Optical storage for 0.53 seconds in a solid-state atomic frequency comb memory using dynamical decoupling
Adrian Holzäpfel, Jean Etesse, Krzysztof T. Kaczmarek, Alexey Tiranov, Nicolas Gisin, Mikael Afzelius
OOptical storage for 0.53 seconds in a solid-stateatomic frequency comb memory using dynamicaldecoupling
Adrian Holz¨apfel, Jean Etesse, Krzysztof T. Kaczmarek,Alexey Tiranov, Nicolas Gisin, Mikael Afzelius
Department of Applied Physics, University of Geneva, CH-1211 Geneva 4,SwitzerlandE-mail: [email protected]
27 April 2020
Abstract.
Quantum memories with long storage times are key elements in long-distance quantum networks. The atomic frequency comb (AFC) memory in particularhas shown great promise to fulfill this role, having demonstrated multimode capacityand spin-photon quantum correlations. However, the memory storage times have so-far been limited to about one millisecond, realized in a Eu doped Y SiO crystalat zero applied magnetic field. Motivated by studies showing increased spin coherencetimes under applied magnetic field, we developed a AFC spin-wave memory utilizinga weak 15 mT magnetic field in a specific direction that allows efficient optical andspin manipulation for AFC memory operations. With this field configuration theAFC spin-wave storage time increased to 40 ms using a simple spin-echo sequence.Furthermore, by applying dynamical decoupling techniques the spin-wave coherencetime reaches 530 ms, a 300-fold increase with respect to previous AFC spin-wavestorage experiments. This result paves the way towards long duration storage ofquantum information in solid-state ensemble memories.
1. Introduction
A future quantum internet relies on the capability of remotely sharing quantuminformation, and last years have seen rapid progresses in increasing the distancesover which it can be distributed. Recent experiments have demonstrated fiber-basedquantum communication of over 400 km [1, 2], but reaching continental distancesusing fiber networks will require quantum repeaters [3]. These will require multiplexedquantum memories, i.e. devices that allow storage of quantum states of light indifferent modes in time, space or frequency [4]. Another key feature is the ability tostore multiplexed quantum states on timescales of at least several hundred millisecondswithout a significant deterioration of storage efficiency over time [5].Optical quantum memories based on spin-states in atomic ensembles have shownthe potential to fulfill these requirements, both in laser-cooled alkali vapors [6, 7, 8, 9, 10] a r X i v : . [ qu a n t - ph ] A p r ptical storage for 0.53 seconds in a solid-state atomic frequency comb memory using dynamical decoupling doped Y SiO crystals [16, 20, 24]. In these experiments the spinstorage is realized on a zero-field nuclear quadrupole resonance where, at zero appliedmagnetic field, each quadrupole state is composed of two degenerate nuclear Zeemanstates.In this article we show that by lifting the degeneracy using a weak external field, theAFC spin-storage time can be extended to 40 ms using a simple spin-echo sequencecomposed of two π pulses. The direction of the magnetic field is carefully chosen basedon previous spectroscopy studies [25, 26, 27], in order to achieve efficient coherent opticaland spin manipulation without cross talk between transitions. Furthermore, the spin-wave storage time can be extended to up to 0.53 s using a dynamical decoupling (DD)sequence [28], two orders of magnitude longer than previous AFC storage experiments.The article is organized as follows: in section 2, we remind the principle of the AFCprotocol and the phenomena that impact its performances in terms of storage time,and describe the technique that we use to push the limitations further in our particularsystem. In section 3, experimental results are presented, and a spectral diffusion modelis developed to explain the observed behaviors. Finally, discussions on discrepanciesbetween the model and the experimental data are made in section 4.
2. Spin-wave AFC and radio frequency manipulations
First, let us introduce the general principle of the AFC protocol, sketched in figure 1.It is based on the creation of a frequency grating (the comb) on an optical transition | g (cid:105) ↔ | e (cid:105) in an inhomogeneously broadened ensemble. The method we use for creatingan AFC is detail in Ref. [19]. An input photon that is absorbed on this transition willcreate a single delocalized excitation, bringing the ensemble into a Dicke state: | ψ (cid:105) ∝ N (cid:88) j =1 e − i πn j ∆ t | g , ..., e j , ...g N (cid:105) , (1)where n j is the number of the comb tooth to which the atom j belongs, and ∆ is thecomb’s periodicity. The ensemble will then naturally undergo dephasing. However, dueto the periodic comb structure, after a time t = 1 / ∆, the atoms will all rephase andsubsequently re-emit the photon as an echo (denoted output in figure 1). In order toallow for an increased storage time and on demand read out, the coherence is mappedonto the long-lived spin transition | g (cid:105) ↔ | s (cid:105) . This is done by applying a π pulse (denotedcontrol in figure 1) on the | e (cid:105) ↔ | s (cid:105) transition before the re-emission has occurred. Toretrieve the excitation, a second identical π pulse is applied to map the coherence back ptical storage for 0.53 seconds in a solid-state atomic frequency comb memory using dynamical decoupling | g (cid:105) ↔ | e (cid:105) transition where it rephases due to the AFC structure. In thisarticle, we will note the duration between the two control pulses T spin , such that thetotal storage time is T spin + 1 / ∆. The efficiency of the whole process can be expressedas: η tot = η AFC ( η ctrl ) η spin , (2)where η AFC is the efficiency of the AFC protocol without spin storage, η ctrl is theefficiency of a single control pulse and η spin quantifies the degree to which coherencecan be preserved while storing in the spin transition. There are two main mechanismsthat contribute to loss of coherence during the spin storage, and consequently to adecrease of η spin . I npu t O u t pu t RF C on t r o l Figure 1.
The spin-wave AFC protocol, in a lambda scheme.
The first mechanism stems from the inhomogeneous broadening of the spintransition Γ inh : as the different ions have slightly different resonance frequencies, theywill dephase with respect to each other in a characteristic time given by ∼ / Γ inh . Thiseffect can be undone by applying well known spin-echo techniques: two radio-frequency(RF) pulses are applied for performing π rotations on the | g (cid:105) ↔ | s (cid:105) transition (seefigure 1). Thanks to this technique, in [24] we were able to push the storage time fromtens of microseconds to milliseconds, while preserving a good signal-to-noise ratio at thesingle photon level.The second mechanism comes from spectral diffusion, which is the variations of thespin transition frequencies over time, due to fluctuations in the ion’s environment. Thisinevitably leads to a dynamical dephasing of the collective Dicke state (1), and thus toa decrease of the memory efficiency as a function of storage time.The detrimental effect of spectral diffusion can be reduced by performing DD of thespin transition: a large number of π pulses is applied on the | g (cid:105) ↔ | s (cid:105) transition at arate that is fast enough to decouple the spins from the fluctuations of the environment.In this case, the ions will spend as much time in the ground state | g (cid:105) as in the excitedstate | s (cid:105) of the spin transition, leading to a compensation of the dephasing and a longereffective coherence time. This spin echo technique has already been successfully appliedto push the spin coherence times in RE ion doped crystals under a particular magneticfield condition, the ZEFOZ (ZEro First Order Zeeman) point, bringing the effective ptical storage for 0.53 seconds in a solid-state atomic frequency comb memory using dynamical decoupling doped Y SiO , reachingZEFOZ points require magnetic field intensities of the order of 1 T [29, 26], makingthis configuration challenging to implement. We have chosen a different approach, forwhich DD is performed under a weak magnetic field. Previous studies have shown thatthe application of a weak field can also enhance the observed coherence time [30]. Inthe following we will detail the different considerations that were made in choosing themagnetic field direction. Our memory is based on a europium doped yttrium orthosilicate crystal( Eu :Y SiO ) in a non-zero magnetic field configuration. The level structure ofthis material is shown in figure 2(a), and consists of an optical transition between theelectronic states F and D connecting two nuclear spin manifolds I = 5 / H spin = ˆI · Q · ˆI + B · M · ˆI . (3)The effective-quadrupolar term ˆI · Q · ˆI is responsible for the zero-field energy levelsplittings of the order of ∼
10 MHz that are shown in the right part of figure 2(a).Three levels are then at our disposal both in the optical ground and excited states, andalready allowed us to implement the previously mentioned spin-wave AFC protocol byusing states | g (cid:105) = | / (cid:105) g , | s (cid:105) = | / (cid:105) g and | e (cid:105) = | / (cid:105) e [16, 20, 24]. As the splittingbetween | g (cid:105) and | s (cid:105) is 34.54 MHz, the spin manipulation could be performed by usinga RF field, but unfortunately decays of the echos proved to be too short to applydynamical decoupling techniques. Previous studies have found that the observed decayof the coherence can be enhanced by applying a weak magnetic field [30]. In addition,such a field will obviously lift the degeneracy of each Zeeman doublet, due to the B · M · ˆI term in Eq.(3). In our case, the strength of the split is of the order of ∼
10 MHz/T (seeright part of figure 2(a)).We have decided to apply the magnetic field at 65 ◦ relative to the D axis in theplane spanned by the D and D polarization axis of the crystal [33]. Three majorpoints motivated this choice of orientation: the reduced number of levels regarding sitedegeneracy, the branching ratio state-selection, and the equal ground states splittings δ g = δ s := δ [25, 26, 27] (see notations of figure 2(a)). The precise considerationsand implications for the choice of this field direction are found in the appendix; inshort, operating with this bias field essentially allows us to profit from the increasedspin coherence time while retaining the same optical depth as well as efficiency of spincontrol that we would have if no external field was applied.The strength of the magnetic field was set to the maximum possible value of themagnetic coils used for this experiments, which was 15 mT. At this field strength wenote that: ptical storage for 0.53 seconds in a solid-state atomic frequency comb memory using dynamical decoupling (a) (b)(c) { { Cryostat m ag n e t i c b i a s f i e l d x zy Control PreparationInput DetectionRF source C on t r o l P r e p a r a ti on I npu t C on t r o l O u t pu t ... X Y YX X Y YX
Dynamical Decoupling
Figure 2.
The AFC protocol, and spin manipulation. (a) Lambda schemes forspin-wave AFC. The splits are all of the order of δ i (cid:39)
10 MHz/T, leadings to splitsof δ i (cid:39)
100 kHz at 15 mT magnetic field. (b) Experimental setup used. (c) Timesequence for spin-wave AFC with DD. δ = 210 kHz > Γ inh ∼
30 kHz (4 a ) δ = 210 kHz > Ω RF / (2 π ) = 23 kHz (4 b ) δ e = 300 kHz > Γ AFC = 160 kHz . (4 c )On one hand, inequalities (4 a ) and (4 b ) simply ensure that the RF field only excitesthe desired transition | +1 / (cid:105) g ↔ | +3 / (cid:105) g or |− / (cid:105) g ↔ |− / (cid:105) g without exciting thecrossed transitions | +1 / (cid:105) g ↔ |− / (cid:105) g or | +1 / (cid:105) g ↔ |− / (cid:105) g . Inequality (4 c ), on theother hand, puts a limit onto the AFC bandwidth Γ AFC which stems from the AFCpreparation and ensures that no additional optical depth is lost in the comb prepara-tion process.The AFC bandwidth is limited here by the strength of the applied magnetic field,which in turn limits the shortest possible input pulse duration and hence the temporalmultimode capacity [17]. However, by using a stronger field of about 100 mT one canrecover the AFC bandwidth that can be achieved at zero applied field. It should benoted that such fields are more readily produced in the lab with respect to the > doped Y SiO . The experimental setup is shown in figure 2(b). The present AFC spin-wave memoryis implemented in a 1000 ppm isotopically pure Eu doped Y SiO crystal (peakoptical absorption α = 2 . − , optical inhomogeneous broadening Γ optinh ∼ . ptical storage for 0.53 seconds in a solid-state atomic frequency comb memory using dynamical decoupling ◦ relative to D , we use spectral hole burning tech-niques as described in [25]. At this angle and with a field intensity of 15 mT, theground state splittings are both equal to δ = 210 kHz and the excited state splitting is δ e = 300 kHz. Given that the spin inhomogeneous broadening of our crystal is of theorder of Γ inh ∼
30 kHz, this field intensity allows us to fulfill condition (4 a ). To opticallyaddress the ions we use a laser at 580.04 nm that is reference locked to another laserstabilized on a high finesse cavity. From spectral hole burning experiments we estimateits linewidth to less than 1 kHz. The laser is split into three optical spatial modes, asshown in figure 2(b): the input (orange in the figure, with a waist w in0 = 33 µ m in thecrystal), the control (yellow in the figure, w ctr0 = 350 µ m) and the preparation mode(red in the figure, w prep0 = 480 µ m). The different modes are used because they allow usto reduce detection noise through leakage and because we observe a different optimumfor AFC preparation and control mode regarding the trade-off between intensity andhomogeneity over the storage volume. All optical modes propagate close to the b axisof the crystal, i.e. orthogonal to the D and D axis, and are polarized along D tomaximize absorption [35, 36].The experimental sequence is illustrated in figure 2(c). We prepare an AFCstructure of inverse periodicity 1 / ∆ = 17 µ s over a bandwidth of Γ AFC = 160 kHz(which satisfies condition (4 c )), using the preparation mode and a preparation schemeas described in [37]. The input field is a classical coherent state (Gaussian, FWHM= 7 µ s) and the control fields are intense Fourier-limited π -pulses (square, FWHM =4 µ s). For the RF control we use a simple resonant LC circuit (Q-factor Q = 25)consisting of a coil wrapped around the crystal (inductance), a parallel and a serialcapacitor, as sketched in figure 2(b). The resonance can easily be tuned to 34.54 MHz byadjusting the two capacitances placed outside of the cryostat. With an input RF powerof several tens of watts we are able to drive the spin transition with a Rabi frequencyΩ RF = 2 π ×
23 kHz. As we want to address ions efficiently over the whole inhomogeneousbroadening Γ inh , quasi-monochromatic pulses would be insufficient given our limitedRabi frequency. Fortunately, efficient π rotations over the entire spin linewidth can beachieved by using adiabatic pulses [38]. Here we used hyperbolic secant profiles withFWHM of 80 µ s and a chirp in frequency of 60 kHz.
3. Experimental results
In order to characterize DD in our system, we performed the storage experimentas described previously while varying several different parameters of our decouplingscheme. The different parameters are illustrated in figure 3. In general, we perform thedecoupling by applying n s sequences, each consisting of an even number of π -pulses n p . ptical storage for 0.53 seconds in a solid-state atomic frequency comb memory using dynamical decoupling π -pulses there is a delay τ , such that we apply a total of n = n s n p pulses over an overall spin storage time of T spin = nτ . ... C on t r o l C on t r o l T spin Sequence 1 Sequence 2 Sequence n s ° ° ° ° ° ° XY ° ° ° ° ° ° ° ° XY ° ° ° ° XX ° ... ° Figure 3. left:
Illustration of a DD sequence for the case of decoupling with an XX-sequence. right:
Phase-relations between the decoupling pulses for the three differentsequences that we implemented in this experiment.
We start by characterizing the AFC spin-wave memory performance using a singlesequence consisting of two identical π -pulses ( n s = 1, n p = 2), which is the minimumnumber of required pulses in an optical storage experiment in order to compensatethe inhomogeneous broadening of the spin transition (see e.g. [11, 24]). The memoryefficiency was measured while changing τ . In figure 4 the resulting memory efficiencyis plotted as a function of T spin = nτ . For the shortest storage time ( T spin = 2 ms), theefficiency reaches η tot = 3 . ± . η AFC = 10 . ± .
7% and η ctrl = 61 ±
2% (both measured with independentmethods), Eq.(2) suggests that the spin wave efficiency η spin is close to unity, withinthe error. Note that η spin takes into account both population transfer errors and phasecoherence errors due to the spin manipulation. These results indicate that the ionsare efficiently manipulated over the whole inhomogeneous spin linewidth, and that nounaccounted experimental inefficiency affects the memory performance. We estimatethe population transfer error of individual π -pulses to be about 2%, see the discussionin section 3.2.The non-exponential shape of the efficiency curve shown in figure 4 indicatesdecoherence due to spectral diffusion [39, 40] that might be mitigated by DD techniques[28, 29, 41, 42, 43]. To demonstrate that DD can indeed extend the coherence time weperformed storage experiments with increasing number of sequences n s = 1 , , , n p = 2 (the global number of pulses is then n = 2 , , , τ was varied. As seen in figure 5(a), one clearly observes an increase of the storage timeas the number of pulses is increased, which indicates that DD is effective in reducing ptical storage for 0.53 seconds in a solid-state atomic frequency comb memory using dynamical decoupling Spin storage time n (ms) E ff i c i e n c y ( % ) Figure 4.
Efficiency as a function of storage time with two identical rephasing π -pulses ( n s = 1, n p = 2, such that n = 2). For the shortest spin storage time of T spin = nτ = = 2 ms the memory efficiency was η tot = 3 . ± . the rate of dephasing due to spectral diffusion. Following Ref. [44], we fit all curves toa stretched exponential (SE): η SEspin ( n, τ ) = exp (cid:34) − (cid:32) nτT ( n ) (cid:33) α (cid:35) . (5)We underline here that this particular form only aims at extracting a characteristictime T ( n ) for the decay, which only loosely depends on α [44]. The resulting T ( n ) asa function of the number of pulses n is shown in figure 5(b), and it follows a power-lawscaling T ( n ) = T (1) n γ where T (1) = 25 ± γ = 0 . ± .
02. This value of γ isactually close to the one that we expect in the case of spectral diffusion governed by anOrnstein-Uhlenbeck (OU) process [39], for which γ OU = 2 / τ c into a Gaussian steady state distribution of spectral width σ , leading to a dephasingrate of ‡ [41]: η OUspin ( n, τ ) = exp [ − n, τ )] , with (6 a )Γ( n, τ ) = ( στ c ) (cid:32)(cid:20) τ c − τ tanh (cid:18) τ τ c (cid:19)(cid:21) nτ − (cid:20) − sech (cid:18) τ τ c (cid:19)(cid:21) (cid:33) . (6 b )To gain a more quantitative understanding of the dephasing process, the decay curveswere then fitted to this model with σ and τ c as global, free parameters, and the resultsare shown as solid lines in figure 5(a). The model fit is good for low n , but does notfit as well the data for high n , particularly for n = 16. As will be discussed later wesuspect this to be caused by some technical error appearing for short pulse separations.However, overall the model fit is rather satisfactory and yields the spectral diffusion OUparameters σ/ (2 π ) = 15 . ± . τ c = 9 . ± . S (cid:39)
17 MHz/T [25, 26, 27] the equivalent ‡ Note that this expression has been slightly approximated with respect to Eq. (A15) in [41] ptical storage for 0.53 seconds in a solid-state atomic frequency comb memory using dynamical decoupling -2 -1 Spin storage time (s) E ff i c i e n c y ( % ) (a) n T ( s ) (b) Figure 5.
AFC spin wave storage with a fixed number of rephasing pulses. (a)Efficiency as a function of storage time for different numbers n of identical π -pulsesacting on the spins, for n = 2 , , ,
16. The solid lines represent the fitted OU spectraldiffusion model of eq. (6 b ) as discussed in the text. (b) Characteristic time scales ofthe decay curves, as a function of number of pulses n . The solid-line shows the fit toa theoretical power law, see text for details. fluctuation of the magnetic field at the position of the europium ion is of the order of∆ B = σ/S ∼ µ T, which is of the same order of magnitude as what has been foundin other studies [29, 47]. We finally note that, knowing σ and τ c , one can theoreticallycalculate the T (1) value appearing in the power-law discussed in the previous section.Indeed, in the limit τ << τ c , (6 a ) simplifies to: η OUspin ( n, τ ) (cid:39) exp (cid:34) − σ τ n τ c (cid:35) , (7)leading to a theoretical dependance of T ( n ) = T (1) n γ with γ = 2 / T (1) = (cid:113) τ c /σ ≈
23 ms [46], in good agreement with the value obtained using the power-lawfit above.
In an optical storage application experiment where on-demand readout of the memory isto be performed, for instance conditioned on some external signal, then it is preferableto keep the pulse separation τ constant. In the case of ideal π -pulses, working withthe smallest possible τ would yield the best decoupling effects (see (7)). In practice,however, the application of many inversion pulses introduces errors as the RF pulsesdo not perform perfect inversions [48, 49, 50]. This is already seen in figure 5(a),where the efficiency when applying n = 16 is clearly lower for short storage times,although a longer overall memory time is reached. Therefore, we expect a maximumeffective coherence time of the memory device for some finite optimal value of the pulseseparation. Furthermore, in addition to using sequences of identical RF pulses, knownas a Carr-Purcell or XX sequence [51], we also studied more complex sequences that ptical storage for 0.53 seconds in a solid-state atomic frequency comb memory using dynamical decoupling τ for each of them.For each storage experiment with fixed τ the memory efficiency was measuredwhile increasing the number of pulses n . As all recorded decay curves were exponentialwithin experimental errors, each of them was fitted to an exponential function η DD spin =exp( − nτ /T DD ), allowing us to extract the effective coherence time T DD of the memory.Note that an exponential decay is expected from the OU spectral diffusion model, see(6 b ), (7) when τ is kept constant while n is varied. (ms) T DD ( s ) XXXY4XY8 (a)
Spin storage time (s) E ff i c i e n c y ( % ) ( s)00.010.020.03 S i gn a l ( V ) (b) Figure 6.
AFC spin-wave storage using different DD sequences with constant durationbetween RF inversion pulses. (a) Effective memory coherence time T DD as a functionof decoupling pulse spacing τ , for DD pulse sequences XX, XY-4 and XY-8. Solidand dashed line come from modeling explained in the main text. (b) The memoryefficiency decay curve for the longest achieved memory time of T DD = 0 .
53 s using theXY8 sequence with τ = 2 . The experimental results of the effective memory coherence measurements arepresented in figure 6(a). As expected the coherence time reaches a maximum valuefor an optimum pulse separation, and the use of more error-resilient pulse sequencesallows one to use shorter pulse separation which results in longer coherence times. Thelongest coherence time was achieved using a XY8 sequence with τ = 2.5 ms, and thecorresponding efficiency curve is shown in figure 6(b). The resulting coherence time of0 .
53 s ± .
03 s represents a more than 10-fold increase compared to storage withoutdecoupling, see figure 5(a), and a 300-fold increase compared to the previous state ofthe art in AFC spin-wave storage [16, 20, 24]. The echo at one second is also shown ininset of figure 6(b) and shows that even at this delay it can be well discriminated.In order to analyse the effective coherence time more quantitatively, we use twomodels that describe the observed dependence in two complementary, asymptoticregimes. For short pulse separation τ we expect coherence time to be predominantlylimited by the inversion error that the pulses introduce into our system. For long pulseseparations, i.e. when there are few inversion pulses applied, the error that is introduced ptical storage for 0.53 seconds in a solid-state atomic frequency comb memory using dynamical decoupling π inversion pulse with an area error (cid:15) (pulse area θ = π + (cid:15) ). Withthis model, we expect an effective coherence time T DD ,(cid:15) = (cid:113) /αn p τ . The pulse errordependence enters into the α parameter, which is α = (cid:15) , (cid:15) / (cid:15) / T DD ,(cid:15) = 2 √ τ /(cid:15) , shown as a dashed line in figure 6(a), which resultedin (cid:15) = 0 . ± .
004 rad. This value is consistent with other measurements of thepopulation error that we have performed, and close to the error previously measured ina similar RF set-up [24].The XY-4 and XY-8 coherence times also follow an approximately lineardependence for the shortest pulse separations, although the few number of points do notallow a quantitative comparison with the model. However, given the 1 /(cid:15) (XY-4) and1 /(cid:15) (XY-8) scaling of the coherence time, significantly longer coherence times shouldbe observed for short τ . For instance, based on the (cid:15) = 0 .
154 value fitted to the XXsequence, we would expect a XY-4 coherence time of about 340 ms for τ = 1 ms, whilethe experiment yielded about 230 ms. The discrepancy is even more significant for theXY-8 sequence. In summary, the coherence time at short delays can be enhanced byusing error-compensating DD sequences, however, it appears that an additional factorlimits the achievable coherence time. This could be due to errors not accounted for inthe model, or due to some other dephasing process that does not follow a OU spectraldiffusion model, see also the discussion in section 4.In the opposite regime of long pulse separations, where the number of π pulsesis smaller, we expect the spectral diffusion to limit the the achievable coherencetime. The OU model predicts an exponential decay with an effective coherence time T DD ,OU = 12 τ c / ( σ τ ), see eq. (7). This model, with the OU parameters we extractedin section 3.1, results in the dotted line in figure 6(a), which is in reasonably goodagreement with the observed coherence times for the XY-4 and XY-8 sequences at largepulse separations where the effect of pulse errors is negligible. We note that it is stronglydependent on the σ parameter, as shown by the blue shaded area in figure 6(a) whichrepresents the prediction within the ± ptical storage for 0.53 seconds in a solid-state atomic frequency comb memory using dynamical decoupling
4. Discussion
The presented results show the applicability and the effectiveness of the DD sequencesto spin-wave AFC protocols in rare-earth ion doped crystals. In the course of its imple-mentation we uncovered some open questions.The first is related to the error-resistant decoupling sequences. As explained in theprevious section, we find no angular error for the RF rephasing pulses that is both con-sistent with the observed improvement from XX to XY4 as well as the improvementfrom XY4 to XY8. This might indicate that our sequences suffer from more complexerrors like phase errors or global errors due to heating. This could mean that cross-talk between the crossed RF transitions might still be present even if conditions (4 a )and (4 b ) are fulfilled. A way to minimize this contribution would be to work at evenhigher magnetic field amplitudes in order to push the crossed transitions further awayin frequency. This problem should be solved first if we want to aim at more complexsequences like KDD [48].The second open question is related to the model itself. Indeed, the OU process seemsto describe the decoherence process in our system quite well, but there are a few dis-crepancies that suggest that not all our assumptions about the system are fulfilled. Inparticular, as we mentioned previously, the fit of the OU model for the four curvesof figure 5(a) cannot be adjusted such that low and high n values can be well fitted.Further, even if we just consider the isolated 16 pulse curve, we do not find a set ofparameters that describes both the behavior in the short τ as well as the long τ regimewell. This might either signify that the pulse density affects their efficiency, as it wouldbe true for heating effects of the RF circuit, or it might suggest that the OU processis not the only form of decoherence that our system experiences. Another process, forexample, could be the coupling of the spins to some electromagnetic background ACfield in the laboratory environment. Even in the presence of these discrepancies, we notethat the OU model allows for a globally satisfying description of our system, and givesvaluable information about the ion and his environment, that are in good agreementwith previous studies.A future important step would be to characterize any additional noise that isintroduced by the decoupling process. While we are confident that we can store andretrieve classical light pulses without any major background contribution, DD withimperfect pulses is expected to introduce noise that is not scaling with the amplitudeof the input [50]. Such noise might be a notable limitation in the storage at the singlephoton level, and would deserve further investigation. Another point that should alsobe addressed is the efficiency of the memory, currently limited to a few percent. Twomain limitations play a role here. The first is the low optical depth of the memory. Acommon method to increase it is letting the input pass several times through crystal.Such a multi-pass configuration, however, requires interferometric stability over thewhole duration of storage, which, in the presence of the vibrating cryostat, is technically ptical storage for 0.53 seconds in a solid-state atomic frequency comb memory using dynamical decoupling
5. Conclusion and outlook
We have presented and experimentally shown the application of a dynamic decouplingsequence under a weak magnetic field to the spin-wave atomic frequency comb protocol.A spectral diffusion model based on the Ornstein-Uhlenbeck process has proven toexplain our experimental observations with good accuracy, and with parameters that arein good agreement with previous studies. Robust spin echo sequences have then beenused to demonstrate storage over durations of the order of a second, with a characteristicdecay time of more than 0.5 s. As next steps, thorough study of noise induced bythe dynamical decoupling sequence [24], and efforts towards increasing the efficiency ofthese memories would allow significant advance in the development of efficient long livedquantum memories.
6. Acknowledgments
We would like to thank Claudio Barreiro for technical support. This work was financiallysupported by the European Union via the Quantum Flagship project QIA (GA No.820445), and by the Swiss FNS Research Project No. 172590.
Appendix A. Considerations regarding the static magnetic bias field
In the present appendix, we detail the different considerations that we have made tochoose the direction and the magnitude of the magnetic field that we apply to ourcrystal, and discuss the resultant constraints on our experiment.
Appendix A.1. Choosing the direction of the field
There are two magnetically inequivalent sub-sites of Europium in Y SiO that we areaddressing in this experiment. These two subsites behave the same under an externalmagnetic bias field, only when the latter is oriented along the b symmetry axis of thecrystal and in the plane orthogonal to this symmetry axis - the plane that is spannedby the D and D polarisation axis of the crystal [33]. We restrict ourselves to thisparticular plane to avoid further complications caused by the sub-sites.Within this plane we choose an angle of φ = φ mag = 65 ◦ relative to the D -axis ofthe crystal. For this configuration two particularly convenient properties emerge.The first property is linked with the branching ratios. Applying a magnetic field will spliteach of the optical transitions into four, but for a certain range of angles (including φ mag ), ptical storage for 0.53 seconds in a solid-state atomic frequency comb memory using dynamical decoupling Figure A1. (a) , (b) Branch-ing ratios for transitions betweenstates | / (cid:105) e and, respectively, | / (cid:105) g and | / (cid:105) g as a functionof the angle of the applied mag-netic field relative to the D -axisin the D -D -plane. For bothcases at an angle of 65 ◦ (redline) transitions, where the signof the quantum number changes,are strongly preferred. (c) Transition strength and fre-quency dependence of the transi-tions | / (cid:105) g ↔ | / (cid:105) g as a func-tion of the angle of the magneticbias field. At 65 ◦ two of the tran-sitions coincide in frequency andhave a strong dipole moment,while the other two are spec-trally well-separated and have aweaker dipole moment. two of the transitions are strongly preferred regarding their respective branching ratio(see figure A1(a), (b)). In consequence, the strong transitions form two independentLambda systems. Each Zeeman state in the | / (cid:105) g manifold is connected to exactly oneZeeman state in the | / (cid:105) e manifold which in turn is connected to exactly one Zeemanstate in the | / (cid:105) g manifold, as illustrated in figure A2 (upper part).The other particularity at this angle is that the ground state splitting δ g is equalto the storage state splitting δ s (see figure 2 and figure A2(a)). This means thattwo of the four transitions | / (cid:105) g ↔ | / (cid:105) g are degenerate in frequency and can bedriven simultaneously in an efficient manner with the RF field, while the other two aresignificantly detuned and weaker (figure A1(c)). Incidentally, the former two transitionsclose the two independent optical Lambda systems we address in our experiment (seefigure A2).The optical fields in the experiment have a bandwidth that is smaller than all ofthe relevant Zeeman splittings. As illustrated in figure A2, the light addresses only onesingle Lambda system at a time for any given ion. The inhomogeneous broadening ofthe respective transition, on the other hand, is much larger than the Zeeman splittings.Consequently, for some ions we are resonant with one of the strong lambda systems andfor other ions the other one - two different classes of ions are being addressed, as shownin the lower part of figure A2. Making use of these two classes simultaneously ensuresthat we do not lose any optical depth compared to the zero bias field scenario. Further,the radio frequency of the respective relevant spin transition is the same for both classes. ptical storage for 0.53 seconds in a solid-state atomic frequency comb memory using dynamical decoupling { { , class I class II Figure A2.
In the upper part we indicate all the transi-tions of an ion, that we maydrive strongly within the exper-iment. They form two indepen-dent, closed Lambda systems. Inthe optical domain these tran-sition are selected through theoptical branching ratios - thespin transitions we select by fre-quency. De facto only one tran-sition of each color will be drivenfor each given ion, such that ev-ery ion will belong to one of twopossible classes that are sketchedin the lower part.
Working with a bias field at angle φ mag allows for profiting from a significantly increasedspin storage time without sacrificing optical depth or increasing the bandwidth of spinsthat we need to manipulate. Appendix A.2. Additional constraints for working with the magnetic bias field
While operating the experiment in this particular configuration is favorable in severalregards, it does require the fulfillment of three additional constraints compared tooperation at zero bias field.The first two are connected to the fact that we want to address the two central nucleartransitions at 65 ◦ in figure A1(c), but not the two crossed ones. Any ion that istransferred to the wrong Zeeman state will no longer contribute to the collective re-emission of the ensemble - either because it is no longer in resonance with the controlfield, or because it acquires an additional phase relative to the rest of the ensemble.Selective transfer requires that the transition are well resolved - in other words thebias field must be sufficiently strong such that the undesired crossed transitions aremuch more separated from the central ones than the inhomogeneous linewidth of the ptical storage for 0.53 seconds in a solid-state atomic frequency comb memory using dynamical decoupling class 1 class 2 . Figure A3. left:
In an inhomogeneously broadened ensemble with a split excitedstate optical pumping at a frequency f not only produces increased transparency at f , but also at the frequencies f + δ e and f − δ e - the so called side-holes. upperright: Hole-burning spectrum in our system at a bias field of 10 mT (field orientationas in main text) with visible side-holes at ±
200 kHz. lower right:
Comb that isdeteriorated by the side-holes (red solid line) vs a comb that is not affected by theside-holes (blue dotted line individual transitions.Even in the case that these transitions are well separated, we could still drive themoff-resonantly. In order to exclude this possibility, the detuning of our RF-field withregard to the unwanted transitions must be much bigger than the Rabi-frequency of thedriving field, which corresponds to constraint (4 b ) that we mention in the body of thearticle.The third constraint is related to the atomic frequency comb preparation - while thecontribution of the two weak optical transitions can be ignored in the storage processitself, this is not true for the AFC preparation. The comb is prepared by frequencyselectively removing ions from the transition by optically pumping them into an auxiliarystate that is not resonant to any of the light of the experimental sequence. In orderto maximize the efficiency we apply many cycles of optical excitation and relaxation,meaning that even ions that are only resonant with a weak optical transition may beremoved from the absorption. By this process, ions that otherwise might be contributingto the AFC are removed and therefore the optical depth (and consequently the efficiencyof the memory) is reduced. The loss is associated with spectral side-holes that occur foroptical pumping in inhomogeneous ensembles with split excited states (see figure A3).This means that in our experiment accidental removal takes place when the transitionsto both Zeeman states in the | / (cid:105) e manifold are within the memory bandwidth. Inother words, accidental removal of ions can be avoided if the memory bandwidth is ptical storage for 0.53 seconds in a solid-state atomic frequency comb memory using dynamical decoupling δ e in | / (cid:105) e , as shown with constraint (4 c ). In thecase of our system the excited state splitting at 15 mT is approximately 300 kHz whilethe memory bandwidth is 160 kHz - so for our experiment the constraint is well-fulfilled.Alternatively, this detrimental effect could be avoided if the comb periodicity is matchedto the excited state splitting. If the excited state splitting is a multiple of the combperiodicity, the position of the side-holes coincide with the position of neighboring holes,meaning that the ions that are removed over a weak transition are the ones that aresupposed to be removed anyway so that there is no accidental loss of optical depththrough this process. Appendix A.3. Discussion of the storage decay curves for the XX sequence
As discussed in Sec. 3.2, the pulse area error model predicts a Gaussian decay of theoutput signal, while the OU spectral diffusion model predicts an exponential decay.Therefore we expect that for the XX sequence the storage decay curves would go frombeing more Gaussian to exponential as the pulse separation τ is increased. In sucha situation one can use a stretched exponential as a model function, see Eq. 5. Inaddition, from the numerical calculations of the effect of pulse area errors presented in[50], one can also expect that the XX sequence produces decay curves with oscillationsaround a stationary value at long delays. A possible model function for the memoryefficiency decay function for the XX sequence is then a stretched exponential with anoffset: η ( t ) = η exp (cid:20) − (cid:18) tT (cid:19) α (cid:21) + c, (A.1)which can describe both exponential ( α = 1) and Gaussian ( α = 2) behavior, as well asa smooth transition in-between the two. storage time (ms) e ff i c i e n c y ( % ) = 2 ms storage time (ms) e ff i c i e n c y ( % ) = 4 ms storage time (ms) e ff i c i e n c y ( % ) = 10 ms Figure A4.
Examples of three memory efficiency curves recorded using the XXsequence. All curves were fitted to the stretched exponential model with an offset, seeEq. A.1. The revivals for lower values of τ have been predicted in [50] in the case ofpulse area error, but are not considered in our very simple fitting model. The fitted α parameters of the stretched exponential can be found in Table A1. If the efficiency decay curves of the XX sequence are fitted with the stretchedexponential, we find that α tends towards larger values for shorter pulse separations ptical storage for 0.53 seconds in a solid-state atomic frequency comb memory using dynamical decoupling τ and approaches one as we move to longer τ (see Figure A4 and Table A1). Thisis qualitatively in agreement with the theoretical modeling that we suggest for thosetwo asymptotic regimes. However, as seen in Table A1 the error bar of the fitted α parameter is particularly high where a more Gaussian decay is expected. In general thestretched exponential results in larger uncertainties in all estimated parameters, as anadditional parameter is added to the fitting algorithm. τ α ± ± ± ± ± ± ± ± Table A1.
Dependence of the stretching coefficient α on the pulse separation τ inthe XX dynamical decoupling scheme. We also studied the robustness of the estimation of the decay constant using anexponential, stretched exponential, or a Gaussian decay. The fitted coherence time forthe XX sequence was rather independent of the choice in model, the mean deviationbetween the models being at most 6%. As the XY-4 and XY-8 sequences could be bestfitted to an exponential function, which is expected due to the low impact of pulse errorsfor those sequences, we decided to fit all decay curves using an exponential function.We emphasize that the main conclusions presented in Sec. 3.2 does not depend on thechoice of the fit model.
References [1] Boaron A, Boso G, Rusca D, Vulliez C, Autebert C, Caloz M, Perrenoud M, Gras G, Bussi`eresF, Li M J, Nolan D, Martin A and Zbinden H 2018
Phys. Rev. Lett. (19) 190502 URL https://link.aps.org/doi/10.1103/PhysRevLett.121.190502 [2] Lucamarini M, Yuan Z L, Dynes J F and Shields A J 2018
Nature
Rev. of Mod. Phys. Journal ofModern Optics Phys. Rev. Lett. http://link.aps.org/abstract/PRL/v98/e060502 [6] Radnaev A G, Dudin Y O, Zhao R, Jen H H, Jenkins S D, Kuzmich A and Kennedy T A B 2010 Nature Physics http://dx.doi.org/10.1038/nphys1773 [7] Nicolas A, Veissier L, Giner L, Giacobino E, Maxein D and Laurat J 2014 Nature Photonics Nat. Photon. http://dx.doi.org/10.1038/nphoton.2016.51 [9] Pu Y F, Jiang N, Chang W, Yang H X, Li C and Duan L M 2017 Nature Communications [10] Tian L, Xu Z, Chen L, Ge W, Yuan H, Wen Y, Wang S, Li S and Wang H 2017 Phys. Rev. Lett. (13) 130505 URL https://link.aps.org/doi/10.1103/PhysRevLett.119.130505 [11] Longdell J J, Fraval E, Sellars M J and Manson N B 2005
Phys. Rev. Lett. http://link.aps.org/abstract/PRL/v95/e063601 [12] Usmani I, Afzelius M, de Riedmatten H and Gisin N 2010 Nat. Com. [13] Heinze G, Hubrich C and Halfmann T 2013
Phys. Rev. Lett. (3) 033601 URL http://link.aps.org/doi/10.1103/PhysRevLett.111.033601 ptical storage for 0.53 seconds in a solid-state atomic frequency comb memory using dynamical decoupling [14] Ferguson K R, Beavan S E, Longdell J J and Sellars M J 2016 Phys. Rev. Lett. (2) 020501URL http://link.aps.org/doi/10.1103/PhysRevLett.117.020501 [15] Seri A, Lenhard A, Riel¨ander D, G¨undo˘gan M, Ledingham P M, Mazzera M and de RiedmattenH 2017
Phys. Rev. X (2) 021028 URL https://link.aps.org/doi/10.1103/PhysRevX.7.021028 [16] Laplane C, Jobez P, Etesse J, Gisin N and Afzelius M 2017 Phys. Rev. Lett. (21) 210501 URL https://link.aps.org/doi/10.1103/PhysRevLett.118.210501 [17] Afzelius M, Simon C, de Riedmatten H and Gisin N 2009
Phys. Rev. A [18] Sinclair N, Saglamyurek E, Mallahzadeh H, Slater J H, George M, Ricken R, Hedges M P,Oblak D, Simon C, Sohler W and Tittel W 2014 Phys. Rev. Lett. https://journals.aps.org/prl/accepted/fd07aYcaZ0d1114d423415468e37be0c2fae8946a [19] Jobez P, Timoney N, Laplane C, Etesse J, Ferrier A, Goldner P, Gisin N and Afzelius M 2016
Phys. Rev. A (3) 032327 URL http://link.aps.org/doi/10.1103/PhysRevA.93.032327 [20] Laplane C, Jobez P, Etesse J, Timoney N, Gisin N and Afzelius M 2016 New Journal of Physics http://stacks.iop.org/1367-2630/18/i=1/a=013006 [21] Sabooni M, Beaudoin F, Walther A, Lin N, Amari A, Huang M and Kr¨oll S 2010 Phys. Rev. Lett. (6) 060501 URL https://link.aps.org/doi/10.1103/PhysRevLett.105.060501 [22] Jobez P, Usmani I, Timoney N, Laplane C, Gisin N and Afzelius M 2014
New Journal of Physics http://stacks.iop.org/1367-2630/16/i=8/a=083005 [23] Saglamyurek E, Sinclair N, Jin J, Slater J A, Oblak D, Bussi`eres F, George M, Ricken R, SohlerW and Tittel W 2011 Nature http://dx.doi.org/10.1038/nature09719 [24] Jobez P, Laplane C, Timoney N, Gisin N, Ferrier A, Goldner P and Afzelius M 2015
Phys. Rev.Lett. (23) 230502 URL http://link.aps.org/doi/10.1103/PhysRevLett.114.230502 [25] Zambrini Cruzeiro E, Etesse J, Tiranov A, Bourdel P A, Fr¨owis F, Goldner P, Gisin N and AfzeliusM 2018
Phys. Rev. B (9) 094416 URL https://link.aps.org/doi/10.1103/PhysRevB.97.094416 [26] Longdell J J, Alexander A L and Sellars M J 2006 Phys. Rev. B Journal of Luminescence
32 – 37 ISSN0022-2313 URL [28] Viola L, Knill E and Lloyd S 1999
Physical Review Letters et al. Nature
Phys. Rev. Lett. http://link.aps.org/abstract/PRL/v72/p2179 [31] Teplov M A 1968 Soviet Physics JETP
Phys. Rev. B (3) 035101 URL https://link.aps.org/doi/10.1103/PhysRevB.66.035101 [33] Li C, Wyon C and Moncorge R 1992 IEEE Journal of Quantum Electronics Review of Scientific Instruments http://dx.doi.org/10.1063/1.5080086 [35] K¨onz F, Sun Y, Thiel C W, Cone R L, Equall R W, Hutcheson R L and Macfarlane R M 2003 Phys. Rev. B http://link.aps.org/abstract/PRB/v68/e085109 [36] Ferrier A, Tumino B and Goldner P 2016 Journal of Luminescence
406 – 410 ISSN 0022-2313[37] Jobez P, Usmani I, Timoney N, Laplane C, Gisin N and Afzelius M 2014
New Journal of Physics Phys. Rev. A http://link.aps.org/abstract/PRA/v71/e062328 [39] Klauder J R and Anderson P W 1962 Phys. Rev. (3) 912–932 URL https://link.aps.org/doi/10.1103/PhysRev.125.912 [40] Mims W B 1968
Phys. Rev. ptical storage for 0.53 seconds in a solid-state atomic frequency comb memory using dynamical decoupling [41] Pascual-Winter M F, Tongning R C, Chaneli`ere T and Le Gou¨et J L 2012 Phys. Rev. B http://link.aps.org/doi/10.1103/PhysRevB.86.184301 [42] Arcangeli A, Lovri´c M, Tumino B, Ferrier A and Goldner P 2014 Phys. Rev. B (18) 184305URL http://link.aps.org/doi/10.1103/PhysRevB.89.184305 [43] Heinze G, Hubrich C and Halfmann T 2014 Phys. Rev. A (5) 053825 URL http://link.aps.org/doi/10.1103/PhysRevA.89.053825 [44] Medford J, Cywi´nski (cid:32)L, Barthel C, Marcus C M, Hanson M P and Gossard A C 2012 PhysicalReview Letters [45] de Sousa R 2009 Electron spin as a spectrometer of nuclear-spin noise and other fluctuations
Topicsin Applied Physics (Springer Berlin Heidelberg) pp 183–220[46] de Lange G, Wang Z H, Rist`e D, Dobrovitski V V and Hanson R 2010
Science [47] Fraval E, Sellars M J and Longdell J J 2004
Phys. Rev. Lett. (7) 077601 URL https://link.aps.org/doi/10.1103/PhysRevLett.92.077601 [48] Souza A M, ´Alvarez G A and Suter D 2011 Phys. Rev. Lett. (24) 240501 URL http://link.aps.org/doi/10.1103/PhysRevLett.106.240501 [49] Wang Z H, de Lange G, Rist`e D, Hanson R and Dobrovitski V V 2012
Phys. Rev. B (15) 155204URL http://link.aps.org/doi/10.1103/PhysRevB.85.155204 [50] Zambrini Cruzeiro E, Fr¨owis F, Timoney N and Afzelius M 2016 Journal of Modern Optics https://doi.org/10.1080/09500340.2016.1204472 [51] Carr H Y and Purcell E M 1954 Phys. Rev. (3) 630–638 URL http://link.aps.org/doi/10.1103/PhysRev.94.630 [52] Sabooni M, Li Q, Kr¨oll S S and Rippe L 2013 Phys. Rev. Lett. http://link.aps.org/doi/10.1103/PhysRevLett.110.133604 [53] Seri A, Lenhard A, Riel¨ander D, G¨undo˘gan M, Ledingham P M, Mazzera M and de RiedmattenH 2017
Physical Review X7