Generation and Characterization of Multimode Quantum Frequency Combs Olivier Pinel,1Pu Jian,1Renne´Medeiros de Arau´jo,1Jinxia Feng,1,2Benoıˆt Chalopin,1,3
Claude Fabre,1,*and Nicolas Treps1
1Laboratoire Kastler Brosl,Universite´Pierre et Marie Curie–Paris6,ENS,CNRS;4place Jussieu,75252Paris,France 2State Key Laboratory of Quantum Optics and Quantum Optics Devices,Institute of Opto-Electronics,Shanxi University,
Taiyuan030006,People’s Republic of China
3Max Planck Institute for the Science of Light,Universita¨t Erlangen-Nu¨rnberg,IOIP,Staudtstras7/B2,91058Erlangen,Germany
(Received11April2011;published23February2012)
Multimode nonclassical states of light are an esntial resource in quantum computation with
continuous variables,for example,in cluster state computation.We report in this Letter thefirst
experimental evidence of a multimode nonclassical frequency comb in a femtocond synchronously
pumped optical parametric oscillator.In addition to a global reduction of its quantum intensityfluctua-
tions,the system features quantum correlations between different parts of its frequency spectrum.This
allows us to show that the frequency comb is compod of veral uncorrelated eigenmodes having
specific spectral shapes,two of them at least being squeezed,and to characterize their spectral shapes.
DOI:10.1103/PhysRevLett.108.083601PACS numbers:42.50.Dv,42.50.Lc,42.65.Yj
Optical frequency combs are perfect tools for high pre-cision metrological applications[1,2].The extension of their extraordinary properties to the quantum domain may lead to significant progress in different areas of quan-tum physics,in particular,in quantum metrology and parameter estimation[3,4]but also in quantum computa-tion with continuous variables[5,6].Indeed,one of the main challenges of experimentally implementing quantum computers in the continuous variable regime,for example, in cluster state computation[6,7],is the generation of highly multimode nonclassical states of light and the scal-ability of this generation.As the difficulty of linearly mixing dis
tinct squeezed light sources[8,9]increas as the number of modes increas,it can be more interesting to u instead a single highly multimode source which directly produces nonclassical resources shared between many modes within the same beam.In this perspective, optical frequency combs,which span over thousands of different frequency modes,are a very promising system for scalable generation of spectral or temporal multimode quantum states.We report in this Letter thefirst experi-mental evidence of a multimode nonclassical frequency comb generated by an optical parametric oscillator(OPO) in the femtocond regime,which opens the way to the generation of the highly multimode states. Multimode nonclassical light has been already experi-mentally generated with spatial multimode beams pro-duced by OPOs[10,11]and,very recently,with the longitudinal modes of an OPO[12,13].In the domain of temporal modes,single mode squeezing of short puls has been obrved in various experiments starting from Ref.[14]in the nanocond regime.Nonclassical states of single femtocond puls are the subject of many recent studies(for example,[15]).Multimode squeezed solitons have been generated in an opticalfiber[16].Single mode quantum noi reduction in picocond frequency combs
has already been achieved with a synchronously pumped
optical parametric oscillator(SPOPO)[17],which is an
OPO pumped by a train of ultrashort puls that are syn-
chronized with the puls making round trips inside the
push it
optical cavity[18–20].
It has recently been shown[21,22]that such SPOPOs
generate squeezed frequency combs which are multimode.
We give in this Letter the experimental confirmation of
the predictions.The different squeezed modes are ac-
tually frequency combs having different spectral profiles
or,equivalently in the time domain,trains of puls having
越狱 第二季
different temporal profiles.
In our experiment,the frequency comb is produced by a singly resonant SPOPO pumped with puls in the femto-
cond range,which are described in the frequency domain
by a superposition of at least105longitudinal modes of
frequencies!p n located around the carrier frequency2!0
and equally spaced by a repetition rate!r:!p n¼2!0þn!r.This huge number of pump modes leads to a great complexity of the parametric down-conversion process
taking place in the intracavity nonlinear crystal.Indeed, each pumping frequency!p n is coupled through pha-matched parametric interaction to many pairs of cavity-resonant frequencies!s‘and!s nÀ‘,where!s‘¼!0þ‘!r,since they satisfy!p n¼!s‘þ!s nÀ‘. However,it has been demonstrated[21]that this interac-tion can be described by a substantially reduced number of modes which are the eigenmodes of the nonlinear interac-tion,named supermodes.The are well-defined coherent superpositions of longitudinal modes characterized by their spectral amplitude and pha profiles.The mode profiles of the supermodes in the frequency domain,or in the time domain,are predicted to be clo to Hermite-Gaussian functions[22].
The experimental tup is shown in Fig.1.The SPOPO is eded by 120fs puls (spectrum width of
7nm)at 795nm with a repetition rate of 76MHz produced by a Ti:sapphire mode-locked lar (where the carrier-envelope pha is not stabilized)pumped by a Nd:YVO 4lar at 532nm.Its cond harmonic at 397nm is ud to pump the 350 m long intracavity BiB 3O 6(BIBO)crystal [23].The SPOPO cavity length is locked by using a Pound-Drever-Hall lock.Above a threshold of typically $50mW ,the SPOPO generates a single signal-idler frequency comb having the same mean frequency as the Ti:sapphire lar (degenerate configuration).Below threshold,we obrve pha-nsitive amplification of the ed.We lock the relative pha between the ed and the pump in the deamplification regime by using another Pound-Drever-Hall lock.The state of the output light is evaluated by using a two-port balanced detector (with quantum effi-ciency above 90%and 30dB noi extinction between the two detectors):The sum of the photocurrent fluctua-tions reprents the intensity fluctuations Án 2of the output beam,and the difference reprents the standard-quantum-limited fluctuations of a beam of the same power Án 2shot ¼h n i .We measure a normalized intensity noi up to Án 2=Án 2shot ¼0:76Æ0:02at $1:5MHz ,corresponding to 1:2Æ0:1dB of noi reduction on the amplitude quad-rature (e Fig.2).This experimentally demonstrates for the first time the nonclassicality of the field generated by a femtocond SPOPO below threshold.The low amount of squeezing may be explained by the fact that the ed is not optimized in the prent status of the experiment:It does not coincide with the SPOPO supermode with the highest achievabl
e squeezing (later referred as the first super-mode).Indeed,the spectrum of this supermode,which
depends on the spectrum of the pump and the length of the crystal,is theoretically 8.3times broader than the spectrum of the ed.The field we measure with the bal-anced detection therefore corresponds to a superposition of different supermodes,which results in a higher intensity noi than in the first supermode.In addition,as the cavity frequency bandwidth at half maximum is 2.5MHz,more squeezing should be obrved at a lower noi frequency,which is prently not possible becau of the prence of excess technical noi (due to the relaxation oscillation of the lar)at low frequencies of analysis.
In order to investigate the multimode nature of the out-put beam [24],we have studied the distribution of the quantum intensity fluctuations of the frequency comb over its optical spectrum,in a way similar to Refs.[16,25].To this aim,the frequency components of the output beam were parated by using two prisms (only one prism is shown in Fig.1for the sake of simplicity).A slit of variable position and width allowed us to record intensity fluctuations of different parts of the spectrum.The spectral resolution of the filter was 1.8nm.More precily,we divided the frequency comb spectrum into four concutive frequency pixels of equal mean intensities and symmetric with respect to the central frequency (e the int in Fig.3).We recorded the intensity noi and the shot noi level for differe
nt frequency intervals:In addi-tion to the measurements of pixels 1;...;4and of the whole spectrum,we also measured combinations of two pixels f 1;2g ,f 2;3g ,and f 3;4g and of three pixels f 1;2;3g and f 2;3;4g ,leading to a total of 10frequency intervals.For each frequency interval,we measured successively the intensity noi and the shot noi during 5s,corresponding
FIG.1(color online).Schematic of the experimental tup.The OPO is synchronously pumped by a frequency comb cen-tered at 397nm.The delay line ensures the temporal overlap between the puls from the pump and ed at the cavity input.At the cavity output,the deamplified frequency comb centered at 795nm is spectrally filtered.A balanced detection is ud to measure the intensity noi.
necps
0.5
1 1.5
2 2.53
−2−1.5−1
−0.500.511.522.5
3Frequency (MHz)
N o r m a l i z e d i n t e n s i t y n o i s e (d B )
FIG.2(color online).Spectral intensity noi (in decibels)as a function of the analysis frequency on the spectrum analyzer.Dashed line:Standard quantum limit.The spike below 1MHz is due to the relaxation oscillation noi of the lar.The data are not corrected from the electronic dark noi of the photodiodes.Resolution bandwidth:30kHz;video bandwidth:10Hz.
to 1000data points for each measurement.The SPOPO remained locked during all the measurement
process.In order to maintain the stability of the lock,we needed to take a small part (around 10%)of the squeezed beam for the locking system,so the SPOPO could be locked inde-pendently from the measurement system.By doing so,the squeezing was reduced to a normalized intensity noi of 0:84Æ0:02at 1.5MHz.The mean values were ud to calculate the normalized intensity noi for each frequency interval.
For each frequency interval formed by the sum of pixels f i 1;...;i m g ,where i 1and i m !i 1are integers between 1and 4,the detected intensity fluctuations are
Á X i ¼i 1;...;i m
n i 2
¼X
i;j ¼i 1;...;i m
cov ðn i ;n j Þ;(1)
injection molding
where cov ðn i ;n j Þ¼h n i n j i Àh n i ih n j i is the photon num-ber correlation function between the different pixels when
n i Þn j and the variance when n i ¼n j .From the mea-surements,the intensity correlation matrix
C ði;j Þ¼cov ðn i ;n j ÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÁn 2i Án 2j
q À ij
Án 2i;shot
Án i (2)
can be reconstructed.The mean matrix is plotted in Fig.3.
As the four pixels have the same mean power,it is simple to show that,in the ca of a single mode state,all the correlation functions should be equal to À0:035.The prence of one diagonal element ð1;1Þwith less local squeezing and of one off-diagonal element ð1;4Þwith more anticorrelation than the other coefficients suggests that the output of the SPOPO is indeed a multimode non-classical field.However,the uncertainty on the off-diagonal elements does not allow us to unambiguously state that the generated state is multimode.Indeed,the standard deviation of the diagonal elements was 0.014,and the standard deviations of the off-diagonal elements spread from 0.022to 0.133.
In order to demonstrate the multimode character,we performed another analysis of the output of the SPOPO which consists in calculating the eigenmodes involved in the prent experiment and to show if the modes are indeed excited.To this purpo,we reconstructed the covariance matrix in the basis of the four frequency pixels:
V x i ;x j ¼12h x i x j þ x j x i i ;
(3)
where x i ¼a i þa y i is the amplitude quadrature and
x ¼x Àh x i .The pha quadrature information cannot be recovered from our technique,as we measure only intensity nois.We make therefore the simple assumption that the different frequency components of the output field have the same pha and can all be taken as real.Using the derivation discusd in Ref.[25],one finds cov ðn i ;n j Þ%h x i ih x j i V x i ;x j .The mean photon number in one zone is taken as h n i i %h x i i 2,leading to the following expression for the elements of the covariance matrix:
V x i ;x j ¼cov ðn i ;n j Þ
所以的英文
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÁn 2i;shot Án 2
j;shot
q :
(4)
The diagonalization of the covariance matrix V allows us to find that the light generated by the SPOPO is made of a t of four uncorrelated modes (the eigenvectors of V ),with given amplitude noi variances (the eigenvalues of V ).We find that,among the four eigenmodes S ‘,two have eigenvalues smaller than the standard quantum limit 1,and one has an eigenvalue greater than 1,as shown in Fig.4.The uncertainty in the measurement is small enough to conclude that the first and the third modes are amplitude squeezed and that the cond has excess noi compared to vacuum but not to conclude whether the fourth mode is excited.This shows that the output of the SPOPO is described by at least 3independent modes,two of them at least being in a squeezed state;the fourth mode may be considered as a vacuum.The uncertainty on the intensity noi of the eigenmodes was estimated in the following way:We generated 10000photon number covariance ma-trices with elements randomly picked among our experi-mental data points.After rotation in the S ‘basis,the off-
diagonal elements of the matrices fluctuate around 0Æ0:03,showing that the modes S l are effectively eigen-modes of the randomly generated fields.The diagonal elements give the normalized intensity noi of the eigen-modes,and their spread gives the spread of the normalized intensity noi.
Furthermore,the eigenmode S 1,within which lies most of the power of the light beam,has approximately the same amount of squeezing as the mean field.The spectrum of S 1is slightly broader than the mean field mode,which is consistent with the theoretical prediction [22]for the first supermode in our experimental conditions.We also ob-rve that the profile of S 2rembles a Hermite-Gaussian mode of the order of 1and shows excess noi;this is
also
FIG.3.Mean normalized photon number correlation matrix
C ði;j Þ.Int:Schematic of the frequency pixels compared to the spectrum.
consistent with the theory,which predicts that the cond supermode of the SPOPO is a first-order Hermite-Gaussian mode squeezed in pha and,therefore,antisqueezed in amplitude.In the same way,the profile of eigenmodes S 3appears like a Hermite-Gaussian mode of the order of 2,and S 3is squeezed,as predicted by its even order.Finally,one notices a slight asymmetry in the modes,which may be related to group-velocity dispersion effects.
The procedure that we have described demonstrates that at least three orthogonal modes are necessary to describe the output field of the SPOPO,two at least being in squeezed states.However,a complete description of the field generated by the SPOPO may involve more modes:Separating the spectrum into more than four pixels may reveal intensity correlations which would require more than three modes to be described.Moreover,this derivation does not take into account the possibility of correlations with the pha fluctuations p i .A complete characterization of the quantum state of the output of the SPOPO requires access to V p i ;p j and V x i ;p j ,which requir
es one to perform a homodyne detection with appropriately shaped local oscil-lator puls [26].
In conclusion,we have experimentally demonstrated that femtocond SPOPOs generate below threshold multi-mode quantum frequency combs that can be precily characterized and analyzed in terms of squeezed uncorre-lated eigenmodes of experimentally determined shapes.Indeed,for Gaussian states,multimode entanglement and multimode squeezing are equivalent given that one can英语语感
choo the measurement basis [27].The low level of noi reduction and of quantum correlation makes it so far a proof-of-principle experiment.Noi reduction,and there-fore possible entanglement,can be certainly drastically incread to much higher levels by working at lower noi frequencies with a less noisy pump lar and eding it with a mode-shaped beam clo to the first supermode.In addition,it can be shown that the exact multimode quan-tum state of the generated light can be adjusted by con-trolling the pump pul shape and duration.This experiment therefore opens the way to the production of quantum frequency combs that are tailored to fit the re-quirements of its numerous applications in quantum infor-mation processing and quantum metrology.
We acknowledge the financial support of the Future and Emerging Technologies (FET)program withi
n the Seventh Framework Program for Rearch of the European Commission,under the FET-Open grant agreement HIDEAS,No.FP7-ICT-221906,and of the ANR project
QUALITIME.
*claude.fabre@upmc.fr
[1]R.Holzwarth et al.,Phys.Rev.Lett.85,2264(2000).
neworiental[2]Th.Udem,R.Holzwarth,and T.W.Ha
¨nsch,Nature (London)416,233(2002).
[3] B.Lamine,C.Fabre,and N.Treps,Phys.Rev.Lett.101,
123601(2008).
FIG.4(color online).Wavelength profile of the eigenmodes S ‘.The pixel modes are smoothed by taking into account the 1.8nm spectral resolution of the bandpass filters.Solid lines:Eigenmodes;das
hed line:mean field mode.Power:Percentage in optical power of the output beam;NIN:normalized intensity
noi.
[4]O.Pinel et al.,Phys.Rev.A85,010101(R)(2012).
romingo
[5]S.Lloyd and S.L.Braunstein,Phys.Rev.Lett.82,1784
(1999).
[6]N.C.Menicucci et al.,Phys.Rev.Lett.97,110501(2006).
[7]J.Zhang and S.L.Braunstein,Phys.Rev.A73,032318
(2006).
[8]M.Yukawa,R.Ukai,P.van Loock,and A.Furusawa,
Phys.Rev.A78,012301(2008).
[9]J.L.O’Brien, A.Furusawa,and J.Vucˇkovic´,Nature
Photon.3,687(2009).
[10]J.Janouk et al.,Nature Photon.3,399(2009).
[11] B.Chalopin,F.Scazza,C.Fabre,and N.Treps,Phys.Rev.
thespringfestival
A81,061804(2010).
[12]N.C.Menicucci,S.T.Flammia,and O.Pfister,Phys.Rev.
Lett.101,130501(2008).
[13]M.Pysher et al.,Phys.Rev.Lett.107,030505(2011).
[14]R.E.Slusher et al.,Phys.Rev.Lett.59,2566(1987).
国庆的诗歌
[15]J.Wenger, A.Ourjoumtv,R.Tualle-Brouri,and P.
Grangier,Eur.Phys.J.D32,391(2004).
[16]S.Spa¨lter et al.,Phys.Rev.Lett.81,786(1998).[17]R.M.Shelby and M.Ronbluh,Appl.Phys.B55,226
(1992).
[18] E.C.Cheung and L.M.Liu,J.Opt.Soc.Am.B7,1385
(1990).
[19] D.C.Edelstein,E.S.Wachman,and C.L.Tang,Appl.
Phys.Lett.54,1728(1989).
[20]H.M.Van Driel,Appl.Phys.B60,411(1995).
[21]G.de Valca´rcel,G.Patera,N.Treps,and C.Fabre,Phys.
Rev.A74,061801(2006).
[22]G.Patera,N.Treps,C.Fabre,and G.J.De Valcarcel,Eur.
Phys.J.D56,123(2009).
[23]H.Hellwig,J.Liebertz,and L.Bohaty´,Solid State
Commun.109,249(1998).
[24]N.Treps,V.Delaubert,A.Maıˆtre,J.M.Courty,and C.
Fabre,Phys.Rev.A71,013820(2005).
[25]T.Opatrny´,N.Korolkova,and G.Leuchs,Phys.Rev.A66,
053813(2002).
[26]S.T.Cundiff and A.M.Weiner,Nature Photon.4,760
(2010).
[27]S.L.Braunstein,Phys.Rev.A71,055801(2005).
