Higher Order Mode Entanglement in a Type II Optical Parametric Oscillator Jun Guo, Chunxiao Cai, Long Ma, Kui Liu, Hengxin Sun, and Jiangrui Gao† State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, China Nonclassicalbeamsinhighorderspatialmodeshaveattractedmuchinterestbuttheyexhibitmuch less squeezing and entanglement than the fundamental spatial modes, limiting their applications. We experimentally demonstrate the relation between pump modes and entanglement of first-order HGmodes(HG10 entangledstates)inatypeIIOPOandshowthatthemaximumentanglementof highorderspatialmodescanbeobtainedbyoptimizingthepumpspatialmode. Toourknowledge, thisisthefirsttimetoreportthis. Utilizingtheoptimalpumpmode,theHG10 modethresholdcan 7 bereachedeasilywithoutHG00 oscillationandHG10 entanglementisenhancedby53.5%overHG00 1 pumping. The techniqueis broadly applicable to entanglement generation in high order modes. 0 2 INTRODUCTION n a J Continuousvariable(CV) squeezedandentangledstates areimportantinprocessessuchas quantumcomputation, 9 quantum communication and quantum metrology. Since the 1985 observation of CV squeezing by Slusher et al. [1], 1 much research has followed on the generation and optimization of squeezing and entanglement in different systems. These include the optical parametric oscillator (OPO) [2, 3], four-wave mixing (FWM) [1], and the in-fiber optical ] h Kerr effect [4, 5]. Among these tools, the OPO is the most widely used. In recent years, squeezing of up to 15 dB in p type I OPOs [6] and entanglement of 8.4 dB in type II OPOs [7] were realized. - t Traditionally most OPOs operate in the fundamental mode. However, higher order modes such as Hermite-Gauss n (HG) and Laguerre-Gauss (LG) modes contain more spatial degrees of freedom and can give more information in a applications than the fundamental mode. They can be used to enhance measurement precision of some physical u q quantities, such as lateral displacement [8] and transverse rotation angle of an optical beam [9]. They can also be [ appliedinquantumimaging[10],quantumstorage[11],quantumsuper-densecoding[12],andbiologicalmeasurement [13]. In recent years, squeezing and entanglement have been expanded to higher order modes in OPOs. Lassen et 1 v al. generated quadrature squeezing of HG00, HG10 and HG20 modes separately with a type I OPO in 2006 [14, 15] 3 and quadrature entanglement of first-order LG modes with a type I OPO in 2009 [16]. Multimode squeezing and 3 entanglement can also be generated in a specially designed OPO [17–19]. Recently, a CV hyperentanglement state, 3 wherein both spin and orbital angular momenta are entangled, was realized in a multimode type II OPO [20, 21]. 5 To date the degree of squeezing and entanglement produced in higher order modes has been much lower than for 0 . the fundamental mode, which limits their applications. Almost all the above cited work adopted the fundamental 1 mode as the pump for the higher order signal modes. This lead to low pump conversion efficiencies and crucially 0 much higher oscillation thresholds than for the fundamental spatial mode, severely limiting the attainable squeezing 7 1 and entanglement levels. : Lassenetal. presentedthe idealpumpforoscillationofthe HG mode, asuperpositionofHG andHG modes, v 10 00 20 but synthesizingthe multi-mode is experimentallyverychallenging[14–16]. Inthis Expresspaper,we experimentally i X demonstratetherelationbetweenpumpmodesandentanglementoffirst-orderHGmodes(HG entangledstates)ina 10 r typeIIOPOandshowthatthemaximumentanglementofhighorderspatialmodescanbeobtainedbyoptimizingthe a pumpspatialmode. Toourknowledge,thisisthefirsttimetoreportthis. Usingtheoptimalpump,theentanglement inseparability for HG mode is enhanced by 53.5% and the threshold is reduced by 66.7% relative to using HG in 10 00 our result. THEORETICAL MODEL For a type II OPO with an HG signal mode, we define vp(~r) as the transverse distribution of the pump mode 10 where~r =(x,y) denotes the transverse coordinates. This can be expanded into a series of HG modes as ∞ vp(~r)= c v (~r), (1) n n0 X n=0 where v (~r) denotes the transverse profile of the nth order HG mode and c is its corresponding coefficient. The n0 n transverse profiles of the signal and idler modes can be described by us(~r) and ui(~r). The full Hamiltonian of the 2 system can be written as Hˆ =i¯hεp aˆp† aˆp +i¯hχΓ aˆp aˆs† aˆi† aˆp†aˆs aˆi , (2) − − (cid:0) (cid:1) (cid:0) (cid:1) where χ is the nonlinear coefficient of the crystal, aˆp, aˆs and aˆi are the annihilation operators of the pump, signal and idler fields, and εp is the pump parameter. Γ is the coupling coefficient of the three intracavity fields given by +∞ Γ= vp(~r)us∗(~r)ui∗(~r)d~r. (3) Z −∞ Additionally considering the quantum vacuum noise caused by the extra losses, the Langevin equations of motion for the intracavity fields can be given by τˆ˙ap(t)=−γpaˆp(t)−χΓaˆs(t)aˆi(t)+εpe−iθp + 2µpˆbpin(t), (4a) p τˆ˙as(t)= γ′aˆs(t)+χΓaˆp(t)aˆi†(t)+ 2γ aˆs (t)+ 2µ ˆbs (t), (4b) − s s in s in p p τˆ˙ai(t)= γ′aˆi(t)+χΓaˆp(t)aˆs†(t)+ 2γ aˆi (t)+ 2µˆbi (t). (4c) − i i in i in p p Here γ (k = p,s,i) are the transmission losses through the output coupler and µ are all other extra losses, k k γ′ =γ +µ (k =s,i)arethe totallosses. τ isthe round-triptime ofthe threemodes inthe cavity,θ is the phaseof k k k p the pump field, aˆl (t) (l =s,i) are the input signaland idler fields, andˆbm(t) (m=p,s,i)are the quantum vacuum in in noise of the three fields induced by the extra losses. Assuming the loss factors γ =1, γ =γ =γ, µ =µ =µ and p s i s i γ′ =γ′ =γ′, then the oscillation threshold is obtained as s i εpth =γ′/(χΓ). (5) aˆlin =αline−iθl(l=s,i), where θl are the phases of the input signal and idler fields. We introduce the amplitude qu(cid:10)adra(cid:11)ture Xˆ = aˆ+aˆ† 2 and phase quadrature Yˆ = i aˆ aˆ† 2. When the relative phase between the pump − − and the seed ϕ =(cid:0) θ ((cid:1)θ(cid:14) +θ ) = 0, the system is in a p(cid:0)aramet(cid:1)r(cid:14)ic amplification state, and the correlation noise p s i − spectra can be given by 4σ V =V =1 η , (6) Xˆs−Xˆi Yˆs+Yˆi − esc(1+σ)2+Ω2 where η = γ/γ′ is the escape efficiency, σ = εp/εpth is the normalized pump parameter, and Ω = ωτ/γ′ is the esc normalizedanalyzing frequency. When the relative phase betweenthe pump andthe seedϕ=θ (θ +θ )=π, the p s i − system is in a parametric deamplification state, and the correlation noise spectra can be given by 4σ V =V =1 η , (7) Xˆs+Xˆi Yˆs−Yˆi − esc(1+σ)2+Ω2 Considering the total detection efficiency of the system, η , Eq. (7) can be rewritten as det 4 p/p th V =V =1 η η , (8) Xˆs+Xˆi Yˆs−Yˆi − det esc p 2 1+ p/p +Ω2 th (cid:16) p (cid:17) where η = η η η , η is the propagation efficiency, η is the homodyne detection efficiency and η is det prop hd phot prop hd phot the quantum efficiency of the photodiode. The normalized pump power is given by p/p =σ2, where p is the actual th pump power and p =γ′2 χ2Γ2 is the threshold pump power. th The inseparability criter(cid:14)io(cid:0)n can(cid:1)be expressed as [22] 8 p/p th V =V +V =2 η η <2. (9) Xs+Xi Ys−Yi − det esc p 2 1+ p/p +Ω2 th (cid:16) p (cid:17) From Eq. (3), (4) and (5), different pump modes correspond to different coupling coefficients and thus different nonlinear efficiencies, leading to different pump thresholds. The coupling coefficient for the HG signalmode u (~r) 00 00 with HG pump mode is 00 +∞ 2 Γ= v (~r)[u (~r)] d~r =1, (10) Z 00 00 −∞ 3 so the oscillation threshold for the HG signal mode with HG pump is p00→00 = γ′2 χ2Γ2 = γ′2 χ2. For the 00 00 th HG signal mode u (~r) generation with all possible pump, we have the expression from(cid:14)(cid:0)Eq. (1(cid:1)) (cid:14) 10 10 ∞ +∞ ∞ 2 Γ= c v (~r)[u (~r)] d~r = c Γ , (11) nZ n0 10 n n nX=0 −∞ nX=0 +∞ 2 where Γ = v (~r)[u (~r)] d~r denotes the coupling coefficient of the nth order HG pump mode. These are n −∞ n0 10 R +∞ 2 Γ = v (~r)[u (~r)] d~r =1/2, (12) 0 Z 00 10 −∞ +∞ Γ = v (~r)[u (~r)]2d~r =1 √2, (13) 2 Z−∞ 20 10 . and Γ = 0 for all other n. The HG signal mode threshold with an HG pump mode (c =1) is p00→10 = n 10 00 0 th γ′2 χ2Γ2 =4γ′2 χ2, and with an HG pump mode (c =1) it is p20→10 =γ′2 χ2Γ2 =2γ′2 χ2. 0 20 2 th 2 F(cid:14)o(cid:0)r the(cid:1)optima(cid:14)l pump mode, Γ = c Γ + c Γ = (1/2)c + 1 √2 c , a(cid:14)n(cid:0)d c2(cid:1)+ c2 =(cid:14)1. The maximum 0 0 2 2 0 2 0 2 value of Γ is √3 2, with c = 1/3 and c = 2/3, so the opti(cid:0)ma(cid:14)l pu(cid:1)mp mode is vp = 1/3v + 2/3v , 0 2 00 20 a superposition o(cid:14)f HG and HpG modes. Thpe HG signal mode threshold with the optpimal pumppmode is 00 20 10 popt→10 =γ′2 χ2Γ2 =4γ′2 3χ2. th .(cid:0) (cid:1) (cid:14) 2.0 HG20 pump HG00 pump 1.5 HGopt pump V 00 00 opt 10 20 10 1.0 pth pth pth 0.5 0.0 0.0 0.5 1.0 1.5 2.0 00 00 p/p th FIG.1: TheoreticalinseparabilitiesV againstnormalizedpumppowerp/p0th0→00 forthreepumpmodes,HG00 (bluesolidline), HG20 (green dashed line) and the optimal pump mode HGopt (red dotted line) under ideal conditions. The parameters are ηdet =1, ηesc =1, Ω=0. Fig. 1givesthetheoreticalcurvesoftheinseparabilitiesversusnormalizedpumppowerforthethreedifferentpump modes HG , HG , and the optimal superposition under ideal conditions. Under HG pumping, the HG signal 00 20 00 00 mode threshold is p00→00, which is one-quarter that of the HG signal mode p00→10 = 4p00→00. When the pump th 10 th th power reaches the HG threshold p00→00, the system starts to oscillate in the HG mode, so the maximum HG 00 th 00 10 entanglementcannotbeobtained. However,withHG pumping,theHG signalmodewillnotbeexcited. TheHG 20 00 10 signal mode threshold p20→10 =2p00→00 can be reached with enough pump power in theory, so the maximum HG th th 10 entanglement can be obtained using an HG pump. With optimal superposition mode pumping, the HG pump 20 00 mode comprises 1/3 the total pump power. The threshold for the HG signal mode is popt→10 =4p00→00 3. Hence 10 th th the maximum power of the HG component of the pump is 4p00→00 9, which is much smaller than the HG(cid:14) signal 00 th 00 modethresholdp00→00. TheHG signalmodewillthereforenotoscil(cid:14)lateinunderoptimalmodepumping. Moreover, th 00 since the HG signal mode threshold is much lower than for pure HG pumping, the maximum entanglement can 10 20 be obtained at lower pump power. 4 EXPERIMENT FIG.2: Schematicoftheexperimentalsetup. NOPA:non-degenerateoptical parametricamplifier, KTP:typeIIKTPcrystal, M1 and M2: cavity mirrors, BS: beam splitter, PBS: polarizing beam splitter, MCs: mode converters, SG: signal generator, Servo: servo amplifier circuit for feedback system, PZTs: piezoelectric transducers, DBS: dichroic beam splitter, Local: local oscillator, BHDs: balanced homodynedetectors, +/-: positive/negative power combiner, and SA: spectrum analyzer. The experimental setup is depicted in Fig. 2. A continuous wave all solid state laser source emits both infrared at 1080 nm and green light at 540 nm. The infrared beam passes through a mode converter (MC1), which converts the HG mode into the HG mode. A part of the HG mode is injected into a non-degenerate optical parametric 00 10 10 amplifier (NOPA) as the seed beam, and the rest of it is used as the local oscillator for homodyne detection. The green beam is used as the pump beam. It is split into two, one beam pass through the mode converter MC2, which converts HG mode into HG mode, the other beam is still HG mode, then the two beams are combined by a 00 20 00 beamsplitter,generatingthe superpositionpump mode. Bythis arrangement,wecanchooseto passeitherthe HG , 00 the HG , or the superposition pump mode. 20 To lock the relative phase between the HG and HG modes, we use an iris aperture to pass only the center of 00 20 the beam profile to a photodiode. With a lock-inamplifier, the relative phase is lockedto zero. The mode converters and the NOPA cavity are locked using the standard Pound-Drever-Hall(PDH) technique [23]. The NOPA cavity consists of two 30 mm radius of curvature plano-concave mirrors and a 3 3 10 mm3 type II × × KTP crystalin the center. The seed beam input mirror M1 is highly reflective (R¿99.95%)at both 1080nm and 540 nm. The transmittance T of the output coupler M2 is 6% at 1080 nm and T¿95% at 540 nm. The cavity is nearly concentric with a length of 62.5 mm and has a waist of 41 µm in the infrared and 29 µm in the green. The NOPA has a finesse of 84 for the signal beam with a free spectral range of 2.4 GHz and a bandwidth of 28 MHz. We lock the relative phase between the seed and the pump beam in the parametric deamplification regime with PZT2. The NOPAoutput beamsandthe greenbeampassthroughadichroicbeamsplitter (DBS), whichreflectsonly the infraredbeamtobemeasured. ThisisdividedintotwopartsbyaPBS.Theyaredetectedbytwobalancedhomodyne detectors(BHDs). ThephotocurrentsfromthetwoBHDsfeedapositive/negativecombiner(+/-),andthoseoutputs are recorded by a spectrum analyzer (SA). The correlation noise spectra of the amplitude sum and phase difference of the signal and idler beams are measured by scanning the phase of the local infrared beam using a mirror mounted on piezoelectric transducer PZT3. 5 EXPERIMENTAL RESULTS The experimental parameters in our experiment are as follows. The analyzing frequency is 5 MHz, the resolution bandwidth (RBW) is 300 kHz, and the video bandwidth (VBW) is 1 kHz. The bandwidth of the NOPA is 28 MHz (from which Ω = 5 MHz/28 MHz = 0.18). The various efficiencies are η = 0.89 0.02, η = 0.90 0.01, η prop phot hd ± ± = 0.81 0.02, and η = 0.79 0.01, thus the total efficiency η = 0.51 0.04. The pump threshold for the HG esc total 00 ± ± ± signal mode with an HG pump is p00→00 = 510 mW. From theoretical prediction, the oscillation threshold for the 00 th HG signalmode is p00→10 = 2.04W with HG pumping, it is p20→10 = 1.02W with HG pumping, and with the 10 th 00 th 20 optimal superposition mode pumping popt→10 = 680 mW. th The measured entanglement inseparabilities V are plotted against the normalized pump power p/p00→00 for the th three different pump modes in Fig. 3. The corresponding theoretical curves in experimental conditions are also depicted. 2.0 HG pump 00 HG pump 1.8 20 HG pump opt 1.6 V 00 00 opt 10 pth pth 1.4 1.2 1.0 0.0 0.5 1.0 1.5 00 00 p/p th FIG. 3: The inseparabilities V versus normalized pump power p/p0th0→00, where p0th0→00 = 510 mW. Data points from the experiment are bluesquares for HG00 pumping,green circles for HG20 pumpingand red triangles for theoptimal pump mode HGopt. Thesolid curvesare thetheoretical valuesin experimental conditions for thethree pumpmodes. From Fig. 3, the entanglement increases with the increasing pump power for the three pump modes and there is goodagreementbetweentheoryandexperiment. Atagivenpumppower,theoptimalpumpmodeHG outperforms opt the other two modes and the HG pump mode outperforms HG . However, the minimum value of V is not close 20 00 to zero as fig.1 due to the nonideal cavity and detection system. The maximum pump power for HG mode in our 00 experimentis500mW,since the oscillatingthresholdofthe HG signalmode is510mW,athigherpower,the OPO 00 will oscillate onthe HG mode, so the maximumentanglement ofHG mode can notbe obtained with HG pump 00 10 00 mode. However, with HG pump mode or the optimal pump mode, the maximum entanglement of HG mode can 20 10 be obtained. Moreover, with the optimal pump mode HG , the maximum entanglement can be obtained at lower opt pump power compared with HG pumping. 20 Fig. 4 gives the HG mode correlation noise powers with the three different pump modes. For HG pumping at 10 00 500 mW, the amplitude sum power was 2.36 0.07 dB and the phase difference power was 2.56 0.06 dB. For HG 20 ± ± pumping at 670 mW these were 2.92 0.08 dB and 2.76 0.10 dB. For HG pumping at 670 mW, the powers were opt ± ± 3.28 0.18dB and 2.92 0.15 dB. The HG mode entanglement inseparabilities for the three pump modes are 10 ± ± V = ∆2 Xˆs +Xˆi + ∆2 Yˆs Yˆi =1.13 0.02<2 (14) 00 D (cid:16) 10 10(cid:17)E D (cid:16) 10− 10(cid:17)E ± 6 Noise Power (dB)12048 (a1) Noise Power (dB)12048 (b1) -4 2.36–0.07 dB -4 2.56–0.06 dB 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 Time (s) Time (s) Noise Power (dB)12048 (a2) Noise Power (dB)12048 (b2) -4 2.92–0.08 dB -4 2.76–0.10 dB 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 Time (s) Time (s) Noise Power (dB)12048 (a3) + 1Noise Power (dB)2048 (b3) + -4 3.28–0.18 dB -4 2.92–0.15 dB 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 Time (s) Time (s) FIG.4: TheHG10 modecorrelation noisepowersfortheamplitudesum(a1-a3)andthephasedifference(b1-b3). Thetoprow was taken using 500 mW of HG00 pumping, the middle row with 670 mW of HG20 pumping, and the bottom row with 670 mW of thesuperposition HGopt pumping. V = ∆2 Xˆs +Xˆi + ∆2 Yˆs Yˆi =1.04 0.02<2 (15) 20 D (cid:16) 10 10(cid:17)E D (cid:16) 10− 10(cid:17)E ± V = ∆2 Xˆs +Xˆi + ∆2 Yˆs Yˆi =0.98 0.04<2 (16) opt D (cid:16) 10 10(cid:17)E D (cid:16) 10− 10(cid:17)E ± Considering the total detection efficiency η = η η η = 0.65 0.04, the inseparabilities of Eqs. (14-16) det prop phot hd ± become 0.66 0.03, 0.52 0.03 and 0.43 0.06. Compared with HG pumping, the inseperability is enhanced by 00 ± ± ± η =53.5% using the optimal pump mode. Summarizingthe experimentalresults,wecannotobtainthe maximumentanglementofthe HG modewithHG 10 00 pumping because of the low HG threshold. With HG pumping, this is not the case. Theoretically, the HG 00 20 10 signal mode threshold can be reached and the maximum entanglement can be obtained, but in our experiment the laser-limited pump power is insufficient. With the optimal pump mode, the HG signal mode threshold is lower 10 and the maximum entanglement can be obtained with lower pump power. Experimentally however, generating the optimal pump mode is relatively complicated and somewhat difficult. Using HG pumping is operationally much 20 easier and with sufficient power we can obtain the same degree of entanglement as the optimal pump mode. CONCLUSION We experimentally studied HG mode entanglement in a type II OPO with three pump modes, HG , HG , and 10 00 20 a superposition of the two modes. The superposition mode, a one-third HG and two-thirds HG combination, is 00 20 theoretically optimal and experimentally shown to be able to obtain a higher entanglement at lower pump power. The experimentalresultsmatchthe theoreticalpredictionverywell. The degreeofentanglementisstillrelativelylow resulting from extra losses and various inefficiencies in our experiment. The technique holds promise to obtain more than 10 dB squeezing for applications in quantum imaging [24, 25]. It is an efficient way to improve the squeezing of high-order spatial modes. 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