Abstract

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 g(2)(0) 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 [35]. In its many applications, phonons play a significant role. Thus, the research on phonons has been a hot topic in quantum information processing [619]. 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 [20] 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 [21]. 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 [22]. 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 [2631]. 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 [3239]. 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 [40]. The strong coupling between the NV center and the mechanical mode can be realized, either through the magnetic field gradient [12,4143], 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 |Ey and ground state |1, |0, |1 of the NV center. An optical field and a phonon mode together are used to dive the transition |Ey|0. A microwave field drives the transition between spin states |0 and |+1 with Rabi frequency Ωd. The coupling between |1 and |+1 is magnetic dipole forbidden. However, it can be induced through phonon mode b. For convenience, we re-label spin states |1, |0, |+1, and |Ey as |1, |2, |3, and |4, respectively. We define operator σij=|ij| and energy difference ωij=ωiωj with ωi,j frequency of energy level, where i,j=1,2,3,4. The Hamiltonian of the whole system is

Ho=ωmbb+ω1σ11+ω2σ22+(ϵ+ω3)σ33+(ω4+δ)σ44+Ωd(eiω23tσ32+H.c)+Ωc(eiωctσ42+H.c)+g24(b+b)σ44+g13(bσ31+H.c),
where b and b are the creation and annihilation operators for the acoustic field with frequency ωm; g13 and g24 are the electron–phonon coupling rate; ω23 and ωc are the frequency of the driving field and optical field with the Rabi frequency of Ωd and Ωc, respectively. Considering the large detuning between the optical field and acoustic field, the optical field can be eliminated, and its effect can be absorbed into the coupling strength between |4 and |2. Applying the Schrieffer-Wolff transformation U=exp[g24(bb)σ44/ωm] and under the condition ω24ωcg242/ωm=ωm, the Hamiltonian in Eq. (1) can be approximated as [17,18]
Haδσ44+εσ33g˜24(bσ42+H.c.)+g˜13(bσ31+H.c.)+Ωd(σ32+H.c.),
where g˜24=g24Ωc/ωm. The electron–phonon coupling rate g24 can be very strong up to 2 MHz, and Ωc is easy to be tuned. This parameter has good controllability [17]. Here, we consider the interaction between the surface phonon and NV center ensemble. The coupling strength between phonon b and energy level σ31 has been enhanced by factor N, namely, g13g˜13=Ng13, with N being the number of NV centers.

 figure: Fig. 1.

Fig. 1. (a) Schematic diagram of the phononic crystal: the NV center ensembles are located near the surface. The phonon is coupled to NV center ensembles driven by the laser field and micro-magnetic field. (b) Energy levels of NV centers. In this structure, we regard |1, |0, |+1, and |Ey as |1, |2, |3, and |4, respectively. δ and ε are the detuning frequency. ωm, ωd, and ωc are the frequencies of the phonon, optical laser driving, and micro-magnetic driving, respectively.

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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 Ha=H0+H1, with

H0=εσ33+Ωd(σ32+H.c.)+g˜13(bσ31+H.c.),H1=δσ44g˜24(bσ42+H.c.).

The first part H0 describes the interactions among three spin states of ground state, where a three-level Λ system is constructed. The second part H1 depicts the coupling between the excited state and ground state, and we analyze this term based on eigenstates of H0. We express eigenstates of H0 with polariton operators as [38,39]

P0=(g˜13σ21Ωdb)/B,P+=μσ31+ν(Ωdσ21+g˜13b)/B,P=νσ31+μ(Ωdσ21+g˜13b)/B,
where B=Ωd2+g2 and |μ|2+|ν|2=1. These operators satisfy the commute formula, [Pi,Pj]=δij,[Pi,Pj]=[Pi,Pj]=0 with i,j=0,±, which means that these polaritons are bosons. The polariton operator P0 corresponds to the dark state, which is responsible for the electromagnetically induced transparency. Using these polaritons, the Hamiltonian H0 can be expressed as
H0=μ0P0P0+μ+P+P++μPP,
where μ0,μ±=(ε±A)/2 with A=ε2+4B2. Now, we reconsider the Hamiltonian H1 and express the Hamiltonian using the new introduced operators. When condition g˜24Ωd/4g˜132μ+,2μ,μ++μ is satisfied, the coupling interaction with level 4 can be approximated as
g˜24(σ41σ12b+bσ21σ14)g˜24g˜132Ωd[σ41(P0)2+(P0)2σ14].

Of all parameters, Ωd is most easy to adjust. In order to obtain Eq. (6), it seems that we can adjust Ωd infinite small, but this is not the case. The higher-order nonlinearity of P0 requires g˜24g˜13/2Ωdδ, and only then can the energy shift for the level |4 be obtained using the 2nd perturbation theory as [3539]

HE=g(P0)2(P0)2,
which is the effective giant nonlinear effect of dark polariton operator P0 with g=g˜242g˜132/4δΩd2, the effective coupling strength. The existence of one polariton in the system will block the absorbing of the second polariton. In this process, the dissipation of polariton P0 should also be considered. The dissipation mainly includes two parts; γ=γa+γp. γa comes from the spontaneous decay of the level |4, which can be estimated as γa=(γ4/2δ)g. γp is caused by the dissipation of phonon in the phononic crystal. In this part, we can see that δ should be large enough to suppress the dissipation γa of polariton P0 and, however, the nonlinear strength g requires that the value of δ should be small. Therefore, choosing balanced parameters is important in our scheme.

In order to produce an obvious large nonlinear effect, a small value of parameter Ωd is required and, thus, only single polariton P0 is prepared at first. Then, adjusting parameter Ωd adiabatically [44,45] until Ωdg˜13, the polariton will evolve into the form of a phonon according to the form of dark state polariton P0 as Eq. (4). Now, we first focus on the preparation of single dark state polariton P0. The microwave is introduced to driving the transition between states |1 and |2 of NV centers with strength Ω. Since σ21=Ωd(ν2P++μ2P)/B+g˜13P0/B, the driving Ω is actually applied to driving polaritons. We choose parameters to satisfy conditions Ωdg˜13 and Ω<g˜242g˜132/4δΩd2μ±, which guarantees that the driving effect on polaritons P± can be neglected, and the driving is mainly to excited dark state polariton P0. The Hamiltonian is

H=g(P0)2(P0)2+Ω˜(P0+P0),
where Ω˜=Ωg˜13/B is driving strength applied on dark state polariton P0. The statistical properties of the single phonon sources can be measured through the second-order correlation function
g(2)(0)=P0(t)P0(t)P0(t)P0(t)/P0(t)P0(t)2.
g(2)(0)>1 means that radical sources tend to emit polaritons in bunches with super-Poisson distributed statistics, while g(2)(0)<1 indicates that emitting polaritons are one by one well separated in time from each other with antibunching, a unique quantum characteristic of the field.

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 ρ(t), and its dynamical process can be depicted as

ρ˙(t)=i[H,ρ(t)]+γ2(1+nth)L[P0]ρ(t)+γ2nthL[P0]ρ(t),
with L[o]ρ=2oρoooρρoo, γ being the dissipation of the system, and nth=P0P0 being the mean thermal polariton number. The initial polariton state is in the thermal state ρ(0)=n=0pn|nn|, where pn=nthn/(1+nth)n+1 is the probability of state |n. We initialize all NV centers in level |1; the mean thermal number of polariton P0 only depends on the mean thermal phonon, namely, P0P0=Ωd2bb/B2.

Now, we will present a numerical simulation of a dynamical process as Eq. (10). We choose the parameters as N=40,000, g13/2π=1kHz [16], Ωd/2π=20kHz, g˜24/2π=200kHz, ϵ/2π=200kHz, δ/2π=40MHz, and ωm/2π=800M. The strength of nonlinear interaction can be calculated as g/2π=25kHz immediately. At a temperature of about 0.5 K, the mean number of polariton is P0P0=0.1. As for the dissipation γ of polariton, γ4/2π=10MHz yields γa=g/8 and γp/2π=800Hz with the quality factor of the phononic crystal Q=106 [20]. Since γp=ωm/Qγa, the dissipation γ of polariton can be approximated as γ=γa. Hilbert space is chosen as {|n}n=0nmax, where |n is the Fock state of polaritons, and nmax=20 is the upper cutoff in our calculation.

The second-order correlation function g(2)(0) and the population of Fock state |1,P1 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 g(2)(0)=2. With time going on, g(2)(0) decreases until g(2)(0)<1, which means that the statistic of our photon source changes from super-Poisson distribution to sub-Poisson distribution, and Fock state |1 becomes dominated. When g(2)(0) evolves to the minimum point, P1 reaches its peak, and the obvious reason for this is that the minimal g(2)(0) 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 g(2)(0) increases with the increasing of the mean thermal number of polaritons. In addition, from Fig. 3, we can see that the time for g(2)(0) 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.

 figure: Fig. 2.

Fig. 2. (a) Evolution of the second correlation function g(2)(0). (b) Evolution of the population of Fock state |1. The coupling strength g/2π=25kHz, the dissipation of system γ=g/8, and the driving strength is Ω˜=g/5. The mean thermal polariton number of the blue dashed line, the red dotted dashed line, and the yellow line are P0P0=0.1,0.3,0.5, respectively.

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 figure: Fig. 3.

Fig. 3. (a) Evolution of the second correlation function g(2)(0). (b) Evolution of the population of Fock state |1. The coupling strength g/2π=25kHz, the dissipation of system γ=g/8, and the mean thermal polariton number is P0P0=0.1. The driving strength of the blue dashed line, the red dotted dashed line, and the yellow line are Ω˜=g/8,g/5,g/2, respectively.

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When the minimum point of g(2)(0) is reached, the driving Ω˜ should be turned off immediately. The Fock state |1 of polariton P0 dominates at this time. Then, we adjust the driving strength of microwave Ωdexp(vt) with velocity v. The velocity should satisfy the adiabatic condition vμ+,μ, which makes sure polaritons P± cannot be excited in this process [44,45]. The dissipation mainly comes from the dissipation of phonon γp, and the evolution time is limited by 1/γp. The proportion of phonon increases with the increasing of Ωd(t)=Ωdexp(vt). When Ωd(t)g˜13, this polariton transforms into a phonon, which is the essence of a single phonon source. In our proposal, we choose v=g˜13/5 and, at time t=25/g˜13, Ωd(t)=Ωdexp(5)g˜13; meanwhile, t1/γp. 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 g(2)(0) numerically, and found that g(2)(0)<1 for practical parameters. Finally, the effect of thermal noise and external driving strength on g(2)(0) 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.

Funding

National Natural Science Foundation of China (NSFC) (20141300566, 61435007, 61771278, 61727801); China Postdoctoral Science Foundation (2016M600999); National Science and Technology Major Project (2017YFA0303700).

Acknowledgment

Rui-Xia Wang acknowledges the support from the China Postdoctoral Science Foundation.

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References

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    [Crossref]
  2. Y.-D. Wang and A. A. Clerk, Phys. Rev. Lett. 108, 153603 (2012).
    [Crossref]
  3. L. Tian, Phys. Rev. Lett. 108, 153604 (2012).
    [Crossref]
  4. R. Andrews, R. Peterson, T. Purdy, K. Cicak, R. Simmonds, C. Regal, and K. Lehnert, Nat. Phys. 10, 321 (2014).
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  5. C. Dong, V. Fiore, M. C. Kuzyk, L. Tian, and H. Wang, Ann. Phys. 527, 100 (2015).
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  6. K. Y. Yasumura, T. D. Stowe, E. M. Chow, T. Pfafman, T. W. Kenny, B. C. Stipe, and D. Rugar, J. Microelectromech. Syst. 9, 117 (2000).
    [Crossref]
  7. D. Rugar, R. Budakian, H. Mamin, and B. Chui, Nature 430, 329 (2004).
    [Crossref]
  8. M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, Nature 462, 78 (2009).
    [Crossref]
  9. I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, Phys. Rev. Lett. 104, 083901 (2010).
    [Crossref]
  10. A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. M. Alegre, A. Krause, and O. Painter, Phys. Rev. Lett. 108, 033602 (2012).
    [Crossref]
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    [Crossref]
  12. P. Rabl, P. Cappellaro, M. G. Dutt, L. Jiang, J. Maze, and M. D. Lukin, Phys. Rev. B 79, 041302 (2009).
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  17. D. A. Golter, T. Oo, M. Amezcua, K. A. Stewart, and H. Wang, Phys. Rev. Lett. 116, 143602 (2016).
    [Crossref]
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    [Crossref]
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    [Crossref]
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2017 (7)

S. Hong, R. Riedinger, I. Marinkovic, A. Wallucks, S. G. Hofer, R. A. Norte, M. Aspelmeyer, and S. Groblacher, Science 358, 203 (2017).
[Crossref]

R.-X. Wang, K. Cai, Z.-Q. Yin, and G.-L. Long, Opt. Express 25, 30149 (2017).
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S. Guan, W. P. Bowen, C. Liu, and Z. Duan, Europhys. Lett. 119, 58001 (2017).
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Y. Chu, P. Kharel, W. H. Renninger, L. D. Burkhart, L. Frunzio, P. T. Rakich, and R. J. Schoelkopf, Science 358, 199 (2017).
[Crossref]

K. Cai, R. Wang, Z. Yin, and G. Long, Sci. China Phys. Mech. Astron. 60, 070311 (2017).
[Crossref]

Y. Ma, T. M. Hoang, M. Gong, T. Li, and Z.-Q. Yin, Phys. Rev. A 96, 023827 (2017).
[Crossref]

X. Shao, D. Li, Y. Ji, J. Wu, and X. Yi, Phys. Rev. A 96, 012328 (2017).
[Crossref]

2016 (4)

Y. Ma, Z.-Q. Yin, P. Huang, W. L. Yang, and J. Du, Phys. Rev. A 94, 053836 (2016).
[Crossref]

Q. Hou, W. Yang, C. Chen, and Z. Yin, J. Opt. Soc. Am. B 33, 2242 (2016).
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D. A. Golter, T. Oo, M. Amezcua, K. A. Stewart, and H. Wang, Phys. Rev. Lett. 116, 143602 (2016).
[Crossref]

D. A. Golter, T. Oo, M. Amezcua, I. Lekavicius, K. A. Stewart, and H. Wang, Phys. Rev. X 6, 041060 (2016).
[Crossref]

2015 (2)

J. D. Cohen, S. M. Meenehan, G. S. MacCabe, S. Groblacher, A. H. Safavi-Naeini, F. Marsili, M. D. Shaw, and O. Painter, Nature 520, 522 (2015).
[Crossref]

C. Dong, V. Fiore, M. C. Kuzyk, L. Tian, and H. Wang, Ann. Phys. 527, 100 (2015).
[Crossref]

2014 (4)

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev. Mod. Phys. 86, 1391 (2014).
[Crossref]

R. Andrews, R. Peterson, T. Purdy, K. Cicak, R. Simmonds, C. Regal, and K. Lehnert, Nat. Phys. 10, 321 (2014).
[Crossref]

Y. Tao, J. Boss, B. Moores, and C. Degen, Nat. Commun. 5, 3638 (2014).
[Crossref]

J. Zhou, Y. Hu, Z.-Q. Yin, Z. D. Wang, S.-L. Zhu, and Z.-Y. Xue, Sci. Rep. 4, 6237 (2014).
[Crossref]

2013 (1)

S. Bennett, N. Y. Yao, J. Otterbach, P. Zoller, P. Rabl, and M. D. Lukin, Phys. Rev. Lett. 110, 156402 (2013).
[Crossref]

2012 (3)

A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. M. Alegre, A. Krause, and O. Painter, Phys. Rev. Lett. 108, 033602 (2012).
[Crossref]

Y.-D. Wang and A. A. Clerk, Phys. Rev. Lett. 108, 153603 (2012).
[Crossref]

L. Tian, Phys. Rev. Lett. 108, 153604 (2012).
[Crossref]

2011 (2)

O. Arcizet, V. Jacques, A. Siria, P. Poncharal, P. Vincent, and S. Seidelin, Nat. Phys. 7, 879 (2011).
[Crossref]

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. Fink, A. Abdumalikov, M. Baur, S. Filipp, M. da Silva, and A. Blais, Phys. Rev. Lett. 106, 243601 (2011).
[Crossref]

2010 (2)

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, Nature 464, 697 (2010).
[Crossref]

I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, Phys. Rev. Lett. 104, 083901 (2010).
[Crossref]

2009 (2)

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, Nature 462, 78 (2009).
[Crossref]

P. Rabl, P. Cappellaro, M. G. Dutt, L. Jiang, J. Maze, and M. D. Lukin, Phys. Rev. B 79, 041302 (2009).
[Crossref]

2008 (4)

M. Hofheinz, E. Weig, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. Oonnell, H. Wang, J. M. Martinis, and A. Cleland, Nature 454, 310 (2008).
[Crossref]

B. Dayan, A. Parkins, T. Aoki, E. Ostby, K. Vahala, and H. Kimble, Science 319, 1062 (2008).
[Crossref]

A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vuckovic, Nat. Phys. 4, 859 (2008).
[Crossref]

N. Na, S. Utsunomiya, L. Tian, and Y. Yamamoto, Phys. Rev. A 77, 031803 (2008).
[Crossref]

2007 (1)

M. J. Hartmann and M. B. Plenio, Phys. Rev. Lett. 99, 103601 (2007).
[Crossref]

2006 (2)

J. Wrachtrup and F. Jelezko, J. Phys. Condens. Matter 18, S807 (2006).
[Crossref]

M. J. Hartmann, F. G. Brandao, and M. B. Plenio, Nat. Phys. 2, 849 (2006).
[Crossref]

2005 (2)

K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, Nature 436, 87 (2005).
[Crossref]

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, Rev. Mod. Phys. 77, 633 (2005).
[Crossref]

2004 (1)

D. Rugar, R. Budakian, H. Mamin, and B. Chui, Nature 430, 329 (2004).
[Crossref]

2002 (1)

S. Rebic, A. Parkins, and S. Tan, Phys. Rev. A 65, 043806 (2002).
[Crossref]

2000 (1)

K. Y. Yasumura, T. D. Stowe, E. M. Chow, T. Pfafman, T. W. Kenny, B. C. Stipe, and D. Rugar, J. Microelectromech. Syst. 9, 117 (2000).
[Crossref]

1999 (3)

M. J. Werner and A. Imamoglu, Phys. Rev. A 61, 011801 (1999).
[Crossref]

S. Rebic, S. Tan, A. Parkins, and D. Walls, J. Opt. B 1, 490 (1999).
[Crossref]

J. Kim, O. Benson, H. Kan, and Y. Yamamoto, Nature 397, 500 (1999).
[Crossref]

1997 (1)

A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch, Phys. Rev. Lett. 79, 1467 (1997).
[Crossref]

1996 (1)

1932 (1)

C. Zener, Proc. R. Soc. London Ser. A 137, 696 (1932).
[Crossref]

Abdumalikov, A.

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. Fink, A. Abdumalikov, M. Baur, S. Filipp, M. da Silva, and A. Blais, Phys. Rev. Lett. 106, 243601 (2011).
[Crossref]

Alegre, T. P. M.

A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. M. Alegre, A. Krause, and O. Painter, Phys. Rev. Lett. 108, 033602 (2012).
[Crossref]

Amezcua, M.

D. A. Golter, T. Oo, M. Amezcua, K. A. Stewart, and H. Wang, Phys. Rev. Lett. 116, 143602 (2016).
[Crossref]

D. A. Golter, T. Oo, M. Amezcua, I. Lekavicius, K. A. Stewart, and H. Wang, Phys. Rev. X 6, 041060 (2016).
[Crossref]

Andrews, R.

R. Andrews, R. Peterson, T. Purdy, K. Cicak, R. Simmonds, C. Regal, and K. Lehnert, Nat. Phys. 10, 321 (2014).
[Crossref]

Ansmann, M.

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, Nature 464, 697 (2010).
[Crossref]

M. Hofheinz, E. Weig, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. Oonnell, H. Wang, J. M. Martinis, and A. Cleland, Nature 454, 310 (2008).
[Crossref]

Aoki, T.

B. Dayan, A. Parkins, T. Aoki, E. Ostby, K. Vahala, and H. Kimble, Science 319, 1062 (2008).
[Crossref]

Arcizet, O.

O. Arcizet, V. Jacques, A. Siria, P. Poncharal, P. Vincent, and S. Seidelin, Nat. Phys. 7, 879 (2011).
[Crossref]

Aspelmeyer, M.

S. Hong, R. Riedinger, I. Marinkovic, A. Wallucks, S. G. Hofer, R. A. Norte, M. Aspelmeyer, and S. Groblacher, Science 358, 203 (2017).
[Crossref]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev. Mod. Phys. 86, 1391 (2014).
[Crossref]

Baur, M.

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. Fink, A. Abdumalikov, M. Baur, S. Filipp, M. da Silva, and A. Blais, Phys. Rev. Lett. 106, 243601 (2011).
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S. Bennett, N. Y. Yao, J. Otterbach, P. Zoller, P. Rabl, and M. D. Lukin, Phys. Rev. Lett. 110, 156402 (2013).
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Benson, O.

J. Kim, O. Benson, H. Kan, and Y. Yamamoto, Nature 397, 500 (1999).
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Bialczak, R. C.

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, Nature 464, 697 (2010).
[Crossref]

M. Hofheinz, E. Weig, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. Oonnell, H. Wang, J. M. Martinis, and A. Cleland, Nature 454, 310 (2008).
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Birnbaum, K. M.

K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, Nature 436, 87 (2005).
[Crossref]

Blais, A.

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. Fink, A. Abdumalikov, M. Baur, S. Filipp, M. da Silva, and A. Blais, Phys. Rev. Lett. 106, 243601 (2011).
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K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, Nature 436, 87 (2005).
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Boozer, A. D.

K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, Nature 436, 87 (2005).
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Boss, J.

Y. Tao, J. Boss, B. Moores, and C. Degen, Nat. Commun. 5, 3638 (2014).
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Bowen, W. P.

S. Guan, W. P. Bowen, C. Liu, and Z. Duan, Europhys. Lett. 119, 58001 (2017).
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C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. Fink, A. Abdumalikov, M. Baur, S. Filipp, M. da Silva, and A. Blais, Phys. Rev. Lett. 106, 243601 (2011).
[Crossref]

Brandao, F. G.

M. J. Hartmann, F. G. Brandao, and M. B. Plenio, Nat. Phys. 2, 849 (2006).
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Budakian, R.

D. Rugar, R. Budakian, H. Mamin, and B. Chui, Nature 430, 329 (2004).
[Crossref]

Burkhart, L. D.

Y. Chu, P. Kharel, W. H. Renninger, L. D. Burkhart, L. Frunzio, P. T. Rakich, and R. J. Schoelkopf, Science 358, 199 (2017).
[Crossref]

Cai, K.

R.-X. Wang, K. Cai, Z.-Q. Yin, and G.-L. Long, Opt. Express 25, 30149 (2017).
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K. Cai, R. Wang, Z. Yin, and G. Long, Sci. China Phys. Mech. Astron. 60, 070311 (2017).
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Camacho, R. M.

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, Nature 462, 78 (2009).
[Crossref]

Cappellaro, P.

P. Rabl, P. Cappellaro, M. G. Dutt, L. Jiang, J. Maze, and M. D. Lukin, Phys. Rev. B 79, 041302 (2009).
[Crossref]

Chan, J.

A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. M. Alegre, A. Krause, and O. Painter, Phys. Rev. Lett. 108, 033602 (2012).
[Crossref]

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, Nature 462, 78 (2009).
[Crossref]

Chen, C.

Chow, E. M.

K. Y. Yasumura, T. D. Stowe, E. M. Chow, T. Pfafman, T. W. Kenny, B. C. Stipe, and D. Rugar, J. Microelectromech. Syst. 9, 117 (2000).
[Crossref]

Chu, Y.

Y. Chu, P. Kharel, W. H. Renninger, L. D. Burkhart, L. Frunzio, P. T. Rakich, and R. J. Schoelkopf, Science 358, 199 (2017).
[Crossref]

Chui, B.

D. Rugar, R. Budakian, H. Mamin, and B. Chui, Nature 430, 329 (2004).
[Crossref]

Cicak, K.

R. Andrews, R. Peterson, T. Purdy, K. Cicak, R. Simmonds, C. Regal, and K. Lehnert, Nat. Phys. 10, 321 (2014).
[Crossref]

Cleland, A.

M. Hofheinz, E. Weig, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. Oonnell, H. Wang, J. M. Martinis, and A. Cleland, Nature 454, 310 (2008).
[Crossref]

Cleland, A. N.

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, Nature 464, 697 (2010).
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Y.-D. Wang and A. A. Clerk, Phys. Rev. Lett. 108, 153603 (2012).
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J. D. Cohen, S. M. Meenehan, G. S. MacCabe, S. Groblacher, A. H. Safavi-Naeini, F. Marsili, M. D. Shaw, and O. Painter, Nature 520, 522 (2015).
[Crossref]

da Silva, M.

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. Fink, A. Abdumalikov, M. Baur, S. Filipp, M. da Silva, and A. Blais, Phys. Rev. Lett. 106, 243601 (2011).
[Crossref]

Dayan, B.

B. Dayan, A. Parkins, T. Aoki, E. Ostby, K. Vahala, and H. Kimble, Science 319, 1062 (2008).
[Crossref]

Degen, C.

Y. Tao, J. Boss, B. Moores, and C. Degen, Nat. Commun. 5, 3638 (2014).
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Deutsch, M.

A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch, Phys. Rev. Lett. 79, 1467 (1997).
[Crossref]

Dong, C.

C. Dong, V. Fiore, M. C. Kuzyk, L. Tian, and H. Wang, Ann. Phys. 527, 100 (2015).
[Crossref]

Du, J.

Y. Ma, Z.-Q. Yin, P. Huang, W. L. Yang, and J. Du, Phys. Rev. A 94, 053836 (2016).
[Crossref]

Duan, Z.

S. Guan, W. P. Bowen, C. Liu, and Z. Duan, Europhys. Lett. 119, 58001 (2017).
[Crossref]

Dutt, M. G.

P. Rabl, P. Cappellaro, M. G. Dutt, L. Jiang, J. Maze, and M. D. Lukin, Phys. Rev. B 79, 041302 (2009).
[Crossref]

Eichenfield, M.

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, Nature 462, 78 (2009).
[Crossref]

Eichler, C.

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. Fink, A. Abdumalikov, M. Baur, S. Filipp, M. da Silva, and A. Blais, Phys. Rev. Lett. 106, 243601 (2011).
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Englund, D.

A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vuckovic, Nat. Phys. 4, 859 (2008).
[Crossref]

Faraon, A.

A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vuckovic, Nat. Phys. 4, 859 (2008).
[Crossref]

Filipp, S.

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. Fink, A. Abdumalikov, M. Baur, S. Filipp, M. da Silva, and A. Blais, Phys. Rev. Lett. 106, 243601 (2011).
[Crossref]

Fink, J.

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. Fink, A. Abdumalikov, M. Baur, S. Filipp, M. da Silva, and A. Blais, Phys. Rev. Lett. 106, 243601 (2011).
[Crossref]

Fiore, V.

C. Dong, V. Fiore, M. C. Kuzyk, L. Tian, and H. Wang, Ann. Phys. 527, 100 (2015).
[Crossref]

Fleischhauer, M.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, Rev. Mod. Phys. 77, 633 (2005).
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Frunzio, L.

Y. Chu, P. Kharel, W. H. Renninger, L. D. Burkhart, L. Frunzio, P. T. Rakich, and R. J. Schoelkopf, Science 358, 199 (2017).
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Fushman, I.

A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vuckovic, Nat. Phys. 4, 859 (2008).
[Crossref]

Golter, D. A.

D. A. Golter, T. Oo, M. Amezcua, I. Lekavicius, K. A. Stewart, and H. Wang, Phys. Rev. X 6, 041060 (2016).
[Crossref]

D. A. Golter, T. Oo, M. Amezcua, K. A. Stewart, and H. Wang, Phys. Rev. Lett. 116, 143602 (2016).
[Crossref]

Gong, M.

Y. Ma, T. M. Hoang, M. Gong, T. Li, and Z.-Q. Yin, Phys. Rev. A 96, 023827 (2017).
[Crossref]

Groblacher, S.

S. Hong, R. Riedinger, I. Marinkovic, A. Wallucks, S. G. Hofer, R. A. Norte, M. Aspelmeyer, and S. Groblacher, Science 358, 203 (2017).
[Crossref]

J. D. Cohen, S. M. Meenehan, G. S. MacCabe, S. Groblacher, A. H. Safavi-Naeini, F. Marsili, M. D. Shaw, and O. Painter, Nature 520, 522 (2015).
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I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, Phys. Rev. Lett. 104, 083901 (2010).
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S. Guan, W. P. Bowen, C. Liu, and Z. Duan, Europhys. Lett. 119, 58001 (2017).
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M. J. Hartmann and M. B. Plenio, Phys. Rev. Lett. 99, 103601 (2007).
[Crossref]

M. J. Hartmann, F. G. Brandao, and M. B. Plenio, Nat. Phys. 2, 849 (2006).
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Hill, J. T.

A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. M. Alegre, A. Krause, and O. Painter, Phys. Rev. Lett. 108, 033602 (2012).
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Y. Ma, T. M. Hoang, M. Gong, T. Li, and Z.-Q. Yin, Phys. Rev. A 96, 023827 (2017).
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S. Hong, R. Riedinger, I. Marinkovic, A. Wallucks, S. G. Hofer, R. A. Norte, M. Aspelmeyer, and S. Groblacher, Science 358, 203 (2017).
[Crossref]

Hofheinz, M.

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, Nature 464, 697 (2010).
[Crossref]

M. Hofheinz, E. Weig, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. Oonnell, H. Wang, J. M. Martinis, and A. Cleland, Nature 454, 310 (2008).
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Hong, S.

S. Hong, R. Riedinger, I. Marinkovic, A. Wallucks, S. G. Hofer, R. A. Norte, M. Aspelmeyer, and S. Groblacher, Science 358, 203 (2017).
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Hou, Q.

Hu, Y.

J. Zhou, Y. Hu, Z.-Q. Yin, Z. D. Wang, S.-L. Zhu, and Z.-Y. Xue, Sci. Rep. 4, 6237 (2014).
[Crossref]

Huang, P.

Y. Ma, Z.-Q. Yin, P. Huang, W. L. Yang, and J. Du, Phys. Rev. A 94, 053836 (2016).
[Crossref]

Imamoglu, A.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, Rev. Mod. Phys. 77, 633 (2005).
[Crossref]

M. J. Werner and A. Imamoglu, Phys. Rev. A 61, 011801 (1999).
[Crossref]

A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch, Phys. Rev. Lett. 79, 1467 (1997).
[Crossref]

H. Schmidt and A. Imamoglu, Opt. Lett. 21, 1936 (1996).
[Crossref]

Jacques, V.

O. Arcizet, V. Jacques, A. Siria, P. Poncharal, P. Vincent, and S. Seidelin, Nat. Phys. 7, 879 (2011).
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Figures (3)

Fig. 1.
Fig. 1. (a) Schematic diagram of the phononic crystal: the NV center ensembles are located near the surface. The phonon is coupled to NV center ensembles driven by the laser field and micro-magnetic field. (b) Energy levels of NV centers. In this structure, we regard | 1 , | 0 , | + 1 , and | E y as | 1 , | 2 , | 3 , and | 4 , respectively. δ and ε are the detuning frequency. ω m , ω d , and ω c are the frequencies of the phonon, optical laser driving, and micro-magnetic driving, respectively.
Fig. 2.
Fig. 2. (a) Evolution of the second correlation function g ( 2 ) ( 0 ) . (b) Evolution of the population of Fock state | 1 . The coupling strength g / 2 π = 25 kHz , the dissipation of system γ = g / 8 , and the driving strength is Ω ˜ = g / 5 . The mean thermal polariton number of the blue dashed line, the red dotted dashed line, and the yellow line are P 0 P 0 = 0.1 , 0.3 , 0.5 , respectively.
Fig. 3.
Fig. 3. (a) Evolution of the second correlation function g ( 2 ) ( 0 ) . (b) Evolution of the population of Fock state | 1 . The coupling strength g / 2 π = 25 kHz , the dissipation of system γ = g / 8 , and the mean thermal polariton number is P 0 P 0 = 0.1 . The driving strength of the blue dashed line, the red dotted dashed line, and the yellow line are Ω ˜ = g / 8 , g / 5 , g / 2 , respectively.

Equations (10)

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H o = ω m b b + ω 1 σ 11 + ω 2 σ 22 + ( ϵ + ω 3 ) σ 33 + ( ω 4 + δ ) σ 44 + Ω d ( e i ω 23 t σ 32 + H.c ) + Ω c ( e i ω c t σ 42 + H.c ) + g 24 ( b + b ) σ 44 + g 13 ( b σ 31 + H.c ) ,
H a δ σ 44 + ε σ 33 g ˜ 24 ( b σ 42 + H.c. ) + g ˜ 13 ( b σ 31 + H.c. ) + Ω d ( σ 32 + H.c. ) ,
H 0 = ε σ 33 + Ω d ( σ 32 + H.c. ) + g ˜ 13 ( b σ 31 + H.c. ) , H 1 = δ σ 44 g ˜ 24 ( b σ 42 + H.c. ) .
P 0 = ( g ˜ 13 σ 21 Ω d b ) / B , P + = μ σ 31 + ν ( Ω d σ 21 + g ˜ 13 b ) / B , P = ν σ 31 + μ ( Ω d σ 21 + g ˜ 13 b ) / B ,
H 0 = μ 0 P 0 P 0 + μ + P + P + + μ P P ,
g ˜ 24 ( σ 41 σ 12 b + b σ 21 σ 14 ) g ˜ 24 g ˜ 13 2 Ω d [ σ 41 ( P 0 ) 2 + ( P 0 ) 2 σ 14 ] .
H E = g ( P 0 ) 2 ( P 0 ) 2 ,
H = g ( P 0 ) 2 ( P 0 ) 2 + Ω ˜ ( P 0 + P 0 ) ,
g ( 2 ) ( 0 ) = P 0 ( t ) P 0 ( t ) P 0 ( t ) P 0 ( t ) / P 0 ( t ) P 0 ( t ) 2 .
ρ ˙ ( t ) = i [ H , ρ ( t ) ] + γ 2 ( 1 + n th ) L [ P 0 ] ρ ( t ) + γ 2 n th L [ P 0 ] ρ ( t ) ,

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