Yong-Hong Ma(馬永紅) Yuan Xu(許媛) Quan-Zhen Ding(丁全振) and Yu-Sui Chen(陳予遂)

1School of Science,Inner Mongolia University of Science and Technology,Baotou 014010,China

2Department of Physics,and Center for Quantum Science and Engineering,Stevens Institute of Technology,Hoboken,NJ 07030,USA

3Department of Physics,New York Institute of Technology,Old Westbury,NY 11568,USA

Keywords: spin squeezed state,SiV center,diamond waveguide

1. Introduction

A spin squeezed state is from an ensemble of spins whose fluctuation in one collective spin direction to the mean spin direction is smaller than the classical limit.[1]As an important quantum resource, spin squeezing has many potential applications,such as quantum information,quantum measurement,quantum computing,[2,3]and quantum network.[4,5]Not only particle entanglement can be studied, but also the accuracy of quantum computation can be greatly improved by preparing the spin-squeezing state.[6,7]Thus it promotes the development of quantum information.[8]

Two types of spin squeezing called one-axis twisting model and two-axis twisting models, were provided by Kitagawaet al.[1]and Winelandet al.,[9]respectively.At present,a larger number of spin squeezing schemes were proposed with the development of quantum information, such as in Bose–Einstein condensates,[10–17]quantum non-demolition measurements,[18–20]nuclear magnetic resonance quadrupolar system,[21]and a standing-wave optical lattice.[22]

However, in experiments, the generation of spin squeezing is quite challenging due to the decoherence effects. The vacancy centers[23]may be promising candidates to generate spin squeezing because of their versatile quantum properties. A number of schemes to generate spin squeezed states were proposed recently in nitrogen-vacancy (NV) center systerms.[3,24–28]According to recent studies, silicon vacant color centers (SiV) are more superior than other vacant centers.[29–32]SiV center is formed by a silicon atom replacing a carbon atom in the diamond and then trapping a hole around it.[33]Since the atoms do not bond with the vacancy,it makes the electrons extremely sensitive to small changes in the electrical,magnetic,and optical properties in the surrounding environment[4,29,31,34]and is often used in the related research of spin-mechanical hybrid devices.[35]The neutral SiV in diamond is a candidate material for quantum network due to optical properties and long spin coherent time.[36]Besides,negatively charged SiV, which has good spectral characteristics and prepares spin states easily, is used in quantum communication, quantum information, and sensor areas.[31,36–39]Therefore, spin squeezed states in SiV systems may play an important role in the progress of science and technology.

In this paper, we propose an effective scheme to prepare the spin squeezed states by coupling SiV centers into onedimensional diamond waveguides.[35,40–42]We derive the evolutions of spin squeezing in this system. The performance of spin squeezing with the spin numbers,the microwave field and other related parameters are studied. The results show that the perfect spin squeezing can be created even in the presence of spin relaxation induced by mechanical dissipation. The arrangement of this paper is as follows. We propose the model under study and give a brief explanation in Section 2. Then,we derive the Hamiltonian of the whole system and its complete dynamic description. The spin squeezing of the scheme is numerically studied in Section 3. We summarize this work and draw conclusions in Section 4. The mean values of the relevant mechanical quantities are presented in Appendix A.

2. The model and master equation

The theoretical model is shown in the Fig.1(a),whereNSiV centers are uniformly arranged in a one-dimensional diamond waveguide. Due to SiV orbital states’inversion symmetry,the lack of the permanent magnetic moment for an electric couple decreases its response to local environmental charge fluctuations.[5,43,44]The optical ground state contains two distinct electronic configurations corresponding to the presence of the unpaired hole in one of the even-parity orbitals eg+and eg?. Physically, the two branches in the optical ground state correspond to the occupation. In a zero magnetic field,the degeneracy of these orbitals is broken by spin–orbit coupling leading to frequency splittingΔ= 2π×46 GHZ.[38]Upon applying a magnetic field, the degeneracy between the spin–orbit eigenstates is further broken and split into two sublevels within each orbital branch corresponding to different spin states of the unpaired hole. Therefore, SiV centers are equivalent to the four-level systems, consisting of two doublets. The energy gap is determined by spin–orbit interactions and a weak Jahn–Teller effect to the item, as shown in Fig. 1(b).|eg+↓〉～|1〉,|eg?↑〉～|2〉,|eg?↓〉～|3〉and|eg+↑〉～|4〉.

Fig.1. (a)The SiV color centers are uniformly placed in a one-dimensional diamond waveguide. The length, width and height of the waveguide are l,m and n, respectively, with l ?{m,n}. (b) Splitting of energy levels in a magnetic field Bz.

The spin mechanical dynamics for the system can be described by Hamiltonian[33]

where the first term in ?His the dynamic description ofNSiV color centers in the waveguide, the second term is the interaction between SiV centers and the one-dimensional diamond waveguide, and the last term is the Hamiltonian of the onedimensional diamond waveguide. These three terms are denoted by

Since the environment will inevitably influence the system,to fit the actual situation, we consider the decoherence and spin relaxation of the system and write the Markovian Lindblad master equation as

wherenth=(e?ωm/kBT ?1)?1is the number of thermal phonon at the environment temperatureTwith Boltzmann constantkB.The super operatorDsatisfiesD(?A)?ρ= ?A?ρ?A??(1/2)?A??A?ρ ?(1/2)?ρ?A??Afor a given operator ?A. Collective spin relaxation induced by mechanical dissipation is described in the last two terms withΓm=γmg2e/Δ2withγmthe decay of mechanical modes. We will demonstrate that the spin squeezing will survive under such dissipative conditions.

3. Analysis of spin squeezing

The detailed calculation of the mean values used inξ2is shown in Appendix A. When the spin squeezing parameter is equal to 1, it corresponds to the coherent state; When the parameter satisfiesξ2<1,it is a spin squeezed state.

Firstly,we consider the time evolution of spin squeezing with different spin numberNwhen the microwave field is absent(c=0). According to the simulation results in Fig.2(a),the degree of spin squeezing becomes strong as the spin numberNincreases. However, the change will not be obvious whenNreaches 300. This kind of spin squeezing can occur with the absence of microwaves, which was confirmed in previous studies.[35]Then,we pay attention to the case in the presence of a microwave field(c/=0). As shown in Fig.2(b),the existence of a microwave field has a noticeable influence on the spin squeezing of the system. We observe that the degree of spin squeezing mounts significantly with varying growth values of the enhancive microwave fieldc(c=2Ωeff).Moreover,the simulation results show that long time and highintensity spin squeezing appear with the increase of the ratioc/a, untilc/a=1. This is because the spin direction of SiV centers changes under the action of an external microwave field, which results in a stronger coupling effect and a higher degree of squeezing. Therefore,it is more significant that the desired spin squeezing effect can be achieved by applying and adjusting the external microwave field’s intensity and direction. In particular, due to theD3danti-symmetry structure of SiV,we can reasonably adjust the magnetic field direction according to the symmetry axis ofC3.[23]

Fig.2. Time evolution of spin squeezing parameter ξ2 with(a)different spin number N and (b) different value of microwave field c. The parameters are(a) a=b=4, c=0, nth =100,Γm =0.05 and γs =0.1 with different spin number(N=50,100,150,200,250,300 for the solid blue line,dashed cyan line,solid magenta line,dashed green line,solid yellow line and dashed red line,respectively);(b)N=100,a=b=4,Γm=0.05 and γs=0.1 with different value of microwave field(c=0,0.8,1.6,2,4,3.2,4.0 for the solid blue line, dashed cyan line, solid magenta line, dashed green line, solid red line and dashed yellow line,respectively).

Next,let’s focus on the related parameters in the effective Hamiltonian for Eq.(7). In Fig.3,we plot the time evolution of the spin squeezing parameterξ2with differentaandb. As shown in Fig.3(a), long time and high-intensity spin squeezing can be obtained with the downtrend of ratioa/c, that is,the preferable spin squeezing states will be obtained with the smaller ratioa/c. However, the ratioa/cshould not be less than 1,because the degree of spin squeezing does not change significantly in this case. Figure 3(b) shows that the period of spin squeezing decreases,the maximum spin squeezing increases corresponding to the minimum value ofξ,asb/cdecreases.

Inspired by the results in Figs. 2 and 3, we are aware of the strength and survival time of spin squeezing reach optimal values whena=b=c.In Fig.4(a),we plot the time evolutions of spin squeezing with different numbers of SiV centers for a finite temperature. The results show that the relatively large spin numbers not only increase the maximum spin squeezing but also prolong the survival time of the whole process because the degree of spin squeezing is positively correlated with the number of particlesN. The more SiV centers in a onedimensional quasi-diamond waveguide,the stronger coupling between different SiV centers and the easier spin squeezing generating. If the number of spinNcontinues to increase,ξ2will approach 0. It indicates that the squeezing amount can be increased indefinitely by increasing the total number of SiV centers. However, the more SiV centers there are, the higher loss rate will be increased in the experiment.

Fig.3.Time evolution of spin squeezing parameter ξ2 with(a)different value of a and(b)different value of b. The parameters are(a)b=c=4,nth=100,N =100, γs =0.1, and Γm =0.05 with different a (a=4,12,20,28,36 for the solid blue line, dashed cyan line, solid magenta line, dashed green line and solid red line,respectively);(b)a=c=4,nth=100,N=100,γs=0.1,and Γm=0.05 with different b(b=0,0.8,1.6,2,4,3.2,4.0 for the solid blue line, dashed cyan line, solid magenta line, dashed green line and solid red line,respectively).

Also,we need to consider the effect of the thermal phonon number related to the selection of ambient temperature on the system. In Fig.4(b),the high-intensity spin squeezing with a longer period will be obtained with the decrease of the thermal phonon number or the decrease of the environment temperature. The reason is that the decoherence effect will be weakened with the decrease of temperature. So the coupling between spins will be enhanced,and the stronger spin squeezing will be reflected simultaneously.

Fig. 4. (a) Time evolution of spin squeezing parameter ξ2 with different spin number N. The parameters are a=b=c=4, nth =100, Γm =0.05,γs = 0.1 with different spin number N (N = 50,100,150,200,300 for the solid blue line, dashed cyan line, solid magenta line, dashed green line and solid red line,respectively). (b)Time evolution of spin squeezing parameter ξ2 with different thermal phonon number nth. The parameters are N =50,a=b=c=4,Γm=0.05 and γs=0.1 with different thermal phonon number nth(nth=100,150,200,250,300 for the solid blue line,solid green line,solid magenta line,solid cyan line and solid red line,respectively).

Since the system inevitably interacts with the environment, we need to consider the effect of decoherence on the system. Based on a complete description of the dynamic evolution of the system, the influences of the dephasing (γs) of the SiV spins,and the decay of the mechanical modes(Γm)on spin squeezing are analyzed respectively, as shown in Fig. 5.As we would expect, the time range of spin squeezing may be prolonged as the decay rateΓmdecreases in Fig.5(a). The weaker of the decay rates are,the longer the system’s dynamics will maintain. Moreover,by choosing different dephasing(γs)in Fig.5(b),it shows that the time range and spins squeezing degree almost remain the same. This is because the lower dephasing(γs),the less affected of the system,and the trend of squeezing time and squeezing degree are increasing. As such,to achieve the maximum spin squeezing,the dephasing(γs)of the SiV spins and the decay of the mechanical modes should be as small as possible.

Fig. 5. Time evolution of spin squeezing parameter ξ2 with (a) different value of effective mechanical decay rate Γm and (b) different value of spin dephasing rate γs. The parameters are(a)a=b=c=4,nth=100,N=50,γs =0.1 with different effective mechanical decay rate Γm (Γm =0.1, 0.08,0.06,0.05 for the solid blue line,solid cyan line,solid magenta line and solid green line, respectively); (b) a=b=c=4, nth =100, N =50, Γm =0.05 with different spin dephasing rate γs (γs=0.1,0.01,0.001 for the solid blue line,solid red line and dashed green line,respectively).

4. Conclusion

One major advantage of building this kind of quantum device with SiV center rather than other solid-state systems is that SiV systems are typically more easily integrated into nanofabricated electrical and optical structures.[47,48]For example, a NV center in a diamond causes coupling to nearby electric field noise,which shifts its optical transition frequency as a function of time. However,SiV center is immune to this spectral diffusion to first order by its inversion symmetry and undoubtedly becomes an ideal candidate for integration into diamond waveguides. Another advantage is the excellent optical coherence property for SiV centers. We can characterize the optical coherence properties of SiV centers in nanostructures by photoluminescence excitation spectroscopy.[47]SiV centers have narrow transitions withΓn/2π=410±160 MHz,which is about 4 times than the lifetime limited linewidthγ/2π=94 MHz. The ratiosΓn/γfor SiV centers are much lower than the values for NV centersΓn/γ100–200.[47]

In summary, it is feasible to prepare spin squeezing by coupling SiV centers into diamond waveguides. Under the condition of large detuning, to obtain high-strength spin squeezing with a long survivable time, we can decrease the environment temperature or increase the number of spin numbers. The application of the external microwave field effectively increases the degree of spin squeezing and prolongs the duration of the spin squeezing state. Furthermore,the stronger the degree of spin squeezing,the more difficult it is to maintain spin squeezing in the decoherence environment, which is expressed asΓmin the theoretical study. Therefore,the influence of decoherence on the system should be minimized. In practice,we can reduce the effect of decoherence on the system by improving the quality and purity of the materials.

Appendix A:Expectation of collective operators with time evolution