1 Introduction

The storage of quantum states and information in long-lasting modes is crucial for integrating quantum computing and communication devices in quantum networks [1–4]. Significant progress has been made in realizing these memories using rare-earth ion-doped crystals [5–11], atomic ensembles [12], single atoms [13, 14], nitrogen-vacancy centers [15], optomechanical systems [16–21], and electromechanical systems [22–25]. Furthermore, the magnon, which is a collective excitation of magnetization, is emerging as a crucial component in hybrid quantum networks [26–31]. Much of its appeal is derived from the strong magnon–photon coupling and the easily-reached nonlinear regime in microwave cavities [31]. The magnon mode can serve as a transducer between microwave and optical photons [26–30], and experimental demonstrations of coherent coupling between magnons, superconducting qubits, and diamond spins have been achieved [32–34]. Direct implementation of storage in magnonic systems has also garnered attention [35, 36]. Magnon dark modes have been utilized to demonstrate a coherent, broadband, and multimode gradient memory [35]. Nonetheless, the coherence time of the storage is restricted to hundreds of nanoseconds due to the limited lifetime of magnons. As the magnetostrictive force facilitates the coupling between magnons and phonons, cavity magnomechanical storage has been theoretically proposed, taking advantage of the prolonged coherence time of phonons, and has shown great potential to achieve long storage time, high efficiency and high fidelity quantum memory [37].

In this work, we report experimental studies of magnomechanical microwave storage for the first time. In cavity magnomechanical systems, we use the magnomechanical induced transparency (MMIT) effect to achieve coherent microwave storage, which can be subsequently extracted by employing a retrieval pulse. In this way, microwave photons can be stored for times longer than the magnon lifetime, due to the lower damping rate of the phononic mode. When the ambient temperature of the sample is reduced to 8 K, we can obtain a mechanical mode with a narrow linewidth as low as 6.4 Hz and the storage lifetime of 24.8 ms. Furthermore, we adopt Ramsey’s method of separated oscillatory fields to study the coherence of the magnomechanical memory [38, 39]. The mechanical interference can then be used to assess the decoherence lifetime of 19.5 ms. Thus our proposed scheme offers a possibility of using magnomechanical systems as quantum memory for photonic quantum information.

2 Theoretical model

For a YIG-sphere placed under a biased magnetic field, the magnon mode of the sphere can couple with the deformation phonon modes via a magnetostrictive force and the single magnon-phonon coupling strength is g as shown in Fig. 1(a). Here, we consider the magnomechanical coupling between a magnon mode and a mechanical oscillator, with the coupling driven by a strong external microwave field at the red sideband, i.e., one mechanical frequency \(\omega _{b}\) below the magnon resonance \(\omega _{m}\), as shown schematically in Fig. 1(b). The magnetostrictive forces lead to a radiation pressure-like interaction between the magnon mode and the mechanical mode, represented by the Hamiltonian:

$$ \begin{aligned} H&=\omega _{m}m^{\dagger}m+\omega _{b}b^{\dagger}b+g \bigl(b^{\dagger}+b \bigr)m^{\dagger}m \\ &\quad {}+i\sqrt{\kappa _{in}}\varepsilon _{d} \bigl(m^{\dagger}e^{-i\omega _{d}t}-me^{i \omega _{d}t} \bigr) \end{aligned} $$
(1)

where m and b are the annihilation operators of the magnon and mechanical modes, respectively. \(\varepsilon _{d}\) is the drive field of the microwave with the frequency of \(\omega _{d}\), and the \(\kappa _{in}\) is the transforming rate from microwave photon to magnon. In the frame rotation of \(H_{0}=\omega _{d}m^{\dagger}m\), we can make a unitary transformation \(m\rightarrow m+ \langle m \rangle \). Therefore, we can approximately treat the pump field as a constant field: \(\langle m \rangle = \frac{\sqrt{\kappa _{in}}\varepsilon _{d}}{-i (\omega _{d}-\omega _{m} )+\frac{\kappa _{m}}{2}}\), where \(\kappa _{m}\) represent the dissipation of magnon. When the magnon mode is excited via a strong pump fixed at one mechanical frequency below the magnon resonance (\(\omega _{m}=\omega _{d}+\omega _{b}\)), and further lead to an interaction \(G(b^{\dagger}m+bm^{\dagger})\), where \(G= \langle m \rangle g\) is the enhanced coupling strength between magnon and phonon. As shown in Eq. (1), the external driving field couples the signal field to the mechanical displacement, which can induce a coherent interconversion between the microwave and motional states.

Figure 1
figure 1

(a) Schematic of the magnon-phonon interaction in a magnomechanical resonator. (b-c) Spectral position and the schematic of the pulse sequence used in microwave photons coherent storage experiments

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For the storage of magnomechanical microwave photons, external microwave pulses applied on the red sideband facilitate the conversion between the microwave signal field and the mechanical excitation during the “writing” and “readout” processes, which are similar to the interactions in cavity optomechanical systems [40–50]. In the storage process, the writing pulse couples a signal pulse resonant with the magnon resonance to the mechanical oscillator, generating a coherent mechanical excitation, as shown in Fig. 1(b-c). In the retrieval process, a readout pulse interacts with the stored mechanical excitation, converting the mechanical excitation back into a microwave pulse resonant with the magnon resonance.

In the magnomechanical systems, the control field at the red sideband couples to a signal field at the magnon resonance (\(\omega _{m}=\omega _{d}+ \omega _{b}\)), this results in the generation of a coherent mechanical excitation through the interconversion process discussed above. A microwave field is generated by anti-Stokes scattering of the control field at the magnon resonance. The excitation of the magnon mode is prevented by destructive interference between the intracavity field generated by the anti-Stokes scattering and that generated by the input signal, which lead to MMIT.

Considering the magnon and mechanical damping processes, the dynamics of the magnomechanical system excited by an input field nearly resonant with the magnon resonance can be described by the following equations of motion:

$$\begin{aligned}& \frac{dm}{dt} = \biggl[i(\omega _{s}-\omega _{m})- \frac{\kappa _{m}}{2}\biggr]m-iGb+ \sqrt{\kappa _{in}}E, \end{aligned}$$
(2)
$$\begin{aligned}& \frac{db}{dt} = \biggl[i(\omega _{s}-\omega _{d}-\omega _{b})- \frac{\kappa _{b}}{2}\biggr]b-iGm, \end{aligned}$$
(3)

where \(\kappa _{b}\) represent the dissipation of phonon. \(\omega _{s}\) and E are the frequency and amplitude of the input microwave signal field, respectively.

3 Experimental results

Figure 2(a) shows a diagram of the experimental setup for the studies of both microwave storage and MMIT under the same or similar experimental conditions. A YIG microsphere with a diameter of \(600~\mu \text{m}\) is attached to a fiber stem with a diameter near \(125~\mu \text{m}\). Firstly, a uniform magnon mode can be supported in the YIG microsphere by a bias magnetic field \(H_{ex}\), which is parallel to the equatorial plane of the YIG microsphere. The relationship between frequency of magnon and magnetic field intensity satisfies \(\omega _{m}=\gamma H_{ex}\), where \(\gamma =2\pi \times 2.8\text{ MHz/Oe}\) is the gyromagnetic ratio. The magnon mode is excited by an antenna placed near the YIG microsphere. In our experiments we put the YIG microsphere and the antenna in a Montana cryogenic chamber. Figure 2(b) shows the normalized microwave reflection spectrum \(S_{21}\) via a network analyzer at 8 K temperature. In our experiment, the magnon mode frequency is \(\omega _{m}/2\pi =4.007935\text{ GHz}\), the frequency of the mechanical mode is \(\omega _{b}/2\pi =7.935\text{ MHz}\), as shown in Fig. 2(b). Based on these interactions, the theoretical calculations are achieved with the parameters of \(\kappa _{m}/2\pi =6.6\text{ MHz}\), \(\kappa _{in}/2\pi =1.6\text{ MHz}\), \(\kappa _{b}/2\pi =6.4\text{ Hz}\). Then we measure the temperature dependence of the mechanical frequency and linewidth from 8 K to 250 K, as shown in Fig. 2(c-d). When the temperature drops from 250 K to 8 K, the center frequency of the mechanical mode increases by 73 kHz, and the linewidth decreases from 25 Hz to 7 Hz.

Figure 2
figure 2

(a) Schematic of the experimental setup for measurement of magnomechanical microwave storge. SRC: microwave source; VNA: vector network analyzer; ESA: electrical spectrum analyzer. (b) Measured microwave reflection spectrum around 4 GHz at 8 K temperature. The linewidth is 6.4 Hz according to the theoretical fitting result. (c-d) The frequency shift and linewidth of mechanical mode versus temperature from 8 K to 250 K

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Then we present experimental results on magnomechanical microwave photons storage. Two microwave sources are used to generate the control pulse and signal pulse with frequency of 4 GHz and 4.007935 GHz, respectively. The duration of the writing and readout pulses are \(\tau _{1}=20\text{ ms}\) and \(\tau _{2}=1\text{ ms}\) with a delay time ΔT. And the writing and signal pulses are synchronized with a repetition rate of 10 Hz. The control pulse and signal pulse are amplified by a microwave amplifier, then coupled by an antenna into the YIG microsphere cavity and reflected into the ESA. The result shown in Fig. 3(a-c), the energy of the retrieved pulse decreases with increasing separation between the writing and readout pulse, representing the decay of the coherent mechanical excitation induced by the signal and the writing pulse. The storage lifetime is determined by the damping time of the mechanical mode. Next, we plot the retrieved pulse intensity as a function of delays from 1.5 ms to 79.5 ms, and the storage lifetime is fitted to 24.8 ms, corresponding to a linewidth of 6.4 Hz, as shown in Fig. 3(d). It is consistent with the mechanical linewidth measured through the reflection spectrum, as shown in Fig. 2(b). When the temperature rises to 100 K, as shown in Fig. 3(e), the memory lifetime has been reduced to 7.8 ms, and the linewidth of the corresponding mechanical mode is fitted to 17.4 Hz by Lorentz fitting, as shown in the illustration of Fig. 3(e). We can also find that as the temperature rises, the retrieved pulses intensity also decrease.

Figure 3
figure 3

(a-c) The measured retrieved signal when the writing pulse and the read pulse have different time delays (5.5 ms, 39.5 ms, 77.5 ms) at 8 K temperature. (d) At a temperature of 8 K, the intensity of the retrieved pulse with various delays from 1.5 ms to 79.5 ms. The solid line is the fitting result with the lifetime parameter of \(T=24.8\text{ ms}\). (e) The intensity of the retrieved pulse with various delays from 1.5 ms to 35.5 ms at the temperature of 100 K. The solid line is the fitting result with the lifetime T of 7.8 ms. Inset: The illustration shows the mechanical mode obtained by the VNA and the solid line is the theoretical result of Lorentz linewidth of 17 Hz

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We have demonstrated the magnomechanical memory with an energy decay time of 24.8 ms. We then study the decoherence lifetime of the memory, which is an important figure of merit for coherent application. This coherent charactrization can be achieved through Ramsey interferometry, which has so far been used in the study of coherence in optomechanical systems [38]. In order to realize a magnomechanical Ramsey interferometry, we achieve magnomechanical coupling in two time-separated regions. During this process, a pair of microwave pulses, including both a driving pulse and a probe pulse, is introduced into the cavity, as illustrated in Fig. 4(a-b). We denote the widths of the pulses by \(\tau _{1}\) and \(\tau _{2}\) and the separation by ΔT. The probe pulse, with frequency \(\omega _{p}\), is near the magnon resonance \(\omega _{m}\) with detuning \(\delta =\omega _{m}-\omega _{p}\), and the driving pulse, with frequency \(\omega _{d}\), is at the red sideband of the magnon resonance \(\omega _{m}=\omega _{d}+\omega _{b}\), with \(\omega _{b}\) being the mechanical frequency. These two pairs of pulses are directed into the YIG cavity. While a pulse pair resides within the cavity, two distinct processes occur: (i) the control and probe microwave photons merge to create coherent phonons, and (ii) the coherent phonons interact with the control microwave photons to generate an anti-Stokes sideband near the magnon resonance. Before the second pulse pair is applied, the mechanical mode barely decays but gathers a phase \(\omega _{b}\Delta T\). These two paths coherent character leads to Ramsey fringes in the microwave output field. In our experiment, the first pulse pair with \(\tau _{1}=20\text{ ms}\) prepares the initial coherent phonons, which interfere with the coherent phonons generated by the second pulse pair with \(\tau _{2}=1\text{ ms}\). The detuning δ between probe frequency and magnon resonance is around 200 Hz, and the Ramsey signal intensity is measured by ESA by changing different ΔT as shown in Fig. 4(c). The solid line is the fitting result with the coherent lifetime of 19.5 ms.

Figure 4
figure 4

(a-b) The spectral position and the sketch of the pulse sequence applied to the magnomechanical Ramsey interferometry. (c) Detected probe intensity as a function of the delay time ΔT and the solid line is the fitting result with the coherent lifetime parameter of \(T=19.5\text{ ms}\) at the detuning δ of 200 Hz

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4 Discussion

In this work, we have carried out a experimental study on magnomechanical microwave photons storage. This study has demonstrated the coherent nature of the microwave photons storage process and has shown that the cryogenic cooling can narrow the linewidth of the mechanical mode. Since the mechanical damping is reduced at low temperatures, we can obtain an energy storage lifetime of microwave photons up to 24.8 ms and decoherence lifetime of 19.5 ms, which is currently limited by the linewidth of the mechanical mode. In addition, thermal phonons causing the destruction of quantum states hinder the utilization of our present system as a quantum memory. However, successful coherent coupling between 10 GHz phonons and magnons has been demonstrated in YIG thin films [51], the integrated magnomechanical quantum storage is achievable by leveraging such high phonon frequency and maintaining ambient temperatures as low as 1 K [37]. By designing the mechanical system with the phonon shield structures [24], the long-lifetime integrated quantum memory is accomplishable and would be an important element for the quantum photonic chips [52].