Acoustic field induced nonlinear magneto-optical rotation in a diamond mechanical resonator

We study the nonlinear magneto-optical rotation (MOR) of a linearly polarized microwave probe field passing through many nitrogen-vacancy (NV) centers embedded in a high-Q single-crystal diamond mechanical resonator. On the basis of the strain-mediated coupling mechanism, we establish a three-level closed-loop system in the ground states of the NV center in the presence of a static magnetic field. It is shown that by applying an acoustic field, the birefringence is induced in the system through the cross-Kerr effect, so that the probe field is transmitted with a high intensity and rotated polarization plane by 90 degrees. In addition, we demonstrate that the acoustic field has a major role in enhancing the MOR angle to 90 degrees. Moreover, it is shown that the MOR angle of the polarization plane after passing through the presented system is sensitive to the relative phase of the applied fields. The physical mechanism of the MOR enhancement is explained using the analytical expressions which are in good agreement with the numerical results. The presented scheme can be used as a polarization converter for efficient switching TE/TM modes in optical communication, the depolarization backscattering lidar, polarization spectroscopy and precision measurements.

system. This makes a limitation in establishing the closed-loop schemes in degenerate levels of an atomic system. The limitation has been removed by using the strain-mediated coupling mechanism along with microwave field in the NV centers in a diamond mechanical resonator (DMR). Diamond mechanical resonator includes a diamond cantilever with many embedded NV centers, which can interact with the resonant phonon modes of a mechanical resonator through the crystal strain. It is a device with micron-scale dimensions, which can be achieved with excellent nanofabrication techniques in experiment 33 . The strain field is produced by the lattice vibration via a piezoelement used on the surface of the diamond layer and transferred through the DMR to couple the lattice strain field and the NV center spins. Many investigations have demonstrated the coherently coupling of the ground triplet state of the NV centers with the strain field in the DMR [34][35][36][37] . Transparency for the acoustic field using the ground triplet state of the NV center driven by the strain and microwave fields has been reported in either V-or Δ-type configuration 38 . Evangelou showed the phase dependent of the transparency of the acoustic field by regarding the Δ-type configuration of the NV center ground state as a closed-loop system 39 . Earlier, Fuchs et al. 36 had demonstrated the sensitivity of Δ-system of the NV center ground state to the relative phase of the applied fields and strain field made in the DMR. Magneto-optical rotation in a tripod four-level NV centers has been previously reported in which the ground triplet state 3 A is coupled to the excited state 3 E via visible optical fields 40 . Here, we use the DMR as a source of the strain field to excite the electric dipole forbidden transition of the system to generate the complete MOR. Now, we take advantage of coupling the ground triplet state of NV center with the strain field and study the MOR of the polarization plane of a linearly polarized microwave probe field in a three-level closed-loop system of NV center's ground state in the DMR. This work is aimed to bring the optical phenomena into the acoustic field's domain. It is presented that by applying an acoustic strain field in the presence of a static magnetic field, difference between the refractive indices of the circular components of the probe field increases and the birefringence is induced in the system. We show that the linearly polarized microwave probe field is transmitted through the system with high intensity while its polarization plane is rotated by 90 degrees. It is demonstrated that applying the acoustic field enhances the MOR angle of the polarization plane of the probe field due to the cross-Kerr effect. In addition, we show that the MOR angle of the polarization plane of the probe field is sensitive to the relative phase of the applied fields and the perfect rotation of the polarization plane happens for the special values of the relative phase. Our analytical results show that the nonlinear cross-Kerr effect is the responsible for the MOR enhancement. The obtain results can be used in the TE/TM polarization modes converters in optical communication, the depolarization backscattering lidar, polarization spectroscopy and precision measurements.

Theoretical framework
The system under study is composed of a high-Q single-crystal DMR with many embedded NV centers which is shown in Fig. 1(a). The NV centers are sensitive to the deformation of the surrounding lattice. When the DMR is vibrated by a piezoelectric film, the strain field is formed and transferred to wherever the NV centers are located. It is noteworthy that the strain in the DMR can be controlled by an external voltage applied to the piezoelectric film. The NV centers considered in this work are negatively charged with two unpaired electrons located at the vacancy. Thus, their ground state has a spin-triplet form. The schematic of the three-level closed-loop system of the NV center's ground state is shown in Fig. 1 is applied to the system parallel to the static magnetic field known as Faraday geometry. Since a linearly polarized field is a combination of right-and left-circularly polarized field, the right-and left-circular components of the The transitions |1 to |2 and |3 are driven by the left-and right-circularly polarized microwave probe field with Rabi frequency Ω − p and Ω + p , respectively. In addition, the electric dipole forbidden transition |2 to |3 is coupled by the strain field with Rabi frequency Ω s . E E / 2 p and µ µ → = → 31 32 . Since displacements of the electrons density of lattice induced by strain wave lead to generate a local electric field, we model the strain field as an effective electric field. By applying a static magnetic field, the degeneracy of the ± 1 can be removed by 2 where µ B and g s are Bohr magneton and Land ′ e factor, respectively. m s stands for magnetic quantum number of the states.
The Hamiltonian describing the interaction between two circularly polarized fields and strain field in the rotating wave and dipole approximations is given by 41 32 are the detunings of the applied fields from the corresponding transitions.
Using the von Neumann equation 41 , the density matrix equations of motion for the three-level closed-loop system in the electric-dipole and rotating-wave approximations can be written as where Γ i 3 (i = 1,2) and Γ 21 are the spontaneous emission from the state |3 and |2 to i and |1 , respectively. In addition, γ γ = Γ + d is the dephasing rate of the state |3 (|2 ).
s is the multi-photon resonance detuning. The response of the quantum system to the rightand left-circularly polarized probe field is described by the susceptibility, which is given by 5  is the field absorption at resonance in which l, k p and N are the length of the atomic medium, probe field wave number and density of atoms, respectively. S ± are the normalized susceptibilities which are given by where ρ i1 (i = 3,2) is the probe field transition coherence, which can be obtained from Eq. (2) The imaginary and real parts of S ± represent the absorption and dispersion of the circularly polarized probe field, respectively. The polarization direction of the input linearly polarized probe field is assumed in x direction, which may be affected and rotated after passing through a medium. If a component of the polarization direction of the output probe field is observed in ŷ direction, the polarization plane of the probe field has been rotated. In experiment, a y-polarized analyzer is used to transmit the light with polarization in ŷ direction. Thus, the intensity of transmission in ŷ direction is used to calculate the polarization rotation of the output probe field. The intensity of transmission of the light with ŷ and x polarization directions are given by 5 www.nature.com/scientificreports www.nature.com/scientificreports/ The MOR angle of the probe field polarization is defined as The polarization rotation of a laser field after passing through a medium can occur due to the birefringence or dichroism induced in the system. When the dispersions (absorptions) of the right-and left-circular components of the linearly polarized probe field are different, while their absorptions (dispersions) are equal, the birefringence (dichroism) is dominant in the system. By separating the real and imaginary parts of + S and − S in Eqs. (5) and (6) x l 2 When β is positive but negligible, what we were looking for in our results, the rotation of the polarization direction of the probe field is merely due to the birefringence induced in the system without any attenuation of the intensity of the probe field after passing through the medium. The negative values of β show the amplification of the intensity of the probe field.

Results and Discussion
In this section, we are going to investigate the MOR in the three-level closed-loop quantum system established in the NV center's ground state affected by the strain field in the DMR. Throughout the results, it is assumed that Δ = Δ = Δ − + p , Ω = Ω = Ω − + p p p and Δ = 0. Also, the parameters are scaled by γ Γ = Γ = = Γ d 31 32 3 , which is equal to . MHz 2 2 36 . Figure 2 shows the imaginary (a) and real (b) parts of + S (dotted) and − S (dashed) describing the behavior of the absorption and dispersion of the right-and left-circular components of the probe field and their difference (solid) versus the detuning of the probe field. The taken parameters are Ω = . Γ 0 01 is corresponding to Gauss 2 static magnetic field. An investigation on Fig. 2(a) shows that the absorption of the circular components of the probe field is equal and negligible around the probe field resonance. Figure 2(b) shows that the corresponding dispersions difference is noticeable at Δ = 0 p . It represents that the difference between the normalized susceptibilities, + S and − S , happens only due to the difference between the refractive indices (dispersions) of the right-and left-circular components of the probe field. Thus, the birefringence generated by the difference of the real parts of the normalized susceptibilities is a merely dominant phenomenon in the system. In Fig. 3, the x-(dashed) and ŷ-(solid) components of the transmitted field (a) as well as the MOR angle of the polarization direction of the probe field (b) are presented versus the detuning of the probe field. The parameters www.nature.com/scientificreports www.nature.com/scientificreports/ are those used in Fig. 2. It is seen in Fig. 3(a) that the intensity of the transmitted field with rotated polarization, T y , increases by 0.91 at Δ = 0 p , while T x , the intensity of the transmitted field with primary polarization direction, attenuates extremely. Since the primary polarization of the probe field is considered in x direction, measuring a large transmission in ŷ direction indicates a complete rotation of the polarization direction of the probe field passing through the medium. Moreover, it is seen in Fig. 3(b) that the MOR angle value reaches 90 degrees at Δ = 0 p , which means that the polarization plane of the probe field has been completely rotated after passing through the quantum system. It is worth noting that the maximum MOR happens merely due to the induced birefringence as shown in Fig. 2.
Another parameter for controlling the MOR angle is the static magnetic field which causes the Zeeman splitting of the energy levels. The static magnetic field can be applied to the NV centers by placing the DMR in center of the Helmholtz coils and controlled by the input DC current. The effect of the static magnetic field on the MOR is displayed in Figs. 4 and 5. Figure 4 shows the absorption (a) and dispersion (b) of the right-(dotted), left-(dashed) circular polarization of probe field and their difference (solid) versus Zeeman splitting Δ B in the probe field resonance condition. The used parameters are Ω = . Γ 0 01 32 3 , γ Γ = = . Γ 0 01 d 21 2 , α = Γ l 107 , Δ = 0 c and Φ = 0. It is seen in Fig. 4 that the dichroism is dominant in the system in the absence of the static magnetic field. By increasing the static magnetic field, one can see that the absorption of the circular components of the probe field and their difference dramatically decrease, while the dispersion of the right-and left-circular polarization of the probe field and their difference grow to the maximum value. It is expected that the contribution of dichroism and birefringence in the MOR is completely dependent on the magnitude of the static magnetic field.
In Fig. 5, the intensity of transmission of the output probe field (a) in x (dashed) and ŷ (solid) directions and the MOR angle of the polarization direction of the probe field (b) are plotted versus Δ B . It is shown in  www.nature.com/scientificreports www.nature.com/scientificreports/ Fig. 5(a) that in the absence of the static magnetic field, the y-component of the transmitted probe field is negligible. Also, the transmission of the probe field with primary polarization, accompanied by higher absorption, is smaller than 0.7. It is illustrated that by increasing the static magnetic field, T y increases and reaches a maximum value at Δ = Γ 17 B , while T x attenuates. According to Fig. 5(b), by increasing the static magnetic field, the MOR angle of the polarization direction of the probe field increases and experiences a rotation by 90 degrees at . It can be said that increasing the static magnetic field enhances the MOR angle by increasing the difference between the refractive indices of the circular components of the probe field.
The next scenario is controlling the MOR via the acoustic field. Noting that the strain field is produced by means of a piezoelement attached to the DMR, the intensity of the acoustic field can be well controlled by the external voltage applied to the piezoelement. Figure 6 displays the effect of the acoustic field on the MOR. In Fig. 6, the behavior of the T x (dashed), T y (solid) (a) and MOR angle (b) of the probe field are plotted versus the intensity of the acoustic field Ω s at Δ = 0 p . The used parameters are Ω = . Γ 0 01 21 2 , α = Γ l 107 , Δ = 0 c and Φ = 0. A bird's eye view of Fig. 6(a) shows that the ŷ-component of the output probe field is zero in the absence of the acoustic field, but T y enhances by increasing the intensity of the acoustic field, while T x decreases. It is noteworthy that the birefringence due to the dispersion difference of two probe field components has a major role in establishing the local maximum in T x . Figure 6(b) demonstrates the effective role of the acoustic strain field in enhancing the MOR of the polarization plane of the probe field. It is seen that by switching off the acoustic field, the MOR angle due to the linear response of the medium becomes negligible. By increasing the intensity of the acoustic field, the nonlinear optical effects enhance the MOR angle and brings it to the maximum value.
The simultaneous effect of the static magnetic field and the acoustic field on the MOR angle of the polarization plane of the linearly polarized field at Δ = 0 p is shown in Fig. 7  , α = Γ l 107 , Δ = 0 c and Φ = 0. This figure shows that the proper choices of the intensity of the acoustic field and the static magnetic field prepare the system to transmit the probe field with different polarization plane rotation. It is seen that the polarization direction of the transmitted field can cover a wide range of the MOR angle from zero to 90 degrees for different values of Δ B and Ω s .
Another parameter for controlling the MOR angle is the relative phase of the applied fields, which can be simply changed by electro-optical phase modulators. It is well known that the optical properties of a closed-loop system, in multi-photon resonance condition, depend on the relative phase of the applied fields 42 . Thus, it is expected that the relative phase of the applied field becomes a useful tool to impose desired rotation on the polarization direction of the probe field. In Fig. 8, the MOR angle of the probe field is presented versus Δ p for different values of the relative phase of the applied fields. The taken parameters are Ω = . Γ 0 01 32 3 , γ Γ = = . Γ 0 01 d 21 2 , α = Γ l 107 and Δ = 0 c . It is seen that the polarization plane of the probe field takes different angles for different values of the relative phase so that at Φ = 0 and π Φ = , it is rotated by 90 degrees at the probe field resonance.
To have a good insight into the effect of the parameters, we present the analytical solution for the transition coherences ρ 21

Conclusion
In summary, the nonlinear MOR of a linearly polarized microwave probe field was investigated after passing through many NV centers embedded in a high-Q single-crystal DMR. We established a three-level closed-loop system from the ground states of the NV center by the mechanism of the strain-mediated coupling in exciting the optically dipole forbidden transition in the presence of a static magnetic field. It was shown that by applying an acoustic field and a static magnetic field, difference between the refractive indices of the circular components of the probe field increases and the birefringence due to the cross-Kerr effect is induced in the system. We obtained a large intensity for the transmitted probe field and it was demonstrated that the acoustic field can enhance the MOR angle to 90 degrees through the cross-Kerr effect. Moreover, we showed that the MOR is sensitive to the