Protecting a quantum memory for a photonic polarization qubit in a cold atomic ensemble by dynamical decoupling

We report an experimental demonstration of storage of photonic polarization qubit (PPQ) protected by dynamical decoupling (DD). PPQ’s states are stored as a superposition of two spin waves by electromagnetically-induced-transparency (EIT). Carr-Purcell-MeiboomGill (CPMG) DD sequences are applied to the spin-wave superposition to suppress its decoherence. Thus, the quantum process fidelity remains better than 0.8 for up to 800μs storage time, which is 3.4-times longer than the corresponding storage time of ~180μs without the CPMG sequences. This work is a key step towards the storage of single-photon polarization qubit protected by the CPMG sequences. ©2014 Optical Society of America OCIS codes: (270.5585) Quantum information and processing; (140.3550) Lasers, Raman; (020.3320) Laser cooling; (020.7010) Laser trapping. References and links 1. L. M. Duan, M. D. Lukin, J. I. Cirac, and P. 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Introduction
Photons with their polarization states encoding in quantum information are good for transmitting quantum information.For realizing long-distance quantum communication [1] and scalable quantum computing [2], long-lived quantum memories for a photonic polarization qubit are crucial [3][4][5].Many physical processes have been exploited to store quantum states of light.The absorptive electromagnetically-induced-transparency [6] and emissive spontaneous-Raman-emission processes [1] in atomic ensembles provide promising storage schemes.In both schemes, photonic states are stored as spin waves and then retrieved after a demand time.However, due to decoherence, the spin waves will decay with time and will be only effectively retrieved within storage lifetimes.There are several factors that limit the lifetimes of quantum memories in atomic ensembles.For the storages of a certain polarization state of light, only a single spin wave is required and the decoherence for such storages includes two main factors: atomic motion and inhomogeneous Zeeman broadening.For storages of photonic polarization qubit (PPQ), two spin waves (SWs) [7,8] are required.Thus, besides the above-mentioned two decoherence mechanisms, random phase between the two SWs caused by magnetic field fluctuations will degrade polarization fidelity and then lead to decoherence [9].In the past several years, many studies on suppressions of spin-wave decoherence have been done and several methods have been proved to be effective.The decoherence resulting from atomic motion can be significantly suppressed by using cold atoms in magneto-optical traps [10] or ultra-cold atoms in optical lattice [11] as storage media and the decoherence caused by spatial gradient of magnetic fields and their temporal fluctuations can be obviously decreased by storing PPQ's states as two magnetic-fieldinsensitive coherences [12,13] or two spatially distinct SWs [14] associated with the clock coherence.Another method to suppress the random phase in PPQ's storages is to compensate the magnetic field noise by using an open-loop feed-forward technique [9].In the past few years, a powerful strategy, known as dynamical decoupling (DD) technique, has been developed to protect qubit memories in single atoms [15,16], array of ions [17], spin ensembles [18], NV-center systems [19] and so on.In very recent years [20][21][22], the optical quantum storages based on electromagnetically induced transparency (EIT) has also been introduced.By applying DD sequences to rotate the two states of a spin wave, the longest storage lifetimes for a fixed polarization light are 16 seconds in atoms confined in optical lattice [20] and 1 minute in solid-state ensembles [22].However, the experiments of suppressing random phase between the two spin waves in the PPQ's storages by DD have not been demonstrated.
In this paper, we experimentally demonstrate the PPQ's storages protected by Carr-Purcell-Meiboom-Gill (CPMG) DD pulse sequences.Based on EIT effect in a cold atomic ensemble, the PPQ is stored as two spin waves, which are associated with the magnetic-fieldinsensitive and magnetic-field-sensitive coherences, respectively.The CPMG sequences containing multiple Raman π pulses are applied to suppress decoherence of the spin waves and thus the quantum process fidelity remains better than 0.8 for up to 800μs storage time. .The writing/reading light field is left circular polarization and the input signal-light field is an arbitrary polarization state.The signal-light field can be written as ( )

Theoretical discussion for protecting PPQ's storage by CPMG DD sequences
where The quantum axis defined by a bias magnetic field z B is along z-direction, so, the σ ± polarized components of the signal field couple to , respectively, are defined by [23], ( ) is the number of atoms between z and z dz + , l is the length of the atomic ensemble.The conversion between the arbitrarily polarized signal-light fields and the two spin waves are described by the theory of two dark-state polaritons [24].According to the theory, the σ + and σ − polarized signal light fields will be converted into spin waves Ŝ− and Ŝ+ , respectively, if the writing beam is turned off over a very short time interval [ ] 0 1 , t t .Such conversion will create a superposition: ( ) .During the storage time, the spin waves Ŝ+ and Ŝ− will suffer from decoherence and undergo Larmor processes, respectively.
For the presented storage scheme, the time interval 1 0 t t − is close to zero, so we have 0 1 t t ≈ .
Assuming 0 0 t = , the superposition at time t will evolve into: where, d τ is the lifetime limited by atomic motions, τ ± are the lifetimes for the magnetic- field-sensitive and magnetic-field-insensitive spin waves Ŝ+ and Ŝ− , respectively, which are limited by inhomogeneous Zeeman broadening [11], ( ) ) are the Larmor frequencies for the transitions If the reading beam is switched on at this time t , the spin-wave supposition will be transferred into the retrieved light field according to [24]: where, R is the retrieval efficiency, ) is the difference of Larmor frequency between the SWs S + and S − , which are induced by guiding magnetic field z B (magnetic field noise ( ) ).The calculated Stokes parameters for the retrieved light field are: where, where 0 1/ (0) is the coherence lifetime [26].The quantum state fidelity is defined by where, are the input (output) density matrices, ˆi σ are the Pauli spin operator.We calculate the quantum state fidelities for six input states based on the Eqs.( 7)-( 9), the results are: , where , , , , H V R L D and A denote horizontal, vertical, right circular, left circular, diagonal (45°), and antidiagonal (−45°) polarizations, respectively.The quantum process fidelity can be expressed as [26]: where is the average quantum state fidelity.Based on Eqs. ( 10)-( 12), we calculate the quantum process fidelity, the result is: where, t θ = Ω is the relative phase between the SWs Ŝ+ and Ŝ− .According to the above equation, we may find that two factors cause decoherence of the qubit memory and make quantum process fidelity decrease with time.One factor is magnetic-field fluctuations, which causes a random phase between two spin waves and then makes the normalized coherence ( ) W t exponentially decay with the storage time.Another is difference between the lifetimes τ + and τ − , which also degrade the quantum process fidelity after a storage time t.The relative phase t θ = Ω changes with the storage time, which will lead to an oscillation of process fidelity p F .However, it can be compensated by using linear optical elements or selecting the reading times at j t j T = Δ (where j is an integer and 2 / T π Δ = Ω is the time interval) to make 2 j θ π = .In this case, cos 1 θ = can be remained and the oscillation of process fidelity will be avoided.When the storage time is very long, ( ) 0 W t → and the quantum process fidelity p F will be close to the boundary of 1 2 . Such boundary corresponds to the case where the retrieval signal is unpolarized light when the input signal is a linearlypolarized light.For suppressing decoherence caused by the two factors, we can apply CPMG DD sequences to decouple the spin-wave superposition from environment noise.The CPMG sequences contains n instantaneous π pulses applied at times 1 t , 2 t , …, n t , with time delays: . Assuming that each π pulse is ideal (means that it doesn't cause any spin-wave decay), it can swap spin waves in a superposition according to: Such swapping can be viewed as a π rotation of the spin-wave superposition.Considering the case where the number of the π-pulses applied within the storage time t is even, i.e., n = 2m ( m is an integer), the evolution of the superposition ( , ) , where, H t e τ ω δ ω τ τ τ τ , the normalized coherence for or for when τ → ∞ , the normalized coherence

S z t S z t H t s z t e s z t
According to the Eq. ( 15), we derive the quantum process fidelity: where t ϑ δ = Ω is the relative phase, which may cause an oscillation of quantum process fidelity with the time interval t δ .Similar to the above discussion for θ , we may take cos 1 ϑ = by compensating the relative phase or selecting appropriate retrieval times.In this case, quantum process fidelity 1 p F → , which means that the CPMG DD sequences may effectively suppress the two decoherence factors.
If the number of the π-pulses applied within the storage time t is odd ( 2 1 n m = − ) and the period T is short enough, the time evolution of the superposition at time , can be written as: where .Comparing with the superposition manipulated by 2m π pulses (See Eq. ( 15)), we find that this is a swapped superposition.For correcting this superposition, we have to apply an additional π-pulse to it after (2m-1)-th π-pulse and then it will be changed into the one that is the same with Eq. ( 15).The involved atomic level scheme of 87 Rb for optical pumping.

Experimental set-up and results
The experimental setup is shown in Fig. 3 (a).A cold 87 Rb atomic cloud in a magneto-optical trap serves as the memory medium.A polarization-insensitive beam-splitter (BS) is used to combine the input signal and writing/reading light beams.Before arriving BS, the signal beam goes through a quarter-wave plate (QWP 1 ) and a half-wave plate (HWP 1 ).By adjusting QWP 1 and HWP 1 , the polarization state of the signal light can be arbitrarily set.For suppressing the dephasing effect resulting from atomic motion, we make the signal and writing/reading light beams collinearly go through the cold-atom cloud along z-direction.The powers (diameters) of the writing-reading and signal beam are 3mW (2mm) and 34μW (1mm), respectively.The horizontally-polarized Raman laser beam (with a ~4.5mm diameter) passes through the cold atoms with a deviation angle of ~0.2° from ẑ .It is tuned to the transitions a e ↔ with a blue detuning 90 (see Fig. 1 (b)).We use an analogue acousto-optic modulator (AOM) to modulate Raman laser amplitude and then obtain a rectangular pulse with a time length of 150ns.By carefully set the input power of the Raman laser beam, we realize a π-rotation between the SW Ŝ+ and Ŝ− .As shown in Fig. 3(b), the σ − polarized pumping laser P1 and σ + polarized pumping laser P2 collinearly propagate through the atoms with a deviation angle ~2° from z-direction, which drive the transitions respectively.The π-polarized pumping laser P3 propagates through the atoms along xdirection, which drives the transition The powers and diameters of the lasers P1, P2 in the center of cold atoms are approximately equal, which are ~10 mW and ~7 mm, respectively, while that of the laser P3 is ~400 µW and ~8.8 mm, respectively.For each experimental circle, the durations for the preparation of the cold atoms and storage-retrieval experiments are ~98 ms and 2 ms, respectively.During the preparation stage, 87 Rb atoms are loaded into magneto-optical trap (MOT) and then performed a Sisyphus cooling.The evaluated magneto-optical trap temperature is ~200 µK, the number of the atoms in the MOT .After the Sisyphus cooling we apply a dc magnetic field 0 375mG B s = along ẑ to define the quantization axis.Then the pump P1, 2, 3 as well as the writing laser beams are switched on to prepare the atoms into the desired ground state with the same population.After about 20μs, the pump P1, 2, 3 are turned off and the signal pulse (with a pulse length of 200ns) is injected into the atom ensemble.At the falling edge of the signal pulse (corresponding to 0 0 t = ), the writing laser beam is ramped to zero and thus the signal pulse is mapped into the spin-wave superposition 0 ( , ) S z t .We then apply CPMG pulse sequences to manipulate the spin waves.By switching on the reading light beam, we retrieve optical signals from the spin waves.Since the reading and retrieval signal beams will propagate collinearly along z-axis, we have to block the reading beam by using a set of filters [13].Then only the retrieved signals enter a polarization analyzing and measuring (PAM) system.The PAM system consists of a quarter-wave plate (QWP 3 ), a half-wave plate (HWP 3 ), a Glan-laser polarizer (GLP) and detectors D 1,2 (Hamamatsu C5331).With QWP 2 and HWP 3 , we can select the polarization basis H V − , L R − or D A − in turn for the polarization analyzing and measuring.We first measure the retrieval efficiencies as the function of time t .The square (black) and circle (red) dots in Fig. 4 are the measured results for ( Ι ) σ + -polarized and ( ΙΙ ) σ −polarized fields, respectively.The solid (black) and dash (red) curves are the fits to these data based on  We next perform the storage and retrieval of H , V , D and R polarization light with and without CPMG pulse sequences, respectively.Figures 5(a , respectively.Based on the polarized analysis for the measured data of retrieval signals, we reconstruct the density matrix out ρ and then obtain quantum process fidelities.Figure 5(c) shows the quantum process fidelities as the function of the storage time.According to the measured Larmor frequency difference Ω , we determine the time interval T Δ and then select reading at times of j t j T = Δ (For details see Sec.2).In 5(c), the square dots correspond to the results when CPMG pulse sequence is not applied.It shows that the quantum process fidelity reduces to ~80% at 180μs.At 500 s t μ ≈ , the quantum process fidelity reaches ~25%, which is less than the boundary of 1/2.We attribute this to inaccurate determination of the time interval T Δ .The circle and triangle dots are quantum process fidelities when CPMG pulse sequences with frequencies of 5 are applied, respectively.In the measurements for the two cases, if the stored optical signals are retrieved after (2m-1)-th π pulse, we apply an additional π-pulse to correct the spin-wave superposition before the retrieval of the optical signals.δ and the difference between the lifetimes τ + for the spin wave (SW) Ŝ+ and τ − for the SW Ŝ− .Both the two factors make the quantum process fidelity degrade with the storage time t.
In the case where the CPMG pulse sequences are not applied, the quantum process fidelity keeps a monotonous decrease with the storage time t.However, at storage time 100 t s μ < , the decrease of the quantum process fidelity caused by the two decoherence factors is not significantly since the time interval is short.While, the decrease of the quantum process fidelity will become significant at . Figure 6 shows the quantum process fidelity at storage time of 800μs as the function of the π-pulse number n (lower) and the frequency dd F (upper), respectively.The measured quantum process fidelity is better than 0.8 for 4 n = and 0.77 for 6 n = .

Discussions
We perform the polarization quantum memory with the classical signal-light pulses of different polarization.However, our storage scheme will also hold for the single-photon polarization states because the attenuation of the signal light in the storage process is linear [28] [29] [30].In other words, the measured polarization fidelities for the single photons will be close to that for the classical light pulses.However, there might exist the background noise photons directed into the single-photon detectors, which will degrade the quantum fidelity of the single-photon polarization states.The background photons may come from the leakage photons from reading beam or spontaneous emissions from the atomic ensemble.For blocking the leakage photons from the reading beam, we may insert more number of optical filters before single-photon detectors.The reason for the generation of the spontaneous emissions is explained in the following.In the atomic ensemble, there may exist small number of atoms which are in the excited state b but are not associated with the spin waves.
When the reading beam is switched on to drive the transition b e ↔ , these atoms will be excited to e and then generate spontaneous emissions.Many processes, for example, thermally excitations, etc., may induce incoherently transfer from the state a into the state b .In the presented scheme, the Raman π-rotations is one process which induce incoherently transfer from the state a into b [26].By taking large enough detuning R Δ = , the number of the transferred atoms will be decreased [26] and then the background noise photons coming from such process may be significantly suppressed.

Conclusions
We demonstrate PPQ's storages protected by CPMG DD sequences.The process fidelity better than 0.8 is measured for up to 800μs storage time with a 10kHz -frequency CPMG sequence being applied, while, it is measured for up to 180μs storage time without CPMG sequence.Compared with the previously reported PPQ's storages [12,13], an advantage of the presented long-lived PPQ's storage is that single-qubit operations can be easily implemented during storage.We believe that this advantage can be used to improve the success probability of the quantum teleportation based on atomic ensembles [31] and find more effective applications in quantum information processing.
1/ f α -type noise[27], so its power spectrum lim ( period T is short enough, we have ( / this case, the two factors causing decoherence are eliminated and spin-wave superposition is changed into:

Fig. 6 .
Fig. 6.Quantum process fidelity p F at storage time of 800μs as the function of the number of π-pulses n (lower) and the frequency dd F (upper). ) In this case, the CPMG π-pulse sequence can timely protect the qubit memory from decoherence and then the quantum process fidelity keeps a slow monotonous decrease with the storage time t.At 800μs, the quantum process fidelities reach ~80% for the cases of 5