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

$$\begin{aligned} \hat{H}_{0}=-D\hat{S}_{z}^{2}+ \hat{H}_{\mathrm{tr}}-g\mu _{B}\hat{S}_{x}H_{0}, \end{aligned}$$

(1)

where \(\hat{H}_{\mathrm{tr}}\) is the transverse anisotropic energy operator. D is the longitudinal anisotropy constant, *g* is the Landé 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

$$\begin{aligned} E_{m}=-Dm^{2}. \end{aligned}$$

(2)

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

$$\begin{aligned} E_{m}^{\pm }\simeq E_{m}\pm \frac{1}{2} \triangle E_{m}, \end{aligned}$$

(3)

where \(\triangle E_{m}\) is the splitting of the *m*th level. Its expression is

$$\begin{aligned} \triangle E_{m}\simeq \frac{2D(S+m)!}{[(2m-1)!]^{2}(S-m)!} \biggl( \frac{g\mu _{B}H_{0}}{2D} \biggr)^{2m}, \end{aligned}$$

(4)

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

$$\begin{aligned} &\psi _{m}^{s} = \frac{1}{\sqrt{2}}(\psi _{m}+ \psi _{-m}), \end{aligned}$$

(5a)

$$\begin{aligned} &\psi _{m}^{a} = \frac{1}{\sqrt{2}}(\psi _{m}- \psi _{-m}), \end{aligned}$$

(5b)

where \(\psi _{\pm m}\) are eigenfunctions of the spin operator \(\hat{S}_{z}\), which can be obtained from the eigen-equation \(\hat{S}_{z}\psi _{\pm m}=\pm m\psi _{\pm m}\). For \(\mathrm{Fe} _{8}\) cluster, we obtain \(S=10\), \(D=0.31\) 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 \(H_{0}=14\) 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. 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 four-wave-mixing process. We assume that the propagation direction of four waves is along the *z*-axis, and \(H_{c}\) and \(\hat{H'}_{p}\) polarizes along the *y*-axis, while \(H_{d}\) and \(\hat{H'}_{m}\) polarizes along the *x*-axis as shown in Fig. 2. The magnetic field expression of two strong magnetic fields are \((H_{j}/2)e^{-i\omega _{j}(t-z/c)+i\phi _{j}}+c.c.\ (j=c,d)\) 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 \(\wp _{p1}=\sqrt{\frac{\hbar \omega _{13}}{2\varepsilon _{0}V}}\). *â* 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, 31–33]

$$\begin{aligned} &\hat{H}=\hat{H}_{0}+\hat{H}_{F}+\hat{H}_{L}+ \hat{H}_{C}, \end{aligned}$$

(6a)

$$\begin{aligned} &\hat{H}_{F}= \int\, \hbar \omega _{m}\hat{a}^{\dagger }_{m} \hat{a}_{m}\,d \omega _{m}+ \int\, \hbar \omega _{p}\hat{a}^{\dagger }_{p} \hat{a}_{p}\,d \omega _{p}, \end{aligned}$$

(6b)

$$\begin{aligned} &\hat{H}_{L}=-\frac{g\mu _{B}}{2}\sum_{j} \hat{\vec{S}}\cdot \vec{H}_{j}e^{-i \omega _{j}(t-z/c)+i\phi _{j}}+H.c., \end{aligned}$$

(6c)

$$\begin{aligned} &\hat{H}_{C}=-\frac{g\mu _{B}}{2}\sum_{k} \hat{\vec{S}}\cdot \vec{\hat{H'}}_{k}+H.c., \end{aligned}$$

(6d)

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]

$$\begin{aligned} \bigl|\psi (t)\bigr\rangle = \bigl\vert \psi _{1}(t)\bigr\rangle + \bigl\vert \psi _{2}(t)\bigr\rangle + \bigl\vert \psi _{3}(t) \bigr\rangle + \bigl\vert \psi _{4}(t)\bigr\rangle , \end{aligned}$$

(7)

with

$$\begin{aligned} \bigl|\psi _{1}(t)\bigr\rangle = {}& \int\, d\omega _{p}f_{\omega _{p}}(t)\hat{a}^{ \dagger }_{p} \vert 0\rangle _{p} \vert 0\rangle _{m} \vert 1 \rangle \\ &{} + \int\, d\omega _{m}f_{\omega _{m}}(t)\hat{a}^{\dagger }_{m} \vert 0\rangle _{p} \vert 0\rangle _{m} \vert 1 \rangle, \end{aligned}$$

(8a)

$$\begin{aligned} \bigl|\psi _{2}(t)\bigr\rangle ={}& \sum g(t)\hat{\sigma }_{21} \vert 0\rangle _{p} \vert 0 \rangle _{m} \vert 1 \rangle, \end{aligned}$$

(8b)

$$\begin{aligned} \bigl\vert \psi _{3}(t)\bigr\rangle ={}& \sum b_{1}(t)\hat{ \sigma }_{31} \vert 0\rangle _{p} \vert 0 \rangle _{m} \vert 1\rangle, \end{aligned}$$

(8c)

$$\begin{aligned} \bigl\vert \psi _{4}(t)\bigr\rangle ={}& \sum b_{2}(t)\hat{ \sigma }_{41} \vert 0\rangle _{p} \vert 0 \rangle _{m} |1\rangle. \end{aligned}$$

(8d)

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 Schrödinger equation and obtain

$$\begin{aligned} &i\partial _{t}f_{\omega _{p}}(t)=\omega _{p}f_{\omega _{p}}(t)- \frac{1}{\hbar }Ng_{p} b_{1}(t)e^{-i\omega _{p}z/c}, \end{aligned}$$

(9a)

$$\begin{aligned} &i\partial _{t}f_{\omega _{m}}(t)=\omega _{m}f_{\omega _{m}}(t)- \frac{1}{\hbar }Ng_{m} b_{2}(t)e^{-i\omega _{m}z/c}, \end{aligned}$$

(9b)

and

$$\begin{aligned} &i\partial _{t}g(t)=\omega _{2}g(t)- \bigl[ \bigl(\Omega _{c}^{0}(z,t) \bigr)^{*}b_{1}(t)+ \bigl( \Omega _{d}^{0}(z,t) \bigr)^{*}b_{2}(t) \bigr], \end{aligned}$$

(10a)

$$\begin{aligned} &i\partial _{t}b_{1}(t)=\omega _{3}b_{1}(t)-g(t) \Omega _{c}^{0}(z,t)- \frac{g_{p}}{\hbar } \int\, d\omega _{p}f_{\omega _{p}}(t)e^{i\omega _{p}z/c}, \end{aligned}$$

(10b)

$$\begin{aligned} &i\partial _{t}b_{2}(t)=\omega _{4}b_{2}(t)-g(t) \Omega _{d}^{0}(z,t)- \frac{g_{m}}{\hbar } \int\, d\omega _{m}f_{\omega _{m}}(t)e^{i\omega _{m}z/c}. \end{aligned}$$

(10c)

We have defined \(\Omega _{c}^{0}(z,t)=\Omega _{c}e^{-i\omega _{c}(t-z/c)+i\phi _{c}}\) and \(\Omega _{d}^{0}(z,t)=\Omega _{d}e^{-i\omega _{d}(t-z/c)+i\phi _{d}}\). \(\Omega _{c}\) 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. (9a)–(9b) by \(e^{i\omega _{p}z/c}\), the quantum field amplitudes propagation equation can be obtained as

$$\begin{aligned} &\biggl(\frac{1}{c}\partial _{t}+\partial _{z} \biggr)H_{p}(z,t)=i \kappa _{12}\beta _{1}(z,t), \end{aligned}$$

(11a)

$$\begin{aligned} &\biggl(\frac{1}{c}\partial _{t}+\partial _{z} \biggr)H_{m}(z,t)=i \kappa _{14}\beta _{2}(z,t), \end{aligned}$$

(11b)

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. (11a)–(11b). With these definitions, Eq. (10a)–(10c) read

$$\begin{aligned} &\partial _{t}g(z,t)=i \bigl[e^{-i\phi _{c}}\Omega _{c}^{*}\beta _{1}(z,t)+e^{-i \phi _{d}}\Omega _{d}^{*}\beta _{2}(z,t) \bigr]-i\omega _{2}g(z,t), \end{aligned}$$

(12a)

$$\begin{aligned} &\partial _{t}\beta _{1}(z,t)=i\triangle _{1} \beta _{1}(z,t)+iH_{p}(z,t)+ig(z,t) \Omega _{c}e^{i\phi _{c}}, \end{aligned}$$

(12b)

$$\begin{aligned} &\partial _{t}\beta _{2}(z,t)=i\triangle _{2} \beta _{2}(z,t)+iH_{m}(z,t)+ig(z,t) \Omega _{d}e^{i\phi _{d}}, \end{aligned}$$

(12c)

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. (12a)–(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. (12a)–(12c) and obtain

$$\begin{aligned} &\tilde{g}(z,\omega )= \frac{e^{-i\phi _{c}}\Omega _{c}^{*}\tilde{\beta }_{1}(z,\omega )+e^{-i\phi _{d}}\Omega _{d}^{*}\tilde{\beta }_{2}(z,\omega )}{\omega _{2}-\omega }, \end{aligned}$$

(13a)

$$\begin{aligned} &\widetilde{\beta }_{1}(z,\omega )=\kappa _{12} \frac{- \vert \Omega _{d} \vert ^{2}+(\omega -\omega _{2})(\Delta _{2}+\omega )}{D(\omega )} \widetilde{H}_{p}(z, \omega ) \\ &\phantom{\widetilde{\beta }_{1}(z,\omega )=}{}+\kappa _{12} \frac{e^{i\phi _{cd}}\Omega _{c}\Omega _{d}^{*}}{D(\omega )} \widetilde{H}_{m}(z, \omega ), \end{aligned}$$

(13b)

$$\begin{aligned} &\widetilde{\beta }_{2}(z,\omega )=\kappa _{14} \frac{- \vert \Omega _{c} \vert ^{2}+(\omega -\omega _{2})(\Delta _{1}+\omega )}{D(\omega )} \widetilde{H}_{p}(z, \omega ) \\ &\phantom{\widetilde{\beta }_{2}(z,\omega )=}{}+\kappa _{14} \frac{e^{i\phi _{dc}}\Omega _{d}\Omega _{c}^{*}}{D(\omega )} \widetilde{H}_{m}(z, \omega ), \end{aligned}$$

(13c)

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 *ω* is the Fourier variable. We substitute Eq. (13a)–(13c) into Fourier-transformd Eq. (11a)–(11b) to obtain

$$\begin{aligned} &\partial _{z}\widetilde{H}_{p}(z, \omega )- \frac{i \omega }{c} \widetilde{H}_{p}(z, \omega ) \\ &\quad= iK_{1}(\omega )\widetilde{H}_{p}(z, \omega )+iK_{2}(\omega ) \widetilde{H}_{m}(z, \omega ), \end{aligned}$$

(14a)

$$\begin{aligned} &\partial _{z}\widetilde{H}_{m}(z, \omega )- \frac{i \omega }{c} \widetilde{H}_{m}(z, \omega ) \\ &\quad = iK_{3}(\omega )\widetilde{H}_{m}(z, \omega ) +iK_{4}(\omega ) \widetilde{H}_{p}(z, \omega ), \end{aligned}$$

(14b)

with

$$\begin{aligned} &K_{1}(\omega )=\kappa _{12} \frac{- \vert \Omega _{d} \vert ^{2}+(w-\omega _{2})(\Delta _{2}+\omega )}{D(\omega )}, \end{aligned}$$

(15a)

$$\begin{aligned} &K_{2}(\omega )=\kappa _{12} \frac{e^{i\phi _{cd}}\Omega _{c}\Omega _{d}^{*}}{D(\omega )}, \end{aligned}$$

(15b)

$$\begin{aligned} &K_{3}(\omega )=\kappa _{14} \frac{- \vert \Omega _{c} \vert ^{2}+(w-\omega _{2}) (\Delta _{1}+\omega )}{D(\omega )}, \end{aligned}$$

(15c)

$$\begin{aligned} &K_{4}(\omega )=\kappa _{14} \frac{e^{i\phi _{dc}} \Omega _{d}\Omega _{c}^{*}}{D(\omega )}, \end{aligned}$$

(15d)

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. (14a)–(14b) can be obtained as follows,

$$\begin{aligned} \widetilde{H}_{p}(z, \omega )={}& \bigl(R_{1}e^{i\lambda _{1}(\omega )z}c+R_{2}e^{i \lambda _{2}(\omega )z} \bigr)\widetilde{H}_{p}(0, \omega ) \\ &{}+R_{3} \bigl(e^{i\lambda _{2}(\omega )z}-e^{i\lambda _{1}(\omega )z} \bigr) \widetilde{H}_{m}(0,\omega ), \end{aligned}$$

(16a)

$$\begin{aligned} \widetilde{H}_{m}(z, \omega )={}&R_{4} \bigl(e^{i\lambda _{2}(\omega )z}-e^{i \lambda _{1}(\omega )z} \bigr)\widetilde{H}_{p}(0, \omega ) \\ &{}+ \bigl(R_{1}e^{i\lambda _{2}(\omega )z}+R_{2}e^{i\lambda _{1}(\omega )z} \bigr) \widetilde{H}_{m}(0, \omega ), \end{aligned}$$

(16b)

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

$$\begin{aligned} \lambda _{1(2)}{(\omega )}=\frac{w}{c}+ \frac{1}{2} \bigl(K_{1}(\omega )+K_{3}( \omega )\mp K_{5}(\omega ) \bigr), \end{aligned}$$

(17)

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, 32–37]. Hence, the inverse Fourier Transform of Eq. (16a)–(16b) is given by

$$\begin{aligned} H_{p}(z,t)={}& \bigl[(A_{1}H_{p}(\eta _{1})-A_{3}H_{m}(\eta _{1}) \bigr]e^{i\alpha _{1}z} \\ &{}+ \bigl[(A_{2}H_{p}(\eta _{2})+A_{3}H_{m}( \eta _{2}) \bigr]e^{i\alpha _{2}z}, \end{aligned}$$

(18a)

$$\begin{aligned} H_{m}(z,t)={}& \bigl[(A_{2}H_{m}(\eta _{1})-A_{4}H_{p}(\eta _{1}) \bigr]e^{i\alpha _{1}z} \\ &{}+ \bigl[(A_{4}H_{p}(\eta _{2})+A_{1}H_{m}( \eta _{2}) \bigr]e^{i\alpha _{2}z}, \end{aligned}$$

(18b)

where \(\eta _{1(2)}=t-z/V_{1(2)}\), \(\alpha _{1(2)}=\frac{1}{2}(K_{1}(0)+K_{3}(0)\mp K_{5}(0))\) and the group velocities of two modes are defined as

$$\begin{aligned} \frac{1}{V_{1(2)}}= \frac{1}{c}+\frac{1}{2} \bigl(K_{1m}(0)+K_{3m}(0)\mp K_{5m}(0) \bigr), \end{aligned}$$

(19)

where \(K_{1m}(0), K_{3m}(0)\) and \(K_{5m}(0)\) are given in the Appendix. From equations (18a)–(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 Δ*τ* which is the ratio of \(\delta _{1}\) to the \(\tau _{0}(10^{7}s^{-1})\) in Fig. 3. From Fig. 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 Δ*τ* is small. Therefore, we can obtain two sets of speed-matched probe-mixing magnetic field pairs that reach the detector after time of delay. When the Δ*τ* 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)–(18b), the expressions for \(|H_{p}(z,t)|^{2}\) and \(|H_{m}(z,t)|^{2}\) can be given as

$$\begin{aligned} \bigl\vert H_{p}(z,t) \bigr\vert ^{2}={}& \vert a \vert ^{2} \bigl\vert H_{p}(\eta ) \bigr\vert ^{2}+ \vert b \vert ^{2} \bigl\vert H_{m}( \eta ) \bigr\vert ^{2}+ab^{*}H_{p}( \eta )H_{m}^{*}(\eta ) \end{aligned}$$

(20a)

$$\begin{aligned} &{}+a^{*}bH_{p}^{*}(\eta )H_{m}(\eta ), \\ \bigl\vert H_{m}(z,t) \bigr\vert ^{2}={}& \vert c \vert ^{2} \bigl\vert H_{p}(\eta ) \bigr\vert ^{2}+ \vert d \vert ^{2} \bigl\vert H_{m}( \eta ) \bigr\vert ^{2}+cd^{*}H_{p}( \eta )H_{m}^{*}(\eta ) \\ &{}+c^{*}dH_{p}^{*}(\eta )H_{m}(\eta ), \end{aligned}$$

(20b)

where the coefficient \(a,b,c\) and *d* are defined as

$$\begin{aligned} &a=A_{1}e^{i\alpha _{1}z}+A_{2}e^{i\alpha _{2}z}, \end{aligned}$$

(21a)

$$\begin{aligned} &b=A_{3} \bigl(e^{i\alpha _{2}z}-e^{i\alpha _{1}z} \bigr), \end{aligned}$$

(21b)

$$\begin{aligned} &c=A_{4} \bigl(e^{i\alpha _{2}z}-e^{i\alpha _{1}z} \bigr), \end{aligned}$$

(21c)

$$\begin{aligned} &d=A_{1}e^{i\alpha _{2}z}+A_{2}e^{i\alpha _{1}z}. \end{aligned}$$

(21d)

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

$$\begin{aligned} \begin{aligned} H_{p}(z,t)H_{m}^{*}(z,t)={}& ac^{*} \bigl\vert H_{p}(\eta ) \bigr\vert ^{2}+bd^{*} \bigl\vert H_{m}( \eta ) \bigr\vert ^{2} \\ &{} +ad^{*}H_{p}(\eta )H_{m}^{*}(\eta )+c^{*}bH_{p}^{*}(\eta )H_{m}( \eta ). \end{aligned} \end{aligned}$$

(22)

Afterwards, by analyzing expressions in Eq. (20b)–(21a) and Eq. (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

$$\begin{aligned} I_{p(m)}(z)= \frac{ \vert H_{p(m)}(z,t_{c}) \vert ^{2}}{ \vert H_{p(m)}(z,t_{c}) \vert ^{2}+ \vert H_{m(p)}(z,t_{c}) \vert ^{2}}. \end{aligned}$$

(23)

We plot the relative phase and intensity of probe and mixing magnetic field as a function of \(z/z_{0}\) in Fig. 4 and Fig. 5. Figure 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

$$\begin{aligned} \Phi _{12}(z)=\operatorname{Arg} \bigl[H_{p}(z,t_{c})H_{m}^{*}(z,t_{c}) \bigr]. \end{aligned}$$

(24)