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Storage and retrieval of microwave pulse in a crystal of molecular magnets based on fourwavemixing
EPJ Quantum Technology volumeÂ 9, ArticleÂ number:Â 10 (2022)
Abstract
In this paper, we consider a crystal of molecular magnets interacting with four alternating magnetic fields. When a DC magnetic field applies to molecular magnets, the energy levels of molecular magnets can be recognized as a fourlevel system. We consider four wave mixing process in the crystal of molecular magnets, By solving the SchrÃ¶dinger equation, the analytic solutions of the probe and mixing magnetic fields are obtained. We have also numerically investigated the dynamical evolution of the probe and mixing magnetic fields. The results show that probe and mixing filed periodically oscillate in the crystal of molecular magnets medium. By adjusting the frequency detuning and the intensity of the coupled magnetic fields, the storage and retrieval of microwave field can be achieved in molecular magnets. In the end, the secondorder correlation function is calculated and the antibunching effect can be achieved in this magnetic medium.
1 Introduction
In recent years, the optical quantum storage has become an important focus of research activity [1â€“10]. Due to the importance of quantum optical memory, there are more and more researchers focusing on quantum storage. Based on balanced twochannel electromagnetically induced transparency, Zhu et al. [1] achieve a quantum memory for singlephoton polarization qubits in lasercooled rubidium atoms and the fidelity is higher than 99%. In particular, the efficiency of storing and retrieving of singlephoton temporal waveforms can be reached as 90.6% in the singlechannel quantum memory. By using timereversal method, Phillips et al. [10] have shown that a light pulse can be trapped and stored in an optically thick medium and the maximum efficiency of storing and retrieving of optical pulses is obtained.
Besides, the nanoscale molecular magnets has aroused great interest because of its quantum magnetic properties [11â€“21]. Petukhov et al. [13] have studied the spin dynamics of molecular magnets in timeresolved magnetization experiments. They found that the phonon bottleneck with 10â€“100Â ms characteristic relaxation time which had a great influence on the magnetization kinetics of molecular magnets. Misiorny et al. [15] have proposed a magnetic switch process implementation scheme based on molecular magnets spin inversion. With the latest advances in nanotechnology, the transmission characteristics of molecular magnets have been studied experimentally and theoretically. The oscillation and wave propagation in noninteracting molecular magnet systems have been studied extensively [22â€“26]. When molecular magnets is subject to a DC magnetic field, the energy level of a singlemolecule magnet split. By adjusting the parameters, we can obtain the corresponding energy level. The corresponding transition frequency is in the range of microwave. Take \(\mathrm{Fe} _{8}\) as an example, its transition frequency \(\omega _{31}\) is about \(10^{11}s^{1}\) when \(H_{0}=14\)Â koe. We propose a scheme to achieve storage and retrieval of microwave pulse in a crystal of molecular magnets. It is noted that molecular magnets have distinguished features, such as long spin decoherence time, lowdensity and sensitivity to external magnetic field, temperature, and so on. Takahashi et al. [27] measured the spin decoherence time \(T_{2}\sim 0.7\) microseconds for \(\mathrm{Fe} _{8}\) by highfrequency pulsed electron paramagnetic resonance. Takahashi et al. [28]also find that the decoherence time varies strongly as a function of temperature and magnetic field, and the optimal decoherence time rises to \(T_{2}=500\)Â ms. So the spin decoherence time may be relevant to engineering molecular magnets for quantum information processing applications. It is important to achieve the storage of microwave photons in a singlemolecule magnet because highfrequency waves support very high bandwidth. Besides, their high directionality reduces the possibility of eavesdropping, so our research may have potential applications in quantum communications. Shvetsov and coworkers [22] consider the electromagnetically induced transparency (EIT) in a crystal of molecular magnets which is driven by two (weak and strong) magnetic fields. The results show that EIT effect can be achieved in a crystal of molecular magnets and the imaginary part of the susceptibility is slightly effected by the distribution of longitudinal anisotropy constant D, whereas the effect of EIT is not suppressed. In the present study, we extend the former research on EIT [22] and fourwave mixing [29] to investigate the properties of quantum magnetic field in a crystal of molecular magnets. Firstly, we obtain the analytical solution for the probe and mixing quantum field. Secondly, we present an alternative scheme for achieving storage and retrieval of probe pulse in crystals of molecular magnets. The frequency of pulse is in range of microwave and the range of application is different from optical pulse in atomic medium. Thirdly, we give an alternative scheme for generating microwave pairs photons in crystals of molecular magnets and achieving the antibunching effect for microwave photons.
The paper is structured as follows. In Sec.II, we first describe the physical model and obtain the propagation equation of the probe and mixing magnetic field; then we obtain group velocities, absorption coefficient and relative phase by solving the dynamic evolution of the field equation. In Sec.III, we propose a scheme to achieve the storage and retrieval of microwave photons. In Sec.IV, we prepare a pair of photon and calculate the secondorder correlation function between them. Eventually, we summarize the results of this work and give the conclusion in Sec. V.
2 Theoretical model and Hamiltonian
We consider a crystal of noninteracting molecular magnets which is subject to the DC magnetic field \(H_{0}\). If z axis is the easy anisotropy axis of magnetic molecule and a DC magnetic field \(H_{0} \) perpendicular to the z axis is applied to the crystal. Ten the Hamiltonian of this system can be written as
where \(\hat{H}_{\mathrm{tr}}\) is the transverse anisotropic energy operator. D is the longitudinal anisotropy constant, g is the LandÃ© factor, and \(\mu _{B}\) is the Bohr magneton. \(\hat{S}_{x}\), \(\hat{S}_{y}\), and \(\hat{S}_{z}\) are the \(x, y \), and z components of the spin operator. For \(\mathrm{Fe} _{8}\) molecules, the operator of the transverse anisotropy energy is \(\hat{H}_{\mathrm{tr}} =K\hat{S}_{y}^{2}\), where K is the transverse anisotropy constant. The transverse anisotropy may be considered as a small perturbation relative to the longitudinal anisotropy, so we can ignore transverse anisotropy. Then the energy level of molecules without considering DC magnetic field can be written as
Due to the DC magnetic field, the molecule energy levels split. Using the method in [30], the expression of the energy level can be written as
where \(\triangle E_{m}\) is the splitting of the mth level. Its expression is
with S the molecule spin. We denote the eigenfunctions corresponding to \(E_{m}^{}\) and \(E_{m}^{+}\) by \(\psi _{m}^{s}\) and \(\psi _{m}^{a}\), respectively. \(\psi _{m}^{s}\) (\(\psi _{m}^{a}\)) is a symmetric (antisymmetric) function with the expression
where \(\psi _{\pm m}\) are eigenfunctions of the spin operator \(\hat{S}_{z}\), which can be obtained from the eigenequation \(\hat{S}_{z}\psi _{\pm m}=\pm m\psi _{\pm m}\). For \(\mathrm{Fe} _{8}\) cluster, we obtain \(S=10\), \(D=0.31\)Â K, \(g=2\) and \(K/D=0.4\). We specify \(\varepsilon _{1}=E_{10}^{}\), \(\varepsilon _{2}=E_{10}^{+}\), \(\varepsilon _{3}=E_{9}^{}\) and \(\varepsilon _{4}=E_{9}^{+}\) with \(\omega _{ef}=\omega _{e}\omega _{f}/\hbar \) \((e\neq f=1\sim 4)\) denoting the corresponding transition frequencies. For \(H_{0}=14\)Â koe, we get \(\omega _{21}=0.58\times 10^{6}s^{1}\), \(\omega _{43}=2.3\times 10^{8}s^{1}\), \(\omega _{31}=8.1\times 10^{11}s^{1}\). The results show that the frequency of probe magnetic field is in the range of microwave. Next, we discuss the property of microwave field in this magnetic medium. As shown in Fig.Â 1, we can only consider the four lowest energy levels. The strong magnetic fields \(H_{c}\) (\(H_{d}\)) couple the state \(2\rangle \) to an excited state \(3\rangle \) \((4\rangle )\) with the frequency \(\omega _{c} (\omega _{d})\) while a quantum field \(\hat{H'}_{p}\) drives the transition from the ground state \(1\rangle \) to an excited state \(3\rangle \) at the frequency \(\omega _{p}\). Then a mixing magnetic field \(\hat{H'}_{m}\) appears at the frequency \(\omega _{m}\) and couples the ground state \(1\rangle \) to an excited state \(4\rangle \) by a fourwavemixing process. We assume that the propagation direction of four waves is along the zaxis, and \(H_{c}\) and \(\hat{H'}_{p}\) polarizes along the yaxis, while \(H_{d}\) and \(\hat{H'}_{m}\) polarizes along the xaxis as shown in Fig.Â 2. The magnetic field expression of two strong magnetic fields are \((H_{j}/2)e^{i\omega _{j}(tz/c)+i\phi _{j}}+c.c.\ (j=c,d)\) and the weak magnetic fields read \(\hat{H'}_{k}=\int (\sqcap (\omega _{k})\wp _{k1}/2)\hat{a}_{k}e^{i \omega _{k}z/c}\,d\omega _{k}\ (k=m, p)\), where c is the speed of light in a vacuum and \(\sqcap ({\omega })\) a boxcar function, \(\wp _{m1}=\sqrt{\frac{\hbar \omega _{14}}{2\varepsilon _{0}V}}\) and \(\wp _{p1}=\sqrt{\frac{\hbar \omega _{13}}{2\varepsilon _{0}V}}\). Ã¢ is the annihilation field operator, V is quantized volume and \(\varepsilon _{0}\) is the dielectric constant. Under dipole approximation, the Hamiltonian of this system is [29, 31â€“33]
where the symbol \(H.c\). means the Hermitian conjugate, \(\hat{H}_{F}\) represents two weak magnetic fields, \(\hat{H}_{L}\) denotes the interaction between the magnetic molecule and two strong magnetic fields, while \(\hat{H}_{C}\) represent the interaction between the magnetic molecule and two quantum fields. For simplicity, we assume the population of the system initially stay in the ground state. The state of the system has the general form [34]
with
The notation \(n_{1}\rangle _{p}n_{2}\rangle _{m}\) means the number of photons in modes \(\omega _{p}\) and \(\omega _{m}\), \(n\rangle \) represent corresponding eigenstates of the molecule level. \(b_{1}(t)\) and \(b_{2}(t)\) stand for the probability amplitudes of the state \(3\rangle \) and \(4\rangle \), and \(g(t)\) denotes the probability amplitude of state \(2\rangle \). \(f_{\omega _{p}}(t)\) and \(f_{\omega _{m}}(t)\) are the wave packet envelope functions of the probe and mixing magnetic fields, respectively. Those functions give a complete description of the state of this system [34]. In order to find their evolution, we insert the \(\langle 0_{m}\langle 1_{p}\langle 1\), \(\langle 1_{m}\langle 0_{p}\langle 1\), \(\langle 0_{m}\langle 0_{p}\langle 2\), \(\langle 0_{m}\langle 0_{p}\langle 3\), \(\langle 0_{m}\langle 0_{p}\langle 4\) into the SchrÃ¶dinger equation and obtain
and
We have defined \(\Omega _{c}^{0}(z,t)=\Omega _{c}e^{i\omega _{c}(tz/c)+i\phi _{c}}\) and \(\Omega _{d}^{0}(z,t)=\Omega _{d}e^{i\omega _{d}(tz/c)+i\phi _{d}}\). \(\Omega _{c}\) and \(\Omega _{c}\) are the Rabi frequencies of the strong magnetic fields which are defined as \(\Omega _{c}=\frac{g\mu _{B}H_{c}}{2}\langle 3\hat{S}_{y}2\rangle \) and \(\Omega _{d}=\frac{g\mu _{B}H_{d}}{2}\langle 4\hat{S}_{x}2\rangle \). If we set \(\kappa _{12}=\frac{Ng_{p}^{2}}{\hbar ^{2}c}\), \(\kappa _{14}=\frac{Ng_{m}^{2}}{\hbar ^{2}c}\), where N is the total number of molecules, \(g_{p}\) and \(g_{m}\) are the coupling constants which are defined as \(g_{p}=\frac{g\mu _{B}\wp _{p1}}{2}\langle 3\hat{S}_{y}1\rangle \) and \(g_{m}=\frac{g\mu _{B}\wp _{m1}}{2}\langle 4\hat{S}_{x}1\rangle \). After multiplying Eq.Â (9a)â€“(9b) by \(e^{i\omega _{p}z/c}\), the quantum field amplitudes propagation equation can be obtained as
We have set \(\beta _{1}(z,t)=b_{1}(t)e^{i\omega _{p}(t\frac{z}{c})}\), \(\beta _{2}(z,t)=b_{2}(t)e^{i\omega _{m}(t\frac{z}{c})}\), \(H_{p}e^{i\omega _{p}(t\frac{z}{c})}=\frac{g_{p}}{\hbar }\int\, d \omega _{p} f_{\omega _{p}}(t)\times e^{i\omega _{p}z/c}\) and \(H_{m}e^{i\omega _{m}(t\frac{z}{c})}=\frac{g_{m}}{\hbar }\int\, d \omega _{m}f_{\omega _{m}}(t)e^{i\omega _{m}z/c}\) in Eq.Â (11a)â€“(11b). With these definitions, Eq.Â (10a)â€“(10c) read
where \(\Delta _{1}=\delta _{1}+i\frac{\gamma _{1}}{2}\) and \(\Delta _{2}=\delta _{2}+i\frac{\gamma _{2}}{2}\), the \(\gamma _{1}\) and \(\gamma _{2}\) are the decay rate of level \(3\rangle \) and \(4\rangle \), respectively. Here all the energy differnces in Eq.Â (12a)â€“(12c) are taken relative to the energy of ground state \(1\rangle \). In order to obtain the solution of these equations, we first take the Fourier transform of Eq.Â (12a)â€“(12c) and obtain
where \(\phi _{cd}= \phi _{c} \phi _{d}\) is the phase difference between the coupling fields. \(\tilde{g}(z,\omega )\), \(\widetilde{H}_{p(m)}(z, \omega )\) and \(\widetilde{\beta }_{1(2)}(z, \omega )\) are the Fourier transforms of \(g(z,t)\), \(H_{p(m)}(z,t)\) and \(\beta _{1(2)}(z,t)\), respectively, and Ï‰ is the Fourier variable. We substitute Eq.Â (13a)â€“(13c) into Fouriertransformd Eq.Â (11a)â€“(11b) to obtain
with
where \(D(\omega )=(\Delta _{1}+\omega )\Omega _{d}^{2}+(\Delta _{2}+ \omega )\Omega _{c}^{2}(\Delta _{1}+\omega )(\Delta _{2}+\omega )( \omega \omega _{2})\). The solution of Eq.Â (14a)â€“(14b) can be obtained as follows,
where \(\widetilde{H}_{p}(0,w)\) and \(\widetilde{H}_{m}(0,w)\) are the initial condition at the \(z=0\), and the expression of \(\lambda _{1(2)}\) are defined as
where the expression of \(K_{5}(\omega )\), \(R_{1}\), \(R_{2}\), \(R_{3}\) and \(R_{4}\) are given in the Appendix. We only focus on the adiabatic regime, where \(\lambda _{1(2)}\) can be extended to the fast convergence power series of dimensionless transformation variables, i.e., \(\lambda _{1(2)}=(\lambda _{1(2)})_{w=0}+w/V_{1(2)}+\mathcal{O}(w^{2})\), \(\alpha _{1(2)}=(\lambda _{1(2)})_{w=0}\), \(R_{i}=R_{i}(\omega )_{w=0}+\mathcal{O}(\omega )\) and \(A_{i}=R_{i}(\omega )_{w=0}\ (i=1,2,3,4)\) [29, 32â€“37]. Hence, the inverse Fourier Transform of Eq.Â (16a)â€“(16b) is given by
where \(\eta _{1(2)}=tz/V_{1(2)}\), \(\alpha _{1(2)}=\frac{1}{2}(K_{1}(0)+K_{3}(0)\mp K_{5}(0))\) and the group velocities of two modes are defined as
where \(K_{1m}(0), K_{3m}(0)\) and \(K_{5m}(0)\) are given in the Appendix. From equations (18a)â€“(18b), we obtain that the probe and mixing magnetic fields contain two propagation modes for the general frequency component, and the two modes of magnetic fields have independent group velocities. We plot the group velocity of probe and mixing magnetic field as a function of dimensionless Î”Ï„ which is the ratio of \(\delta _{1}\) to the \(\tau _{0}(10^{7}s^{1})\) in Fig.Â 3. From Fig.Â 3, we see that one wave packet mode propagates with negative group velocity and another wave packet mode propagates with positive group velocity in the range of \(2<\Delta \tau < 2\). We also obtain slow light when Î”Ï„ is small. Therefore, we can obtain two sets of speedmatched probemixing magnetic field pairs that reach the detector after time of delay. When the Î”Ï„ is small, the group velocities of the two propagation modes are equal. As a result, a pair of fields with matching group velocity can be obtained.
In what follows, we consider the group velocities \(V_{1} =V_{2}=V\), i.e., \(\eta _{1}=\eta _{2}=\eta \). We can then obtain the intensity of probe and mixing magnetic field. From Equations (18a)â€“(18b), the expressions for \(H_{p}(z,t)^{2}\) and \(H_{m}(z,t)^{2}\) can be given as
where the coefficient \(a,b,c\) and d are defined as
If we define the relative phase between probe and mixing magnetic fields as \(\Phi =\operatorname{Arg}[H_{p}(z,t)H_{m}^{*}(z,t)]\), so we can find the relative phase from the following equation
Afterwards, by analyzing expressions in Eq.Â (20b)â€“(21a) and Eq.Â (22), we discuss different propagation of probe and mixing magnetic field by fixing the reference system at the peak of the weak photon pulse \((t_{c} = z/v)\), Therefore, we only need to show the intensity of probe and mixing magnetic field changes in the space dimension z. The normalized intensity of probe and mixing magnetic fields are defined as
We plot the relative phase and intensity of probe and mixing magnetic field as a function of \(z/z_{0}\) in Fig.Â 4 and Fig.Â 5. FigureÂ 4 indicates that the intensity and the relative phase of probe and mixing magnetic field periodically oscillate in the crystal of molecular magnets medium. This implies that one can recover probe field at the output of the medium by properly choosing the parameter. The relative phase between two modes is
3 Numerical simulations and analysis
In this section, we show the microwave probe pulse propagation by numerically integrating Eq.Â (13a)â€“(13c). Moreover, we also present the process of storage and retrieval of a microwave probe pulse. To simulate the pulse propagation in time and distance, we adopt Gaussian profiles of temporal width \(\tau =10\ \mu s\) centered at \(t_{c}=7.5\tau \). For the singlemolecularmagnets medium, we select \(z_{0} = 0.5\)Â mm, \(\gamma _{1}\tau = 50\), \(\gamma _{2}\tau = 60\), \(\kappa _{12} \tau z_{0} =2\times 10^{4}\), \(\kappa _{14}\tau z_{0} =3\times 10^{4}\), \(\Omega _{c}\tau =\Omega _{d}\tau =1000 \), \(\delta _{1} = 0\) and \(\delta _{2}\tau = 1.5\times 10^{2}\), With these parameters, the intensity of the probe pulse and FWM pulse as a function of position and time is shown in Fig.Â 6. We can see that the intensity of microwave probe pulse and microwave FWM pulse exhibit complementary periodic oscillations in the process of propagation by the evolution equation of the numerical integrated system. Namely, we can obtain output photon with frequency of \(\omega _{m}\), when the input photon frequency is \(\omega _{p}\) The behavior of the intensities of probe pulse and FWM pulse fits well with the theoretical model.
In order to achieve the storage and retrieval process, we use the control beams of form
with \(\Omega _{c}(0)\tau =\Omega _{d}(0)\tau =1000\), \(\delta _{1}=0\), \(\delta _{2}\tau =1.5\times 10^{2}\), \(\sigma =1/\tau \), \(t_{1}=11\tau \), \(t_{2}=24\tau \), and other parameter are the same in the plot Fig.Â 3. We show an example of the microwave probe pulse can be recovered by appropriately choosing the control fields in Fig.Â 7(a), the storage time of the probe pulse is approximately given by \(t_{2}t_{1}=13\tau \). Shown in Fig.Â 7(b) is the temporal profile of the control field(blue solid line) and initial microwave probe pulse(red dashed line) of the medium.
4 Generating paired photons and twophoton intensity correlation function
In this section, we consider when the injected quantum state is a singlephoton wave packet of the specific form follows [38â€“41]
Here, \(P_{1}\) means amplitude and satisfies \(\int _{\infty }^{\infty }\,d\omega ^{\prime }P_{1}^{2}=1\), the Ï– is the central frequency of wave packet. We assume that only one photon with central frequency \(\omega _{p}\) and no phonon with central frequency \(\omega _{m}\), Hence, the initial state for the system is
In general, at any given time, the state of a molecular magnet system can be written as
Using the identity \(\sum_{n m} \vert \alpha _{n m}(t) \vert ^{2}=1\), we can work out the coefficients \(\alpha _{n m}(t)\) according to the following equation:
where \(\hat{H}_{p}\) and \(\hat{H}_{m}\) denote the probe and mixing field operators and F denotes the combinations of products of the field operators. In the case of a small gain, there is one photon generated with the frequency \(\omega _{p}\) when a singleprobe photon is injected the crystal of molecular magnets. At the same time, a mixing photon is produced at the frequency \(\omega _{m}\) by stimulated Raman process. We can write the final state of the system as
The physical meaning of each is clear. \(\alpha _{10}\) indicates the probability with which the injected probe photon is in the ground state. \(\alpha _{20} \) describes the photons generated by the excitation emission in the probe mode, but no mixing photon is produced. \(\alpha _{20} \) is the probability amplitude of the photons produced by the stimulus in the probe mode and the photon generated in the mixing mode. The second term in the equation exists when the pumping field is very weak, which can excite the FWM process. When the pump field is strong enough, this term \(\alpha _{20}\) will disappear from the equation. In this case, the state of the system is simplified to the following form
With the help of Eq.Â (29), the expression of \(\alpha _{21}^{2}\) and \(\alpha _{10}^{2}\) can be obtained as
It is worth noting that for high gain media, higher order terms such as \(\alpha _{32}  3 \rangle _{p}  2 \rangle _{m}\) will appear in Eq.Â (31), then we can obtain photon pairs with frequencies \(\omega _{p}\) and \(\omega _{m}\) by theoretical calculations. In order to show the timedependent properties of the generated photon pairs, we work out the correlation function between the \(\omega _{p}\) and \(\omega _{m}\) with a time delay \(\tau _{d}\),
Using the Eq.Â (29), and the intensity correlation function \(G_{H_{1}H_{2}}^{(2)} (\tau _{d} )\) can be expressed as
The secondorder correlation function \(g_{H_{1}H_{2}}^{(2)} (\tau _{d} )\) is defined as \(g_{H_{1}H_{2}}^{(2)} (\tau _{d} )=G_{H_{1}H_{2}}^{(2)} (\tau _{d} ) / G_{H_{1}}^{(1)}(0) G_{H_{2}}^{(1)}(0)\). In order to discuss the quantum properties of probe and mixing magnetic fields, we also plot the secondorder correlation function as a function of delay time \(\tau _{d}\) in Fig.Â 8. The evolution of secondorder correlation function is similar to that of the damped Rabi oscillations. The antibunching effect of weak magnetic field is achieved when \(g_{H_{1}H_{2}}^{(2)} (\tau _{d} )<1\). When a photon of frequency \(\omega _{p}\) is generated, then a photon of frequency \(\omega _{m}\) is generated by fourwave mixing after a delay \(\tau _{d}\). Thus, a pair of photons of frequencies \(\omega _{p}\) and \(\omega _{m}\) is obtained in the crystal of molecular magnets system.
5 Conclusion
In this paper, we study the propagation dynamics of probe and mixing magnetic field in the crystal of molecular magnets. In the adibatic regime, we obtain the analytic solution for the probe and mixing magnetic field with two different modes. For suitable parameter range, the results show that two modes have similar group velocity and can retain the initial shape for propagating in the magnetic medium. The intensity and relative phase of probe and mixing magnetic field in the crystal of molecular magnets medium exhibit periodic oscillation in propagation. Also, we have demonstrated that, at certain positions in the magnetic medium, the microwave single photon superposition state can be stored and recovered by adjusting the intensity of coupling magnetic field. Finally, we calculate the secondorder correlation function of probe photon and mixing photon, yielding microwave correlated photons pair and achieving the antibunching effect for microwave photons in the crystal of molecular magnets.
Availability of data and materials
The data sets supporting the results of this article are included within the article.
Abbreviations
 EIT:

electromagnetically induced transparency
 DC:

directcurrent
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Acknowledgements
We first thank Dr. Chen Li for revising the English of this article. This project is supported by the Program for Innovative Teams of Outstanding Young and Middleaged Researchers in the Higher Education Institutions of Hubei Province (Grant No. T2020014), and in part by the program of Outstanding Innovation Team of Hubei Normal University (No: T201902).
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This work is supported by HBNU.
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JC, JL together conceived the idea for the study. JC and NL performed the numerical calculation and plotted the figures. All the authors discussed and got the conclusion of the manuscript. HL and JL wrote the manuscript. All authors reviewed the manuscript. All authors read and approved the final manuscript.
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Appendix
Appendix
Here we give expression of relevant terms in Eqs.Â (16a)â€“(16b), (17) and (19) in the main article:
and
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Chen, J., Liu, N., Shan, C. et al. Storage and retrieval of microwave pulse in a crystal of molecular magnets based on fourwavemixing. EPJ Quantum Technol. 9, 10 (2022). https://doi.org/10.1140/epjqt/s4050702200128z
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DOI: https://doi.org/10.1140/epjqt/s4050702200128z