Broadband coherent optical memory based on electromagnetically induced transparency
Yan-Cheng Wei, Bo-Han Wu, Ya-Fen Hsiao, Pin-Ju Tsai, Ying-Cheng Chen
BBroadband coherent optical memory based on electromagnetically inducedtransparency
Yan-Cheng Wei , , Bo-Han Wu , Ya-Fen Hsiao , Pin-Ju Tsai , , and Ying-Cheng Chen , ∗ Institute of Atomic and Molecular Sciences, Academia Sinica, Taipei 10617, Taiwan Department of Physics, National Taiwan University, Taipei 10617, Taiwan and Center for Qauntum Technology, Hsinchu 30013, Taiwan (Dated: September 22, 2020)Quantum memories, devices that can store and retrieve photonic quantum states on demand,are essential components for scalable quantum technologies. It is desirable to push the memorytowards the broadband regime in order to increase the data rate. Here, we present a theoreticaland experimental study on the broadband optical memory based on electromagnetically-induced-transparency (EIT) protocol. We first provide a theoretical analysis on the issues and requirementsto achieve a broadband EIT memory. We then present our experimental efforts on EIT memory incold atoms towards the broadband or short-pulse regime. A storage efficiency of ∼ with a pulseduration of 30 ns (corresponding to a bandwidth of 14.7 MHz) is realized. Limited by the availableintensity of the control beam, we could not conduct an optimal storage for the even shorter pulsesbut still obtain an efficiency of larger than with a pulse duration of 14 ns (31.4 MHz). Theachieved time-bandwidth-product at the efficiency of is 1267. I. INTRODUCTION
Quantum memories are crucial components in linear-optics-based quantum computation and long-distancequantum communication based on quantum repeaterprotocol[1, 2]. There have been significant efforts andprogress on quantum memories in the past two decades.Parameters to evaluate the performance of a quantummemory include fidelity, efficiency, storage time, band-width or capacity, and noise level. It is still a great chal-lenge to develop a quantum memory with a high perfor-mance on all aspects. In this work, we focus our discus-sion on achieving a broad bandwidth while maintaininga high efficiency for coherent optical memories based onEIT protocol.Optical memory based on the off-resonant Ramantransition is generally considered to be advantageous inachieving a high bandwidth. Based on Raman memoryprotocol, there have been many works pushing the band-width towards ∼
100 MHz-1 GHz range with an efficiencyof ranging from 10-30 %[3–8]. With the same protocol,one recent work achieved an efficiency of 82 % at a band-width of ∼
100 MHz[9].However, the on-resonant two-photon transition in a Λ -type atomic system also allows one to implement thebroadband optical memory, as long as the optical depthof the media is high and the intensity of the control beamis strong enough[10–13]. The memory character is quitedifferent depending on the ratio of the spectral band-width of the probe pulse (denoted as B ) to the EITtransparent bandwidth ( ∆ ω EIT ). If B is larger than ∆ ω EIT , the memory operation involves coherent absorp-tion of the probe pulse by the two Autler-Townes absorp-tion peaks and this is called the Autler-Townes-splitting ∗ Corresponding author: [email protected] (ATS) protocol[11–13]. The light-matter interaction un-der such a condition is non-adiabatic which involves thetransfer between the optical coherence and the collec-tive ground-state (spin) coherence. In the opposite case(
B < ∆ ω EIT ), the memory operation is the EIT proto-col which relies on adiabatic elimination of the absorp-tion and coherent transfer between the optical field andthe collective atomic spin wave. The features and dif-ferences of the EIT and ATS protocols have been wellstudied[10, 12, 13]. A bandwidth of 14.7 MHz withan efficiency of 8.4 % using the ATS protocol has beendemonstrated[12]. An efficiency of up to 92% using theEIT protocol has been demonstrated but the bandwidthis only up to 2.2 MHz[14–16].In this paper, we explore the issues and requirementsin order to extend the bandwidth of adiabatic EITmemories. Except efficiency, we define another impor-tant figure of merit for a memory, waveform likeness(WL)[14, 16, 17], which quantifies the degree of distor-tion of a probe pulse. In the single-photon regime, wave-form likeness is related to the overlap of the temporalmode of photons with and without storage, which is cru-cial in some applications of quantum memory. We showthat waveform likeness can be considered as an exper-imental parameter to evaluate the degree of adiabatic-ity of a memory. For a steady-state EIT spectrum, weidentify that it can be categorized into two regimes inwhich the EIT transparent linewidth is proportional tothe intensity or field amplitude of the control field at lowand strong control field, respectively. We term these tworegimes as the intensity-linear or field-linear EIT band-width regime. For a pulsed probe case, we show thatEIT memory can be categorized into the narrowband orbroadband regime, with
B < Γ or B > Γ , respectively,where Γ is the spontaneous decay rate of the probe tran-sition. EIT memory is limited by efficiency in the nar-rowband regime and is limited by waveform likeness inthe broadband regime. We clarify that EIT storage can a r X i v : . [ phy s i c s . a t o m - ph ] S e p FIG. 1. (a) Transition scheme of the Λ -type EIT system. | g (cid:105) and | s (cid:105) are the two ground states and | e (cid:105) is the excitedstate. Ω p,c stand for the Rabi frequency of the probe andcontrol field, respectively. (b) Photonic storage through theEIT protocol. In both graphs, the yellow shaded area denotesthe spin coherence in the two ground states. maintain a high efficiency and a high waveform likenesseven in the broadband regime. The quantitative require-ments to achieve such a high-performance storage aregiven. Our theoretical study provides essential physicalinsights in implementing a broadband EIT memory.We then present our experimental efforts toward abroadband EIT memory. We achieved a storage effi-ciency of ∼ for probe pulses with a full-width-half-maximum (FWHM) temporal duration ( T p ) of 30 ns,which corresponds to a bandwidth of 14.7 MHz. Lim-ited by the available control intensity, we cannot achievethe optimized efficiency for T p <
30 ns but still obtain anefficiency of above for T p = 14 ns (31.4 MHz). Thetime-bandwidth product (TBP), defined as the ratio ofthe storage time at storage efficiency to the FWHMduration ( T p ), is an important figure of merit for memoryapplication. We achieve a TBP of 1267. II. THEORETICAL STUDY ON BROADBANDEIT STORAGE
The transition scheme of the Λ -type system for EITstorage is shown in Fig. 1 (a). The population is assumedto be prepared in the ground state, | g (cid:105) . The weak probefield, to be stored and retrieved on demand, drives the | g (cid:105) ↔ | e (cid:105) transition with a Rabi frequency of Ω p . Thecontrol field, with a Rabi frequency of Ω c , drives the | s (cid:105) ↔ | e (cid:105) transition.Under the rotating-wave approximation, the systemHamiltonian can be described by, ˆ H = − δ p ˆ σ ee − δ δ ˆ σ ss + 12 (Ω p ˆ σ eg + Ω c ˆ σ es + h.c. ) , (1)where ˆ σ ij ≡ | i (cid:105) (cid:104) j | denotes the atomic operator, δ p,c de-notes the one-photon detunings of the probe and controlfield, respectively, and δ = δ p − δ c denotes the two-photon detuning.Theoretical analysis is based on the Maxwell-Schrödinger equation (MSE) and the optical Bloch equa-tions (OBE). Under the weak-probe perturbation, the relevant equations are ∂ t σ eg = ( iδ p − γ ge ) σ ge + i c σ sg + i p ∂ t σ sg = ( iδ − γ sg ) σ sg + i c σ eg (2) ( 1 c ∂ t + ∂ z )Ω p = i D Γ2 L σ eg , (3)where D and L denotes the optical depth and the lengthof the atomic media, respectively[15]. A. Spectral Response
Eqs. 2 and 3 can be easily solved in the frequencydomain ( ω -space) by Fourier transformation[15]. The ω -space probe field can be obtained as, W p ( ω, z ) = W p ( ω,
0) exp[ ik ( ω ) z ] , (4)where k ( ω ) = ωc + D Γ4 L ωiω ( i ( ω + δ p ) − Γ2 ) + Ω c ≡ k ( ω ) + k ( ω ) , (5), where k ( ω ) = ω/c is the free-space wavenumberand k ( ω ) is the spectral response function of the EITmedium, W p ( ω, is the spectral amplitude of the in-put probe pulse. For simplicity, we assume δ c = 0 and the ideal case with γ sg = 0 . For the steady-state probe transmission, we can set ω = 0 and obtain T ( δ p ) = Exp ( − Re ( k ( δ p )) L ) . The FWHM bandwidth ofthe EIT transparent window ( ∆ ω EIT ) is one importantparameter and can be obtained as, ∆ ω EIT = (cid:115) D − ln 2 + 2 ln 2(Ω c / Γ ) − (cid:112) g ( D, Ω c )2 ln 2 Γ , (6)where g ( D, Ω c ) = ( D − ln 2)( D − ln 2 + 4 ln 2(Ω c / Γ )) . (7)The relationship between ∆ ω EIT and Ω c is depicted inFig.2 (a). In the weak-control regime, the EIT transpar-ent bandwidth is[15], ∆ ω EIT ≈ √ ln2 Ω c √ D Γ . (8)The EIT transparent bandwidth is linearly proportionalto Ω c or the control intensity such that we term it asthe intensity-linear bandwidth regime. In the strong-control regime, ∆ ω EIT ≈ Ω c , which scales linearly withthe control field amplitude such that we term it as thefield-linear bandwidth regime. By setting ∆ ω EIT equalto each other for both regimes, the transition Ω c betweenthese two regimes can be estimated to be Ω c ≈ . √ D Γ .This explains the trend in Fig.2 (a) why the intensity-linear regime is wider for a larger optical depth. It shouldbe pointed out that intensity-linear or field-linear regimedoes not necessarily correspond to the EIT or ATS pro-tocol for the memory[33]. As mentioned in the intro-duction, the relative ratio of the bandwidth of the probepulse ( B ) and the EIT transparent bandwidth ( ∆ ω EIT )determines the EIT or ATS regime. Also, the intensity-linear or field-linear bandwidth regime does not necessar-ily determines the broadband or narrowband EIT storageregime either. The broadband or narrowband EIT stor-age depends on the relative ratio of B and Γ , as will beexplained later. B. Pulse
In real applications, one needs to store optical probepulses of a finite bandwidth, instead of of a continuouswave. We consider the probe pulses with a temporalGaussian profile with an intensity FWHM duration of T p . In the frequency domain, the Rabi frequency of theprobe pulse reads, W p ( ω,
0) = T p Ω p √ (cid:32) − ω T p (cid:33) , (9)where Ω p is the peak Rabi frequency of the input probepulse in time domain. The FWHM bandwidth B of theprobe pulse in ω -space is B = ln T p . We consider theon-resonance case for the probe and control fields ( δ p = δ c = 0 ) and the ideal case with γ sg = 0 and γ ge = Γ2 .The general case can be referred to Ref. [15].Using Eqs 4, 5, and 9 and Fourier transform the probepulse back to the time domain, the solution of the slowlight pulse in the time domain can be obtained as [15], Ω p ( t, z = L ) = 1 √ π Ω p T p √ ln2 (cid:90) ∞−∞ dωe − iωt W p ( ω, e ik ( ω ) L . (10)Applying Taylor expansion of k ( ω ) with respect to ω ,one obtains the dispersion terms of an EIT medium, k ( ω ) = (cid:88) n a n ω n a = iD ΓΩ c , a = − D Γ Ω c ,a = 4 iD Γ(Ω c − Γ )Ω c , a = − D Γ (2Ω c − Γ )Ω c , (11)where the explicit form of dispersion terms up to O ( ω ) are written down. The dispersion term O ( ω ) , O ( ω ) and O ( ω ) is related to the group delay, pulse broadening, andpulse asymmetry, respectively[15, 18]. The ratio of theabsolute value of O( ω ) to O( ω ) for a pulse of bandwidth B is, | O ( ω ) O ( ω ) | = | Ω c − Γ | Ω c B Γ . (12)If this ratio is negligible then one can keep up to O ( ω ) term only and obtain an analytic form of the probe pulsesolution, which reads, Ω p ( t, z = L ) = Ω p β Exp [ − ln2 ( ( t − T d ) βT p ) ] , (13)where β = (cid:115) ln2 D Γ T p Ω c , (14)and T d = Lv g = Lc + D ΓΩ c , (15)where v g is the group velocity of the probe pulse in theEIT medium. It should be noted that the term L/c in T d may not be negligible in the short pulse regime. Theenergy transmission of the probe pulse is, T = 1 β = 1 (cid:113) ln2 η D , (16)where η ≡ D ΓΩ c T p . (17)The parameter η is approximately equal to T d /T p if v g (cid:28) c . In general case, the slow light transmission efficiency(TE) can be exactly calculated by,TE = (cid:82) ω dω | W p ( ω, | exp ( − z Im [ k ( ω )]) (cid:82) ω dω | W p ( ω, | . (18)We focus our discussion on EIT memory in the highoptical depth regime (e.g. D > ), where Ω c (cid:29) Γ is valid and thus | O ( ω ) /O ( ω ) | ≈ B Γ . In the narrow-band regime (i.e. B (cid:28) Γ ), the formulation for keep-ing the dispersion up to the O ( ω ) term is already agood approximation. On the other hand, in the broad-band EIT storage regime ( B > Γ ), O ( ω ) term couldbe larger than O ( ω ) term and should be considered,which may lead to the pulse distortion. We next con-sider the contribution of even higher order dispersionterms in the broadband EIT regime. We note thatthe ratio of the higher order even and odd dispersionterms, O ( ω n +2 ) /O ( ω n ) and O ( ω n +3 ) /O ( ω n +1 ) with n = 1 , , ... , are ≈ ( B Ω c ) . Since we consider the storagein the EIT protocol, the condition B < ∆ ω EIT is valid.From Fig. 2 (a), we know that ∆ ω EIT < Ω c is alwaysvalid. Thus, ( B Ω c ) < ( B ∆ ω EIT ) < is valid. Neglecting
60 70 80 90 10022.533.5
Transmission efficiency (%) η s t o r e (b) Ω c ( Γ ) ∆ ω E I T ( Γ ) (a) − B ( Γ ) T r an s m i ss i on ( % ) (c) − B ( Γ ) W L ( % ) (d) F=95.5%F=97.5%F=98.5%F=99.5%
D=10D=100D=1000 Ω c2 − linear linewidth for D=1000 Ω c − linear linewidth D=10000D=3000D=1000D=300D=100 D=10000D=3000D=1000D=300D=100
FIG. 2. (a) Solid black, blue and pink curves denote theFWHM EIT transparent bandwidth ( ∆ ω EIT ) versus Ω c foroptical depth D of 10, 100 and 1000, respectively. The greenand red dotted line are the approximate formula of ∆ ω EIT inthe intensity-linear bandwidth regime for D =1000 and field-linear bandwidth regime, respectively. (b) The η store for therequired TE is plotted. F = 99 . , . , . , . forthe curves from top to bottom. (c) and (d) depict TE andWL versus different bandwidth of the input probe pulse ( B )for various optical depths. Here, Ω c for different D are deter-mined by η = the dispersion term of O ( ω ) and above is a reasonableapproximation.In the EIT memory protocol, one adiabatically turnsoff the control field to convert the optical probe field intocollective atomic spin wave and then retrieve the opti-cal field back by turning on the control field on demand.In the dark-state polariton picture, the storage and re-trieval process can be considered as a coherent conver-sion between the optical field and atomic spin wave[23].However, even if the control field is non-adiabatically (in-stantly) turn off and on, the additional energy loss is onthe order of v g /c , which is very small for most of thesituations[40, 41]. In the storing process, one needs toselect a suitable Ω c such that the group velocity is slowenough and almost the whole probe pulse can be com-pressed into the media[15]. Also, the timing when thecontrol field is turned off needs to be suitable such thatthe leakage of the front and rear tails of the probe pulseare minimized [15].To quantify this leakage loss, we define an operationalefficiency, termed as F . Then, the overall storage effi-ciency (SE) of a memory can be estimated through theproduct of the transmission efficiency (TE) of the slowlight and F , SE = TE × F . We emphasize that theremay exist additional losses during storing and retrievingperiods due to the finite γ sg , which we set to zero for sim-plicity in the current discussion[40]. In order to compressalmost the whole probe pulse into the atomic medium,choosing a suitable Ω c to be smaller than a certain value(and thus η larger than a certain value denoted as η store through Eq. 17) to enable F approaching to near unityis the prerequisite. The value of η store should be around2-3[15]. Fig.2 (b) depicts the relationship of the TE and η store for some given F s ( F = 95 . ∼ . ). For asmaller optical depth and thus a smaller TE, the broad-ening effect is more severe [15] such that η store needs tobe larger to minimize the tail leakage. More quantita-tive details about how to calculate these curves can bereferred to Appendix A. We only concern with the highSE case, e.g. SE > ∼ , and in that case η store ranges ≈ . ∼ . .In Fig. 2 (c), we calculate the slow light transmis-sion via Eq. 18 versus the bandwidth of the probe pulseunder the constraint of η = 2 . , which corresponds to F = 98 . . The storage efficiency is thus mainly de-termined by TE with an uncertainty of less than . .From Fig. 2 (c), TE is nearly a constant depending onthe optical depth ( D ) at low pulse bandwidth. With alarger D , the bandwidth of constant TE extends to awider value, roughly proportional to √ D Γ . This trendof nearly a constant TE below a certain bandwidth whenEIT is in the regime of intensity-linear bandwidth regimeis understandable and explained below. In the intensity-linear regime, η can be rewritten as η ∝ √ D B ∆ ω EIT byEq. 8 and Eq. 17. For a fixed η and a fixed D , the ratioof pulse bandwidth to EIT bandwidth is fixed such thatthe slow light transmission stays a constant. From Fig.2 (c), one also observes that the bandwidth range of theTE plateau is larger for a larger D . With a constant η , alarger D means that the ratio B ∆ ω EIT is smaller. There-fore, more frequency components of the probe pulse lie inthe central region of the EIT transparent window, whichleads to a larger transmission.In Fig.2 (c), when the bandwidth of the probe pulse B keeps increasing, the transmission drops and a dip struc-ture appears and then it increases toward unity. Thisdip structure appears when the pulse bandwidth is onthe order of the spectral separation of the two Aulter-Towens absorption peaks, which is ∼ Ω c , such that itsabsorption is maximum. At an even higher B , a sig-nificant portion of the pulse spectral component are be-yond the Aulter-Towens absorption peaks and lie in thefar-detuned and near transparent regime. This is whythe transmission rises again. Although TE is high underthis circumstance, the probe pulse experiences a severedistortion because high order dispersion terms are notnegligible. As an example, Fig. 3 (a) and (b) depictstwo slow light pulses with a T p of 4 and 0.04 / Γ , re-spectively, after passing through an EIT medium with D = 1000 and η = 2 . . Fig. 3 (c) and (d) depictthe corresponding EIT transmission, phase shift and thepulse spectrum. For the long pulse case, its spectrumis relatively narrow and lies within the transparent win-dow. Also, the pulse spectrum lies in a spectral rangeof nearly linear phase shift dependence on the detuning.Thus, the group velocity is well-defined and the slow lightpulse resembles the input pulse with a certain group de-lay. For the short pulse case, however, its spectrum ex-tends over the two Aulter-Townes absorption doubletsand also lies in the complicated profile of the phase shiftspectrum. The pulse not only experiences a significant Time (1/ Γ ) T r an s m i ss i on Time (1/ Γ ) T r an s m i ss i on − −
20 0 20 40 − Probe Detuning ( Γ ) −
50 0 50 − Probe Detuning ( Γ ) TransmissionPhase shiftPulse spectrum (a)(c) (b)(d)
FIG. 3. (a) and (b) the input (blue) and output probe (red)pulses with a T p of 4 and 0.04 / Γ , respectively, for D = 1000 and η = 2 . . Transmission and waveform likeness for theslow light pulse is . and . in (a) and . and . in (b), respectively. In (c) and (d), the EIT trans-mission spectrum, phase shift and probe pulse spectrum areplotted, corresponding to the parameters of those in (a) and(b), respectively. The maximum phase shift is normalized tounity for clarity. absorption but also a significant dispersion such that theoutput pulse has a significant distortion. In the quan-tum memory application, this serious distortion in thetemporal mode may introduce some complications in theapplication. Except the efficiency, it seems necessary todefine another figure of merit to quantify this distortionfor evaluating the performance of an optical memory. C. Waveform Likeness
Inspired by the aforementioned discussion, we define afigure of merit, waveform likeness (WL), to quantify theextent of distortion, which reads as,WL ≡ (cid:12)(cid:12)(cid:12)(cid:82) ∞−∞ Ω ∗ p,in ( t − T d )Ω p,out ( t ) dt (cid:12)(cid:12)(cid:12) (cid:82) ∞−∞ | Ω p,in ( t ) | dt (cid:82) ∞−∞ | Ω p,out ( t ) | dt . (19)Waveform likeness reflects the similarity between the in-put and slowed (or retrieved) probe pulses[14, 15]. Fig. 2(d) depicts the WL of different input pulse bandwidths,corresponding to the TE of Fig. 2 (c). The severe pulsedistortion at high bandwidth can be manifested by thedegradation of WL. For clarity, we do not show the WLbelow in Fig. 2 (d). Similar to the TE, there isa plateau for WL at the low B regime. For a larger D , this plateau extends to a wider range, but is slightlysmaller than that of the TE. This suggests that WL isa more stringent parameter than the TE at high pulsebandwidth.In the example shown in Fig.3 (b) for the broadbandcase, the output pulse profile splits over a long periodof time. Although TE remains high ( . ), the WL isvery low ( . ) and this highlights the need to introducethe parameter WL. In the case of Fig.3 (b), one needs to collect the trifling signals over a long time window tomaintain the desired SE. It may affect the feasibility ofstorage, especially in the single-photon regime for quan-tum memory application since the noises makes it ardu-ous to collect signal over the long time window. Besides,the group delay is not well-defined in the broadband case.A certain portion of the pulse is delayed with sufficienttime but there exists a front tail virtually without anydelay. It is therefore not feasible to store the probe pulsewith a negligible amount of leakage, or F ≈ .In the single-photon regime, waveform likeness is re-lated to the overlap of the temporal mode of the output tothe input pulse. A high WL is important since it is essen-tial to preserve a good single-photon waveform to reacha high non-classical property in quantum storage[17]. Inthe application of quantum memory, it may involves withthe HongâĂŞOuâĂŞMandel (HOM) two-photon interfer-ence. The contrast of HOM interference depends on thedegree of indistinguishability of the temporal [36] andspectral mode [37, 38] for the two photons. The severedistortion increases the difficulty of creating two indistin-guishable photons in both temporal and spectral mode.Besides, with a better WL, the arrival time of the re-trieved signal can be well-controlled, instead of spreadingout over a long time period. This temporal control is alsoimportant to quantum communication through time-binprotocol [32]. In the following subsections, we presentfurther analysis of the WL.
1. Broadband Limit
Examine the numerator in the definition of WL in Eq.19, one can simplify it to,WL ∝ | (cid:90) dω | W p ( ω, | e i ( k ( ω ) L − D Γ ω Ω2 c ) | . (20)The expression of k ( ω ) of Eq.5 can be written as, k ( ω ) L = D ΓΩ c ω − i ω ΓΩ c − ω Ω c . (21)For a probe pulse of bandwidth B , B Ω c < in the broad-band EIT memory regime since ∆ EIT (cid:39) Ω c . Also, wehave B > Γ such that ΓΩ c < B Ω c < . Under such con-ditions, the term ω ΓΩ c (cid:28) in the denominator of Eq.21and thus it can be neglected. In that situation, k ( ω ) isalmost a real number and reads as, k ( ω ) L (cid:39) D Γ ω Ω c − ω . (22)Thus, the transmission efficiency of the output probepulse is near unity, as can be seen from Eq. 18 becauseIm ( k ( ω )) is near zero. This is understandable since theEIT transparent window is very wide and has a flat-topprofile in the broadband regime, such as that shown inFig. 3 (d). If B (cid:28) Ω c , almost all the pulse spectrum FIG. 4. (a) Solid curves denote simulated WLs with different bandwidths when fixing ( ξ c , η ) = (5 . , . . (b) WL of different ( ξ c , η ) when B/ Γ (cid:29) . In the simulation, B/ Γ = 880 . Note that WL will converge to the value in this figure in the broadbandlimit. (c) The cross section of (b) along the ξ c axis for η = 2 . . lies in the near-unity transparent window and thus thetransmission is near unity.Because WL is actually determined by the relative in-terplay between W p and k ( ω ) , we can consider all spec-tral behaviors normalized to B . We define (cid:101) k ( (cid:101) ω ) = k ( (cid:101) ωB ) and (cid:102) W p ( (cid:101) ω ) = W p ( (cid:101) ωB ) , where (cid:101) ω = ωB . Whenconsidering the different bandwidth B , (cid:102) W p ( (cid:101) ω ) remainsthe same and the WL of spectral integral Eq.20 is onlydetermined by (cid:101) k ( (cid:101) ω ) , which reads, (cid:101) k ( (cid:101) ω ) L = D Γ B (cid:101) ω ( Ω c B ) − (cid:101) ω . (23)Here, we introduce two parameters: ξ c and ξ D , definedas ξ c ≡ Ω c Bξ D ≡ D Γ B . (24)Therefore, Eq.23 can be rewritten (cid:101) k ( (cid:101) ω ) L ≈ ξ D (cid:101) ωξ c − (cid:101) ω (25)If we keep ξ c and ξ D stay at the same values when vary-ing B , the normalized response, (cid:101) k ( (cid:101) ω ) L , can be main-tained exactly at the same value through adjusting the Ω c and D according to Eq. 24 with Ω c = ξ c B and D = ξ D B Γ .Since the same normalized spectrum maintains the sameWL, the values of ξ c and ξ D therefore determine the cri-terion for the waveform likeness. These two parametersconnect to η by η = ln ξ D ξ c . For simplicity, we con-sider hereafter ξ c and η as the two independent variables,while ξ D can be determined by the other two parameters.Thus, the whole picture of WL in the broadband limit canbe summarized by η and ξ c . It should be pointed out that ξ c denotes the ratio of EIT bandwidth to the bandwidthof input probe pulse in the broadband EIT regime. Con-ceivably, a larger ξ c means that the probe spectrum is well-located in the central EIT window, which results ina high WL.Note that once fixing ( ξ c , η ) , WL converges to a con-stant value in the large bandwidth limit, as depicted inthe example of Fig.4 (a). An numerical simulation of theconvergent WL for different ( ξ c , η ) with a given large B of880 Γ is shown in Fig.4 (b). In the broadband EIT regime,one can use Fig.4 (b) to determine WL and thus estimatethe required experimental parameters, ( D, Ω c ) by Eq. 24.In other words, under the condition of B/ Γ (cid:29) , all in-formation about WL has been mapped in Fig.4 (b). Thetrend of WL can be easily captured by Fig.4 (b): decreas-ing η and increasing ξ c is favorable to WL. However, η has a minimum requirement such that the storage of al-most the whole pulse is possible. For a fixed η of 2.5,WL is determined solely by ξ c , as depicted in Fig.4 (c).In the high-WL condition, the pulse distortion majorlycomes from the third order dispersion term. Under sucha condition for WL ≈
1, the approximate analytical formof WL can be obtained, which reads,WL ≈ (1 − . × ( ηξ c ) ) . (26)With a given parameter set of ( ξ c , η ) , the approximateWL is also determined.
2. Adiabatic Storage
Experimentally, the three variable parameters are D, Ω c and T p (or B ). By putting the relation of ξ c intothat of η and reducing Ω c , one gets, DT p Γ = ηξ c × ( ) . (27)In Ref.[19], the parameter DT p Γ has been identified as ameasure of degree of adiabaticity in EIT storage and alarger value in this parameter indicates a higher degreeof adiabaticity. Therefore, we term it as the adiabaticityparameter. As mentioned before, the parameter η has tobe larger than a certain value (e.g. ∼ FIG. 5. (a) WL versus DT p Γ . Here, we fix D = 1000 andvary T p . The solid curve is based on the numerical simulationand the dashed curve is calculated through Eq.26. (b) WL asa function of T p and D in logarithmic-logarithmic scale. Inboth graphs, η = 2 . is assumed. almost the whole probe pulse into the media. The adi-abaticity parameter DT p Γ is thus mainly determined bythe parameter ξ c . Once both η and ξ c are determined, theadiabaticity parameter also determine the WL throughEq. 26.The dash line in Fig.5 (a) depicts an example of thecalculated WL through Eq. 26 versus the adiabaticityparameter for D = 1000 and η = 2 . . The solid line isWL from the numerical calculation. The analytic for-mula matches the numerical calculation at high WLs butshows certain deviation at lower WLs. Fig.5 (b) depictsa numerical calculation of the two-dimensional contourplot of WL versus T p Γ and D for η = 2 . . It is evidentto observe that WL is nearly a constant for a constantproduct of DT p Γ . This is understandable since the adia-baticity parameter DT p Γ is related to the parameter set ( ξ c , η ) by Eq. 27. And a given parameter set of ( ξ c , η ) di-rectly determine the WL through Eq. 26. As mentioned,the parameter DT p Γ can be considered as a measure ofthe degree of adiabaticity[19]. Due to its direct relationwith WL, one can reverse the logic and consider WL asan experimentally observable parameter useful for evalu-ating the degree of adiabaticity for the memory.In comparison with the ATS protocol [12, 39], whichutilizes the non-adiabatic process to convert the polar-ization coherence into spin coherence, the value DT p Γ locates in ∼ and therefore the WL is not high. Inthe ATS protocol, the pulse distortion comes from theoscillation between the polarization and spin coherenceand one extracts one of the pulse out but not all of thephotonic signal in the temporal domain. If one only fo-cus on the extracted pulse, it is not severely distortedbut remains a Gaussian shape [12, 39]. But it certainlysacrifices the efficiency. In the EIT protocol, we includeall optical signal in the temporal domain in the evalua-tion of WL. It is possible to reach a high WL and a highSE simultaneously for adiabatic EIT storage. However,there is a high demand on D and Ω c for EIT protocol, aswill be discussed below. FIG. 6. (a) and (b) depicts the TE and WL in a given D and T p Γ = 10 ( . ) or B/ Γ ≈ . ( . ). Here, Ω c is determined by the relation η = 2 . . (c) The requiredoptical depth ( D req ) versus the bandwidth of probe pulse ( B ).The red (blue) markers denote the required optical depth tosatisfy TE ≥ (WL ≥ ). Optical depth above theshaded area can satisfy both TE ≥ and WL ≥ .(d) The corresponding required Ω c for D req in c, which meetsboth TE ≥ and WL ≥ . D. Experimental Requirements
In this section, we want to provide a guide on the re-quired optical depth D and control intensity (character-ized by Ω c ) when implementing a broadband EIT mem-ory with a value of SE and WL above a given threshold.Fig. 6 (a) and (b) depict two representative examples ofTE and WL versus D for the narrowband and broadbandcases with B/ Γ of 0.277 and 693.15, respectively, with η = 2 . to satisfy the storage requirement. In the nar-rowband case, the WL is larger than the TE, while thesituation is opposite for the broadband case. In otherwords, TE (WL) is the bottleneck when operating thenarrowband (broadband) EIT storage. From these fig-ures, one can determine the minimum required opticaldepth in order to obtain the TE and WL of larger thana certain value (e. g. with TE ≥ and WL ≥ )simultaneously. From Eq. 17, one can then determinesthe required Ω c for a fixed η . Considering the similar cal-culations for different bandwidth B , one can reach Fig.6 (c), which depicts the required D versus B for the re-quest with TE ≥ and WL ≥ . The optical depthabove the shaded region will satisfy both the thresholdconditions on TE and WL. It is evident that the requiredoptical depth (denoted as D req ) is constrained by TE orWL when B (cid:46) Γ or B (cid:38) Γ , respectively. We term bothregimes as the TE-limited or WL-limited regime, respec-tively. This trend is related to the fact mentioned insection II B that one has to consider the dispersion up toO( ω ) for the broadband EIT memory regime ( B > Γ )but is enough to O( ω ) for the narrowband ( B < Γ )regime.The TE-limited feature in the narrowband EIT regimealso explains why the previous researches for narrow-band EIT memories did not bothered by the distortionissue since SE is a bottleneck in that regime, not theWL[14, 15]. On the contrary, upon pushing towards thebroadband regime, one steps into the WL-limited regimeand the pulse distortion is a must to consider. From Fig.6 (c), it can be observed that the required optical depth islinearly proportional to the bandwidth in the WL-limitedregime. This trend is understandable if we look at Eq.27 and modify it to, D req = ηξ c B Γ . (28)Fig. 6 (d) depicts the required Ω c to meet the request onTE and WL. Clearly, the demand on D and Ω c is veryhigh for EIT protocol to achieve a high TE and WL. Asa reference, a B of ∼ requires D ∼ (10 ) and Ω c ∼ (10 ) for a WL ≥ . Experimentally,an optical depth of larger than has been achievedfor cold atoms in free space or in a hollow-core fiber[21].With cold atoms inside a cavity, an effective optical depthof 7600 has been achieved[20]. Another way to achievea high TE and a high WL using the EIT protocol butwith a much lower optical depth is to utilize the forwardstorage and backward retrieval scheme[14, 44], althoughsome experimental complexities are needed.While the broadband EIT memory requires a high op-tical depth and a strong control intensity, we expect thatsome complicated effects may appear in realistic situationwhich are out of the scope of this paper. For example,nonlinear optical effect such as the photon switching ef-fect due to the off-resonant excitation of the coupling fieldto the nearby transition [42, 43] or the four-wave mixing[35] may become a serious issue. But there are also somemethods that can reduce these effects[15]. The influenceof cooperative effect due to the resonant dipole-dipoleinteraction on the EIT memory may become significantand await further study[15].With these theoretical studies in mind, we then presentour experimental efforts towards broadband EIT mem-ory. III. EXPERIMENTAL SETUP
We utilize a cesium magneto-optical trap (MOT) witha cigar-shaped atomic cloud to implement the EIT-basedoptical memory. To increase the optical depth of atomicmedia, we employ the temporally dark and compressedMOT, as well as Zeeman-state optical pumping[21].Pumping population towards the single Zeeman substate( | F = 3 , m = 3 (cid:105) ) also makes the storage performanceless sensitive to the stray magnetic field [15, 22], whichis desirable for the long-time storage. Efforts are madeto reduce the ground-state decoherence rate γ , such asminimizing the stray dc and ac magnetic fields and us-ing the near co-propagating probe and control beams toreduce the residual Doppler broadening. Details of theMOT setup can be refereed to[15, 21]. EtalonPMTFC 50:50
BS FCFiberAOM (cid:3043) FC FEOM
L1 L2 Black dotFCProbe beam
M MProbe
Control
QWP Iris x preparationPort (cid:1832) (cid:4593) (cid:3404) 46 S P WriteSpinwave ReadSpinwave (cid:1865) (cid:3404) 3(cid:1832) (cid:3404) 4(cid:1832) (cid:3404) 3 (cid:1865)′ (cid:3404) 4 (cid:1865)′ (cid:3404) 4(cid:1865) (cid:3404) 3|1(cid:1767) |3(cid:1767)|2(cid:1767) |1(cid:1767)|2(cid:1767) |3(cid:1767) (a)(b)
FIG. 7. (a) Energy levels and laser excitations of Cs for EIT memory. (b) Experimental setup for EIT memory.AOM: acousto-optic modulator; BS: beam splitter. FEOM: fiber electro-optic modulator; RF: radio-frequency signal;PMT: photomultiplier tube; M: mirror; L: lens; QWP: quar-ter waveplates; FC: fiber coupler; Port 1 is transmitted by apolarized-maintaining fiber. The EIT optical memory is operated at the D linewith the probe beam driving the | F = 3 (cid:105) → | F (cid:48) = 4 (cid:105) σ + -transition and control beam driving the | F = 4 (cid:105) →| F (cid:48) = 4 (cid:105) σ + -transition, as shown in Fig.7 (a). As pointedout in [15], operating the EIT at D transition can re-duce the control-intensity-dependent ground-state deco-herence rate due to the off-resonant excitation of the con-trol beam to the nearby transition. The detailed setup isplotted in Fig.7 (b). The probe beam from a laser sourcedoubly passes one acousto-optic modulator (AOM1) foradjusting its frequency with a minimal spatial movement.It is then sent to another AOM (AOM2) for switchingwith a 160-ns square pulse. The probe beam is then cou-pled into a fiber electro-optic modulator (EOM) to shapethe probe pulse into a Gaussian waveform with a FWHMof larger than 10 ns. Due to the finite extinction ratio ( ∼
18 dB) of the fiber EOM, adding AOM2 as an additionalswitch is to minimize the probe leakage during storage.The probe beam is then coupled with the control beamthrough a
50 : 50 beam splitter. Both beams are sent intothe cold atomic ensembles. Before coming into MOT cell,the probe beam is focused by a lens (L1) to an intensity e − diameter of ∼ µ m around the atomic clouds whilethe coupling beam is collimated by the same len (L1)with a diameter of ∼ µ m. After going out from theMOT cell, the control beam is focused by another lens(L2) and is then blocked by a black dot. Probe beam,on the other hand, is collimated by the lens L2 and thencoupled into a fiber before passing through three iridesand an etalon filter, which filters out unwanted controllight upon detection. The probe beam is then detectedby a photomultiplier tube (Hamamatsu R636-10). FIG. 8. In (a), the red, green and blue curves represent theinput, slowed (efficiency . ), and stored-and-retrieved (ef-ficiency . ) probe pulses. The parameters D and Ω c are392 and 16.48 Γ , respectively. (b) depicts the Ω c versus thecontrol power. The solid blue line is a linear fit to the data.(c) depicts the optimum Ω c which maximize the efficiency ver-sus the optical depth. The blue solid line is a linear fit to thedata. (d) depicts the efficiencies of the slowed (blue square)and stored-and-retrieved (red circle) probe pulses versus theoptical depth. The solid blue line is a theoretical calculationof the slow light efficiency, assuming γ gs = 0 , γ ge = 0 . and η = 2.5. IV. RESULTS AND DISCUSSIONS
Here, we present our experimental results on broad-band storage using EIT protocol. We first discuss theefficiency versus the optical depth at a fixed temporalwidth for the input probe pulse. Then we vary the tem-poral width (or bandwidth) of the input probe pulse andstudy its efficiency dependence. Finally, we study theefficiency dependence on the storage time.
A. Efficiency dependence on optical depth
In a previous work[15], we have studied the storage ef-ficiency versus the optical depth for probe pulses of T p =
200 ns. Because the pulse bandwidth B = 2 π × Γ = 2 π × T p of 30 ns, correspondingto a bandwidth of B = 2 π × D ) and adjust the control intensity foreach D to obtain an optimized efficiency. A representa-tive raw data showing the input, slowed, and stored-then-retrieved probe pulses is shown in Fig. 8 (a). In this case,the parameters D and Ω c are 392 and 16.48 Γ , respec-tively. The Ω c is determined by the spectral separation ofthe Aulter-Townes splitting in the EIT spectrum, takenat a very low optical depth (e. g. D < ) such that theAulter-Townes doublets are clear. The optical depth isdetermined by the spectral fitting of the probe transmis-sion spectrum with the control field off. In the fitting,we set γ ge = 0 . which takes the finite laser linewidth P o w e r ( a r b . un i t s ) (a) Input LightSlow LightStored Light0 50 100406080100 E ff i c i en cy o r W L ( % ) T p (ns) (c) EfficiencyWaveform Likeness 0 10 20 30 400100200300 Ω c ( Γ ) B/2 π (MHz) (d) P o w e r( a r b . un i t s ) (b) Input LightSlow LightStored Light
FIG. 9. Fig. (a) depicts one example of the raw datawith T p = 14 ns, D = 356 and Ω c = 17 . . The efficiencyand waveform likeness for the stored-then-retrieved pulse is . and . , respectively. Fig. (b) depicts one ex-ample of the raw data to demonstrate the distortion in theslow light pulse. In this case, the parameters { T p , D, Ω c } are { . ns , , . } , respectively. The efficiency and WL forthe stored-then-retrieved pulse are 31.8 % and 64.7%, respec-tively. (c) The efficiency (blue circle) and waveform likeness(red square) versus the FWHM pulse duration. For T p < ns, we are limited by available control power, as also shown in(d), such that the efficiency and WL goes down. (d) The used Ω c versus the pulse bandwidth for the data corresponding tothose of (c). The blue solid line is a linear fitting curve forthe five date points with narrower bandwidth (or larger T p ). and the laser frequency fluctuation into account[15].As a consistency check of the determined Ω c , the opti-mized Ω c for each D versus the control power is shown inFig. 8 (b). The data fit very well to a linear relation asexpected. Fig. 8 (c) depicts the optimized Ω c versus D ,which reasonably follows a linear relation. We roughlychoose η to be nearly a constant to obtain the optimizedefficiency. Due to Eq.17, this implies that Ω c is linearlyproportional to D for a fixed T p . The corresponding ef-ficiencies of the slowed and stored-then-retrieved probepulses are shown in the blue squares and red circles ofFig. 8 (d), respectively. The blue solid line is a theo-retical calculation of the slow light efficiency based onEq. 16 with γ ge = 0 . and η = 0 . . The theoret-ical curve matches well with the slow light efficiencies.At the highest D of 392, the achieved efficiency of thestored-then-retrieved pulse is . . . B. Efficiency dependence on the temporal width
Keeping at a large D of 356, we then vary T p fromlarge to small values (or B from small to large values)and adjust the Ω c to obtain an optimized efficiency. Un-fortunately, the control intensity is technically limitedto a certain level (corresponding to Ω c = 17 . ) suchthat when T p < ns we cannot obtain the optimizedefficiency. The results are shown in Fig. 9 (c). The cor-responding Ω c versus B are shown in Fig. 9 (d). Forthe data with T p >
30 ns, the Ω c versus B is well fit0by a linear relation, which means that η is maintainedat a constant value. The corresponding efficiencies for T p >
30 ns are pretty much a constant when η is keptat a constant. The efficiency goes down when T p < Ω c . Based on the fit curveof the input and retrieved probe pulses, we can calculatethe waveform likeness. The WL are shown in the redsquare data in Fig. 9 (c). With our current experimentalparameters, the EIT storage is in the TE-limited regimesuch that WL is larger than TE. In Fig. 9 (a), the rawdata for T p =
14 ns with Ω c = 17 . are shown. Dueto the non-optimized Ω c , the slowed and retrieved pulseshave a significant broadening which leads to a reducedWL. In Fig. 9 (b), we intentionally show an examplewith a significant pulse distortion or even splitting forthe slowed pulse, in which B < ∆ ω EIT is not satisfied.It is not surprised that both the efficiency and WL of theretrieved pulse are low under such a situation, which are31.8% and 64.7 %, respectively.
C. Storage time
We then study the efficiency versus the storage timewith T p =
30 ns for the input pulses. We have mini-mized the stray magnetic field by prolonging the stor-age time through three pairs of compensation coils. Fig.10 depicts an example of efficiency versus the storagetime, ranging up to 70 µ s. The data is fit to a curveof A ∗ Exp [ − ( t/τ ) ] with fitting parameters A = 80 . and τ = 54 . µ s. The time-bandwidth product (TBP),defined as the ratio of the storage time at efficiencyto the FWHM input pulse duration ( T p ), is an importantfigure of merit in quantum memory application. The de-termined TBP based on Fig. 10 is 1267, slightly higherthan our previous work of 1200 with T p =200 ns[15]. Thestorage time may still be limited by the residual mag-netic field and the residual Doppler broadening due tothe atomic motion and the finite angle ( ∼ ) betweenthe probe and control beams[45]. V. CONCLUSION
In summary, we explore the EIT-based storage to-wards broadband regime. The requirements for high-performance broadband EIT memory have been theoreti-cally discussed. Large optical depth is necessary to reachthe high-performance storage for short pulses, and wave-form likeness becomes the limit when reaching the broad-band regime. We experimentally demonstrate the broad-band EIT memory with a storage efficiency of ∼ (WL 92.6 %) for a 30-ns pulse and of > (WL 79.7%) for a 14-ns pulse. The achieved time-bandwidth prod-uct is 1267. Our work clarify that it is possible to obtaina high-efficiency and a high-bandwidth for adiabatic EITmemory, provided one can achieve a high optical depthand a strong control intensity. FIG. 10. Efficiency versus the storage time. The bluesolid line is a fitting curve to the data with the fit function: A ∗ Exp [ − ( t/τ ) ] , where the fitting parameters A = 80 . and τ = 54 . µ s. The grey dashed line represents the efficiency. Appendix A: Leakage-induced Loss
In this appendix, we consider the relation between thechoice of the parameter η and the optimized operationalefficiency F due to the pulse leakage during storage pro-cess for a given transmission efficiency (TE) of the slowlight pulse, as plotted in Fig. 2 (b). We assume the EITstorage is in the regime that the dispersion up to O( ω )is a good approximation. As mentioned in the theoreticalsection, the slow light pulse is broadened by a factor of β (Eq. 14) and the relation TE = β is satisfied in the ideallimit of γ gs = 0 . The overall operating efficiency F isthe product of two terms, F i and F o , due to the leakageof the front and real tail of the pulse, respectively, whichread[15], F i = 12 (1 + erf (2 √ ln κ )) F o = 12 (1 + erf (2 √ ln η − κ ) /β )) F = F i × F o (A1)Here, κ denotes the ratio of the turned-off time of thecontrol field ( T c ) to the temporal width of the input pulse,i.e. κ ≡ T c T p .To search for the optimal κ that minimizes the leakage-induced loss, we take the derivative of F with respect to κ , ∂ κ F = 0 e − (2 √ ln κ ) ( 12 (1 + erf (2 √ ln η − κ ) /β ))) − β e − (2 √ ln η − κ ) /β ) ( 12 (1 + erf (2 √ ln κ ))) = 0 (A2)If the broadening is not too severe such that β (cid:39) , thenwhen κ = β ( η − κ ) the derivative is approximately zero.This assumption is valid if TE is not much less thanunity (see Sec.II B). Under such an approximation, the1optimized κ is, κ = η β = η TE , (A3)and F i = F o . 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