re shot-noise limited Ṽ Ain = nd the qudrture vrinces reduce to =Ṽ Bin Ṽ A Ṽ A 4 = j= = j in + j + j sc + j s 4 j= j in + j + j sc 74 + j s. 75 We now turn to the discussion of limiting cse to otin simple nlytic forms y mking the following ssumptions tht re pplicle to most prcticl cses of squeezing: i the pump noise is comprle to the seed noise ii the fundmentl intrcvity field is much weker thn the secondhrmonic intrcvity field iii there is no intrcvity scttering iv there re no verge cvity detunings t oth the fundmentl nd second-hrmonic frequencies v the frequency of interest is within the linewidth of the OPO cvity nd vi we re only interested in mximum phse squeezing =0 or mximum mplitude squeezing = so tht ny term proportionl to * is zero. Under these ssumptions we cn mke the following pproximtions. For the mplitude qudrture vrince

9 PHOTOTHERMAL FLUCTUATIONS AS A FUNDAMENTAL PHYSICAL REVIEW A in tot + * + * in tot in i in * * tot 3 in i T i s in ā * C tot + * + ā * C * tot + * tot tot 4 in 0 tot + tot + * + * + tot 3 i T i i * * tot s ā * C tot + * + ā * C * tot + * tot tot 4 0 s tot + * + * s tot s i s * * tot 3 i s tot ā * C tot + * + ā * C * tot + * s i T tot tot 4 s 0 sc 0 sc 0 3 sc 0 4 sc For the phse qudrture vrince in i in * * in tot * * tot in tot 3 in i T s in ā * C tot * ā * C * tot * tot tot 4 in 0 i * * tot 3 tot + tot * + * + tot i T s ā * C tot * ā * C * tot * tot tot

10 KEISUKE GODA et l. PHYSICAL REVIEW A s i s * * s tot * * tot s tot 3 s tot ā * C tot * ā * C * tot * s i T tot tot 4 s 0 sc 0 sc 0 3 sc 0 4 sc Note tht in this limiting cse the noise coupling constnts 3 in 3 3 s 3 in 3 nd 3 s hve the frequency dependence of / i T nd therefore they increse s the frequency decreses for T degrding the squeezing level t low frequencies. In Sec. V we will discuss which couping constnts re dominnt t high frequencies low frequencies nd intermedite frequencies. V. RESULTS In this section we discuss vrious cses of the influence of the phototherml noise on squeezed qudrture vrinces in oth mplitude nd phse qudrtures. The most significnt phototherml effects re seen in the phse qudrture nd therefore we minly discuss qudrture vrinces in the phse qudrture. Section V A presents such results. Section V B discusses the effect of squeezing with the phototherml noise on conventionl grvittionl wve interferometer when phototherml-noise-limited squeezed field is injected into it. The following plots re otined from the exct normlized qudrture vrinces in Eqs. 7 nd 73 with relistic vlues for OPO prmeters which re listed in Tle I. The effect of green-induced infrred sorption 6 is not considered in this pper. A. Normlized qudrture vrinces with the phototherml effect The mplitude qudrture is reltively immune to the phototherml noise for the following resons. The intrcvity fundmentl field is demplified in the degenerte prmetric oscilltion nd thus the noise coupling is smller thn in the phse squeezing cse. Moreover in the idel cse the system is held on resonnce nd operted t the phse-mtched temperture. The detuning fluctutions do not couple into the mplitude qudrture for cvity on resonnce since the frequency derivtive of the mplitude response of the cvity is zero. Figure compres the effect of the phototherml noise etween the mplitude nd phse squeezing cses. Note tht they re not otined simultneously from the OPO; different pump phses re required. The phototherml noise is therefore significnt in the phse qudrture nd reltively unimportnt in the mplitude qudrture in most prcticl cses. In the sence of the phototherml noise the normlized qudrture vrince would e flt within the OPO linewidth. As cn e seen in Figs. 5 in which the normlized qudrture vrinces versus frequency re plotted the squeezing level is cut off t frequencies elow 0 khz depending on the OPO cvity prmeters due to the phototherml noise which hs / + T roll-off in vrince. Prmeter vlues used for the figures re summrized in Tle I. The high-frequency cutoff is due to the linewidth of the OPO cvity elow which the seed field is squeezed. At frequencies ove the high cutoff frequency nd dominte in the mplitude nd phse qudrtures respectively. At frequencies etween the two cutoff frequencies in s in nd s dominte depending on the OPO cvity prmeters in the mplitude nd phse qudrtures respectively. The phototherml cutoff frequency is greter thn the ditic limit T in most prcticl cses nd therefore t low frequencies ove T 3 in 3 s 3 in nd 3 s tht hve the frequency dependence of / dominte depending on the OPO cvity prmeters in the mplitude nd phse qudrtures respectively s cn esily e seen in Eqs. 76 nd 77. At frequencies elow the ditic limit T the qudrture vrinces ecome flt. The domintion of these s in ech frequency nd is vlid regrdless of which qudrture vrince is squeezed or ntisqueezed. Figure 3 shows the normlized qudrture vrince of phse squeezed stte s function of frequency for vrious pump powers. As the pump power pproches the OPO threshold the squeezing level t high frequencies increses ut higher pump power lso increses the phototherml noise contriution. The phototherml noise is lrgest t low frequencies nd limits squeezing to occur only t higher frequencies where the phototherml noise is smll. The increse in the phototherml noise s the pump power pproches the OPO threshold is lso ttriutle to the increse in the fundmentl field mplitude vi high prmetric gin which increses the phototherml noise coupling. Squeezing t lower frequencies cn e cquired t the expense of the rodnd level of squeezing

11 PHOTOTHERMAL FLUCTUATIONS AS A FUNDAMENTAL PHYSICAL REVIEW A TABLE I. OPO cvity prmeters. Prmeter Symol Vlue Units Fundmentl wvelength 064 nm Second-hrmonic wvelength 53 nm Reflectivity of input coupler t fundmentl in R % frequency Reflectivity of put coupler t fundmentl R 95.6 % frequency Reflectivity of input coupler t second-hrmonic in R 4.0 % frequency Reflectivity of put coupler t second-hrmonic R % frequency Asorption rte t fundmentl frequency s 0. %/cm Scttering rte t fundmentl frequency sc 0.0 %/cm Asorption rte t Second-Hrmonic Frequency s 4.0 %/cm Scttering rte t Second-Hrmonic Frequency sc 0.5 %/cm Crystl length z 7.5 mm Nonliner coupling strength /m/s Phse-mtched refrctive index n.33 Specific het of crystl C 633 J/ kg/ K Density of crystl g/cm 3 Therml conductivity of crystl 4 W/K/m Rdius of nonliner interction r 0 36 m Phse mismtch constnt 749 /m/k Therml expnsion constnt in ordinry xis /K Therml expnsion constnt in extrordinry xis /K Photorefrctive constnt in ordinry xis dn /dt /K Photorefrctive constnt in extrordinry xis dn /dt /K Temperture offset T 0.00 K Cvity detuning t fundmentl frequency det 0 Hz Cvity detuning t second-hrmonic frequency det 0 Hz Figure 4 shows the normlized qudrture vrince of phse squeezed stte s function of frequency for vrious seed powers. Since the coupling of the phototherml effect to the qudrture vrinces is proportionl to the seed power the phototherml noise contriution limits squeezing to higher frequencies s the seed power increses. Hence the seed power should e set s smll s possile to void the phototherml noise. However the reduction of the seed power leds to difficulties in otining n opticl signl for controlling the phse of the seed. Therefore in prctice the seed power should e properly chosen such tht it optimizes control stility nd the frequency of interest is ove the phototherml cutoff frerequency. A crystl with smller sorption rte cn lso reduce the pump noise coupling nd therefore the phototherml noise. Its effect ppers similr to the effect of lower seed power s shown in Fig. 4. Figure 5 shows the normlized qudrture vrince of phse squeezed stte s function of frequency for vrious pump mplitude noise levels. As the pump noise increses the overll squeezing level lso decreses due to the direct coupling of the pump noise to the qudrture vrinces 7 9. At the sme time the phototherml noise induced y FIG.. Color online The comprison of normlized mplitude nd phse qudrture vrinces reltive to the shot noise vs frequency with the phototherml effect. The mplitude qudrture is reltively immune to the phototherml noise in most prcticl cses. The seed power is mw. The pump power is 0.5P th. The input fields re shot-noise limited

12 KEISUKE GODA et l. PHYSICAL REVIEW A FIG. 3. Color online The normlized phse qudrture vrince reltive to the shot noise vs frequency with the phototherml effect for different pump powers. The phototherml cutoff frequency is higher for higher level of squeezing wheres it is lower for lower level of squeezing. The seed power is mw. The input fields re shot-noise limited. the pump noise lso increses driving up the phototherml noise limited frequency. Figure 6 shows the qudrture vrince of phse squeezed stte s function of pump or seed power for vrious frequencies. With the phototherml effect the mximum squeezing would e chieved t the OPO threshold. In the presence of the phototherml noise the squeezing level strts to degrde s the pump power pproches the OPO threshold. The phototherml noise cn lso e minimized y reducing the seed power. FIG. 5. Color online The normlized phse qudrture vrince reltive to the shot noise vs frequency with the phototherml effect for different pump noise levels. The phototherml cutoff frequency is higher for higher pump noise level. The seed power is 0 mw. The pump power is 0.5P th. In summry the undesirle consequences of the phototherml noise cn e minimized y stisfiying the following conditions: i low seed power ii quiet pump field iii crystl with low sorption rte nd iv squeeze the mplitude qudrture vrince rther thn the phse qudrture vrince. B. Effect of the phototherml noise on grvittionl wve interferometers One primry purpose of low-frequency squeezing is to improve the sensitivity of GW interferometers in the GW nd which is typiclly Hz 4. To implement it low-frequency squeezed field needs to e prepred for injection to the drk port of the GW interferometers. However s discussed in Sec. V A the squeezing level of phse-squeezed light is limited y the phototherml noise t low frequencies nd hence it plces n importnt limit on the use of squeezed light in the GW interferometers. For conventionl GW interferometer with rm lengths L nd mirror msses m the spectrl density of the GW noise when relistic squeezed field is injected to the drk port is given y 5 S h = h SQL K + K Ṽ + A 78 where FIG. 4. Color online The normlized phse qudrture vrince reltive to the shot noise vs frequency with the phototherml effect for different seed powers. The phototherml cutoff frequency is higher for higher seed power. The pump power is 0.5P th. The input fields re shot-noise limited. Here Ṽ A + = Ṽ A cos + + Ṽ A sin

13 PHOTOTHERMAL FLUCTUATIONS AS A FUNDAMENTAL PHYSICAL REVIEW A FIG. 6. Color online Top: The normlized phse qudrture vrince reltive to the shot noise level vs pump power with the phototherml effect for different frequencies. The seed power is mw. Bottom: The normlized phse qudrture vrince reltive to the shot-noise level vs seed power with the phototherml effect for different frequencies. The pump power is 0.5P th. The input fields re shot-noise limited in oth grphs. h SQL 8 m L 80 is the noise spectrl density of the dimensionless GW strin t the stndrd quntum limit SQL for GW interferometer with uncorrelted rdition pressure noise nd shot noise K = I 0/I SQL is the effective coupling constnt tht reltes the put signl to the motion of the GW interferometer mirrors Ṽ A Ṽ A re the mplitude nd phse qudrture vrinces of the input squeezed field with squeeze ngle respectively nd cot K 8 is the effective ponderomotive squeeze ngle of the interferomter. Here is the linewidth of the rm cvities I 0 is the FIG. 7. Color online The spectrl noise density normlized y the stndrd quntum limit SQL for conventionl GW interferometer with i no squeezed input Unsqueezed ii squeezed light injected with the phototherml noise Frequency-Independent iii squeezed light injected with frequency-dependent squeeze ngle nd the phototherml noise Frequency-Dependent +Phototherml iv squeezed light injected with the phototherml noise Frequency-Independent+Phototherml nd v mplitudefiltered squeezed light injected with the phototherml noise Amplitude Filter+Phototherml. With perfect squeeze ngle rottion the photothermlly noisy mplitude qudrture does not couple into the spectrl noise density nd therefore the spectrl noise density with squeezed light injected with frequencydependent squeeze ngle nd with the phototherml noise Frequency-Dependent with Phototherml looks identicl to iii. The filter linewidth is 400 Hz. The input squeezed source is chosen to e phototherml-noise-limited mplitude squeezed light since the mplitude qudrture is reltively immune to the phototherml noise. The seed nd pump powers re 0 mw nd 0.5P th respectively. The seed nd pump fields re shot-noise limited. The ntisqueezed phse qudrture vrince hs the phototherml cutoff frequency t khz nd the ditic limit t 00 Hz. opticl power to the emsplitter of the GW interferometer nd I SQL is the opticl power to rech the SQL. Squeezed stte sources re generlly frequency independent ut the desired ngle my e produced y using opticl cvities plced etween the squeezed stte source nd the interferomter s filters. Both filters tht rotte the squeeze ngle to mtch the ponderomotive squeeze ngle 57 nd filters tht ttenute the ntisqueezing in desired nd 9 hve een proposed. Figure 7 shows the noise spectrl density for the conventionl GW interferometer in vrious cses when phototherml-noise-limited mplitude-squeezed field is injected into the interferometer. The effect of squeezing with the phototherml noise on the sensitivity of the GW interferometer is plotted in the figure. Since the mplitude qudrture is reltively insensitive to the phototherml noise we choose the squeeze ngle such tht the mplitude qudrture vrince is squeezed nd the phse qudrture vrince is ntisqueezed. The seed nd pump powers to the OPO cv

14 KEISUKE GODA et l. ity re 0 mw nd 0.5P th. If such squeezed light field is used for injection with rottion of the squeeze ngle it degrdes the spectrl noise density t 00 Hz compred with the unsqueezed cse lthough it reduces shot noise t high frequencies. If set of two filter cvities is used to give the frequency-dependent squeeze ngle such tht = 5 the sensitivity is improved t ll frequencies. This is ecuse the second term in Eq. 78 ecomes zero nd the phse qudrture vrince does not couple into the sensitivity curve. We note tht implementtion of such squeeze ngle rottion requires the filter cvities to e long on the order of kilometers in order to minimize the effect of losses tht destroy squeezing in the process 58. Alterntively if squeeze mplitude filter is used efore injecting the squeezed light into the GW interferometer 9 such tht for filter linewidth f where S h = h SQL K Ṽ + K Ṽ + +K = f + = f f PHYSICAL REVIEW A the phse qudrture vrince contining the phototherml noise cn e reduced t frequencies elow 00 Hz lthough the level of squeezing t frequencies ove 00 Hz is slightly decresed. Here f = 400 Hz is used. Note tht phototherml-noise-limited phse-squeezed input field with similr experimentl prmeters the seed power= mw the pump power=0.5p th does not enle quntum noise reduction elow 400 Hz even if the optiml frequency-dependent squeeze ngle rottion is pplied. VI. CONCLUSIONS We hve derived nd solved the field evolution equtions in the degenerte opticl prmetric oscilltor OPO with the phototherml noise through the photorefrctive effect nd therml expnsion of nonliner crystls. We lso hve discussed vrious cses the effect of the phototherml noise on mplitude nd phse qudrture vrinces. We hve found tht the phototherml noise in the OPO introduces significnt mount of noise on phse squeezed ems mking them less thn idel for low-frequency pplictions such s GW interferometers wheres mplitude squeezed ems re less sensitive to the phototherml noise nd my provide etter choice for low-frequency pplictions. This prolem cn e solved y reducing the seed power nd pump noise nd using nonliner crystl with low sorption rte in order to decrese the phototherml noise. ACKNOWLEDGMENTS We would like to thnk our collegues nd collortors t the LIGO Lortory nd Center for Grvittionl Physics t the Austrlin Ntionl University especilly Thoms Coritt Dvid Ottwy Stnley Whitcom nd Kentro Somiy for vlule discussions. We lso thnk our collegue Sergey Vytchnin t Moscow Stte University for correcting n error nd Romn Schnel t Universität Hnnover for clrifying n importnt point. We grtefully cknowledge support from Ntionl Science Foundtion Grnt Nos. PHY nd PHY nd the Austrlin Reserch Council. M. Xio L-A. Wu nd H. J. Kimle Phys. Rev. Lett C. Fre J. B. Fouet nd A. Mitre Opt. Lett E. S. Polzik J. Crri nd H. J. Kimle Appl. Phys. B: Photophys. Lser Chem C. M. Cves Phys. Rev. D H. J. Kimle Y. Levin A. B. Mtsko K. S. Thorne nd S. P. Vytchnin Phys. Rev. D K. McKenzie D. A. Shddock D. E. McClellnd B. C. Buchler nd P. K. Lm Phys. Rev. Lett K. Wodkiewicz nd M. S. Zuiry Phys. Rev. A D. D. Crouch nd S. L. Brunstein Phys. Rev. A J. Ge-Bncloche nd M. S. Zuiry Phys. Rev. A P. K. Lm T. C. Rlph B. C. Buchler D. E. McClellnd H-A. Bchor nd J. Go J. Opt. B: Quntum Semiclssicl Opt K. McKenzie N. Grosse W. P. Bowen S. E. Whitcom M. B. Gry D. E. McClellnd nd P. K. Lm Phys. Rev. Lett W. P. Bowen R. Schnel N. Treps H-A. Bchor nd P. K. Lm J. Opt. B: Quntum Semiclssicl Opt R. Schnel H. Vhlruch A. Frnzen S. Chelkowski N. Grosse H-A. Bchor W. P. Bowen P. K. Lm nd K. Dnzmnn Opt. Commun J. Lurt T. Coudreu G. Keller N. Treps nd C. Fre Phys. Rev. A V. B. Brginsky M. L. Gorodetsky nd S. P. Vytchnin Phys. Lett. A Y. T. Liu nd K. S. Thorne Phys. Rev. D M. Cerdonio L. Conti A. Heidmnn nd M. Pinrd Phys. Rev. D P. K. Lm T. C. Rlph B. C. Buchler D. E. McClellnd H-A Bchor nd J. Go J. Opt. B: Quntum Semiclssicl Opt M. J. Collett nd C. W. Grdiner Phys. Rev. A B. Yurke Phys. Rev. A S. Vytchnin personl communiction. M. De Ros L. Conti M. Cerdonio M. Pinrd nd F. Mrin Phys. Rev. Lett R. C. Echhrdt C. D. Nors W. J. Kozlovsky nd R. L

15 PHOTOTHERMAL FLUCTUATIONS AS A FUNDAMENTAL Byer J. Opt. Soc. Am. B A. E. Siegmn Lsers University Science Books Mill Vlley Cliforni A. G. White M. S. Tumn T. C. Rlph P. K. Lm D. E. McClellnd nd H-A. Bchor Phys. Rev. A Y. Furukw K. Kitmur A. Alexndrovski R. K. Re M. M. Fejer nd G. Foulon Appl. Phys. Lett PHYSICAL REVIEW A J. Hrms R. Schnel nd K. Dnzmnn Phys. Rev. D J. Hrms Y. Chen S. Chelkowski A. Frnzen H. Vhlruch K. Dnzmnn nd R. Schnel Phys. Rev. D T. Coritt N. Mvlvl nd S. Whitcom Phys. Rev. D