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

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 τ. Once the coherence is transferred to a spin wave, for example by a single-frequency π-pulse, the dipole evolution is effectively frozen. To read out this spin wave, a second transfer pulse is applied after a time T_S, resulting in a three-level echo (3LE) with a total storage time of τ + T_S. If each transfer pulse has an efficiency of η_T, then the total efficiency is given by [39]

$$\begin{equation} \eta_{\rm 3LE} = \eta_{\rm AFC} \, \eta_{\rm T}^2, \end{equation} \tag{ 1 }$$

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

$$\begin{equation} \eta_{\rm AFC} \approx {\skew4\tilde{d}}^2 \, \mathrm{e}^{-\frac{7}{F^2}} \, \mathrm{e}^{-\skew4\tilde{d}}\, \mathrm{e}^{-d_0}, \end{equation} \tag{ 2 }$$

where F = Δ/γ is the finesse of the comb with γ being the peak width, $\skew4\tilde{d} = {d}/{F}$ 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.

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 μs and the coherence time with zero magnetic field is T₂ = 111 μs [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].

Zoom In Zoom Out Reset image size

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 α = 23 cm⁻¹ for the optical transition at 605.977 nm. The inhomogeneous linewidth is measured to be 5 GHz. Half-wave plates (λ/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 μm 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.

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 1/2_g–3/2_e transition, while the coherent transfer of the optical excitation to and from the ground state (dashed arrows) is tuned to the 3/2_g–3/2_e transition. The echo is emitted on the 1/2_g–3/2_e transition (dotted arrow). The remaining 5/2_g 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 5/2_g 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 Δ, 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,  1/2_g–3/2_e of class I and 1/2_g–1/2_e of class II).

The burn-back procedure also populates the 3/2_g 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 3/2_g–3/2_e 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, T_Prep from figure 1(b), is 200 ms, of which 50 ms are used to create the comb. This includes 5 × 100 μs 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.

Figure 2(a) shows an example of a two-level AFC echo (dashed curve) for an excited state storage time of τ = 2 μs. In the inset an example of comb with a periodicity Δ = 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 η_AFC = 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 γ = 125 (167) kHz, then F = 4 (3) and the inferred optical depths are d = 4.12 (3.66) and d₀ = 0.45 (0.26).

Zoom In Zoom Out Reset image size

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 3/2_g ground state. The effect of the first transfer pulse is clearly seen from the significant reduction of the AFC echo. A second transfer pulse is applied after a time T_S = 4 μs 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 π₁, π₂ or the input itself.

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

$$\begin{equation} \eta(T_{\rm S})_{\rm 3LE} = \eta(0)_{\rm 3LE}\times\exp\left[{\frac{-(\gamma_{\mathrm{IS}} T_{\rm S})^2}{{2 \log(2)}/{\pi^2}}}\right], \end{equation} \tag{ 3 }$$

where γ_IS is the spin inhomogeneous broadening and η(0)_3LE is the 3LE efficiency at zero delay. Fitting the data with this equation and extrapolating to T_S = 0 where the spin inhomogeneity has no effect, the 3LE efficiency (η(0)_3LE) for the area (amplitude) is 5.6 ± 0.1% (5.0 ± 0.1%). The corresponding inhomogeneity is γ_IS = 25.6 ± 0.2 kHz (25.7 ± 0.4 kHz), 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 η_T = 73% for T_S = 4 μs. 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.

We now characterize the transfer efficiency η_T as a function of the power in the control pulse and thus the Rabi frequency (Ω_R). The experimental conditions for this measurement are identical to those outlined in figure 2(a). A time T_S of 4 μs 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 $\mathcal {A}_{\mathrm { AFC}}$ 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:

$$\begin{equation} \mathcal{A}_{\rm AFC}(\bar{\Omega}_{\mathrm{R}} t) = \mathcal{A}_{\mathrm{IN}}\frac{\eta_{\rm AFC}}{2}\,\left[ 1 + \cos(\bar{\Omega}_{\mathrm{R}} t)\right], \end{equation} \tag{ 4 }$$

$$\begin{equation} \eta_{\rm 3LE}(\bar{\Omega}_{\mathrm{R}} t) = \frac{\eta_{\rm AFC}}{4}\,\left[ 1 - \cos(\bar{\Omega}_{\mathrm{R}} t)\right]^2, \end{equation} \tag{ 5 }$$

where $\mathcal {A}_{\mathrm {IN}}$ is the area of the input pulse and we introduce $\bar {\Omega }_{\mathrm {R}}$ 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 $\bar {\Omega }_{\mathrm {R}} = 2\pi \times 340\,\mathrm {kHz}$ for a power of 5.7 mW at the crystal.

Zoom In Zoom Out Reset image size

For a comparison, we measure the Rabi frequency with coherent hole burning techniques [54] using only the control mode, resulting in 2π × 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 2π × 430 kHz. This Rabi frequency is independent of spectral inhomogeneity and spatial mode overlap, and is therefore higher than the effective Rabi frequency.

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 (I_e and I_l) separated by t_s and with a relative phase difference ϕ. 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 (π₂ and π₃) separated also by t_s (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 (E_l1 + E_e2).

Zoom In Zoom Out Reset image size

Figure 4(b) shows an interference fringe of the central output mode as the relative phase difference ϕ is changed for τ = 5 μs, T_S = 12 μs, t_s = 1 μs and pulse durations of 700 ns. For each phase, a total of ten trials is made for statistics. The detection window width t_v 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 T_S. Shown in the lower panel of figure 4(a), the visibility remains at a constant value of $\bar {V} = 81.7\pm 1.4\%$ as T_S 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%.

The observed visibilities would result in storage conditional fidelities exceeding 90% (F_C = (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.

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 (N_max) is limited by the number of teeth (N_teeth) 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 π-pulse, and thus the bandwidth of the memory, is determined by the available Ω_R. For a limited Ω_R, 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 × 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 τ = 7 μs (corresponding to comb teeth separated by Δ = 133 kHz) was programmed, with a bandwidth of 2 MHz (N_teeth = 15). A total of 1 × 10⁵ (2 × 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 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.

Zoom In Zoom Out Reset image size