Abstract
Biphotons with narrow bandwidth and long coherence time can enhance lightatominteraction, which leads to strong coupling between photonic and atomic qubits.Such strong coupling is desirable in quantum information processing, quantumstorage and communication. In particular, paired photons with a long coherencetime over submicroseconds facilitate the direct manipulation of biphoton wavefunction.In this paper, we report the narrowband biphotons with a coherence time of2.34 μs generated from spontaneous fourwave mixing (SFWM)in a dense cold atom cloud, in which the antiStokes photons go through anarrow electromagneticallyinduced transparency (EIT) window. In our knowledge,this is the best record of coherence time for paired photons achieved so far.A number of factors limiting the coherence time are analyzed in detail. Wefind the EIT coherence plays an essential role in determining the coherencetime for paired photons. The EIT dephasing rate is the ultimate limit to thecoherence time and an ultralong coherence time above ten microseconds ispossible by further improvement of the dephasing rate below 100 kHz.
Introduction
Nonclassical paired photons play an important role in quantum informationprocessing, quantum storage and telecommunication. They are highly correlatedin pairs and their secondorder cross correlation function violates the CauchySchwartzinequality. These paired photons constitute heralded single photon source,i.e., when one of them is detected, its counterpart can be considered as asingle photon in Fock state. Traditional schemes, spontaneous parametric downconversion (SPDC) in nonlinear crystal, generate broadband paired photonswith linewidth up to the order of THz^{1,2,3,4,5}. The pairedphotons produced in the SPDC scheme are usually considered to be simultaneous,with their coherent time separation, i.e., coherence time, of the order ofpicosecond. However, efficient quantum state transfer at lightatom interfacerequires narrowband paired photon source or narrowband single photons. SPDCwith high finesse cavity^{6,7,8,9} efficiently generates pairedphotons with linewidths around 10 MHz and further suppression belowatomic natural linewidth (e.g., 6 MHz for rubidium atoms) demands delicatecavity fabrication^{10}. Writeread sequenced fourwave mixingprocess provide an access to the generation of narrowband paired photonsin atomic gas^{11,12} and the paired photons produced throughatomic spin states storing and retrieving are shown to be highly correlated^{13}. Instead of timeseparate writing and reading processes, spontaneousfourwave mixing (SFWM) with both controlling beams present simultaneouslyin a generation cycle is utilized to produce timefrequency entangled pairedphotons, i.e., biphotons, in cold atomic ensembles^{14,15,16}.Therefore in timefrequency domain, the twophoton state can be expressedby a temporal correlation function, or a probability wavefunction^{17,18}.
The coherence time of the biphotons is defined as the temporal length ofthe biphoton correlation function and it represents the largest time distanceof the biphotons emitted from a source. SPDC scheme produce wideband biphotonswith the temporal wavefunction too short to be detected directly by the fastestsingle photon counter, or to be modulated directly in time domain. Moreover,the ultrashort coherence time complicates the future quantum networking basedon the entangled paired photons emitted from independent sources, which requireaccurate synchronization of the excitation laser pulses^{19,20}.To largely increase the biphoton coherence time, in this report, we demonstratethe generation of biphotons from a cloud of laser cooled ^{85}Rbatoms prepared in a twodimensional magnetooptical trap (MOT), with the coherencetime exceeding far above nanoseconds. When electromagnetically induced transparency(EIT) is introduced into the SFWM process, the linewidth and correspondingtemporal length of the biphoton correlation function can be manipulated bycontrolling the slow light effect of the produced photons. More interestingly,if one could produce a single photon with temporal waveform up to tens of μs,demanding 10 kHz modulation bandwidth, the single photon source couldthen be easily manipulated as it is an optical pulse. These ultranarrowbandpaired photons have shown great advantage in heralded single photon waveformmanipulation^{17,21,22,23}, single photon quantum storage^{24} and imaging^{25}.
We would like to start with the fourlevel doubleΛ atomic system forbiphoton generation, as shown in Figure 1(b). For theexcited states 3〉 and 4〉 of ^{85}Rb atoms, the populationdecay rates are Γ_{3(4)} = 2π × 6 MHz.The dephasing rates of the excited states are γ_{13(14)}= (1/2)Γ_{3(4)} for cold atomic ensemble. The states 1〉,2〉 and 3〉 with a resonant strong coupling beam ω_{c}constitute a EIT configuration. Though the transition between the state 1〉and 2〉 is prohibited, the nonzero ground states dephasing rate γ_{12}is caused by atomic collisions, atomic thermal motions and Lamer precessionof atoms in nonzero B field. In the MOT system we build, the Lamer precessionof atoms are expected to be the main reason. Further we estimate the dephasingrate of the EIT coherence by measuring the probe beam transmission spectrumwith the resonant coupling beam and thus obtain γ_{12}= 2π × 0.03 MHz between 5S_{1/2}F= 2 and 5S_{1/2}F = 3. Figure 1(a)shows the backward geometry of fourwave mixing process, in which the oppositearrangedpump ω_{p} and coupling beams ω_{c}generate paired Stokes (ω_{s}) and antiStokes (ω_{as})emitted in opposite directions. The biphotons, Stokes and antiStokes photons,are simultaneously generated from the SFWM nonlinear process in the atomicensemble. The Stokes photons are usually fardetuned from atomic transitions,and therefore they propagate always with the speed of light in vacuum c.On the other hand, the antiStokes photons are produced with a resonant couplinglaser beam, which creates a coherent EIT window for the photons to propagatethrough with a slow group velocity V_{g}. In this regime, theEIT coherence determines the coherence of the paired photons. There are threeimportant parameters to determine the slow light feature. The first one isthe optical depth (OD) of the medium, defined as OD = Nα_{0}L,where N denotes the atom density, α_{0} is the resonantabsorption coefficient and L is the effective length of the atom cloud.The second parameter is the coupling field Rabi frequency Ω_{c},representing the coupling field strength. The last but not the least factoris the dephasing rate γ_{12} of the EIT coherence. Whenignoring the dephasing rate, we could increase coherence time T_{c}of the biphoton simply by prolonging the EIT delay time τ_{g}= L/V_{g}, which is proportional to the ratio OD/Ω_{c}^{2}.For example, Du et al.^{26} reported timefrequency entangledpaired photons with 0.9 μs coherence time, generated froma lasercooled atomic cloud with optical depth (OD) of 53. From a darklinetwodimensional MOT^{29} with OD as high as 130, Zhao et al.^{27} reported that the coherence time of these narrowband entangledpaired photons reaches 1.7 μs. By phase locking two spatialsymmetrical paths of the SFWM process, Liao et al.^{28}show polarization entangled narrowband paired photon with the coherence timeof 300 ns, which can be prolonged to 900 ns when they graduallyreduce the coupling laser power. However, with a very low laser field coupling,the dephasing rate between two hyperfine states becomes important. Reducingthe coupling laser power does not always prolong the coherence time T_{c},but instead results in a maximum T_{c} at optimum Ω_{c}.In the following sections, we demonstrate that the limit of T_{c}is corporately determined by OD and the dephasing rate of the EIT groundstatecoherence. In the case of extremely large OD, the coherence time of the biphotonfinally approaches the ultimate limit determined by the dephasing rate. Withan optimized dephasing rate and OD = 100 currently available in thesystem, we can realize a coherence time T_{c} up to 2.34 μs,which is the longest temporal length for biphoton reported to the best ofour knowledge.
The twophoton state generated in the SFWM described in Figure1 can be described by a temporal biphoton wavefunction^{18},
in which, ψ is the wavefunctionamplitude and τ = t_{as} − t_{s}denotes the delay time. is the nonlinearparametric coupling coefficient and χ^{(3)} is thethirdorder nonlinear susceptibility. The longitudinal function , where Δk = k_{as} − k_{s} −(k_{c} − k_{p}) cos θ is thephase mismatching for the backward geometry shown in Figure1. Also, we include the intensity distribution f(z) ofthe pump or coupling beam along the z axis. According to experimentsetup, we define a Gaussian beam distribution with a e^{−2}full width of d = 1.2 cm for both pump and coupling laser beams,and thus .
From equation(1), we obtain the numerical results forthe jointdetection probability. Furthermore, equation(1)indicates that the spectrum of twophoton wavefunction is determined by theproduct of the κ(ω) and Φ(ω). Nextwe take some approximations to obtain an analytic solution for the wavefunctionamplitude. Firstly, we assume that the EIT slow light effect play a dominantrole to decide the twophoton wavefunction. Therefore we consider the casewhere Δω_{g} < Ω_{e}, in which denotes the full width half maximumbandwidth of the function Φ(ω) and is the spectral separation of the two symmetric peak value of κ(ω)^{26}. Now, Φ(ω) decides the twophoton wavefunctionenvelope, while is approximatedas a constant κ_{0}. With this assumption, the coherencetime of the paired photons is solely ascribed to the fact that, the antiStokesphotons propagate through the atomic cloud with a slow velocity compared tothe paired Stokes photons. Secondly, we assume θ → 0 or and thus f(z) ≈ 1 andthe z integration term becomes a sinc function .With the above assumption, we can simplify the wavefunction amplitude as,
where, the EIT loss term is included. The EIT loss term αL is normallya small value close to 0. Through Taylor expansion at this small value, equation(2) gives an analytical solution after integral:
In other regions τ < 0and τ > L/V_{g}, Ψ(τ)= 0. If we ignore EIT loss term αL completely, equation(3)becomes a rectangular function Π(τ: 0, τ_{g}= L/V_{g}) as Refs. 18,and thus the 1/e coherence time is directly equals to EIT delay time τ_{g}.Otherwise, the twophoton waveform amplitude exhibits a linear decay endingat τ = τ_{g}(1/αL + 1/2). When αLtends to be infinity, the coherence time is bounded by τ_{g}/2.
Results
We prepare the cold atom cloud with optical depth OD = 100, andfor SFWM process we set the coupling Rabi frequency as Ω_{c}= 4.2γ_{13}, 1.7γ_{13} and 1.2γ_{13},calculated from . To guarantee thelow gain limit condition for SFWM process, we use a low pump power correspondingto Ω_{p} = 0.36γ_{13}. Figure2 shows the coincidence measurement results over these three cases,along with the numerical results of the two photon wavefunction amplitude equation(1). Figure 2(a) shows the caseof Ω_{c} = 4.2γ_{13}. Apart from thebiphoton optical precursor emerging immediately after τ = 0 ns,the twophoton joint detection waveform exhibits a Gaussian shape impressedfrom the controlling beams transverse beam profile f(z). At τ= 300 ns, the normalized crosscorrelation function . In contrast, we have and . Thus the CauchySchwartz inequality is violated by a factor of 506. Therefore,the generated photon pair source is proved to be nonclassical. Further, withthe Stokes photon as reference, its paired antiStokes photon constitutesa heralded single photon. We measure the conditional secondorder correlationfunction for window lengths of 500 ns,and hence we verify the Fockstate single photon nature of the heralded antiStokesphotons. The 1/e coherence time of the twophoton waveform is 553 ns,and correspondingly the spectral bandwidth of the heralded single photon (alsofor the photon pair) is estimated as Δω_{g} = 2π ×1.5 MHz, which is smaller than the atomic natural linewidth 2π ×6 MHz of ^{85}Rb 5P state. Taking into account all thedetection efficiencies and 10% of duty cycle, the photon pairs generated rateis 6436 pairs/s. Figure 2(b) shows the case of Ω_{c}= 1.7γ_{13} with the same OD. The 1/e coherencetime 2340 ns and corresponding bandwidth 2π × 0.38 MHz.The photon pair generation rate is 2043 pairs/s, with the maximum ofthe normalized crosscorrelation function .Therefore, the CauchySchwartz inequality is still violated by a factor of12. With a lower pump beam power Ω_{p} = 0.21γ_{13},we obtain for a 2400 ns coincidencewindow. Further decreasing the coupling power diminishes the visibility as Figure 2(c), Ω_{c} = 1.2γ_{13},and distorts the twophoton waveform as a linear decay tail. The dash linein Figure 2(c) denotes the analytical solution of equation(3) and it matches the experimental data and numericalcalculation well. The EIT loss term αL largely reduces the wavefunctionvisibility and maximum is around3, with photon pair generation rate about 826 pair/s only. Consideringthe visibility, the nearrectangular waveform envelope in Figure2(b) is the best we obtain with our present experimental setup.
The second important result is concerning the lower bound for couplinglaser power. Figure 3 plots the measured 1/ecoherence time (denoted as T_{c} below) and as a function of Ω_{c} in four cases: OD= 22, 35, 50, 100. All of them satisfy the condition Δω_{g} < Δω_{tr},in which denotes the spectral widthof the EIT window. Accordingly, in Figure 3(a) we plotthe numerical results simulated from equation(1), andfor comparison, we also calculate the numerical curves for OD = 300,1000. If we take the atomic ground states dephasing rate as γ_{12}= 2π × 45 kHz, the experimental data perfectly matchthe numerical curves, except that the data point at OD = 100, Ω_{c}= 1γ_{13} is measured as 1900 ns, much lower thanthe calculated value. This is because the biphoton waveform visibility dropsand some part of the coincidence counts are buried into the accidental counts.As is discussed above, at a low level of coupling field, the EIT loss termseriously damages the waveform shape, which instead exhibits a linear decaytail as equation(3). Therefore, Figure3 shows that the numerical lines for various ODs all indicatea drop of T_{c} at small Ω_{c} and theoptimum Ω_{c} value increase with OD. On the other hand, Figure 3(b) shows the measured maximum of at these four OD cases. When Ω_{c} < γ_{13}, is below 2 and does not violate CauchySchwartzinequality. The limit of the coherence time is corporately determined by ODand the atomic ground state dephasing rate γ_{12}. Therefore,in Figure 4(a) we simulate the maximum T_{c}as a function of OD when keeping γ_{12} and in Figure4(b) vary γ_{12} when maintaining OD. Firstly, Figure 4(a) indicates extreme ODs bring the maximum T_{c}to approach 3.5 μs, which is determined by 1/γ_{12}. T_{c}glows very slow after OD > 200. On the other hand, Figure4(b) shows a sensitive improvement of T_{c} when reducing γ_{12}.Hence, γ_{12} is the ultimate limiting factor. To furtherimprove the biphoton coherence time, we need to minimize the dephasing rate γ_{12}.
In conclusion, we have demonstrated that optical depth OD, the couplingfield and the EIT dephasing rate corporately determine the biphoton coherencetime. In a dense cold atom cloud, the biphoton coherence time is efficientlymanipulated by the EIT slow light effect and therefore can be increased withhigher OD and lower coupling field amplitude. However, when the EIT couplingstrength is reduced further, the dephasing rate γ_{12}becomes important and the EIT loss diminishes the twophoton waveform visibility,and thus ruin the nonclassical nature. The dephasing rate γ_{12}is the ultimate limiting factor. In a twodimensional MOT system, γ_{12}is mainly caused by the quadruple field produced by the trapping coil. Asin our experiment, through balancing the cooling beams power to position theatomic cloud along the field center, we can reduce γ_{12}to 283 kHz and thus produce paired photons with temporal coherencetime 2.34 μs with a reasonable OD of 100. The measured violationfactor of the CauchySchwartz inequality is 12. The conditional secondorderautocorrelation function of antiStokes photons with reference to Stokesphotons is measured as , which verifiesthe quantum Fock state nature. To push the limit even further, it is provedthat shutting down the trapping magnetic field in the measurement window andmaintaining a zero magnetic field at the atomic cloud is an efficient methodto reduce γ_{12} to below 100 kHz^{30}.According to our calculation shown in Figure 4(b), onecould obtain T_{c} = 10 μs if γ_{12}= 0.004γ_{13} ≈ 75 kHz.
Methods
The cold atomic ensemble is prepared in a twodimensional ^{85}Rbmagnetooptical trap (MOT) with length L = 1.5 cm. The coolinglaser beam is reddetuned from the atomic transition 5S_{1/2}F= 3 → 5P_{3/2}F = 4 by 20 MHz. The totalpower of the six cooling beams is 130 mW and each beam has an intensityof 6.42 mW/cm^{2}. The repump beam is onresonanceto the atomic transition 5S_{1/2}F = 2 → 5P_{3/2}F= 2, which collects the atoms back to the cooling circulation. The total powerof the repump beam is 30 mW and it is applied onto the MOT along ±xaxis shown in Figure 1. The system is run periodicallywith a MOT preparation time of 4.5 ms followed with a measurement windowof 0.5 ms. The repump beam switches off 300 μs beforethe cooling beam is off and therefore over 99% of trapped atoms are pumpedto the ground state 5S_{1/2}F = 2. The center gradientof the 2D MOT quadrature magnetic field is 10 G/cm and remains alwayson. To minimize the Zeeman shift for atomic energy state, we carefully balancethe six cooling beams to align the laser cooled atoms along the positionsof zero transverse B field.
After the atoms are cooled and trapped, we apply continuous pump (ω_{p})and coupling (ω_{c}) lasers within the measurement window.Within the measurement window, the pump and coupling beam are present simultaneouslyon the cold atom cloud, with polarization σ^{+} and σ^{−}respectively. The paired Stokes (ω_{s}) with polarization σ^{+}and antiStokes (ω_{as}) photons σ^{−}are emitted in opposite directions, to satisfy the angular momentum conservationin the FWM process. We collect the paired photons along the longitudinal axis(z axis as denoted), along which we obtain the largest OD for the cloud.To minimize the scattered optical noise from the strong laser beams, the pumpcouplingaxis is deviated from the z axis by θ = 3°. The pumpbeam is bluedetuned to the transition 1〉 → 4〉 by Δ_{p}= 146 MHz, to avoid strong excitation and maintain low gain limit.The coupling beam is onresonance to the transition 2〉 → 3〉,and therefore constitute a threelevel Λ EIT scheme for antiStokes photons.
Either of the collection end is composed of a quarter wave plate followedby a polarization beamsplitter, a single mode fiber (SMF, fiberfiber couplingefficiency 70%) and a FabryPerot filter (each with 70% transmission), andfinally a single photon counting module (SPCMAQRH14FC from Excelitas, eachwith 50% detection efficiency). The coincidence measurements are carried outwith Time to Digital Converter (DPC230 from Becker & Hickl GmbH).
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Acknowledgements
This work is supported by the National Natural Science Foundation of China(NSFC) through the major research plan (Precision Measurement Physics GrantNo. 91436211). Also, the work is supported by NSFC research programs, includingGrant No. 11204086, No. 11374003, No. 11234003, No. 11129402, National BasicResearch Program of China (973 Program Grants No. 2011CB921604) and ShanghaiScience and Technology Committee (Grant No.13PJ1402100).
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J.F.C. designed the experiment. Z.H. and P.Q. built up the system. Z.H.performed the experiment and analyzed the data. L.Z. worked out the analyticalsolution. J.F.C. wrote the manuscript and all authors contributed to the finalmanuscript. The whole program is under the supervision of W.P.Z.
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Han, Z., Qian, P., Zhou, L. et al. Coherence time limit of the biphotons generated in a dense cold atomcloud. Sci Rep 5, 9126 (2015). https://doi.org/10.1038/srep09126
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DOI: https://doi.org/10.1038/srep09126
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