We report on the first experimental demonstration of purification of
nonclassical continuous variable states. The protocol uses two copies of
phase-diffused states overlapped on a beam splitter and provides Gaussified,
less mixed states with the degree of squeezing improved. The protocol uses only
linear optical devices such as beam splitters and homodyne detection, thereby
proving these optical elements can be used for successful purification of this
type of state decoherence which occurs in optical transmission channels

DiGuglielmo, J.

Fiurasek, J.

Franzen, A.

Hage, B.

Schnabel, R.

English

arXiv

Alexander Franzen

Boris Hage

James DiGuglielmo

Jaromír Fiurás̆ek

Roman Schnabel

Crossref

Experimental Demonstration of Continuous Variable Purification of Squeezed States

We report on the first experimental demonstration of purification of nonclassical continuous variable states. The protocol uses two copies of phase-diffused states overlapped on a beam splitter and provides Gaussified, less mixed states with the degree of squeezing improved. The protocol uses only linear optical devices such as beam splitters and homodyne detection, thereby proving these optical elements can be used for successful purification of this type of state decoherence which occurs in optical transmission channels

MPG.PuRe

Experimental Demonstration of Continuous Variable Purification of Squeezed StatesAlexander Franzen,1 Boris Hage,1 James DiGuglielmo,1 Jaromı´r Fiura´s˘ek,2 and Roman Schnabel11Institut fu¨r Gravitationsphysik, Leibniz Universita¨t Hannover and Max-Planck-Institut fu¨r Gravitationsphysik (Albert-Einstein-Institut), Callinstrasse 38, 30167 Hannover, Germany2Department of Optics, Palacky´ University, 17. listopadu 50, 77200 Olomouc, Czech Republic(Received 28 April 2006; published 13 October 2006)We report on the first experimental demonstration of purification of nonclassical continuous variablestates. The protocol uses two copies of phase-diffused states overlapped on a beam splitter and providesGaussified, less mixed states with the degree of squeezing improved. The protocol uses only linear opticaldevices such as beam splitters and homodyne detection, thereby proving these optical elements can beused for successful purification of this type of state decoherence which occurs in optical transmissionchannels.DOI: 10.1103/PhysRevLett.97.150505 PACS numbers: 03.67.a, 03.65.Ud, 42.50.DvPurification of quantum states is one of the key tech-niques for quantum-information processing and for longdistance quantum communication networks with the pur-pose to counteract decoherence and to regain the nonclass-ical properties of Fock states, entangled single-photonstates as well as squeezed and entangled continuous vari-able states [1,2]. First purification protocols for single-photon states have been demonstrated experimentally, re-lying on coincidence measurements, destructive post-selection, and photon counters [3].Continuous variable states are of particular interest. Thesubclass of so-called Gaussian states are easily accessibleexperimentally and still provide the resource for manyquantum-information protocols, such as quantum telepor-tation [4,5], entanglement swapping [6], and dense coding[7]. They also provide a resource to improve high precisionoptical measurements, e.g., in gravitational-wave detection[8]. Therefore, much attention has been attracted by thediscovery that entangled Gaussian states cannot be purifiedwith local Gaussian operations [9]. Whereas the Gaussiansetting solely builds on parametric oscillation, beam split-ters, phase shifters, and homodyne detectors, additionalnon-Gaussian operations like single-photon detectionseemed to be essential for the purification. In a moregeneral investigation it was then theoretically shown thatthe Gaussian setting has to be left at a single point to indeedbreak this no-go theorem [10,11].Non-Gaussian states can be purified by an iterativeGaussification protocol whose single step involves mixingtwo copies of the state on balanced beam splitters andconditioning on Gaussian measurements on one of theoutput copies [10–13]. In a recent paper [13] it was shownthat an experimentally feasible purification protocol of thiskind can counteract the effect of phase fluctuations thatoccur when squeezed or entangled states are generated ortransmitted through a noisy channel. The outstanding ad-vantage of the protocol considered in Ref. [13] lies in thefact that homodyne detection which can be performed withvery high efficiency is used as a conditioning measure-ment. It has been proven that the protocol can be appliediteratively and meets all criteria of a proper purificationprotocol. In particular, it has been numerically shown thatthe logarithmic negativity [14] of phase-diffused entangledstates indeed increases through the purification protocol.In this Letter we report the experimental demonstrationof a purification protocol as proposed in Ref. [13]. Twocopies of single-mode squeezed states were sent throughtwo channels that exhibited uncorrelated quasirandomphase noise and subsequently they were purified. It wasespecially shown that the protocol can restore squeezingthat was completely lost due to the phase noise, as sug-gested by the theory. To counteract optical loss, which isanother decoherence effect, has not been subject of thework presented here.Consider a single-mode squeezed state whose Wignerfunction reads WSMS  2VxVpq1 exp x22Vx p22Vp; (1)where Vx and Vp are variances of the x and p quadratures,respectively, with VxVp  1=4. Assuming Vx  Vp thestate is squeezed if Vx < 1=2. Under the influence of therandom phase fluctuations this state will evolve into amixed generally non-Gaussian state with Wigner functiongiven by Wx; p  12VxVpp Z exp x22Vx p22Vpd;(2)where x  x cos p sin, p  p cos x sin,and  denotes the probability distribution of the ran-dom phase shift,Rd  1. The purification protocolas proposed in Ref. [13] is largely independent of theparticular form of the fluctuations and qualitatively holdsfor any . However, in this work we realized Gaussianphase fluctuations of the form   221=2exp 222, being completely described by the single pa-PRL 97, 150505 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending13 OCTOBER 20060031-9007=06=97(15)=150505(4) 150505-1 © 2006 The American Physical Societyrameter . Figure 1 shows the calculated Wigner functionof a squeezed state with random Gaussian phase fluctua-tions. It is clearly visible that phase fluctuations transformpure Gaussian squeezed states into mixed states with areduced degree of squeezing. However, one also realizesthat the phase-diffused state exhibits a non-GaussianWigner function. Phase fluctuations, therefore, naturallylead to a non-Gaussian setting and the no-go theorem forpurification no longer holds. In Fig. 1 the strength of thefluctuation (  0:523) and the squeezing assumed(10 dB) was higher than in our experiment to highlightthe formation of the non-Gaussian Wigner function.Figure 2 shows the experimental setup that was used todemonstrate purification of phase-diffused squeezed states.The laser source was a continuous wave nonplanarNd:YAG ring laser with 300 mW of output power at1064 nm and 800 mW at 532 nm. The latter was used topump two optical parametric amplifiers (OPAs) to producetwo amplitude-squeezed light beams with an approxi-mate power of 0.06 mW at 1064 nm. Both OPAs wereconstructed from type I noncritically phase-matchedMgO:LiNbO3 crystals inside hemilithic resonators, similarto the design that previously has been used in Ref. [15].Each resonator was formed by a high-reflection (HR)coated crystal surface (reflectivity r2 > 0:999) and a metalspacer mounted outcoupling mirror (reflectivity r2 0:957). The intracavity crystal surface was antireflectioncoated for both the fundamental wavelength (1064 nm,r2 < 0:05%) and the second harmonic (532 nm, r2 <0:2%). The outcoupling mirror had a small reflectivity ofr2  0:15	 0:02 for 532 nm. The OPAs were seededthrough the HR surface with a coherent laser beam of15 mW power and pumped through the outcoupling mirrorwith 100 mW (not shown in Fig. 2), resulting in a classicalgain of about 10. The lengths of both OPA cavities as wellas the phase of the second harmonic pump beams werecontrolled using radio-frequency modulation or demodu-lation techniques. All control error-signals were derivedfrom the seed fields reflected from the OPA cavities. Anonclassical noise power reduction of slightly more than3.5 dB was directly observed with a homodyne detector incombination with a spectrum analyzer. Subtracting thevariance of the electronic noise yielded squeezing of upto 5 dB. The squeezing spectrum was almost white fromapproximately 5 to 15 MHz. Both squeezed fields freelypropagated from the OPAs passing high-reflection mirrorsthat were quasirandomly moved by piezo-electric trans-ducers (PZTs) to mimic the effects of noisy optical trans-mission channels. The voltages applied to the PZTs wereproduced as follows. Two independent random numbergenerators produced data strings with Gaussian distribu-tion. Both strings were digitally filtered to limit the fre-quency band to 2–2.5 kHz. The output interface was acommon PC sound card with a signal to noise ratio of110 dB.Homodyne detection confirmed that for both beamssqueezing degraded in the same way when phase noisewas increased. In order to demonstrate the purificationprotocol the two squeezed beams were subsequently over-lapped at a 50=50 beam splitter (labeled BSPUR in Fig. 2).The visibility of the squeezed beams’ mode matching was97.4% and was limited by OPA crystal inhomogeneities.The phase of the two squeezed beams at the purificationbeam splitter was controlled at frequencies below thefrequency band of the impressed phase noise. The outputfields from the purification beam splitter were then sent totwo homodyne detectors as shown in Fig. 2. Homodynevisibilities were at 93.5% for homodyne detector HD I and96.2% for HD II. Both homodyne detectors were servo-−5 −4 −3 −2 −1 0 1 2 3 4 5−3−2−10123Phase quadrature  pAmplitude quadrature xFIG. 1 (color online). Numerically simulated Wigner functionfor a phase-diffused squeezed state (in shot-noise units). Herewe assumed 10 dB squeezing, an initially pure state, andGaussian phase fluctuations with standard deviation of  0:523 (  30), providing a mixed state with clearly visiblenon-Gaussian statistics.LOHD IHD IIPhase noise I Phase noise II  OPA I OPA II BSPURLODetection andprocessingSqueezinggenerationNoisy opticaltransmission................................................................................................................................................FIG. 2 (color online). Schematic experimental setup. Twoamplitude-squeezed beams were produced by two length-controlled OPA cavities. Quasirandom band-limited Gaussianwhite phase noise is applied to both squeezed beams to mimicphase noise in optical transmission channels. The two beams aresubsequently overlapped at a 50=50 beam splitter (BSPUR). Timeseries of quadrature values for both beam splitter outputs wererecorded using balanced homodyne detection (HD) and a dataacquisition system. OPA: optical parametric amplifier; LO: localoscillator; : piezo-electric transducer.PRL 97, 150505 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending13 OCTOBER 2006150505-2controlled (with loop bandwidth smaller than the noisefrequencies) to detect the amplitude quadratures. The de-tector difference currents were electronically mixed with a7 MHz local oscillator. The demodulated signals were thenfiltered with steep low-pass filters at 30 kHz and synchro-nously sampled with 100 kHz.Both time series of quadrature values xI and xII from thetwo homodyne detectors were post-processed to accom-plish and confirm the purification protocol. The quadraturevalues xI measured at the first homodyne-detector (HD I)were used to produce a trigger signal for squeezing puri-fication on the second beam. A trigger signal was simplygiven when a measured quadrature value jxIj was belowsome chosen threshold X. The second homodyne detectorwas not part of the purification scheme itself but served as averification device, confirming that purification had ac-tually occurred.Figure 3 shows the success of our purification protocolversus phase-noise strength. Measurement points on theupper curve represent quadrature variances of a single copybefore purification, whereas the lower curve provides themeasured variances of xII after two-copy purification. Inagreement with the theory, purification becomes moreefficient for larger standard deviations of the phase noise. The variances plotted correspond to directly observedsqueezing; electronic dark noise of the detector was nottaken into account. For no additional phase noise (  0)slightly more than 3.5 dB of squeezing was observed whichwas completely lost for noise with > 0:28.To quantify how effective the purification protocol wasthe variance of the purified states needs to be comparedwith the variance of a single copy before purification. If thetwo states are not copies with identical variances, such asituation might appear in realistic experiments, then thelower variance provides the reference. In Fig. 3 the uppercurve represents the variances of both individual copieswhich were identical within the measurement error barsgiven, nevertheless the purified state exhibits lowervariance.Figure 4 further characterizes our purification experi-ment focusing on the setting with a phase noise of standarddeviation   0:304. It is shown how the purified outputvariance (left graph) and the rate of surviving data points(right graph) depend on trigger threshold X. For a triggerthreshold that retains still 50% of the data, a noise varianceabove vacuum noise was purified to a value clearly belowvacuum noise reference.Figure 5 shows how the variance depends on the rate ofretained data points. Without any purification no state isSigma of noiseVariance normalized to shot-noiseFIG. 3 (color online). Amplitude quadrature variance of thesingle copy squeezed field versus additional phase-noise strength(upper curve). Squeezing of 3.55 dB was directly observedwithout impressed phase noise (  0) and completely lost forhigh levels of phase noise (> 0:28). The lower curve illustratesthe amount of squeezing after purification. As predicted by thetheory purification becomes more efficient for higher values of. For  values of about 0.3 squeezing was purified from a fieldthat showed a higher than vacuum noise variance before.Trigger threshold Trigger thresholdVariance normalized to shot-noiseRate of survivorsFIG. 4 (color online). Purified output variance (left graph) andthe rate of surviving data points (right graph) versus triggerthreshold X. Both curves correspond to   0:304.Rate of survivorsVariance normalized to shot-noiseFIG. 5 (color online). Measured amplitude quadrature varian-ces of purified states versus state survival rate for four differentstrengths of phase noise . The shot-noise reference is indicatedby the horizontal line. The ripples at small survival rates origi-nate from correspondingly larger uncertainty in the variancesobtained.PRL 97, 150505 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending13 OCTOBER 2006150505-3discarded and 100% of data points are still available. In allcases choosing the threshold such that half the states werekept provided a significantly reduced variance.Successful purification should provide not only animproved strength of nonclassical correlation and Gaussifi-cation but also a decreased mixedness of the states. Botheffects were clearly observed in our experiment. Consideragain the purification experiment with phase noise 0.304 inFig. 4. Without purification the product of the amplitudeand phase quadrature variances was 7.6, after purificationwe measured 6.0. Lastly, fitting Gaussian profiles to themeasured data manifested a reduction in the reduced 2value by a factor of 4. Increased squeezing strength,Gaussification, and decreased mixedness can physicallyall be well understood by realizing that the two outputfields from the purification beam splitter are correlatedand (possibly) entangled. Two squeezed fields overlappedon a 50=50 beam splitter have been used before to generateentangled states [16–18]. For optimal entanglement gen-eration the phase between the two squeezed inputs isadjusted in such a way that the Wigner functions of thebeam splitter outputs are circular. In case of the purificationof phase-diffused squeezed states, the situation is different.The Wigner functions of both beam splitter outputs arenon-Gaussian, as shown in Fig. 1. If a high amplitudequadrature value xI is measured, it is likely that also xIIexhibits a large value. Exactly those events are discarded inthe purification protocol with an appropriately adjustedtrigger threshold. Since such events are likely to representthe outer parts of the two convexities, the effect of therandom phase fluctuations with all its consequences iscounteracted. Note that the purified state need not bemonitored by HD II and may remain available for furtheruse.In conclusion, we have successfully demonstrated acontinuous variable purification protocol using squeezedstates which was able to suppress non-Gaussian sources ofdecoherence such as phase fluctuations. To the best of ourknowledge this is the first experimental two-copy purifica-tion of nonclassical continuous variable states reported.Our result is a significant step forward toward continuousvariable entanglement purification which appears to be astraightforward extension of the work presented here [13].Furthermore, in combination with a single non-Gaussianoperation (such as single-photon detection) our protocolcould be used to purify even states suffering from losses,forming a general CV purification protocol. Squeezed statepurification (or distillation) has also been reported inRef. [19]. However, our work provides the following dis-tinct features. First, we considered a rather natural noisesource that is indeed expected to be a problem in quantumcommunication and squeezed mode generation. Second,our scheme involved two copies of a state and it has beenproven to work iteratively if more copies of the state areavailable.We acknowledge financial support from the DeutscheForschungsgemeinschaft (DFG), Project No. SCHN 757/2-1. J. F. acknowledges financial support from MSMT(No. MSM6198959213 and No. LC06007) and from theEU under Project COVAQIAL (No. FP6-511004).[1] C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher,J. A. Smolin, and W. K. Wootters, Phys. Rev. Lett. 76, 722(1996).[2] D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S.Popescu, and A. Sanpera, Phys. Rev. Lett. 77, 2818(1996).[3] P. G. Kwiat, S. Barraza-Lopez, A. Stefanov, and N. Gisin,Nature (London) 409, 1014 (2001); J.-W. 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Franzen, Alexander

Hage, Boris

DiGuglielmo, James

Fiurasek, Jaromir

Schnabel, Roman

Max Planck Society eDoc Server

Experimental demonstration of continuous variable purification of
  squeezed states

Experimental demonstration of continuous variable purification of squeezed states

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MPG.PuRe

MPG.PuRe

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