Echo protocols of an optical quantum memory

Abstract

Based on new obtained analytical results, the main properties of photon echo quantum memory protocols are analysed and discussed together with recently achieved experimental results. The main attention is paid to studying the influence of spectral dispersion and nonlinear interaction of light pulses with resonant atoms. The distinctive features of the effect of spectral dispersion on the quantum storage of broadband signal pulses in the studied echo protocols are identified and discussed. Using photon echo area theorem, closed analytical solutions for echo protocols of quantum memory are obtained, describing the storage of weak and intense signal pulses, allowing us to find the conditions for the implementation of high efficiency in the echo protocols under strong nonlinear interaction of signal and control pulses with atoms. The key existing practical problems and the ways to solve them in realistic experimental conditions are outlined. We also briefly discuss the potential of using the considered photon echo quantum memory protocols in a quantum repeater.

Graphical abstract

Keywords

optical quantum memory / photon echo / crystals with rare earth ions / quantum repeater

1 Introduction

Optical quantum memory (QM) is a device that is designed to store the quantum state of light, e.g., photonic qubits, for a given time. Optical QM is considered as a crucial element in many quantum technologies, for example, in a quantum repeater [1], a universal optical quantum computer [2], quantum light sources [3-5] as well as in experiments on fundamental tests of quantum theory [6-8]. The optical QM must provide high efficiency (low energy losses), high fidelity of the recalled quantum states, a sufficiently long lifetime and high information capacity. Practically important physical parameters of the QMs are the operating wavelength of light fields and the possibility of integration into existing devices. Performing additional operations with quantum states of light [9] and quantum addressing [10, 11] is becoming increasingly important.

In the last two decades, significant progress has been achieved in the development of optical QM based on the photon echo effect [12-20]. The interest in echo-protocols QM is caused by their ability to store a large number of photonic qubits in a single QM cell. Such a multi-mode QM has great versatility, allowing quantum storage of both single-photon and intense light fields in arbitrary quantum states, while preserving their quantum properties, which is of great importance for quantum communications and quantum computing.

Generalizing the ideas of spin echo [21] and photon echo [22, 23], echo-based QM protocols ensure the reversibility of quantum dynamics not only of an ensemble of spins or atoms, but also of signal fields resonantly interacting with them. Conceptually, the full reversal of the quantum dynamics in an atom-field system makes the echo based QM protocols a variant of the Loschmidt echo [24, 25] with a key difference in presence of an intermediate state in which an input light pulse is completely absorbed by atoms. The possibility of implementing such a variant of photon echo was proposed and justified in Refs. [26-29] for an optical depth atomic ensemble with controlled reversible inhomogeneous broadening (CRIB) of the resonant atomic detunings called later by the CRIB protocol in Refs. [30-32] and then used in the development of modified echo protocols adaptable for easier practical implementation [13, 15, 16].

In this article, based on obtaining a number of new analytical results, we analyze the basic physical properties and conditions for the effective implementation of photon echo QM protocols of weak and intense light fields, discover their specific unknown features, advantages and difficulties in practical implementation. Using the concepts of the developed theory, the most important experimental results obtained in the studied QM echo protocols and the existing problems in achieving basic parameters are also discussed.

Following the original works of the CRIB protocol [26-29] and its further generalizations [30, 33-36], we first consider the basic scenario and physical properties of the QM echo protocol. Then we study the physical features that arise in its modified versions caused by the use of various inhomogeneous broadening of the atomic transition and methods of controlling the excited optical quantum coherence. Here we show new manifestations of spectral dispersion and the resulting problems in the implementation of broadband QM echo protocols and their relationship to the presence or absence of temporal reversibility, and discuss ways to solve the problems that have arisen. The analysis is summarized by discussing the properties of the echo protocols while storage of intense signal pulses. Nonlinear patterns in the storage of signal pulses are studied on the basis of obtaining closed analytical solutions for the pulse area of the stored signals, which allows us to find more general conditions for achieving the maximum efficiency of the echo-protocols. Analyzing the properties of the studied QM echo protocols, we focus on the best experimental results achieved and ways of its further improvement. At the end of the article, we briefly discuss the potential of using the considered photon echo quantum memory protocols in a quantum repeater.

2 CRIB/GEM protocol

The CRIB protocol proposed and justified in Refs. [26-29] is based on the use of an ensemble of N three-level atoms having an inhomogeneous broadening G(\mathrm{\Delta } /{\mathrm{\Delta }}_{in}) with a linewidth {\mathrm{\Delta }}_{in} at optical transition |1⟩\to |3⟩. All atoms are prepared in the state |1⟩.

Fig.1 shows a diagram of atomic levels, the time sequence of the signal and two controlling laser \pi pulses followed by photon echo emission in backward geometry. The implementation of optical QM with intermediate storage of information about the signal pulse on the long-lived spin coherence of the atomic ground state is called the spin-wave protocol. At stage of echo emission, the initial frequency detuning \mathrm{\Delta } of atoms at the atomic transition |1⟩\leftrightarrow |3⟩ is reversed -\mathrm{\Delta } to restore the macroscopic atomic coherence excited by the signal pulse. In the first works, the CRIB protocol was proposed to be implemented using the Doppler effect in a gas [26] and controlling hyperfine interactions of rare earth ions (REI) by changing the orientation of the neighboring nuclear spins [28]. Experimentally, it is convenient to implement CRIB protocol in crystal with REIs by creating a narrow line within inhomogeneous broadening resonant transition by using a laser hole burning technique [37]. The Stark (Zeeman) effect can then be applied in an external electric (magnetic) field, the gradient of which is oriented perpendicular to the propagation of the signal light field and to change its polarity on demand [31, 38].

The signal light pulse is launched into the QM cell with a carrier frequency that is resonant with the center of the atomic transition line |1⟩\to |3⟩. The signal duration \delta t_s is assumed to be shorter than the phase relaxation time of the optical transition (\delta t_s\ll \gamma =1/T_2), and its spectral width is less than the inhomogeneous broadening of the optical transition . Weak signal pulses does not significantly change the population difference of atomic states at the resonant transition |1⟩\to |3⟩. Taking this into account, the quantum Maxwell−Bloch (MB) equations are linearized. The equations for slowly varying atomic coherence {\sigma }^j(\tau ) and field annihilation operator a_s(Z,\tau ) ([ {\hat{a}}_s (Z,\tau ),{\hat{a}}_s^{†}( Z^{\prime },\tau )]=\delta (z- z^{\prime })) in the co-moving coordinate system Z=z, \tau =t-z/v_g (v_g is the group velocity) have the form:

(1)\begin{array}{rl}& v_g\frac{\mathrm{\partial }}{\mathrm{\partial }Z}{\hat{a}}_s(Z,\tau )=\frac{\mathrm{i}}{2}\sum _{j=1}^N\mathrm{\Omega }({\mathit{r}}_{\perp }^j){\hat{\sigma }}_{-}^j\delta (Z-Z_j),\\ & \frac{\mathrm{\partial }{\hat{\sigma }}_{-}^j}{\mathrm{\partial }\tau }=-\mathrm{i}{\mathrm{\Delta }}_j{\hat{\sigma }}_{-}^j+\frac{\mathrm{i}}{2}{\mathrm{\Omega }}^{\ast }({\mathit{r}}_{\perp }^j){\hat{a}}_p(Z_j,\tau ),\end{array}

where {\mathrm{\Delta }}_j is a frequency offset of j-th atom. In Eq. (1) the effect of relaxation and Langevin forces during interaction with the signal pulse are neglected. By introducing the interaction constant \mathrm{\Omega }( {\mathit{r}}_{\perp }^j), we also included echo protocols in a single-mode waveguide [39] (\mathrm{\Omega }( {\mathit{r}}_{\perp }^j)={\mathrm{\Omega }}_{0}f({\mathit{r}}_{\perp }^j) is a coupling constant between j-th atom and waveguide mode with {\mathrm{\Omega }}_0=E_0 ⟨\mathit{d}\cdot \mathit{e}⟩/\hslash being a single photon Rabi-frequency of j-th atom located at {\mathit{r}}^j). The membrane function f({\mathit{r}}_{\perp }) determines the dependence of the interaction constant of an atom on its position {\mathit{r}}_{\mathbf{\perp }} in the transverse plane of the waveguide.

The solution of Eq. (1) describes the resonant absorption by atoms of a signal field, the Fourier components of which are attenuated by Behr’s law {\hat{a}}_s (Z,\omega )={\hat{a}}_s(Z,0){\mathrm{e}}^{-\alpha (\omega )Z/2} with an absorption coefficient

(2)\alpha (\omega )=\beta \chi (\omega )=\beta {\displaystyle \int \mathrm{d}\delta {\displaystyle \frac{G(\delta /{\mathrm{\Delta }}_{in})}{1/T_2+\mathrm{i}(\delta -\omega )}},}

where \beta =\frac{N⟨|\mathrm{\Omega }({\mathit{r}}_{\perp }){|}^2⟩}{2L{v}_g}, N is a number of atoms, ⟨|\mathrm{\Omega } ({\mathit{r}}_{\perp }){|}^2⟩=\frac{1}{S}\iint | \mathrm{\Omega }( {\mathit{r}}_{\perp }){|}^2\mathrm{d}x\mathrm{d}y, S and L are the cross-section area and the length of the waveguide, respectively, T_2 is a time of phase relaxation. For example, for the Lorentzian shape of the line G(\delta /{\mathrm{\Delta }}_{in})\to G_L(\delta /{\mathrm{\Delta }}_{in})=\frac{{\delta }_{in}}{\pi ({\mathrm{\Delta }}_{in}^2+{\delta }^2)} and \chi (\omega )={\chi }_L(\omega )=\frac{1}{{\mathrm{\Delta }}_{in}-\mathrm{i}\omega } where \alpha (\omega ) is characterized by a spectrally dependent absorption coefficient and dispersion.

After an absorption of the signal (t>\delta t_s), the excited optical coherence of atoms passes into free evolution

(3){\hat{\sigma }}_{-}^j(\tau )=\mathrm{i}\sqrt{\pi /2}{\mathrm{\Omega }}^{\ast }({\mathit{r}}_{\perp }^j){\hat{a}}_s(0,{\mathrm{\Delta }}_j){\mathrm{e}}^{-\alpha ({\mathrm{\Delta }}_j)Z_j-\mathrm{i}{\mathrm{\Delta }}_j\tau }.

This behavior demonstrates the characteristic properties of QM echo protocols: the spectral components of light pulses are recorded in the corresponding spectral components of atomic coherence and the condition Re\{\alpha (\mathrm{\Delta })\} L\gg 1 determines the maximum achievable spectral width of the QM. The spatial distribution of coherence {\hat{\sigma }}_{-}^j(\tau >\delta t_s) does not depend on the temporal shape of the signal pulse. The signal field can contain a large number of pulses, which is limited only by the ratio {\mathrm{\Delta }}_{in}/\gamma \gg 1. However, short signal pulses (\delta {\omega }_s\sim {\mathrm{\Delta }}_{in}) acquire undesirable phase modulation in the depth of the medium due to the growing influence of spectral dispersion \alpha (\mathrm{\Delta }). Below, we discuss the negative effect of the spectral dispersion on the properties of echo protocols and the possibilities of its suppression.

To increase the QM storage time, it is common to apply a short control laser \pi pulse that propagates parallel to the signal pulse {\mathit{k}}_1 \uparrow \uparrow {\mathit{k}}_s (\delta t_1\ll \delta t_s) and being in resonance with the frequency of the atomic transition |2⟩\leftrightarrow |3⟩ [26]. The \pi pulse, without experiencing absorption, converts the optical coherence {\hat{\sigma }}_{-}^j(t_1 ) at time t_1 to the long-lived spin coherence {\hat{\sigma }}_{12}^j(t_1 ) between transition |1⟩\to |2⟩ which completes the mapping of the quantum state of the signal pulse into the QM cell. During the storage time T, spin coherence is negatively affected by static and dynamic local fluctuations of the magnetic field, dipole-dipole interactions with neighboring electron and nuclear spins, which lead to irreversible decay in spin coherence. Highly efficient suppression of these effects can be achieved by using additional rephasing radio-frequency pulses [40-44], as well as placing the crystal in an external magnetic field [45-47], in which the frequencies of spin transitions are not sensitive to weak local fluctuations of the magnetic field. These methods open up ways to create QM with a lifetime reaching minutes and hours [48, 49].

In the CRIB protocol, before applying a short second laser \pi pulse (\delta t_2\ll \delta t_s), the frequency detuning of each atom changes to the opposite ({\mathrm{\Delta }}_j\to -{\mathrm{\Delta }}_j). The second \pi pulse propagates in the opposite direction to the signal pulse ({\mathit{k}}_2\uparrow \downarrow {\mathit{k}}_s) and, being resonant to the atomic transition |2⟩\to |3⟩, transfers the spin coherence {\hat{\sigma }}_{12}^j(t_2 ) at time t_2=T+t_1 to the optical coherence {\hat{\sigma }}_{-}^j(t_2 ) of the atomic transition |1⟩\to |3⟩. At the time t_e=T+2 t_1, the coherence of atoms is phasing when {\hat{\sigma }}_{-}^j(t)=- \mathrm{i}\sqrt{\pi /2}{\mathrm{\Omega }}^{\ast }({\mathit{r}}_{\perp }^j ){\hat{a}}_s(0,{\mathrm{\Delta }}_j)\cdot {\mathrm{e}}^{-\alpha ({\mathrm{\Delta }}_j)Z+i{\mathrm{\Delta }}_j(t-t_e)}, which leads to the generation of a photon echo in the opposite direction to the signal pulse {\mathit{k}}_e \uparrow \downarrow {\mathit{k}}_s (see Fig.1).

Here we should note that in a waveguide, the exact realization of laser \pi pulses becomes impossible for all the atoms located at different points of the waveguide cross-section. In this case, as shown in a recent work [39], it is still possible to obtain analytical solutions for strong nonlinear interaction of the control laser pulses with atoms when analysing the pulse areas of the studied light pulses. However, the QM protocol efficiency is becoming less and we will conduct further studies under the assumption of plane waves for the control laser pulses having the same intensity in the cross section of the signal and echo fields. The echo signal emission is described by a system of linear equations for slow variables of atomic coherence and amplitude of the echo signal in a moving coordinate system Z=z, \tilde{\tau }=t+z /v_g. The equations differ from Eq. (1) only in replacing \frac{\mathrm{\partial }}{\mathrm{\partial }Z} \to -\frac{\mathrm{\partial }}{\mathrm{\partial }Z}, {\mathrm{\Delta }}_j\to -{\mathrm{\Delta }}_j, and non-zero initial conditions for the atomic coherence. The solution is written using the Fourier components of the echo signal {\hat{a}}_e (Z=0 ,t)= \frac{1}{\sqrt{2\pi }}\int \mathrm{d}\omega {\tilde{a}}_e(0,\omega ){\mathrm{e}}^{\mathrm{i}\omega (t-t_e)} on the output medium (Z=0):

(4)\begin{array}{rl}{\tilde{a}}_e(0,\omega )=& \mathrm{\Gamma }(t_e){\tilde{a}}_s(0,-\omega )\\ & \times \frac{2\pi \beta G(-\omega /{\mathrm{\Delta }}_{in})}{[\alpha (\omega )+\alpha (-\omega )]}\{1-{\mathrm{e}}^{-[\alpha (\omega )+\alpha (-\omega )]L}\},\end{array}

where \mathrm{\Gamma }( t_e)=⟨ {\mathrm{e}}^{-\mathrm{i}\delta {\varphi }_j(t_e)}⟩ takes into account the phase relaxation of atomic coherence at the time of the echo emission [\delta {\varphi }_j( t_e) is a random phase acquired by a atom during the time the signal is stored in the QM cell]. Taking into account the symmetric inhomogeneous broadening, due to which

(5){\displaystyle \frac{1}{2}}[\alpha (\omega )+\alpha (-\omega )]= \pi \beta G(\omega /{\mathrm{\Delta }}_{in})\equiv {\alpha }_R(\omega ),

where for the Lorentzian inhomogeneous broadening {\alpha }_R(\omega )\equiv {\alpha }_{RL}(\omega )=\beta \frac{{\mathrm{\Delta }}_{in}}{{\mathrm{\Delta }}_{in}^2+{\omega }^2} is an absorption coefficient.

At the output of the medium, we get [27, 50]

(6){\tilde{a}}_e (0,\omega )=\mathrm{\Gamma }(t_e){\tilde{a}}_s(0,-\omega )\{1-{\mathrm{e}}^{-{\alpha }_R(\omega )L}\}.

According to Eq. (6), the effect of spectral dispersion is fully compensated in the backward CRIB protocol [27]. In the presence of sufficient large optical depth ({\alpha }_R(\omega )L\gg 1) and weak atomic phase relaxation (\mathrm{\Gamma } (t_e)\cong 1) the photon echo ({\tilde{a}}_e (0,\omega )={\tilde{a}}_s(0,-\omega )) completely restores the signal pulse in the form of its time-reversed copy {\tilde{a}}_e(0,t) ={\tilde{a}}_s(t_e-t) [26, 27]. The maximum spectral width {\omega }_s of the signal pulse is determined only by the optical depth of the medium {\alpha }_R(\delta {\omega }_s)L in Eq. (6).

It is worth noting that the absence of the influence of spectral dispersion on the properties of the CRIB protocol in the backward geometry is due to the accuracy of the reversibility of MB equations describing the stages of signal absorption and photon echo emission when performing transformations: {\mathrm{\Delta }}_j \to -{\mathrm{\Delta }}_j, \tau \to -\tilde{\tau }, a_s(Z, \tau )\to -a_e(Z, \tilde{\tau }), which is valid even in the case of intense signal pulses [30, 35]. Using this property, it is possible to find the wave function of echo signals even for the complex (entangled) multiphoton states of the input signal pulses [51]. The use of a superposition of quantum states of the initial signal pulse and a photon echo with spectrally inverted wave function parameters provides an interesting resource for the preparation of new entangled states of multiphoton fields. It was also shown [36] that using the symmetry of MB equations of the both stages in CRIB protocol on off-resonant Raman transitions makes it possible to manipulate the temporal duration and carrier frequency of echo signal. The possibilities of the wavelength conversion in the photon echo QM technique are also considered in the work [52]. The concept of backward geometry of echo retrieval proposed in the CRIB protocol was also applied in the effective implementation of QM based on electromagnetically induced transparency [53], Autler−Townes splitting [54, 55] and QM on off-resonant Raman scattering [56]. From an engineering point of view, the implementation of such geometry is not difficult. Similar propagation of signal and control laser fields has been experimentally implemented in such works as [57, 58].

If the \pi-pulses of equal wave vectors {\mathit{k}}_1 \uparrow \uparrow {\mathit{k}}_2 are applied, then the echo signal will be emitted in the same direction {\mathit{k}}_e\uparrow \uparrow {\mathit{k}}_s. In this forward geometry, the solution of the MB equations for the echo signal in the medium has the form {\hat{a}}_e(Z,t)=\frac{1}{\sqrt{2\pi }}\int \mathrm{d}\omega {\tilde{a}}_e (Z,\omega ){\mathrm{e}}^{\mathrm{i}\omega (t-t_e-Z/v_g)}, where

(7)\begin{array}{rl}{\tilde{a}}_e(Z,\omega )=& \mathrm{\Gamma }(t_e){\tilde{a}}_s(0,-\omega )\\ & \times \frac{2\pi G(-\omega /{\mathrm{\Delta }}_{in})}{[\chi (\omega )-\chi (-\omega )]}[{\mathrm{e}}^{-\alpha (-\omega )Z/2}-{\mathrm{e}}^{-\alpha (\mathrm{\Delta })Z/2}].\end{array}

Considering Lorentzian inhomogeneous broadening in Eq. (7), we get

(8)\begin{array}{rl}{\tilde{a}}_e(Z,\omega )=& \mathrm{\Gamma }(t_e){\tilde{a}}_s(0,-\omega a)\\ & \times \frac{\sin [\frac{\omega }{2{\mathrm{\Delta }}_{in}}{\alpha }_{RL}(\omega )Z]}{\omega /(2{\mathrm{\Delta }}_{in})}{\mathrm{e}}^{-{\alpha }_{RL}(\omega )Z/2}.\end{array}

According to Eq. (8), the spectral dispersion contributes to a decrease in spectral efficiency \eta (\omega ,Z)= \frac{⟨I_e(Z,\omega )⟩}{⟨I_e(0,\omega )⟩} (where ⟨I_{e,s}(z ,\omega )⟩=⟨{\hat{a}}_{e,s}^{†}(z,\omega ){\hat{a}}_{e,s}(z,\omega )⟩) as follows:

(9)\eta (\omega ,Z)\sim {\mathrm{e}}^{-{\alpha }_{RL}(\omega )Z}{\displaystyle \frac{{\sin }^2\left[{\displaystyle \frac{\omega }{2{\mathrm{\Delta }}_{in}}}{\alpha }_{RL}(\omega )Z\right]}{[\omega /(2{\mathrm{\Delta }}_{in}){]}^2}}.

Note that the exponential factor in the expression (9) reflects the effect of the absorption coefficient, and the second factor describes the influence of spectral dispersion. For narrowband signal pulse , the maximum {\eta }_m(\omega ) reaches at {\alpha }_R(\omega )L\approx 2 and Eq. (5) reduces to the well-known solution for \frac{\omega }{2{\mathrm{\Delta }}_{in}} {\alpha }_R(\omega )Z\ll \pi /2:

(10){\tilde{a}}_e (0,\omega )\cong \mathrm{\Gamma }( t_e){\tilde{a}}_s(0, -\omega ) {\alpha }_R(\omega )Z{\mathrm{e}}^{-{\alpha }_R(\omega )Z/2},

where the spectral efficiency no longer depends on the dispersion of the medium, but is determined only by the spectral properties of the absorption coefficient \eta (\omega )\sim {\alpha }_R^2(\omega )Z^2{\mathrm{e}}^{{\alpha }_R(\omega )Z} shown in Fig.2. The maximum of spectral efficiency \eta (\omega ) reaches the same value of 54%, but at the optimal distance L(\omega )=2/{\alpha }_R(\omega ) that increases with increasing of the frequency detuning \frac{|\omega |}{{\mathrm{\Delta }}_{in}}. The spread of optimal distances reduces the overall efficiency of the protocol while storage signal fields with a larger spectral width. However, the spectral efficiency of storing broadband signals is even more influenced by spectral dispersion. The effect of spectral dispersion on spectral efficiency depending on the optical depth of the medium is shown in top graphic images of Fig.3 ({\eta }_n:{\alpha }_R(0)L=n ). The efficiency \eta (\omega ,Z) reaches its maximum {\eta }_m( \omega ,L) at optical depth {\alpha }_{RL}( \omega )L=\frac{{\mathrm{\Delta }}_{in}}{\omega }\arctan (\frac{2\omega }{{\mathrm{\Delta }}_{in}}) where {\eta }_m(\omega ,L)= \frac{4}{1+4(\frac{\omega }{{\mathrm{\Delta }}_{in}}{)}^2} \cdot \exp \{-{\alpha }_{RL}(\omega )L\}.

In Fig.3 it is evident that the influence of spectral dispersion also leads to a decrease in the maximum spectral efficiency with increasing optical depth {\alpha }_R(0)L. Accordingly, due to the influence of spectral dispersion, the storage of short light pulses will be also accompanied by their stronger attenuation and an additional narrowing of their spectrum (an increase in their durations) (see Fig.4). Thus, unlike the backward geometry, spectral dispersion strongly affects the efficiency and fidelity of the implementation of the forward CRIB protocol. It should be noted that the negative manifestation of spectral dispersion in the quantum storage of broadband signal pulses is characteristic of all QM echo protocols that do not have full temporal reversibility of the interaction of light pulses with resonant media. This problem becomes especially important when using optically depth media.

Nonlinear effects. To study the efficiency of CRIB protocol (and other QM echo protocols) for intense narrowband signal pulses (\delta {\omega }_s \ll \frac{\pi {\mathrm{\Delta }}_{in}}{{\alpha }_R(0)L}), the equation for the pulse area of the photon echo can be used [59-62]:

(11)\begin{array}{rl}\pm \frac{\mathrm{\partial }}{\mathrm{\partial }z}{\theta }_e(z)=& \frac{{\alpha }_R(0)}{2}[2P_e(0,z){\cos }^2\frac{{\theta }_e(z)}{2}\\ & +W_e(0,z)\sin {\theta }_e(z)],\end{array}

where signs “+” and “-” correspond to the forward and backward geometry of the echo signal emission, P_e(0, z) is the amplitude of the phasing coherence component at a resonant frequency. For example, for a two-pulse photon echo P_e(0,z)=\mathrm{\Gamma }(t_e )\sin {\theta }_1(z){\sin }^2\frac{{\theta }_2(z)}{2} and W_e(0, z) is a component of the atomic inversion which slowly changes from frequency detuning, taken at the central frequency. For the two-pulse (primary) echo W_e(0, z)=- \cos {\theta }_1(z)\cos {\theta }_2(z) [62]. Eq. (11) describes the pulse area of the generated echo signal in the two-level medium. Assuming P_e(0, z)=0, and W_e(0,z)=-1, Eq. (11) reduces to the well-known McCall−Hahn area theorem [63].

Everywhere below we focus on the case of plane waves of signal radiation and controlling laser pulses. The well-known solution for the pulse area of the signal (first) pulse [63] {\theta }_s(z)= 2\arctan \left\{\tan \left(\frac{{\theta }_s(0)}{2}\right){\mathrm{e}}^{-{\alpha }_R(0)L/2}\right\}. By applying two laser \pi-pulses at an adjacent atomic transition and inverting the frequency offsets of atoms, we obtain the following expressions for phasing polarization P_{crib}(0,z )=\mathrm{\Gamma }(t_e)\sin {\theta }_s(z) and atomic inversion W_{crib}(0,z)=-\cos {\theta }_s(z) corresponding to the formation of photon echo emission in the backward CRIB protocol. Analytical solution of Eq. (11) for the pulse area of echo signal in the medium {\theta }_e(0\le z\le L) is

(12)\begin{array}{rl}\tan \left(\frac{{\theta }_e(z)}{2}\right)=& \mathrm{\Gamma }(t_e)\tan \left(\frac{{\theta }_s(0)}{2}\right)\\ & \times \frac{{\mathrm{e}}^{-{\alpha }_R(0)z/2}[1-{\mathrm{e}}^{-{\alpha }_R(0)(L-z/2)}]}{1+{\tan }^2\left({\displaystyle \frac{{\theta }_s(0)}{2}}\right){\mathrm{e}}^{-{\alpha }_R(0)L}}.\end{array}

Below we will focus on the solution (12) describing the photon echo at the medium output ({\theta }_e(z=0)):

(13)\begin{array}{rl}{\theta }_e(0)=& 2\arctan \{\mathrm{\Gamma }(t_e)\tan \left(\frac{{\theta }_s(0)}{2}\right)\\ & \times \frac{1-{\mathrm{e}}^{-{\alpha }_R(0)L}}{1+{\tan }^2\left({\displaystyle \frac{{\theta }_s(0)}{2}}\right){\mathrm{e}}^{-{\alpha }_R(0)L}}\}.\end{array}

The solution (13) shown in Fig.5 generalizes Eq. (6) to an arbitrary pulse area {\theta }_s(0) of the signal pulse. As can be seen in Fig.5, increasing the pulse area of the input signal pulse {\theta }_s(0) close to \pi leads to additional requirement for the optical depth: {\alpha }_R(0)L\gg 1 and {\alpha }_R(0)L \gg \ln \left\{{\tan }^2\left(\frac{{\theta }_e(0)}{2}\right)\right\} to ensure sufficiently high efficiency. The last inequality expresses the manifestation of nonlinear interaction of light with atoms (since \ln \left\{{\tan }^2\left(\frac{{\theta }_e(0)}{2}\right)\right\}\gg 1 when {\theta }_s(0)\to \pi). Two limiting cases are interesting:

1) {\alpha }_R( 0)L\ll 1

{\theta }_e(0)=\mathrm{\Gamma }(t_e)\sin {\theta }_s(0){\alpha }_R(0 )L, which shows that an echo signal is emitted by the atomic polarization (\sim \sin {\theta }_s(0)) excited by a signal pulse. This solution describes typical modes of the photon echo emission in the optically thin media [21-23].

2) {\alpha }_R( 0)L\gg 1

{\theta }_e(0)=2\arctan \left\{\mathrm{\Gamma }(t_e)\tan \left(\frac{{\theta }_s(0)}{2}\right)\right\}. The solution shows a complete reconstruction of an arbitrary input signal pulse area {\theta }_e(0)= {\theta }_s(0) at \mathrm{\Gamma } (t_e)\cong 1.

Thus, the solutions (12) and (13) make it possible to understand the effect of phase relaxation of atoms and the optical depth of the medium on the reconstruction of an echo signal in conditions of nonlinear resonant interaction with atoms. It should be noted that the study of these effects cannot be carried out using the well-known method of the inverse scattering problem [64], which is aimed at finding analytical solutions for light pulses (solitons) arising in the depth of the medium without taking into account the influence of phase relaxation of atoms [65].

Solution (13) reproduces the reversibility of the nonlinear Maxwell−Bloch equations in the CRIB protocol [30], demonstrating that the photon echo area theorem (11) plays the role of a time-reversed version of the McCall−Hahn area theorem [63]. Thus the photon echo area theorem [59, 62] provides important tool for analytical study of photon echo at the conditions of nonlinear light-atoms interaction and can be useful for studies of different echo signals [62, 66] and QM echo-protocols, as it will be used below.

Substituting the same values of phasing polarization P_{crib}(0 ,z) and atomic inversion W_{crib}(0,z ) in Eq. (11), but taking into account the propagation of the photon echo in the forward direction, we find the following solution for its pulse area {\theta }_e(z) in the medium:

(14)\begin{array}{rl}\tan \left(\frac{{\theta }_e(z)}{2}\right)=& \mathrm{\Gamma }(t_e)\tan \left(\frac{{\theta }_s(0)}{2}\right)\\ & \times \frac{{\alpha }_R(0)z{\mathrm{e}}^{-{\alpha }_R(0)z/2}}{1+{\tan }^2\left({\displaystyle \frac{{\theta }_s(0)}{2}}\right){\mathrm{e}}^{-{\alpha }_R(0)z}}.\end{array}

For weak input signal pulse {\theta }_s(0)\ll 1 in Eq. (14), we obtain {\theta }_e(z)={\mathrm{\Gamma }}_R( t_e){\theta }_s(0) {\alpha }_R(0)z {\mathrm{e}}^{-{\alpha }_R(0)z/2} corresponding to the above linear solutions for the photon echo of weak spectrally narrow signal pulse (8). From Eq. (14), we found that with increasing {\theta }_s(0) the efficiency reaches its maximum at a larger optical depth {\alpha }_R(0)L=2/|\cos {\theta }_s(L)|\ge 2. If we evaluate the overall performance behavior of the protocol in Eq. (14) using a seemingly acceptable ratio {\eta }_t({\theta }_e(z) )=| \frac{\tan \left({\theta }_e(z)/2\right)}{\tan \left({\theta }_s(0)/2\right)}{|}^2, we get fast decrease of {\eta }_t({\theta }_e(z) ) with increasing {\theta }_s( 0) and {\alpha }_R(0)L (see Fig.6).

It is more correct to use a measure {\eta }_{\theta }({\theta }_e(z ))=|{\theta }_e (z)/{\theta }_s(0){|}^2, which is confirmed in experiments with light pulses, the spectral width of which is much smaller than the inhomogeneous broadening of the resonant transition [39, 67]. Under these conditions, the time profile of photon echo signals can reproduce the time profile of signal pulses even under conditions of strong nonlinear interaction, but not at too large optical depth of the resonant medium, where optical solitons have not yet had time to form. Using the efficiency measurement {\eta }_{\theta }({\theta }_e(z )), we find that the efficiency can increase asymptotically approaches unity at {\theta }_s(0)\to \pi (see Fig.7), reflecting the nonlinear nature of the self-induced transparency effect near the bifurcation point \theta =\pi, but then falling to zero over long distances (see Fig.7).

Thus, the enhancement of the nonlinear interaction of the signal pulse with a two-level medium leads both to an increase in the optimal optical depth and to maximum protocol efficiency. The counter intuitive enhancement of the efficiency is obviously associated with an increase of the penetration depth of an intense light pulse into the medium caused by the effect of self-induced transparency [63]. However, these efficiency measures based on the estimation of the echo pulse area, require further clarification when switching to using on an energy efficiency measure.

The study of quantum properties in the storage of intense signal pulses is already gaining interest for optical quantum communications [68] and it can be carried out using the capabilities of the inverse scattering method [64, 65] and its quantum generalization [69]. This, however, will require its further development in obtaining non-soliton solutions.

The discussed properties of CRIB protocol are basically repeated in studied below AFC and ROSE protocols, with the advent of some new features and practically significant advantages. However, the departure from the exact reversibility of the light−atoms equations of motion in some cases lead to the additional problems in effective implementation of these protocols. We discuss some of them in the next sections.

The Gradient Echo Memory (GEM) protocol has been proposed as a way to implement the CRIB protocol [31], where the CRIB procedure ({\mathrm{\Delta }}_j \to -{\mathrm{\Delta }}_j) is provided using a switchable electric field gradient along the propagation of a signal light pulse (\mathrm{\Delta } =\chi z) causing a linear Stark effect of narrow homogeneously broadened optical transition in REI ensemble, where \chi L is the spectral width of the GEM cell that satisfies the condition \chi L >\delta {\omega }_s.

The GEM protocol has been demonstrated in solids and atomic gases [32, 38, 70]. In Ref. [70], an efficiency of 69% was achieved on a Pr{}^{3+}:Y{}_2SiO{}_5 crystal. Due to the application of the Zeeman effect with a switchable magnetic field gradient in Refs. [38, 71, 72], using GEM protocol with Raman transition (\mathrm{\Lambda }-scheme) of rubidium in a vapor gas, the efficiency of 87% was achieved for quantum states of light, which remains the best among optical quantum memory protocols.

The GEM protocol coincides with the CRIB protocol when signal is retrieved in the backward direction, respectively, while having full temporal reversibility of the Maxwell−Bloch equations corresponding to the stage of absorption of the signal pulse and the stage of its retrieval in the photon echo. However, as it was firstly shown numerically [73] in the GEM protocol it is possible to achieve efficiency close to 100%, not only in the backward but also in the forward direction of the photon echo emission (see Fig.8). The efficiency of GEM protocol with forward and backward echo emission is described by the same equation (6) where {\alpha }_R(0)L is replaced by an effective optical depth {\varkappa }_{eff}=2\beta /\chi [38, 74]. This useful property arises due to the fact that resonant atoms are always located at one point in the medium, which does not lead to reabsorption of echo signals and gives great advantages in using this protocol. In particular, the retrieval of the signal field in the forward direction makes GEM protocol very attractive for use in quantum optical holography.

At the same time, due to the violation of temporal reversibility, the echo signal emitted in the forward direction acquires additional nonlinear phase modulation factor {\mathrm{e}}^{\mathrm{i}{\varphi }_n(t)} [74]:

(15){\tilde{a}}_e (t)\sim \mathrm{\Gamma } (t_e){\tilde{a}}_s(t_e- t){\mathrm{e}}^{\mathrm{i}{\varphi }_n(t)},

where the phase {\varphi }_n(t)= {\varkappa }_{eff}L n\{1 +\frac{t-t_e}{t_1+t_m}\}, t_m={\varkappa }_{eff}/(\chi L ), t_e is the time moment of photon echo emission. Phase chirping caused by the nonlinear behavior of the phase {\varphi }_n(t) is weakened if the interaction time t_1 of a signal pulse with the atoms is longer than the pulse duration (t_1\gg \delta t_s\sqrt{{\varkappa }_{eff}/\pi }, for t_m\ll t_1). Therefore, after the signal has entered the medium, it is necessary to wait a certain time t_1 before turning off the external gradient. This requirement arises because in the space region of the exact resonance, the signal pulse experiences a significant slowdown in its propagation due to the manifestation of strong spectral dispersion.

Characterizing the specific properties of GEM protocol, it is also important to note that the signal pulse is stored in the spatial shape of the excited atomic coherence. Unlike the slow-light QM protocol [75], the spatial shape is determined by the shape of the signal pulse spectrum. Accordingly, unlike slow-light QM, a large number of signal pulses can be stored in the same area of the QM cell space, so the number of which is only limited by the ratio of inhomogeneous broadening to the homogeneous linewidth of the resonant transition (\chi L/\gamma \gg 1). At the same time, the specific nature of the inhomogeneous broadening of the resonant transition of the GEM protocol does not allow the application of the pulse area theorem in the study of general nonlinear patterns of interaction of light pulses with atoms, the understanding of which remains an unsolved theoretical task.

The ability to control the magnitude and sign of the gradient of the external electric (or magnetic) field gives researchers additional opportunities to control the operation of GEM protocol. These are particularly relevant to the GEM protocol on the Raman transitions. The following can be noted:

1) Typically, the CRIB/GEM protocol is characterized by obtaining a photon echo with a time-reversed shape compared to the time shape of the signal pulse. Using Raman transitions in GEM protocol, it is possible to restore the photon echo, which will have the temporal shape of a signal pulse. To do this, after absorption of a signal pulse we first reverse the frequency detuning of atoms (\mathrm{\Delta } \to -\mathrm{\Delta }) without using a control laser pulse, ensuring complete dephasing of the spin coherence of atoms with a sufficiently long time of evolution of spin coherence. Then change the frequency offsets again (-\mathrm{\Delta }\to \mathrm{\Delta }) in the presence of the control laser pulse will lead to the photon echo emission with a time profile of the signal pulse.

2) The possibility of reducing the inhomogeneous broadening of the optical transition by switching off the gradient of the applied electric (magnetic) field makes it possible to realize enhanced off-resonant interaction of the quantum coherence stored in the medium with probe quantum light fields. Such interactions are used to provide nonlinear Kerr interactions of single-photon wave packets in order to deterministically implement the conditional cross-phase shift gates [76-80]. Recently demonstrated GEM protocol [81] appears to be a promising alternative to the slow-light protocol [76, 82] in the deterministic implementation of two-photon qubit gates, which can be free from the limitations inherent in the use of nonlinear interaction of two propagating photon wave packets detailed in Refs. [83, 84].

3) It is also worth noting that by changing the magnitude of the gradient of the electric (magnetic) fields, the duration of the photon echo signal can be manipulated while maintaining high efficiency [71, 85], which is impossible when using a transverse inhomogeneous broadening to the propagation of the signal field (that is, for the usual CRIB protocol [26]).

4) Another promising area of Raman GEM protocol development is the creation of a quantum light generation using reservoir engineering with a help of four-wave interaction [86].

5) The record efficiency achieved in the GEM protocol on the Raman transition in atomic gas stimulated work on the implementation of QM echo protocols in crystals with REIs [87, 88]. The solution to this problem is possible by using a resonator schemes with crystals having a sufficiently narrow inhomogeneous broadening of the optical transitions. The search for such crystals continues [89-92].

6) It is worth noting that the inverted time form of the photon echo signal in CRIB/GEM protocol allows the implementation of a NOT gate plus phase gate with a signal pulse prepared in the time-bin state of the photonic qubit [85]. The implementation of CNOT gate is a big challenge for further development of the CRIB/GEM protocol. Among the possible ways to solve it, one can try to use a two-resonator scheme of time-bin quantum RAM scheme [10].

A significant limitation of the CRIB/GEM protocol is the difficulty of achieving the required high optical depth of the resonance transition in a wide spectral region. This problem can be solved to some extent by additional using methods that allow significantly increasing the constant of photon-atom interaction. For example, this is possible when implementing this protocol in a high-Q resonator [93], by using surface plasmon polaritons [94, 95] instead of flying light fields and applying other nanooptics methods [96].

3 AFC protocol

The atomic frequency comb (AFC) protocol uses an ensemble of atoms with equidistantly spaced narrow resonant lines {\mathrm{\Delta }}_n =n{\mathrm{\Delta }}_{afc} with {\mathrm{\Delta }}_{afc} is the distance between the lines (n = 0, ±1, ±2, ...). Such inhomogeneous broadening was first proposed for photon echo in 1985 and demonstrated on a system of nonlinear oscillators in Refs. [97, 98].

After excitation by a resonant pulse at time t=0, the n-th group of oscillators goes into free oscillation, acquiring a phase \sim {\mathrm{e}}^{-\mathrm{i}n{\mathrm{\Delta }}_{afc}t}. When the phases of all oscillators coincide again at time t_n=2 \pi n/{\mathrm{\Delta }}_{afc}, the echo signal is emitted. In 2008, this variant of inhomogeneous broadening was proposed for echo-based QM in Ref. [99]. During the interaction time of a short light pulse with AFC structure \delta t_s\ll {\mathrm{\Delta }}^{-1} of the inhomogeneous line broadening is characterized by an averaged (effective) optical density {\alpha }_{afc}(\omega )=\frac{\mathrm{{\rm Y}}}{{\mathrm{\Delta }}_{afc}}\alpha (\omega ), where \mathrm{{\rm Y}} is the spectral width of a single peak, \alpha (\omega ) is determined from Eq. (4), f={\mathrm{\Delta }}_{afc}/\mathrm{{\rm Y}} is a finesse of AFC structure. Due to this property, the MB equations can be solved independently for each non-overlapping input light pulse at absorption and echo emission stages. A solution for the Fourier component of an echo signal {\hat{a}}_e (Z,t )=\frac{1}{\sqrt{2\pi }}\int \mathrm{d}\omega {\tilde{a}}_e(Z,\omega ){\mathrm{e}}^{-\mathrm{i}\omega (t-t_e-Z/v_g)} emitted in parallel to the signal pulse after its absorption in the medium has the form:

(16)\begin{array}{rl}{\tilde{a}}_e(Z,\omega )=& {\mathrm{\Gamma }}_{afc}(t_e){\tilde{a}}_s(0,\omega )\\ & \times {\alpha }_{R,afc}(\omega )Z{\mathrm{e}}^{-{\alpha }_{afc}(\omega )Z/2},\end{array}

where {\alpha }_{R,afc} (\omega )={\alpha }_R(\omega )/f is an effective absorption coefficient, {\mathrm{\Gamma }}_{afc}( t_e=1 /{\mathrm{\Delta }}_{afc})={\mathrm{e}}^{-7/2f^2} [100], which requires the creation of a high finesse AFC structure with {\mathrm{\Delta }}_{afc}/\mathrm{{\rm Y}}> 20 to get {\mathrm{\Gamma }}_{afc}(1/{\mathrm{\Delta }}_{afc})\cong 1. Since {\alpha }_{afc}(\omega )={\alpha }_{R,afc}( \omega )(1+\mathrm{i}\frac{\omega }{{\mathrm{\Delta }}_{in}}) (for Lorentzian inhomogeneous broadening), and \alpha (\omega )={\alpha }_R(\omega )(1+ \mathrm{i}\frac{\omega }{{\mathrm{\Delta }}_{in}}) the echo signal in Eq. (16) acquires phase modulation [see in comparison (6)], which obviously distorts the time profile of the broadband echo signal, but not affects the spectral efficiency of the AFC protocol {\eta }_L(\omega ,Z)\sim [{\alpha }_{R,afc}( \omega )Z {]}^2\exp \{ -{\alpha }_{R,afc}( \omega )Z\}.

The maximum of \eta (\omega ,Z) remains equals to 52% at {\alpha }_{R,afc}(0)L=2 for narrowband signal pulse (\delta {\omega }_s\ll {\mathrm{\Delta }}_{in}) and {\mathrm{\Gamma }}_{afc}( 1/ {\mathrm{\Delta }}_{afc})\cong 1, that is similar to the CRIB protocols. However, the echo pulse {\hat{a}}_e (Z,t ) of Eq. (16) reproduces the temporal shape of the signal pulse that indicates a violation of the temporal reversibility in the AFC protocol. Taking into account the spectral dispersion in {\alpha }_{afc}(\omega ), we get that narrowband echo signal propagates with modified group velocity {\tilde{v}}_g =v_g\frac{1}{1-\frac{v_g{\alpha }_{R,afc}(0)}{2{\mathrm{\Delta }}_{in}}}, which leads to an additional (negative) time delay in the emission of the echo signal \delta t_e=- \frac{{\alpha }_{R,afc}(0)L}{2{\mathrm{\Delta }}_{in}} caused by the interaction of the light pulses with resonant atoms. For the maximum efficiency (i.e., at {\alpha }_{R,afc} (0)L=2) we have the same time delay \delta t_e=-{\mathrm{\Delta }}_{in}^{-1}.

Now we consider AFC protocol in a 3-level scheme with echo emission in the backward direction. This is implemented similarly to the CRIB protocol ([26] and see above) by applying two counter propagating laser \pi pulses at an adjacent atomic transition after signal pulse absorption (see backward geometry in Fig.1). After similar calculations, we get

(17)\begin{array}{rl}{\tilde{a}}_e(0,\omega )=& {\mathrm{\Gamma }}_{afc}(t_e){\tilde{a}}_s(0,\omega )\\ & \times \frac{{\alpha }_R(\omega )}{\alpha (\omega )}[1-{\mathrm{e}}^{-{\alpha }_{afc}(\omega )L}].\end{array}

At low optical depth ({\alpha }_{R,afc}L\ll 1) in Eq. (17), we have {\tilde{a}}_e (0,\omega )={\mathrm{\Gamma }}_{afc}(t_e ){\tilde{a}}_s(0,\omega ){\alpha }_{R,afc}L, this indicates no influence of spectral dispersion to the echo pulse parameters at low efficiency, which is observed experimentally. At a high optical depth ({\alpha }_{R,afc}L\gg 1), we get by assuming the Lorentzian inhomogeneous broadening:

(18){\tilde{a}}_e (0,\omega )={\mathrm{\Gamma }}_{afc}(t_e ){\tilde{a}}_s(0,\omega ){\displaystyle \frac{1-\mathrm{i}{\displaystyle \frac{\omega }{{\mathrm{\Delta }}_{in}}}}{1+{\displaystyle \frac{{\omega }^2}{{\mathrm{\Delta }}_{in}^2}}}}.

It is seen in Eq. (18) that the spectral efficiency {\eta }_L(\omega ) of the AFC protocol decreases with an increase in the spectral width of the signal pulse {\eta }_L( \omega )= {\left(1+\frac{{\omega }^2}{{\mathrm{\Delta }}_{in}^2}\right)}^{-1}, which leads to a narrowing of the spectrum of the restored signal

(19)| {\tilde{a}}_e (0,\omega ){|}^2\sim {\displaystyle \frac{|{\tilde{a}}_s(0,\omega ){|}^2}{1+{\displaystyle \frac{{\omega }^2}{{\mathrm{\Delta }}_{in}^2}}}}.

By taking into account that (1- \mathrm{i}\frac{\omega }{{\mathrm{\Delta }}_{in}})\cong \exp \{ -\mathrm{i}\frac{\omega }{{\mathrm{\Delta }}_{in}}\} for narrowband signal pulse (\delta {\omega }_s\ll {\mathrm{\Delta }}_{in}), we get from Eqs. (17) and (18) that the spectral dispersion only leads to an additional time delay in the emission of the echo signal \delta t_e=- {\mathrm{\Delta }}_{in}^{-1}. However in the case of broadband photon echo signal (\delta {\omega }_s \sim {\mathrm{\Delta }}_{in}), the spectral dispersion causes too large phase distortions that reduce the efficiency and fidelity of the signal pulse retrieval [101]. The sensitivity of a high-performance AFC protocol to the spectral dispersion distinguishes it from the CRIB protocol, which does not acquire additional phase distortions. The need to suppress the spectral dispersion caused by interaction with resonant atoms becomes important already when an efficiency of more than 60% is realized [101]. The increasing the operation spectral range {\mathrm{\Delta }}_{QM} while maintaining high efficiency and fidelity when using AFC protocol requires an additional suppression of the spectral dispersion.

The existing experiments on AFC protocol in crystals with RE ions are being implemented by creating the AFC structure of a certain spectral interval inside an inhomogeneously broadened resonant transition [18, 19]. Taking into account such a spectral design of AFC structure, it is possible to propose an optimal way to create such a structure in which spectral dispersion will be maximally suppressed in a given operating spectral range of AFC protocol [102, 103]. The approach [102, 103] is based on the creation of AFC structure with optimal spectral parameters inside a inhomogeneously broadened line for the given operating spectral range {\mathrm{\Delta }}_{QM} inside this AFC-structure and necessary protocol efficiency [102].

Full consideration of the effect of the spectral dispersion of atoms when creating an AFC structure in a limited frequency range {\mathrm{\Delta }}_0 (see Fig.9-Fig.11) of an inhomogeneously broadened transition includes two terms. A detailed analysis of their influence is the subject of a separate study, taking into account the real parameters of the optical transition [104]. One of them is due to the frequency dispersion of the AFC structure itself, and the other is caused by the influence of the remaining two spectral regions (side “wings”). Atoms whose frequencies belong to the side wings of the resonant transition change the frequency dispersion without practically affecting the resonant absorption coefficient, leading to a specific change in {\alpha }_{afc}(\omega ) to a form \mathrm{i}{\alpha }_{R,afc}( \omega )\chi (\omega ) [101], where

(20)\begin{array}{rl}\chi (\omega )=& {\chi }^{\prime }(\omega )+\mathrm{i}{\chi }^{″}(\omega )\\ =& {\chi }_0(\omega )+{\chi }_1^{\prime }(\omega )+{\chi }_2^{\prime }(\omega ).\end{array}

The first term in \chi (\omega ) is caused by the from AFC structure {\chi }_0(\omega )={\chi }_0^{\prime }(\omega )+\mathrm{i}{\chi }_0^{″}(\omega ), where by taking into account Gaussian shape of the inhomogeneous broadening

(21)\begin{array}{rl}{\chi }_0^{\prime }(\omega )=& -\frac{1}{f}\exp (-\zeta {\omega }^2/{\mathrm{\Delta }}_{in}^2)Erf\mathrm{i}(-\sqrt{\zeta }\omega /{\mathrm{\Delta }}_{in}),\\ {\chi }_0^{″}(\omega )=& -\frac{1}{f}\exp (-\zeta {\omega }^2/{\mathrm{\Delta }}_{in}^2),\end{array}

where \zeta =4Ln2, {\alpha }_{R,afc}(\omega ){\chi }_0^{″}( \omega ) describes the average value of the absorption coefficient within the AFC structure.

From the left and right “wings” of the inhomogeneously broadened line we have dispersion terms

(22){\chi }_1^{\prime }(\omega )=\left(1-{\displaystyle \frac{1}{f}}\right){\displaystyle \frac{1}{\pi }}{\displaystyle \underset{-\mathrm{\infty }}{\overset{-{\mathrm{\Delta }}_0/2{\mathrm{\Delta }}_{in}}{\int }}{\displaystyle \frac{\exp (-\zeta x^2)}{{\displaystyle \frac{\omega }{{\mathrm{\Delta }}_{in}}}-x}}\mathrm{d}x,}

(23){\chi }_2^{\prime }(\omega )=\left(1-{\displaystyle \frac{1}{f}}\right){\displaystyle \frac{1}{\pi }}{\displaystyle \underset{{\mathrm{\Delta }}_0/2{\mathrm{\Delta }}_{in}}{\overset{\mathrm{\infty }}{\int }}{\displaystyle \frac{\exp (-\zeta x^2)}{{\displaystyle \frac{\omega }{{\mathrm{\Delta }}_{in}}}-x}}\mathrm{d}x.}

The amplitude of photon echo in backward AFC protocol is given by [102, 103]

(24){\tilde{a}}_e (0,\omega )={\mathrm{\Gamma }}_{afc}(t_e ){\tilde{a}}_s(0,\omega )\mathcal{D}(\omega ),

where

(25)\mathcal{D}( \omega )= {\displaystyle \frac{1-\mathrm{e}\mathrm{x}\mathrm{p}[\mathrm{i}{\omega }_0\chi (\omega )L/c]}{1-\mathrm{i}{\chi }^{\prime }(\omega )/{\chi }^{″}(\omega )}}.

The behavior of the dispersion {\chi }^{\prime }(\omega ) depending on the spectral width {\mathrm{\Delta }}_0 of the AFC structure is shown in Fig.10. It is seen that there is a plateau where {\chi }^{\prime }(\omega )\approx 0 and \frac{\mathrm{d}}{\mathrm{d}\omega }{\chi }^{\prime }(\omega )\approx 0 within spectral range 0.25{\mathrm{\Delta }}_{in} for {\mathrm{\Delta }}_0/{\mathrm{\Delta }}_{in}\cong 1. The efficiency of AFC protocol ({\eta }_L(\omega )=|\mathcal{D}(\omega ){|}^2) is shown in Fig.11, where its high value (>0.93) exists in the same spectral range with the used parameters. Moreover the spectral scheme [103], which provides strong suppression of spectral dispersion, allows achieving protocol efficiency of more than 95% for broadband light pulses as shown in Fig.11.

The discussed effects of spectral dispersion are due to the lack of precise temporal reversibility of the quantum memory protocol. Note that the strong negative effect of dispersion effects becomes significant when trying to achieve high efficiency in a wide spectral range compared with the inhomogeneous broadening {\mathrm{\Delta }}_{in} of the resonant transition. A large influence of spectral dispersion also occurs when implementing the protocol in a resonator, the study of the specific properties of which requires special consideration [93].

The AFC protocol was experimentally implemented at different wavelengths in several crystals with different REI [105-110]. In these studies, the AFC structure are usually created in a small spectral range {\mathrm{\Delta }}_0 of an inhomogeneously broadened optical transition. The magnitude of {\mathrm{\Delta }}_0 and finesse f have not yet been chosen optimally to ensure strong suppression of spectral dispersion. An efficiency of 62% was achieved [111], a storage time (T_{st}) of 1 hour [49], fidelity of > 99% [112, 113] (for efficiency of 11 and 7%), the storage of more than 1000 temporal light modes [114-116] and a large operating spectral width ({\mathrm{\Delta }}_0 > 6 GHz) were demonstrated [115, 117, 118] but with efficiency of <10%. In these experiments, the efficiency of AFC protocol was severely limited by the small finesse. Increasing finesse to values f > 10 with appropriate suppression of the negative effect of spectral dispersion will require an increase in the initial optical depth of the atomic transition {\alpha }_R(0) > 30. This will be possible when using a crystal that combines the presence of a high concentration of ions and a long optical coherence time.

So far, practically significant results have been obtained in different experiments and on different crystals. In most of the above experiments, AFC memory acted as a passive delay line. For practical applications, on-demand reading is required. This can be achieved in two ways. Firstly, the Stark effect can be applied as it was implemented in the works [119, 120]. However, this method has a limitation on storage time associated with a finite AFC peak width (\approx 100 kHz). Another way to implement on demand signal reading is transferring optical coherence using a \pi pulse to an additional pre-emptied spin sub-level [see discussion of Eq. (10)]. At this point, it is also possible to use dynamic decoupling pulses to increase storage time. Such schemes were implemented in the following works [105, 121-125].

The real way to increase the efficiency of AFC protocol is associated with its implementation in an optical resonator. The AFC protocol was already demonstrated in an optical resonator [111, 126, 127], in an integrated waveguide circuit [128-130] and in a photonic-crystal cavity [120, 131]. The maximum efficiency \eta \approx 60% was achieved for storage time of \approx 1 μs [111, 126]. The results demonstrate a great practical potential of this protocol even in conditions of low efficiency. In this case, it will be necessary to resolve the issue of increasing the working bandwidth, since it cannot become large when using high-quality resonators [93].

4 ROSE protocol

The use of a natural inhomogeneous broadening of atomic transition would significantly facilitate the achievement of high efficiency and a large spectral width. In 2011, QM echo protocols were proposed using the natural inhomogeneous broadening of the atomic transition [132-135], among which the protocol using the revival of silenced echo (ROSE) signal received the most attention [133].

The original scheme of ROSE protocol was proposed for a system of two-level atoms. The retrieval of the stored signal light pulse is implemented by using two subsequent rephasing \pi pulses. The ROSE echo signal is the result of the primary echo retrieval, the radiation of which is blocked by suppression of wave synchronism, or controlled dephasing of atomic coherence by external fields [132, 136, 137]. Thus, the ROSE protocol turns out to be the closest to the primary photon echo, which is most often used in optical echo spectroscopy. In this case, the echo signal is emitted in an uninverted atomic medium in a similar way to the CRIB protocol, without the appearance of additional quantum noise. An important advantage of ROSE protocol in comparison with other QM echo-protocols is the possibility of using a long natural lifetime of optical coherence and a higher optical density of the resonant transition. These properties of the ROSE protocol make it promising for further development.

The ROSE protocol was experimentally implemented in bulk crystals [133, 138, 139], in a waveguide [39, 140] and in an impedance-matched cavity [67, 141]. The retrieval efficiency of \ge 40% was achieved in the works [57, 58]. However, a significant problem of the ROSE protocol is the difficulty of implementing \pi pulses in atomic media, which causes noise in the emitted photon echo signal [58, 67, 142, 143].

Recent work [144] demonstrated the possibility of significantly suppressing the optical noise in ROSE protocol through the use of a 4-level atomic scheme. In this protocol, called by noiseless photon-echo (NLPE) protocol, laser \pi pulses acted on an adjacent optical transition, and the resulting optical noise was suppressed using a filter crystal (see Fig.12). Thus, it was possible to increase the SNR to 42.5 with a storage time of 22.5 μs and efficiency \eta \approx 6.4%. Despite a significant increase in SNR, in order to achieve a practically significant value of QM efficiency, it is necessary to improve the implementation of rephasing control laser pulses, so that their pulse area will be close to \pi.

At the same time, it should be noted that the ROSE protocol differs significantly from the CRIB, GEM and AFC protocols by using intense control laser pulses at resonant atomic transitions of high optical depth. Presence of high optical depth leads to nonlinear propagation effects of control laser pulses, leading to considerable changes of its parameters and quantum noises in ROSE signal. Studying these effects and emerging problems is very important to find optimal conditions and ways to implement ROSE protocols.

The nonlinear effects of coherent control and propagation of control pulses are enhanced when the pulse areas of these pulses become different from \pi. The consequence of these effects may be a significant deterioration of the properties of the ROSE protocol, even when working with weak signal pulses. It should be noted that this situation remains typical for the current level of experimental implementation of optical QM due to the difficulties of implementing laser \pi pulses.

By using photon echo area theorem (11), we focus below on the study of nonlinear propagation effects of controlling laser pulses in experimentally implemented versions of the ROSE protocol [58, 133, 142] using a two-level system of atoms (see Fig.12). It is also worth noting that the approach based on the use of the area theorem allows us to obtain basic information about the properties of photon echo signals when the spectral width of the signal pulses is much smaller than the inhomogeneous broadening of the atomic transition. It has been demonstrated in recent works for ROSE protocol in a two-level medium with a forward geometry of the echo signal emission [58, 142], in a cavity [67] and in a single mode waveguide [39].

Here we are particularly interested in the nonlinear properties of the two-level ROSE protocol, which are manifested by the use of control laser pulses, the pulse areas of which begin to differ from \pi. The studied ROSE protocols [58, 144] were implemented in the forward geometry of the echo signal emission. In this case, the control laser pulses propagate in the opposite direction to the signal pulse, which makes it possible to noticeably attenuate the optical noise caused by the action of laser pulses. In experiments [58, 144], weak coherent signal pulse were used which can be characterize by small input pulse area {\theta }_s(z)={\theta }_s(0 ){\mathrm{e}}^{-\frac{\alpha }{2}z} (where {\theta }_s(0)\ll 1). Therefore, ignoring the population of the excited optical level after the absorption of the signal pulse, we get the solutions for pulse areas of intensive first and second control laser pulses by using McCall−Hahn area theorem [63, 145]:

(26)\begin{array}{rl}& \tan \left[\frac{{\theta }_1(z)}{2}\right]\cong {\beta }_1{\mathrm{e}}^{-\alpha z/2},\\ & \tan \left[\frac{{\theta }_2(z)}{2}\right]\cong {\beta }_2\left(\frac{1+{\beta }_1^2}{{\mathrm{e}}^{\alpha z}+{\beta }_1^2}\right){\mathrm{e}}^{\alpha z/2},\end{array}

where {\beta }_{1,2}=\tan [\frac{{\theta }_{1,2}(0)}{2}] and as can be seen from the solution (26), if the pulse area of the first pulse is close to \pi and {\beta }_1^2 \gg {\mathrm{e}}^{\alpha z}, the second pulse experiences amplification during its propagation in the medium.

The solutions (26) can be used for studies of the protocol efficiency in the case where the spectrum of laser pulses is much narrower than inhomogeneous broadening of the atomic transition, but wider than the spectrum of the signal pulse [67]. Moreover, the solutions remain manageable when using chirped laser pulses [146], which allow a significant increase in the working spectral range for the perfect laser control of atomic coherence. We interested by the case of weak signal pulse pulse and intensive two pulse sequence of control laser pulses (see Fig.12), where we get the following relations for the resonant atomic coherence and inversion [67]:

(27)\begin{array}{rl}{P}_e(0,z)=& \mathrm{\Gamma }(t_e){\mathrm{e}}^{-{\alpha }_R(0)z/2}{\theta }_s(0)\\ & \times {\sin }^2\frac{{\theta }_1(z)}{2}{\sin }^2\frac{{\theta }_2(z)}{2},\end{array}

(28)W_e(0,z)=-\cos {\theta }_1(z)\cos {\theta }_2(z).

The formal solution of Eq. (11) for the pulse area {\theta }_R(z) of forward ROSE protocol is

(29)\begin{array}{rl}{\theta }_R(z)=& 2\arctan \{\frac{1}{2}\mathrm{\Gamma }(t_e){\theta }_s(0){\alpha }_R(0)\\ & \times {\int }_0^z\mathrm{d}{z}^{\prime }{P}_e(0,z^{\prime })\exp \{\frac{{\alpha }_R(0)}{2}{\int }_{z^{\prime }}^z\mathrm{d}{z}^{″}{W}_e(0,z^{″})\}\},\end{array}

which we use in further analysis.

To study the influence of deviation of control pulse areas from \pi, in the theoretical description we limit ourselves to the case when the two control laser pulses propagate parallel to the first signal pulse and the photon echo. Substituting expressions (26) in Eq. (29) after calculating the integrals, we get

(30){\theta }_e(z)= 2\arctan \{{\displaystyle \frac{1}{2}}\mathrm{\Gamma }( t_e){\theta }_s(0)\mathrm{\Phi }(z)A( z)\},

where

(31)\begin{array}{rl}A(z)=& {\alpha }_R(0)z+\frac{(1-{\mathrm{e}}^{{\alpha }_R(0)z})}{(1+{\beta }_1^2)[1+{\beta }_1^{-2}{\mathrm{e}}^{{\alpha }_R(0)z}]}\\ & +\ln [\frac{1+{\beta }_1^{-2}}{(1+{\beta }_1^{-2}{\mathrm{e}}^{{\alpha }_R(0)z})}],\end{array}

(32)\mathrm{\Phi }( z)={\displaystyle \frac{{\mathrm{e}}^{\frac{1}{2}{\alpha }_R(0)z}{\beta }_2^2(1+{\beta }_1^2{)}^2({\beta }_1^2+{\mathrm{e}}^{{\alpha }_R(0)z})}{{\beta }_1^2[{\beta }_1^4+M{e}^{{\alpha }_R(0)z}+{\mathrm{e}}^{2{\alpha }_R(0)z}]}}.

Here M={\beta }_2^2(1+ {\beta }_1^4 )+2{\beta }_1^2(1+{\beta }_2^2) and we assumed that the primary echo signal is suppressed by the additional controlled dephasing [124].

Together with the backward CRIB protocols (13), (14), the solution (30) is the third example of an exactly solvable task of photon echo QM protocols under the conditions of strong nonlinear light-atoms interactions. The question arises − what will be the behavior of the ROSE signal intensity with deviation of the pulse area of control laser pulses from \pi in the optically depth medium? The depicted in Fig.13 analytical solution (30) shows that with a significant deviation in the areas of control pulses from \pi can lead to strong amplification of the echo signal in the optical depth medium {\alpha }_R(0)z >2.5. The maximum growth occurs at {\theta }_c\sim 0.8\pi. However the signal amplification is observed even under conditions of the enlightenment of the atomic medium, which cannot yet act as an amplifier. The increase in the echo signal amplitude is a manifestation of the nonlinear coherent interaction of light with atoms in such a medium. At the same time, the appearance of optical quantum noise is the result of spontaneous transitions of atoms to the ground state. The level of such quantum noise will remain weak with a small deviation of the pulse area of the control laser pulses from \pi. A detailed study of the level of quantum noise will require special research based on the application of strict quantum theory. It is also seen that by choosing an optical depth {\alpha }_R(0)L =2, there is a maximum echo signal amplitude with a minimum gain for a noticeable deviation of the pulse areas of the control laser fields from \pi, which ensures a low level of optical quantum noise, respectively.

Thus, the solution obtained using the area theorem, makes it possible to determine the required accuracy of choosing the pulse area of control pulses used in an optically dense medium. A similar theoretical study can be carried out for a 4-level NLPE scheme, where, however, obtaining a closed analytical solution requires a special study, and this also concerns the analysis of the backward geometry of ROSE protocol schemes.

It is also important when implementing the spin-wave protocol, in particular, directly using the four-level ROSE protocol. Currently, different groups use pulses of a special time and frequency dependence, for example complex hyperbolic secant pulse [147, 148] and hyperbolic-square-hyperbolic pulse [149] which are especially effective given the pulse duration limitation. In experimental works [58, 67, 125, 133, 143, 144], a pulse area in the range from 0.7\pi to 0.9\pi was achieved. In this case the area theorem approach can be useful as well.

5 Multimode repeater with absorptive memory

In quantum communication channels, the error probability scales exponentially with the channel length. To overcome this limitation a quantum repeater scheme was proposed in the work [150]. Later it was theoretically shown [151, 152] that the use of a multimode optical quantum memory can increase the rate of generation of quantum entanglement by several orders of magnitude between quantum repeater nodes [1, 153] and reduce the requirements for the entanglement storage time. Instead of generating entanglement between an atom and light, as in the DLCZ protocol [154], it was proposed to use an external source of photon pairs (biphotons) in an entangled quantum state. The requirement for the QM in this version of the repeater is the ability to record, store and retrieve several independent modes, for example, temporal, spatial or frequency modes, to ensure the appropriate multiplexing.

Another advantage of the multimode quantum repeater is the use of two-photon interference in the Bell state measurement nodes, in contrast to the single-photon interference used in the DLCZ protocol. Single-photon interference imposes additional requirements for strict phase stability of the control lasers and the length of the fiber (interferometer), which is extremely difficult in practical applications using long distances to transmit quantum information. An external source of biphotons is often a nonlinear crystal in which the spontaneous parametric down conversion (SPDC) effect is realized. This effect is a nonlinear process of interaction of radiation with an anisotropic crystal, in which a pair of photons is spontaneously born from one photon of intense pump radiation [155]. In this case, quantum entanglement can be encoded into various degrees of freedom of biphotons, such as polarization, frequency, spatial or temporal modes, orbital angular momentum, as well as into temporal bins (wave packets of single-photon states) [155].

So far the main optical QM scheme used for quantum teleportation of the quantum state of a photon and a qubit in crystals doped with REIs is the AFC protocol. In the work [156] teleportation of the polarization state of a telecom-wavelength (\lambda \sim 1.3 μm) photon onto the state of a polarization preserving QM cell. Entanglement is established between a Nd{}^{3+}:Y{}_2SiO{}_5 crystal (\lambda \sim 883 nm, {\eta }_{QM}= 5%, storage time 50 ns), which stores a single photon whose polarization is entangled with a flying photon of telecommunication wavelength. The latter is jointly measured with another flying polarization qubit in the weak coherent state to be teleported, which heralds the teleportation. The fidelity of the qubit retrieved from the memory is shown to be F> 80%.

Quantum entanglement between two solid-state QM cells was experimentally demonstrated in Ref. [118]. Entangled pairs of photons in time-bin modes were used, with photon wavelengths in each pair of 794 nm and 1535 nm. They were stored in a Tm{}^{3+}:LiNbO{}_3 crystal (\lambda = 794 nm) and in an optical fiber doped with erbium ions Er{}^{3+} (\lambda = 1535 nm). Quantum entanglement between two different QM cells was obtained with a fidelity of ~93%. Due to the low efficiency of such QM cells, which was 0.1% in the erbium-doped fiber with storage time (T_{st}) of 6 ns and 0.4% for the Tm{}^{3+}:LiNbO{}_3 crystal (T_{st} = 32 ns), the probability of entanglement of two QM cells was about 4 × 10{}^{-6}, which is not sufficient for practical use and stimulates further improvement of this QM.

In Ref. [157] quantum teleportation over a distance of 1 km was experimentally realized between an enegy-time entangled photonic qubits at a telecommunication wavelength (\lambda \sim 1.43 μm) and a photonic qubit with a wavelength of 606 nm, which was stored in a Pr{}^{3+}:Y{}_2SiO{}_5 crystal. In the case of successful interference of two photons at a telecommunication wavelength with subsequent measurement of the Bell state, the probability of teleportation of the qubit after the QM cell for a short distance of 5 m was 1.2 × 10{}^{-2}, while for a distance of 1 km it decreased slightly and was 7.5 × 10{}^{-3}. The frequency of successful measurement of the Bell state was 1 Hz at {\eta }_{QM} \approx 18%. In this experiment, a feedback system was additionally used, implementing a conditional phase shift of the qubit extracted from the QM, as required by the quantum teleportation protocol and time multiplexing was also demonstrated. Note that it is possible to improve the parameters of implemented QM and the whole system, such as on-demand readout of photonic qubit [122], fiber integration [129], heralded entanglement between two spatially separated QMs [158] and increase in efficiency of QM cell up to 60% [159] by application of impedance-matched cavity.

Heralded distribution of quantum entanglement between two QM cells was experimentally implemented in the work [160]. The experimental setup consisted of two separate sources of polarization-entangled photon pairs that interfered at a beam splitter and were then projected onto Bell states, resulting in polarization entanglement of the two states that were stored in separate QM cells at a distance of 3.5 m. The QM cell was a Nd{}^{3+}:YVO{}_4 crystal (\lambda = 880 nm). The AFC protocol were implemented in these crystals with a storage time of 56 ns, a bandwidth of 1 GHz, a signal photon recovery efficiency of {\eta }_{QM} \approx 13%, and a quantum state recovery fidelity F\approx 96%. Bell states were detected at a frequency of 100 Hz, which predicted entanglement of the two QM cells with each other. The final entanglement distribution frequency, taking into account all losses, was 1.1 per hour with a quantum state recovery fidelity F = 80.4%. So far this experiment is the closest to a scalable quantum repeater node.

The above works used only one degree of freedom (DOF) of photons for entanglement creation. They can also be entangled in more than one of their DOF, i.e., hyperentangled, and can share more bits of entanglement [161, 162]. In the work [163] the storage of energy-time and polarization hyperentanglement. One of the photons were stored in a Nd{}^{3+}:Y{}_2SiO{}_5 crystal (\lambda \sim 883 nm), while the other photon has a telecommunication wavelength suitable for transmission in optical fiber.

It should be noted that recently the components of quantum repeater nodes have also been actively developed with importance for practical application. In the work [164] storage in Er{}^{3+}:Y{}_2SiO{}_5 crystal of the entangled state of two telecom photons generated from an integrated photonic chip was implemented. The erbium doped fiber was used as a quantum memory cell in the work [115, 165] to store photons at telecommunications wavelengths, making it easier to integrate into fiber communication lines. Another way is to connect fibers to waveguides in crystals in which QM is implemented as was demonstrated in Refs. [128, 129, 166].

Simultaneous achievement of the required values of the QM characteristics (telecom operating wavelength, high storage bandwidth and storage efficiency, long storage time and noiseless) has not yet been implemented in any of the protocols. Below we summarize the best achieved parameters in the experimental implementation of photon echo based optical quantum memory. In the ROSE QM a relatively high efficiency (>40%) at telecom wavelength can be achieved with a storage time of tens of μs [57, 58]. The ROSE protocol compares favorably with other QM protocols in its relative simplicity, but the complexity of creating effective rephasing pulses as close as possible to \pi leads to a decrease in efficiency and the appearance of optical quantum noise greatly exceeding the single-photon level. For example, in a modified ROSE protocol with four control pulses and the use of two additional levels [144], it was possible to increase the signal-to-noise ratio to \sim42 with a storage time of 22.5 μs of a signal pulse containing an average of 1.17 photons. However, the use of additional \pi pulses led to a decrease in efficiency to 6.4%.

CRIB, GEM and AFC protocols of QM are low-noise, but their implementation is complicated by the necessity of preparation of a narrow spectral component or a comb of such components inside the inhomogeneously broadened line of the optical transition. In Ref. [70] an efficiency of ~69% was achieved on a Pr{}^{3+}:Y{}_2SiO{}_5 crystal for storing quantum states of light with a low noise level. The implementation of the GEM protocol by using the Zeeman effect in a switchable magnetic field gradient was demonstrated in works in three-level atomic gases on a non-resonant Raman transition [38, 71, 72, 167]. In these works, rubidium vapor gas was used and an efficiency of 87% was achieved, which remains the best efficiency in the implementation of optical QM. At the same time, the spectral width of such memory was only about 100 kHz.

AFC protocol allows a significant increase in the working spectral width and, accordingly, an increase in the information capacity of the QM. The AFC protocol was experimentally implemented for different crystals with REIs with different optical transition wavelengths [105-107, 110]. In AFC protocol an efficiency of 62% was achieved [111], a storage time (T_{st}) of 1 hour [49], fidelity of > 99% [112, 113] (for efficiency of 11 and 7%), the storage of more than 1000 temporal light modes [114-116] and a large operating spectral width ({\mathrm{\Delta }}_0> 6 GHz) were demonstrated [115, 117, 118] but with efficiency of <10%.

We also draw the reader’s attention to recent reviews devoted to the issues on theoretical and experimental implementation of quantum repeater nodes in other physical systems [168], as well as an analysis of requirements of qunatum memory efficiency and storage time [169].

6 Conclusion

We have considered the basic photon echo QM protocols, analyzed their main properties and key problems in achieving high efficiency for broadband signal fields. Using the new obtained analytical results we analyzed effects of spectral dispersion on the efficiency of QM echo-protocols and nonlinear effects of the atomic interactions with intensive signal and control light pulses. The presented analytical results and performed analysis made it possible to find requirements for the atomic and light parameters in the efficient storage of signal fields. Basic properties and current problems of QM echo-protocols were discussed together with recent experimental results demonstrated significant progress in improving the performance of the studied echo-protocols. Concluding the analysis of basic echo protocols of QM, it is worth noting the need for further development of the approaches to their implementation. In this regard, it is of interest to use pre-created long-lived macroscopic coherence in the system of atoms [170], in which the possibility of suppressing quantum noise and dynamic programming of storage time appears.

Description of optical quantum memory on atomic ensembles requires taking into account a large number of parameters, including both the parameters of the macroscopic system and the microscopic parameters of atoms, without which it is difficult to expect to achieve high efficiency of the implemented protocols. At present, the fidelity in the implementation of quantum memory protocols remains not as high as necessary for the implementation of quantum computing, which is due to the complexity of describing all the factors involved in the dynamics of the functioning of quantum memory. The description of existing protocols is still carried out using model approaches to describing the dynamics of the behavior of atoms and light, which is in satisfactory agreement with existing experimental data within the framework of experimental accuracy. It should be expected that subsequent progress in improving experimental studies of optical quantum memory will lead to the need for a more detailed and more accurate description of the quantum dynamics of light and atoms interacting with it. These issues currently arise when it comes to the need for a more detailed description of the behavior of atoms experiencing strong interactions with neighboring atoms and the environment. To describe the quantum dynamics of such atomic systems, the use of a quantum computer will soon become an urgent task. It is also necessary to expect the receipt of experimental data that claim to be of higher accuracy, which will make it important to build more complex theories and the need for powerful computing resources for their use and verification.

Among the most important tasks of the near future, which it is important to solve on the way to improving the basic parameters of broadband QM on photon echo, we would like to draw the attention of researchers to the solution of the following tasks.

i) Ensuring a high degree of temporal reversibility of protocols and compensation for the influence of spectral dispersion to achieve high efficiency and fidelity of signal pulse retrieval.

ii) Development of experimental methods for the realization of laser \pi pulses in the excitation of optically dense atomic ensembles.

iii) Improvement of methods for the preparation of the initial state (high-precision depopulation of spin sublevels) of rare earth ions.

iv) Further development of dynamical-decoupling rf- and microwave pulse sequences providing high preserving of coherency and techniques of depopulation of spin sub-levels for high enough SNR. Now the best SNR is 7.4 for 20 ms storage time using AFC protocol [123].

At the same time, further improvement of the QM echo protocols should be expected using optical resonators [171, 172]. Cavity assisted QMs will significantly reduce the requirements for the optical depth of the resonant transition, involve off-resonant Raman interaction for direct mapping to long-lived states of REI [87, 88], as well as its use in various integrated circuits [19, 173, 174]. Along this path, there are still many tasks to be solved in the development of new QM resonator schemes [93, 175] for controlling the interaction of photons with coherent atomic ensembles. The development of broadband QM using multi-resonator systems is also of a large interest here [176-178].

When implementing optical quantum memory protocols in concentrated atomic ensembles, significant development of the theory is already important problem, which takes into account the influence of interatomic interactions, the resulting features of atomic relaxation, the manifestation of the local Lorentz field in the dynamics of atoms, the propagation of light fields in waveguide structures, etc. These questions arise in specific experiments, the detailed analysis of which will require further development of the theory, the application of which will require the development of numerical methods for analyzing the studied quantum memory protocols. We also draw the reader’s attention to recent reviews devoted to the issues of experimental implementation of optical QM [18, 20, 179] and to the reviews on first experimental results obtained in integrated QM using REI doped crystals [19, 174], where the influence of spectral dispersion and nonlinear interaction of light pulses with atomic systems also require special study [93].

References

[1] N. Sangouard, C. Simon, H. de Riedmatten, and N. Gisin, Quantum repeaters based on atomic ensembles and linear optics, Rev. Mod. Phys. 83(1), 33 (2011)

[2] P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, Linear optical quantum computing with photonic qubits, Rev. Mod. Phys. 79(1), 135 (2007)

[3] R. N. Stevenson, M. R. Hush, A. R. R. Carvalho, S. E. Beavan, M. J. Sellars, and J. J. Hope, Single photon production by rephased amplified spontaneous emission, New J. Phys. 16(3), 033042 (2014)

[4] A. D. Manukhova, K. S. Tikhonov, T. Y. Golubeva, and Y. M. Golubev, Noiseless signal shaping and cluster-state generation with a quantum memory cell, Phys. Rev. A 96(2), 023851 (2017)

[5] D. L. Chen, Z. Q. Zhou, C. F. Li, and G. C. Guo, Nonclassical photon-pair source based on noiseless photon echo, Phys. Rev. A 107(4), 042619 (2023)

[6] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Proposed experiment to test local hidden-variable theories, Phys. Rev. Lett. 23(15), 880 (1969)

[7] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Bell nonlocality, Rev. Mod. Phys. 86(2), 419 (2014)

[8] J. M. Mol, L. Esguerra, M. Meister, D. E. Bruschi, A. W. Schell, J. Wolters, and L. Worner, Quantum memories for fundamental science in space, Quantum Sci. Technol. 8(2), 024006 (2023)

[9] M. Hosseini, B. M. Sparkes, G. T. Campbell, P. K. Lam, and B. C. Buchler, Storage and manipulation of light using a Raman gradient-echo process, J. Phys. At. Mol. Opt. Phys. 45(12), 124004 (2012)

[10] E. S. Moiseev and S. A. Moiseev, Time-bin quantum RAM, J. Mod. Opt. 63(20), 2081 (2016)

[11] K. C. Chen, W. Dai, C. Errando-Herranz, S. Lloyd, and D. Englund, Scalable and high-fidelity quantum random access memory in spin-photon networks, PRX Quantum 2, 030319 (2021)

[12] A. I. Lvovsky, B. C. Sanders, and W. Tittel, Optical quantum memory, Nat. Photonics 3(12), 706 (2009)

[13] W. Tittel, M. Afzelius, T. Chaneliére, R. Cone, S. Kröll, S. Moiseev, and M. Sellars, Photon‐echo quantum memory in solid state systems, Laser Photonics Rev. 4(2), 244 (2010)

[14] F. Bussières, N. Sangouard, M. Afzelius, H. de Riedmatten, C. Simon, and W. Tittel, Prospective applications of optical quantum memories, J. Mod. Opt. 60(18), 1519 (2013)

[15] K. Heshami, D. G. England, P. C. Humphreys, P. J. Bustard, V. M. Acosta, J. Nunn, and B. J. Sussman, Quantum memories: Emerging applications and recent advances, J. Mod. Opt. 63(20), 2005 (2016)

[16] T. Chanelière,G. Hétet,N. Sangouard, Chapter two-quantum optical memory protocols in atomic ensembles Advances in Atomic, Molecular, and Optical Physics, Vol. 67, Eds.: E. Arimondo, L. F. DiMauro, and S. F. Yelin, Academic Press, 2018, pp 77–150

[17] Y. L. Hua, Z. Q. Zhou, C. F. Li, and G. C. Guo, Quantum light storage in rare-earth-ion-doped solids, Chin. Phys. B 27(2), 020303 (2018)

[18] M. Guo, S. Liu, W. Sun, M. Ren, F. Wang, and M. Zhong, Rare-earth quantum memories: The experimental status quo, Front. Phys. 18(2), 21303 (2023)

[19] Z. Q. Zhou, C. Liu, C. F. Li, G. C. Guo, D. Oblak, M. Lei, A. Faraon, M. Mazzera, and H. de Riedmatten, Photonic integrated quantum memory in rare-earth doped solids, Laser Photonics Rev. 17, 1 (2023)

[20] Y. Lei, F. Kimiaee Asadi, T. Zhong, A. Kuzmich, C. Simon, and M. Hosseini, Quantum optical memory for entanglement distribution, Optica 10(11), 1511 (2023)

[21] E. L. Hahn, Spin echoes, Phys. Rev. 80(4), 580 (1950)

[22] U. Kopvillem and V. Nagibarov, Luminous echo of paramagnetic crystals, Fiz. Metal. i Metalloved. 15, 313 (1963)

[23] N. A. Kurnit, I. D. Abella, and S. R. Hartmann, Observation of a photon echo, Phys. Rev. Lett. 13(19), 567 (1964)

[24] T. Gorin, T. Prosen, T. H. Seligman, and M. Znidaric, Dynamics of Loschmidt echoes and fidelity decay, Phys. Rep. 435(2−5), 33 (2006)

[25] A. Wisniacki, Loschmidt echo, Scholarpedia J. 7(8), 11687 (2012)

[26] S. A. Moiseev and S. Kröll, Complete reconstruction of the quantum state of a single-photon wave packet absorbed by a Doppler-broadened transition, Phys. Rev. Lett. 87(17), 173601 (2001)

[27] S. A. Moiseev and M. I. Noskov, The possibilities of the quantum memory realization for short pulses of light in the photon echo technique, Laser Phys. Lett. 1(6), 303 (2004)

[28] S. A. Moiseev, V. F. Tarasov, and B. S. Ham, Quantum memory photon echo-like techniques in solids, J. Opt. B 5(4), S497 (2003)

[29] S. A. Moiseev and B. S. Ham, Photon-echo quantum memory with efficient multipulse readings, Phys. Rev. A 70, 063809 (2004)

[30] B. Kraus, W. Tittel, N. Gisin, M. Nilsson, S. Kröll, and J. I. Cirac, Quantum memory for nonstationary light fields based on controlled reversible inhomogeneous broadening, Phys. Rev. A 73(2), 020302 (2006)

[31] A. L. Alexander, J. J. Longdell, M. J. Sellars, and N. B. Manson, Photon echoes produced by switching electric fields, Phys. Rev. Lett. 96(4), 043602 (2006)

[32] B. Lauritzen, J. Minář, H. de Riedmatten, M. Afzelius, N. Sangouard, C. Simon, and N. Gisin, Telecommunication-wavelength solid-state memory at the single photon level, Phys. Rev. Lett. 104(8), 080502 (2010)

[33] N. Sangouard, C. Simon, M. Afzelius, and N. Gisin, Analysis of a quantum memory for photons based on controlled reversible inhomogeneous broadening, Phys. Rev. A 75(3), 032327 (2007)

[34] A. V. Gorshkov, A. André, M. D. Lukin, and A. S. Sørensen, Photon storage in Λ-type optically dense atomic media. I. Cavity model, Phys. Rev. A 76(3), 033804 (2007)

[35] S. A. Moiseev and W. Tittel, Optical quantum memory with generalized time-reversible atom–light interaction, New J. Phys. 13(6), 063035 (2011)

[36] E. S. Moiseev and S. A. Moiseev, Scalable time reversal of Raman echo quantum memory and quantum waveform conversion of light pulse, New J. Phys. 15(10), 105005 (2013)

[37] M. Nilsson, L. Rippe, S. Kröll, R. Klieber, and D. Suter, Hole-burning techniques for isolation and study of individual hyperfine transitions in inhomogeneously broadened solids demonstrated in Pr³⁺:Y₂SiO₅, Phys. Rev. B 70(21), 214116 (2004)

[38] M. Hosseini, B. Sparkes, G. Campbell, P. Lam, and B. Buchler, High efficiency coherent optical memory with warm rubidium vapour, Nat. Commun. 2(1), 174 (2011)

[39] S. A. Moiseev, M. M. Minnegaliev, E. S. Moiseev, K. I. Gerasimov, A. V. Pavlov, T. A. Rupasov, N. N. Skryabin, A. A. Kalinkin, and S. P. Kulik, Pulse-area theorem in a single-mode waveguide and its application to photon echo and optical memory in Tm³⁺: Y₃Al₅O₁₂, Phys. Rev. A 107(4), 043708 (2023)

[40] B. Merkel, P. Cova Fariña, and A. Reiserer, Dynamical decoupling of spin ensembles with strong anisotropic interactions, Phys. Rev. Lett. 127(3), 030501 (2021)

[41] S. A. Moiseev and V. A. Skrebnev, Symmetric-cycle pulse sequence for dynamical decoupling of local fields and dipole–dipole interactions, J. Phys. At. Mol. Opt. Phys. 48(13), 135503 (2015)

[42] S. A. Moiseev and V. A. Skrebnev, Short-cycle pulse sequence for dynamical decoupling of local fields and dipole–dipole interactions, Phys. Rev. A 91(2), 022329 (2015)

[43] A. M. Waeber, G. Gillard, G. Ragunathan, M. Hopkinson, P. Spencer, D. A. Ritchie, M. S. Skolnick, and E. A. Chekhovich, Pulse control protocols for preserving coherence in dipolar-coupled nuclear spin baths, Nat. Commun. 10(1), 3157 (2019)

[44] M. M. Minnegaliev, R. V. Urmancheev, V. A. Skrebnev, and S. A. Moiseev, Investigation of a sequence of dynamical decoupling pulses for dipole-coupled spin systems with inhomogeneous broadening, Opt. Spectrosc. 126(1), 1 (2019)

[45] G. Heinze, C. Hubrich, and T. Halfmann, Stopped light and image storage by electromagnetically induced transparency up to the regime of one minute, Phys. Rev. Lett. 111(3), 033601 (2013)

[46] M. Zhong, M. P. Hedges, R. L. Ahlefeldt, J. G. Bartholomew, S. E. Beavan, S. M. Wittig, J. J. Longdell, and M. J. Sellars, Optically addressable nuclear spins in a solid with a six-hour coherence time, Nature 517(7533), 177 (2015)

[47] M. Rančić, M. P. Hedges, R. L. Ahlefeldt, and M. J. Sellars, Coherence time of over a second in a telecom-compatible quantum memory storage material, Nat. Phys. 14, 50 (2017)

[48] M. Hain, M. Stabel, and T. Halfmann, Few-photon storage on a second timescale by electromagnetically induced transparency in a doped solid, New J. Phys. 24(2), 023012 (2022)

[49] Y. Ma, Y. Z. Ma, Z. Q. Zhou, C. F. Li, and G. C. Guo, One-hour coherent optical storage in an atomic frequency comb memory, Nat. Commun. 12(1), 2381 (2021)

[50] N. Sangouard, C. Simon, M. Afzelius, and N. Gisin, Analysis of a quantum memory for photons based on controlled reversible inhomogeneous broadening, Phys. Rev. A 75(3), 032327 (2007)

[51] S. A. Moiseev, Photon-echo-based quantum memory of arbitrary light field states, J. Phys. At. Mol. Opt. Phys. 40(19), 3877 (2007)

[52] B. S. Ham, A wavelength-convertible quantum memory: Controlled echo, Sci. Rep. 8(1), 10675 (2018)

[53] A. V. Gorshkov, A. André, M. Fleischhauer, A. S. Sørensen, and M. D. Lukin, Universal approach to optimal photon storage in atomic media, Phys. Rev. Lett. 98(12), 123601 (2007)

[54] S. H. Autler and C. H. Townes, Stark effect in rapidly varying fields, Phys. Rev. 100(2), 703 (1955)

[55] E. Saglamyurek, T. Hrushevskyi, A. Rastogi, K. Heshami, and L. J. LeBlanc, Coherent storage and manipulation of broadband photons via dynamically controlled Autler–Townes splitting, Nat. Photonics 12(12), 774 (2018)

[56] P. Vernaz-Gris, A. D. Tranter, J. L. Everett, A. C. Leung, K. V. Paul, G. T. Campbell, P. K. Lam, and B. C. Buchler, High-performance Raman memory with spatio-temporal reversal, Opt. Express 26(10), 12424 (2018)

[57] J. Dajczgewand, J. L. Le Gouët, A. Louchet-Chauvet, and T. Chanelière, Large efficiency at telecom wavelength for optical quantum memories, Opt. Lett. 39(9), 2711 (2014)

[58] M. M. Minnegaliev, K. I. Gerasimov, T. N. Sabirov, R. V. Urmancheev, and S. A. Moiseev, Implementation of an optical quantum memory protocol in the ¹⁶⁷Er³⁺: Y₂SiO₅ crystal, JETP Lett. 115(12), 720 (2022)

[59] S. A. Moiseev, Some general nonlinear properties of photon-echo radiation in optically dense media, Opt. Spectrosc. 62, 180 (1987) (English translation of Optika i Spektroskopiya)

[60] S. A. Moiseev, Quantum memory for intense light fields based on the photon echo, Bull. Russ. Acad. Sci. Phys. 68, 1260 (2004)

[61] R. Urmancheev, K. Gerasimov, M. Minnegaliev, T. Chanelière, A. Louchet-Chauvet, and S. Moiseev, Two-pulse photon echo area theorem in an optically dense medium, Opt. Express 27(20), 28983 (2019)

[62] S. A. Moiseev, M. Sabooni, and R. V. Urmancheev, Photon echoes in optically dense media, Phys. Rev. Res. 2(1), 012026 (2020)

[63] S. L. McCall and E. L. Hahn, Self-induced transparency, Phys. Rev. 183(2), 457 (1969)

[64] M. J. Ablowitz, D. J. Kaup, and A. C. Newell, Coherent pulse propagation, a dispersive, irreversible phenomenon, J. Math. Phys. 15(11), 1852 (1974)

[65] A. Maimistov, A. Basharov, S. Elyutin, and Y. Sklyarov, Present state of self-induced transparency theory, Phys. Rep. 191(1), 1 (1990)

[66] S. A. Moiseev and R. V. Urmancheev, Photon/spin echo in a Fabry–Perot cavity, Opt. Lett. 47(15), 3812 (2022)

[67] M. M. Minnegaliev, K. I. Gerasimov, R. V. Urmancheev, A. M. Zheltikov, and S. A. Moiseev, Linear Stark effect in Y₃Al₅O₁₂: Tm³⁺ crystal and its application in the addressable quantum memory protocol, Phys. Rev. B 103(17), 174110 (2021)

[68] N. T. Vinh, D. V. Tsarev, and A. P. Alodjants, Coupled solitons for quantum communication and metrology in the presence of particle dissipation, J. Russ. Laser Res. 42(5), 523 (2021)

[69] V. I. Rupasov, Contribution to the Dicke superradiance theory. Exact solution of the quasione-dimensional quantum model, Sov. Phys. JETP 56, 989 (1982) [Zh. Eksp. Teor. Fiz. 83, 1711 (1982)]

[70] M. P. Hedges, J. J. Longdell, Y. Li, and M. J. Sellars, Efficient quantum memory for light, Nature 465(7301), 1052 (2010)

[71] M. Hosseini, B. M. Sparkes, G. Hétet, J. J. Longdell, P. K. Lam, and B. C. Buchler, Coherent optical pulse sequencer for quantum applications, Nature 461(7261), 241 (2009)

[72] Y. W. Cho, G. T. Campbell, J. L. Everett, J. Bernu, D. B. Higginbottom, M. T. Cao, J. Geng, N. P. Robins, P. K. Lam, and B. C. Buchler, Highly efficient optical quantum memory with long coherence time in cold atoms, Optica 3(1), 100 (2016)

[73] I. Saidasheva,N. Arslanov,S. Moiseev, Modeling of photon echo with controlled external field gradient: possibility of high efficient quantum memory, in: Proceedings of the VI-th International Congress Basic Problems of Optics, Ed.: Prof. S. A. Kozlov, St. Petersburg State Univ. of Information Technologies, 16−20 October, 2006, St.-Petersburg, Russia, 2006, p. 127

[74] S. A. Moiseev and N. M. Arslanov, Efficiency and fidelity of photon-echo quantum memory in an atomic system with longitudinal inhomogeneous broadening, Phys. Rev. A 78(2), 023803 (2008)

[75] M. D. Lukin, Trapping and manipulating photon states in atomic ensembles, Rev. Mod. Phys. 75(2), 457 (2003)

[76] M. D. Lukin and A. Imamŏglu, Nonlinear optics and quantum entanglement of ultraslow single photons, Phys. Rev. Lett. 84(7), 1419 (2000)

[77] Z. B. Wang, K. P. Marzlin, and B. C. Sanders, Large cross-phase modulation between slow copropagating weak pulses in ⁸⁷Rb, Phys. Rev. Lett. 97(6), 063901 (2006)

[78] K. P. Marzlin, Z. B. Wang, S. A. Moiseev, and B. C. Sanders, Uniform cross-phase modulation for nonclassical radiation pulses, J. Opt. Soc. Am. B 27(6), A36 (2010)

[79] B. He and A. Scherer, Continuous-mode effects and photon‒photon phase gate performance, Phys. Rev. A 85(3), 033814 (2012)

[80] P. Bienias and H. P. Büchler, Two photon conditional phase gate based on Rydberg slow light polaritons, J. Phys. At. Mol. Opt. Phys. 53(5), 054003 (2020)

[81] A. C. Leung, K. S. I. Melody, A. D. Tranter, K. V. Paul, G. T. Campbell, P. K. Lam, and B. C. Buchler, Observation of cross phase modulation in cold atom gradient echo memory, New J. Phys. 24(9), 093011 (2022)

[82] A. Feizpour, M. Hallaji, G. Dmochowski, and A. M. Steinberg, Observation of the nonlinear phase shift due to single post-selected photons, Nat. Phys. 11, 905 (2015)

[83] J. H. Shapiro, Single-photon Kerr nonlinearities do not help quantum computation, Phys. Rev. A 73(6), 062305 (2006)

[84] J. Gea-Banacloche, Impossibility of large phase shifts via the giant Kerr effect with single-photon wave packets, Phys. Rev. A 81(4), 043823 (2010)

[85] S. A. Moiseev and W. Tittel, Temporal compression of quantum-information-carrying photons using a photon-echo quantum memory approach, Phys. Rev. A 82(1), 012309 (2010)

[86] E. S. Moiseev, A. Tashchilina, S. A. Moiseev, and A. I. Lvovsky, Darkness of two-mode squeezed light in Λ-type atomic system, New J. Phys. 22(1), 013014 (2020)

[87] S. A. Moiseev, Off-resonant Raman-echo quantum memory for inhomogeneously broadened atoms in a cavity, Phys. Rev. A 88(1), 012304 (2013)

[88] A. Kalachev and O. Kocharovskaya, Multimode cavity-assisted quantum storage via continuous phase-matching control, Phys. Rev. A 88(3), 033846 (2013)

[89] P. Goldner,A. Ferrier,O. Guillot-Nӧel, Rare Earth-Doped Crystals for Quantum Information Processing Handbook on the Physics and Chemistry of Rare Earths, Vol. 46, Elsevier B. V., 2015, pp 1–78

[90] M. N. Popova, S. A. Klimin, S. A. Moiseev, K. I. Gerasimov, M. M. Minnegaliev, E. I. Baibekov, G. S. Shakurov, M. Bettinelli, and M. C. Chou, Crystal field and hyperfine structure of ¹⁶⁷Er³⁺ in YPO₄:Er single crystals: High-resolution optical and EPR spectroscopy, Phys. Rev. B 99(23), 235151 (2019)

[91] K. I. Gerasimov, T. N. Sabirov, S.A. Moiseev, E. I. Baibekov, M. Bettinelli, M. Chou, Y. C. Yen, and M. N. Popova, Spectroscopy and photon echo at the Er³⁺ transition with a small inhomogeneous broadening and telecommunication wavelength in a YPO₄ crystal, Opt. Spectrosc. 131(5), 607 (2023)

[92] K. Gerasimov, E. Baibekov, M. Minnegaliev, G. Shakurov, R. Zaripov, S. Moiseev, A. Lebedev, and B. Malkin, Magneto-optical and high-frequency electron paramagnetic resonance spectroscopy of Er³⁺ ions in CaMoO₄ single crystal, J. Lumin. 270, 120564 (2024)

[93] E. S. Moiseev, A. Tashchilina, S. A. Moiseev, and B. C. Sanders, Broadband quantum memory in a cavity via zero spectral dispersion, New J. Phys. 23(6), 063071 (2021)

[94] M. S. Tame, K. R. McEnery, S. K. Özdemir, J. Lee, S. M. Maier, and S. Kim, Quantum plasmonics, Nat. Phys. 9(6), 329 (2013)

[95] S. A. Moiseev,E. S. Moiseev, 2010 Multimode nano scale Raman echo quantum memory, Vol. 26 of NATO Science for Peace and Security Series - D: Information and Communication Security, Eds.: J. Kowalik, R. Horodecki, and S. Y. Kilin, Quantum Cryptography and Computing: Theory and Implementation, IOS Press BV, 2010

[96] L. Novotny,B. Hecht, Nano-Optics, Cambridge University Press, 2012

[97] B. Y. Dubetskii and V. P. Chebotaev, Echoes in classical and quantum ensembles with determinate frequencies, JETP Lett. 41, 328 (1985)

[98] B. Y. Dubetskii and V. P. Chebotaev, Imaginary echo in a gas in a Doppler expanded transition, Bull. Acad. Sci. USSR Phys. Ser. 50, 70 (1986)

[99] H. de Riedmatten,M. Afzelius,M. U. Staudt,C. Simon,N. Gisin, A solid-state light−matter interface at the single-photon level, Nature 456(7223), 773 (2008)

[100] M. Afzelius, C. Simon, H. De Riedmatten, and N. Gisin, Multimode quantum memory based on atomic frequency combs, Phys. Rev. A 79(5), 052329 (2009)

[101] S. A. Moiseev and J. L. Le Gouët, Rephasing processes and quantum memory for light: Reversibility issues and how to fix them, J. Phys. At. Mol. Opt. Phys. 45(12), 124003 (2012)

[102] N. Arslanov and S. Moiseev, Optimal periodic frequency combs for high-efficiency optical quantum memory based on rare-earth ion crystals, Quantum Electron. 47(9), 783 (2017)

[103] N. M. Arslanov and S. A. Moiseev, Maps of broadband quantum memory based on an atomic frequency comb, Optics and Spectroscopy 126(1), 29 (2019)

[104] N. Arslanov,S. Moiseev, Quantum storage of spectrally multimode fields in the AFC protocol, 2024 (in progress)

[105] A. Holzäpfel, J. Etesse, K. T. Kaczmarek, A. Tiranov, N. Gisin, and M. Afzelius, Optical storage for 0.53 seconds in a solid-state atomic frequency comb memory using dynamical decoupling, New J. Phys. 22, 063009 (2020)

[106] J. S. Stuart, M. Hedges, R. Ahlefeldt, and M. Sellars, Initialization protocol for efficient quantum memories using resolved hyperfine structure, Phys. Rev. Res. 3(3), L032054 (2021)

[107] E. Z. Cruzeiro, A. Tiranov, J. Lavoie, A. Ferrier, P. Goldner, N. Gisin, and M. Afzelius, Efficient optical pumping using hyperfine levels in ¹⁴⁵Nd³⁺:Y₂SiO₅ and its application to optical storage, New J. Phys. 20(5), 053013 (2018)

[108] R. A. Akhmedzhanov, L. A. Gushchin, A. A. Kalachev, S. L. Korableva, D. A. Sobgayda, and I. V. Zelensky, Atomic frequency comb memory in an isotopically pure ¹⁴³Nd³⁺:Y⁷LiF₄ crystal, Laser Phys. Lett. 13, 015202 (2016)

[109] D. Rieländer, K. Kutluer, P. M. Ledingham, M. Gündogan, J. Fekete, M. Mazzera, and H. de Riedmatten, Quantum storage of Heralded single photons in a praseodymium-doped crystal, Phys. Rev. Lett. 112(4), 040504 (2014)

[110] R. A. Akhmedzhanov, L. A. Gushchin, A. A. Kalachev, N. A. Nizov, V. A. Nizov, D. A. Sobgayda, and I. V. Zelensky, Memory for polarization state of light based on atomic frequency comb in a ¹⁵³Eu: Y₂SiO₅ crystal, Laser Phys. Lett. 20(1), 015204 (2023)

[111] S. Duranti,S. Wengerowsky,L. Feldmann,A. Seri,B. Casabone,H. de Riedmatten, Efficient cavity-assisted storage of photonic qubits in a solid-state quantum memory, Opt. Express 32(15), 26884 (2024)

[112] T. X. Zhu, C. Liu, M. Jin, M. X. Su, Y. P. Liu, W. J. Li, Y. Ye, Z. Q. Zhou, C. F. Li, and G. C. Guo, On-demand integrated quantum memory for polarization qubits, Phys. Rev. Lett. 128(18), 180501 (2022)

[113] Z. Q. Zhou, W. B. Lin, M. Yang, C. F. Li, and G. C. Guo, Realization of reliable solid-state quantum memory for photonic polarization qubit, Phys. Rev. Lett. 108(19), 190505 (2012)

[114] M. Businger, L. Nicolas, T. S. Mejia, A. Ferrier, P. Goldner, and M. Afzelius, Non-classical correlations over 1250 modes between telecom photons and 979-nm photons stored in ¹⁷¹Yb³⁺:Y₂SiO₅, Nat. Commun. 13(1), 6438 (2022)

[115] S. H. Wei, B. Jing, X. Y. Zhang, J. Y. Liao, H. Li, L. X. You, Z hen Wang, Y. Wang, G. W. Deng, H. Z. Song, D. Oblak, G. C. Guo, and Q. Zhou, Quantum storage of 1650 modes of single photons at telecom wavelength, npj Quantum Inf. 10, 19 (2024)

[116] M. Bonarota, J. L. Le Gouët, and T. Chanelière, Highly multimode storage in a crystal, New J. Phys. 13(1), 013013 (2011)

[117] E. Saglamyurek, N. Sinclair, J. Jin, J. A. Slater, D. Oblak, F. Bussières, M. George, R. Ricken, W. Sohler, and W. Tittel, Broadband waveguide quantum memory for entangled photons, Nature 469(7331), 512 (2011)

[118] M. li G. Puigibert,M. F. Askarani,J. H. Davidson,V. B. Verma,M. D. Shaw,S. W. Nam, T. Lutz,G. C. Amaral,D. Oblak,W. Tittel, Entanglement and nonlocality between disparate solid-state quantum memories mediated by photons, Phys. Rev. Res. 2(1), 013039

[119] S. P. Horvath, M. K. Alqedra, A. Kinos, A. Walther, J. M. Dahlström, S. Kröll, and L. Rippe, Noise-free on-demand atomic frequency comb quantum memory, Phys. Rev. Res. 3(2), 023099 (2021)

[120] I. Craiciu, M. Lei, J. Rochman, J. G. Bartholomew, and A. Faraon, Multifunctional on-chip storage at telecommunication wavelength for quantum networks, Optica 8(1), 114 (2021)

[121] G. Corrielli, A. Seri, M. Mazzera, R. Osellame, and H. de Riedmatten, Integrated optical memory based on laser-written waveguides, Phys. Rev. Appl. 5(5), 054013 (2016)

[122] J. V. Rakonjac, D. Lago-Rivera, A. Seri, M. Mazzera, S. Grandi, and H. de Riedmatten, Entanglement between a telecom photon and an on-demand multimode solid-state quantum memory, Phys. Rev. Lett. 127(21), 210502 (2021)

[123] A. Ortu, A. Holzäpfel, J. Etesse, and M. Afzelius, Storage of photonic time-bin qubits for up to 20 ms in a rare-earth doped crystal, npj Quantum Inf. 8, 29 (2022)

[124] M. K. Alqedra, S. P. Horvath, A. Kinos, A. Walther, S. Kröll, and L. Rippe, Stark control of solid-state quantum memory with spin-wave storage, Phys. Rev. A (Coll. Park) 109(1), 012607 (2024)

[125] M. Businger, A. Tiranov, K. T. Kaczmarek, S. Welinski, Z. Zhang, A. Ferrier, P. Goldner, and M. Afzelius, Optical spin-wave storage in a solid-state hybridized electron-nuclear spin ensemble, Phys. Rev. Lett. 124(5), 053606 (2020)

[126] M. Sabooni, Q. Li, S. Kröll, and L. Rippe, Efficient quantum memory using a weakly absorbing sample, Phys. Rev. Lett. 110(13), 133604 (2013)

[127] P. Jobez,I. Usmani,N. Timoney,C. Laplane,N. Gisin, M. Afzelius, Cavity-enhanced storage in an optical spin-wave memory, New J. Phys. 16, 083005 (2014)

[128] D. C. Liu, P. Y. Li, T. X. Zhu, L. Zheng, J. Y. Huang, Z. Q. Zhou, C. F. Li, and G. C. Guo, On-demand storage of photonic qubits at telecom wavelengths, Phys. Rev. Lett. 129(21), 210501 (2022)

[129] J. V. Rakonjac, G. Corrielli, D. Lago-Rivera, A. Seri, M. Mazzera, S. Grandi, R. Osellame, and H. de Riedmatten, Storage and analysis of light−matter entanglement in a fiber-integrated system, Sci. Adv. 8(27), eabn3919 (2022)

[130] E. Saglamyurek, J. Jin, V. B. Verma, M. D. Shaw, F. Marsili, S. W. Nam, D. Oblak, and W. Tittel, Quantum storage of entangled telecom-wavelength photons in an erbium-doped optical fibre, Nat. Photonics 9(2), 83 (2015)

[131] I. Craiciu, M. Lei, J. Rochman, J. M. Kindem, J. G. Bartholomew, E. Miyazono, T. Zhong, N. Sinclair, and A. Faraon, Nanophotonic quantum storage at telecommunication wavelength, Phys. Rev. Appl. 12(2), 024062 (2019)

[132] D. L. McAuslan, P. M. Ledingham, W. R. Naylor, S. E. Beavan, M. P. Hedges, M. J. Sellars, and J. J. Longdell, Photon-echo quantum memories in inhomogeneously broadened two-level atoms, Phys. Rev. A 84(2), 022309 (2011)

[133] V. Damon,M. Bonarota,A. Louchet-Chauvet,T. Chanelière,J. L. Le Gouët, Revival of silenced echo and quantum memory for light, New J. Phys. 13, 093031 (2011)

[134] S. A. Moiseev, Photon-echo quantum memory with complete use of natural inhomogeneous broadening, Phys. Rev. A 83(1), 012307 (2011)

[135] S. E. Beavan, P. M. Ledingham, J. J. Longdell, and M. J. Sellars, Photon echo without a free induction decay in a double-Λ system, Opt. Lett. 36(7), 1272 (2011)

[136] A. Arcangeli, A. Ferrier, and P. Goldner, Stark echo modulation for quantum memories, Phys. Rev. A 93(6), 062303 (2016)

[137] B. S. Ham, A controlled ac Stark echo for quantum memories, Sci. Rep. 7(1), 7655 (2017)

[138] K. I. Gerasimov, M. M. Minnegaliev, S. A. Moiseev, R. V. Urmancheev, T. Chanelière, and A. Louchet-Chauvet, Quantum memory in an orthogonal geometry of silenced echo retrieval, Opt. Spectrosc. 123(2), 211 (2017)

[139] J. Liu, J. Liu, J. Cui, L. Wang, and G. Zhang, Light pulse storage in Pr: YSO crystal based on the revival of silenced echo protocol, Opt. Express 32(5), 6986 (2024)

[140] C. Liu, Z. Q. Zhou, T. X. Zhu, L. Zheng, M. Jin, X iao Liu, P. Y. Li, J. Y. Huang, Y u Ma, T. Tu, T. S. Yang, C. F. Li, and G. C. Guo, Reliable coherent optical memory based on a laser-written waveguide, Optica 7, 192 (2020)

[141] M. M. Minnegaliev, K. I. Gerasimov, R. V. Urmancheev, S. A. Moiseev, T. Chanelière, and A. Louchet-Chauvet, Realization of the revival of silenced echo (ROSE) quantum memory scheme in orthogonal geometry, AIP Conf. Proc. 1936, 020012 (2018)

[142] M. M. Minnegaliev, K. I. Gerasimov, and S. A. Moiseev, Implementation of a quantum memory protocol based on the revival of silenced echo in orthogonal geometry at the telecommunication wavelength, JETP Lett. 117(11), 865 (2023)

[143] M. Bonarota, J. Dajczgewand, A. Louchet-Chauvet, J. L. Le Gouët, and T. Chanelière, Photon echo with a few photons in two-level atoms, Laser Phys. 24(9), 094003 (2014)

[144] Y. Z. Ma, M. Jin, D. L. Chen, Z. Q. Zhou, C. F. Li, and G. C. Guo, Elimination of noise in optically rephased photon echoes, Nat. Commun. 12(1), 4378 (2021)

[145] E. Hahn, N. Shiren, and S. McCall, Application of the area theorem to phonon echoes, Phys. Lett. A 37(3), 265 (1971)

[146] D. J. Kaup, Coherent pulse propagation: A comparison of the complete solution with the McCall−Hahn theory and others, Phys. Rev. A 16(2), 704 (1977)

[147] M. S. Silver, R. I. Joseph, and D. I. Hoult, Selective spin inversion in nuclear magnetic resonance and coherent optics through an exact solution of the Bloch−Riccati equation, Phys. Rev. A 31(4), 2753 (1985)

[148] I. Roos and K. Mølmer, Quantum computing with an inhomogeneously broadened ensemble of ions: Suppression of errors from detuning variations by specially adapted pulses and coherent population trapping, Phys. Rev. A 69(2), 022321 (2004)

[149] M. Tian, T. Chang, K. D. Merkel, and W. Randall, Reconfiguration of spectral absorption features using a frequency-chirped laser pulse, Appl. Opt. 50(36), 6548 (2011)

[150] H. J. Briegel, W. Dür, J. I. Cirac, and P. Zoller, Quantum repeaters: The role of imperfect local operations in quantum communication, Phys. Rev. Lett. 81(26), 5932 (1998)

[151] C. Simon, H. de Riedmatten, M. Afzelius, N. Sangouard, H. Zbinden, and N. Gisin, Quantum repeaters with photon pair sources and multimode memories, Phys. Rev. Lett. 98(19), 190503 (2007)

[152] N. Sangouard, C. Simon, B. Zhao, Y. A. Chen, H. de Riedmatten, J. W. Pan, and N. Gisin, Robust and efficient quantum repeaters with atomic ensembles and linear optics, Phys. Rev. A 77(6), 062301 (2008)

[153] J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, Multiphoton entanglement and interferometry, Rev. Mod. Phys. 84(2), 777 (2012)

[154] L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, Long-distance quantum communication with atomic ensembles and linear optics, Nature 414(6862), 413 (2001)

[155] A. Anwar, C. Perumangatt, F. Steinlechner, T. Jennewein, and A. Ling, Entangled photon-pair sources based on three-wave mixing in bulk crystals, Rev. Sci. Instrum. 92(4), 041101 (2021)

[156] F. Bussières, C. Clausen, A. Tiranov, B. Korzh, V. B. Verma, S. W. Nam, F. Marsili, A. Ferrier, P. Goldner, H. Herrmann, C. Silberhorn, W. Sohler, M. Afzelius, and N. Gisin, Quantum teleportation from a telecom-wavelength photon to a solid-state quantum memory, Nat. Photonics 8(10), 775 (2014)

[157] D. Lago-Rivera,J. V. Rakonjac,S. Grandi,H. de Riedmatten, Long distance multiplexed quantum teleportation from a telecom photon to a solid-state qubit, Nat. Commun., 14, 1889 (2023)

[158] D. Lago-Rivera, S. Grandi, J. V. Rakonjac, A. Seri, and H. de Riedmatten, Telecom-heralded entanglement between multimode solid-state quantum memories, Nature 594(7861), 37 (2021)

[159] S. Duranti, S. Wengerowsky, L. Feldmann, A. Seri, B. Casabone, and H. de Riedmatten, Efficient cavity-assisted storage of photonic qubits in a solid-state quantum memory, Opt. Express 32(15), 26884 (2024)

[160] X. Liu, J. Hu, Z. F. Li, X. Li, P. Y. Li, P. J. Liang, Z. Q. Zhou, C. F. Li, and G. C. Guo, Heralded entanglement distribution between two absorptive quantum memories, Nature 594(7861), 41 (2021)

[161] C. X. Huang, X. M. Hu, Y. Guo, C. Zhang, B. H. Liu, Y. F. Huang, C. F. Li, G. C. Guo, N. Gisin, C. Branciard, and A. Tavakoli, Entanglement swapping and quantum correlations via symmetric joint measurements, Phys. Rev. Lett. 129(3), 030502 (2022)

[162] H. Zeng, M. M. Du, W. Zhong, L. Zhou, and Y. B. Sheng, High-capacity device-independent quantum secure direct communication based on hyper-encoding, Fundamental Research 4(4), 851 (2024)

[163] A. Tiranov, J. Lavoie, A. Ferrier, P. Goldner, V. B. Verma, S. W. Nam, R. P. Mirin, A. E. Lita, F. Marsili, H. Herrmann, C. Silberhorn, N. Gisin, M. Afzelius, and F. Bussières, Storage of hyperentanglement in a solid-state quantum memory, Optica 2(4), 279 (2015)

[164] M. H. Jiang,W. Xue,Q. He, Y. Y. An,X. Zheng,W. J. Xu,Y. B. Xie,Y. Lu, S. Zhu,X. S. Ma, Quantum storage of entangled photons at telecom wavelengths in a crystal, Nat. Commun. 14, 6995 (2023)

[165] E. Saglamyurek,M. Grimau Puigibert,Q. Zhou,L. Giner,F. Marsili,V. B. Verma, S. Woo Nam,L. Oesterling,D. Nippa,D. Oblak,W. Tittel, A multiplexed light−matter interface for fibre-based quantum networks, Nat. Commun. 7(1), 11202 (2016)

[166] X. Zhang,B. Zhang,S. Wei,H. Li,J. Liao, C. Li,G. Deng,Y. Wang,H. Song,L. You, B. Jing,F. Chen,G. Guo,Q. Zhou, Telecom-band integrated multimode photonic quantum memory, Sci. Adv. 9(28), adf4587 (2023)

[167] B. M. Sparkes, M. Hosseini, C. Cairns, D. Higginbottom, G. T. Campbell, P. K. Lam, and B. C. Buchler, Precision spectral manipulation: A demonstration using a coherent optical memory, Phys. Rev. X 2, 021011 (2012)

[168] S. H. Wei,B. Jing,X. Y. Zhang,J. Y. Liao,C. Z. Yuan, B. Y. Fan,C. Lyu,D. L. Zhou,Y. Wang,G. W. Deng, H. Z. Song,D. Oblak,G. C. Guo,Q. Zhou, Towards real-world quantum networks: A review, Laser Photonics Rev. 16, 2100219 (2022)

[169] V. Semenenko, X. Hu, E. Figueroa, and V. Perebeinos, Entanglement generation in a quantum network with finite quantum memory lifetime, AVS Quantum Science 4(1), 012002 (2022)

[170] S. A. Moiseev,K. I. Gerasimov,M. M. Minnegaliev,E. S. Moiseev, Optical quantum memory on macroscopic coherence, arXiv: 2408.09991 (2024)

[171] S. A. Moiseev, S. N. Andrianov, and F. F. Gubaidullin, Efficient multimode quantum memory based on photon echo in an optimal QED cavity, Phys. Rev. A 82(2), 022311 (2010)

[172] M. Afzelius and C. Simon, Impedance-matched cavity quantum memory, Phys. Rev. A 82(2), 022310 (2010)

[173] R. R. Meng, X. Liu, M. Jin, Z. Q. Zhou, C. F. Li, and G. C. Guo, Solid-state quantum nodes based on color centers and rare-earth ions coupled with fiber Fabry–Pérot microcavities, Chip (Wurzbg.) 3(1), 100081 (2024)

[174] L. Labonté, O. Alibart, V. D’Auria, F. Doutre, J. Etesse, G. Sauder, A. Martin, E. Picholle, and S. Tanzilli, Integrated photonics for quantum communications and metrology, PRX Quantum 5(1), 010101 (2024)

[175] N. S. Perminov, D. Y. Tarankova, and S. A. Moiseev, Superefficient cascade multiresonator quantum memory, Laser Phys. Lett. 15(12), 125203 (2018)

[176] E. S. Moiseev and S. A. Moiseev, All-optical photon echo on a chip, Laser Phys. Lett. 14(1), 015202 (2017)

[177] A. R. Matanin, K. I. Gerasimov, E. S. Moiseev, N. S. Smirnov, A. I. Ivanov, E. I. Malevannaya, V. I. Polozov, E. V. Zikiy, A. A. Samoilov, I. A. Rodionov, and S. A. Moiseev, Toward highly efficient multimode superconducting quantum memory, Phys. Rev. Appl. 19(3), 034011 (2023)

[178] N. S. Perminov and S. A. Moiseev, Integrated multiresonator quantum memory, Entropy (Basel) 25(4), 623 (2023)

[179] R. M. Pettit,F. H. Farshi,S. E. Sullivan,A. Veliz-Osorio,M. K. Singh, A perspective on the pathway to a scalable quantum internet using rare-earth ions, Appl. Phys. Rev. 10(3), 031307 (2023)

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