Geeratio of picosecod pulsed coheret state superpositios Ruifag Dog, Aders Tipsmark, Amie Laghaout, Leoid A. Krivitsky, Miroslav Ježek ad Ulrik L. Aderse Departmet of Physics, Techical Uiversity of Demark, Fysikvej, 8 Kgs. Lygby, Demark Quatum Frequecy Stadards Divisio, Natioal Time Service Ceter (NTSC), Chiese Academy of Scieces, 16 Litog, Shaaxi, Chia Data Storage Istitute, Agecy for Sciece Techology ad Research (A*STAR), 68 Sigapore Departmet of Optics, Palacký Uiversity, 1. listopadu 1, 16 Olomouc, Czech Republic Abstract: We preset the geeratio of approximated coheret state superpositios - referred to as Schrödiger cat states - by the process of subtractig sigle photos from picosecod pulsed squeezed states of light at 8 m. The squeezed vacuum states are produced by spotaeous parametric dow-coversio (SPDC) i a periodically poled KTiOPO crystal while the sigle photos are probabilistically subtracted usig a beamsplitter ad a sigle photo detector. The resultig states are fully characterized with time-resolved homodye quatum state tomography. Varyig the pump power of the SPDC, we geerated differet states which exhibit o-gaussia behavior. 1 Optical Society of America Refereces ad liks 1. L. A. Wu, H. J. Kimble, J. Hall ad Huifa Wu, "Geeratio of Squeezed States by Parametric Dow Coversio", Phys. Rev. Lett. 5, 5 5(1986), http://lik.aps.org/doi/1./physrevlett.5.5.. H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Fraze, N. Lastzka, S. Goßler, K. Dazma ad R. Schabel, "Observatio of Squeezed Light with 1-dB Quatum-Noise Reductio", Phys. Rev. Lett. 1, 6(8), http://lik.aps.org/doi/1./physrevlett.1.6.. Y. Takeo, M. Yukawa, H. Yoezawa, ad A. Furusawa. "Observatio of -9 db quadrature squeezig with improvemet of phase stability i homodye measuremet", Opt. Express, 1(), http://www.opticsifobase.org/abstract.cfm?uri=oe---1.. R. Dog, J. Heersik, J. F. Corey, P. D. Drummod, U. L. Aderse ad G. Leuchs, "Experimetal evidece for Rama-iduced limits to efficiet squeezig i optical fibers", Opt. Lett., 6 8(8), http://www.opticsifobase.org/abstract.cfm?uri=ol---6. 5. M. G. A. Paris, "Displacemet operator by beam splitter", Phys. Lett. 1, 8 8(1996), http://likighub.elsevier.com/retrieve/pii/59619698. 6. S. Lloyd, "Coheret quatum feedback", Phys. Rev. A 6, 18(), http://lik.aps.org/doi/1./physreva.6.18.. S. Scheel, W. Muro, J. Eisert, K. Nemoto ad P. Kok, "Feed-forward ad its role i coditioal liear optical quatum dyamics", Phys. Rev. A, 1(6), http://lik.aps.org/doi/1./physreva..1. 8. H. M. Wisema ad G. J. Milbur, "Quatum theory of optical feedback via homodye detectio", Phys. Rev. Lett., 58 551(199), http://lik.aps.org/doi/1./physrevlett..58.
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produced [1,,, ]. Moreover, Gaussia projectors ca be implemeted usig homodye detectio, which is capable of reachig ear-uity detectio efficiecy [1]. Fially, Gaussia displacemet operatios combied with low-oise liear feedback have bee implemeted with high quality [5, 6,, 8]. This progress has lead to the implemetatio of various Gaussia protocols such as quatum teleportatio [9], quatum key distributio [1], quatum cloig [], quatum secret sharig [1] ad quatum computatio [1, 1]. However, several o-go theorems exist for systems cosistig of purely Gaussia states ad Gaussia operatios. With this costraied set of states ad operatios it is impossible to perform etaglemet distillatio [, 16, 1], quatum error correctio [18], uiversal quatum computig [19, ], quatum bit commitmet [1], ad to violate Bell s iequality []. To realize these protocols, o-gaussia approaches are required. This o-gaussiaity ca be ijected ito the system at differet stages. It ca eter through a o-gaussia measuremet strategy [,, 5], o-gaussia oise characteristics [6], or it ca be icorporated through a o-gaussia state preparatio strategy [, 8, 9,, 1, ]. Importat examples of a pure o-gaussia state are the photo umber eigestates, the Fock states,(=1,...). Such states have bee prepared ad fully characterized i optical systems usig SPDC followed by a o-gaussia heraldig measuremet [,, 5, 6,, 8]. Aother family of o-gaussia states, which has gaied much iterest i recet years, are the Schrödiger cat states which are superpositios of two coheret states of differet phase, α ± α. Despite the costituets beig Gaussia, the superpositio exhibits strog o-gaussiaity which is sufficiet for the realizatio of various protocols, examples beig the realizatio of quatum computatio[, 9,, 1], error correctio of Gaussia oise[] ad the violatio of Bell s iequality [1]. It has bee demostrated that such coheret state superpositios (CSS) with a moderate amplitude α 1 ca be well approximated by a photo-subtracted squeezed state [], or equivaletly, a squeezed sigle-photo state. Fidelities betwee the ideal CSS state ad the photo-subtracted squeezed state as a fuctio of the excitatio, α, of the CSS ad the degree of squeezig of the squeezed state are show i Fig. 1. The moderate amplitude of the CSS ca be o-determiistically amplified to a larger amplitude CSS by meas of liear iterferece ad heraldig based o photo coutig or homodye detectio evets [8,,, 5]. A large CSS ca also be prepared through a coditioal homodye measuremet o a Fock state i which the amplitude of the CSS state scales with the umber of photos i the Fock state [6]. Ispired by these ideas for the geeratio of CSS ad motivated by the potetial applicatios, various groups have realized photo subtractio with squeezed states. These implemetatios have bee carried out either with cotiuous-wave (CW) or pulsed light sources, ad with wavelegths ragig from the ear-ifrared to the telecommuicatio regime. The photo subtractio has bee carried out usig a asymmetric beamsplitter that reflects a small portio of the light i which a photo is measured ad thus subtracted from the squeezed state. The measuremet has bee realized with sigle-photo avalache photodiodes (APD) as well as with photo-umber-resolvig trasitio edge sesors [5, 6,, 8, 9, 5, 51, 5]. The largest directly measured value of the Wiger fuctio egativity is. ad it is obtaied usig cotiuous wave (CW) squeezed states [5]. Usig pulsed istead of CW squeezed light, the reported egativities as well as the purities of the geerated o-gaussia states are much lower. However, despite the lower quality of the geerated states, there has bee much iterest i pulsed experimets due to the relative simplicity
Fig. 1. Fidelity betwee a photo-subtracted squeezed state ad a ideal coheret state superpositio, α α for differet degrees of squeezig. It ca be see that the fidelity remais high, F>.9, for α up to 1, provided the squeezig degree is ot too large. of the experimetal setup ad the iheret temporal cofiemet of the geerated states. Previous pulsed experimets o geeratig CSS have employed femtosecod pulsed lasers ad very short (sub-millimeter log) oliear crystals [5, 5]. Here we preset experimets o CSS geeratio usig picosecod laser pulses ad a -mm log quasi-phase matched PPKTP crystals for squeezed light productio. By usig picosecod istead of femtosecod pulses, the group velocity dispersio is small ad thus a loger crystal for squeezed light productio could be used.. Experimet A CSS state with a small amplitude ca be approximated by a photo subtracted squeezed state, ad it ca be fully characterized by meas of its Wiger fuctio which is obtaiable by homodye tomography. I the followig we preset the differet parts of our experimetal setup to geerate ad characterize a CSS. We itroduce the two required parametric processes (up-coversio ad dow-coversio), the photo subtractio setup, ad fially the homodye detector. We also briefly discuss a simple model for predictig the performace of the experimet. The experimetal setup is show i Fig.. We used a cavity-dumped Titaium- Sapphire pulsed laser, which produced early Fourier-trasform-limited pulses with duratio.6 ps at 8 m with a average eergy up to J ad a repetitio rate of 8 khz. The spectral width of the pulses was measured to be.16 m, correspodig to a spectral width of GHz with a ceter wavelegth of 89. m. A fractio of about 1% was used as a local oscillator (LO) for homodye detectio, a weak seed beam was directed to the parametric dow-coversio process for aligmet ad the remaiig
Laser APD Heraldig output Data processig ABS1 SHG PPKTP1 Seed DM IF BF OPA RF PPKTP SMF PBS1 ABS LO HWP PBS Fig.. Schematic of the experimetal setup. The cavity dumped laser emits.6 ps optical pulses at 8 m with a repetitio frequecy of 8 khz. ABS1: 9/1 beamsplitter, BF: blue filter, DM: dichroic mirror, RF: red filter, ABS: Asymmetric beamsplitter with R =.%, SMF: sigle-mode fiber, IF: iterferece filter, APD: avalache photodiode, PBS: polarizig beamsplitter, HWP: half-wave plate. part was directed to a frequecy doublig process..1. Frequecy doublig For frequecy doublig a -mm log periodically poled KTP crystal (PPKTP1) was used. The crystal polig-period was chose for 1st order quasi phase-matchig correspodig to a polig period of Λ.8 μm with all fields polarized alog the crystal z-axis. The legth of the crystal was chose to be mm which was a compromise betwee havig a large iteractio legth ad avoidig phase mismatch as a result of group velocity dispersio. SHG efficiecy, [%] 5 5 1 5 Efficiecy (measured) Efficiecy (Fit) Secod harmoic output power 5 1 5 5. Iput power, [mw]. 1.5. 1.5 1..5 5..5 Secod harmoic power, [mw] Fig.. (a) Blue diamods correspod to the secod-harmoic geeratio coversio efficiecy for differet iput powers of the fudametal beam, P F. These poits are fitted with η = η tah (g P F )withη = 5% ad g =.18. Red triagles are the total geerated SHG power for differet power levels of the fudametal beam. The beam waist was set to w 9 μm, thus achievig a weak focusig coditio
with the depth-of-focus (z 6 mm) beig times loger tha the crystal legth [5, 55]. The secod harmoic coversio efficiecy, η was ivestigated as a fuctio of iput power of the the fudametal beam ad the result is displayed i Fig.. A maximum frequecy doublig efficiecy of % was achieved for a icidet average power of mw. The spectral properties of the resultig frequecy doubled light was ivestigated by a wavelegth meter ad it was measured to have a ceter wavelegth of 1.8 m ad a badwidth of.1 m, correspodig to a spectral width of about 1 GHz... Parametric dow-coversio After the frequecy doublig crystal, the residual light at 8 m was filtered out usig a series of filters. The filtered blue light was the focused ito a secod PP- KTP crystal (PPKTP) used to geerate squeezed vacuum through the SPDC. It has bee show that for sigle pass pulsed SPDC experimets the gai-product g mi g max, where g mi (g max ) is the atteuatio(amplificatio) factor is ehaced by defocussig the pump [55, 56]. This was cofirmed i our setup, ad the waist of the pump was set to w,p μm ad a depth-of-focus of z mm. The pump ca thus be regarded as a plae wave withi the legth of the crystal, leadig to a improvemet of the degree of squeezig [56]. After the SPDC, the remaiig pump was filtered out usig a series of filters. The geerated squeezed vacuum was directed to a asymmetric beamsplitter (ABS) with a reflectivity of R.%. The reflected part was directed to a avalache photodiode, while the trasmitted part was subjected to full quatum state tomography by meas of time-domai balaced homodye detectio (TD-BHD) [5, 58]. I pulsed experimets, squeezig is ofte geerated i a sigle pass cofiguratio without the use of ehacemet cavities. Thus the squeezig is geerated i may differet spatial ad temporal modes [59, 6, 61]. The mode (ad thus the degree of squeezig) beig measured by the homodye detector depeds o the spatio-temporal profile of the local oscillator: the mode of the squeezig spectrum that spatially ad temporally overlaps with the LO will be measured by the homodye detector. The amout of measured squeezig ca be optimized by ijectig a weak seed beam, correspodig to the mode of the LO, ito the SPDC crystal ad studyig the classical parametric (de-)amplificatio of the seed beam. Depedig o the relative phase betwee the seed ad the pump, the seed ca be either amplified or de-amplified. As described above the beam waist of the pump i the PPKTP was set to w,p = μm. Optimal phasematchig betwee the two waves is obtaied for w,p /w,s = [55], ad thus we set the beam waist of the seed (w,s = 16 μm). We achieved a optimal de-amplificatio of g mi =. ad a amplificatio of gmax =.6 for a pump power of 9 mw. A full characterizatio of the gais, g {mi,max}, as a fuctio of pump power is show i Fig.. Usig a simple model we fid [6], g {mi,max} = ɛexp({+r, r})+(1 ɛ) (1) where exp(±r) is the itrisic gai ad ɛ is a parameter describig the spatial overlap betwee the seed ad the pump. ɛ was experimetally estimated to be. ±.1 (see below). Through power-shape fittig, the depedece of r o the the pump power is foud to be r =.8 P p. This measuremet of the classical gai also shows that withi the gai rage of our experimet, the effect of gai-iduced diffractio is egligible [56, 6].
Fig.. Classical parametric gai versus average pump power, P p. The blue squares correspod to the measured deamplificatio values whereas the blue diamods are the measured amplificatio values. The red ad gree dotted lies are fits usig Eq. (1), with the itrisic parametric gai r =.8 P p ad ɛ =. ±.1... Time-Domai balaced homodye detectio The measuremet setup for homodye detectio is show i Fig.. The trasmitted squeezed vacuum state was superimposed with the LO at PBS1, the combied optical pulses were split at PBS ad the resultig two beams were focused oto a pair of PIN photodiodes (Hamamatsu, S88, quatum efficiecy of η ph =95± %). Usig a HWP, the splittig ratio of PBS was tued to balace the homodye detector. The detector used i the experimet is based o the desig by Hase et al. [58]. Its output was recorded by a digital oscilloscope (LeCroy, LTL) usig the cavity dumper sigal from the laser as a trigger. The fial quadrature value is the extracted by itegratig the sigal over the idividual pulses. The itegratio requires a well defied pulse widow, Tw, which is determied by the repetitio rate of the laser f Rep = 8 khz yieldig Tw 1. μs. The detector has a badwidth of MHz which is cofirmed by the geeratio of 5 s wide electroic pulses resultig from the detectio of the picosecod optical pulses. Sice electroic pulse is shorter tha the pulse widow, oly a fractio of the pulse widow cotais valuable iformatio. As a result, oly a part of the measured pulse cotributes to the itegratio, used to extract the quadrature values. We ivestigated the sigal-to-oise ratio (SNR) (shot-oise variace to electroic oise variace) of the detectio scheme for various choices of a weight fuctio folded with the measured pulse. It was foud that a simple boxcar-average, ecompassig about % of the measuremet widow, was a optimal choice. The shot oise referece is obtaied by measurig a vacuum iput state. The referece level is kow to icrease liearly with the local oscillator (LO) power. To verify that the system was ideed shot-oise limited, we measured the shot-oise as
a fuctio of the LO power, see Fig. 5(a). The electroic oise was measured to be. mv which correspods to 5 electros/pulse. I Fig. 5(a) it is clearly see that the shot-oise depeds liearly o the LO power, ad the gai of the detector was foud to be 1.6 mv /1 6 photos per pulse. I Fig. 5(b) the ratio betwee the shot oise ad the electroic oise (electroic oise clearace) is plotted as a fuctio of the LO power. It ca be see that the oise clearace surpasses db whe the LO pulse cotais more tha 1 6 photos (correspodig to a power of 1μW). This correspods to a electroic oise equivalet quatum efficiecy of η el 99.5 ±.5% [6]. (a) (b) SNR, [db] 5 1 5 1 5 P LO, N photo /pulse, [mio] Fig. 5. (a) The quadrature variace measuremet of the shot oise as a fuctio of the LO power. (b) the sigal-to-oise ratio (SNR) of the shot oise variace as a fuctio of the LO power. The overall homodye detectio efficiecy η hd is give by η hd = η op η mmη ph η el () where η op is the propagatio efficiecy of the state through optical compoets ad η mm is the degree of mode-matchig betwee the LO ad the squeezed pulse. These values were measured to be η op =.9 ±. ad η mm =.95 ±., givig a total homodye efficiecy of η hd =. ±.... Photo subtractio The reflected photos from ABS were detected by a fiber coupled APD (Perki-Elmer SPCM-AQR-1-FC). Usig this sigal as a trigger we coditioally prepared a photo subtracted squeezed state. To reduce the effect of detector dark couts, the trigger sigal for the homodye measuremet is derived by correlatig the APD sigal with the cavity dumper sigal (see Fig. ). By settig the coicidece widow to 1 s - correspodig to 1/1 of the total measuremet widow - we achieved a 1-fold decrease of the detector dark couts, resultig i a dark cout rate of. ±.5 s 1. To esure that the APD detectio evets are spatially ad spectrally matched with the optical mode of the LO it is ecessary to employ filterig i the APD arm. The spectral filterig was carried out usig a Fiber-coupled Tuable Fabry-Perot (FP) Filter Cavity (Micro Optics, FFP-TF-8-5). It was coupled via two 1.5-meter-log sigle mode fiber pigtails for spatial filterig. The badwidth of the filter was GHz correspodig to.5 m, ad the cetral wavelegth could be tued usig a voltage supply. The total detectio efficiecy of the heraldig chael was estimated to be about 1 ± 5 % icludig the
couplig efficiecy to the fiber, the peak trasmissio of the FP filter ad the detectio efficiecy of the APD. The total gated sigal trigger cout rate varied from s 1 up to s 1 for pump powers of 1 mw up to 8 mw..5. Gaussia model for estimatio of photo-subtracted squeezed state I order to predict the performace of the photo-subtractio experimet, we derived a simple model Refs. [5, 8, 65]. The model takes ito accout various experimetal imperfectios, which could compromise the quality of the prepared output states. The aalysis is broke ito three parts. The first part is the geeratio of the squeezed state. The secod is the tap-off o the asymmetric beamsplitter, ad projectio oto the o-off click detector with a filter i frot. Fially, the third part is the imperfect homodye detector used for characterizatio. We choose to work with the Wiger quasi-probability distributios sice they provide a coveiet framework for such types of models. I the Wiger picture the vacuum state is give by a simple Gaussia distributio i the quadrature variables, ˆX =[x,p] T. W ( ˆX) = e x p () π A squeezed state ca be writte i the same way with the variables rescaled accordig to the quadrature variaces, V x,p /: x W s ( ˆX) = e Vx p Vp π () V x V p where the Heiseberg ucertaity priciple costrais the variaces as V x V p 1. The squeezed state is split o a asymmetric beamsplitter with reflectivity R, ad oe part is measured usig the positive operator value measure (POVM) elemet, ˆΛ=ˆ1, W out ( ˆX 1 ) = π W ( ˆX 1, ˆX )W Λ ( ˆX ) d ˆX (5) where W Λ ( ˆX ) is the Wiger fuctio for the POVM elemet ad W ( ˆX 1, ˆX )isthe Wiger fuctio of the state after the beamsplitter. I Sec.. we described how proper filterig i the heraldig arm was ecessary i order to esure that idetical modes were detected by the APD ad the homodye detector. However, i practice, some false modes will be detected by the APD. This will be modeled by the modal purity parameter Ξ. It describes the probability that the photo detected from the APD came from the targeted optical mode. The output from the system ca the be expressed as follows, W out,ξ ( ˆX) = ΞW out ( ˆX)+(1 Ξ)W s ( ˆX) (6) where W s (x,p) is the state we see whe the APD detectios are ucorrelated with the optical mode used i the experimet.. Experimetal results.1. Squeezed vacuum The squeezed vacuum produced by the SPDC was characterized by usig a digital oscilloscope with which we acquired 65 quadrature values. Usig a sample rate
of 1 MS/s for 8 ms we acquired 1 pts. per pulse. The cavity dumper sigal (rate of 8 khz) was used to set the time widow for each pulse acquisitio, see Sec... The relative phase betwee the LO ad the quatum state was scaed by a saw-tooth modulatio applied to a piezocrystal attached to a highly reflectig mirror placed i the LO arm. The miimum ad maximum quadrature variaces of the squeezed pulses were measured as a fuctio of pump power ad the measured values are show i Fig. 6. Fig. 6. The measured squeezig ad ati-squeezig as a fuctio of the average pump power, P p. The blue diamods are the measured maximum variaces (atisqueezig), the blue squares are the measured miimum variaces (squeezig), ad the gree aed red dotted lies are fittigs to Eq. (1) where r =.8 P p ad η =.6 ±.1. Assumig a loss model, the squeezig ad ati-squeezig variaces, V {mi,max},ca be fitted to a simple relatio, V {mi,max} = (η(g mi,g max )+(1 η))v vac () where g {mi,max} is the parametric gai, η is the total detectio efficiecy, ad V vac is the quadrature variace of the vacuum state. By settig the parametric gai to the values foud i Sec.., the expressios ca be fitted to the experimetal data usig the efficiecy η as the fittig parameter. We fid reasoable fits to both series of data for η =.6. This value is lower tha the experimetally accessed oe of %, see Sec... Part of the discrepacy is caused by the tap-off beamsplitter which adds 8% loss to the squeezed states (ot accouted for i sectio.). The remaiig discrepacy of 9%, we speculate to be resultig from a mismatch betwee the temporal modes of the LO ad the squeezed vacuum [5, 66]. However, as this loss effect has ot be carefully studied ad localized, we will be usig η =. as the estimated detectio loss i the followig sectios. This value will be used to correct the experimetal data for losses. Usig the experimetal parameters i Fig. 6 as well as the formalism give i Sec..5, the expected properties of the photo-subtractio squeezed state is theoretically predicted. We calculate the fidelity of the CSS state, F odd, its amplitude, α, adthe
egativity of its Wiger fuctio at the origi, W (,), at four pump power levels. The results are listed i Table 1 ad Table (corrected for losses). It is clear that ay oe of the photo subtracted squeezed states are expected to exhibit strog o-gaussiaity with relatively large egativities of the Wiger fuctios. P p [mw] F odd α W(,)..6.8 -.9..58 1.5 -.6 6..55 1. -. 8..5 1. -. Table 1. Estimatio of the parameters characterizig the photo subtracted squeezed state for differet pump powers. The predictios are based o the measured values for the squeezed vacuum states corrected for losses associated with the detectio (η =.) as well as the asymmetric beamsplitter (8% loss). I these estimates we have set the mode match parameter Ξ to uity for all power levels. P p [mw] F odd α W(,)..8(.99).99(1.6) -.1(-.)..(.89) 1.18(1.) -.16(-.6) 6..69(.8) 1.(1.) -.1(-.) 8..6(.6) 1.5(1.56) -.(-.) Table. Estimatio of the parameters characterizig the photo subtracted squeezed state for differet pump powers ad after correctig for the homodye losses of.. For compariso, the umbers corrected for.6/.9 efficiecy are also give i brackets... Photo-subtracted squeezed vacuum Next, we prepared photo-subtracted squeezed states for differet pump powers ragig from to 8 mw. I this rage, the photo detectio rate of the APD varied from s 1 to s 1. Each time a detectio evet from the APD was correlated with the syc sigal from the laser, a trigger sigal was derived ad set to the oscilloscope, see Sec..1. For every trigger sigal, the digital oscilloscope sampled the homodye sigal for 1 μs with a samplig rate of 5 MS/s, makig up a sigle measuremet segmet. The segmets were stored cosecutively util data segmets filled up the memory of the oscilloscope. The quadratures were extracted i the same way as for the squeezig measuremet. Durig oe measuremet series the relative phase betwee the LO ad the quatum state was scaed over a rage of π. We used maximum likelihood estimatio to recostruct the prepared quatum state [6, 68, 69]. Before the recostructio, the acquired quadrature files were cocateated, ad the etire batch of quadratures measuremets was used for recostructio. I order to recostruct the quatum state, a estimatio of the phase referece was required. Sice the phase was scaed, we did ot have a stable phase referece. I order to extract the phase iformatio the quadrature data was stored i bis of 1 quadratures ad the variace of each bi was evaluated. The phase of bi i ca be estimated
by comparig its variace to the miimum ad maximum variaces usig the relatio V i (θ i )=V mi cos θ i + V max si θ i, where θ i is the phase associated with bi i. From this expressio we obtaied a estimate for the phases i each bi. Without loss of geerality, the assiged phase was chose betwee ad π/. Havig assiged a phase to the quadratures, we performed maximum likelihood recostructio. We recostructed desity matrices ad from those we calculated the Wiger quasi-probability distributios, see Fig. (without loss-correctio) ad Fig. 8 (corrected for losses). (a) mw.6 W(X,P).1.6 Projected W(X,P).1.8. Desity matrix.6.. Photo distributio...6 X P. X 5 1 (b) mw.6 W(X,P).1.6 Projected W(X,P).1.8. Desity matrix.6.. Photo distributio...6 X P. X 5 1 (c) 6 mw.6 W(X,P).1.6 Projected W(X,P).1.8. Desity matrix.6.. Photo distributio...6 X P. X 5 1 (d) 8 mw.6 W(X,P).1.6 Projected W(X,P).1.8. Desity matrix.6.. Photo distributio...6 X P. X 5 1 Fig.. Plot of the Wiger fuctios, the projected Wiger fuctios, the desity matrices ad the photo distributios of the recostructed states for pump powers from mw to 8 mw. There are o correctios for losses i these plots. From Fig. we see that the geerated states are o-gaussia ad o-classical. Moreover, it is evidet that the Wiger fuctios become more squeezed as the pump
(a) mw.6 W(X,P).1.6 Projected W(X,P).1.8. Desity matrix.6.. Photo distributio...6 X P. X 5 1 (b) mw.6 W(X,P).1.6 Projected W(X,P).1.8. Desity matrix.6.. Photo distributio...6 X P. X 5 1 (c) 6 mw.6 W(X,P).1.6 Projected W(X,P).1.8. Desity matrix.6.. Photo distributio...6 X P. X 5 1 (d) 8 mw.6 W(X,P).1.6 Projected W(X,P).1.8. Desity matrix.6.. Photo distributio...6 X P. X 5 1 Fig. 8. Plot of the Wiger fuctios, the projected Wiger fuctios, the desity matrices ad the photo distributios of the loss-corrected recostructed states forpumppowersfrommwto8mw. power icreases while the dip aroud the origi retais its structure. The dip attais a egative value for all realizatios if the measuremet results are corrected for losses. The maximum measured egativity for the ucorrected data is W (,) =. which correspods to a egativity of W (, ) =.6 after loss correctio. The fidelities betwee the experimetally produced states ad the ideal cat states (maximized over the excitatios α) are summarized i Table. By comparig the results i Table with the predictios i Table 1 we see that the fidelities, as well as the egativities, are geerally smaller tha predicted. The discrepacy, however, gets smaller at higher pump powers. This effect is caused by slow
P p [mw] F odd α W(,) Ξ..6..16....88..86 6..9 1.8.18.91 8..9 1.19..96 Table. Parameters characterizig the prepared photo subtracted squeezed states. istabilities of the experimetal setup which become sigificat for loger measuremet rus as is the case for low pump powers where the ru time is about 1s (to acquire quadrature values). For high powers, however, the measuremet time (1-s) is shorter ad thus the ifluece of istabilities is less proouced. The mai source of istability is a mechaical drift of the filterig Fabry-Perot cavity which was ot actively stabilized durig the measuremet. A drift of the cavity results i the detectio by the APD of the frequecy modes, which are differet from the oes measured at the homodye detector, which results i degradatio of the performace. This correspods to a lower value of the parameter Ξ i Eq. (6). To estimate values for the mode match parameter Ξ for the differet power levels, we fit the theoretical predictios to the actual measuremet results by usig Ξ as a fittig parameter. The obtaied values of Ξ for which the theoretical fidelities ad Wiger fuctio egativities match the experimetal oes are show i Table, which shows that the mode matchig parameter is icreasig for icreasig pump powers. Icorporatig the geeralized Beroulli trasformatio ito the maximum likelihood algorithm, the homodye detectio iefficiecy ca be corrected [69]. As metioed above, we used the coservative estimate of the detectio efficiecy of % for the correctio i order to avoid overestimatig the egativities of the corrected Wiger fuctios. The Wiger fuctios after correctio are displayed i Fig. 8, ad the results for the fidelities, egativities ad sizes are summarized i Table. P p [mw] F odd α W(,)..55.8...5 1.1. 6..5 1.1.5 8..56 1..6 Table. Parameters characterizig the prepared photo subtracted squeezed states after correctio for imperfect detectio.. Coclusio We have preseted a method of the preparatio of photo-subtracted squeezed states i a system based o picosecod pulsed laser pulses. It is based o geeratig squeezed vacuum from SPDC i a PPKTP crystal followed by sigle photo subtractio, eabled by the reflectio of a sigle photo o a asymmetric beamsplitter ad its detectio by the APD. Various states were produced with varyig degree of squeezig.
The resultig states were fully characterized by homodye tomography with which the Wiger fuctios ad desity matrices were recostructed. We foud a maximum egativity of W (,) =. without ay loss-correctios ad W (,) =.6 after loss-correctio. The egativity appeared to be largest for the largest degrees of squeezig. It is attributed to the shorter measuremet time associated with larger squeezig ad thus greater robustess to istabilities of the setup. Usig a simple theoretical model, we performed a compariso betwee theoretical ad experimetal results. A discrepacy is attributed to a mismatch betwee the modes measured by the APD ad the homodye detector. By fittig the experimetal results, we estimated a value for the mismatch. It was foud to deped o the degree of squeezig ad therefore o the measuremet time: the mismatch was foud to become icreasigly severe as the degree of squeezig was lowered correspodig to a loger measuremet time. As metioed above, the exteded measuremet time resulted i istabilities durig the measuremet time, thus degradig the performace. I particular, the temporal filter prior to sigle photo detectio was driftig thereby resultig i the detectio of differet temporal modes durig the acquisitio time. I order to improve the performace of the setup, the temporal filter cavity must be actively stabilized.