Long Light Storage Time in an Optical Fiber
Wui Seng Leong, Mingjie Xin, Chang Huang, Zilong Chen, Shau-Yu Lan
LLong Light Storage Time in an Optical Fiber
Wui Seng Leong, Mingjie Xin, Chang Huang, Zilong Chen and Shau-Yu Lan ∗ Division of Physics and Applied Physics, School of Physical and Mathematical Sciences,Nanyang Technological University, Singapore 637371, Singapore (Dated: September 14, 2020)Light storage in an optical fiber is an attractive component in quantum optical delay line technolo-gies. Although silica-core optical fibers are excellent in transmitting broadband optical signals, it ischallenging to tailor their dispersive property to slow down a light pulse or store it in the silica-corefor a long delay time. Coupling a dispersive and coherent medium with an optical fiber is promisingin supporting long optical delay. Here, we load cold Rb atomic vapor into an optical trap inside ahollow-core photonic crystal fiber, and store the phase of the light in a long-lived spin-wave formedby atoms and retrieve it after a fully controllable delay time using electromagnetically-induced-transparency (EIT). We achieve over 50 ms of storage time and the result is equivalent to 8.7 × − dB µ s − of propagation loss in an optical fiber. Our demonstration could be used for buffering andregulating classical and quantum information flow between remote networks. Optical delay lines or optical buffers play importantroles in long-distance quantum communication networksfor storing, delaying, and, thus, exchanging informationbetween different quantum nodes. The superb perfor-mance of optical fibers as transmission lines has driventhe consideration of integrating these functionalities intothe optical fibers themselves [1, 2]. One of the challengesis maintaining the coherence of the medium in the fiber inwhich the information of light is encoded. The direct useof solid-core materials of telecommunication band fibersvia stimulated Brillouin scattering and doped erbium ionsto store light has been demonstrated to tens of nanosec-onds [3–6]. Despite their tens of GHz bandwidth andtelecom wavelength operation, the short-lived acousticwaves in the materials and spin coherence of doped er-bium in the respective settings limit the performance ofthe light storage.Alternatively, interfacing long-lived atomic spin stateswith optical fibers could provide a route to achievinglong storage time. Loading room temperature atomic va-por into a hollow-core photonic crystal fiber [7] and coldatoms near the surface of an optical nanofiber [8] haveachieved storage times of about tens of nanoseconds andfew microseconds, respectively. They are limited by thetransit time of atoms through the optical modes. Confin-ing atoms in the evanescent field of an optical nanofiber[9, 10] and guided field of a hollow-core fiber [11, 12]could eliminate decoherence from transit time but in-troduce additional decoherence from the confining fields.Both systems have shown storage time of about a few mi-croseconds. To overcome the decoherence, we use a stateinsensitive optical potential to confine Rb atoms in theguiding mode of a 4-cm-long hollow-core photonic crystalfiber [13, 14]. The phase of the optical pulse is mappedonto the atomic spin wave formed by a pair of long-livedhyperfine ground states and retrieved with a controllabledelay using EIT. We demonstrate light storage over 50ms with a bandwidth of 1 MHz. ∗ [email protected] ProbeControlDelayed probe
𝐹 = 3, 𝑚 = 0𝐹 = 2, 𝑚 = 0 𝐹 ′ = 3 Gold mirrorGold mirror MW antenna90R:10T NPBSQWPPBS PM fiberHWP PM fiber
Dipole beam(821 nm) Depump,
Optical pump,
Control,Probe
APD
QWP
PBSPM fiber 50R:50T NPBS Etalon
Dipole beam(821 nm)
PBS
Control
Probe
FIG. 1. Details of the experimental setup. NPBS: non-polarizing beam splitter. PBS: polarizing beam splitter.HWP: half-wave plate. QWP: quarter-wave plate. PM fiber:polarization-maintaining fiber. APD: avalanche photodiode.The QWPs are used to adjust the ellipticity of the dipolebeams to induce the vector light shift. In order to combinethe linearly polarized EIT beams and the elliptically polar-ized dipole beam, we use a 90% reflection and 10% transmis-sion NPBS. A 50:50 NPBS is then used to separate the EITbeams from the counter-propagating dipole beam. We use atemperature-stabilized solid-state etalon to minimize leakageof the control beam to the APD. The depump light on the F =3 to F ’=3 D2 line is used to prepare the atoms in the F =2 state. The optical pump beam is resonant on the F =2to F ’=2 D1 line. The first control pulse is for converting theprobe pulse into an atomic spin-wave and the second is forretrieving it. A cold Rb atomic ensemble is prepared by amagneto-optical trap (MOT) ∼ a r X i v : . [ phy s i c s . op ti c s ] S e p - 5 0 5 1 00 . 00 . 10 . 20 . 3 - 1 5 - 1 0 - 5 0 5 1 0 1 5 2 00 . 00 . 20 . 40 . 60 . 81 . 0 - 1 0 0 1 0 2 0 3 0 4 0 5 00 . 0 20 . 0 40 . 0 60 . 0 8 T i m e t ( 1 0 - 7 s ) I n p u t S l o w e d p u l s e
Intensity (arb. units) ( a )
N o n - u n i f o r m U n i f o r m I n p u t
Normalized intensity
T i m e t ( 1 0 - 7 s ) ( b ) T i m e T ( 1 0 - 7 s ) S l o w e d p u l s e D e l a y e d p u l s e L e a k e d p u l s e
Intensity (arb. units) ( c )
FIG. 2. Input, slowed, and delayed light pulses through the hollow-core fiber. (a) Measured slowed light when the controlbeam is permanently on. The input pulse (black squares) is fitted by a Gaussian function with a FWHM of 304 ns, and theslowed pulse (blue circles) is fitted by a Gaussian function with a FWHM of 415 ns. The peak of the fitted Gaussian functionof slowed light is delayed by 173 ns. (b) Numerical calculations of slowed pulses. The blue circles are calculations of the slowedpulse with non-uniform profile of the control beam, the probe beam, and the atom density in the fiber. The green squares arethe calculations of the slowed light with uniform EIT beam profiles and atom density distribution. (c) The measured temporalprofiles of the leaked and delayed light pulses. The control pulse is switched off when the probe pulse is propagating inside theatomic medium. The bandwidth of the delayed pulse is defined from its FWHM of 160 ns. ensemble is loaded into an optical trap guided by the fiberwith a temperature of 10 µ K. The wavelength of the op-tical dipole beam is 821 nm with 250 mW power, andthe measured optical depth ( D ) is about 40 by probingall the Zeeman states of the Rb D2 line F =3 to F ’=4transition. When the atoms are in the fiber, we opticallypump them into the F =2, m =0 state by a π -polarizedlight and a repump light coupled through the fiber. Thequantization axis is defined by a 2 G magnetic field alongthe fiber axis and the magnetic field is also used to liftthe degeneracy of the Zeeman sublevels.Our EIT scheme is formed by a three-level Λ config-uration [15] operating on the hyperfine clock states of Rb for long coherence times. The probe beam is res-onant on | (cid:105) = | F =2, m =0 (cid:105) to | (cid:105) = | F ’=3, m = ± (cid:105) andthe control beam is resonant on | (cid:105) = | F =3 m =0 (cid:105) to | (cid:105) .A 795 nm diode laser is tuned to the control beam fre-quency, and a small portion of the power is split to set upthe probe beam by an electro-optical modulator around3 GHz followed by an etalon to filter out the undesiredsidebands. Both beams are perpendicularly linearly po-larized, and co-propagate inside the fiber. After exitingthe fiber, the probe beam is separated from the controlbeam by polarization and etalon filtering and detectedwith an avalanche photodiode. The control and probebeams power inside the fiber are 300 nW and 40 nW,respectively, and the frequencies and the power of bothbeams are controlled by two independent and phase co-herent acousto-optical modulators (AOMs). The 80 MHzradio frequency for the probe AOM is modulated by aGaussian pulse from a waveform synthesizer. The 80MHz radio frequency for the control AOM is modulatedby a square pulse.When the atoms are under free fall inside the fiber, wemeasure the group delay of the probe pulse, as shownin Fig. 2(a). The intensity of the input probe pulse can be approximated with a Gaussian temporal profile of I p ( t )= I p0 exp(-4(ln2) t / T ), where I p0 is the maximumintensity, T p =304 ns is the FWHM duration, and t isthe temporal coordinate. A Gaussian function is fittedto the transmitted probe pulse to determine the groupdelay T d =174 ns with 24% efficiency of transmission.We numerically calculate the group delay and the ef-ficiency of the transmitted light by integrating over theinhomogeneous profiles of the atomic cloud in the dipoletrap, the probe beam, and the control beam inside thefiber from the center of the fiber to the wall of the fiber W =31.5 µ m [16]. The intensity of the probe pulse aftertransmitting through the atomic ensemble is obtained as (cid:90) W πr × ( Ω p β × exp( − γ D ΓΩ + 4 γ γ ) × exp( − t − T d βT p ) )) dr, (1)where the factor β = (cid:115) D Γ( γ Ω + 2 γ γ + 4 γ ) T (Ω + 4 γ γ ) , (2)and the group delay T d = D Γ(Ω c +4 γ ) / (Ω +4 γ γ ) .The normalized Rabi frequency of the probe pulse is as-sumed to have a Gaussian distribution Ω p =exp(- r / R ),where r is the radial coordinate of the fibre core and R =22 µ m is the 1/ e mode field radius. The groundstate coherence γ , which is mainly due to the inho-mogeneous differential ac Stark shift of the control anddipole beams, is fixed at 2 π × (2 × Hz) based on themeasurement of EIT linewidth and ground and excitedstates coherence γ is fixed at half of the Rb D1
H C F w i t h a t o m s S o l i d - c o r e f i b e r
Efficiency (%)
T i m e T ( m s ) ( a ) H C F w i t h a t o m s S o l i d - c o r e f i b e r
Efficiency (%)
T i m e T ( m s ) ( b ) F r e e - f a l l i n g w i t h c a n c e l l a t i o n
Normalized efficiency
T i m e T ( m s ) ( c ) FIG. 3. Storage efficiency as a function of time. (a) The grey circles are the measurements of the storage efficiency withoutlight shift cancellation. The function A (1+( T / T c ) ) − / is fitted to the data with T c =823(10) µ s, where A is the efficiencyat T =0. The decay time is limited by the differential ac Stark shift of the dipole potential on the atomic spin coherence.For comparison, the curve in purple represents the 0.03 dB µ s − =0.15 dB km − loss of 1550 nm light propagating in a solidcore fiber. The error bars are the standard errors of 40 experimental runs. (b) Measurements of the efficiency with light shiftcancellation (orange circles). An exponential decay function is fitted to the data with a 1/ e time of 20(1) ms. The curve inpurple is the same as in (a). The error bars are the standard errors of 40 experimental runs. (c) Measurements of normalizedstorage efficiency for a free-falling atomic spin-wave with light-shift cancellation. The control and probe pulses are sent whenatoms are in motion inside the fiber. line P state decay rate Γ= 2 π × (5.75 × Hz). Theoptical depth D then follows the Gaussian distribution D = L × n × exp ( − r /R ) × σ × exp ( − r /R ), wherethe length of the ensemble L along the fiber axis is ap-proximated with 5 mm [17], the number density n isapproximated with 1.3 × cm − from the maximummeasured D =40, and σ =0.65 × − cm is the scatter-ing cross-section of the D1 line F = 3, m = 0 to F ’ = 3, m = ± γ is chosen as 2 π × (10 Hz)from the coherence time measurement of the microwaveRamsey interferometer before light shift cancellation ofthe optical lattice described in the following section.Figure 2(b) compares the group delay of the probepulse in the fiber with the free space scenario of uniformcontrol beam, probe beam, and atoms density profiles.The inhomogeneous profiles of the control and the atomdensity in the fiber result in more than a factor of 2 re-duction of the group delay and the efficiency. Figure 2(c)shows the temporal profile of the probe pulse after turn-ing off the control beam when it is propagating insidethe medium and turning on the control beam again af-ter a storage time T . The retrieved pulse has a FWHMduration of 160 ns, which corresponds to about 1 MHzbandwidth. We are able to store and retrieve 10% of theinput probe pulse in the fiber.For the delayed light measurements, we use a movingoptical lattice to guide the atoms into the hollow-corefiber and stop them inside the fiber with a stationary optical lattice. This is to avoid loss and decoherence ofthe atoms due to their motion in the fiber. The movingoptical lattice beams are formed by a pair of counter-propagating fields (100 mW each) in the fiber with avelocity v =∆ f × λ/ − , where ∆ f =50 kHz isthe frequency detuning of the two lattice fields, and λ =821 nm is the lattice wavelength. We transport about10% of the atoms into the fiber compared to the loadingwith just a single optical dipole beam. The dominantloss is due to heating of the atoms during the transport-ing process which results in escape of atoms trapped inthe higher vibrational energy levels of the moving opticallattice. Figure 3(a) shows the storage efficiency of thelight as a function of time. We plot the efficiency of 1550nm light propagating in a solid-core fiber loop for com-parison. Our result exceeds the performance of 1550 nmlight after 600 µ s of delay time.We further improve the performance by minimizing thedifferential ac Stark shift of | (cid:105) and | (cid:105) states which isthe primary decoherence source of the atomic spin coher-ence [17]. This is carried out by introducing an elliptical-polarized dipole beam and an external magnetic field [18]to create a vector light shift to cancel the scalar lightshift. Figure 3(b) shows the storage efficiency as a func-tion of the delay time. We apply an external magneticfield of 2.08 G and optimize the efficiency by changing theellipticity of lattice beams through quarter-wave plates.An exponential decay function is fitted to the data withthe 1/ e decay time of 20 ms, which is limited by theinhomogeneous stray magnetic field of the experimentalapparatus [17]. Figure 3(c) shows experimental results ofthe normalized efficiency of light storage in a free-fallingatomic spin-wave for comparison. The storage time isshorter than for the atoms trapped in a stationary opti-cal lattice. We attribute this to the inhomogeneity of the T d d = 1 m s D = 5 6 7 8 H z a = 0 . 1 5 E x p e r i m e n t S i m u l a t i o n
Normalized efficiency
T i m e T ( m s ) ( a ) T d d = 2 . 5 m s D = 5 4 7 8 H z a = 0 . 1 6 E x p e r i m e n t S i m u l a t i o n
Normalized efficiency
T i m e T ( m s ) ( b ) R a m e s e y w it h D D t = 1 0 2 ( 8 ) m s Contrast
T im e T ( m s ) T d d = 5 m s D = 5 9 3 4 H z a = 0 . 1 4 E x p e r i m e n t S i m u l a t i o n
Normalized efficiency
T i m e T ( m s ) ( c ) FIG. 4. Efficiency versus delay time with different dynamic decoupling (DD) pulse separation times. The black squares representan average of 40 experimental runs of light delay efficiency and the error bars indicate the standard errors. The red circles arethe fit of the simulation to the experimental data. The lines are used to guide the eyes. The efficiencies are normalized to thefirst points for each T dd . The ratio of the Gaussian waists of the ensemble to the control beam in radial direction α and thefrequency difference of the microwave and the EIT two-photon detuning ∆ are set as free parameters to fit the data. Due tothe radial distribution of the EIT beams and atoms, the Rabi frequencies of the control and the probe pulses are set with aGaussian distribution in the radial direction in the simulation, where their maximum Rabi frequencies excite a π /2 pulse with250 ns duration. The duration of the microwave π pulses is 37 µ s and the Rabi frequency is assumed uniform across all theatoms. The inset shows the measurement of contrast versus time using microwave Ramsey interferometer with DD sequence.The curve is an exponential fit to the experimental data. magnetic field and dipole potential along the trajectoryof atoms that perturb the spin coherence and the relativemotion of atoms, respectively.One way to prolong the delay time is to execute thedynamic decoupling (DD) method [19, 20], a series of N microwave population inverting π pulses separated bycycling time T dd is applied to the atoms during the stor-age period to rapidly rephase the spin coherence, where N is an even integer. We study the DD method underour fiber delay line condition experimentally and theo-retically. Figure 4 shows the delay efficiency versus timemeasurements with T dd =1, 2.5, and 5 ms. We observethe oscillation of the efficiency over time due to the fre-quency difference ∆ between the microwave frequencyand the two-photon detuning between the control andthe probe pulses. We confirm the oscillation by simulat-ing the rotation of the Bloch vectors of the atoms underDD.In the simulation, we set the residual inhomogeneousbroadening of the dipole beam δ =2 π × (50 Hz) from thedata obtained in Fig. 3(b) and introduce α as the ratioof Gaussian waists of the atomic ensemble to the controlbeam in radial direction and set it as a free parameter to-gether with ∆. We fit the simulation to the data and findthe parameters α and ∆ for each T dd . The decay of theefficiency is due to the spatial inhomogeneity of the con-trol beam propagating in the fiber, which is determinedby α . Further details can be found in the Appendix.In the Bloch sphere picture, the inhomogeneous controlpulse prepares Bloch vectors on the Bloch sphere with adistribution for the DD sequence. To verify it, we usethe same DD sequence on a microwave Ramsey interfer-ometer, replacing the EIT probe and control pulses withmicrowave pulses which are assumed to be homogeneous across the whole atomic ensemble. We find that the 1/ e decay time τ of the contrast of the pure microwave se-quence is two times longer than the storage time underthe same DD; see the inset of Fig. 4(b).Light storage in the hollow-core fiber shows three or-ders of magnitude slower decay rate than the propaga-tion loss of light in the solid-core fiber. Moreover, thedelay time could be varied precisely by merely turningon and off the control pulse as an optical switch withoutmodifying any hardware as in the solid-core fiber loopdelay line. The low initial storage efficiency ( ∼ × , a figureof merit for quantifying the performance of the opticalstorage capacity. The solid-core fiber systems [3–6] haveachieved TBP to about 800 with a few percent of effi-ciency and atomic systems [7–12] have achieved TBP toabout 10 with about 30% efficiency. Our efficiency can beimproved by increasing the optical depth [11, 16] or us-ing other light storage methods such as Autler − Townessplitting to ease the demand of high optical depth [21].The near-infrared wavelength of our light can be con-verted into the telecom band in the same medium using ahigher level of atomic transitions [22, 23] and be extendedto single photons for quantum network applications [24].We thank Alex Kuzmich and Yi-Hsin Chen for read-ing the manuscript. This work is supported by the Sin-gapore National Research Foundation under Grant No.NRFF2013-12, Nanyang Technological University understart-up grants, and Singapore Ministry of Education un-der Grants No. Tier 1 RG107/17.
Appendix: Details of simulations of the light storageefficiency under dynamical decoupling1. Assumptions
We assume the following for the dynamical decoupling(DD) simulations:1. The effective Rabi frequency Ω R ( r ) of the EIT con-trol and probe pulse for an atom at radial position r is Gaussian distributed according the fundamen-tal mode field profile of the hollow-core fiber asΩ R ( r ) = Ω maxR e − r /R , where R = 22 µ m is the1 /e mode field radius and Ω maxR = π/ (500 ns).2. The EIT fields are two-photon resonant on the un-perturbed atomic transition.3. The Rabi frequency Ω M of the microwaves is uni-form across the ensemble with a π -pulse time of37 µ s.4. The microwave detuning from the unperturbedtransition frequency is ∆. ∆ is used as one of thefit parameters to the data.5. The inhomogeneous broadening δ ( r ) caused by theresidual optical lattice differential ac Stark shift isalso Gaussian distributed as δ ( r ) = δ max e − r /R ,where δ max = 2 π × (50 Hz) from the data in Fig.3(b) of the main paper.6. Initially, at t = 0, the phase of the microwaves isthe same as the EIT fields, i.e., over time t , themicrowave phase relative to an atom at position r is (∆ + δ ( r )) t .7. The atoms are Gaussian distributed according to ρ ( r ) = exp (cid:0) − r / R atom (cid:1) /R atom where R atom isthe rms radius of the atomic ensemble. The prob-ability distribution is normalized: (cid:82) ∞ ρ ( r ) rdr = 1.In the simulation, the atom positions are fixed forthe duration of the dynamical decoupling sequence.The ratio α of the ensemble size to the mode fieldradius, i.e., α ≡ R atom /R , is used as the other fitparameter to the data.
2. Simulation on a single atom
The goal is to determine the collective state of theatoms after the EIT pulse and microwave dynamical de-coupling pulses with the atoms evolving phase in the op-tical lattice.Beginning with an atom at position r in the groundstate, represented by the Bloch vector u = (0 , , − π/ π/ R ( r ) and detuning δ ( r ). In the Bloch sphere picture, the inhomogeneous EIT pulseprepares Bloch vectors on the Bloch sphere with a distri-bution for the DD sequence.After the EIT pulse, the atom evolves phase δ ( r ) T dd in the optical lattice for DD cycling time T dd . In theBloch sphere picture, the Bloch vector rotates about the z -axis by angle δ ( r ) T dd . A microwave π -pulse is then ap-plied, with Rabi frequency Ω M , detuning ∆ + δ ( r ), andphase (∆ + δ ( r )) t , where t is the time at which the π -pulse is applied. Again, the radial dependence encodesthe inhomogeneity of the ensemble with respect to themicrowaves. The DD sequence repeats the elementary se-quence [free-evolution T dd – π -pulse] for an even numberof times. The Bloch vector of the atom J = ( J x , J y , J z ) isnumerically determined after each repeating unit of theDD sequence.
3. Weighting over ensemble
The ensemble has a radial distribution across the fibermode profile leading to a distribution of Rabi frequencyΩ R ( r ) and detuning δ ( r ). As there is a one-to-one map-ping between the radial position r and the normalizedfiber mode intensity I ( r ) = e − r /R , the single atomsimulation J is actually a function J ( I ) of the normal-ized mode intensity I .To simulate the effect of this inhomogeneous distribu-tion, we weight the simulated single atom Bloch vectors J ( I ) by the intensity distribution ρ ( I ) that the ensemblesamples as in the following (cid:104) J (cid:105) ≡ (cid:90) J ( I ) ρ ( I ) dI. (A.1)This defines the collective Bloch vector (cid:104) J (cid:105) that we useto calculate the retrieval efficiency. Using the Gaussianatom distribution provided in the assumptions, it can beshown that the equivalent intensity distribution is givenby ρ ( I ) = 14 α I α − . (A.2)For the numerical simulations, we approximate the inte-gral using the trapezium rule.
4. Retrieval efficiency
The retrieval efficiency is determined by the collectivecoherence of the ensemble, which corresponds to the com-ponent of the collective Bloch vector in the equatorialplane J ⊥ ≡ (cid:104) J x (cid:105) + (cid:104) J y (cid:105) , where (cid:104) J x (cid:105) is the x -componentof the collective Bloch vector (cid:104) J (cid:105) ; similarly for (cid:104) J y (cid:105) . Thenormalized efficiency is then ( J ⊥ /J ⊥ , ) where J ⊥ , is thecollective coherence of the ensemble right after the EITstorage process.
5. Fitting simulation to data
To fit the simulation to the data, we use the method ofleast squares. Specifically, we define the sum of squaresdeviation between the simulation and experimental data points asSS = (cid:88) i (simulation(∆ , α ) i − data i ) error i (A.3)where error i is the standard error of the i -th experimentaldata point. We then numerically minimize SS over thefree parameters ∆ and α to obtain fits to the data sets. [1] L. Th´evenaz, Slow and fast light in optical fibres. Nat.Photonics , 474 (2008).[2] G. M. Gehring, R. W. Boyd, A. L. Gaeta, D. J. Gauthier,and A. E. Willner, Fiber-based slow-light technologies. J.Lightwave Technol. , 3752 (2008).[3] Z. Zhu, D. J. Gauthier, and R. W. Boyd, Stored light inan optical fiber via stimulated Brillouin scattering. Sci-ence , 1748 (2007).[4] E. Saglamyurek, J. Jin, V. B. Verma, M. D. Shaw, F.Marsili, S. W. Nam, D. Oblak, and W. Tittel, Quantumstorage of entangled telecom wavelength photons in anerbium-doped optical fibre. Nat. Photonics , 83 (2015).[5] J. Jin, E. Saglamyurek, M. G. Puigibert, V. Verma, F.Marsili, S. W. Nam, D. Oblak, and W. Tittel, Telecom-wavelength atomic quantum memory in optical fiberfor heralded polarization qubits. Phys. Rev. Lett. ,140501 (2015).[6] E. Saglamyurek, M. G. Puigibert, Q. Zhou, L. Giner,F. Marsili, V. B. Verma, S. W. Nam, L. Oesterling, D.Nippa, D. Oblak, and W. Tittel, A multiplexed light-matter interface for fibre-based quantum networks. Nat.Commun. , 11202 (2016).[7] M. R. Sprague, P. S. Michelberger, T. E. F. Champion,D. G. England, J. Nunn, X.-M. Jin, W. S. Kolthammer,A. Abdolvand, P. St. J. Russell, and I. A. Walmsley,Broadband single-photon-level memory in a hollow-corephotonic crystal fiber. Nat. Photonics , 287 (2014).[8] B. Gouraud, D. Maxein, A. Nicolas, O. Morin, and J.Laurat, Demonstration of a memory for tightly guidedlight in an optical nanofiber. Phys. Rev. Lett. ,180503 (2015).[9] C. Sayrin, C. Clausen, B. Albrecht, P. Schneeweiss,A. Rauschenbeutel, Storage of fiber-guided light in ananofiber trapped ensemble of cold atoms. Optica , 353(2015).[10] N. V. Corzo, J. Raskop, A. Chandra, A. S. Sheremet,B. Gouraud, and J. Laurat, Waveguide-coupled singlecollective excitation of atomic arrays. Nature , 359(2019).[11] F. Blatt, L. S. Simeonov, T. Halfmann, T. Peters, Sta-tionary light pulses and narrowband light storage in alaser-cooled ensemble loaded into a hollow-core fiber.Phys. Rev. A , 043833 (2016).[12] T. Peters, T.-P. Wang, A. Neumann, L. S. Simeonov, andT. Halfmann, Stopped and stationary light at the single-photon level inside a hollow-core fiber. Opt. Express , 5340 (2020).[13] B. Debord, M. Alharbi, T. Bradley, C. Fourcade-Dutin,Y. Y. Wang, L. Vincetti, F. G´erˆome, and, F, Benabid,Hypocycloid-shaped hollow-core photonic crystal fiberPart I: Arc curvature effect on confinement loss. Opt.Express , 28597 (2013).[14] M. Alharbi, T. Bradley, B. Debord, C. Fourcade-Dutin,D. Ghosh, L. Vincetti, F. G´erˆome, and, F, Benabid,Hypocycloid-shaped hollow-core photonic crystal fiberPart II: Cladding effect on confinement and bend loss.Opt. Express , 28609 (2013).[15] M. Fleischhauer, A. Imamoglu, and J. P. Marangos, Elec-tromagnetically induced transparency: optics in coherentmedia. Rev. Mod. Phys. , 633 (2005).[16] Y.-F. Hsiao, P.-J. Tsai, H.-S. Chen, S.-X. Lin, C.-C.Hung, C.-H. Lee, Y.-H. Chen, Y.-F. Chen, I. A. Yu,and Y.-C. Chen, Highly efficient coherent optical mem-ory based on electromagnetically induced transparency.Phys. Rev. Lett. , 183602 (2018).[17] M. Xin, W. S. Leong, Z. Chen, and S.-Y. Lan, Trans-porting long-lived quantum spin coherence in a photoniccrystal fiber. Phys. Rev. Lett. , 163901 (2019).[18] A. Derevianko, Theory of magic optical traps for Zeemaninsensitive clock transitions in alkali-metal atoms. Phys.Rev. A , 051606(R) (2010).[19] Y. Sagi, I. Almog, and N. Davidson, Process tomographyof dynamical decoupling in a dense cold atomic ensemble.Phys. Rev. Lett. , 053201 (2010).[20] Y. O. Dudin, L. Li, and A. Kuzmich, Light storage onthe time scale of a minute. Phys. Rev. A , 031801(R)(2013).[21] E. Saglamyurek, T. Hrushevskyi, A. Rastogi, K. Hes-hami, and L. J. LeBlanc, Coherent storage and manipu-lation of broadband photons via dynamically controlledAutlerTownes splitting. Nat. Photonics , 774 (2018).[22] T. Chaneli`ere, D. N. Matsukevich, S. D. Jenkins, T. A.B., Kennedy, M. S. Chapman, and A. Kuzmich, Quan-tum telecommunication based on atomic cascade transi-tions. Phys. Rev. Lett. , 093604 (2006).[23] A. G. Radnaev, Y. O. Dudin, R. Zhao, H. H. Jen, S. D.,Jenkins, A. Kuzmich, and T. A. B. Kennedy, A quan-tum memory with telecom wavelength conversion. Nat.Physics , 894 (2010).[24] H. J. Kimble, The quantum internet. Nature453