Scholarly article on topic 'Coherent storage of temporally multimode light using a spin-wave atomic frequency comb memory'

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Academic research paper on topic "Coherent storage of temporally multimode light using a spin-wave atomic frequency comb memory"

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Coherent storage of temporally multimode light using a spin-wave atomic frequency comb memory

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New Journal of Physics

The open access journal for physics

Coherent storage of temporally multimode light using a spin-wave atomic frequency comb memory

M Gundogan1, M Mazzera1, P M Ledingham1,3, M Cristiani1 and H de Riedmatten

1 ICFO-Institut de Ciencies Fotoniques, Avenida Carl Friedrich Gauss 3, 08860 Castelldefels, Barcelona, Spain

2 ICREA-Institucio Catalana de Recerca i Estudis Avancats, 08015 Barcelona, Spain

E-mail: patrick.ledingham@icfo.es

New Journal of Physics 15 (2013) 045012 (16pp)

Received 11 January 2013 Published 19 April 2013 Online at http://www.njp.org/

doi:10.1088/1367-2630/15/4/045012

Abstract. We report on the coherent and multi-temporal mode storage of light using the full atomic frequency comb memory scheme. The scheme involves the transfer of optical atomic excitations in Pr3+ : Y2SiO5 to spin waves in hyperfine levels using strong single-frequency transfer pulses. Using this scheme, a total of five temporal modes are stored and recalled on-demand from the memory. The coherence of the storage and retrieval is characterized using a time-bin interference measurement resulting in visibilities higher than 80%, independent of the storage time. This coherent and multimode spin-wave memory is promising as a quantum memory for light.

3 Author to whom any correspondence should be addressed.

^ I Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence.

Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

New Journal of Physics 15 (2013) 045012 1367-2630/13/045012+16$33.00

© IOP Publishing Ltd and Deutsche Physikalische Gesellschaft

Contents

1. Introduction

1.1. Three-level atomic frequency comb scheme

1.2. Praseodymium ...............

10 12 13

2. Experimental description

2.1. Optical setup . . . .

2.2. Memory preparation

3. Results

3.1. Atomic frequency comb storage

3.2. Spin-wave storage.......

3.3. Transfer characterization . . .

3.4. Coherent storage.......

3.5. Multimode storage......

4. Discussion

5. Conclusions Acknowledgments References

1. Introduction

The coherent, efficient and reversible mapping between quantum light and matter represents a fundamental challenge in the field of quantum information science. Overcoming this challenge would enable the realization of a quantum memory for light, a device that can store and, on-demand, recall quantum states of light with high efficiency and fidelity. Such memories have potential applications for use in a quantum repeater [1-3], a resource allowing for quantum communication over long distances. Other applications include linear optics quantum computation [4], deterministic single-photon sources [5] and multi-photon quantum state engineering [6]. For practical applications, it is likely that a quantum memory with a high multimode capacity would be necessary.

One example of a quantum memory for light is a single atom trapped in a cavity [7]. An alternative candidate is the use of atomic ensembles [8]. The benefit of using such systems is that the light can be absorbed collectively, enhancing the coupling between light and matter. Indeed, extensive research effort has been put into single-atom and atomic ensemble-based memories alike. The realizations of ensemble-based memories include cold [9-14] and hot atomic gases [15-18] and solid state systems [19-37].

A promising ensemble-based quantum memory for light is the atomic frequency comb memory (AFC) [38] that is based on inhomogeneously broadened media, such as cryogenically cooled rare-earth ion-doped crystals. The AFC memory requires fine spectral tailoring of the inhomogeneously broadened absorption line into a series of equally spaced, narrow absorbing peaks. A resonant input pulse whose bandwidth matches that of the comb is collectively absorbed. This atomic coherence initially dephases, but due to the periodic structure of the comb, it rephases and coherently re-emits an echo. This light is in the same spatial mode as the input and at a delayed time of t = x/a, where A is the spectral distance between the peaks.

Recent progress on solid state quantum memories using the AFC scheme includes the storage of weak coherent pulses at the single-photon level [19, 22-24] and the storage of multiple temporal modes in a crystal [25, 26]. Storage of quantum light has also been demonstrated. The use of non-classical light generated by spontaneous parametric down conversion has enabled entanglement between one photon and one collective atomic excitation stored in a crystal [27, 28], entanglement between two crystals [29] and time-bin qubit storage [30]. Recently, the versatility of these memories was extended to the quantum storage of polarization qubits [31-33].

1.1. Three-level atomic frequency comb scheme

All of the above realizations used AFC storage in the excited state. However, strictly speaking, it is not an on-demand memory for light but rather a pre-programmed delay. In addition, the delay is limited to the excited state lifetime. On-demand retrieval of light, which is necessary for applications in quantum information science, can be achieved with a full AFC scheme which, as proposed in [38], transfers coherently the optical atomic excitation to and from a spin excitation. This proposal requires the use of three ground states. One is for the initial state (comb), one for the spin-wave excitations and a third one is needed as an auxiliary state for unnecessary ions that are not part of the AFC. The coherence needs to be transferred before the re-emission time t . Once the coherence is transferred to a spin wave, for example by a single-frequency n-pulse, the dipole evolution is effectively frozen. To read out this spin wave, a second transfer pulse is applied after a time TS, resulting in a three-level echo (3LE) with a total storage time of t + TS. If each transfer pulse has an efficiency of nT, then the total efficiency is given by [39]

where ^afc is the two-level AFC echo efficiency. Assuming Gaussian absorbing peaks, the efficiency in the forward direction is approximated well by

where F = A/y is the finesse of the comb with y being the peak width, d = d/ F is the effective optical depth experienced by the absorbed pulse with d being the optical depth and d0 the absorbing background [38]. The transfer efficiency can be optimized by the use of chirped pulses as theoretically discussed in [40].

An additional benefit of the full AFC scheme is given by the presence of an extra degree of freedom related to the propagation direction of the control pulse. This allows for backward retrieval of the echo, enabling in theory a storage and retrieval efficiency of 100% [38]. Also, spatial separation of the control mode from the input mode is available.

A proof of principle of the full AFC scheme has been demonstrated using praseodymium-doped yttrium oxyorthosilicate (Pr3+: Y2SiO5) with a total efficiency of 0.5-1% (transfer efficiency of 30-45%), with a multimode capacity of two temporal modes [39]. More recently, the full AFC scheme has been implemented in a different rare-earth ion sample (Eu3+ : Y2SiO5) [41] with a lower efficiency due to the low oscillator strength of the Eu3+ transition compared to Pr3+. In this paper, using Pr3+ : Y2SiO5, we confirm for the first time the coherent nature of the storage explicitly by way of a time-bin interference measurement. Furthermore, we extend the efficiency and multimode capacity beyond what has been demonstrated previously. For this demonstration, we use bright pulses; however, the scheme is, in principle, extendable to the use of single photons as we will discuss later.

n3LE — nAFC ,

PP-KTP

Control Mode

Pr3+ :Y2 SiO5

AOM Input Mode (a)

±5/2e, 1D2(1) ±3/2e-±1/2e'

3 H4 (1)

±1/2g. ±3/2g-

±5/2g.

4.8 MHz 4.6 MHz

Control Mode

10.2 MHz Input Mode

17.3 MHz

^ Input

TS + T

Output Det.

Control Det.

Figure 1. (a) Experimental setup. See the text for details. A/2 (A/4): half (quarter) wave plate; L: 100 mm lens; AOM: acousto-optic modulator in doublepass configuration; EOM: electro-optic modulator driven at 12.5 MHz; PDH: the Pound-Drever-Hall module for laser frequency locking. (b) Level scheme and pulse sequence for the spin-wave AFC memory. The comb is prepared in a time TPrep = 200 ms. A waiting time of TW = 1 ms is used between the end of the preparation and the start of the 3LE sequence. Ts is the time between the control pulses n1 and n2, and t is the AFC delay time. The total storage time is ts + t .

1.2. Praseodymium

Cryogenically cooled Pr3+ : Y2SiO5 is an attractive medium for solid state quantum memories. Just as all the rare-earth ions, Pr3+ is characterized by a partially filled 4f shell spatially located within the full 5s and 5p ones. This feature allows Pr3+ to maintain an atomic-like energy level scheme, with homogeneous linewidth of the order of 2 kHz [42, 43], even when embedded in a crystalline matrix. Furthermore, the low nuclear magnetic moment of the neighbouring ions (Si and Y) prevents spin flips and magnetic ion-host interactions. A second-order hyperfine interaction is able to split the crystal field singlets into three sublevels, providing the threefold ground state required by the full AFC scheme. The optical transition used for the storage protocol connects the lowest-lying levels (labelled as (1)) of the ground 3H4 and excited 1D2 manifolds (see figure 1(b)), characterized by a wavelength of 605.977 nm. In samples doped with 0.02% Pr3+, it has been reported to exhibit an inhomogeneous bandwidth of ~ 5 GHz, which allows for tailoring the absorption profile into a frequency comb structure. The excited state has a lifetime of T1 = 164 fzs and the coherence time with zero

magnetic field is T2 = 111 fzs [42]. With the use of specific external magnetic fields [44] and dynamical decoupling sequences [45], light has been stored using electromagnetically induced transparency in Pr3+ : Y2SiO5 for times exceeding 1 s [46] and a hyperfine ground state coherence time exceeding 30 s has been demonstrated using similar techniques [47].

2. Experimental description

2.1. Optical setup

Figure 1(a) shows the experimental setup. Our laser source is based on sum frequency generation of 987 and 1570 nm light. These wavelengths are combined in a periodically poled potassium titanyl phosphate (PPKTP). With input power of 429 and 1138 mW and coupling efficiency into the waveguide of about 37.5 and 26.9% for the 987 and 1570 nm lasers, respectively, we achieve an output power of 190 mW at 606 nm, corresponding to a conversion efficiency of 385% W-1.

Acousto-optic modulators (AOM) are used in double-pass configuration to create the necessary pulsed light for the echo experiments. This configuration allows to scan the frequency of the light by several MHz while keeping the intensity flat and spatial mode the same. The RF signals used to drive the AOMs are generated using an arbitrary waveform generator (Signadyne). We use two different optical paths to interact with the sample, the control and input modes, each having a double-pass AOM. The advantage of this choice is twofold. On the one hand, having a mode for the preparation and strong control light spatially separated from the input and weak echo light, prevents noise from the strong control mode polluting the echo mode. On the other hand, a strong control pulse might cause a free induction decay due to off-resonant excitation or an incorrectly prepared transparency window (see section 2.2). The echo would be hidden by this additional noise if the two modes were not spatially separated. These advantages become important from the perspective of single-photon level inputs.

After each AOM, the light is coupled to a single-mode fibre and then out-coupled on an isolated optical bench, where the closed cycle cryogenic cooler (Oxford V14) used for cooling the crystal to 2.8 K is located. Our Pr3+ : Y2SiO5 sample is 3 mm thick with 0.05% doping. The absorption coefficient is measured to be a = 23 cm-1 for the optical transition at 605.977 nm. The inhomogeneous linewidth is measured to be 5 GHz. Half-wave plates (A/2) ensure that the polarization is aligned parallel to the optical D2-axis of the crystal, in order to maximize the absorption [43]. A glass plate is placed in the input mode allowing a reference signal on the input detector. An f = 100 mm lens is used to focus both modes on to the crystal, resulting in a beam width size of around 95 ^m for both the control and input modes. The maximum power in the control mode before the cryostat window is measured to be 7 mW. The maximum power available in the input mode is 1.5 mW. An additional f = 100 mm lens is placed after the crystal for spatial mode matching. The echo mode is steered to a detector via a single-mode fibre with 50% coupling efficiency.

The 606 nm light is frequency locked to a passively stabilized cavity using the Pound-Drever-Hall locking technique [48]. This method of locking provides the desirable laser linewidth for the fine spectral tailoring of the AFC, as well as long-term frequency stability. The setup schematic is shown in figure 1(a). The cavity has a free spectral range of 1 GHz and a linewidth of 1.45 MHz leading to a finesse of 690. A temperature stabilized Invar spacer is used to hold the cavity mirrors. Light used for locking is picked off before the AOMs and

directed through an electro-optic modulator (EOM). The EOM is driven by an RF source (Toptica Digilock module) resulting in phase-modulated sidebands at 12.5 MHz. The error signal is created using in-house electronics and is fed-back to the piezo and current drivers of the 987 nm laser via the Toptica Digilock and fast analogue linewidth control (FALC 110) modules, respectively. Spectral hole-burning experiments reveal the laser linewidth to be around 100 kHz for ms timescales.

2.2. Memory preparation

The scheme of the full AFC protocol realized in this work is depicted in figure 1(b). The input pulse (solid arrow) is resonant with the comb on the 1 /2g-3/2e transition, while the coherent transfer of the optical excitation to and from the ground state (dashed arrows) is tuned to the 3/2g-3/2e transition. The echo is emitted on the 1/2g-3/2e transition (dotted arrow). The remaining 5/2g ground level is exploited as the auxiliary state.

The preparation of the memory follows the approach described in [49]. A transparency window is first created inside the absorption profile with the procedure hereby described. The laser frequency is swept by 12 MHz using an AOM, thus pumping all the ions that are resonant with this light to a non-resonant ground state. The sweep is repeated 100 times, resulting in a transparency window (the 'pit') with a width (full-width at half-maximum) of 18 MHz [49, 50]. To tailor the AFC, the burn-back method is used. It consists of the pumping of ions from the 5/2g auxiliary state, some of which will decay in the ground states previously emptied, giving rise to isolated narrow peaks in the transparency window. By sending several burn-back pulses at frequencies differing by A, the AFC structure is created. The finesse of the comb, and thus the efficiency of the AFC echo, can be optimized with a proper choice of duration, power and the number of the burn-back pulses.

It is important to note that, since the hyperfine level splittings MHz) are smaller than the inhomogeneous width GHz), different classes of ions will be resonant with the burn-back pulses [50]. In the frequency window of interest for the comb, transitions associated with two classes of ions are indeed present (namely, 1 /2g-3/2e of class I and 1 /2g-1 /2e of class II).

The burn-back procedure also populates the 3/2g ground state, while an efficient full AFC scheme requires it to be empty. As a matter of fact, if some ions remained in the ground state addressed by the control pulse, they would be driven to the excited state and give noise in the echo mode due to spontaneous emission. To avoid this additional noise, a further 'clean' sweep is performed in the spectral region of the 3/2g-3/2e transition whose bandwidth matches that of the comb. This clean sweep offers also the ulterior advantage of emptying the ground states of the unwanted classes.

In this paper, the total preparation time, TPrep from figure 1(b), is 200 ms, of which 50 ms are used to create the comb. This includes 5 x 100 ^s burn-back pulses for the five comb peaks repeated every 500 ns for 100 times and a 2 MHz clean sweep centred at the transfer pulse frequency repeated 1000 times.

For the experimental results presented in the next section, we use different detection methods. For the results presented in sections 3.1-3.4 a photodiode (Thorlabs PDB150) is used and a memory is prepared for each pulse that is stored. For section 3.5 a single-photon counter (Laser Components, model Count) is used and for each memory prepared, 500 pulse trains are used.

iop Institute of Physics <j)deutsche physikalische Gesellschaft (b)

Time [ps]

CD | 2

Ts [ ps]

Figure 2. (a) The full AFC memory scheme. The blue dashed curve shows the two-level echo from an AFC programmed to give a delay of 2 fzs (A = 500 kHz). The green solid curve shows the effect of the transfer pulses. The AFC echo is reduced and the 3LE is observed. The dark blue dotted trace shows a reference of the input pulse, with a full-width at half-maximum duration of 840 ns. For clarity, this curve is divided by 5. The dot-dashed trace indicates the temporal location of the 800 ns rephasing pulses separated by TS = 4 ^s. The inset shows a singleshot measurement of a comb with A = 500 kHz. (b) 3LE efficiency versus TS for the same conditions as in (a). All curves and data points are the average of ten experimental runs.

3. Results

3.1. Atomic frequency comb storage

Figure 2(a) shows an example of a two-level AFC echo (dashed curve) for an excited state storage time of t = 2 ^s. In the inset an example of comb with a periodicity A = 500 kHz is represented for illustrative purposes. It is worth noting that the read method and the limited detection response time lead to underestimation of the comb optical density and finesse. Using input pulses of 840 ns duration, an AFC efficiency of nAFC = 15.6 ± 0.5% is observed. The efficiency is measured by taking the area of the echo in the blue dashed trace and comparing this with the area of the input pulse when sent through a spectral pit (dark blue dotted trace). When measuring the input pulse area, its polarization is rotated to be perpendicular to the optical D2-axis, ensuring weak interaction with any residual ions in the pit. To estimate the parameters of the comb, we compare the area of the transmitted pulse to that of the echo [19]. Assuming a tooth width of y = 125 (167) kHz, then F = 4 (3) and the inferred optical depths are d = 4.12 (3.66) and d0 = 0.45 (0.26).

3.2. Spin-wave storage

Figure 2(a) shows an example of the full AFC scheme of spin-wave storage (solid curve). A strong single-frequency control pulse is used to transfer the excited state coherence to and from the 3/2g ground state. The effect of the first transfer pulse is clearly seen from the

Figure 3. Two-level AFC echo area Aafc (dots) and three-level efficiency n3LE (crosses) as a function of the square root of power of the control pulses. The dashed and solid lines represent a fit to the data as described in the text. Each data point is the average of ten trials. Error bars indicate the maximum standard error in the mean measured for each data set.

significant reduction of the AFC echo. A second transfer pulse is applied after a time TS = 4 ^s and a 3LE is clearly seen above the noise. It is a direct result of the application of both control pulses, as confirmed by the fact that it vanishes on removing either ni, n2 or the input itself.

Figure 2(b) shows the 3LE efficiency, both amplitude and area (inset), as a function of TS. Due to the inhomogeneous broadening of the spin state, the efficiency of the echo decays as a function of TS. This inhomogeneity is known to be Gaussian with the following form [41]:

-(yis ts )2 1

n(Ts)3LE = n(0)3LE X exp

|_2log(2)/n 2J

where yIS is the spin inhomogeneous broadening and n(0)3LE is the 3LE efficiency at zero delay. Fitting the data with this equation and extrapolating to TS = 0 where the spin inhomogeneity has no effect, the 3LE efficiency (n(0)3LE) for the area (amplitude) is 5.6 ± 0.1% (5.0 ± 0.1%). The corresponding inhomogeneity is yIS = 25.6 ± 0.2 kHz (25.7 ± 0.4kHz), in agreement with previous realizations [39, 51]. Comparing the AFC efficiency with this 3LE efficiency, the transfer efficiency is calculated to be 57-60% using equation (1).

An alternative method for assessing the transfer efficiency is to compare the area of the AFC echo to that reduced by the first control pulse. From figure 2(a), we get nT = 73% for TS = 4 ^s. This is higher than what is obtained from the 3LE. The disagreement could be due to the imperfect spatial mode overlap between the control and input modes.

This is the full AFC scheme with the highest efficiency reported so far. Nevertheless, in view of applications in quantum communication, further improvements are necessary as we will discuss later.

3.3. Transfer characterization

We now characterize the transfer efficiency nT as a function of the power in the control pulse and thus the Rabi frequency (^R). The experimental conditions for this measurement are identical to those outlined in figure 2(a). A time TS of 4 ^s is chosen as a time fast compared to the spin inhomogeneous dephasing time so as not to limit the efficiency.

Figure 3 shows the 3LE efficiency and the area of the AFC echo Aafc as a function of the square root of power of the control pulse. Intuitively, as the power is increased the AFC

echo decreases and the 3LE increases. This behaviour can be modelled simply with a two-level atomic system being driven resonantly [52, 53], resulting in the following equations:

Aafc(¿2rt) = Ain^ [1 + cos(22Rt)], (4)

n3LE(¿2rt) = ^ [1 - cos(22rt)]2 , (5)

where AIN is the area of the input pulse and we introduce ¿2R as the effective Rabi frequency. This takes into account that the input and control pulses have inhomogeneous spectral and spatial profiles and also the imperfect spatial mode overlap between the two modes. The data in figure 3 are fit simultaneously with the above equations and show good agreement. The effective Rabi frequency is found to be ¿2R = 2n x 340kHz for a power of 5.7 mW at the crystal.

For a comparison, we measure the Rabi frequency with coherent hole burning techniques [54] using only the control mode, resulting in 2n x 420 kHz. Finally, knowing the oscillator strength of the transition [49, 50] and the laser intensity, we calculate the expected Rabi frequency to be about 2n x 430 kHz. This Rabi frequency is independent of spectral inhomogeneity and spatial mode overlap, and is therefore higher than the effective Rabi frequency.

3.4. Coherent storage

In order to implement a quantum memory, it is crucial that the phase between two modes is preserved during storage and retrieval. For quantum communication schemes, information is often encoded into the amplitude and phase of time-bin qubits, as they are known to be robust against decoherence in optical fibres [55]. We confirm for the first time the coherent storage of time-bin information using the spin-wave AFC memory.

To verify the coherence of the storage, we send a pair of pulses (Ie and I) separated by ts and with a relative phase difference 0. The pair of pulses is stored and analysed directly in the memory. This requires the implementation of partial readouts [56], achieved by sending two readout pulses (n2 and n3) separated also by ts (see the upper panel of figure 4(a)). The resulting output of the memory is three temporal modes, where the central mode shows coherent interference between the echo of the late pulse from the first readout and the echo of the early pulse induced by the second readout (Ex1 + Ee2).

Figure 4(b) shows an interference fringe of the central output mode as the relative phase difference 0 is changed for t = 5 ^s, TS = 12 ^s, ts = 1 ^s and pulse durations of 700 ns. For each phase, a total of ten trials is made for statistics. The detection window width tv is chosen to be 500 ns. The data clearly show coherent behaviour with a weighted sinusoidal fit giving a visibility of 84.0 ± 5.7%.

The coherent nature of the storage is probed further by investigating the fringe visibility as a function of TS. Shown in the lower panel of figure 4(a), the visibility remains at a constant value of V = 81.7 ± 1.4% as TS is increased, i.e. is independent of the storage efficiency. This behaviour is typical of ensemble-based quantum memories [57], where atomic decoherence decreases the efficiency of recall but not the conditional fidelity. This is true as long as the signal is much greater than the background noise, which may prove challenging to achieve for single photon level inputs. For a comparison, we repeated the experiment while changing the relative phase of the readout pulses, and also obtained an interference fringe with V = 84%.

(a) n i Ie I

n 2 n 3

Eli + Ee2

1 0.8 3 0.6

t Time

**.....* * *

12 16 20

Ts [ßs]

£ £ 0.6

I; \ j IV

lit'/'j

-2 -1 0

Time [ßs]

0 50 100 150 200 250 300 350 400

0 [Degrees]

Figure 4. (a) Upper panel: the pulse sequence used for the interference experiment. Lower panel: visibility and maximum peak area versus TS. The dots (crosses) correspond to the measured visibility (area). For each TS, a curve like (b) is measured from which the visibility is extracted via a weighted fit. The error bar is a confidence interval of this fit of one standard deviation. The horizontal dotted line is the mean of the visibilites, V = 81.7 ± 1.4%. The dashed line is a fit using equation (3) with a corresponding spin inhomogeneity of YIS = 26.8 ± 1.7 kHz. (b) Echo area in the interference window versus phase difference between the two input pulses. Each data point is the average over ten trials with the error bar representing one standard deviation of the mean. The dashed line is a weighted fit of the data to a sine. The visibility is 84.0 ± 5.7%. Inset: time-bin qubit interference for a phase difference of 195° (solid line) and 375° (dotted line). The corresponding data points in the main figure are labelled with circles. The vertical dashed lines indicate the interference window (iv = 500 ns).

The observed visibilities would result in storage conditional fidelities exceeding 90% (FC = (1 + V)/2), which, if this could be demonstrated with single photon fields, would be sufficient for application in quantum communication. The maximal possible visibility should however be 100%. Note that, besides the possible decoherence in the crystal, the visibility is directly affected by the phase noise between the two input time bins and between the two control pulses. To test this phase noise, we carried out an interference measurement using two pulse photon echoes (i.e. without transfer to the ground state). The resulting visibilities were similar, confirming that storing in the ground state does not affect the coherence. Also, this indicates that the main source of decoherence is the limited coherence of the laser itself and not the storage and retrieval process.

3.5. Multimode storage

An important feature of a quantum memory is its multimode capacity. A memory exhibiting this property allows for multiplexing, which, for example, can increase the success rate of a

quantum repeater by a factor given by the multimode capacity [58]. Temporal multimodality has been demonstrated with the two-level AFC [25, 26], but most implementations have used systems that do not have three ground states and are thus not extendible to the full scheme.

As stated in [39], the maximum multimode storage capacity (Nmax) is limited by the number of teeth (Nteeth) in the comb. This can be understood in terms of the ratio between the duration of each input mode and the total duration of the pulse train. The former is limited by the comb bandwidth. The latter is limited by the AFC storage time and thus by the width and the number of teeth that can fit in the comb bandwidth.

For the two-level AFC experiments, a great deal of effort has been made to increase the comb width [25], up to GHz bandwidth [26, 28]. However, this is not possible for three-level AFC experiments, since the comb bandwidth is determined by the hyperfine splitting, which ranges from a few MHz for Pr-doped solids to tens of MHz for Eu-doped solids. In order to increase the multimode capacity for spin-wave storage, the difficulty is then to reduce as much as possible the AFC peak widths, which requires a narrow line preparation laser. For our system, the limiting hyperfine splitting is the excited state one 5 MHz), while our tooth width is fixed by the laser linewidth of ~ 100 kHz. The ultimate limit on the tooth width is given by the homogeneous width of the praseodymium ion 2 kHz) [42].

Another parameter affecting the storage efficiency that should be taken into account is the temporal duration of the control pulses. In the ideal case, their bandwidth matches that of the comb. In fact, the duration of the n -pulse, and thus the bandwidth of the memory, is determined by the available £2R. For a limited £2R, efficient transfer between optical and spin-wave atomic excitations requires long control pulses, which in turn requires long input pulses. This limits the number of modes that can be stored. With this in mind, extension to an efficient multimode spin-wave memory is non-trivial.

The application of several strong input modes to a single-class comb can either induce the emission of a two-pulse photon echo, accumulated echo, or in the worst case, destroy the comb itself. To avoid the occurrence of these undermining events, the storage of multiple modes is carried out with low-power input pulses 2 x 104 photons per pulse) and the recalled echoes are detected with a single-photon detector (SPD; model Count, Laser Components). Neutral density filters (OD = 6.5) are placed after the crystal to avoid saturating the detector. Moreover, a double-pass AOM is used to temporally gate any noise from the control mode polluting the output mode. For each memory prepared, a total of 500 pulse trains are sent. The electronic signal from the SPD is sent to a time-stamping card which collects the arrival times in a histogram.

Figure 5 shows an example of five temporal modes stored and retrieved using the full AFC scheme. An AFC storage time t = 7 ^s (corresponding to comb teeth separated by A = 133 kHz) was programmed, with a bandwidth of 2 MHz (Nteeth = 15). A total of 1 x 105 (2 x 106) pulse trains were used to observe the AFC (3LE) echo. The requirements of a long AFC storage time and of short input pulses contribute to the decrease of the 3LE efficiency with respect to the optimized case of a single mode shown in figure 2(a). Indeed, both the two-level AFC echo and the transfer efficiency are decreased compared to figure 2(a). In fact, the total efficiency for the storage and retrieval of five modes is here limited to 0.1%. A confirmation of the fact that the overall efficiency strongly depends on the AFC storage time comes from the inset of figure 5, which shows the 3LE efficiency as a function of the temporal mode number. As the number of modes is decreased, a shorter AFC storage time is used and

Time [^s]

Figure 5. Three-level storage of five temporal modes. The AFC delay is t = 7 ^s with TS = 7 ^s. The blue bars from t = 0-6 ^s show five distinct temporal modes whose height are divided by 200. The green bars show the two-level AFC echo divided by 5, while the red bars show the three-level echo. The echoes show clearly the five distinct temporal modes. Inset: three-level efficiency versus mode number. For one mode t = 3 ^s and for each increase in temporal mode number an additional ^s is programmed for the AFC storage time.

higher efficiencies can be achieved. Despite the low efficiency observed, this is to the best of our knowledge the first reported demonstration of spin-wave storage of more than two temporal modes in a doped crystal.

4. Discussion

Although spin-wave storage with an echo efficiency of n3LE = 5.6% is, to our knowledge, the highest efficiency reported so far, a significant increase in this efficiency is required to be viable for quantum communication. To this end, we have strategies for further efficiency improvement. An increase in optical depth leads to an increase in efficiency. This can be obtained by increasing the physical length of the crystal. Experimentally, preparing a memory with an optically and physically thick crystal is feasible, as demonstrated in [20]. An alternative method for increasing the optical depth is to increase the doping concentration at the risk of a reduced coherence time due to ion-ion interactions.

The efficiency of the AFC scheme is limited to 54% in the forward direction due to reabsorption of the emitted light. However, in principle, the efficiency can be increased to unity by retrieving the echo in the backward direction [38]. It is realized by the use of counter-propagating control pulses, which for the present setup is readily achievable.

In this paper, the coherent transfer was mediated by the use of fixed-frequency, temporally square transfer n-pulses. The transfer efficiency is given by the available Rabi frequency, which is limited by the available laser intensity. Thus, one option to increase the Rabi frequency is to directly increase the control intensity by increasing the laser power or by using smaller beams. However, as described in [40], the transfer efficiency can be optimized with the use of control pulses with a hyperbolic tangent frequency chirp and a hyperbolic secant temporal shape. The benefit is that chirped pulses require less intensity than fixed-frequency n-pulses, for a given transfer efficiency.

An alternative method for increasing the efficiency is to place the crystal in an 'impedance matched' cavity, as proposed in [59, 60] and recently demonstrated in [61]. In principle, the input light can be completely absorbed with a readout efficiency limited only by atomic dephasing.

Our memory time was limited to around 20 ^s due to the inhomogeneous spin broadening. The storage time can be improved by the use of RF spin echo techniques. Such techniques counteract the effect of dephasing that is introduced by the inhomogeneity of the spin state. Dynamical decoupling techniques have been implemented successfully in Pr3+ : Y2SiO5 [46, 47], Pr3+ : La2(WO4)3 [62] and Tm3+ : YAG [63]. A further increase in storage time can result from the elimination of decoherence sources. For solid state systems, the coherence time can be increased by the application of an external magnetic field of a specific magnitude and angle [44, 64]. In doing so, the first-order Zeeman shift becomes vanishingly small, reducing the sensitivity of the Pr3+ ions to the host spins. However, this lifts the degeneracy of the hyperfine levels, which could reduce the available bandwidth for the memory.

We note that, for our experiment, the excited state efficiency nAFC for longer storage times and hence the multimode capacity will benefit from a narrower band laser. Our laser was limited to producing AFC echoes efficiently 1%) for storage times of up to 7 ^s.

The next experimental milestone regarding this memory is the storage of single-photon level light and non-classical light into a single spin-wave excitation. The results reported here were obtained with the use of bright input pulses, but, in principle, can be extended to the quantum light regime as AFC echo memories scale linearly with input power and are, in principle, noise-free [19]. As well as increasing the efficiency, the major experimental challenge is to filter out pollutive light from the echo mode emitted from other hyperfine levels. This is particularly difficult in Pr3+ : Y2SiO5 as the hyperfine levels occur only 10 MHz away from the signal frequency. For this, narrow band filtering would be required, for which a Pr3+ : Y2SiO5 crystal is ideally suited, as demonstrated in [35, 65].

5. Conclusions

In conclusion, we have reported on coherent and multi-temporal mode storage of light using the full AFC memory scheme. Using this scheme, a total efficiency of n3LE = 5.6% was observed, the highest efficiency reported so far. The coherent nature of the spin-wave storage is shown for the first time using a time-bin interference measurement resulting in an average visibility of 81.7%, independent of the storage time. A total of five temporal modes was stored and recalled on-demand from the memory, the highest number of modes stored as a spin wave in a doped crystal. We discussed ways to improve the efficiency and storage time further and the extendibility to the single-photon regime.

Acknowledgments

Financial support from the European projects CHIST-ERA QScale and FP7-CIPRIS (MC ITN-287252), as well as from the ERC starting grant QuLIMA, is acknowledged. MM acknowledges the Beatriu de Pinos program (2010BP B0014) for financial support.

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