Storage and retrieval of microwave pulse in a crystal of molecular magnets based on four-wave-mixing

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 four-level system. We consider four wave mixing process in the crystal of molecular magnets, By solving the Schrödinger 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 second-order correlation function is calculated and the anti-bunching 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 [1–10]. Due to the importance of quantum optical memory, there are more and more researchers focusing on quantum storage. Based on balanced two-channel electromagnetically induced transparency, Zhu et al. [1] achieve a quantum memory for single-photon polarization qubits in laser-cooled rubidium atoms and the fidelity is higher than 99%. In particular, the efficiency of storing and retrieving of single-photon temporal waveforms can be reached as 90.6% in the single-channel quantum memory. By using time-reversal 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 nano-scale molecular magnets has aroused great interest because of its quantum magnetic properties [11–21]. Petukhov et al. [13] have studied the spin dynamics of molecular magnets in time-resolved magnetization experiments. They found that the phonon bottleneck with 10–100 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 [22–26]. When molecular magnets is subject to a DC magnetic field, the energy level of a single-molecule 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 $H_{0}=14$ 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, low-density 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 high-frequency 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 $T_{2}=500$ 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 single-molecule magnet because high-frequency 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 four-wave 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 anti-bunching 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 second-order 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

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

(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 mth 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 Δτ 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 Δτ 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 Δτ 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.

Figure 1

Schematic of energy level in a system of molecular magnets. Two strong magnetic fields $H_{c}$ and $H_{d}$ drive the transition from $|2\rangle$ to $|3\rangle, |4\rangle$, respectively. A quantum electromagnetic wave $\hat{H'}_{p}$ of frequency $\omega _{p}$ drives transition from $|1\rangle$ to $|3\rangle$, then a mixing field is generated with frequency $\omega _{m}$. $\delta _{1}$ and $\delta _{2}$ are the single photon frequency detuning which are defined as $\delta _{1}=\omega _{31}-\omega _{p}$, $\delta _{2}=\omega _{41}-\omega _{m}$

Figure 2

Sketch of the probe, mixing and control beam spatial configuration in a system of molecular magnets

Figure 3

Group velocities $\operatorname{Re}[V_{1}/c]$ (blue solid line) and $\operatorname{Re}[V_{2}/c]$ (red dashed line) as function of the dimensionless photon detuning Δτ with $z_{0} = 0.5$ mm, $\tau =10\ \mu s$ $\gamma _{1}\tau = 50$, $\gamma _{2}\tau = 60$, $\kappa _{12} \tau z_{0} =2\times 10^{4}$, $\kappa _{14}\tau z_{0}= 3\times 10^{4}$

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)

Figure 4

(a) Intensities of the microwave probe magnetic field pulse $I_{p}$ (solid line) and mixing magnetic field pulse $I_{m}$ (dashed line) as a function of $z/z_{0}$. We set $\phi _{cd}=\pi /4$, $H_{p}(t_{c})=H_{m}(t_{c})$ and $\Delta \tau =40$, the other parameters are chosen as those used in the plot of Fig. 3

Figure 5

The relative phase $\Phi _{12}$ as a function of $z/z_{0}$, other parameters are chosen as those used in the plot of Fig. 4

3 Numerical simulations and analysis

In this section, we show the microwave probe pulse propagation by numerically integrating Eq. (13a)–(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 single-molecular-magnets medium, we select $z_{0} = 0.5$ 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. 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.

Figure 6

(a) Intensity of the microwave probe pulse as a function of position and time. (b) Intensity of the microwave FWM pulse as a function of position and time. These parameters are explained in the text

In order to achieve the storage and retrieval process, we use the control beams of form

$$\begin{aligned} \Omega _{c}(t)/\Omega _{c}(0)=1-\tanh \bigl[\sigma (t-t_{1}) \bigr]+\tanh \bigl[ \sigma (t-t_{2}) \bigr], \end{aligned}$$

(25)

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. 3. We show an example of the microwave probe pulse can be recovered by appropriately choosing the control fields in Fig. 7(a), the storage time of the probe pulse is approximately given by $t_{2}-t_{1}=13\tau$. Shown in Fig. 7(b) is the temporal profile of the control field(blue solid line) and initial microwave probe pulse(red dashed line) of the medium.

Figure 7

(a) Storage and retrieval of microwave probe pulse as a function of position and time. (b) Temporal profile of the coupling beams. These parameters are explained in the text

4 Generating paired photons and two-photon intensity correlation function

In this section, we consider when the injected quantum state is a single-photon wave packet of the specific form follows [38–41]

$$\begin{aligned} |1\rangle _{\varpi }= \int _{-\infty }^{\infty }\,d\omega ^{\prime }P_{1} \bigl(\varpi +\omega ^{\prime } \bigr)\hat{a}^{\dagger } \bigl( \omega ^{\prime } \bigr)|0\rangle, \end{aligned}$$

(26)

Here, $P_{1}$ means amplitude and satisfies $\int _{-\infty }^{\infty }\,d\omega ^{\prime }|P_{1}|^{2}=1$, the ϖ 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

$$\begin{aligned} |\psi _{\mathrm{in}}\rangle = \vert 1\rangle _{p} \vert 0\rangle _{m} \end{aligned}$$

(27)

In general, at any given time, the state of a molecular magnet system can be written as

$$\begin{aligned} |\psi _{\mathrm{out}} \rangle =\sum_{n m} \alpha _{n m}(t) \vert n \rangle _{p} \vert m \rangle _{m}. \end{aligned}$$

(28)

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:

$$\begin{aligned} \begin{aligned} & \bigl\langle \psi _{\mathrm{out}} \bigl\vert F \bigl( \hat{H}_{p}^{\dagger }, \hat{H}_{m}, \hat{H}_{p}^{\dagger } \hat{H}_{m}, \ldots \bigr) \bigr\vert \psi _{\mathrm{out}} \bigr\rangle \\ &\quad = \bigl\langle \psi _{\mathrm{in}} \bigl\vert F \bigl(\hat{H}_{p}^{\dagger }, \hat{H}_{m}, \hat{H}_{p}^{\dagger } \hat{H}_{m}, \ldots \bigr) \bigr\vert \psi _{\mathrm{in}} \bigr\rangle , \end{aligned} \end{aligned}$$

(29)

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 single-probe 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

$$\begin{aligned} \bigl|\psi (t) \bigr\rangle =\alpha _{10} \vert 1 \rangle _{p} \vert 0 \rangle _{m}+ \alpha _{20} \vert 2 \rangle _{p} \vert 0 \rangle _{m}+\alpha _{21} \vert 2 \rangle _{p} \vert 1 \rangle _{m}. \end{aligned}$$

(30)

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

$$\begin{aligned} \bigl| \psi (t)\bigr\rangle = \vert \alpha _{10} 1 \rangle _{p} \vert 0 \rangle _{m}+ \vert \alpha _{21} \Vert 2 \rangle _{p} \vert 1 \rangle _{m}. \end{aligned}$$

(31)

With the help of Eq. (29), the expression of $|\alpha _{21}|^{2}$ and $|\alpha _{10}|^{2}$ can be obtained as

$$\begin{aligned} & \vert \alpha _{21} \vert ^{2}=2 \vert c \vert ^{2}P_{1}^{2}(\eta ), \end{aligned}$$

(32a)

$$\begin{aligned} & \vert \alpha _{10} \vert ^{2}= \bigl( \vert a \vert ^{2}+ \vert b \vert ^{2}-4 \vert c \vert ^{2} \bigr)P_{1}^{2}(\eta ), \end{aligned}$$

(32b)

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. (31), then we can obtain photon pairs with frequencies $\omega _{p}$ and $\omega _{m}$ by theoretical calculations. In order to show the time-dependent properties of the generated photon pairs, we work out the correlation function between the $\omega _{p}$ and $\omega _{m}$ with a time delay $\tau _{d}$,

$$\begin{aligned} G_{H_{1}-H_{2}}^{(2)} (\tau _{d} )= \bigl\langle \hat{H}_{1}^{ \dagger }(t) \hat{H}_{2}^{\dagger } (t+ \tau _{d} ) \hat{H}_{2} (t+\tau _{d} ) \hat{H}_{1}(t) \bigr\rangle . \end{aligned}$$

(33)

Using the Eq. (29), and the intensity correlation function $G_{H_{1}-H_{2}}^{(2)} (\tau _{d} )$ can be expressed as

$$\begin{aligned} \begin{aligned}G_{H_{1}-H_{2}}^{(2)}(\tau _{d})={}& \biggl( \int\, d\omega P_{1}^{2}( \omega ) \vert a \vert ^{2}+ \int\, d \omega \vert c \vert ^{2} \biggr) \int\, d\omega \vert b \vert ^{2} \\ & {}+ \int\, d \omega P_{ 1}^{2} \vert b \vert ^{2} \int\, d \omega \vert c \vert ^{2}+ \biggl\vert \int\, d \omega e^{i \omega \tau _{d}} a^{*} b \biggr\vert ^{2} \\ & {}+ \int\, d \omega e^{-i \omega \tau _{d}} P_{1}^{2}(\omega ) a^{*} b \int\, d \omega e^{-i \omega \tau _{d}} cd^{*} \\ &{} + \int\, d \omega e^{i \omega \tau _{d}} P_{ 1}^{2}(\omega ) a b^{*} \int\, d \omega e^{i \omega \tau _{d}} c^{*} \,d. \end{aligned} \end{aligned}$$

(34)

The second-order 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 second-order correlation function as a function of delay time $\tau _{d}$ in Fig. 8. The evolution of second-order correlation function is similar to that of the damped Rabi oscillations. The anti-bunching 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 four-wave 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.

Figure 8

The second-order correlation function with the delay time $\tau _{d}$. Other parameters are chosen as those used in the plot of Fig. 6

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 second-order correlation function of probe photon and mixing photon, yielding microwave correlated photons pair and achieving the anti-bunching effect for microwave photons in the crystal of molecular magnets.

Appendix

Here we give expression of relevant terms in Eqs. (16a)–(16b), (17) and (19) in the main article:

$$\begin{aligned} &R_{1}(\omega ) = \frac{K_{3}(\omega )-K_{1}(\omega )+iK_{5}(\omega )}{2iK_{5}(\omega )}, \\ &R_{2}(\omega )= \frac{K_{1}(\omega )-K_{3}(\omega )+iK_{5}(\omega )}{2iK_{5}(\omega )}, \\ &R_{3}(\omega )=\frac{K_{2}(\omega )}{i K_{5}(\omega )}, \\ &R_{4}(\omega )=\frac{K_{4}(\omega )}{i K_{5}(\omega )}, \\ &K_{5}(\omega ) = \sqrt{ \bigl[K_{1}(\omega )-K_{3}(\omega ) \bigr]^{2}+4K_{2}( \omega )K_{4}(\omega )} \end{aligned}$$

and

$$\begin{aligned} &D_{m}(0)= \vert \Omega _{d} \vert ^{2}+ \vert \Omega _{c} \vert ^{2}+\Delta _{1}\omega _{2}+ \Delta _{2}\omega _{2}-\Delta _{1} \Delta _{2}, \\ &K_{1m}(0)=\kappa _{12} \frac{( \vert \Omega _{d} \vert ^{2}-\Delta _{2}+\omega _{2})D_{m}(0)+(\Delta _{2}-\omega _{2})D(0)}{D^{2}(0)}, \\ &K_{2m}(0)=-\kappa _{12} \frac{e^{i\phi _{cd}}\Omega _{c}\Omega _{d}^{*}D_{m}(0)}{D^{2}(0)}, \\ &K_{3m}(0)=\kappa _{14} \frac{( \vert \Omega _{c} \vert ^{2}-\Delta _{1}+\omega _{2})D_{m}(0)+(\Delta _{1}-\omega _{2})D(0)}{D^{2}(0)}, \\ &K_{4m}(0)=-\kappa _{14} \frac{ e^{i\phi _{dc}}\Omega _{d}\Omega _{c}^{*}D_{m}(0)}{D^{2}(0)}, \\ &K_{5m}(0)= \frac{(K_{1}(0)-K_{3}(0))(K_{1m}(0)-K_{3m}(0))}{\sqrt{K_{5}(0)}} \\ &\phantom{K_{5m}(0)= }{} + \frac{2K_{2m}(0)K_{4}(0) +2K_{2}(0)K_{4m}(0)}{\sqrt{K_{5}(0)}}. \end{aligned}$$