# Optimal absorption and emission of itinerant fields into a spin ensemble memory

Linda Greggio, Tristan Lorriaux, Alexandru Petrescu, Mazyar Mirrahimi, Audrey Bienfait

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Linda Greggio, Tristan Lorriaux, Alexandru Petrescu, Mazyar Mirrahimi, Audrey Bienfait. Optimal absorption and emission of itinerant fields into a spin ensemble memory. 2025. hal-05345345

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# Optimal absorption and emission of itinerant fields into a spin ensemble memory

Linda Greggio,¹, * Tristan Lorriaux,² Alexandru Petrescu,¹ Mazyar Mirrahimi,¹ and Audrey Bienfait², §

¹Laboratoire de Physique de l'Ecole Normale Supérieure, Mines Paris, Inria, CNRS, ENS-PSL, Sorbonne Université, PSL Research University, Paris, France

²Laboratoire de Physique de l'Ecole Normale Supérieure de Lyon, France

(Dated: June 30, 2025)

Quantum memories integrated in a modular quantum processing architecture can rationalize the resources required for quantum computation. This work focuses on spin-based quantum memories, where itinerant electromagnetic fields are stored in large ensembles of effective two-level systems, such as atomic or solid-state spin ensembles, embedded in a cavity. Using a mean-field framework, we model the ensemble as an effective spin communication channel and develop a cascaded quantum model to describe both absorption and emission processes. We derive optimal time-dependent modulations of the cavity linewidth that maximize storage and retrieval efficiency for finite-duration wavepackets. Our analysis yields an upper bound on efficiency, which can be met in the narrow bandwidth regime. It also shows the existence of a critical bandwidth above which the efficiency severely decreases. Numerical simulations are presented in the context of microwave-frequency quantum memories interfaced with superconducting quantum processors, highlighting the protocol's relevance for modular quantum architectures.

## I. INTRODUCTION

Quantum memories - devices capable of faithfully absorbing, storing and retrieving one or more quantum states - are fundamental components in a range of quantum technologies [1]. Their ability to preserve quantum information can enhance long-distance communication [2-4] through quantum repeaters [5], increase the sensitivity of quantum sensors [6, 7] and reduce the resource requirements for quantum computation [8-10]. Various physical systems can serve as quantum memories, including atomic ensembles [11], solid-state defects [12, 13], molecular gases [14], single-mode oscillators [15, 16], atoms and ions [17, 18]. These systems rely on interactions such as microwave [15], mechanical [16], electronic [11], magnetic [19], or vibrational coupling [12] to store quantum information. This article focuses on one particular approach: storing itinerant electromagnetic fields within a large ensemble of effective two-level systems (spins) embedded in a cavity, such as atoms or solid-state defects. Early proposals [20-22] for such cavity-based quantum memories often emphasized operating in a “perfectly matched” regime, where to achieve efficient absorption, the losses induced by the ensemble on the cavity precisely balance the decay rate of the cavity due to its coupling to the input line. Practical quantum memories must also integrate seamlessly with quantum processors [10]. This requires high-fidelity quantum state transfer (QST) using flying qubits — quantum signals that travel through a communication channel between processor and memory. One key capability for many itinerant QST protocols is the ability to dynamically control the coupling strength between the communication channel and the emitter node, as well as the

receiver node [23-25]. To address this challenge, we develop a quantum cascaded model [26] that describes both the absorption and emission stages of the memory operation. This framework allows us to determine the maximum achievable storage efficiency as a function of the memory's physical parameters and the bandwidth of the incoming quantum signal. Furthermore, we derive the optimal time-dependent modulation of the cavity bandwidth required during each step of the process to reach such an efficiency.

The spin ensemble is characterized by an inhomogeneous linewidth Γ, determined by the frequency distribution of the spins. This linewidth sets both the spectral bandwidth of the memory and the upper bound on its operational speed [27]. Upon absorption, a quantum state becomes rapidly scattered across the internal degrees of freedom of the ensemble. Retrieval protocols are typically based on spin-echo strategies [12, 22, 28], in which an initial magnetization created in the ensemble is refocused by applying pulse sequences that counteract dephasing from inhomogeneous broadening. Successful retrieval is limited by the ensemble's coherence time $T_2$. Depending on the spin and its dominant decoherence mechanisms, a variety of refocusing pulse sequences — ranging from simple Hahn echoes to more elaborate multi-pulse schemes — have been developed to extend coherence and thus storage time [29]. The ratio of the temporal duration of the stored signal to the total coherence time offers one metric for a key figure of merit: the quantum memory capacity, which quantifies how many distinct temporal modes can be reliably stored.

In contrast to these refocusing pulses, which impose their own constraints on the physical characteristics of the memory and its operation, optimizing the absorption and emission efficiency primarily depends on achieving sufficient coupling between the spin ensemble and the cavity. In such hybrid systems, the loss rate induced by the spins on the cavity is given by $\kappa_s = 4g_{\text{ens}}^2/\Gamma$ [30],

<footnote>* linda.greggio@inria.fr</footnote>

<footnote>§ audrey.bienfait@ens-lyon.fr</footnote>
where $g_{ens}$ is the collective coupling strength between the cavity and the ensemble. Efficient absorption of slow or continuous signals — those with bandwidths smaller than the inhomogeneous broadening — can be achieved by matching $\kappa_s$ to the cavity’s decay rate $\kappa$, assuming this rate to be solely governed by the cavity coupling to the input channel, with no additional intrinsic losses. This has been shown in earlier mean-field analyses [21, 22]. In this article, we extend this line of work to explore the storage and retrieval efficiency of finite-duration wave packets with two primary objectives. First, we aim to increase the storage capacity of the memory. Second, we seek to better characterize the absorption and emission timescales of the memory to compare them to the timescale of typical processing quantum nodes, in the context of modular architectures incorporating quantum memories. Finally, we also include the intrinsic cavity loss in this analysis, which turns out to be an important limiting factor for the efficiency of the protocol.

As shown in [31] for the absorption step, efficient storage of finite-duration pulses requires dynamic modulation of the cavity linewidth, mirroring strategies used in itinerant QST for optimal efficiency. Here, we remain within the mean-field theoretical framework developed in earlier works [21, 22], and leverage this description to model the ensemble as an effective spin communication channel. The channel interacts with a fictitious spin bosonic field with coupling rate $\Gamma$, which in turn couples to the cavity with strength $g_{ens}$. This model reduces the absorption process to a system of two coupled bosonic modes with one mode able to decay into a spin communication channel. In particular, it allows us to formulate and solve an optimization problem to maximize absorption efficiency. The emission step naturally emerges as a quantum cascade of the absorption step, with the emitted field into the spin channel driving a future instance of the same two-mode system. This enables a similar optimization of the emission process.

The article is organized as follows: in a first section, we recall briefly the physics of the itinerant absorption-refocus-retrieve protocol, and introduce the physical system and its interaction structure. In a second part, we map the evolution of this system during the protocol into a quantum cascaded model. From this model, we derive the protocol efficiency in a steady-state operation regime. In Sec. V, we demonstrate that these steady-state efficiencies serve as upper bounds for the more realistic case of finite-length wavepackets, and derive the optimal modulation functions needed to maximize efficiency for a given input waveform. Finally, we perform numerical simulations to compute these optimal modulation profiles and analyze the corresponding decrease in efficiency as a function of the incoming signal bandwidth and the cavity’s intrinsic loss. We do so in the particular context of quantum memories implemented at microwave frequencies, and possibly coupled to superconducting quantum processors.

## II. PHYSICAL MODEL AND STORAGE PROTOCOL

An ensemble of $N$ spins is embedded in a cavity with intrinsic loss rate $\kappa_0$, see Fig. 1(a). The cavity field is described by its annihilation $\hat{\epsilon}$ and creation $\hat{\epsilon}^{\dagger}$ operators, and is coupled to a measurement waveguide at a tunable coupling rate $\kappa(t)$. Any spin $j$ of the ensemble, of Larmor frequency $\omega_j$, interacts with the cavity at a strength $g_j$ with an interaction Hamiltonian $\hat{H}_j/\hbar = g_j(\hat{\epsilon}\hat{\sigma}_+^j + \hat{\epsilon}^{\dagger}\hat{\sigma}_-^j)$, where $\hat{\sigma}_-^j$ and $\hat{\sigma}_+^j$ are the spin lowering and raising operators. The ensemble coupling rate to the cavity is given by $g_{ens}^2 = \int p(g)g^2dg$. We consider the spin spectral distribution $n$ to be a Lorentzian centered at the mean spin Larmor frequency $\omega_s$, i.e. $n(\Delta_j = \omega_j - \omega_s) = \frac{\Gamma}{2\pi}\frac{1}{\Gamma^2+\Delta_j^2}$, and uncorrelated with the spin coupling distribution $p(g_j)$. All spins are assumed to be in their ground state at the beginning of the protocol.

We consider a simple echo-based protocol proposed in earlier works [21] illustrated in Fig. 1(b). During the first stage of the protocol, an incoming signal of envelope $\varepsilon_{in}(t)$, at mean Larmor frequency $\omega_s$, is absorbed by the spin ensemble. Two refocusing pulses applied at $t = \tau_a$ and $2\tau_a+\tau_e$ let us retrieve this absorbed quantum state at time $T_E = 2\tau_a+2\tau_e$. The echo occurring between the two refocusing pulses, at time $2\tau_a$, is suppressed by detuning the cavity between the refocusing pulses $\tau_a < t < 2\tau_a + \tau_e$. During absorption and emission, the equations of motion (EOMs) of the system in the frame rotating at $\omega_s$, read [21]

$$ \frac{d}{dt}\varepsilon = -\frac{\kappa_0 + \kappa(t) + 2i\Delta_{cs}}{2}\varepsilon + i\sum_{j=1}^{N}g_j\sigma_{-}^{j} + \sqrt{\kappa(t)}\varepsilon_{in}(t), \quad (1) $$

$$ \frac{d}{dt}\sigma_{-}^{j} = -i\Delta_{j}\sigma_{-}^{j} + ig_{j}\varepsilon, \quad (2) $$

where $\sigma_{-}^{j}(t)$ and $\varepsilon(t)$ are the expectation values of the operators $\hat{\sigma}_{-}^{j}$ and $\hat{\varepsilon}$, while $\Delta_{cs} = \omega_c - \omega_s$ is the detuning between the cavity and the spin central frequency. During emission, there is no input field, i.e. the envelope $\varepsilon_{in}(t) = 0$. In the above EOMs, we include intrinsic losses for the cavity field without considering any population or phase decays of individual spins as their coherence time is assumed to be much longer than the protocol duration. We also do not consider any spin-spin interaction. Finally, in Eq. (2), we have already implemented a mean-field approximation. Indeed, its last term should write as $-ig_j\langle\hat{\sigma}_z^j\hat{\varepsilon}\rangle$. It can be simplified through the Holstein-Primakoff approximation [21, 22, 32] to a factorized product $-ig_j\langle\hat{\sigma}_z^j\rangle\langle\hat{\varepsilon}\rangle$. Since we are considering the incoming field $\varepsilon_{in}$ only carries a few photons compared to the large number of spins ($N > 10^4$) in the ensemble, we expect $\langle\hat{\sigma}_z^j\rangle \sim -1$ during absorption. This mean population is also recovered after the two refocusing pulses, yielding the simplified form of Eq. (2) we are using.
<figure><img src="image_2.png" /><figcaption>FIG. 1. (a) Spin ensemble of central frequency $\omega_s$ and of inhomogeneous broadening $\Gamma$ coupled with strength $g_{\text{ens}}$ to a cavity driven by the input field $\varepsilon_{\text{in}}(t)$. The cavity is subject to intrinsic losses $\kappa_0$ and interacts with the transmission line through a tunable coupling $\kappa(t)$. The intra-cavity field $\varepsilon(t)$ and the field in the spin ensemble $\Sigma(t)$ describe the dynamics of the stored field. (b) Memory protocol: once the incoming field is absorbed, the spins start to dephase. Two refocusing pulses are used to rephase the spins, and an echo is emitted at time $T_E$. To avoid the emission of the first noisy echo at $2\tau_a$, the cavity is detuned between the two pulses. $\varepsilon_{\text{out}}(t)$ represents the retrieved field. (c) Quantum cascaded formalism: the memory system is represented as two cascaded copies of a two-mode bosonic system. In the absorption copy the incoming field $\varepsilon_{\text{in}}$ is stored in the spin ensemble, while in the emission copy the spin ensemble is driven by a feedback term $\Sigma_{\text{e,in}}$, representing a filtered version of the absorption output field $\Sigma_{\text{a,out}}$, with filter function specified in the box.</figcaption></figure>

### III. ABSORPTION AND EMISSION: A QUANTUM CASCADED MODEL

We now cast the interaction of the cavity with the spin ensemble as an interaction with a bosonic mode $\Sigma$ coupled to a communication channel. To introduce this bosonic mode, we replace the term $\sum_{j=1}^N g_j \sigma_{-}^j$ in Eq. (1) with its integrated version $\int_g \int_{\Delta} g \sigma_{-}^{\Delta,g} p(g) n(\Delta) dgd\Delta$. By integrating the EOM of $\sigma_{-}^j$ (Eq. (2)) during the absorption step, we can proceed to the elimination of $\sigma_{-}^{\Delta,g}$ in the above mentioned term, so that it becomes solely dependent

on $\varepsilon_a$, the cavity mean field during the absorption step (see Appendix A, Eq. (A3))

$$ \sum_{j=1}^{N} g_j \sigma_{-}^{j,a} \rightarrow \\ ig_{\text{ens}}^2 \int_{-\infty}^{t} ds \varepsilon_a(s) \int d\Delta n(\Delta) e^{-i\Delta(t-s)}. \qquad (3) $$

By recognizing the Fourier transform of the Lorentzian function $n(\Delta)$, we can equalize this term to $ig_{\text{ens}} \Sigma_a(t)$, where $\Sigma_a(t) = g_{\text{ens}} \int_{-\infty}^{t} ds e^{-\frac{\Gamma}{2}(t-s)} \varepsilon_a(s)$, with $\Sigma_a(-\infty) = 0$. The variable $\Sigma_a$ can also be defined by an EOM:

$$ \frac{d}{dt} \Sigma_a = -\frac{\Gamma}{2} \Sigma_a + g_{\text{ens}} \varepsilon_a(t). \qquad (4) $$

We can now rewrite the intra-cavity EOM (Eq. (1)) using $\Sigma_a$, and we obtain a new expression for the dynamics of the cavity

$$ \frac{d}{dt} \varepsilon_a = -\frac{\kappa_0 + \kappa_a(t) + i2\Delta_{cs}}{2} \varepsilon_a \\ -g_{\text{ens}} \Sigma_a + \sqrt{\kappa_a(t)} \varepsilon_{\text{in}}(t). \qquad (5) $$

Eqs. (4) and (5) form a closed set of equations describing the full dynamics of the spin-cavity system during absorption $t < \tau_a$. We can also recognize that they correspond to the evolution of the mean-values of two bosonic modes, $\hat{\varepsilon}_a$ and $\hat{\Sigma}_a$, which interact with strength $g_{\text{ens}}$. Moreover, we can identify the decay rates for each mode: respectively $\kappa_0$ and $\kappa_a$ for the cavity field $\varepsilon_a$, and $\Gamma$ for the mode $\Sigma_a$. This decay corresponds to the emission of a fictitious field $\Sigma_{a,out} = \sqrt{\Gamma} \Sigma_a$ into a newly defined spin channel, see Fig. 1(c). One can intuitively understand the nature of $\Sigma_a$ as the field stored in a collective “bright” mode of the spin ensemble, which is maximally coupled to the cavity. However, due to the ensemble broadening, the energy stored in this bright mode rapidly dissipates into “dark” modes, representing the uncoupled degrees of freedom of the spin ensemble. This process corresponds in our model to emission into the spin channel.

Refocusing techniques, acting as time reversal, permit to recover the initial magnetization created on the spin ensemble. The resulting filtered field will then drive the two bosonic modes system during emission, in a quantum cascade arrangement. To highlight this quantum cascade, we introduce a new cavity variable $\varepsilon_e$ and new spin variables $\sigma_{-}^{j,e}$ to describe the system during the emission step ($t > 2\tau_a + \tau_e$), obeying respectively the EOMs Eq. (1) (without the driving term) and Eq. (2). Similarly to absorption, we can integrate the EOM of $\sigma_{-}^{j,e}$ to obtain an expression for the term $\sum g_j \sigma_{-}^{j,e}$ that solely depends on $\varepsilon_a$ and $\varepsilon_e$ (see Appendix A, Eq. (A10)), so that the cavity field evolution is given by

$$ \frac{d}{dt} \varepsilon_e = -\frac{\kappa_0 + \kappa_e(t) + i2\Delta_{cs}}{2} \varepsilon_e - g_{\text{ens}} \Sigma_e, \qquad (6) $$
where $\kappa_e(t)$ is the cavity coupling rate during the emission part. Similarly to the absorption step, we can recognize that the cavity mode interacts at strength $g_{\text{ens}}$ with the bosonic mode $\Sigma_e$. As detailed in Appendix A, we find that this bosonic mode is defined by the following EOM

$$ \frac{d}{dt}\Sigma_e = -\frac{\Gamma}{2}\Sigma_e + g_{\text{ens}}\varepsilon_e(t) + \sqrt{\Gamma}\Sigma_{e,\text{in}}, \quad (7) $$

assuming $\Sigma_e(-\infty) = \varepsilon_e(-\infty) = 0$. Here, $\Sigma_{e,in}$ corresponds to a drive field for this new spin bosonic mode. It corresponds to a filtered version of $\Sigma_{a,out}$ given by $\Sigma_{e,in} = \sqrt{\Gamma}\tilde{\Sigma}_a(T_E - t)$, where $\tilde{\Sigma}_a$ is defined by the following EOM (see Appendix A)

$$ \frac{d}{dt}\tilde{\Sigma}_a = -\frac{\Gamma}{2}\tilde{\Sigma}_a + g_{\text{ens}}\varepsilon_a(-t), \quad (8) $$

with $\tilde{\Sigma}_a(-\infty) = 0$. By exploiting the symmetry in Eqs. (4) and (8), we can express the filter function $H$ between the emitted spin field during absorption $\Sigma_{a,out}$, and the incoming spin drive field during emission $\Sigma_{e,in}$ as

$$ H[\delta] = \frac{\Sigma_{e,in}[\delta]}{\Sigma_{a,out}[\delta]} = \frac{\frac{\Gamma}{2} + i\delta}{\frac{\Gamma}{2} - i\delta}e^{-i\delta T_E}. \quad (9) $$

Using Eqs. (4) to (7), we identify a quantum cascaded system [26, 33] represented in Fig. 1(c): the two interacting bosonic modes $\varepsilon_a$ and $\Sigma_a$, driven by a cavity drive $\varepsilon_{in}$ and emitting a spin field $\Sigma_{a,out}$, govern the dynamics of the two interacting bosonic modes $\varepsilon_e$ and $\Sigma_e$ through the spin drive $\Sigma_{e,in}$. We define the efficiency of the absorption step as the ratio between the energy transferred from the drive field $\varepsilon_{in}$ to the field stored inside the spin ensemble, and the energy of that drive field

$$ \eta_a = \frac{\int dt |\Sigma_{a,out}(t)|^2}{\int dt |\varepsilon_{in}(t)|^2}. \quad (10) $$

Similarly, the efficiency of the emission step is given by the ratio of the energy transferred from the spin field, $\Sigma_{e,in}$, to the energy of the field emitted by the cavity $\varepsilon_{e,out} = \sqrt{\kappa_e(t)}\varepsilon_e$, namely

$$ \eta_e = \frac{\int dt |\varepsilon_{e,out}(t)|^2}{\int dt |\Sigma_{e,in}(t)|^2}, \quad (11) $$

where the integrals are understood to run over $t \in (-\infty, \infty)$ (Appendix A). The overall efficiency of the protocol is given by $\eta = \eta_a \times \eta_H \times \eta_e$, where $\eta_H$ corresponds to the filter effect given by the refocusing pulses Eq. (9), i.e. $\eta_H = \frac{\int dt |\Sigma_{e,in}(t)|^2}{\int dt |\Sigma_{a,out}(t)|^2}$. Here, we consider perfect refocusing pulses, namely $\eta_H = 1$.

Next, in Sec. IV, we derive the absorption and emission efficiencies $\eta_a$ and $\eta_e$ for a slow-pulse limit of the quantum cascaded system, as closed-form expressions in terms of the physical parameters $g_{\text{ens}}$, $\kappa_0$ and $\Gamma$. We will then show that these efficiencies represent an upper bound when considering faster pulses in Sec. V.

## IV. MEMORY EFFICIENCY IN THE CONTINUOUS-DRIVE LIMIT

In this section, we consider the quantum cascaded model driven by a continuous-wave drive, with a constant coupling rate $\kappa$. We are interested in this steady-state behavior to derive the optimal value of $\kappa$, as a function of the physical parameters $g_{\text{ens}}$, $\Gamma$ and $\kappa_0$, which maximizes the energy transmitted to the spins and then remitted by refocusing, or equivalently the efficiency $\eta$ defined above. The limit we derive here is a “slow-pulse” limit, which corresponds to pulses whose bandwidth is much narrower than the system bandwidth. While this limit might not be very useful in practice, as the pulse length should be much shorter than the coherence time of the quantum system generating it, it will provide a useful upper bound for $\eta$ in the case of realistic finite-length pulses, as treated in the next Sec. V.

We define the following transmission coefficients associated to absorption and emission

$$ t_a = \Sigma_{a,out}/\varepsilon_{in}, \quad t_e = \varepsilon_{e,out}/\Sigma_{e,in}, \quad (12) $$

with $t_a$ being the transmission coefficient from the driveline to the spins, while $t_e$ is the transmission coefficient from the spins back to the driveline. Note that these transmission coefficients are related to the transmissivities of the spin-cavity channel $|t_a|^2, |t_e|^2$ [34]. Using the EOMs for absorption and emission written in the frequency domain, these transmission coefficients are, in the Fourier domain, (see Appendix B)

$$ \begin{aligned} t_a[\delta] &= \frac{2\sqrt{\kappa_s\kappa_a}}{\kappa_0 + \kappa_a + \tilde{\kappa}_s + 2i(\tilde{\delta} + \Delta_{cs})}, \\ t_e[\delta] &= -\frac{2\sqrt{\kappa_s\kappa_e}}{\kappa_0 + \kappa_e + \tilde{\kappa}_s + 2i(\tilde{\delta} + \Delta_{cs})}, \end{aligned} \quad (13) $$

with frequency dependence given by $\tilde{\kappa}_s/\kappa_s = 1 - \delta(\delta + \Delta_{cs})/g_{\text{ens}}^2$ and $\tilde{\delta}/\delta = 1 + (\kappa_0 + \kappa_{a/e})/\Gamma$. In absence of modulation, with constant coupling rate to the input line, the absorption and emission are described by a reciprocal linear system, so that the transmission coefficients $|t_a|$ and $|t_e|$ are symmetric under an interchange a $\leftrightarrow$ e.

The efficiencies defined in Eqs. (10) and (11) can then be rewritten as a function of the transmission coefficients, by making use of Eq. (12) and of Parseval’s theorem to pass to the frequency domain

$$ \eta_a = \frac{\int d\delta |t_a[\delta] \varepsilon_{in}[\delta]|^2}{\int d\delta |\varepsilon_{in}[\delta]|^2}, \quad \eta_e = \frac{\int d\delta |t_e[\delta] \Sigma_{e,in}[\delta]|^2}{\int d\delta |\Sigma_{e,in}[\delta]|^2}. \quad (14) $$

From these expressions for the efficiencies, it is clear that an optimization of the memory protocol comes from a maximization of the transmissivities $|t_a|^2$, and $|t_e|^2$. To this end, we first maximize the transmissivities, as obtained from Eq. (13), at each frequency value $\delta$, with respect to $\Delta_{cs}$, finding the optimal $\Delta_{cs}^*[\delta] = (\frac{\kappa_s/\Gamma}{1+4(\delta/\Gamma)^2} - 1)\delta$. Then, we evaluate the transmissivity
at $\Delta_{cs}^*$, and maximize it with respect to $\kappa_{a/e}$, to obtain equal values for emission and absorption $\kappa_{a/e}^*[\delta] = \kappa_0 + \frac{\kappa_s}{1+4(\delta/\Gamma)^2}$, leading then to equal optimal transmissivities $|t_{a/e}[\delta]|^2 = \frac{\kappa_s}{\kappa_s+\kappa_0+4\kappa_0(\delta/\Gamma)^2}$. Finally, these are clearly maximized for $\delta^* = 0$. We thus find that the sweet spot for the quantum memory protocol occurs at zero frequency, i.e. for constant input $\varepsilon_{in}(t) = \varepsilon_{in}$, leading to zero detuning $\Delta_{cs}^* = 0$, and $\kappa_{a/e}^* = \kappa_0 + \kappa_s$, a coupling rate that maximizes both the transmission from the driveline to the spins and the transmission from the spins back to the driveline.

The transmission coefficients at this sweet spot are then constant in time and read

$$t_a = -t_e = \sqrt{\frac{\kappa_s}{\kappa_s + \kappa_0}}, \qquad (15)$$

while the protocol efficiency is

$$\eta = \eta_a^2 = |t_a|^4 = \left( \frac{\kappa_s}{\kappa_0 + \kappa_s} \right)^2. \qquad (16)$$

This expression highlights that the quality of the memory is given by the ratio $\kappa_s/\kappa_0$, which describes how well the cavity is coupled to the spins compared to its coupling to the environment. Fig. 2 shows the transmissivity at zero detuning as a function of the coupling rate $\kappa_{a/e}$ (panel (a)) and of internal losses $\kappa_0$ (panel (b)) for increasing values of $g_{ens}/\Gamma$. In panel (a) the crosses mark the sweet spot $\kappa_{a/e}^* = \kappa_0 + \kappa_s$ at $\kappa_0/\Gamma = 1/30$, while the transmissivity depicted in panel (b) is evaluated at the sweet spot $\kappa_{a/e}^*$. Moreover, panel (b) shows that even in this case of optimal energy transfer, the cavity internal losses have a detrimental effect on the transmissivity, which can be overcome by increasing the coupling rate between the spins and the cavity, i.e. $\kappa_s$. The gain in transmissivity for larger $\kappa_s$ is also clear in the other panel.

Along with the transmitted energy comes the reflected energy, which is associated with the reflection coefficients $r_a = \varepsilon_{a,out}/\varepsilon_{in}$ back to the driveline and $r_e = \Sigma_{e,out}/\Sigma_{e,in}$ back to the spins. Their expressions are presented in Appendix B (see Eq. (B5) and Eq. (B8)), and if we evaluate them at the transmission sweet spot ($\kappa_{a/e}^*, \delta^*$ and $\Delta_{cs}^*$) we find $r_a = 0$, i.e. the incoming drive is perfectly absorbed and $r_e = \frac{\kappa_0}{\kappa_0+\kappa_s}$. This means that even in the case of optimal energy transfer, some energy remains within the spins during emission.

At the optimal operation point for constant coupling rate, the cooperativity of the system is $C^* = \frac{4g_{ens}^2}{\Gamma(\kappa_0+\kappa_{a/e}^*)} = \frac{\kappa_s}{(2\kappa_0+\kappa_s)} \le 1$, giving the efficiency of the protocol as

$$\eta = \left( \frac{2C^*}{C^* + 1} \right)^2. \qquad (17)$$

Unit efficiency is thus achieved under unit cooperativity (the so-called impedance matching condition [21]), whenever the intrinsic losses of the cavity can be neglected.

<figure><img src="image_3.png" /><figcaption>FIG. 2. Transmissivity in the continuous-drive limit, when the cavity, the spins and the drive are all at resonance, as a function of cavity-driveline coupling ((a)), and of internal losses ((b)), for three different values of $g_{ens}/\Gamma$ (0.2 green, 1 blue, 2 red curve). In (a) the internal losses decay rate is fixed to $\kappa_0/\Gamma = 1/30$. The crosses correspond to the optimal coupling $\kappa_{a/e}^* = \kappa_0 + \kappa_s$, which indeed maximizes the transmissivity. In (b) the coupling is set to its optimal value $\kappa_{a/e}^*$, showing that the detrimental effect of intrinsic losses can be mitigated by increasing the coupling between the spins and the cavity.</figcaption></figure>

## V. OPTIMAL MEMORY EFFICIENCY FOR TEMPORAL PULSES

We now wish to derive the temporal modulation of the coupling strengths during absorption, $\kappa_a(t)$, and emission, $\kappa_e(t)$, which optimize the storage efficiency of the memory $\eta = \eta_a \times \eta_e$ for an incoming pulse $\varepsilon_{in}(t)$. Based on the continuous-drive limit we derived above, we assume that the optimal spin-cavity detuning remains similar, and we consider $\Delta_{cs} = 0$ in this section. Consequently, all the field variables are real. Before describing the optimization problem, we perform a time rescaling $\tau = \Gamma t$: $\tilde{\delta} = \delta/\Gamma$, $\tilde{\kappa}_0 = \kappa_0/\Gamma$, $\tilde{\kappa}_s = \kappa_s/\Gamma$, $\mathfrak{g} = g_{ens}/\Gamma$, $\tilde{\kappa}(\tau) = \kappa(\tau/\Gamma)/\Gamma$, $\tilde{T}_E = \Gamma T_E$. The resulting equations are dimensionless and describe the memory protocol dynamics in units of the spin decay rate $\Gamma$, which sets a natural time scale for storage into the memory system. We denote the cavity and spin fields $\varepsilon_{a/e}(\tau/\Gamma)$ and $\Sigma_{a/e}(\tau/\Gamma)$, as well as the input drive fields for absorption $\varepsilon_{in}(\tau/\Gamma)$ and emission $\tilde{\Sigma}_{a}(\tau/\Gamma)$, all functions of the rescaled time $\tau$, by $\mathcal{E}_{a/e}(\tau)$, $\mathcal{S}_{a/e}(\tau)$, and $\sqrt{\Gamma}\mathcal{E}_{in}(\tau)$, $\mathcal{S}_{in}(\tau)$. With these notations, the EOMs describing emission write

$$\begin{aligned} \frac{d}{d\tau}\mathcal{E}_e &= -\frac{\bar{\kappa}_0 + \bar{\kappa}_e(\tau)}{2}\mathcal{E}_e - \mathfrak{g}\mathcal{S}_e, \\ \frac{d}{d\tau}\mathcal{S}_e &= -\frac{1}{2}\mathcal{S}_e + \mathfrak{g}\mathcal{E}_e + \mathcal{S}_{in}(\bar{T}_E - \tau), \qquad (18) \\ \frac{d}{d\tau}\mathcal{S}_{in} &= -\frac{1}{2}\mathcal{S}_{in} + \mathfrak{g}\mathcal{E}_a(-\tau), \end{aligned}$$

while the EOMs for the absorption process are

$$\begin{aligned} \frac{d}{d\tau}\mathcal{E}_a &= -\frac{\bar{\kappa}_0 + \bar{\kappa}_a(\tau)}{2}\mathcal{E}_a - \mathfrak{g}\mathcal{S}_a + \sqrt{\bar{\kappa}_a(\tau)}\mathcal{E}_{in}, \qquad (19) \\ \frac{d}{d\tau}\mathcal{S}_a &= -\frac{1}{2}\mathcal{S}_a + \mathfrak{g}\mathcal{E}_a. \end{aligned}$$
A. Optimizing absorption

Instead of fixing the shape of the incoming pulse $\mathcal{E}_{\text{in}}(\tau)$, for simplicity, we equivalently fix the shape of $S_a(\tau)$, such that the energy emitted into the spin ensemble equals 1. In this way, the shapes of $\mathcal{E}_a$ and $\mathcal{E}_{\text{in}}$ can be found by the absorption EOMs (19) and can be expressed in terms of $S_a(\tau)$ and its derivatives $\dot{S}_a(\tau)$ and $\ddot{S}_a(\tau)$, as well as $\bar{\kappa}_a(\tau)$. This last function is the only remaining degree of freedom that we would like to determine by solving an optimization problem which maximizes the absorption efficiency $\eta_a = \frac{\int d\tau |S_a(\tau)|^2}{\int d\tau |\mathcal{E}_{\text{in}}(\tau)|^2}$.

Specifically, this optimization problem can be formulated as follows. For a given $S_a$ such that $\int d\tau |S_a(\tau)|^2 = 1$, solve

$$ \min_{\bar{\kappa}_a(\tau) \ge 0} \int d\tau |\mathcal{E}_{\text{in}}(\tau)|^2. \qquad (20) $$

Noting that by Eq. (19) $\mathcal{E}_{\text{in}} = A[S_a, \dot{S}_a, \ddot{S}_a]/\sqrt{\bar{\kappa}_a} + \sqrt{\bar{\kappa}_a}B[S_a, \dot{S}_a, \ddot{S}_a]$ with

$$ \begin{aligned} A[S_a, \dot{S}_a, \ddot{S}_a] &= \frac{1}{g}(\ddot{S}_a + \frac{1 + \bar{\kappa}_0}{2}\dot{S}_a + \frac{4g^2 + \bar{\kappa}_0}{4}S_a) \\ B[S_a, \dot{S}_a, \ddot{S}_a] &= \frac{1}{2g}(\dot{S}_a + S_a/2), \end{aligned} \qquad (21) $$

we can re-express the optimization problem as

$$ \min_{\bar{\kappa}_a(\tau) \ge 0} \int d\tau \left( \frac{A(\tau)}{\sqrt{\kappa_a(\tau)}} + B(\tau)\sqrt{\kappa_a(\tau)} \right)^2, \qquad (22) $$

whose optimal solution is clearly given by

$$ \bar{\kappa}_a^*(t) = \left| \frac{A(t)}{B(t)} \right|. \qquad (23) $$

Now, given the optimal solution $\bar{\kappa}_a^*$, depending on the choice of $S_a$, we aim to derive an upper bound on the absorption efficiency, that is, a lower bound for the functional in (20). Using the functions $A(t)$ and $B(t)$ decomposing $\bar{\kappa}_a^*$, we can express the optimal value of the functional as

$$ \min \int |\mathcal{E}_{\text{in}}(\tau)|^2 d\tau = \int \left| \frac{B}{A} \right| \left( A + B \left| \frac{A}{B} \right| \right)^2 d\tau. \qquad (24) $$

The integrand $|\frac{B}{A}|(A + B|\frac{A}{B}|)^2$ can be rewritten as $2|AB| + 2AB$ which is lower bounded by $4AB$. Using the definitions of $A$ and $B$ in Eq. (21), we can show that (see Appendix C)

$$ \min \int |\mathcal{E}_{\text{in}}(\tau)|^2 d\tau \ge \int d\tau \left[ 4 \frac{\bar{\kappa}_0}{\bar{\kappa}_s} \dot{S}_a^2 + \left( 1 + \frac{\bar{\kappa}_0}{\bar{\kappa}_s} \right) S_a^2 \right]. \qquad (25) $$

Using Parseval's theorem, this inequality is equivalent to

$$ \min \int |\mathcal{E}_{\text{in}}(\tau)|^2 d\tau \ge \frac{1}{2\pi} \int \frac{\bar{\kappa}_s + \bar{\kappa}_0(1 + 4\bar{\delta}^2)}{\bar{\kappa}_s} |S_a[\bar{\delta}]|^2 d\bar{\delta}, \qquad (26) $$

which is lower bounded by $\frac{\bar{\kappa}_s + \bar{\kappa}_0}{2\pi\bar{\kappa}_s} \int d\bar{\delta} |S_a[\bar{\delta}]|^2$. We recall that we have assumed $\frac{1}{2\pi} \int d\bar{\delta} |S_a[\bar{\delta}]|^2 = 1$, which implies that the absorption efficiency $\eta_a$ is directly upper-bounded by

$$ \eta_a \le \frac{\kappa_s}{\kappa_0 + \kappa_s}. \qquad (27) $$

As seen before, this upper bound corresponds to the maximal absorption efficiency Eq. (16) found in the case of constant $\kappa_a$ for an adiabatic incoming pulse.

B. Optimizing emission

Now that we have found the profile of $\bar{\kappa}_a$ that realizes the maximal transfer of energy from the driveline to the spins, we aim to find the profile of $\bar{\kappa}_e$ that permits the maximal transfer of energy from the spins back to the driveline, namely we want to maximize $\eta_e = \frac{\int d\tau \bar{\kappa}_e(\tau) |\mathcal{E}_e(\tau)|^2}{\int d\tau |S_{\text{in}}(\tau)|^2}$.

To this end, we first express the denominator $\int d\tau |S_{\text{in}}(\tau)|^2$ using our knowledge of the absorption step. Through Eq. (9), we can show that $S_{\text{in}}(\bar{\delta}) = H[\Gamma\bar{\delta}]S_a(\bar{\delta})$. Given that we have assumed perfect refocusing pulses, i.e. $|H| = 1$, we have $\int d\tau |S_{\text{in}}(\tau)|^2 = \int d\tau |S_a(\tau)|^2 = 1$. Similarly to the absorption problem, maximizing the emission efficiency thus reduces to maximizing $E_{\text{out}} = \int d\tau \bar{\kappa}_e(\tau) \mathcal{E}_e(\tau)^2$.

Rewriting the second Eq. (18) as $g\mathcal{E}_e = \dot{S}_e + \frac{1}{2}\mathcal{S}_e - S_{\text{in}}$, and manipulating Eq. (18) as shown in Appendix C, it is possible to eliminate the dependence of $E_{\text{out}}$ on $\bar{\kappa}_e$, so that $E_{\text{out}}$ is simply expressed as a function of $S_e$, its derivatives, and $S_{\text{in}}$

$$ E_{\text{out}} = \int d\tau [S_{\text{in}}^2 - (S_e - S_{\text{in}})^2 - \frac{\bar{\kappa}_0}{\bar{\kappa}_s}(S_e + 2\dot{S}_e - 2S_{\text{in}})^2]. \qquad (28) $$

Furthermore, note that, following Eq. (18), we have

$$ \bar{\kappa}_e(\tau) = -\bar{\kappa}_0 - 2 \frac{\ddot{S}_e + 1/2 \dot{S}_e - \dot{S}_{\text{in}} + \tilde{g}^2 S_e}{\dot{S}_e + 1/2 S_e - S_{\text{in}}}. \qquad (29) $$

Therefore, the optimization problem can be written as a maximization of the output energy (28) as a function of $S_e$, under the constraint

$$ \bar{\kappa}_0 + 2 \frac{\ddot{S}_e + 1/2 \dot{S}_e - \dot{S}_{\text{in}} + \tilde{g}^2 S_e}{\dot{S}_e + 1/2 S_e - S_{\text{in}}} \le 0. \qquad (30) $$
This constraint can be written in the following form

$$S_e \in \{S_e | h_1(S_e) \le c_1(\tau), h_2(S_e) \ge c_2(\tau)\} \cup \{S_e | h_1(S_e) \ge c_1(\tau), h_2(S_e) \le c_2(\tau)\} \quad (31)$$

with

$$\begin{aligned} h_1(S_e) &= 4\ddot{S}_e + 2(1 + \bar{\kappa}_0)\dot{S}_e + (\bar{\kappa}_s + \bar{\kappa}_0)S_e, \\ h_2(S_e) &= 2\dot{S}_e + S_e, \\ c_1(\tau) &= 4\dot{S}_{in}(\tau) + 2\bar{\kappa}_0 S_{in}(\tau), \\ c_2(\tau) &= 2S_{in}(\tau). \end{aligned}$$

Therefore, the constraint can be seen as the union of two convex sets. It is possible to solve each of these convex optimization problems under constraints using an appropriate iterative algorithm, such as the Uzawa algorithm for finding the saddle point of the associated Lagrangian, and then compare the maximal values in each set to find the global maximum under constraints.

Here, for simplicity, we have decided to relax the constraint, find the unconstrained optimal output energy, and finally check if this optimal solution actually satisfies the constraint. To this end, using Parseval's theorem, we re-express the output energy in Eq. (28) as

$$E_{\text{out}} = \frac{1}{2\pi} \int d\bar{\delta} \left[ |S_{\text{in}}[\bar{\delta}]|^2 - |S_e[\bar{\delta}] - S_{\text{in}}[\bar{\delta}]|^2 - \frac{\bar{\kappa}_0}{\bar{\kappa}_s} |(1 + 2i\bar{\delta})S_e[\bar{\delta}] - 2S_{\text{in}}[\bar{\delta}]|^2 \right]. \quad (32)$$

The corresponding optimal spin field (without constraint) is given by

$$S_e^*[\bar{\delta}] = \frac{2(1 - 2i\bar{\delta})\bar{\kappa}_0 + \bar{\kappa}_s}{(1 + 4\bar{\delta}^2)\bar{\kappa}_0 + \bar{\kappa}_s} S_{\text{in}}[\bar{\delta}]. \quad (33)$$

Using the second equation in (18), we can also deduce the optimal cavity field

$$\mathcal{E}_e^* = \sqrt{\bar{\kappa}_s} \frac{2i\bar{\delta} - 1}{\bar{\kappa}_0(1 + 4\bar{\delta}^2) + \bar{\kappa}_s} S_{\text{in}}[\bar{\delta}], \quad (34)$$

while using the first Eq. (18) we can determine the corresponding optimal cavity bandwidth modulation

$$\bar{\kappa}_e^*(\tau) = -\bar{\kappa}_0 - 2 \frac{\mathcal{F}^{-1}[i\bar{\delta}\mathcal{E}_e^* + g\mathcal{S}_e^*](\tau)}{\mathcal{F}^{-1}[\mathcal{E}_e^*](\tau)}, \quad (35)$$

where $\mathcal{F}^{-1}$ stands for the inverse Fourier transform.

In the absence of internal loss, $\kappa_0 = 0$, we find $S_e^* = S_{\text{in}}$, $\mathcal{E}_e^* = -\mathcal{E}_a(\tau - \bar{T}_E)$, corresponding to an emission efficiency of 1. However, we have to check if the corresponding cavity bandwidth modulation satisfies the positivity constraint, i.e. that $\bar{\kappa}_e^*$ given as follows is positive for all times (for $\kappa_0 = 0$)

$$\bar{\kappa}_e^*(\tau) = \frac{4\ddot{S}_{\text{in}} - 2\dot{S}_{\text{in}} + \bar{\kappa}_s S_{\text{in}}}{S_{\text{in}} - 2\dot{S}_{\text{in}}}. \quad (36)$$

At this point, we consider again the case of non-vanishing internal loss $\bar{\kappa}_0 > 0$ and we derive an upper bound for the emission efficiency $\eta_e$. We plug in the general expression of $S_e^*$ derived in Eq. (33) in the output energy (32), which yields

$$E_{\text{out}} = \frac{1}{2\pi} \int d\bar{\delta} |S_{\text{in}}|^2 \frac{\bar{\kappa}_s}{(1 + 4\bar{\delta}^2)\bar{\kappa}_0 + \bar{\kappa}_s}. \quad (37)$$

This provides an upper bound for the emission efficiency as no positivity constraint on $\bar{\kappa}_e$ is imposed. This bound is actually saturated when Eq. (35) remains positive for every time $\tau$.

Now, let us go even further and note that from the above equation we obtain

$$E_{\text{out}} \le \frac{\bar{\kappa}_s}{\bar{\kappa}_s + \bar{\kappa}_0} = \frac{\kappa_s}{\kappa_s + \kappa_0} \quad (38)$$

with the inequality a consequence of $\frac{1}{2\pi} \int d\bar{\delta} |S_{\text{in}}|^2 = 1$. We thus have $\eta_e \le \frac{\kappa_s}{\kappa_0 + \kappa_s}$, which together with the previously derived bound (27) gives

$$\eta = \eta_a \times \eta_e \le \left( \frac{\kappa_s}{\kappa_0 + \kappa_s} \right)^2. \quad (39)$$

As for absorption, this upper bound on absorption and re-emission efficiency is reached in the limit of adiabatic input drives and with a constant modulation $\kappa_{a/e}$ (see Eq. (16)). This limit is entirely governed by the amount of cavity intrinsic losses compared to the spin-induced losses. In the following section, we perform numerical simulations to evaluate the efficiency cost in using pulses whose bandwidth lies beyond this adiabatic limit.

## VI. NUMERICAL SIMULATIONS

From Sec. V, we can derive not only upper bounds on the efficiency but also determine the optimal modulation of the cavity coupling rates $\bar{\kappa}_a^*$ and $\bar{\kappa}_e^*$ (see Eqs. (23) and (35)), as well as the corresponding input drive $\varepsilon_{\text{in}}(t)$, given a fixed shape for the occupancy of the spin field mode $\Sigma_a$. By considering a particular wavepacket shape $u(\alpha\Gamma t)$, we can then derive the expected efficiency as a function of the wavepacket bandwidth $\alpha\Gamma$. Although this analysis does not allow us to identify an optimal pulse shape for a given bandwidth, it lets us quantify the reduction in efficiency in operating the quantum memory with finite-duration pulses for this specific pulse shape. From this study, we wish to extract the minimal wavepacket duration $\tau_{\text{in}}^*$ beyond which a reduction in efficiency occurs, and compare it with two key metrics for a quantum memory. First, comparing $\tau_{\text{in}}^*$ to the memory storage time (given by the ensemble coherence time $T_2$) provides an estimate of the capacity of the memory, i.e. the number of temporal modes it can potentially store. Second, in the context of a modular architecture, this wavepacket will realistically be emitted by a qubit. For a quantum
memory to be pertinent in such an architecture, $\tau_{in}^*$ must be short compared to the qubit lifetime to not limit the itinerant transfer between qubit to memory.

## A. Physical parameters

We wish to realize this temporal analysis in the particular context of quantum memories operating at microwave frequencies that may eventually be coupled to superconducting quantum bits. The most promising spin candidates for these memories are, for example, donors in silicon [35] or rare-earth ions [36]. These spin systems are attractive candidates since they possess clock transition “sweetspots” where the decoherence created by spin-spin interactions is canceled [37]. For example, at these particular operating points, bismuth donors in silicon achieve $T_2^{\text{Bi}} \approx 1$ s [35], ytterbium ions in yttrium orthosilicate yield $T_2^{171\text{Yb}:\text{Y}_2\text{SiO}_5} = 4$ ms [38] and the same ions in scheelite $T_2^{171\text{Yb}:\text{CaWO}_4} = 0.15$ s [39]. Very strikingly, the last system exhibits this coherence time in the absence of a biasing magnetic field. These spin coherence times compare favorably to present-day record coherence time for superconducting qubits ($\approx 1$ ms), and even more favorably for qubits inserted in large-scale superconducting quantum processors or for qubits in quantum state transfer experiments where the achievable coherence time is rather about $\sim 0.02$ ms [40, 41].

Using an ensemble of these spins for implementing a microwave quantum memory will rely on patterning a superconducting microwave resonator implementing the cavity on top of the crystal containing the spins [28, 42] or placing this crystal on top of another substrate on which the resonator is patterned [29]. In the former case, the strain induced by the superconducting film on the crystal can broaden the spin inhomogeneous linewidth (which may already be sizable) to $\Gamma/(2\pi) \sim 10$ MHz, whereas in the latter geometry one may hope to be closer to its natural values, which can be as low as 5 kHz for $^{171}\text{Yb} : \text{CaWO}_4$ for instance [39]. The particular choice of geometry for the superconducting resonator, combined with the particularities of the spin systems and the concentration of spins, sets the ensemble coupling constant [43]. For example, donors in silicon samples cannot be too heavily doped, so will reach a more limited coupling constant $g_{\text{ens}}/(2\pi) \sim 100$ kHz [42, 44] than for example some rare-earth ions where an ensemble coupling strength in the range of 4 MHz have been observed [30]. Overall, depending on the particular set of experimental conditions, the ratio $g_{\text{ens}}/\Gamma$ can take values between 0.02 and 200. However, let us note that the larger ensemble coupling constants are typically reached using high-spin concentrations, which limit the coherence time achievable at the clock-transition through second-order spin-spin interactions [35], so that a more reasonable upper bound for $g_{\text{ens}}/\Gamma$ would be $\sim 10$. These parameters correspond to spin-induced losses on the cavity of the order of $\kappa_s/(2\pi) = 1$ kHz to 10 MHz.

Let us now focus on the achievable cavity decay rates. For intrinsic losses, when the resonator is patterned on high-grade microwave substrates such as sapphire or silicon, decay rates as low as $\kappa_0/(2\pi) = 10$ kHz may be achieved [45]. When using a spin-doped crystal made of other materials, more realistic values $\kappa_0/(2\pi)$ up to $\sim 300$ kHz should be considered. The ratio $\kappa_0/\Gamma$ can thus take values from $1 \times 10^{-4}$ up to 30. The derivation of Sec. IV makes clear that the coupling rate of the cavity to the input line should at minima match $\kappa_0 + \kappa_s$. We assume in the following that this is the case and that no limitation will come from limited coupling to the measurement line. Experimentally, this would imply modulating $\kappa$ from $\kappa_{\text{min}} < 10$ kHz to $\kappa_{\text{max}} > 10$ MHz. This has already been achieved for superconducting resonators using Josephson junctions [46, 47], and can also be done on a smaller range with kinetic inductance when the resonator should be operated in a magnetic field [48].

## B. Wavepacket: hyperbolic secant case

To dictate the choice of the wavepacket $\Sigma_a(t)$ that we will consider in the rest of the study, we remark that the optimal shape for $\kappa_a$ is given by a ratio of the derivatives of $\Sigma_a(t)$, and by a ratio of Fourier transforms for $\kappa_e$. Using a wavepacket that is not at least twice differentiable, with a well-behaved Fourier transform, is thus not advised. In practice, we consider an hyperbolic secant pulse $\Sigma_a(t) = \sqrt{\alpha/2}/\cosh(\alpha\Gamma t)$ normalized to unity. In Appendix D, we also perform the study for a Lorentzian-shaped wavepacket, which gives a higher requirement for the maximum coupling rate $\kappa_{\text{max}}$, and which performs worse in terms of efficiency compared to the hyperbolic secant pulse. The bandwidth of the input pulse is $\alpha\Gamma$, meaning that the spins are capable of absorbing all the frequency components of the input field within the ensemble linewidth $\Gamma$ if $\alpha \le 1$.

To perform numerical simulations, we use this wavepacket to compute $\bar{\kappa}_a^*$ using Eq. (23). We then compute the input field $\mathcal{E}_{\text{in}}$ and the cavity field $\mathcal{E}_a$ through the EOMs (19).

For the emission process, we compute the dynamics in the frequency domain using Eqs. (33) and (34), and then apply the inverse Fourier transform to obtain the time-domain evolution. The associated modulation of the coupling through Eq. (35) leads to maximal emission efficiency, but does not take into account the positivity constraint on $\bar{\kappa}_e^*(\tau)$. When this constraint is violated, as occurs for pulse speeds $\alpha \ge \alpha_{\text{em}}^*$, the output field $\mathcal{E}_{e,\text{out}} = \sqrt{\bar{\kappa}_e}\mathcal{E}_e$ becomes ill-defined. To address this issue in the fast-pulse regime ($\alpha \ge \alpha_{\text{em}}^*$), where Eq. (35) yields negative values for the modulated coupling, we instead adopt a suboptimal but physical modulation: we mirror the absorption profile, setting $\kappa_e(t) = \kappa_a(T_E - t)$, which aligns with the time-reversed interpretation of emission in the quantum cascade formalism. We then compute the cavity field $\mathcal{E}_e$, the spin field $\mathcal{S}_e$ through the EOMs (18).
<figure><img src="image_4.png" /><figcaption>FIG. 3. Numerical simulations results for a hyperbolic secant pulse, with system parameters $g_{ens}/\Gamma = 1/2$ and $\kappa_0/\Gamma = 1/30$, corresponding to critical speeds $\alpha_{abs}^* = 0.474$ and $\alpha_{em}^* = 0.5$. We compare a slow-adiabatic case ($\alpha = 0.12$), with faster pulses just below ($\alpha = 0.46$) and above $\alpha = 0.69$ these thresholds. (a) Field stored in the spins during absorption and emission (dimensionless). (b) Incoming and retrieved field (in units of $\sqrt{\Gamma}$); evaluated using $\kappa_e(t)^*$ when defined, and $\kappa_a(T_E - t)$ when not (dashed-dot yellow curve). (c)-(d) Optimal modulation for cavity-driveline coupling during absorption (see Eq. (23)) and emission (see Eq. (35)) in units of $\Gamma$. As $\alpha$ is increased, getting away from the “slow-pulse” limit (black-dotted line), efficiency decreases (printed in (b)). At high speed ($\alpha > 1/2$), the denominator of Eq. (23) vanishes near $t = 0$, creating a singularity in $\kappa_a(t)$. A similar divergence occurs for $\kappa_e(t)$ near $T_E$, accompanied by an unphysical modulation (crossing of the zero).</figcaption></figure>

The feedback term $S_{in}$ is evaluated in the frequency domain as $S_{in}(\bar{\delta}) = H[\Gamma\bar{\delta}]S_a(\bar{\delta})$, then Fourier transformed back to the time domain. Finally, we compare the associated emission efficiency with the upper bound given by Eq. (37). This upper bound is saturated for pulse speeds $\alpha < \alpha_{em}^*$.

Numerical results for the cavity and spin field, as well as the modulated coupling rates, are presented in Fig. 3 for four values of the bandwidth $\alpha = 0.12, 0.25, 0.46$ and $0.69$. For this set of parameters, we find $\alpha_{em}^* = 0.5$. In the small bandwidth limit ($\alpha = 0.12$, green), the coupling rate approaches its “slow-pulse” optimal value $\kappa_0 + \kappa_s$ (black dotted line), and the numerically-determined efficiency ($\eta = 93.5\%$) comes close to the adiabatic upper bound ($\eta = 93.7\%$). Even at intermediate bandwidth ($\alpha = 0.25$), there is little loss in efficiency ($\eta = 93.2\%$). The input and outgoing cavity fields also closely resemble hyperbolic secant pulses. As we consider faster pulses, the optimal coupling modulations for absorption and emission lose in smoothness, and their minimum gets closer to 0. Similar to emission, we define the bandwidth $\alpha_{abs}^*$ at which a cancellation point appears in the optimal absorption coupling $\kappa_a^*(t)$. We observe that just below this limit ($\alpha = 0.46$), the efficiency is still quite high ($\eta = 92.0\%$). When switching to higher speed ($\alpha = 0.69$, beyond even the definition region of $\bar{\kappa}_e^*$), the absorption efficiency is largely impacted ($\eta_{abs} = 74.9\%$), evidencing a trade-off between the pulse bandwidth and its storage efficiency even though the optimal modulation function is still defined. We also see that the input pulse is considerably deformed, and is rather akin to a simple exponential wavepacket. For emission, we observe that taking the suboptimal choice $\kappa_e(t) = \kappa_a(T_E - t)$ also leads to a rather limited efficiency ($\eta_{em} = 61.0\%$), leading to a total

efficiency of $\eta = 45.7\%$.

### C. Efficiency as a function of pulse bandwidth and intrinsic losses

Restricting our analysis to hyperbolic secant pulses, we first study the efficiency of the absorption process as a function of the wavepacket bandwidth $\alpha\Gamma$ without (Fig. 4(a)) and with intrinsic losses (Fig. 4(b)). In both cases, we observe that the efficiency lies close to the slow-pulse limit (Eq. (16)) up to some critical bandwidth value after which the efficiency drops considerably. We observe that the intrinsic losses, which irremediably lower the slow-pulse efficiency, seem to have a marginal effect on this bandwidth limitation. As we observed in Fig. 3, this drop in efficiency ties to the optimal modulation shape derived in Eq. (23), which expresses $\kappa_a^*(t)$ as the absolute value of the ratio of A over B, where A and B are linear combinations of derivatives of the considered wavepacket. In our optimization problem, we only consider $A \neq 0$ and $B \neq 0$, which are non-zero at all times. At low speed, we find that $A/B$ is always positive, so that the absolute value does not play any role, and A and B are far from canceling at any point in time. In this case, Eq. (24) has a close form solution, which gives us an analytical expression of the absorption efficiency

$$ \eta_{\text{abs}} = \frac{\kappa_s}{\kappa_s + \kappa_0 (1 + 4\alpha^2/3)} \quad (40) $$
<figure><img src="image_5.png" /><figcaption>FIG. 4. Absorption efficiency analysis as a function of the pulse speed $\alpha$, increasing the ratio $g_{\text{ens}}/\Gamma$ from 0.05 (blue) to 0.5 (dark red), for zero cavity internal losses (a) and when $\kappa_0/\Gamma = 1/30$ (b). When the input pulse is slow enough ($\alpha < \alpha_{\text{abs}}^*$), the absorption efficiency is given by Eq. (40). Above this critical value $\alpha_{\text{abs}}^*$ (black crosses), which increases with $g_{\text{ens}}/\Gamma$, the efficiency drops. Its approximation by Eq. (42) is marked with red crosses. (c) Minimal pulse duration as a function of the spin-cavity coupling strength $g_{\text{ens}}$, for different values of the cavity internal losses $\kappa_0/2\pi$ (printed in the legend, in [MHz]). An increment of $g_{\text{ens}}$ allows for a faster input pulse, up to a point set by the spins' bandwidth $\Gamma$. A similar effect is observed when increasing $\kappa_0$, as you can see from the relative decrease between the different curves. We must recall that an increase of $\kappa_0$ corresponds as well to a decrease in efficiency, meaning that you are allowed to go faster but with an efficiency cost.</figcaption></figure>

Noting that

$$ \kappa_a^*(\tau) = |\kappa_0 + \Gamma + \frac{\Gamma(1 + 4\alpha^2(2 \operatorname{sech}(\alpha\tau)^2 - 1)) - \kappa_s}{2\alpha \tanh(\alpha\tau) - 1}|, \quad (41) $$

The maximum limit for speed comes from having the denominator well-defined and non-zero at all times, which corresponds here to $\alpha < 1/2$. Even below this limit, the nominator may present a cancellation point that would activate the absolute value. We find that after this critical speed $\alpha_{\text{abs}}^*$ (represented by black crosses in Fig. 4(b)), the optimal efficiency drops significantly due to this lack of a smooth control function, and Eq. (40) gives just an upper bound for $\eta_{\text{abs}}$. For hyperbolic secant pulses, we can derive that the cancellation point appears for

$$ \alpha_{\text{abs}}^* \approx \frac{1}{2} \min \left[ \frac{\kappa_s + \kappa_0}{\Gamma + \kappa_0}, 1 \right], \quad (42) $$

marked with red crosses in Fig. 4(b), which stand in the vicinity of the numerically evaluated $\alpha_{\text{abs}}^*$ (black crosses).

In absence of intrinsic losses on the cavity, this result implies that when $\kappa_s > \Gamma$ the optimal bandwidth to operate at full efficiency is at best $\Gamma/2$. Otherwise said, the memory protocol is only efficient for pulses whose bandwidth is below $\Gamma/2$, or equivalently, whose temporal extent is longer than $2/\Gamma$. For smaller couplings, i.e. $\kappa_s < \Gamma$, the admissible bandwidth is limited by $\kappa_s/2$, since $\alpha_{\text{abs}}^* \approx \kappa_s/2\Gamma$.

In Fig. 4(c), we compute the minimal temporal extent $1/\alpha_{\text{abs}}^*\Gamma$ as a function of $g_{\text{ens}}$ for different values of the intrinsic loss rate $\kappa_0$. Interestingly, the optimal operating bandwidth increases with intrinsic losses, but of course at the expense of protocol efficiency. Moreover,

the lower bound for this extent is given by $2/\Gamma$. This result has direct implications for the memory system. First, it limits its capacity in storing multiple temporal modes. Indeed, at best, one would be able to store in the memory $\sim T_2\Gamma/2$ temporal pulses, and in case of a hybrid system with a small ensemble coupling constant, only $\sim T_2\kappa_s/2$ pulses. In the case of second-long coherence time, this metric appears very optimistic since the capacity would then largely exceed the number of spins in the ensemble, which goes beyond the hypothesis of storing few excitations compared to the number of spins laid out in Sec. II. One should then consider another metric, which would account for the breakdown of the Holstein-Primakoff approximation.

This minimal temporal extent also imposes constraints for inserting the memory in a modular architecture. Let us consider the simplest architecture of one qubit able to emit on demand its quantum state in the input waveguide of the quantum memory, with a wavepacket envelope appropriate for perfect absorption into the memory. To be a faithful representation of its past state, the itinerant quantum state should have a temporal extent significantly smaller than the emitting qubit coherence times. However, if the corresponding bandwidth is too large, it may not be able to be absorbed with good efficiency into the memory, independently of its shape. Assuming a coherence time of 15 µs for a qubit connected in a quantum state transfer type architecture [40], it implies that the wavepacket should be emitted within the order of 1.5 µs. Assuming minimal intrinsic losses $\kappa_0/(2\pi) = 20$ kHz and a spin linewidth $\Gamma/(2\pi) = 300$ kHz, the ensemble coupling constant should then be larger than $g_{\text{ens}}/(2\pi) = 120$ kHz to be able to absorb efficiently this wavepacket. This constraint appears very reasonable in light of the
parameters we considered for promising spin systems.

The above analysis only concerns the optimal absorp-
tion efficiency. Concerning the optimal emission effi-
ciency, (37) provides an upper bound. This upper bound
is plotted in Fig. 5(a). This upper bound is attained
for slow enough pulses such that the cavity bandwidth
modulation $\kappa_e^*$ given by (35) satisfies the positivity con-
straint. In Fig. 5(a), this corresponds to $\alpha < \alpha_{em}^*$ de-
noted by magenta crosses. As can be seen in the plot,
these critical values are very close to the critical values
for the absorption process. Above these critical values,
the upper bound is not necessarily saturated and to find
the actual optimal emission efficiency, one needs to solve
a convex optimization problem with constraints as stated
before.

We conclude by evaluating the impact of intrinsic
losses on the total efficiency $\eta$ of the protocol. The nu-
merical result is depicted in Fig. 5(b). The parameter
$g_{\text{ens}}/\Gamma$ is set to 0.5 ($\alpha_{\text{em}}^* \sim \alpha_{\text{abs}}^* \sim 0.5$ for all values of
$\kappa_0$), and we plot the efficiency as a function of $\kappa_0/\Gamma$ for
$\alpha = 0.12, 0.25, 0.46$ and $\alpha = 0.69 > \alpha_{\text{abs}}^*, \alpha_{\text{em}}^*$. In this last
case of pulse speed above the critical point, the optimal
modulation for $\kappa_e$, given by Eq. (35), violates the posi-
tivity constraint. Consequently, we adopt the suboptimal
modulation $\kappa_e(t) = \kappa_a^*(T_E - t)$. This choice is motivated
by the fact that, for slow pulses, this time-reversed mod-
ulation is close to the optimal emission modulation (see
Fig. 7). Fig. 5(b) shows how much the efficiency associ-
ated to this suboptimal modulation (yellow, solid) devi-
ates from its upper bound (yellow, dash-dotted), which
is given by the product of the optimal absorption effi-
ciency and the upper bound for emission efficiency (see
Eq. (37)).

VII. CONCLUSION

In this work, we have presented a quantum cas-
caded model for absorption and emission of an itiner-
ant wavepacket into a spin ensemble embedded in a cav-
ity. This model is derived in a mean-field framework
and allowed us to derive an upper bound for the memory
catch and release efficiency depending on the hybrid sys-
tem parameters. Spin coherence dynamics, for which we
predict a limited impact were excluded from the present
derivation for conciseness, but could be straightforwardly
included. We demonstrate that this efficiency bound is
reached for adiabatic “slow” pulses and that there exists a
critical bandwidth below which the memory performance
remains close to this upper bound. Above this threshold,
the efficiency decreases severely. We also derived the re-
quired modulation of the cavity coupling to the input
waveguide to reach optimal absorption and emission of a
given wavepacket.

Another assumption of our model that could be easily
revisited is the hypothesis of perfect refocusing pulses.
While this allowed us to express simply the feedback oc-
curring between absorption and emission, a more pre-

<figure><img src="image_6.png" /></figure>

FIG. 5. (a) Emission efficiency analysis as a function of the pulse speed $\alpha$ using the same convention as in Fig. 4(b). Dash-dotted line is the theoretical upper bound given by Eq. (37), that is saturated for $\alpha < \alpha_{em}^*$ (magenta crosses), which closely matches $\alpha_{abs}^*$ (black crosses). (b) Efficiency versus cavity internal losses for the four pulse speeds in Fig. 3. For this coupling strength of $g_{ens} = \Gamma/2$, we have $\alpha_{em}^* \approx 0.5$ for all values of $\kappa_0$. Thus, the first three pulses remain below the critical speed and reach the upper bounds shown by green, blue and red solid curves. The fastest pulse exceeds this critical speed, so that the upper bound (dash-dotted) can not be reached. Taking the suboptimal choice of $\kappa_e(t) = \kappa_a(T_E - t)$ yields the solid yellow curve.

cise description of the action of the pulses, depending on
each spin frequency or coupling constant, could be taken
into account, which would allow us to derive a finite-
refocusing efficiency.

We conclude from this analysis that achieving in
practice a high-fidelity quantum memory will require a
bandwidth-tunable cavity with low intrinsic losses. Con-
cerning the spin ensemble and its coupling to the cavity,
the required parameters are within the reach of current
experimental conditions.

In future works, the results presented here could also
be compared to numerical simulations discretizing the
spin ensemble to study its dynamics [22] or to experimen-
tal realizations. Another theoretical development could
be to use a mean-field framework extended to the sec-
ond order to include the noise dynamics of the cavity
and of the spin ensemble in the present analysis, and see
whether this analysis would confirm the mean-field dy-
namics we have described here. Going further, outlining
under which conditions the current mean-field dynamics
would map to a full bosonic mode interaction dynam-
ics, where the spin-cavity system could be described by a
system density matrix, would be extremely helpful in the
context of modeling a modular architecture. Indeed, this
ability would allow for deriving transfer efficiency and fi-
delity in a very straightforward manner despite the high
number of degrees of freedom in the overall system and
its hybrid nature.
ACKNOWLEDGMENTS

L. Greggio, A. Petrescu and M. Mirrahimi acknowl-
edge funding from French ANR grant OCTAVES (ANR-
21-CE47-0007). T. Lorriaux acknowledges the support of
France 2030 project QuanTEdu-France ANR-22-CMAS-
0001, and A. Bienfait and T. Lorriaux acknowledge fund-
ing from the Plan France 2030 through the project ANR-
22-PETQ-0003.

Appendix A: Derivation of the EOMs

In this appendix, we derive the equations of motion
for the intra-cavity field absorbed by the spins and then
re-emitted at T<sub>E</sub> = 2τ<sub>a</sub> + 2τ<sub>e</sub>.

We recall that during absorption ($t \le \tau_a$), the dynam-
ics of the intra-cavity field and of $\sigma^j_-$ of spin $j$ is dictated
by the following EOMs (see [21])

$$
\begin{equation}
\begin{split}
\frac{d}{dt} \epsilon_a = & - \frac{\kappa_0 + \kappa_a(t) + i 2 \Delta_{cs}}{2} \epsilon_a \\
& + i \sum_{j=1}^{N} g_j \sigma_{-}^{j,a} + \sqrt{\kappa_a(t)} \epsilon_{in}(t),
\end{split}
\tag{A1}
\end{equation}
$$

$$
\frac{d}{dt} \sigma_{-}^{j,a} = -i \Delta_{j} \sigma_{-}^{j,a} + ig_{j} \epsilon_{a}. \quad (A2)
$$

We consider $\epsilon_a(-\infty) = \sigma_-^{j,a}(-\infty) = 0$ and we integrate Eq. (A2)

$$
\sigma_{-}^{j,a}(t) = ig_j \int_{-\infty}^{t} ds \, e^{-i\Delta_j(t-s)} \epsilon_a(s). \quad (A3)
$$

In the EOM for the intra-cavity field (A1), we replace
the term $\sum_{j=1}^N g_j \sigma_j^-$ with its integrated version, using
the spin frequency and coupling distributions

$$
\sum_{j=1}^{N} g_j \sigma_{-}^{j} \rightarrow \int_{g} \int_{\Delta} g \sigma_{-}^{\Delta,g} p(g) n(\Delta) dg d\Delta, \quad (A4)
$$

and we plug in the expression for $\sigma_{-}^{\Delta,g}$ given in Eq. (A3),
to obtain

$$
-g_{\text{ens}}^2 \int_{-\infty}^{t} ds \, \varepsilon_a(s) \int d\Delta n(\Delta) e^{-i\Delta(t-s)},
$$

with $g_{\text{ens}}^2 = \int p(g)g^2 dg$, and where we recognize the
Fourier transform of the frequency distribution $n(\Delta)$, i.e.
$n(t-s) = e^{-\Gamma|t-s|/2}$. We can thus rewrite the above ex-
pression as $-g_{\text{ens}}^2 \int_{-\infty}^t ds e^{-\Gamma|t-s|/2}\varepsilon_a(s)$, that we equalize
to $-g_{\text{ens}} \Sigma_a$, where $\Sigma_a(t) = g_{\text{ens}} \int_{-\infty}^t ds e^{-\Gamma(t-s)/2}\varepsilon_a(s)$,
with $\Sigma_a(-\infty) = 0$, representing the field stored in the
cavity.

Consequently, the spin-cavity EOMs during absorption
rewrite as

$$
\begin{equation}
\begin{split}
\frac{d}{dt} \epsilon_a = & - \frac{\kappa_0 + \kappa_a(t) + i 2 \Delta_{cs}}{2} \epsilon_a - g_{\text{ens}} \Sigma_a \\
& + \sqrt{\kappa_a(t)} \epsilon_{\text{in}},
\end{split}
\tag{A5}
\end{equation}
$$

$$
\frac{d}{dt} \Sigma_a = g_{\text{ens}} \epsilon_a - \frac{\Gamma}{2} \Sigma_a.
$$

Now, we present the EOM for $\sigma^j_-$ of spin $j$ in between
the two pulses ($\tau_a < t < T_E = 2\tau_a + \tau_e$), when the cavity
is detuned

$$
\frac{d}{dt} \sigma_{-}^{j,a/e} = +i \Delta_{j} \sigma_{-}^{j,a/e} \quad (A6)
$$

where the change of sign reflects the flipping caused
by the first refocusing pulse. Using the convention
$\sigma_{-}^{j,a/e}(\tau_{\text{a}}^{+}) = \sigma_{-}^{j,a}(\tau_{\text{a}}^{-})$ together with Eq. (A3), we obtain

$$
\begin{align}
\sigma_{-}^{j,a/e}(t) &= \sigma_{-}^{j,a/e}(\tau_a) e^{i\Delta_j(t-\tau_a)} \nonumber \\
&= ig_j \int_{-\infty}^{\tau_a} ds \, e^{i\Delta_j(s+t-2\tau_a)} \epsilon_a(s). \tag{A7}
\end{align}
$$

The EOMs during emission ($t \ge T_E = 2\tau_a + \tau_e$) write as
the ones during absorption (A2), but without the driving
term in the intra-cavity field EOM

$$
\frac{d}{dt} \epsilon_e = - \frac{\kappa_0 + \kappa_e(t) + i 2 \Delta_{cs}}{2} \epsilon_e + i \sum_{j=1}^{N} g_j \sigma_{-}^{j,e}, \quad (A8)
$$

$$
\frac{d}{dt} \sigma_{-}^{j,e} = -i \Delta_{j} \sigma_{-}^{j,e} + ig_{j} \epsilon_{e}. \quad (A9)
$$

We integrate Eq. (A9) using the convention $\sigma_{-}^{j,e}(T_{E}^{+}) = \sigma_{-}^{j,a/e}(T_{E}^{-})$ and we obtain

$$
\begin{align}
\sigma_{-}^{j,e}(t) &= \sigma_{-}^{j,e}(T_E) e^{i\Delta_j(T_E-t)} + ig_j \int_{T_E}^{t} ds \, e^{i\Delta_j(s-t)} \epsilon_e(s) \nonumber \\
&= ig_j \int_{-\infty}^{\tau_a} ds \, e^{i\Delta_j(s+T_E-t)} \epsilon_a(s) \nonumber \\
&\quad + ig_j \int_{T_E}^{t} ds \, e^{i\Delta_j(s-t)} \epsilon_e(s), \tag{A10}
\end{align}
$$

where we have used Eq. (A7). As done for absorption, we
replace the term $\sum_{j=1}^N g_j \sigma_j^-$ with its integrated version
(see Eq. (A4)), we substitute $\sigma_{-}^{\Delta,g}$ with its expression
(A10), we recognize the Fourier transform of $n$, and we
get

$$
-g_{\text{ens}}^2 \left( \begin{aligned} & \int_{-\infty}^{\tau_a} ds e^{-\frac{\Gamma}{2}|t-T_E-s|} \varepsilon_a(s) \\ + & \int_{T_E}^{t} ds e^{-\frac{\Gamma}{2}(t-s)} \varepsilon_e(s) \end{aligned} \right). \quad (A11)
$$

The second term can be expressed as $-g_{\text{ens}} \tilde{\Sigma}_e$,
which introduces a new variable $\tilde{\Sigma}_e(t) =$
<figure><img src="image_7.png" /><figcaption>FIG. 6. Same plots as in Fig. 3, but for a Lorentzian pulse. In (c)-(d) the printed efficiency for different values of the pulse speed is lower with respect to the relative one shown for a hyperbolic secant pulse (see Fig. 3(c)-(d)), especially for fast pulses. This demonstrates the better performance of a hyperbolic secant pulse compared to a Lorentzian pulse.</figcaption></figure>

$g_{\text{ens}} \int_{T_E}^{t} ds e^{-\frac{\Gamma}{2}(t-s)} \varepsilon_e(s)$, with $\tilde{\Sigma}_e(T_E) = 0$. To deal with the first integral we define the variable $\tilde{\Sigma}_a(t) = g_{\text{ens}} \int_{-\tau_a}^{t} ds e^{-\frac{\Gamma}{2}(t-s)} \varepsilon_a(-s)$, with $\tilde{\Sigma}_a(-\tau_a) = 0$; representing a filtered version of the field stored in the spins during absorption. Using these new variables, together with the previously defined $\Sigma_a$, we re-express Eq. (A11) as $-g_{\text{ens}} \Sigma_e$, where the field stored in the spin ensemble during emission is defined as

$$ \Sigma_e(t) = \begin{cases} \Sigma_a(t - T_E) + \tilde{\Sigma}_a(T_E - t) + \tilde{\Sigma}_e(t) & \text{if } t \le T_E + \tau_a \\ \Sigma_a(\tau_a) e^{-\frac{\Gamma}{2}(t - T_E - \tau_a)} + \tilde{\Sigma}_e(t) & \text{if } t > T_E + \tau_a \end{cases} $$

We can now present the EOMs for the field in the cavity and the one stored in the spins during emission

$$ \begin{aligned} \frac{d}{dt}\varepsilon_e &= -\frac{\kappa_0 + \kappa_e(t) + i2\Delta_{cs}}{2}\varepsilon_e - g_{\text{ens}}\Sigma_e, \\ \frac{d}{dt}\Sigma_e &= \begin{cases} g_{\text{ens}}\varepsilon_e - \frac{\Gamma}{2}\Sigma_e + \Gamma\tilde{\Sigma}_a(T_E - t) & \text{if } t \le T_E + \tau_a \\ g_{\text{ens}}\varepsilon_e - \frac{\Gamma}{2}\Sigma_e & \text{if } t > T_E + \tau_a \end{cases} \\ \frac{d}{dt}\tilde{\Sigma}_a &= -\frac{\Gamma}{2}\tilde{\Sigma}_a + g_{\text{ens}}\varepsilon_a(-t), \end{aligned} \quad (A12) $$

with $\tilde{\Sigma}_a(T_E - t)$ representing a feedback term from absorption, appearing when $t \le T_E + \tau_a$.

Concerning $\Sigma_e(t)$, continuity and differentiability are not lost in $t = T_E + \tau_a$. The initial conditions are $\varepsilon_e(T_E) = \varepsilon_a(\tau_a)e^{-\frac{\kappa_0}{2}(\tau_a + \tau_e)}$, $\Sigma_e(T_E) = \Sigma_a(-\tau_e) + \tilde{\Sigma}_a(\tau_e)$.

In our work, we neglect the cut-off of the feedback term since it does not affect the storage time interval, and we consider $\Sigma_e$ undergoing the following EOMs

$$ \dot{\Sigma}_e = g_{\text{ens}} \varepsilon_e - \frac{\Gamma}{2} \Sigma_e + \sqrt{\Gamma} \Sigma_{e,in}, \quad (A13) $$

with $\Sigma_{e,in} = \sqrt{\Gamma} \tilde{\Sigma}_a(T_E - t)$ being the feedback term from absorption.

Finally, we consider $\tau_a$ large enough such that the entire field has left the cavity at the moment of the first

refocusing pulse. This allows us to consider the intracavity field $\varepsilon_a$ to be zero after $\tau_a$, and the intra-cavity field $\varepsilon_e$ to be zero before $T_E$. The same consideration holds for the field stored in the spins since we can consider that before the refocusing pulses the information has been totally washed out by inhomogeneous broadening, namely $\Sigma_a$ and $\Sigma_e$ are zero after $\tau_a$ and before $T_E$ respectively. Under this assumption, we can consider $\varepsilon(\pm\infty) = \Sigma(\pm\infty) = 0$.

## Appendix B: Transmission and reflection coefficients

In this appendix, we aim to derive the expression for the transmission and reflection coefficients in the frequency domain, in the case of constant coupling $\kappa$ and detuning $\Delta_{cs}$.

We recall the definition of the transmission and reflection coefficients associated with absorption and emission

$$ \begin{aligned} t_a &= \Sigma_{a,out}/\varepsilon_{in}, & t_e &= \varepsilon_{e,out}/\Sigma_{e,in}, \\ r_a &= \varepsilon_{a,out}/\varepsilon_{in}, & r_e &= \Sigma_{e,out}/\Sigma_{e,in}, \end{aligned} \quad (B1) $$

with $t_a$ being the transmission coefficient from the driveline to the spins, $t_e$ being the one from the spins back to the driveline, $r_a$ being the reflection coefficient back to the driveline and $r_e$ being the one back to the spins.

We rewrite the absorption EOMs (A5) in the frequency domain, considering a non-zero detuning $\Delta_{cs}$ and a constant $\kappa_a$

$$ \begin{aligned} i\delta\varepsilon_a[\delta] &= -i\Delta_{cs}\varepsilon_a[\delta] - \frac{\kappa_0 + \kappa_a}{2}\varepsilon_a[\delta] - g_{\text{ens}}\Sigma_a[\delta] \\ &\quad + \sqrt{\kappa_a\varepsilon_{in}[\delta]}, \end{aligned} \quad (B2) \\ i\delta\Sigma_a[\delta] = g_{\text{ens}}\varepsilon_a[\delta] - \frac{\Gamma}{2}\Sigma_a[\delta]. \end{aligned} $$

Rearranging the terms we get $\Sigma_a[\delta]$ and $\varepsilon_a[\delta]$ as functions
of $\epsilon_{in}[\delta]$

$$ 
\begin{aligned}
\Sigma_a[\delta] &= \frac{4g_{\text{ens}}}{\Gamma} \sqrt{\kappa_a} \epsilon_{\text{in}}[\delta], \\
\epsilon_a[\delta] &= \frac{2\sqrt{\kappa_a} (1 + i \frac{2\delta}{\Gamma})}{d[\delta, \Delta_{\text{cs}}, \kappa_a]} \epsilon_{\text{in}}[\delta],
\end{aligned} 
\quad (B3) $$

with

$$ d[\delta, \Delta_{\text{cs}}, \kappa] = \kappa_0 + \kappa + \tilde{\kappa}_s + 2i(\tilde{\delta} + \Delta_{\text{cs}}) \quad (B4) $$

where $\kappa_s = 4g_{\text{ens}}^2/\Gamma$, $\tilde{\kappa}_s/\kappa_s = 1 - \frac{\delta(\delta+\Delta_{\text{cs}})}{g_{\text{ens}}^2}$ and $\tilde{\delta}/\delta = 1 + (\kappa_0+\kappa)/\Gamma$. We consider the absorption transmission and reflection coefficients defined in Eq. (B1) in the frequency domain, we substitute $\Sigma_{a,out}$ with $\sqrt{\Gamma}\Sigma_a$ and $\epsilon_{a,out}$ with $\sqrt{\kappa_a(t)}\epsilon_a - \epsilon_{in}$, and we plug in Eq. (B3) to obtain

$$ 
\begin{aligned}
t_a[\delta] &= \frac{2\sqrt{\kappa_s \kappa_a}}{d[\delta, \Delta_{\text{cs}}, \kappa_a]}, \\
r_a[\delta] &= \frac{2\kappa_a (1 + i \frac{2\delta}{\Gamma})}{d[\delta, \Delta_{\text{cs}}, \kappa_a]} - 1.
\end{aligned} 
\quad (B5) $$

We pass to emission and we rewrite the relative EOMs (first eq. in (A12) and Eq. (A13)) in the frequency domain, considering the detuning $\Delta_{cs}$ and a constant $\kappa_e$

$$ 
\begin{aligned}
i\delta\epsilon_e[\delta] &= -i\Delta_{\text{cs}}\epsilon_e[\delta] - \frac{\kappa_0 + \kappa_e}{2}\epsilon_e[\delta] - g_{\text{ens}}\Sigma_e[\delta], \\
i\delta\Sigma_e[\delta] &= g_{\text{ens}}\epsilon_e[\delta] - \frac{\Gamma}{2}\Sigma_e[\delta] + \sqrt{\Gamma}\Sigma_{e,in}[\delta].
\end{aligned} 
\quad (B6) $$

Rearranging the terms we get $\Sigma_e[\delta]$ and $\epsilon_e[\delta]$ as functions of the driving field $\Sigma_{e,in}[\delta]$

$$ 
\begin{aligned}
\Sigma_e[\delta] &= \frac{\frac{2}{\sqrt{\Gamma}}(2i(\delta + \Delta_{\text{cs}}) + \kappa_0 + \kappa_e)}{d[\delta, \Delta_{\text{cs}}, \kappa_e]} \Sigma_{e,in}[\delta], \\
\epsilon_e[\delta] &= -\frac{2\sqrt{\kappa_s}}{d[\delta, \Delta_{\text{cs}}, \kappa_e]} \Sigma_{e,in}[\delta].
\end{aligned} 
\quad (B7) $$

We consider the emission transmission and reflection coefficients defined in Eq. (B1) in the frequency domain, substituting $\epsilon_{e,out}$ with $\sqrt{\kappa_e}\epsilon_e$ and $\Sigma_{e,out}$ with $\sqrt{\Gamma}\Sigma_e - \Sigma_{e,in}$, and we plug in Eq. (B7) to obtain

$$ 
\begin{aligned}
t_e[\delta] &= -\frac{2\sqrt{\kappa_e \kappa_s}}{d[\delta, \Delta_{\text{cs}}, \kappa_e]}, \\
r_e[\delta] &= \frac{2(2i(\delta + \Delta_{\text{cs}}) + \kappa_0 + \kappa_e)}{d[\delta, \Delta_{\text{cs}}, \kappa_e]} - 1.
\end{aligned} 
\quad (B8) $$

Note that the transmission coefficient during emission represents a mirrored version of the one during absorption, i.e. $t_e[\delta] = -t_a[\delta]$, with relative coupling constant $\kappa_e$ and $\kappa_a$.

## Appendix C: Input-output energies

In this appendix, we provide the derivation of the inequality (25) leading to the upper bound for the absorption process, and the identity (28) required for treating the optimization of the emission process.

We recall that the input energy evaluated at the optimal $\tilde{\kappa}_a^*$ reads

$$ \min \int \mathcal{E}_{\text{in}}(\tau)^2 d\tau = \int \left|\frac{B}{A}\right| \left(A + B \left|\frac{A}{B}\right|\right)^2 d\tau, \quad (C1) $$

where $A(\tilde{\kappa}_a)$ and $B(\tilde{\kappa}_a)$ are defined in Eq. (21). The integrand $|\frac{B}{A}|(A+B|\frac{A}{B}|)^2$ can be rewritten as $2|AB| + 2AB$, which is lower bounded by $4AB$. Consequently, a lower bound for the input energy is given by

$$ 
\begin{aligned}
4 \int d\tau AB &= \\
&= \frac{1}{4g^2} \int d\tau [2(4\ddot{S}_a\dot{S}_a + 2(1 + \bar{\kappa}_0)\dot{S}_a^2 + (\bar{\kappa}_0 + \bar{\kappa}_s)S_a\dot{S}_a) \\
&\quad + 4\ddot{S}_aS_a + 2(1 + \bar{\kappa}_0)\dot{S}_aS_a + (\bar{\kappa}_0 + \bar{\kappa}_s)S_a^2] \\
&= \frac{1}{\bar{\kappa}_s} \int d\tau [4\bar{\kappa}_0\dot{S}_a^2 + (\bar{\kappa}_0 + \bar{\kappa}_s)S_a^2],
\end{aligned} 
$$

where we have used the fact that $S_a(\pm\infty) = \dot{S}_a(\pm\infty) = 0$. This proves the inequality (25). Now, for the emission process, we recall the EOMs

$$ 
\begin{aligned}
\frac{d}{d\tau}\mathcal{E}_e &= -\frac{\bar{\kappa}_0 + \bar{\kappa}_e(\tau)}{2}\mathcal{E}_e - g\mathcal{S}_e, \\
\frac{d}{d\tau}\mathcal{S}_e &= -\frac{1}{2}\mathcal{S}_e + g\mathcal{E}_e + \mathcal{S}_{\text{in}}(\bar{T}_E - \tau).
\end{aligned} 
\quad (C2) $$

We multiply the first equation by $\mathcal{E}_e$, and the second equation by $\mathcal{S}_e$, then integrate both of them over time to obtain

$$ 
\begin{aligned}
\int d\tau \mathcal{E}_e \dot{\mathcal{E}}_e &= \int d\tau \left( -\frac{\bar{\kappa}_0 + \bar{\kappa}_e(\tau)}{2} \mathcal{E}_e^2 - g \mathcal{S}_e \mathcal{E}_e \right), \\
\int d\tau \mathcal{S}_e \dot{\mathcal{S}}_e &= \int d\tau \left( g \mathcal{E}_e \mathcal{S}_e - \frac{1}{2} \mathcal{S}_e^2 + \mathcal{S}_e \mathcal{S}_{\text{in}} \right).
\end{aligned} 
\quad (C3) $$

We recall that $\mathcal{E}_e(\pm\infty) = \mathcal{S}_e(\pm\infty) = 0$, we sum the two equations in (C3) and we get

$$ 0 = \int d\tau \left( -\frac{\bar{\kappa}_0 + \bar{\kappa}_e(\tau)}{2} \mathcal{E}_e^2 - \frac{1}{2} \mathcal{S}_e^2 + \mathcal{S}_e \mathcal{S}_{\text{in}} \right), \quad (C4) $$

from which we derive the output energy as

$$ \int d\tau \bar{\kappa}_e(\tau) \mathcal{E}_e^2 = \int d\tau \left( -\bar{\kappa}_0 \mathcal{E}_e^2 - \mathcal{S}_e^2 + 2\mathcal{S}_e \mathcal{S}_{\text{in}} \right). \quad (C5) $$

We rewrite the second equation in (C2) as $\mathcal{E}_e = \frac{1}{g}(\dot{S}_e + \frac{1}{2}\mathcal{S}_e - \mathcal{S}_{\text{in}})$, we insert it in Eq. (C5) and we obtain

$$ 
\begin{aligned}
\int d\tau \bar{\kappa}_e(\tau) \mathcal{E}_e^2 &= \int d\tau \left( -\frac{\bar{\kappa}_0}{g^2} \left( \dot{S}_e + \frac{1}{2} \mathcal{S}_e - \mathcal{S}_{\text{in}} \right)^2 - \mathcal{S}_e^2 + 2\mathcal{S}_e \mathcal{S}_{\text{in}} \right).
\end{aligned} 
\quad (C6) $$

To pass to the frequency domain through Parseval's theorem, we get rid of the cross product $\mathcal{S}_e\mathcal{S}_{\text{in}}$ by rewriting the term $-\mathcal{S}_e^2+2\mathcal{S}_e\mathcal{S}_{\text{in}}$ as $-(\mathcal{S}_e-\mathcal{S}_{\text{in}})^2+\mathcal{S}_{\text{in}}^2$. Consequently, we obtain the identity (28).
<figure><img src="image_8.png" /><figcaption>FIG. 7. Same conventions as in of Fig. 3. Optimal shape for $\kappa_e(t)$ (solid line, as defined in Eq. (35)) compared to the time-reversed absorption profile $\kappa_a(T_E-t)$ (dashed-dotted line), for a slow hyperbolic secant pulse (a) and a slow Lorentzian pulse (b), both not far from the “slow-pulse” limit (black-dotted line). When $\alpha < \alpha_{em}^*$, $\kappa_e(t)$ closely resembles a mirrored version of the time-reversed $\kappa_a$, supporting the choice of using a mirrored coupling profile for emission in the fast-pulse regime ($\alpha \ge \alpha_{em}^*$).</figcaption></figure>

## Appendix D: Further numerical results

In this appendix, we present the simulation results for a Lorentzian pulse, showing its worse performance com-

pared to the chosen hyperbolic secant pulse in terms of efficiency. Moreover, we show that, for $\alpha < \alpha_{em}^*$, the optimal modulation for the coupling during emission is approximately a mirrored version of the one derived for the absorption step, justifying why we are taking this approximation above the critical speed, when the positivity constraint for the optimization problem should be active.

The Lorentzian pulse, normalized to unity, writes
$$ \Sigma_a(t) = \frac{\sqrt{2\alpha/\pi}}{1+(\alpha\Gamma t)^2}. $$
The numerics are implemented the same way as for the hyperbolic secant pulse (see Subsec. VIB). The results are presented in Fig. 6, showing a lower efficiency compared to the one obtained by the corresponding hyperbolic secant pulse.

The optimal $\kappa_e$, given by Eq. (35), for both types of pulses is presented in Fig. 7, together with the time-reversed version of the optimal $\kappa_a$, given by Eq. (23). The first one represents a slightly translated version of the second one, which allows us to perform the approximation.

[1] K. Heshami, D. G. England, P. C. Humphreys, P. J. Burtard, V. M. Acosta, J. Nunn, and B. J. S. and, Journal of Modern Optics **63**, 2005 (2016), pMID: 27695198, https://doi.org/10.1080/09500340.2016.1148212.

[2] H. J. Kimble, Nature **453**, 1023 (2008).

[3] L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, Nature **414**, 413 (2001).

[4] M. K. Bhaskar, R. Riedinger, B. Machielse, D. S. Levo-nian, C. T. Nguyen, E. N. Knall, H. Park, D. Englund, M. Lončar, D. D. Sukachev, and M. D. Lukin, Nature **580**, 60 (2020).

[5] N. Sangouard, C. Simon, H. de Riedmatten, and N. Gisin, Rev. Mod. Phys. **83**, 33 (2011).

[6] S. Zaiser, T. Rendler, I. Jakobi, T. Wolf, S.-Y. Lee, S. Wagner, V. Bergholm, T. Schulte-Herbrüggen, P. Neumann, and J. Wrachtrup, Nature Communications **7**, 12279 (2016).

[7] W. Ding, W. Zhang, and X. Wang, Phys. Rev. A **102**, 032612 (2020).

[8] E. Knill, R. Laflamme, and G. J. Milburn, Nature **409**, 46 (2001).

[9] D. Thaker, T. Metodi, A. Cross, I. Chuang, and F. Chong, in *33rd International Symposium on Computer Architecture (ISCA'06)* (2006) pp. 378-390.

[10] E. Gouzien and N. Sangouard, Phys. Rev. Lett. **127**, 140503 (2021).

[11] A. I. Lvovsky, B. C. Sanders, and W. Tittel, Nature Photonics **3**, 706 (2009).

[12] W. Tittel, M. Afzelius, T. Chaneliére, R. Cone, S. Kröll, S. Moiseev, and M. Sellars, Laser & Photonics Reviews **4**, 244 (2010).

[13] C. Clausen, I. Usmani, F. Bussières, N. Sangouard, M. Afzelius, H. de Riedmatten, and N. Gisin, Nature **469**, 508 (2011).

[14] P. Rabl, D. DeMille, J. M. Doyle, M. D. Lukin, R. J. Schoelkopf, and P. Zoller, Phys. Rev. Lett. **97**, 033003 (2006).

[15] M. Reagor, W. Pfaff, C. Axline, R. W. Heeres, N. Ofek, K. Sliwa, E. Holland, C. Wang, J. Blumoff, K. Chou, M. J. Hatridge, L. Frunzio, M. H. Devoret, L. Jiang, and R. J. Schoelkopf, Phys. Rev. B **94**, 014506 (2016).

[16] A. M. Bozkurt, S. Miles, S. L. D. ten Haaf, C.-X. Liu, F. Hassler, and M. Wimmer, “Interaction-induced strong zero modes in short quantum dot chains with time-reversal symmetry,” (2025), arXiv:2405.14940 [cond-mat.mes-hall].

[17] N. Sangouard, R. Dubessy, and C. Simon, Phys. Rev. A **79**, 042340 (2009).

[18] A. Reiserer and G. Rempe, Rev. Mod. Phys. **87**, 1379 (2015).

[19] C. E. Bradley, J. Randall, M. H. Abobeih, R. C. Berrevoets, M. J. Degen, M. A. Bakker, M. Markham, D. J. Twitchen, and T. H. Taminiau, Phys. Rev. X **9**, 031045 (2019).

[20] M. Afzelius and C. Simon, Phys. Rev. A **82**, 022310 (2010).

[21] M. Afzelius, N. Sangouard, G. Johansson, M. U. Staudt, and C. M. Wilson, New Journal of Physics **15**, 065008 (2013).

[22] B. Julsgaard, C. Grezes, P. Bertet, and K. Mølmer, Phys. Rev. Lett. **110**, 250503 (2013).
[23] K. Jahne, B. Yurke, and U. Gavish, Physical Review A 75, 010301 (2007).

[24] A. N. Korotkov, Physical Review B 84, 014510 (2011).

[25] E. Chatterjee, D. Soh, and M. Eichenfield, Journal of Physics A: Mathematical and Theoretical 55, 105302 (2022).

[26] C. W. Gardiner, Phys. Rev. Lett. 70, 2269 (1993).

[27] K. Shinbrough, D. R. Pearson, B. Fang, E. A. Goldschmidt, and V. O. Lorenz, in Advances In Atomic, Molecular, and Optical Physics, Vol. 72, edited by L. F. DiMauro, H. Perrin, and S. F. Yelin (Academic Press, 2023) pp. 297-360.

[28] J. O'Sullivan, O. W. Kennedy, K. Debnath, J. Alexander, C. W. Zollitsch, M. Šimėnas, A. Hashim, C. N. Thomas, S. Withington, I. Siddiqi, K. Mølmer, and J. J. L. Morton, Phys. Rev. X 12, 041014 (2022).

[29] C. Grezes, B. Julsgaard, Y. Kubo, W. L. Ma, M. Stern, A. Bienfait, K. Nakamura, J. Isoya, S. Onoda, T. Ohshima, V. Jacques, D. Vion, D. Esteve, R. B. Liu, K. Mølmer, and P. Bertet, Physical Review A 92, 020301 (2015).

[30] M. U. Staudt, I.-C. Hoi, P. Krantz, M. Sandberg, M. Simoen, P. Bushev, N. Sangouard, M. Afzelius, V. S. Shumeiko, G. Johansson, P. Delsing, and C. M. Wilson, Journal of Physics B: Atomic, Molecular and Optical Physics 45, 124019 (2012).

[31] J. Z. Bernád, M. Schilling, Y. Wen, M. M. Müller, T. Calarco, P. Bertet, and F. Motzoi, Journal of Physics B: Atomic, Molecular and Optical Physics 58, 035501 (2025).

[32] H. Primakoff and T. Holstein, Phys. Rev. 55, 1218 (1939).

[33] H. J. Carmichael, Phys. Rev. Lett. 70, 2273 (1993).

[34] C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Rev. Mod. Phys. 84, 621 (2012).

[35] G. Wolfowicz, A. M. Tyrshkin, R. E. George, H. Riemann, N. V. Abrosimov, P. Becker, H.-J. Pohl, M. L. W. Thewalt, S. A. Lyon, and J. J. L. Morton, Nature Nanotechnology 8, 561 (2013).

[36] W. Tittel, M. Afzelius, A. Kinos, L. Rippe, and A. Walther, "Quantum networks using rare-earth ions," (2025), arXiv:2501.06110 [quant-ph].

[37] E. Fraval, M. J. Sellars, and J. J. Longdell, Physical Review Letters 92, 077601 (2004).

[38] J. Alexander, G. Dold, O. W. Kennedy, M. Šimėnas, J. O'Sullivan, C. W. Zollitsch, S. Welinski, A. Ferrier, E. Lafitte-Houssat, T. Lindström, P. Goldner, and J. J. L. Morton, Physical Review B 106, 245416 (2022).

[39] A. Tiranov, E. Green, S. Hermans, E. Liu, F. Chiossi, D. Serrano, P. Loiseau, A. M. Kumar, S. Bertaina, A. Faraon, and P. Goldner, "Sub-second spin and lifetime-limited optical coherences in $^{171}$yb$^{3+}$:cawo$_{4}$," (2025), arXiv:2504.01592 [quant-ph].

[40] P. Kurpiers, P. Magnard, T. Walter, B. Royer, M. Pechal, J. Heinsoo, Y. Salathé, A. Akin, S. Storz, J. C. Besse, S. Gasparinetti, A. Blais, and A. Wallraff, Nature 558, 264 (2018).

[41] J. Niu, L. Zhang, Y. Liu, J. Qiu, W. Huang, J. Huang, H. Jia, J. Liu, Z. Tao, W. Wei, Y. Zhou, W. Zou, Y. Chen, X. Deng, X. Deng, C. Hu, L. Hu, J. Li, D. Tan, Y. Xu, F. Yan, T. Yan, S. Liu, Y. Zhong, A. N. Cleland, and D. Yu, Nature Electronics 6, 235 (2023).

[42] V. Ranjan, J. O'Sullivan, E. Albertinale, B. Albanese, T. Chanelière, T. Schenkel, D. Vion, D. Esteve, E. Flurin, J. J. L. Morton, and P. Bertet, Phys. Rev. Lett. 125, 210505 (2020).

[43] C. Grezes, B. Julsgaard, Y. Kubo, M. Stern, T. Umeda, J. Isoya, H. Sumiya, H. Abe, S. Onoda, T. Ohshima, V. Jacques, J. Esteve, D. Vion, D. Esteve, K. Mølmer, and P. Bertet, Phys. Rev. X 4, 021049 (2014).

[44] J. O'Sullivan, O. W. Kennedy, C. W. Zollitsch, M. Šimėnas, C. N. Thomas, L. V. Abdurakhimov, S. Withington, and J. J. Morton, Physical Review Applied 14, 064050 (2020).

[45] M. Kjaergaard, M. E. Schwartz, J. Braumüller, P. Krantz, J. I.-J. Wang, S. Gustavsson, and W. D. Oliver, Annual Review of Condensed Matter Physics 11, 369 (2020).

[46] J. Kerckhoff, R. W. Andrews, H. S. Ku, W. F. Kindel, K. Cicak, R. W. Simmonds, and K. W. Lehnert, Physical Review X 3, 021013 (2013).

[47] J. Grebel, H. Yan, M.-H. Chou, G. Andersson, C. R. Conner, Y. J. Joshi, J. M. Miller, R. G. Povey, H. Qiao, X. Wu, and A. N. Cleland, Physical Review Letters 132, 047001 (2024).

[48] C. Bockstiegel, Y. Wang, M. R. Vissers, L. F. Wei, S. Chaudhuri, J. Hubmayr, and J. Gao, Applied Physics Letters 108, 222604 (2016).