We propose a single phonon source based on nitrogen-vacancy (NV) centers, which are located in a diamond phononic crystal resonator. The strain in the lattice would induce the coupling between the NV centers and the phonon mode. The strong coupling between the excited state of the NV centers and the phonon is realized by adding an optical laser driving. This four-level NV center system exhibits coherent population trapping and yields giant resonantly enhanced acoustic nonlinearities, with zero linear susceptibility. Based on this nonlinearity, the single phonon source can be realized. We numerically calculate of the single phonon source. We discuss the effects of the thermal noise and the external driving strength.
© 2018 Optical Society of America
In the last decade, one of the most active fields in quantum optics has been optomechanics, which studies the coupling between the mechanical mode and the optical or microwave field [1,2]. It has wide applications, including gravitational wave detectors, squeezing of light, and quantum nondemolition measurement. Optomechanics also has applications in quantum information processing, such as an interface between the optical qubits and the superconducting microwaves [3–5]. In its many applications, phonons play a significant role. Thus, the research on phonons has been a hot topic in quantum information processing [6–19]. Besides the above application, there are some unique advantages for a phonon in a low energy scale. The acoustic wavelength can be as small as μm if its frequency is comparable with the microwave photon. On the one hand, the much smaller mode volume can realize individual superconducting qubit control  and, on the other hand, it can support a large number of modes which is beneficial for the storage of quantum information. Besides, its potential applications in the detection of opaque substances can make up for the disadvantages of optics. Until now, an electro-magnetic field-induced acoustics transparent has been proposed based on an NV center ensemble, which can be used to control phonon velocity in diamond . A phonon detector is also put forward in recent works [22,23]. With more and more attention focused on the phonon, studies on a single phonon source become indefensible, but the setup proposed now to produce a single phonon is either too sophisticated [24,25] or based on measurement . Thus, a simple and measurement-free single phonon source is needed.
The strong nonlinear acoustics interaction is the core for our single phonon source proposal, which is similar to the single photon source [26–31]. In an optical cavity, the giant nonlinearity can be produced through coupling between optical field and four-level atoms, where one excitation will cause detuning for another excitation [32–39]. The similar four-level system for a phonon can be obtained in the NV centers systems. The NV center in diamond has many advantages, such as the coherence time of the NV centers is very long at room temperature, and the energy splitting of ground spin states can be adjusted by using a magnetic field . The strong coupling between the NV center and the mechanical mode can be realized, either through the magnetic field gradient [12,41–43], or the strain [16,20]. Assisted with optical laser and microwave diving, the phonon can be coupled to an excited state and ground three spin states of NV centers simultaneously [17,18], which can be regarded as the effective four-level system. This four-level system which exhibits coherent population trapping (CPT) yields giant resonantly enhanced nonlinearities, while the linear susceptibility is zero. Based on this acoustic nonlinearity, the single phonon state can be produced.
The setup of our model is shown in Fig. 1(a). The NV center ensembles are doped on the surface of the phononic crystal made of diamond. The phonon mode in the phononic crystal interacts with the NV centers under the external optical and microwave fields diving. The energy levels and driving in the NV centers are illustrated in Fig. 1(b). We focus on the excited state and ground state , , of the NV center. An optical field and a phonon mode together are used to dive the transition . A microwave field drives the transition between spin states and with Rabi frequency . The coupling between and is magnetic dipole forbidden. However, it can be induced through phonon mode . For convenience, we re-label spin states , , , and as , , , and , respectively. We define operator and energy difference with frequency of energy level, where . The Hamiltonian of the whole system is1) can be approximated as [17,18] 17]. Here, we consider the interaction between the surface phonon and NV center ensemble. The coupling strength between phonon and energy level has been enhanced by factor , namely, , with being the number of NV centers.
The four-level system that exhibits CPT yields giant resonantly enhanced nonlinearities, while the linear susceptibility is identically zero. We divide the Hamiltonian into two parts as , with
The first part describes the interactions among three spin states of ground state, where a three-level system is constructed. The second part depicts the coupling between the excited state and ground state, and we analyze this term based on eigenstates of . We express eigenstates of with polariton operators as [38,39]
Of all parameters, is most easy to adjust. In order to obtain Eq. (6), it seems that we can adjust infinite small, but this is not the case. The higher-order nonlinearity of requires , and only then can the energy shift for the level be obtained using the 2nd perturbation theory as [35–39]
In order to produce an obvious large nonlinear effect, a small value of parameter is required and, thus, only single polariton is prepared at first. Then, adjusting parameter adiabatically [44,45] until , the polariton will evolve into the form of a phonon according to the form of dark state polariton as Eq. (4). Now, we first focus on the preparation of single dark state polariton . The microwave is introduced to driving the transition between states and of NV centers with strength . Since , the driving is actually applied to driving polaritons. We choose parameters to satisfy conditions and , which guarantees that the driving effect on polaritons can be neglected, and the driving is mainly to excited dark state polariton . The Hamiltonian is
To simulate this process, the influence of the environment must be considered. The state of our system is usually in the mixed state, expressed by density matrix , and its dynamical process can be depicted as
Now, we will present a numerical simulation of a dynamical process as Eq. (10). We choose the parameters as , , , , , , and . The strength of nonlinear interaction can be calculated as immediately. At a temperature of about 0.5 K, the mean number of polariton is . As for the dissipation of polariton, yields and with the quality factor of the phononic crystal . Since , the dissipation of polariton can be approximated as . Hilbert space is chosen as , where is the Fock state of polaritons, and is the upper cutoff in our calculation.
The second-order correlation function and the population of Fock state are calculated, and shown in Fig. 2 for different thermal noises and Fig. 3 for different driving strengths, respectively. The initial state is in thermal state, corresponding to . With time going on, decreases until , which means that the statistic of our photon source changes from super-Poisson distribution to sub-Poisson distribution, and Fock state becomes dominated. When evolves to the minimum point, reaches its peak, and the obvious reason for this is that the minimal represents the maximal probability to produce the single polariton. The thermal state will demolish the classical properties of our system, which is shown in Fig. 2, where the minimal value of increases with the increasing of the mean thermal number of polaritons. In addition, from Fig. 3, we can see that the time for to reach the minimal value will decrease when the driving strength increases. Thus, the increasing of the driving strength can save the time to obtain the single polariton which is useful to resist decoherence.
When the minimum point of is reached, the driving should be turned off immediately. The Fock state of polariton dominates at this time. Then, we adjust the driving strength of microwave with velocity . The velocity should satisfy the adiabatic condition , which makes sure polaritons cannot be excited in this process [44,45]. The dissipation mainly comes from the dissipation of phonon , and the evolution time is limited by . The proportion of phonon increases with the increasing of . When , this polariton transforms into a phonon, which is the essence of a single phonon source. In our proposal, we choose and, at time , ; meanwhile, . Therefore, a single phonon source can be realized based on our scheme.
In conclusion, we have proposed a scheme to produce a single phonon based on the nonlinear effect in an interactive process between the phonon and the NV centers. We have shown that the nonlinear coupling strength can be stronger than the phonon decay rate. We have also calculated the second correlation function numerically, and found that for practical parameters. Finally, the effect of thermal noise and external driving strength on has been simulated and discussed. Recently, the studies on phonons have witnessed significant progresses, and we hope that our study stimulates further experimental researches on the application of the phonon in quantum information processing.
National Natural Science Foundation of China (NSFC) (20141300566, 61435007, 61771278, 61727801); China Postdoctoral Science Foundation (2016M600999); National Science and Technology Major Project (2017YFA0303700).
Rui-Xia Wang acknowledges the support from the China Postdoctoral Science Foundation.
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