Quantum properties of nitrogen-vacancy center in diamond coupled to mechanical resonators

The optical absorption spectrum of the hybrid system composed of the nitrogen-vacancy (NV) center coupled to the mechanical resonators is. investigated. Based on the pump-probe technology, the electronically induced transparency (EIT) is observed and the reasonable explanation for this transparency is given. We demonstrate that the position of the transparency is equal to the detuning difference. In addition, a completely new scheme to measure the coupling between the NV center and current carrying carbon nanotube mechanical resonator is obtained. Furthermore, an optical approach to measure the frequency of the vibrating graphene mechanical resonator is proposed. This work can provide some help for the application in the field of high-precision measurement and quantum information processing.

The research of nanomechanical systems, such as current carrying carbon nanotubes [22] and vibrating graphene mechanical resonators, is currently receiving much attention.They have some applications in like quantum information processing [18,[20][21][22][23], quantum measurements [23][24][25] and force detection [26,27].Many efforts have been made to study on the hybrid spin-NR system.A magnetic coupling between the nanomechanical resonator and the NV electronic spin can be induced in a strong magnetic field gradient [28].This hybrid system, which is composed of NV center and resonator, is susceptible to reach the strong coupling regime, as already envisioned in ref. [29,30].Li et al. have proposed a hybrid system in which a single NV center spin in diamond is coupled to a suspended current-carrying carbon nanotube [31].The strong coupling mechanism and coherent manipulation in this hybrid quantum has been realized, which establishes a theoretical foundation for us to study further.Moreover, the optomechanical system based on graphene resonator has also been investigated experimentally [32][33][34][35].
Weber et al. proposed a scheme of coupling graphene resonators with superconducting cavities, which can significantly reduce the resonant frequency of graphene resonators, thus enhancing zero motion and increasing mechanical nonlinearity, which has important prospects for the study of quantum motion [33].
In this work, motivated by these developments, we propose a hybrid system consisting of nitrogenvacancy center in diamond coupled to mechanical resonators.The presented work has some remarkable advantages.Firstly, the EIT has been observed in both diamond NV center coupled to a single graphene resonator or a single carbon nanotube resonator, as well as in both graphene and carbon nanotube resonator.Moreover, the position of EIT can be tuned by controlling the detuning between the NV center and current carrying carbon nanotube resonator.Secondly, an optical method is proposed to measure coupling strength and the frequency of resonator.
The structure of this paper is organized as follows.In the second section, the physical model and the basic numerical calculation for the present system are given.The EIT is observed and the reason for it is given in the third section.Besides, the schemes to measure the coupling strength between the NV center and current carrying carbon nanotube mechanical resonator and the frequency of vibrating graphene mechanical resonator are obtained.Finally, a conclusion is given in the fourth section.The hybrid system is presented in Figure 1, which is composed of NV center in diamond, vibrating graphene resonator and a current carrying carbon nanotube resonator.The system is driven by pump and probe field simultaneously.The graphene membrane is laid in the x-z plane and vibrates along the y-direction in the system [36].The Hamiltonian of the NV center in diamond can be characterized by the pseudospin -1/2 operators S  and z S .So its Hamiltonian can be described by a a describes the annihilation (creation) operator for the current carrying carbon nanotube.The interaction Hamiltonian between the NV center and the vibrating graphene membrane is [29,39] means the interaction Hamiltonian between the NV center and the current carrying carbon nanotube.The coupling strength between the NV center and the vibrating graphene membrane (current carrying nanotube) is  

H S E e S E e S E e S E e
where  is the dipole moment; ( )   denotes the frequency of the pump field (probe field); means the amplitude of the pump field (probe field).So the Hamiltonian of the system is read as follows ( 1 In the rotating frame at the pump field frequency pu  , the Hamiltonian can be rewritten as where is the detuning between the current carrying carbon and pump field, The Heisenberg-Langevin equations can be easily obtained as follows.Here quantum properties of z S , S  and N is ignored, and introduce the damping and noise terms. where where B  and T indicate the Boltzmann constant and the temperature of the reservoir of the coupled system, respectively.  t  means the Langevin noise operators in the system with zero mean value   =0 f t and follows correlation functions To solve Eqs. ( 3)-( 6), we make the ansatz.:

  
( 1 1 ) Then the steady-state solutions are obtained as follows: where .
The above steady-state solutions 0 N , 2 a and 0 S , have a connection with the population inversion of the NV center, which is determined by Bringing the Eqs.( 9)-( 12) into Eqs.( 3)-( 6), the linear optical susceptibility is obtained as where

Results and discussions
In order to study the phenomenon of EIT comprehensively, we depict the absorption spectrum as a function of pr  in Figure 2. The black curve denotes the probe absorption spectrum without any couplings in Figure 2(a).It has the characteristics of symmetry and complete absorption.as the detuning value is 0.5.
Therefore, we can obtain a result that the location of EIT window is equal to the value of c p u    .
Here, we set the  is equal to c p u    .This ascribes to the reason that the mechanically induced coherent population oscillation leads to the occurring transparency window.To verify this result, we draw a diagram of the relationship between the position of dip and the value of  in Figure 4. From this figure, a linear relationship between the position of dip and the value of  is obtained.Undoubtedly, it is known the information that the location of EIT window is equal to the value of .
Figure 5 shows the absorption spectrum as a function of pr  with or without the coupling between diamond NV center and current carrying carbon nanotube.It demonstrates the coupling strength can be obtained by measuring the distance of the peak-splitting.There is only an absorption peak (black solid line) under the condition of the coupling between diamond NV center and current carrying carbon nanotube is zero.But when the coupling is added, the absorption spectrum splits into two peaks and has a zero absorption point at 0 pr   .And as the coupling strength becomes larger and larger, the width of transparency window is wider and wider.The physical mechanism of this result is due to the mechanically induced coherent population oscillation.When the detuning of the detection pump is equal to the frequency of the resonator, it makes the diamond NV center interfere with the beat frequency of the two light fields [41].The splitting distance of two peaks as a function of the coupling strength is shown in the inset of Figure 5.It is found that the splitting distance is linear with the coupling strength.Consequently, the result provides an effective method to measure the coupling strength between diamond NV center and current carrying carbon nanotube.Therefore, a method is proposed to obtain the coupling strength only by measuring the distance of two peaks in the probe absorption spectrum.
The absorption spectrum as a function of pr  for three different frequencies of the vibrating graphene membrane is shown in Figure 6.It is noted that there exist two peaks, a right positive peak and a left negative peak, whatever the frequency is.The two peaks correspond to the resonant absorption and amplification of the mechanical mode, respectively.Besides, the position of the peak is exactly equal to the frequency of the vibrating graphene membrane.Therefore, a new approach to measure the frequency of the vibrating graphene membrane is provided.It is mainly divided into two steps.Firstly, the frequency of the first pump light should be fixed at the frequency of the NV center, and then the second probe light should be swept across the frequency range of the NV center.It is found that there exist two sharp peaks in the absorption spectrum of the probe light, and the positions of the peaks correspond to the vibration frequency of the vibrating graphene membrane.The basic principle of this method is mechanically induced coherent population oscillation.What's more, with the frequency of vibrating graphene membrane becomes larger and larger, the width of EIT window at 0.5 pr    turns narrower and narrower.The reason for this result is that the quantum coherent interaction between NV center and vibrating graphene membrane.

Conclusion
In summary, we have proposed a hybrid system based on the NV center and mechanical resonators.We have found that there is a zero absorption in the absorption spectrum when the NV center couples either the vibrating graphene membrane or the current carrying carbon nanotube.It indicates that the phenomenon of the EIT occurs at that position.Additionly, the relationship between the position of EIT window and the value of detuning difference have been demonstrated.Moreover, an approach to measure the frequency of vibrating graphene membrane is obtained.Our works may have potential applications on the quantum precision measurement.

Figure 1 .
Figure 1.The Physical model of this system.
means the detuning between NV center and pump field, frequency of pump field, and = pr pu     stands for the detuning of the frequency of the probe field and the pump field.

Figure 2 Figure 2 .Figure 3 .
Figure 2. (a) The absorption spectrum as a function of pr  without any couplings.(b) with the coupling 2 0.2  

Figure 4 .
Figure 4.The relationship between the dip position and the value of  .The parameters are same with Figure 3.

6
As illustrated in Figure3, the location of the window of EIT at different values of detuning of NV center and the current carrying carbon nanotube is depicted.Figure3(a) denotes that when the detuning of the NV center and the current carrying carbon nanotube   c

Figure 5 . 3 
Figure 5.The absorption spectrum as a function of pr  for three different coupling strengths between diamond NV center and current carrying carbon nanotube.The inset is the splitting distance of two peaks as a function of the coupling strength.The different parameter used is 1 =0.3  and other parameters are same with Figure 2.

Figure 6 .
Figure 6.The absorption spectrum as a function of pr  for three different frequencies of the vibrating graphene membrane.The parameters are 1 =0.3MHz    .The Hamiltonian of the NV center coupled to the pump laser and probe laser is