Rare-earth quantum memories: The experimental status quo

Abstract

Rare-earth doped crystals carry great prospect in developing ensemble-based solid state quantum memories for remote quantum communication and fast quantum processing applications. In recent years, with this system, remarkable quantum storage performances have been realized, and more exciting applications have been exploited, while the technical challenges are also significant. In this paper, we outlined the status quo in the development of rare-earth-based quantum memories from the point of view of different storage protocols, with a focus on the experimental demonstrations. We also analyzed the challenges and provided feasible solutions.

Graphical abstract

Keywords

solid-state quantum memory / rare-earth crystals / quantum network / quantum communication

1 Introduction

Based on the fundamental principles of quantum mechanics, quantum information science provides people with a brand-new way of information acquisition, processing and transmission [1–6]. One of the ultimate goals of modern quantum information science is to build a global quantum network that is analogous to the internet we are using nowadays [7–9]. The basic element of such a quantum network is a quantum computer in which entangled quantum states are used to perform computations in parallel, allowing some computation tasks to be dramatically more efficient than conventional computers. A quantum network is then realized through interconnecting quantum computers by using particles to distribute entanglement [10].

Among the many material systems, photons are considered the natural information carrier in the quantum network because it is easier to generate and transmit. However, the direct transmission range of light is limited by losses associated with scattering, diffraction, and absorption in optical fibers [11–13]. Restrained by the non-cloning theorem, direct amplification of quantum states encoded on the optical modes is not possible [14]. A feasible solution to overcome this range limitation is the quantum repeater proposal, which involves dividing a distant link to many nodes and distributing entangled photon pairs between these nodes [15–18]. Instead of establishing entanglement simultaneously in all elementary links, quantum memories are stationed at each node such that entanglement can be stored and distributed gradually. Further, for practical quantum computer devices, the gate executions are carried out sequentially, and sometimes the calculation results need to be stored for later computation. Moreover, a viable strategy to synchronize operations is to create a gate execution delay for a qubit. Thus, implementing a quantum memory in a quantum computer system can afford more flexibility to perform processing tasks and sometimes can also reduce the number of required qubits [19–22].

There is significant interest in developing suitable memories based on mapping optical quantum states in the energy levels of a material system, while a significant milestone is the so-called DLCZ protocol (proposed by Duan, Lukin, Cirac & Zoller) [23]. It demonstrated for the first time the fundamental operation of a quantum repeater by integrating atomic ensemble and linear optical techniques [23]. In this proposal, on-demand conversion between the flying qubits encoded in photons and the stationary qubits in atomic ensembles can be conducted with high efficiency. This inspired vast research interest in quantum storage applications based on atomic ensembles, and many impressive experimental advances have been reported in recent years [24–43].

The atomic systems are used for developing quantum memories because states of their internal electronic levels are mostly well characterized and usually coupled weakly to the environment. In contrast, solid-state systems have strong interactions with the dynamic environment which results in short coherence times for their optical or hyperfine transitions, accordingly short quantum storage time. However, there is a unique solid-state system, rare-earth-ion (RE) doped crystals, which can possess long coherence time for both optical and hyperfine transitions [44–47]. In this system, electrons of the optically addressable 4f^N levels are considered to have free-atom-like properties because they are well shielded by the full-filled outer electronic shells from interacting with the crystal environment [see Fig.1(a)] [48, 49]. Secondly, the large ratio between inhomogeneous [see Fig.1(b)] and homogeneous linewidth of RE crystals provides tremendous potential for multi-mode storage capacity. Further, as solid, RE crystals provide advantages from the point of view of engineering feasibility. Thus, this system is one of the most promising candidate for developing quantum storage devices and progress in storage performance based on different protocols has been achieved during the past decades.

In this paper, we aim to review the research progress of quantum-memory development based on RE crystals. The paper is arranged in the following way: in Section 2, we will introduce the basic performance criteria that a quantum memory needs to satisfy in the application of a quantum network. Section 3 and Section 4 introduce the main progress of absorptive quantum memories, including memories based on electromagnetically induced transparency (EIT) and photon-echo protocols. Finally, we finish the paper with a conclusion and outlook.

2 Quantum memory performance

In this section, we will discuss the main performance evaluation criteria concerned with the realization of a quantum memory, such as storage fidelity, storage time, storage efficiency, storage mode capacity and storage bandwidth.

To ensure the validity of the information in the stored procedure, (conditional fidelity) fidelity F is defined as the overlap between the density matrices {\rho }_{in} of the input state and {\rho }_{out}, the density matrices of the retrieved state, corresponding to the following equation:

(1)\begin{array}{l}F=(Tr\sqrt{\sqrt{{\rho }_{in}}{\rho }_{out}\sqrt{{\rho }_{in}}}{)}^2.\end{array}

The fidelity of the ideal stored procedure can reach the maximum value of 1 while in practical experiments, it is often required to exceed 2/3 for a classical memory [51, 52].

In general cases, the storage time is required to be comparable to the time for establishing the entanglement over the entire repeater link [16]. However, when quantum storage was applied to realize synchronization in the high-speed optical quantum network, the requirement for storage time is relatively flexible.

Storage efficiency is defined as the ratio between the energy of the retrieved signal and that of the input signal, which means the probability of re-emitting a photon. Improving storage efficiency can greatly increase the rate of entanglement distribution. For every 1% increase in storage efficiency, the entanglement distribution rate can increase about 10% [16].

Multi-mode storage can also increase the entanglement distribution rate while significantly reducing the stability requirements of the repeater protocol [53, 54]. Storage bandwidth is defined as the maximum frequency bandwidth of the optical signal that can be effectively stored. With the storage bandwidth \delta and storage time \tau, the dimensionless time bandwidth product is defined as B = \delta \times \tau [55]. The maximum number of effective procedures can be stored within a fixed storage time determines the memory’s synchronization capability.

Optical quantum memory is an indispensable part of many applications of quantum information [20]. However, each application has its specific requirements for the performance of the memory, which needs to be optimized according to the actual situation. For example, a memory used to buffer qubits in an optical quantum processor, due to the fast operations, may not require as long storage time as that in long-distance quantum communication, but it may concerns a lot about the bandwidth and multi-mode capacity.

3 EIT memories

In the DLCZ protocol, the material system work not only as a quantum memory but also as source of quantum states. In contrast, so-called absorptive memories work through converting quantum information encoded in light to atomic excitations of a material system, storing and retrieving the quantum information on demand. Historically, absorptive quantum memories developed along two distinct paths, that are optically controlled memories and photon echo memories [56]. In an optically controlled memory, intense control pulses are used to activate absorption and retrieval, two typical examples of which are the Raman memory and the EIT protocol [57–59]. The main difference is that the control field is off-resonant for the Raman scheme and on-resonance for the EIT protocol. To be effective, high excitation rate is required for the off-resonance Raman process while in RE solids, oscillation strength of transitions is usually very weak because they are parity forbidden, thus this protocol has not received that much attention [50]. Hence, in this section, we will introduce the EIT scheme as an example of optically controlled memories. The name EIT was first termed by Harris and his co-workers at Stanford University in 1990 [59] and has been continuously active in the research work on quantum storage.

3.1 EIT slow light

In a three-level of lambda configuration shown in Fig.2(a), the transition between the two lower energy states, |g⟩ and |s⟩, is dipole-forbidden while transitions of both states to an excited state, |e⟩, is allowed. We call a beam that couples |g⟩ and |e⟩ the “signal” beam and a beam that couples |s⟩ and |e⟩ the “control” beam. Both beams can be absorbed by the atoms if they enter the atomic medium alone. However, when the two beams are applied simultaneously, they interfere destructively, which can cause a transparency window in the signal field if the control beam is strong. The response of the atomic ensemble to the input beams can be described in terms of the susceptibility X [60–62]:

(2)\begin{array}{l}X\approx X_R+\mathrm{i}{X}_I,\end{array}

(3)X_R=4{\displaystyle \frac{N}{V}}{\displaystyle \frac{|{\mu }_{ge}{|}^2}{{ϵ}_0\hslash }} {\displaystyle \frac{-\delta (|{\mathrm{\Omega }}_c{|}^2-4{\mathrm{\Delta }}_s\delta )+{\mathrm{\Delta }}_s{\mathrm{\Gamma }}_s^2}{|({\mathrm{\Gamma }}_e+\mathrm{i}2{\mathrm{\Delta }}_s)({\mathrm{\Gamma }}_s+\mathrm{i}2\delta )+|{\mathrm{\Omega }}_c{|}^2{|}^2}},

(4)X_I=2{\displaystyle \frac{N}{V}}{\displaystyle \frac{|{\mu }_{ge}{|}^2}{{ϵ}_0\hslash }} {\displaystyle \frac{{\mathrm{\Gamma }}_s(|{\mathrm{\Omega }}_c{|}^2+{\mathrm{\Gamma }}_s{\mathrm{\Gamma }}_e)+4{\delta }^2{\mathrm{\Gamma }}_e}{|({\mathrm{\Gamma }}_e+\mathrm{i}2{\mathrm{\Delta }}_s)({\mathrm{\Gamma }}_e+\mathrm{i}2\delta )+|{\mathrm{\Omega }}_c{|}^2{|}^2}},

where {\mathrm{\Delta }}_s=w_{eg}-w_s, {\mathrm{\Delta }}_c=w_{es}-w_c are detuning of the signal beam the control beam while \delta = {\mathrm{\Delta }}_s- {\mathrm{\Delta }}_c is the two photon detuning. Also, {\mathrm{\Gamma }}_e and {\mathrm{\Gamma }}_s are the decoherence rates of the |e⟩ \to |g⟩ and |s⟩ \to |g⟩ transitions respectively, N is the total number of atoms in the sample, and {\mathrm{\Omega }}_s and {\mathrm{\Omega }}_c are the Rabi frequency for the signal and the control field.

In a situation when \delta =0, that is {\mathrm{\Delta }}_s={\mathrm{\Delta }}_c= \mathrm{\Delta }, an ideal EIT medium, for which the relaxation rate between the two lower energy states, |g⟩ and |s⟩, is very small, i.e., {\mathrm{\Gamma }}_s \to 0 and X\approx 0 accordingly, we will have Eq. (4) to describe the system. In this equation, X_I determines the dissipation of the field by the ensemble, i.e., absorption coefficient \alpha, while X_R determines the refractive index n of the EIT medium. This indicates that there is neither dispersion nor absorption in the medium, which corresponds to the electromagnetic induction transparency (EIT) phenomenon,

(5)\begin{array}{rl}& n=\mathrm{R}\mathrm{e}\sqrt{1+X}\approx 1+\frac{1}{2}{X}_R,\\ & \alpha =\frac{4\pi }{\lambda }\mathrm{I}\mathrm{m}\sqrt{1+X}\approx \frac{2\pi }{\lambda }{X}_I,\end{array}

According to Eq. (4), the linear susceptibility spectrum of an EIT medium can be drawn as Fig.2(b). In a narrow transparent windows, the refractive index changes steeply, which causes the group velocity {\nu }_g of the signal light to be less than the speed of light traveling in vacuum [see Fig.2(a)], and in the meantime, the medium’s absorption can be significantly reduced. This is the so-called slow light effect.

In RE crystals, Ham et al. [63] reported transmission without absorption in a Y³⁺ doped Y₂SiO₅ (YSO) crystal at temperature of 5 K for the first time. In this work, |g⟩ and |s⟩ are hyperfine states of ³H₄, |e⟩ is ¹D₂. Wavelengths of signal and control beams are about 605.7 nm, splittings of |g⟩ and |s⟩ is about tens of MHz. After this pioneering work, light speeds of 45 m/s and even stopping of the light were also observed in the same system [64–66]. These successful EIT demonstrations inspired vast research interests in conducting EIT quantum storage in RE crystals.

3.2 EIT storage

The EIT storage process can be described by the Bloch−Maxwell equations [57], in which the time-dependent spin wave S_{sg}(z,t) is mapped from polarization P_{eg} (z,t ) and signal field {\mathcal{E}}_{in}(z,t) under the control beam {\mathrm{\Omega }}_c^{\ast }(t- z/ c),

(6)\begin{array}{rl}& \left({\mathrm{\partial }}_t+c{\mathrm{\partial }}_z\right){\mathcal{E}}_{in}(z,t)=\mathrm{i}g\sqrt{N}{P}_{eg}(z,t),\\ & {\mathrm{\partial }}_t{P}_{eg}(z,t)=-(\gamma +\mathrm{i}\mathrm{\Delta })P_{eg}(z,t)+\mathrm{i}g\sqrt{N}{\mathcal{E}}_{in}(z,t)\\ & +\mathrm{i}{\mathrm{\Omega }}_c(t-z/c)S_{sg}(z,t),\\ & {\mathrm{\partial }}_t{S}_{sg}(z,t)=\mathrm{i}{\mathrm{\Omega }}_c^{\ast }(t-z/c)P_{eg}(z,t).\end{array}

For optimal efficiency, the system should be controlled in a time-reversal procedure. The phase matching requires k_{in} -k_{w,c}=k_{out}- k_{r,c}, where the four terms are wave vectors of input pulse, control write pulse, output pulse and control read pulse, respectively. Thus, the maximum storage efficiency of a forward readout is slightly lower than a backward retrieval readout because it does not correspond to a complete reversal of the write procedure [58].

In the EIT process, the group velocity of the signal field is so low that the entire pulse is compressed and stored in the sample. Meanwhile, in order to avoid absorption, the spectral bandwidth of the signal pulse must be narrower than the EIT window {\mathrm{\Gamma }}_{EIT}. This suggests that the signal pulse shape must fulfill a compromise between spectral and spatial conditions for efficient storage, which can be written as [69, 72]

(7)\begin{array}{rl}& {\mathrm{\Gamma }}_{EIT}\approx \frac{{\mathrm{\Omega }}_c^2\sqrt{\mathrm{l}\mathrm{n}2}}{{\mathrm{\Gamma }}_e\sqrt{\alpha L}}\gg \frac{1}{{\tau }_s}\iff \frac{{\mathrm{\Omega }}_c^2{\tau }_s}{{\mathrm{\Gamma }}_e}\gg \sqrt{d},\\ & L\ge v_g{\tau }_s\approx \frac{{\mathrm{\Omega }}_c^2{\tau }_s}{\alpha {\mathrm{\Gamma }}_e}\iff d\ge \frac{{\mathrm{\Omega }}_c^2{\tau }_s}{{\mathrm{\Gamma }}_e}.\end{array}

In this equation, d is the optical depth, {\mathrm{\Omega }}_c is control Rabi frequency and {\tau }_s is the signal pulse duration. The equation implies that high optical depth of a medium is required for EIT-based memories to achieve high storage efficiency. In cold atoms, a feasible solution is to increase the dimension of a magneto-optical trap (MOT) [73, 74], which leads to impressive storage efficiency, 92.0% for a coherent optical memory [75], quantum memory for single-photon polarization qubits with an efficiency of more than 85% [76]. One of the primary concerns with this system is the diffusion of atoms, imposing a limit on the longest achievable storage time.

For RE crystals based memory, a low concentration of RE ions are trapped in crystal lattice, so the atomic diffusion is not a concern but the low optical depth is still a challenge. The first report of EIT storage in RE crystal is performed in \mathrm{P}{\mathrm{r}}^{3+}:YSO, with a storage time longer than 1 second. However, the overall storage efficiency is only 1% due to the low optical density. This is because the relevant transitions are within the 4f^N shell which are parity forbidden but only become weakly allowed as a result of the level mixing caused by the crystal field.

A feasible way to achieve higher optical depth is to extend the light path in the medium. To avoid causing more inhomogeneity in the control field from a longer crystal, an alternative of a multi-pass configuration was applied as illustrated in Fig.3(a), which basically worked by making the control field go through the crystal for multiple times [69, 77, 78[. Fig.3(b) depicts the experimentally determined light storage efficiency {\eta }_{EIT} versus the signal pulse duration {\tau }_p and the control power P_c ( P_c\propto {\mathrm{\Omega }}_c^2) for a single pass (N_{pass}= 1) and a multipass setup (N_{pass}=10). By applying the iterative pulse shaping algorithm developed and first implemented by Novikova et al. [79, 80] for a multi-pass setup, a storage efficiency of 76.3% was achieved N_{pass}=14 [77]. The maximum achievable efficiency is limited by the finite loss from crystal surfaces. Further, errors caused by the environmental perturbation are accumulated in a long beam path, deteriorating the spin wave shape, resulting in a storage efficiency that is lower than the theoretically expected value [68]. As we know, there are yet no reports of EIT storage on single photon level in a doped solid. Rencently, Hain et al. [70] used double pass setup to successfully store a few photons. They achieved a maximum storage efficiency of 1.7% after a storage time of 10s with an average number of photons \overline{n} = 53 [70] [see Fig.3(c)]. In a coherent optical storage experiment, the strong control beam and the stored signal are almost collinear. In this situation, it is difficult to filter the scattered noise and coupled noise photons in the control beam. Meanwhile, the phase mismatching caused by the large angle separation leads to strong four-wave mixing (FWM), significantly degrading the purity of a single photon state [76, 81–83]. With the above limitations, the weakest light signal was supposed to be more than 10 photons to ensure a reasonable SNR.

The time multi-mode capability of the EIT protocol, given by the width of its transparent window, is limited to a few MHz. However, its angular multiplexing in the storage and retrieval light pulse has allowed demonstrations of saving an image, which was realized first in rubidium vapor [84, 85] and then also in RE crystals [86, 87]. A spatially real image was stored in an RE crystal, instead of Fourier image, as was demonstrated in a gas medium. This reduces the loss of information caused by the fact that the control beam does not cover the higher frequency component of the Fourier spectrum [84], leading to superior quality of the retrieved image. In 2013, Heinze et al. [71] extended EIT storage times to one minute by applying dynamic decoupling (DD) pulse sequence to fight decoherence in the relevant spin transition, as shown in Fig.3(d).

In summary, the current progress in RE crystals based EIT protocol mainly achieved on the storage of classical signals. The main challenge in storing quantum information comes from the media’s low optical depth. Cavity enhancement has been proved to be a feasible solution while in practical experiment, designing a suitable cavity involves various complex tasks and increasing the complexity of the experimental process [88]. Errors from an imperfect cavity and a complicated controlling system can be accumulated and cause further decoherence and a worse SNR. In addition, EIT storage scheme is based on homogenous broadening of a media, limiting its maximum achievable storage capacity. In contrast, storage protocols based on the idea of photon echo allows using a media’s inhomogeneous broadening for multi-mode storage under a given optical depth.

4 Photon echo memories

In an RE crystal, a dopant ion inevitably experience static variations in the strain, electric or magnetic field from site to site, causing an inhomogeneous distribution of the dopant’s transition frequencies, which generally ranges from 1−30 GHz, depending on the doping concentration and fabrication processes. A memory protocol using this inhomogeneous broadening allows prospective capacity of multiple mode storage and a typical example is the photon echo memories. In contrast to the EIT protocol, photon echo memories generally work by applying a time-reversed absorption process, instead of making the sample transparent [89].

Coherent manipulation and storage of classical light states using photon echo (PE) techniques dates back to the 1980s [90, 91]. An initially polarized ensemble of atoms, which are inhomogeneously broadened, is excited to a superposition of two levels by applying an optical \pi / 2 pulse, creating a collective dipole moment. The static detuning in the transition frequency among the different subgroups of atoms leads to a decay in the collective dipole moment. An optical \pi pulse after a delay \tau reverse the atom population between the ground state and the excited state, resulting in a refocusing of the collective dipole moment of the system at a delay of 2\tau, irradiating as an echo signal. The phase accumulation due to the static detuning during the dephasing and rephasing intervals cancels, thus the echo amplitude is governed by the dynamic contribution. As the optical version of the Hahn spin echo, the photon echo technique have been widely used in the spectroscopic studies for optical materials and in classical optical memory applications [92].

However, a standard PE scheme is not suitable for storing quantum information. In a quantum regime, the incoming signal is an arbitrarily weak field, whose quantum feature can be easily buried in the noise from the amplified spontaneous emission induced by the applied \pi pulse [93–96]. Hence, various proposals were applied to tackle with this issue, some of which have been experimentally proved to be very effective. In this section, we will briefly introduce two typical proposals and the relevant research progress, including controlled reversible inhomogeneous broadening (CRIB) [97] and atomic frequency comb (AFC) [98].

4.1 Controlled reversible inhomogeneous broadening (CRIB) and gradient-echo memory (GEM)

The fundamental principle of a CRIB scheme is to manipulate the inhomogeneous broadening through a controllable magnetic or electric field, such that the phase of the collective dipole moments is refocused to its initial value at 2\tau. Under a controlled excitation, all atoms will then irradiate light coherently. In contrast to a PE scheme, no rephasing \pi pulse is used to inverse the population.

Typically, CRIB can be realized by using an electric field to induce an extra broadening in an initially homogeneously broadened ensemble of atoms [100, 101]. After a waiting period that is shorter than the system’s coherence time, the electric field is inverted, accordingly the coherence evolution of the system is inverted. Depending on whether the applied field is perpendicular to or along the direction of light propagation, it is called a “transverse” CRIB or a “longitudinal” CRIB. For the former, theoretical model shows that the maximum achievable storage efficiency is limited to 54% due to the re-absorption by RE ions [102]. The “longitudinal” CRIB often refers to a gradient echo memory (GEM) as shown in Fig.4. Applying an electric field gradient along the direction of the light propagation prevents the output echo from being reabsorbed, allowing an ideal echo emission and a storage efficiency upper limit of 100% [99, 103].

In 2006, an experimental result of CRIB storage in a solid-state system was reported for the first time, yet with a low storage efficiency. In this work, the applied material system, an \mathrm{E}{\mathrm{u}}^{3+}:YSO crystal, experienced a linear Stark shift and macroscopic electric field gradient [97, 104]. The efficiency was extended to 15% in a later experiment [103]. In 2010, CRIB storage and on-demand retrieval of a weak beam at telecommunication wavelength was demonstrated in an {\mathrm{E}\mathrm{r}}^{3+}:YSO crystal, with average photon number of one. In this experiment, a second laser beam was used to transfer the states of atoms down to a short-lived ground state from the excited state, which improved the degree of spin polarization and reduced the fluorescence noise arising from the state preparation process [107]. However, due to the imperfect optical pumping [Fig.5(a)], the large background absortion (\sim 80 \mathrm{\%}) and reabsorption process limited the storage efficiency to 0.25%. In 2010, a GEM-based memory demonstrated a storage efficiency of 69% for a weak coherent light signal [108] [the solid line in Fig.5(b)], which was for the first time beyond the no-cloning limit [109]. The experiment is performed in a 0.005% \mathrm{P}{\mathrm{r}}^{3+}:YSO. Given the long ground state lifetime, a 140 dB absorptive top-hat profile was created with a more efficient optical pumping. According to the theoretical model, and considering a perfect optical pumping process, Hedges et al. [106] predicted that it is possible to achieve a peak efficiency of about 93% if the crystal length was 10 cm instead of 14 mm.

We have briefly introduced the two-level CRIB protocol and especially its “longitudinal” variant, namely the GEM protocol. Although the theoretical model shown an efficiency of 100% for a GEM-based quantum memory is possible, it can be challenging to get close to this limit for RE crystals, given by the low absorption depth [110–115]. We can optimize the optical pumping process to reduce the background absorption and increase the number of effective atoms. An alternative way is to use stoichiometric RE crystals to increase the density [116–118]. Although the achieved storage efficiency is as high as 69%, the storage bandwidth of \mathrm{P}{\mathrm{r}}^{3+} is limited to ~5 MHz due to the low nuclear quadrupole interaction. Among the many rare earth crystals, there are some systems with a larger nuclear quadrupole moments, such as \mathrm{E}{\mathrm{u}}^{3+}:YSO. In \mathrm{E}{\mathrm{u}}^{3+}:YSO, the bandwidth is about ~5.7 MHz for {}^{151} \mathrm{E}\mathrm{u}{}^{3+} and ~15 MHz for ¹⁵³Eu³⁺ respectively [119].

In addition to its application in quantum storage, GEM has also been proposed to realize quantum state transfer by performing time reversal operation [120]. The advantage is that the collective excitation of the ensemble is beneficial to the fidelity of the quantum state that is transferred and there is no need for strong coupling between a high-Q cavity and the atoms that make up the nodes of a given quantum network [121]. We will not go into too much detail here.

4.2 Atomic frequency comb (AFC)

In practical large-scale quantum network applications, the multi-mode capacity of a quantum memory is imperative because it can dramatically accelerate communication. RE solid-state ensembles allows the potential for multiplex storage in frequency, time, space and polarization mode. However, it can be experimentally challenging using EIT or CRIB scheme, limited by the optical depth. [123]. In contrast, the atomic frequency comb (AFC) protocol can achieve multi-mode storage with a relatively low optical depth [122].

The AFC protocol was first proposed by de Riedmatten et al. [98] in 2008 and it grounded in the large ratio between the inhomogeneous and homogeneous linewidth of an RE crystal, which is {10}^5−{10}^8 typically. In the broad inhomogeneous profile, a series of absorption peaks, namely a frequency comb, are produced through the hole-burning technique. As shown in Fig.6(a), it works by burning a series of narrow holes in the atomic medium. The ions, pumped to the metastable |aux⟩ level are then no longer couple to the incoming field. This comb-like absorption profile is characterized by two parameters: peak separation {\mathrm{\Delta }}_{tooth} and peak width {\gamma }_{tooth}. With an absorption of an incoming photon (at t=0), whose waveform spanned over the frequency comb, the state of the signal is converted into a collective excitation in the form [127]:

(8)| \mathrm{\Psi }( t)⟩={\displaystyle \frac{1}{\sqrt{N}}\sum _{j=1}^N{c}_j{\mathrm{e}}^{2\pi \mathrm{i}{\mathrm{\Delta }}_{j}t}{\mathrm{e}}^{-\mathrm{i}{k}_j{z}_j}|g_1....e_j...g_N⟩.,}

where {\mathrm{\Delta }}_j\approx m {\mathrm{\Delta }}_{tooth}( m\in \mathbb{Z}, the peak order) and z_j are the frequency and position of the j\mathrm{t}\mathrm{h} ion in the ensemble. Given the periodic property of the medium, after the initial dephasing, the phase of all the excited ions will automatically refocus at t=\frac{n}{{\mathrm{\Delta }}_{tooth}}(n\in \mathbb{Z}) and coherently irradiate light signal. Here high-order echoes (n>1) are in theory possible, but they tend to have a much lower SNR [127], so only the first-order echo was considered. It is worth noting that for a two-level AFC protocol as discussed above, the storage time is predefined so it is not possible for on-demand recall. On demand recall can be realized by transferring the excitation to another metastable state |s⟩ with a control field, namely the spin-wave storage.

There have been impressive AFC demonstrations since it was proposed, including coherent preservation of polarization qubit states [128–133], herald photons memories [134–142], etc. In this section, we will focus on discussing the recent research progress of the AFC-based memory, from the perspective of each storage performance.

4.2.1 Multi-mode storage

If the input signal pulse duration T_m is much shorter than the fixed storage time \frac{1}{{\mathrm{\Delta }}_{tooth}}, multi-mode temporal storage N_t=\frac{1}{{\mathrm{\Delta }}_{tooth}{T}_m} is possible [see Fig.6(b)]. For a short input pulse of a duration time Tm, AFC bandwidth of {\mathrm{\Gamma }}_{AFC}>\frac{2.5}{T_m} is required [119]. Thus, the temporal multi-mode capacity in the fixed storage time \frac{1}{{\mathrm{\Delta }}_{tooth}} is written as

(9)N_t={\displaystyle \frac{1}{{\mathrm{\Delta }}_{tooth}{T}_m}}={\displaystyle \frac{{\mathrm{\Gamma }}_{AFC}}{2.5{\mathrm{\Delta }}_{tooth}}}\approx {\displaystyle \frac{N_{tooth}}{2.5}},

where N_{tooth} =\frac{{\mathrm{\Gamma }}_{AFC}}{{\mathrm{\Delta }}_{tooth}} is the number of the peaks in the AFC profile.

In 2008, de Riedmatten et al. [98] realized four temporal modes storage at single-photon level. After this pineering work, 64 weak coherent pulses in the time domain were coherently mapping in a \mathrm{N}\mathrm{d}{}^{3+}:YSO [124] system [see Fig.7(a)] Tang et al. [143] demonstrated a storage of 100 temporal modes in a hybrid solid state system, where a \mathrm{N}\mathrm{d}{}^{3+}:\mathrm{Y}\mathrm{V}\mathrm{O}{}_4 crystal is used to store the polarization encoded single photons emitted from quantum dots. Further, there were also demonstrations to encode each pair of temporal modes as a time-bin qubit, which was widely used in fiber-based quantum communication due to their resilience against polarization decoherence in fibers [144–146].

In a system with a short coherence time, the decoherence term {\eta }_{T_2}={\mathrm{e}}^{-\frac{4}{\mathrm{\Delta }{T}_2}} need to be included when estimating the multi-mode capacity during the actual storage time \frac{1}{{\mathrm{\Delta }}_{tooth}}, Eq. (9) is then rewritten as [119]

(10)N_t\approx {\displaystyle \frac{\mathrm{l}\mathrm{n}(1/{\eta }_{T_2})}{10}}{\mathrm{\Gamma }}_{AFC}{T}_2.

In this equation, {\mathrm{\Gamma }}_{AFC}{T}_2 represents the time-bandwidth product of a AFC memory. Kramer ions, such as Nd³⁺ or \mathrm{E}{\mathrm{r}}^{3+}, usually possesses a larger bandwidth {\mathrm{\Gamma }}_{AFC} but a shorter coherence time as a result of the non-zero electron spin, which limits the mode capacity. In contrast, non Kramer ions can have longer coherence time. The \mathrm{E}{\mathrm{u}}^{3+}:YSO crystal is a very typical non-Kramer RE system, where Jobez et al. [147] achieved storage of 100 temporal modes and at the same time a long storage time (\frac{1}{{\mathrm{\Delta }}_{tooth}}). However, non-Kramer ions usually have a smaller hyperfine splittings which limits the AFC bandwidths {\mathrm{\Gamma }}_{AFC}. About Kramer RE systems, there is a very unique \mathrm{Y}{\mathrm{b}}^{3+}:YSO system, with hyperfine splittings of hundreds MHz to GHz [148], demonstrated both long optical and spin coherence time at the low-field clock transitions [149].

In a storage experiment, the number of combs required to be prepared increases linearly with mode number. Applying sequential hole burning technique to prepare a large number of combs takes a long time and can raise the temperature of the system, resulting in instantaneous spectral diffusion and extra decoherence [150]. A solution to this issue is to apply a parallel comb preparation technique [147], which works by using a phase and amplitude modulated pulse, appearing as a periodic squarish frequency comb profile in the Fourier domain. The parallel comb preparation technique not only mitigate the heating problem, but also improves the resolution of the combs by modifying the Fourier limitation. Using such a method in \mathrm{Y}{\mathrm{b}}^{3+}:YSO, it realized over 1250 storage modes [151]. Such a high mode capacity can greatly enhances the communication rate in a large-scale quantum networks.

The large inhomogenous broadening of RE doped crystals provides intrinsic advantage for spectral multi-modality of up to N_f= \frac{{\mathrm{\Gamma }}_{inh}}{{\mathrm{\Gamma }}_{AFC}}, which work by tailoring AFC profile at different site across the static broadening to absorb photons of different energies. Sinclair et al. [125] first demonstrated spectral AFC multiplexing based in \mathrm{T}{\mathrm{i}}^{3+}:\mathrm{T}{\mathrm{m}}^{3+}:\mathrm{L}\mathrm{i}\mathrm{N}\mathrm{b}{\mathrm{O}}_3 with each mode encode by a time bin qubit [see Fig.7(b)]. Then heralded single photons were shown to be manipulated for time-frequency multiplexing storage [152–154]. Furthermore, the Orbital Angular Momentum (OAM) of photons have been shown to possess outstanding degree of freedom in carrying high-dimensional entanglement and spatial multi-mode operations [126]. The storage result is sketched out in Fig.7(c).

4.2.2 Efficient storage

The storage and retrieval efficiency of an AFC-based quantum memory can be written as [122]

(11){\eta }_{AFC}={\tilde{d}}^2{\mathrm{e}}^{-\tilde{d}}{\mathrm{e}}^{-d_0}{\mathrm{e}}^{\frac{-7}{F^2}},

where F=\frac{{\mathrm{\Delta }}_{tooth}}{{\gamma }_{tooth}} is defined as the finesse of the comb and \tilde{d}\approx \frac{d}{F} is the effective absorption considering that the number of atoms interacting with the signal is reduced because of the comb profile. The four terms at the right side of the equation represent respectively the effect of atom-light coupling, resorptive effect, background absorption and the dephasing due to the broadening of a peak in the comb. The first term is a square term because both absorption and the reemission of the field is taken into account.

For a forward reemission, the reabsorption process limits the efficiency up to 54%. The backward retrieval does not have this issue but it requires a pair of accurate \pi pulse, which can be experimentally very difficult [158]. A promising strategy to improve efficiency is to apply an impedance matched cavity to enhance light matter interaction. Technically, impedance matching of a Fabry–Pérot (FP) cavity, for instance, requires the reflection of both mirror to satisfy \sqrt{R_1}=R_2{\mathrm{e}}^{-\tilde{d}} [159, 160]. With an optical cavity, Sabooni et al. [155] and Jobez et al. [161] demonstrated AFC storage for classical light with an efficiency of 53% in \mathrm{E}{\mathrm{u}}^{3+}:YSO and 56% in \mathrm{P}{\mathrm{r}}^{3+}:YSO respectively. For the latter, 16% of the energy was not absorbed due to the impedance mismatching as shown in Fig.8(a).

Research of a quantum memory often use input photons produced through spontaneous parametric down conversion (SPDC) process, which can have a broad bandwidth, causing difficulty to impedance matching of a cavity. A recent research in \mathrm{T}{\mathrm{m}}^{3+}:{\mathrm{Y}}_3\mathrm{A}{\mathrm{l}}_5{\mathrm{O}}_{12} has demonstrated enhanced absorption over a wide spectral range by using a low finesse cavity and achieved 90% absorption of the input photons, leading to a storage efficiency of 27.5% as well as 1.6 GHz bandwidth [162]. It was argued that higher efficiency is possible by optimizing impedance matching and filtering the broad SPDC photons. Again, applying cavity enhanced AFC protocol, Duranti et al. [163] reported successful storage of weak coherent states and weak coherent time-bin qubits with efficiency of up to 62% and 51% respectively. It was conducted in a \mathrm{P}{\mathrm{r}}^{3+}:YSO crystal, and the absorption depth was \tilde{d}=0.4, which limited by the intra cavity losses and cavity bandwidth.

As we mentioned, on demand retrieval in an RE system can be realized through spin-wave AFC storage, which works by applying a \pi pulse to transfer the excitation to an auxiliary level, usually meaning another long-lived spin state. Then the total efficiency can be expressed as \eta ={\eta }_{AFC}{\eta }_t^2 [164], where {\eta }_t is the population transfer efficiency, and the decoherence of the spin state is ignored because it is slow in most cases. In order to raise population transfer efficiency, people have designed some special pulse sequences, two typical examples are the complex hyperbolic secant(CHS) pulse [165, 166] and the hyperbolic-square-hyperbolic (HSH) pulse [167]. In theory, a transfer efficiency of 100% is possible [165] and the actual efficiency realized is 90% per HSH pulse [168].

4.2.3 Long-term storage

For a memory to be useful in a quantum network, its storage time must be comparable to the total light transmission times across the communication network. The storage time of a memory is limited by the transition’s coherence time. In RE systems, the application of spin-wave AFC protocol [164], provides not only on-demand storage but also longer storage time, which is given by the hyperfine coherence, and usually longer than the optical coherence time. Storage time of the spin-wave AFC protocol is then written T_{total}=T_w+ \frac{1}{{\mathrm{\Delta }}_{tooth}}, where T_w is the wait time.

In order to obtain a longer storage time, various strategies have been used to extend the hyperfine coherence times. It has been shown that long spin coherence times can be achieved in some Kramers RE ions with strong mixing of electron spin and nuclear spin. Spin coherence time of several millisecond at zero or weak applied magnetic field was observed in such systems [149, 171]. Applying a strong magnetic field of seven Tesla even resulted in a hyperfine coherence time of 1.3 s in \mathrm{E}{\mathrm{r}}^{3+}:YSO [172]. In a non-Kramer RE system, \mathrm{E}{\mathrm{u}}^{3+}:YSO, a combined effect of a ZEFOZ transition and DD pulses extended the hyperfine coherence time from 10 ms to 6 hours [44, 173]. This same ZEFOZ transition was used for a spin-wave AFC storage, which realized storage of classical optical signal for one hour [156] [Fig.8(b)]. Integration of spin-wave AFC storage and DD have resulted in storage of time-bin qubits for 20 ms [157] [see Fig.8(c)].

4.3 Other protocols

Some other PE-based storage protocols have been proposed in RE doped systems. The main issue with a single PE sequence is that the applied \pi pulse resulted in a population-inverted medium, irradiating noise to the echo signal. The fundamental principle of the revival of silenced echo (ROSE) protocol, as illustrated in Fig.9(a), works by applying a second \pi pulse to transfer the population back to the initial state, rephasing the induced dipole for the second time and irradiating the signal coherently without amplification noise [174–176]. It is worth noting that the rephasing \pi pulses are in practice replaced by complex hyperbolic secant (CHS) pulses. In addition, the first echo become a “silenced echo” because special phase matching conditions are designed such that when the first rephasing occurs, no field emission is produced, but the coherent free evolution continues. Using a ROSE protocol, optical storage efficiency of 40% in an \mathrm{E}{\mathrm{r}}^{3+}:YSO was demonstrated [177].

The major challenge in an ROSE protocol is that imperfect \pi pulses lead to a large residual ensemble coherence, the free induction decay (FID), which possess the same spatial and frequency mode as the \pi pulses, destructing SNR for single photon-level storage. As illustrated in Fig.9(b), the “four-level echo” (4LE) protocol employs a double-\mathrm{\Lambda } energy level setup to transfer the input coherence to a different transition such that the FID noise can be frequency filtered from the echo signal [169]. The 4LE pulse sequence also allows off-axis phase matching, implying possibility for spatial filtering for the signal. It demonstrated a PE level of storage efficiency, but the coherence is reduced by the inhomogeneity caused by using four transitions.

The noiseless photon-echo (NLPE) scheme, proposed by Ma et al. [170], can be considered as an integration of ROSE and 4LE. It employs symmetrical 4LE configuration to conduct double rephasing and it satisfies the phase matching condition to silence the echo generated by 4LE. This is illustrated more in Fig.9(c). The NLPE protocol is possible to obtain higher storage efficiency and SNR than the AFC protocol because it does not need the spectral tailoring process. However, due to the limited storage time in the excited state, optical depth and the bandwidth limit caused by the instantaneous spectral diffusion, NLPE still has a lower multi-mode capacity compared to AFC [178, 179]. Further, the symmetrical 4LE configuration involves a considerable number of control pulses, which can introduce more complexity for cavity enhancement.

The last protocol we are to discuss is the Stark-modulated atomic frequency comb (SMAFC) protocol, proposed by Horvath et al. [180–183], which has demonstrated multi-mode storage capability and on-demand retrieval on a two level system. With this protocol, no irreversible dephasing is produced by the two electric field control pulses. Hence, it can be integrated to the cavity enhanced AFC protocol and will allow on-demand storage without additional optical control pulses.

We have presented a brief introduction of the quantum memory protocols, derived from the two pulse photon echo technique. The GEM is remarkable for its efficiency and AFC for its high multimode capacity, both verifies the appealing prospect for a practical quantum memory, if cavity enhancement is optimized. Among all the different RE ions, non-Kramer ions tend to provide longer coherence time while Kramer ions can provide greater bandwidth and mode capacity.

5 Conclusion and outlook

Construction of a quantum network is an important and challenging goal in the field of quantum information, which requires integration of a series of functional quantum devices, including quantum memories, photon pair sources, frequency adapters and quantum sensors, to name a few. In terms of developing quantum memories, a number of different material systems have been investigated, but no system has yet demonstrated all the required performances for a memory to be practical.

In this review, we focused on a particular system, the RE crystals and have briefly summarized its experimental progress on the development of quantum storage devices. Remarkable storage performances has been demonstrated, such as an efficiency of 69%, storage mode number of 1250 and storage time of 20 ms. Recently, RE systems were also demonstrated to be used for storing hyper-entangled and high dimensional quantum states (qudits), which can enable more efficient Bell state measurements and simplify purification procedure [126, 184–187]. However, for majority of the mentioned storage protocols, there is a main limitation for better performances, that is the system’s low optical depth. To preserve good coherence property, storage experiments tend to use a dilutedly doped sample and apply an impedance matched cavity to enhance its optical depth. The reality is, optimizing impedance matching for a cavity is technically not easy, especially in a situation when the input photon source has a large bandwidth, for instance, the SPDC source.

In practical application, interfacing a quantum memory with an entangled photon source is a very common challenge. For RE-based quantum memories, some elegant solutions, based on rephased amplified spontaneous emission (RASE) [188–193] and AFC-DLCZ [194–198], have been successfully demonstrated. Through such strategies, entangled photon pair source can be generated from an RE system that is used as a memory. In most cases, the source and the memory are from different systems and a hybrid configuration is alternative. An RE-based memory has demonstrated successful interfacing to an efficient quantum dot system [143], the latter of which has shown the potential to realize narrower and faster emission through cavity enhancement [199–201].

As solid, RE crystals have also demonstrated its advantage of being easy to integrate with other systems. At present, some groups are carrying out research work in this field and have made many breakthroughs as detailed in the following references [176, 202–209].

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