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Demonstration of reversible phase-insensitive optical amplifier Jun-ichi Yoshikawa,1 Yoshichika Miwa,1 Radim Filip,2 and Akira Furusawa1 1Department of Applied Physics, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan 2Department of Optics, Palacký University, 17. listopadu 1192/12, 772 07 Olomouc, Czech Republic (Dated: January 7, 2011) Weexperimentallydemonstratephase-insensitivelinearopticalamplification whichpreservesthe idler at the output. Since our amplification operation is unitary up to small excess noise, it is reversiblebeyondtheclassicallimit. Theentanglementbetweenthetwooutputmodesistheresource forthereversibility. TheamplificationgainofG=2.0isdemonstrated. Inaddition,combiningthis amplifier with a beamsplitter, we also demonstrate approximate cloning of coherent states where an anticlone is present. We investigate the reversibility by reconstructing the initial state from 1 the output correlations, and the results are slightly beyond the cloning limit. Furthermore, full 1 characterization of the amplifier and cloner is given by using coherent states with several different 0 2 mean values as inputs. Our amplifier is based on linear optics, offline preparation of nonclassical ancillas, and homodyne measurements followed by feedforward. Squeezed states are used as the n ancillas, andnonlinearopticaleffectsareexploitedonlyfortheirgeneration. Theancillas introduce a nonclassicality into theamplifying operation, making entanglement at the output. J 6 PACSnumbers: 03.67.Hk,42.50.Dv,42.65.Yj ] h I. INTRODUCTION aîn is another mode’s annihilation operator in the aux- p idl iliary system. Throughout this paper, the ancilla mode - nt Quantumoptics is governedby rules imposed by com- represented by aîindl is denoted by the term “idler” and a mutation relations which have to be kept during time distinguishedfromother ancilla modes. Eq.(1) becomes u evolution. Optical amplification is no exception to this the input-output relation of optimal PIA when the idler q story. Typically, the amplified output is suffered from is in a vacuum state. The quantum fluctuation of the [ inevitable excess noise. This limitation is quantum- idler contaminates the amplified signal. This is the in- 1 mechanically imposed, thus does not depend on the spe- evitable excess noise of PIA. Note that this penalty pre- v cific realization methods. Caves classified general linear vents amplification from being a loophole of the uncer- 9 amplification into phase-insensitive amplification (PIA) tainty relation in joint measurements [2, 3]. At the limit 3 andphase-sensitiveamplification(PSA)[1]. He alsosys- of high-gain amplification, we can see the famous 3 dB 1 tematically derivedthe quantumlimit ofexcessnoisefor cost of the noise figure for PIA of coherent states. In 1 . suchgenerallinearamplificationwitharbitrarygainfrom addition to this intrinsic excess noise, further nonintrin- 1 therequirementtopreservecommutationrelations. This sic excess noise may be causedby other ancilla modes in 0 excess noise originates from quantum fluctuations in the nonoptimal PIA. 1 1 auxiliary system required to keep energy conservation. There are numerous practical realizations of optical : We concentrate on PIA, supposing the target of am- amplification. Doped fiber amplifiers (DFAs) and semi- v plificationtobeopticalwaveamplitudeofasinglemode, conductor optical amplifiers (SOAs) utilize stimulated i X which is denoted by the term “signal”. Classical coun- emissions [4], and Raman amplifiers (RAs) and optical r terpartof PIAis a conversionof arbitrarycomplex wave parametricamplifiers(OPAs)utilizenonlinearopticalef- a amplitude α C into √Gα, where G 1 is the gain of fects. In principle, there is no quantum-mechanical rea- ∈ ≥ amplification. As is found in ordinary textbooks, anni- sontopreventtheserealizationsfromachievingtheopti- hilationoperatorsin quantumoptics correspondto com- malPIAintheformofEq.(1). However,therealdevices plex amplitudes in classical optics. Therefore, we de- with current technology are accompanied by further ex- scribe the amplifying process by the transformation of cess noises. annihilation operators. Quantum-mechanically optimal Recently, PIA operating almost at the optimal level is PIA in the sense that the excess noise is minimized can experimentally demonstrated by Josse et al. by utiliz- be achieved by the following transformation [1]: ing feedforward [5]. The reasonfor the high efficiency of aôsiugt =√Gaîsnig+eiθ√G−1(aîindl)†, (1) Jsiocsasleo’spPerIaAtioisntshoartnitondcoleasssnicoatlraenqcuiilrlaesi.neItffiucsieesntanvoancculuams- whereaîn andaôut arethesignalmode’sannihilationop- stateasanancillawhichispresenteverywhere,andlinear sig sig erators before and after the amplification, respectively. optics and homodyne measurements followed by feedfor- There is an extra term eiθ√G 1(aîn) which is in- ward which are highly efficient. − idl † troduced in order to meet the commutation relation of Although Josse’s PIA is a good attainment, it is not [aˆsig,aˆ†sig] = 1 for both the input and output signal theendofthestory. ThesignaltransformationinEq.(1) modes. Here, θ R is an arbitrary phase factor, and is an irreversible thermalizing process. Complete PIA ∈ 2 should have unitary realization on an expanded Hilbert signaland idler outputs. For the cloning experiment, we space. In order to unitarize PIA, two-mode description checkbipartiteentanglementbetweeneachcloneandthe is sufficient. The full input-output relation becomes as anticlone, which as a whole proves tripartite entangle- follows: ment of class I [9]. Moreover, for both experiments, the reversibilityis investigatedfrom the output correlations. aôsiugt =√Gaîsnig+eiθ√G−1(aîindl)†, (2a) Our idler-preserving PIA is significant in several re- aôut =√Gaîn +eiθ√G 1(aîn ) . (2b) spects. First of all, the reversibility will pave the way idl idl − sig † to new schemes. Recently, there is a proposal of a CV quantuminterfacethatenablesinprincipleaunitfidelity Note that the roles of the signaland idler are symmetric of transfer using such reversible PIA [10]. Moreover,the in this relation. reversibility in cloning is also advantageous. Cloning of Thesignificanceofunitarizationmustbethereversibil- unknownstatesisdistributionofinformation,anditsre- ity. Theinversetransformationiseasilyderivedwhenwe versibility reserves the option to recover the distributed take notice of the fact that Eq. (2) is equivalent to two- fragments of the information. This will be further dis- mode squeezing operation. A two-mode squeezing oper- cussed in Sec. III. Secondly, our PIA would have some ationparametrizedby (G,θ) is canceledby another two- applicationsastwo-modesqueezingoperation. Notethat mode squeezing operation where the squeezing direction one-mode squeezing operation is already demonstrated is opposite, i.e., (G,θ+π). Nonetheless, in many ampli- successfully in Ref. [11] with similar approach. fication schemes including Josse’s experimental demon- In this introduction, PIA has been described together stration [5], the idler output is lost in the inextractable withabriefhistoricalreview. Especially,thenonclassical environment, making the process irreversible. property of PIA is discussed, which is obscure in many Inordertorealizeidler-preservingandclose-to-optimal amplificationprocessesbecausetheidleroutputislostin PIA, we require some nonclassicality for the amplifier. the inextractable environment. The subsequent contents This is contrastive to Josse’s idler-nonpreserving PIA ofthispaperareasfollows. InSec.II,feedforward-based which does not require any nonclassicality. A typical PIA is described, explicitly showing the excess noise due strategy to introduce nonclassicality into feedforward- to finite squeezing of ancillas. In Sec. III, CV quantum based quantum circuits is to use nonclassical states as state cloning and its connection with PIA are described. ancillas. Continuous-variable (CV) quantum teleporta- InSec.IV,theexperimentalsetupisdescribed. InSec.V, tion [6] and CV error correction [7] are good examples. the experimental results for PIA of coherent states with In these examples, squeezed states are used as ancillas G= 2.0 are shown. In Sec. VI, the experimental results that support the performance beyond the classicallimit, for 1 2 approximate cloning of coherent states are and the complex operations after the state preparation → shown. In Sec. VII, our experimental achievements are stage are efficiently implemented by linear optics. summarized. In this paper, by employing the feedforward-based scheme proposed in Ref. [8], we demonstrate PIA of coherent states which preserve the idler output. The II. FEEDFORWARD-BASED AMPLIFIER scheme basically relies on linear optics including homodyne measurements and feedforward. Squeezed vacuum states are used as ancillas, which inject nonclassicality In our definition, feedforward means that the opera- into our PIA. Only for generating the nonclassical an- tions after some measurements are changed depending cilla states, we resort to nonlinear optical effects. Our on the measurement outcomes which in general are ob- demonstration is for the amplification gain of G = 2.0, tained randomly. In particular,in this paper it indicates whichistunedviapassiveopticaldevicesandfeedforward phase space displacement operations whose amounts are electric circuits. Combining PIA for G=2.0 with a half proportional to the results of homodyne measurements. beamsplitter, we also demonstrate 1 2 approximate We know two specific schemes for feedforward-based → cloning of coherent states, where an “anticlone” remains PIA that preserves the idler at the output. One scheme at the output. (Anticlone will be explained in Sec. III.) is proposed by Filip et al. in Ref. [8], in which PIA is In principle, our amplifier and cloner becomes quantum- composed of two feedforward-based single-mode squeez- mechanically optimum at the limit of infinite squeezing ers proposed in the same paper. The other scheme is of the ancillas. For the case of finite squeezing, as is the proposed by Josse et al. in Ref. [5] as a modification realsituationinexperiments,furtherexcessnoiseinvades of the idler-nonpreserving PIA. Note that Josse’s idler- inaccordancetothelevelofthe squeezing. However,the preserving PIA is just a theoretical proposal and the degradation is small enough to retain nonclassical fea- idler-nonpreserving PIA alone is experimentally demon- tures. Thebehaviorsofouramplifierandclonerarefully strated. characterized by using several coherent states as inputs. Both of Filip’s scheme and Josse’s scheme rely on lin- Furthermore, we also pay much attention to the output ear optics including homodyne measurements and feed- correlations, because nonclassical properties clearly ap- forward, and require offline-prepared nonclassical states pear in them. For the PIA experiment, we check the as ancillas. Moreover, in both schemes, the gain of Einstein-Podolsky-Rosen(EPR) correlationbetween the amplification is accurately and stably determined via 3 the choice ofpassiveopticaldevices andcorrespondingly One-to-onecorrespondenceof1 G< and0<R 1 ≤ ∞ ≤ feedforwardgains. Asforthenonclassicalancillas,Filip’s is easily checked. Note that the feedforward gain is also scheme requires two single-mode squeezed states, on the parameterizedbyR. Itischosensothattheantisqueezed otherhand,Josse’sschemerequiresatwo-modesqueezed noises from the ancillas are canceled out at the output. state. Since twosingle-modesqueezedstatescanbe con- ForthedemonstrationofG=2,ThevalueofRshould verted to a two-mode squeezed state and vice versa by be chosenas3 2√2 0.17. The resultinginput-output − ≈ a half beamsplitter interaction, the amounts of the non- relation becomes as follows: classical resources required for the two distinct schemes are the same. xôut =√2xîn+xîn q√2 1xôut, (5a) For both schemes, the feedforward-based PIA coin- 1 1 2 − − A cides with the quantum-mechanically optimal PIA only pôut =√2pîn pîn+q√2 1pôut, (5b) at the limit of infinite squeezing of the ancillas. For the 1 1 − 2 − B case of finite squeezing, excess noise contaminates the xôut =√2xîn+xîn+q√2 1xôut, (5c) output to some extent. Note that this is a common 2 2 1 − A matter of feedforward-based CV deterministic process- pôut =√2pîn pîn+q√2 1pôut. (5d) ing [6, 7, 12]. The difference between the two schemes 2 2 − 1 − B proposed by Filip and Josse solely arises in this excess noise. ForFilip’sscheme,itappearssymmetricallyinthe III. QUANTUM STATE CLONING signal and idler outputs. On the other hand, for Josse’s scheme, it appears only in the idler output. The better It is known as the no-cloning theorem that an un- choice between different schemes depends on the specific knownquantumstate ψ cannotbe perfectly duplicated application. | i as ψ ψ [13]. However,approximatecloning ispossible, We have chosen the symmetrical one. In the demon- | i| i which can go beyond some classical limit in general. stration in Sec. V, we confirm the symmetry of PIA by In this section, we will discuss CV cloning, and make swapping the roles of the signal and idler. its connection with PIA. Furthermore, its reversibility Fig. 1(a) shows the schematic of our PIA, from which is discussed by introducing the notion of anticlone. In the symmetry of the signal and idler is obvious. Its de- general, cloning can be described as a unitary operation tailswillbedescribedinSec.IV. Herewegivetheinput- supported by ancilla systems. The ancilla output sys- output relation. In the following, the quadrature phase tem generally depends on the cloned state, from which amplitudes of each optical mode are denoted by xˆ and anticlones are obtained. We show the equations for pˆ, which correspond to the real and imaginary parts of 1 2 cloning, which corresponds to the experimental the mode’s annihilation operator aˆ, i.e., aˆ=xˆ+ipˆ. The → demonstration in Sec. VI. However, PIA allows general phase factor θ in Eq. (2) can be arbitrarily changed by K L cloning in principle, which will be described in pre- and post-processing of phase rotation of the idler. → Appendix A. Here, the notation K L means that L Therefore, we consider the case of θ = 0 without loss of → clones are created from K identical originals. generality. Explicitly showing the excess noise coming First, we would like to say that pragmatic cloning for from finitely squeezed ancillas,the input-output relation CV is not the universal cloning [14] with respect to the becomes as follows [8]: infinite-dimensionalHilbertspace,becauseitisanunnat- xô1ut=21(cid:0)√1R+√R(cid:1)xî1n+21(cid:0)√1R−√R(cid:1)xî2n−q1−2RxôAut, (3a) uapraplesairtsuawtiitohnetqhuaatlapllrtohbeabstialitteys. iInntgheenneoranlc,otmhepacchtosicpeacoef pô1ut=12(cid:0)√1R+√R(cid:1)pî1n−21(cid:0)√1R−√R(cid:1)pî2n+q1−2RpôBut, (3b) tisheemabppedrodpedriaintetchleoninefirndietep-ednidmsenosniohnoawl Hthilebeinrtfosrpmaaceti.on xô2ut=12(cid:0)√1R+√R(cid:1)xî2n+21(cid:0)√1R−√R(cid:1)xî1n+q1−2RxôAut, (3c) emTbheeddteydpiacsaal dsiitsupalaticoenmeisntthonatsotmheeCquVanitnufmormstaattieonψis. | i pô2ut=12(cid:0)√1R+√R(cid:1)pî2n−21(cid:0)√1R−√R(cid:1)pî1n+q1−2RpôBut. (3d) Hkneorwe,n|ψoir, uwnhkinchowwne. reTfehrent,o tahseasectoroefsptaotses,ibilse eoitrhigeir- The subscripts‘1’and‘2’representthe twomainmodes. nal states is S = Dˆ(x ,p )ψ (x ,p ) R2 , where d d d d { | i| ∈ } They correspond to the signal and idler, though we do Dˆ(x ,p ) exp[ 2i(x pˆ p xˆ)] is the displacement op- d d d d not specify which is which because the relation is sym- erator. [Th≡e prob−ability d−ensity p(x ,p ) is omitted be- d d metric. xÂ and pˆB denote the squeezed quadratures of causeweconsiderforsimplicitythecasewhere(xd,pd)is the two ancilla modes. At the limit ofinfinite squeezing, uniformly distributed.] As a special case of this, the set thesetermsvanish,andthetransformationabovestrictly S becomes all coherent states when the core state ψ is coincides with the optimal PIA. The amplification gain known to be a vacuum state. This way of embeddi|ngi is G is determined via one parameter R, which is a com- found in ordinary CV quantum key distribution (QKD) mon reflectivity of two beamsplitters BS-A and BS-B in protocols [15]. Fig. 1(a), with the relation of For such protocols, the role of cloning is distribution G= 1 1 +√R 2. (4) oftheinformation,ratherthanduplicationofaquantum 4(cid:0)√R (cid:1) state. Therefore, the measure of the cloning precision 4 shouldberelatedtotheestimationof(x ,p ),insteadof For Gaussian cloning of coherent states, the added d d the traditional fidelity. Furthermore, asymmetric cloner noisevariancenk andthefidelityFk havecorrespondence issignificantaswellassymmetriccloner. Arbitraryshare as Fk =1/(1+nk). By using Eq. (7), the upper limit of ratiooftheinformationisachievedbyclonerwithtunable fidelity is obtained for arbitrarily asymmetric Gaussian asymmetry. cloning. In particular, it becomes F = 2/3 for the sym- We suppose a simple picture of cloning where some metriccase. Thisissignificantlyhigherthantheclassical noise is added to the original state as the penalty of limit of F =1/2,where we regardthe limit of state esti- cloning. Then, the quality of cloning is totally deter- mationastheclassicallimitofsymmetriccloningbecause minedbythis noise. Forsimplicity,weimposerotational the estimated state is classicalinformation which can be symmetry on the noise added to each clone. This is nat- copied any number of times. Note that the sameness of urally justified when the core state ψ is either known state estimationand asymptotic cloning where the num- | i tobesymmetricorunknown. Theaddednoiseischarac- berofclonestendstoinfinityisprovenforageneralsetS terized by its variance nk ≡(∆xncloni-ske)2+(∆pncloni-ske)2 [16], ofpossibleoriginalstates[23]. Werefertothesefidelities where k 1,2 for 1 2 cloning. Note that nk corre- only for the consistency with previous works. We stress ∈ { } → sponds to the mean photon number of thermalization in again that our actual interest is the variances. the k-th clone. We have seen above that the clones are made of the The variances nk are directly connected to the mean signal output of the amplifier. When cloning is unitarily square errors in the estimation of (xd,pd). Therefore, a realized, we still have the idler output, whose state is measure can be constructed from them. Given the de- affected by the original state. Now we pay attention to sired asymmetry, the cost function is determined [16]: this ancilla output system. As mentioned at the beginning of this section, anti- C(n ,n )=c n +c n . (6) 1 2 1 1 2 2 clonesarebyproductsofcloningwhichareobtainedfrom The positive parameters ck determine the asymmetry. the ancilla systems. Especially, for 1 2 cloning, the → (c = c corresponds to symmetric cloning.) The cloner idler output itself is ananticlone. It is anapproximation 1 2 that minimize the cost function is the optimal. ofthe phase-conjugatedoriginalstate,orinotherwords, It is obvious that the optimal cloner is Gaussianwhen the output of an approximate NOT gate. Qubit version the cost function is set as a function of the noise vari- of this gate is demonstrated in Ref. [24]. ances as in Eq. (6). In order to minimize it, the ancillas Anticlones are important when we are concerned with that supportcloningare choseninminimum uncertainty the reversibility. The originals can in principle be per- states,whichareGaussian. Thiscontrastswiththeeval- fectlyreproducedonlywhenalltheclonesandanticlones uationby the fidelity. Non-Gaussiancloning canslightly are present. For the reversibility, the essential resource go beyond the Gaussian fidelity for coherent states [17]. is nonclassical correlations, or entanglement. Concep- We emphasize that the evaluation by the variances is tually, the excess noises in cloning is canceled by using more practical. It is pointed out that optimal attack in the nonclassical correlations. Therefore, we discuss the QKD is Gaussian [18, 19]. existence of entanglement in the three-mode output sys- There is a restriction on the excess noises nk which is tem of 1 2 cloning. Clones are obtained by split- imposed by quantum mechanics [20, 21]: ting the am→plified signal, thus there is no entanglement among clones. However, there is entanglement between n n (1/2)2. (7) 1 2 ≥ eachcloneandtheanticlone. Westressthattheresource The optimal cloner with respect to the cost function in for the recovery is not the anticlones themselves but the Eq.(6)satisfiestheequalityinEq.(7)bynecessity. This entanglement. We can obtain anticlones without entan- noise penalty comes from consistency with the uncer- glement with clones as follows. Suppose the situation tainty relation. The attainable information of the origi- wheretwoindependentclonersarerunning,andthesame nal state does not increase by cloning due to this noise. states are used as the inputs for them. Then, the clones Recall that the inevitable noise in PIA comes from the obtainedfromone clonerdo nothaveentanglementwith same reason. the anticlones from the other. In this case, the originals Indeed, the optimal cloner can be constructed from can not be recoveredfrom the noncorrelated outputs. the optimal phase-insensitive amplifier and beamsplit- For the recovery of the originals, the inverse unitary ters. For example, 1 2 cloning with arbitrary asym- operation is not required. The optimal cloning can be → metry is achieved by putting an amplifier in one of the fully reversed by the Bell measurement on a clone and arms of a Mach-Zehnder interferometer [21]. Especially, an anticlone and subsequent feedforward to the remain- for symmetric cloning, the reflectivity of the first beam- ing single clone [25]. Note that this recovery scheme splitter becomes unity, i.e., it is achieved by first am- works not only for coherent states but also for an ar- plifying the original with G = 2 and then splitting the bitrary core state ψ . This scheme is efficient from two | i amplified signalin half. This procedure can be extended pointsofview. Oneisonatechnicallevelthatthehomo- to K L cloning [16, 22]. The optimality of this re- dyne measurements and feedforward displacement oper- → alization is proven with respect to the cost function in ations are quite efficient with current technology. The Eq. (6) [16]. other is on a conceptual level that the performer of the 5 Bell measurement and the owner of the remaining clone respectively. The subscript ‘idl’ denotes the idler input who is willing to recover the original can be spatially forPIAwhichisinavacuumstate. Theannihilationop- separated. For this case, they only need classical chan- erators with the subscript ‘vac’ indicate another ancilla nels for communication, and never quantum channels. in a vacuum state, which invades from the empty port Note that even partial reversalis possible with a similar of the final half beamsplitter. For the excess noise of the scheme based on local operations and classical commu- two clones, n = n = 1/2 is easily checked. Therefore, 1 2 nication(LOCC), which converts,e.g., symmetric clones this cloner is optimum when evaluated by the cost func- to asymmetric clones [25]. tioninEq.(6)withc =c . WhenPIAisrealizedwitha 1 2 We would like to discuss practical aspects of cloning feedforward-basedscheme as is found in Sec. VI, further and its reversibility assisted by classical communication. excessnoise contaminatesthe output in accordancewith As described above, cloning of a quantum state is re- the squeezing levels of the ancillas. garded as distribution of information among plural par- ticipants. The information of the original is to some ex- tend accessibleto individualparticipants. This situation isclearlydistinguishedfromthatfoundinusualquantum IV. EXPERIMENTAL SETUP errorcorrectingcodes where the quantuminformation is mapped onalargerHilbert spaceso thatno information about the original is accessible from a localized system. Schematic of the experimental setup for PIA is illus- Such share of information would play important roles in trated in Fig. 1(a), and that for approximate cloning is several scenarios, in which the reversibility would give a illustratedin Fig. 1(b). The light source is a Ti:sapphire tactical aspect to information exchange. For example, laser, which has a continuous-wave single-mode output cloning is a possible attack by an eavesdropperin QKD. of 860 nm in wavelength and about 1.5 W in power. We In this example, the reversibility of cloning provides the treat the quantum states of narrowsidebands located at opportunityforthecommunicatorstonegotiatewiththe 1.34 MHz apart from the optical carrier frequency. eavesdropper when they know the attack [25]. Since co- Two main beams that go from in-1 and in-2 to out-1 herent states are a strong candidate for the information and out-2 in Fig. 1(a) carry the quantum states which carrier in quantum communication, cloning of coherent are targets of PIA. The setup has a form of a Mach- states is especially of great significance. Zehnderinterferometerthatholdsasingle-modesqueezer There are several experimental previous works which (squeezer-A or squeezer-B) in each arm. This decompo- demonstrate cloning of coherent states beyond the clas- sition of unitary PIA into squeezers andbeamsplitters is sical limit of F = 1/2 in non-reversible ways, i.e., their derivedfromthebosonicversionofBloch-Messiahreduc- anticlones are lost in the environment. In Ref. [26], us- tionshowninRef.[28]. Wenotethatthissetupisalmost ing feedforward-based PIA of Ref. [5], almost quantum- the same as that for quantum nondemolition (QND) in- limited 1 2 cloning is demonstrated. In Ref. [27], → teraction demonstrated in Ref. [29]. This fact shows the telecloning is demonstrated where the original coherent capabilityofoursetuptorealizemanytypesoftwo-mode state is teleported and cloned at the same time. Gaussian interaction. Combining PIA for G = 2.0 with InSec.VI,wedemonstrate1 2symmetricGaussian → anotherhalfbeamsplitterasis showninFig.1(b),1 2 cloner which preserves an anticlone at the output. As → approximate cloning of coherent states is achieved. is shown in Fig. 1(b), we apply a half beamsplitting to the signal output of the feedforward-basedPIA with the Squeezer-A and squeezer-B are feedforward-based gain G = 2 described in Sec. II. In the demonstration, squeezers,whicharetheoreticallyproposedinRef.[8]and the reversibility is checked from the output correlations. experimentally demonstratedin Ref. [11]. Eachsqueezer Toourknowledge,thereisnopreviousexperimentofthis consumes an ancilla in a squeezed state, which is gener- kind even in qubit regime. Although our demonstration ated by an optical parametric oscillator (OPO). isonlyforcoherentstates,ourclonershouldequallywork Note that several essential optical elements are omit- for arbitrary core state ψ as discussed above. ted from Fig. 1, such as a second harmonic generation | i We close this section by giving the input-output rela- (SHG) cavity to generate pump beams for OPOs, and tion of the optimal 1 2 symmetric cloning. By substi- three spatial-mode cleaning cavities (MCCs). One MCC → tuting G=2 and θ =0 into the input-output relationin is used for local oscillators (LOs) for homodyne mea- Eq.(2)andsplittingthesignaloutputinhalf,weobtain, surements and auxiliary beams for feedforwarddisplace- ments. TheothertwoMCCsareusedforindividualinput aˆcln-1 =aôrg+ √12aˆ†idl+ √12aˆvac, (8a) beams. aˆcln-2 =aôrg+ √12aˆ†idl− √12aˆvac, (8b) FirTsthley,exwpeeripmreepnatarelpirnopcuedtucroehiserdeinvtidsetdaitnetsoathnrdeeasntceipllsa: aˆ =aˆ +√2aˆ , (8c) squeezed vacuum states. Secondly, we implement PIA a-cln †org idl and cloning via feedforward. Finally, the output states where the subscripts ‘org’, ‘cln-1’, ‘cln-2’, and ‘a-cln’ de- arehomodynemeasuredforverification. Inthefollowing, notetheoriginal,firstclone,secondclone,andanticlone, we describe the experimental details of each step. 6 Phase Insensitive Amplifier Phase Insensitive Amplifier Squeezer-A Verification Squeezer-A Verification In-1 LO M LO Original LO M LO EO OPO EO EO OPO EO M 99:1 M 99:1 Clone-1 EOM BS-A Out-1 EOM BS-A HBS-C Clone-2 HBS-P HBS-F HBS-P HBS-F LO Out-2 OM BS-B BS-B AntiClone E 99:1 99:1 M O E In-2 OPO LO EOM LO Idler OPO LO EOM LO Squeezer-B Squeezer-B (a)Setup forPIA. (b)Setup for1→2cloner. FIG. 1: Schematics of experimental setups. OPO: Optical parametric oscillator. EOM: Electro-optic modulator. LO: Local oscillator. HBS-P,HBS-F,HBS-C:Halfbeamsplitter. BS-A,BS-B:BeamplistterwithreflectivityR≈0.17. Fourbeamsplitters (HBS-P, HBS-F, BS-A, and BS-B) are variable, composed of two polarization beamsplitters and one half-wave plate. A. Preparation second harmonic of a fundamental beam, generated by a SHG cavity. Most of the Ti:sapphire laser output is sent to the SHG cavity, whose output of about 300 mW At this step, we generate coherent states which are is divided into two to pump the individual OPOs. The used as inputs, and squeezed vacuum states which are SHG cavityhasalmostthe same configurationas thatof used as ancillas. OPOs,whereasaKNbO (KN)crystalisusedinsteadof The nonzero mean values of the sideband coherent 3 the PPKTP crystal. states at 1.34 MHz are produced by appropriately mod- Modulation sidebands other than 1.34 MHz are ex- ulating the optical carriers. In our setup, the relative ploitedforactivefeedbackcontrolofopticalinterferences. phase of interference at each beamsplitter is designed to Amodulationat13.5MHzisutilizedforlockingcavities, befixedwithactivefeedbackcontrol. Therefore,inorder including the SHG cavity, the two OPOs, and the three tomakeanarbitraryphasespacedisplacementinthein- MCCs. Onthe otherhand, lowerfrequency modulations put modes, both amplitude modulation (AM) and phase at193kHz and333kHz areutilizedatthe OPOsto lock modulation (PM) are utilized. AM and PM make nonzero mean values of xîn and pîn for the first input mode, thephasesofthe pumpbeams. Furthermore,the twoin- 1 1 and those of pîn and xîn for the second input mode, re- putbeamsaremodulatedat108kHzand154kHz. These 2 2 four low-frequencymodulations contribute to the lock of spectively. Eachoffourelectro-opticmodulators(EOMs) thedownstreaminterferometricsysteminthesubsequent before PIA in Fig. 1(a) corresponds to one of these four steps, as mentioned later. quadratures. On the other hand, in Fig. 1(b), only two EOMsaredepictedbeforePIAwhicharebothlocatedat the first input beam path. Therefore, the symmetry of thetwoinputmodesisbrokeninthecloningexperiment. B. Amplifier and Cloner One input mode is the target of cloning, and the other input mode is set in a vacuum state throughout. For The two input beams are combined at a preceding both experiments, the modulations are switched on and half beamsplitter (HBS-P) and then sent to squeezer-A offinordertouseseveralcoherentstatesasinputs. After and squeezer-B. After the squeezing operations, the these EOMs, there are MCCs though they are omitted two beams interfere again at another half beamsplitter from Fig. 1. (HBS-F), which completes PIA. By splitting one of the Squeezed vacuum states are each generated by an twooutputbeamsbyanotherhalfbeamsplitter(HBS-C), OPO which is driven below the threshold. Our OPO 1 2 cloner is obtained. → has a bow-tie shaped configuration with a round-trip The squeezing procedure goes as follows. First, the length of about 500 mm. It contains a periodically- main beam is combined with an ancilla beam coming poledKTiOPO (PPKTP)crystalas a nonlinearoptical from an OPO at a beamsplitter (BS-A or BS-B). Next, 4 medium,whichiscommerciallyavailablefromRaicoland one of the two beams after the beamsplitter is homo- has10mmlengthand1mmby1mmcrosssection. The dyne measured. Finally, the measurement outcome is experimental datails of our OPO squeezing are found in fedforwardtotheremainingbeam. BS-AandBS-Bhave Ref. [30]. The squeezing level with the pump of about the common reflectivity of R. This parameter R deter- 100 mW is about 5 dB relative to the shot noise level mines the degreeofthe feedforward-basedsqueezingand − at 1.34 MHz. The pump beams for the OPOs are the thus the gain of amplification G with the relation shown 7 in Eq. (4). As is already mentioned, R 0.17 for our signalfromahomodynedetectorismixedwiththerefer- ≈ demonstration of G=2.0. ence signal at 1.34 MHz, and then low-pass filtered with The feedforward operation is a phase space displace- the cutoff of 30 kHz. Subsequently it is analog-to-digital ment whose amount is proportional to the random out- (A/D) converted for storage with the sampling rate of come of the homodyne measurement. The electric sig- 300kHzandtheresolutionof14bits(PXI-5122,National nal from the homodyne detector is sent to an EOM to Instruments Corporation). In this analysis, the phase of be converted into an optical signal, where the gain and thehomodynedetectionisslowlyscanned. Thephasein- phase at 1.34 MHz are carefully chosen. The auxiliary formation is stored simultaneously with the quadrature beam which is modulated by the feedforward EOM has values using the same A/D board. From the resulting the power of 150 µW, 1% of which subsequently enters marginal distributions, phase space distributions (i.e., the mainstream via an asymmetric beamsplitter (99:1). Wigner functions) are reconstructed, where we assume The powers of the two input beams are 10 µW, and that all the quantum states obtained in the experiments those of the two ancilla beams are 2 µW. These pow- are Gaussian. The first and second moments are com- ers are considerably smaller than 3 mW of LOs used for puted so that the likelihoods are maximized. homodyne detections. The other way is the power analysis at 1.34 MHz us- ThefourbeamsplittersofPIA(HBS-P,HBS-F,BS-A, ing a spectrum analyzer. In this analysis, the measured and BS-B) are actually composed of two polarization quadraturesaresettoeitherxôrpˆ. Notonlythe powers beamsplittersandahalf-waveplate inthe samemanner ofthe outputquadraturesbutalsothoseoftheir correla- as the QND experiment in Ref. [29]. Their reflectivities tionsaremeasuredforseveralinputcoherentstates. The arevariablebyrotatinghalf-waveplates. Theyenableus resolution bandwidth is 30 kHz, the video bandwidth is to measure the input states as well as the output states 300 Hz, the sweep time is 0.1 s, and 20 times averaging with the same homodyne detectors for verification. The is taken for each trace. propagationlossesoftwomainbeamsaremeasuredtobe Note thatwe caneasilysee the effect ofthe Hermitian 7% on average, which mostly come from these variable conjugate term in Eq. (2b) as a mirror image with the beam splitters. former way of analysis, whereas we can not do this with Inordertocontroltherelativephasesatbeamsplitters the latter. with active feedback, interferences between the carriers and the low-frequency modulations are monitored. This is typically done by picking up 1% of the beam after the V. EXPERIMENTAL RESULTS FOR interference,thoughsuchdetailsareomittedfromFig.1. PHASE-INSENSITIVE AMPLIFIER For each locking point, an appropriate modulation sideband is chosen, and the error signal is extracted from The two main modes are denoted by “mode-1” and the interference between the carrier and the sideband “mode-2”. One of them is the “signal” and the other is by demodulation. However, HBS-P and HBS-F are ex- the “idler”, which are initially in a coherent state and a ceptions,wheretheinterferencebetweentwomodulation vacuum state, respectively. By swapping the role of the sidebands, namely 108 kHz and 154 kHz, is exploited. signal and idler, we check the symmetry of our PIA. The beat frequency of 46 kHz is chosen for the reference We first show the results of the lock-in detection, be- signal of demodulation to obtain the error signals. cause it is intuitively easier to see. Figs. 2 and 4 show the experimental quadrature values at various phases of LOs. ForFig.2,themode-1isthesignalandthemode-2 C. Verification is the idler, whereas for Fig. 4, the mode-2 is the signal and the mode-1 is the idler. There are three sub- PIA is characterized by measuring two-mode input figures corresponging to the signal input (a), the signal states as well as two-mode output states using two ho- output (b), and the idler output (c). Horizontal axes modyne detections. In the cloning experiment, on the are the measurementphases φ which are scanned from 0 otherhand,three-modeoutputstatesarecomparedwith to 2π. The quadrature at φ = 0 corresponds to xˆ and single-mode input states. The input states are measured that at φ = π/2 does to pˆ. Vertical axes are normal- bysettingthereflectivitiesofthefourvariablebeamsplit- ized quadrature values where the standard deviation of ters to unity and disabling the feedforward. The quan- vacuum fluctuation is 0.5. Each set of data is taken for tumefficiencyofahomodynedetectorisabout99%,and about0.2seconds. Quadraturedataareplottedevery10 thedarknoiseisabout17dBbelowtheopticalshotnoise pointsinthefigures,whereasthewholedataareusedfor produced by the LO. The interference visibilities to the theanalysis. Thesinusoidalcurveofthesignalinputrep- LOs are 98% on average. resents the nonzero mean amplitude of a coherent state, Theoutcomesofthefinalhomodynemeasurementsare and the fluctuation around the sinusoid represents the analyzed in either of the two ways below. quantum noise. With regard to the fluctuation, it grows In one way of analysis, the quadrature data are di- uniformlyatboththesignalandidleroutputs. Thisuni- rectly treated, which are obtained by lock-in detection formity is an evidence of the phase-insensitivity of our of 1.34 MHz components of the homodyne outputs. A amplifier. On the other hand, with regard to the sinu- 8 4 4 e 2 e 2 d d u u plit 0 plit 0 m m A-2 A-2 -4 -4 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 Phase (π rad) Phase (π rad) (a)Signal input. (a)Signal input. 4 4 4 4 e 2 e 2 e 2 e 2 d d d d u u u u plit 0 plit 0 plit 0 plit 0 m m m m A-2 A-2 A-2 A-2 -4 -4 -4 -4 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 Phase (π rad) Phase (π rad) Phase (π rad) Phase (π rad) (b)Signaloutput. (c)Idler output. (b)Signal output. (c)Idler output. FIG.2: Quadraturedata. Mode-1for signal andmode-2for FIG.4: Quadraturedata. Mode-2for signal and mode-1for idler. The phases of homodyne measurements are scanned idler. The phases of homodyne measurements are scanned from 0 to 2π, which correspond to horizontal axes. Vertical from 0 to 2π, which correspond to horizontal axes. Vertical axes are quadraturevalues. axes are quadraturevalues. 3 3 3 3 2 2 2 2 1 1 1 1 p 0 p 0 p 0 p 0 -1 -1 -1 -1 -2 -2 -2 -2 -3 -3 -3 -3 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 x x x x (a)Experiment. (b)Theory. (a)Experiment. (b)Theory. FIG.3: Phasespacedistributionscomputedfromthequadra- FIG.5: Phasespacedistributionscomputedfromthequadra- turedatainFig.2. ThefirstandsecondmomentsofGaussian turedatainFig.4. ThefirstandsecondmomentsofGaussian Wigner functions are represented by ellipses. Green: Signal Wigner functions are represented by ellipses. Green: Signal input. Red:Signal output. Blue: Idler output. input. Red: Signal output. Blue: Idler output. soidalcurve,thetwooutputmodesshowdifferentbehav- arethreeellipses. Theellipseingreenisthesignalinput. iors. At the signal output, the amplitude of the sinusoid Itsradiusisalmost0.5whichcorrespondstothestandard is amplified from that of the signal input, maintaining deviationofvacuumfluctuation. Theellipseinredisthe the phase. At the idler output, the amplitude of the si- signal output. Its center is about √2 times farther away nusoidisthesameasthesignalinput, whereasthephase from the origin and its radius is about √3 times larger is flipped. This flip is due to the Hermitian conjugate than those of the signal input. The ellipse in blue is the term in Eq. (2b). For both figures, the same qualitative idler output. Its radius is almostthe same as that of the behaviors as mentioned above are observed. signaloutput, whereasits centeris flipped aroundthe x- Figs.3and5arephasespacediagrams,whicharecom- axesfromthatofthesignalinput,whichagainrepresents puted from the quadrature data shown in Figs. 2 and 4, the Hermitian conjugate term in Eq. (2b). respectively. The experimental results (a) and the theo- In principle, we can fully characterize our PIA with retical calculations for the optimal PIA (b) are depicted only the above way of analysis that treats quadrature next to each other. In the theoretical calculations, the valuesdirectly. However,suchtreatmentrequiresalarge experimental value is used for the amplitude of the sig- amount of data for good accuracy. Thus, in the follow- nal input. The first and second moments are expressed ing, we resort to power measurements using a spectrum by ellipses, which correspondto the cross sections of the analyzer. Notonlyoutputquadratures(showninFigs.6 Wignerfunctions. Notethatthetheoreticalellipsesin(b) and 7) but also their correlations (shown in Figs. 8, 9, are strictly circles. In each phase space diagram, there and10)aretakenforseveralinputstates. Ineachfigure, 9 xout pout xout pout xin xout pout xout pout 8 1 1 2 2 16 1 1 1 2 2 6 12 dB) 4 dB) 8 er ( er ( ow 2 ow 4 P P 0 0 -2 -4 (a)hxîni=6 0. 1 FIG.6: Outputpowersforvacuuminputs. Verticalaxesare powers in dB scale normalized by shot noises. Blue: Output pin xout pout xout pout quadratures. Cyan:Shotnoises. Red:TheoryforoptimalPIA 16 1 1 1 2 2 outputs. Magenta: Theory for our PIA outputs with −5 dB 12 squeezed ancillas. Green: Theory for our PIA outputs with vacuum ancillas. dB) 8 er ( ow 4 P results of eachquadrature are contained in one of boxes. 0 VerticalaxesarepowersindBscalewhicharenormalized by corresponding shot noises. -4 Fig. 6 shows the experimental results for vacuum in- (b)hpîni=6 0. puts (fluctuating traces), together with their theoretical 1 expectations(straightlines). Therearefourboxescorre- xin xout pout xout pout sponding to the four output quadratures, namely, xôut, 16 2 1 1 2 2 1 pôut, xôut, and pôut. The traces in blue are the powers 1 2 2 12 of the output quadratures. The traces in cyan around 0 dB are the powers of the shot noises, which are used dB) 8 fornormalization. Sincetheinputsareinvacuumstates, er ( the powers of the shot noises correspond to those of the ow 4 P input quadratures xîn, pîn, xîn, and pîn. We put three 1 1 2 2 0 kinds oftheoreticallines correspondingto threedifferent conditions. For the optimalPIAofacoherentstate with -4 G = 2.0, the output quadrature variances become three (c)hxîni=6 0. times larger than the initial shot-noise-limited variance, 2 where two from amplification and one from contamina- pin xout pout xout pout tion by the other mode. The corresponding 4.8 dB is 16 2 1 1 2 2 marked by the lines in red. Our PIA with finite ancilla 12 squeezing is suffered from further excess noise. Assum- ing−5dBofsqueezingfortheancillastates,wecalculate dB) 8 theoretical values which are marked by the lines in ma- er ( genta. We also show them with vacuum ancillas by the ow 4 P lines in green. The lines in red, magenta, and green are 0 very close to each other, thus other experimental errors are dominant rather than the ancilla squeezing levels in -4 these results. Therefore, it is hard to discuss nonclassicality of our PIA only from these results. As is shown (d)hpî2ni=6 0. later,the effects ofancilla squeezingmoreclearlyappear in the output correlations. FIG.7: Outputpowers for inputsin severalcoherent states. One of four input quadratures xîn, pîn, xîn, pîn is excited, at Next we use severalcoherent states as inputs. The re- 1 1 2 2 thesametimetheotherthreequadraturesareleftatthevac- sults are shown in Fig. 7. The four input quadratures uumlevel. VerticalaxesarepowersindBscalenormalizedby xîn, pîn, xîn, and pîn are displaced from zero mean val- 1 1 2 2 shot noises. Magenta: Excited input quadratures. Red: Out- ues by turns, leaving the other three quadratures at the put quadratures with excitation in inputs. Blue: Output vacuum level. There are four subfigures labeled from(a) quadratureswithout excitation in inputs. Cyan: Shot noises. to (d) corresponding to such four excitations. For each subfigure, there are five boxes. The trace in magenta in the leftmost box shows the measured power of the ex- 10 citedinputquadrature. Theotherfourboxescorrespond x p to the four output quadratures, namely, xôut, pôut, xôut, 10 1 1 2 and pôut. The traces in red show the output quadrature 8 2 6 powers with the input excitation. They are compared to 4 those without the excitation shown by the blue traces, dB) 2 which are replottings of the blue traces in Fig. 6. The er ( 0 obtained results show the following features. When a ow -2 quadrature xˆ or pôf an input mode is excited, the same P -4 -6 quadratures of both of the output modes are excited, -8 whereas the conjugate quadratures do not change from -10 the nonexcited levels. The two increased output powers differ by about 3.0 dB, where the larger one corresponds FIG. 8: Two-mode squeezing and antisqueezing for vacuum to the amplified signal and the smaller one corresponds inputs. Vertical axes are powers in dB scale normalized by to the phase-conjugatedidler output. These features are summed shot noises of two homodyne detections. Lower exactly what are expected from Eq. (2) for G = 2 and Blue: Two-mode squeezing in xôut − xôut and pôut + pôut. 1 2 1 2 θ = 0. Note that the coefficients √2 correspond to the Upper Blue: Two-mode antisqueezing in xôut + xôut and 1 2 3.0 dB. pôut −pôut. Cyan: Summed shot noises. Lower Red: The- 1 2 TheresultsinFigs.6and7areonlyforthefivespecific ory of two-mode squeezing for optimal PIA outputs. Upper Red:Theoryoftwo-modeantisqueezingforoptimalPIAout- input states. However, the results for other input states puts. Magenta: Theory of two-mode squeezing for our PIA can be predicted on the assumption of linearity. More outputswith−5dBsqueezedancillas. Green:Theoryoftwo- precisely speaking, the absolute values of the coefficients mode squeezing for our PIA outputswith vacuum ancillas. of xîn, pîn, xîn, and pîn in Eq. (2) are determined from 1 1 2 2 these results. On the other hand, the signs of the coefficients are not determined from them. However,they are put modes, we investigate the reversibility of our PIA. checked from the phase space diagrams shown in Figs. 3 For this purpose, we virtually realize the inverse trans- and 5. In the above sense, the results shown so far give formationelectricallyand reconstructthe initial quadra- fullinformationoftheinput-outputrelationwhenoutput tures. Neglecting the excess noise from finite ancillas, modes are separately concerned. PIA that we demonstrate has the input-output rela- Inordertofullycharacterizeouramplifier,theindivid- tion as aôut = √2aîn + (aîn) , aôut = √2aîn + (aîn) , 1 1 2 † 2 2 1 † ualbehaviorsoftheoutputmodesarenotenough. Inthe which is obtained by substituting G = 2 and θ = 0 following,weareconcernedwiththeoutputcorrelations. into Eq. (2). The inverse transformation becomes as SinceunitaryPIAisequivalenttotwo-modesqueezing, aôut = √2aîn (aîn) , aôut = √2aîn (aîn) , or equiva- 1 1 − 2 † 2 2 − 1 † the two output modes should be entangled, and have an lently, EPR type of correlation. The results of EPRcorrelation are shown in Fig. 8. Here the two input modes are both xôut =√2xîn xîn, xôut =√2xîn xîn, (9a) 1 1 − 2 2 2 − 1 in vacuum states. There are two boxes corresponding to pôut =√2pîn+pîn, pôut =√2pîn+pîn. (9b) x and p correlations. The lower traces in blue show the 1 1 2 2 2 1 two-modesqueezingofxôut xôut andpôut+pôut,onthe Therefore, by adding or subtracting the two homo- 1 − 2 1 2 other hand, the upper traces in blue show the two-mode dyne outcomes with 3.0 dB difference of gains, the ini- antisqueezingof xôut+xôut and pôut pôut, respectively. tial quadratures are reconstructed. The reconstructed 1 2 1 − 2 They are compared with the summed shot noises of the quadraturesare denotedby the superscripts“rec”in the two homodyne detections shown by the traces in cyan. following. Note that the initial quantum state is not re- Several theoretical lines are plotted together. The lower covered in the experiment. In addition, note also that and upper lines in red are the theoretical values of two- onlyoneoftwoquadraturesxôrpˆcanbereconstructedin mode squeezing and antisqueezing for the optimal PIA, eachmoment,andneverbothsimultaneously. Therecov- respectively. Our results of two-mode squeezing are de- eryoftheinitialstateispossibleonlywheneitheraquan- gradedfromtheidealcaseduetofinitesqueezingofancil- tum channel is between the signal and idler outputs [25] las. Assuming 5 dB squeezing for ancillas, theoretical or linear pre-processing is applied [10]. However, the − expectation is marked by the lines in magenta. That demonstrated reconstruction of the initial quadratures for vacuum ancillas is also marked by the lines in green, can show that necessary correlations for the recovery of whichisexactlyequaltotheshotnoiselevel. Incontrast, the initial state are present. thetheoreticaltwo-modeantisqueezingisalwaysidealfor In Fig. 9, the results of such reconstruction of initial arbitrary ancillas. The experimental results well agree vacuumfluctuationsareshown. Therearefourboxescor- with the theory assuming 5 dB squeezing of ancillas. responding to the four reconstructed quadratures xˆrec, − 1 Since the lower traces in blue are both below the traces pˆrec, xˆrec, and pˆrec. The powers of the reconstructed 1 2 2 incyan,existenceofentanglementisverifiedbetweenthe quadratures are shown by the traces in blue. The traces twooutput modes via the Duan-Simon criterion[31,32]. in cyan are the powers of the summed shot noises of the Fromthenonclassicalcorrelationbetweenthetwoout- twohomodynedetections,whicharetakenwiththesame