Long Light Storage Time in an Optical Fiber

Optical delay lines or optical buffers play important roles in long-distance quantum communication networks for storing, delaying, and, thus, exchanging information between different quantum nodes. The superb performance of optical fibers as transmission lines has driven the consideration of integrating these functionalities into the optical fibers themselves Th ; Geh . One of the challenges is maintaining the coherence of the medium in the fiber in which the information of light is encoded. The direct use of solid-core materials of telecommunication band fibers via stimulated Brillouin scattering and doped erbium ions to store light has been demonstrated to tens of nanoseconds Zhu ; Sag ; Jin ; Sag2 . Despite their tens of GHz bandwidth and telecom wavelength operation, the short-lived acoustic waves in the materials and spin coherence of doped erbium in the respective settings limit the performance of the light storage.

Alternatively, interfacing long-lived atomic spin states with optical fibers could provide a route to achieving long storage time. Loading room temperature atomic vapor into a hollow-core photonic crystal fiber Spr and cold atoms near the surface of an optical nanofiber Gou have achieved storage times of about tens of nanoseconds and few microseconds, respectively. They are limited by the transit time of atoms through the optical modes. Confining atoms in the evanescent field of an optical nanofiber Say ; Cor and guided field of a hollow-core fiber Bla ; Pet could eliminate decoherence from transit time but introduce additional decoherence from the confining fields. Both systems have shown storage time of about a few microseconds. To overcome the decoherence, we use a state insensitive optical potential to confine ⁸⁵Rb atoms in the guiding mode of a 4-cm-long hollow-core photonic crystal fiber Deb ; Alh . The phase of the optical pulse is mapped onto the atomic spin wave formed by a pair of long-lived hyperfine ground states and retrieved with a controllable delay using EIT. We demonstrate light storage over 50 ms with a bandwidth of 1 MHz.

A cold ⁸⁵Rb atomic ensemble is prepared by a magneto-optical trap (MOT) $\sim$5 millimeters above a 4-cm-long open-end hollow-core photonic crystal fiber, as shown in Fig. 1. After sub-Doppler cooling, the atomic ensemble is loaded into an optical trap guided by the fiber with a temperature of 10 $\mu$K. The wavelength of the optical dipole beam is 821 nm with 250 mW power, and the measured optical depth ($D$) is about 40 by probing all the Zeeman states of the ⁸⁵Rb D2 line $F$=3 to $F$’=4 transition. When the atoms are in the fiber, we optically pump them into the $F$=2, $m$=0 state by a $\pi$-polarized light and a repump light coupled through the fiber. The quantization axis is defined by a 2 G magnetic field along the fiber axis and the magnetic field is also used to lift the degeneracy of the Zeeman sublevels.

Our EIT scheme is formed by a three-level $\Lambda$ configuration Fle operating on the hyperfine clock states of ⁸⁵Rb for long coherence times. The probe beam is resonant on $|$1$\rangle$=$|F$=2, $m$=0$\rangle$ to $|$3$\rangle$=$|F$’=3, $m$=$\pm$1$\rangle$ and the control beam is resonant on $|$2$\rangle$=$|F$=3 $m$=0$\rangle$ to $|$3$\rangle$. A 795 nm diode laser is tuned to the control beam frequency, and a small portion of the power is split to set up the probe beam by an electro-optical modulator around 3 GHz followed by an etalon to filter out the undesired sidebands. Both beams are perpendicularly linearly polarized, and co-propagate inside the fiber. After exiting the fiber, the probe beam is separated from the control beam by polarization and etalon filtering and detected with an avalanche photodiode. The control and probe beams power inside the fiber are 300 nW and 40 nW, respectively, and the frequencies and the power of both beams are controlled by two independent and phase coherent acousto-optical modulators (AOMs). The 80 MHz radio frequency for the probe AOM is modulated by a Gaussian pulse from a waveform synthesizer. The 80 MHz radio frequency for the control AOM is modulated by a square pulse.

When the atoms are under free fall inside the fiber, we measure the group delay of the probe pulse, as shown in Fig. 2(a). The intensity of the input probe pulse can be approximated with a Gaussian temporal profile of $I_{\textrm{p}}$($t$)=$I_{\textrm{p0}}$exp(-4(ln2)$t^{2}$/$T_{\textrm{p}}^{2}$), where $I_{\textrm{p0}}$ is the maximum intensity, $T_{\textrm{p}}$=304 ns is the FWHM duration, and $t$ is the temporal coordinate. A Gaussian function is fitted to the transmitted probe pulse to determine the group delay $T_{\textrm{d}}$=174 ns with 24$\%$ efficiency of transmission.

We numerically calculate the group delay and the efficiency of the transmitted light by integrating over the inhomogeneous profiles of the atomic cloud in the dipole trap, the probe beam, and the control beam inside the fiber from the center of the fiber to the wall of the fiber $W$=31.5 $\mu$m Hsi . The intensity of the probe pulse after transmitting through the atomic ensemble is obtained as

where the factor

and the group delay $T_{\textrm{d}}=D\Gamma(\Omega_{c}^{2}+4\gamma_{21}^{2})/(\Omega_{\textrm{c}}^{2}+4\gamma_{31}\gamma_{21})^{2}$. The normalized Rabi frequency of the probe pulse is assumed to have a Gaussian distribution $\Omega_{\textrm{p}}$=$\exp$⁡(-$r^{2}$/$R^{2}$), where $r$ is the radial coordinate of the fibre core and $R$=22 $\mu$m is the 1/$e^{2}$ mode field radius. The ground state coherence $\gamma_{21}$, which is mainly due to the inhomogeneous differential ac Stark shift of the control and dipole beams, is fixed at 2$\pi\times$(2$\times$10⁵ Hz) based on the measurement of EIT linewidth and ground and excited states coherence $\gamma_{31}$ is fixed at half of the ⁸⁵Rb D1 line $P$ state decay rate $\Gamma$= 2$\pi\times$(5.75$\times$10⁶ Hz). The optical depth $D$ then follows the Gaussian distribution $D=L\times n\times\exp⁡(-2r^{2}/R^{2})\times\sigma\times\exp⁡(-2r^{2}/R^{2})$, where the length of the ensemble $L$ along the fiber axis is approximated with 5 mm Xin , the number density $n$ is approximated with 1.3$\times$10¹¹ cm^-3 from the maximum measured $D$=40, and $\sigma$=0.65$\times$10^-9 cm² is the scattering cross-section of the D1 line $F$ = 3, $m$ = 0 to $F$’ = 3, $m$ = $\pm$1 transition at the center of the dipole trap.

When the cloud size in the radial direction is assumed to have the same distribution as the light in the fiber, the simulated delayed pulse is within 10$\%$ of the experimental delayed pulse in terms of delay time and efficiency. The uniform profile calculation uses the peak intensity of control and probe beams, and the peak optical depth. The ground state coherence $\gamma_{21}$ is chosen as 2$\pi\times$(10³ Hz) from the coherence time measurement of the microwave Ramsey interferometer before light shift cancellation of the optical lattice described in the following section.

Figure 2(b) compares the group delay of the probe pulse in the fiber with the free space scenario of uniform control beam, probe beam, and atoms density profiles. The inhomogeneous profiles of the control and the atom density in the fiber result in more than a factor of 2 reduction of the group delay and the efficiency. Figure 2(c) shows the temporal profile of the probe pulse after turning off the control beam when it is propagating inside the medium and turning on the control beam again after a storage time $T$. The retrieved pulse has a FWHM duration of 160 ns, which corresponds to about 1 MHz bandwidth. We are able to store and retrieve 10$\%$ of the input probe pulse in the fiber.

For the delayed light measurements, we use a moving optical lattice to guide the atoms into the hollow-core fiber and stop them inside the fiber with a stationary optical lattice. This is to avoid loss and decoherence of the atoms due to their motion in the fiber. The moving optical lattice beams are formed by a pair of counter-propagating fields (100 mW each) in the fiber with a velocity $v$=$\Delta f\times\lambda/2$=2.05 cm s^-1, where $\Delta f$=50 kHz is the frequency detuning of the two lattice fields, and $\lambda$= 821 nm is the lattice wavelength. We transport about 10$\%$ of the atoms into the fiber compared to the loading with just a single optical dipole beam. The dominant loss is due to heating of the atoms during the transporting process which results in escape of atoms trapped in the higher vibrational energy levels of the moving optical lattice. Figure 3(a) shows the storage efficiency of the light as a function of time. We plot the efficiency of 1550 nm light propagating in a solid-core fiber loop for comparison. Our result exceeds the performance of 1550 nm light after 600 $\mu$s of delay time.

We further improve the performance by minimizing the differential ac Stark shift of $|1\rangle$ and $|2\rangle$ states which is the primary decoherence source of the atomic spin coherence Xin . This is carried out by introducing an elliptical-polarized dipole beam and an external magnetic field Der to create a vector light shift to cancel the scalar light shift. Figure 3(b) shows the storage efficiency as a function of the delay time. We apply an external magnetic field of 2.08 G and optimize the efficiency by changing the ellipticity of lattice beams through quarter-wave plates. An exponential decay function is fitted to the data with the 1/$e$ decay time of 20 ms, which is limited by the inhomogeneous stray magnetic field of the experimental apparatus Xin . Figure 3(c) shows experimental results of the normalized efficiency of light storage in a free-falling atomic spin-wave for comparison. The storage time is shorter than for the atoms trapped in a stationary optical lattice. We attribute this to the inhomogeneity of the magnetic field and dipole potential along the trajectory of atoms that perturb the spin coherence and the relative motion of atoms, respectively.

One way to prolong the delay time is to execute the dynamic decoupling (DD) method Sagi ; Dud , a series of $N$ microwave population inverting $\pi$ pulses separated by cycling time $T_{\textrm{dd}}$ is applied to the atoms during the storage period to rapidly rephase the spin coherence, where $N$ is an even integer. We study the DD method under our fiber delay line condition experimentally and theoretically. Figure 4 shows the delay efficiency versus time measurements with $T_{\textrm{dd}}$=1, 2.5, and 5 ms. We observe the oscillation of the efficiency over time due to the frequency difference $\Delta$ between the microwave frequency and the two-photon detuning between the control and the probe pulses. We confirm the oscillation by simulating the rotation of the Bloch vectors of the atoms under DD.

In the simulation, we set the residual inhomogeneous broadening of the dipole beam $\delta$=2$\pi\times$(50 Hz) from the data obtained in Fig. 3(b) and introduce $\alpha$ as the ratio of Gaussian waists of the atomic ensemble to the control beam in radial direction and set it as a free parameter together with $\Delta$. We fit the simulation to the data and find the parameters $\alpha$ and $\Delta$ for each $T_{\textrm{dd}}$. The decay of the efficiency is due to the spatial inhomogeneity of the control beam propagating in the fiber, which is determined by $\alpha$. Further details can be found in the Appendix.

In the Bloch sphere picture, the inhomogeneous control pulse prepares Bloch vectors on the Bloch sphere with a distribution for the DD sequence. To verify it, we use the same DD sequence on a microwave Ramsey interferometer, replacing the EIT probe and control pulses with microwave pulses which are assumed to be homogeneous across the whole atomic ensemble. We find that the 1/$e$ decay time $\tau$ of the contrast of the pure microwave sequence is two times longer than the storage time under the same DD; see the inset of Fig. 4(b).

Light storage in the hollow-core fiber shows three orders of magnitude slower decay rate than the propagation loss of light in the solid-core fiber. Moreover, the delay time could be varied precisely by merely turning on and off the control pulse as an optical switch without modifying any hardware as in the solid-core fiber loop delay line. The low initial storage efficiency ($\sim 3\%$) in our experiment compared to the free space case is due to the inhomogeneous profile of the control beam and atom density inside the fiber. We have achieved the time-bandwidth product (TBP) to about 5$\times$10⁴, a figure of merit for quantifying the performance of the optical storage capacity. The solid-core fiber systems Zhu ; Sag ; Jin ; Sag2 have achieved TBP to about 800 with a few percent of efficiency and atomic systems Spr ; Gou ; Say ; Cor ; Bla ; Pet have achieved TBP to about 10 with about 30$\%$ efficiency. Our efficiency can be improved by increasing the optical depth Bla ; Hsi or using other light storage methods such as Autler$-$Townes splitting to ease the demand of high optical depth Sag3 . The near-infrared wavelength of our light can be converted into the telecom band in the same medium using a higher level of atomic transitions Cha ; Rad and be extended to single photons for quantum network applications Kim .

We thank Alex Kuzmich and Yi-Hsin Chen for reading the manuscript. This work is supported by the Singapore National Research Foundation under Grant No. NRFF2013-12, Nanyang Technological University under start-up grants, and Singapore Ministry of Education under Grants No. Tier 1 RG107/17.

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