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Abstract
We propose a scheme to entangle Silicon-Vacancy (SiV) centers embedded in a diamond acoustic waveguide. These SiV centers interact with acoustic modes of the waveguide via strain-induced coupling. Through Morris-Shore transformation, the Hilbert space of this hybrid quantum system can be factorized into a closed subspace in which we can deterministically realize the symmetrical Dicke states between distant SiV centers with high fidelity. In addition, the generation of entangled Dicke states can be controlled by manipulating the strength and frequency of the driving field applied on SiV centers. This protocol provides a promising way to prepare multipartite entanglement in spin-phonon hybrid systems and could have broad applications for future quantum technologies.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Macroscopic (many-qubit) entanglement is one of the fascinating aspects of quantum mechanics [1–3]. It has a wide range of applications in current quantum technologies, such as quantum computation [4–8], quantum key distribution [9–12], and quantum metrology [12–15]. In the past decades, cavity quantum electrodynamics (QED) systems, such as atomic systems, have provided a great platform to propose different schemes to prepare multipartite entanglement [16–19]. In spin systems, entanglement between distant spins has also been studied for a long time [8,20]. For example, one can generate entanglement between distant color-center spins by detecting the emitted photons already entangled with different color centers via couplings between centers and the photon mode [21]. This so-called measurement-based scheme can effectively produce entanglement but not deterministically. An alternative to deterministically generate entanglement of centers is based on the coupling between color-center spins and cavity modes [22–24]. By sharing common modes of a cavity or a waveguide, distant color centers can be coherently coupled and entangled [25–27]. Then, one can manipulate these color centers to realize scalable quantum information processing(QIP).
In these years, many researchers have devoted considerable effort to color centers for their possible applications in future quantum technologies [28–30]. Among many color centers, the NV center plays an essential role in quantum information processing because of its long coherence time and good controllability at room temperature [31–38]. It can also be used as a sensor of ultra-high precise solid magnetometers and thermometers [39–41]. However, the wide zero-phonon-line (ZPL) spectrum of the NV center is still a big challenge in exploring broader applications [31]. Recently, another type of color centers, the Silicon-Vacancy (SiV) center has attracted increasing attention [42–48]. Compared with the NV center, the SiV center has much narrower ZPL resulting from its inversion symmetry [49–56], and one could employ it to manufacture a more efficient single-photon source [57]. Moreover, the SiV center is sensitive to the crystal strain benefiting from its electronical doublet ground state [8,58–61], which could be used to realize strong and controllable coupling between SiV centers and acoustic modes. Therefore, the SiV center is regarded as one of the promising candidates for constructing spin-phonon hybrid quantum systems, which can be used to cool the acoustic modes, transfer quantum states, and so on [8,36,59,60,62–66].
In this paper, we propose a scheme to generate high-fidelity entanglement of distant SiV centers embedded in a quasi-one-dimensional (1D) diamond waveguide, which is a hybrid quantum system that has wide-range applications in quantum information processing [8,62,64–66]. We derive the effective Hamiltonian of this hybrid system at first [8,65]. Then, by factorizing the Hilbert space of the system through Morris-Shore (MS) transformation, we can realize the symmetrical Dicke states, which are highly entangled states [67–72]. While the frequency of the driving field applied on SiV centers is being adjusted, the system may evolve into the target Dicke state, stay at it for a while, and then reverse to its initial state. Such a disentanglement process is caused by twice adiabatic avoid-crossing during the driving field’s monotonic modulation, which is unique to this scheme and has not been discussed before. The whole process and the duration of the target Dicke state can be optimized by appropriately adjusting the intensity and frequency of the driving field. Consequently, these properties allow us to generate many-body entanglement in this spin-phonon hybrid system.
Our paper is organized as follows. In Sec. 2, we introduce the model of a hybrid quantum system consisting of an array of SiV centers embedded in a quasi 1D acoustic waveguide. Section 3 describes the scheme to prepare the symmetrical Dicke states between three SiV centers through MS transformation. In Sec. 4, we numerically simulate the population conversion between different states and analyze the influence caused by the dissipation of hybrid system. Finally, we discuss the experimental feasibility of this scheme and make a summary in Sec. 5.
2. Model
As depicted in Fig. 1(a), an array of $N$ SiV centers is coupled to an acoustic mode of a 1D diamond waveguide via the strain-induced coupling. This coupling arises from the change in Coulomb energy of the electronic states due to the displacement of atoms forming the defect. The total Hamiltonian of this hybrid system contains three parts [8,49,65],
where $H_{\rm SiV}$ and $H_{\rm ph}$ are the Hamiltonians of SiV centers and the acoustic mode, respectively, and $H_{\rm strain}$ describes the strain-induced coupling between the orbital degree of the SiV center and the acoustic mode of the waveguide as shown in Fig. 1(b).
Fig. 1. (a) Sketch of an array of $N$ SiV centers embedded in a 1D diamond waveguide. The length, width, and thickness of the waveguide are L, w, d, respectively. Molecular structure of the SiV center is shown as inset. (b) The level structure of the electronic ground state of the SiV center. Two time-dependent microwave driving fields induce the transitions between levels $\vert 1\rangle \leftrightarrow \vert 4\rangle$ and $\vert 2\rangle \leftrightarrow \vert 3\rangle$, while the transitions between levels $\vert 1\rangle \leftrightarrow \vert 3\rangle$ and $\vert 2\rangle \leftrightarrow \vert 4\rangle$ are caused by the strain-induced coupling.
The SiV center is an interstitial point defect in which a silicon atom is positioned midway between two adjacent missing carbon atoms in the diamond lattice as depicted in the inset of Fig. 1(a). Its ground state is four-fold degenerate, and three interactions can lift the degeneracy: the spin-orbit coupling $H_{\rm SO}$ and Jahn-Teller (JT) effect $H_{\rm JT}$ lift the orbital degeneracy, while the spin degeneracy is canceled by the Zeeman interaction $H_Z$ [8,49]. In the presence of these interactions, the four degenerate states are split into two doublets as shown in Fig. 1(b), with corresponding two lower eigenstates $\vert 1\rangle =\vert e_-,\downarrow \rangle,\vert 2\rangle =\vert e_+,\uparrow \rangle$ and two upper eigenstates $\vert 3\rangle =\vert e_+,\downarrow \rangle,\vert 4\rangle =\vert e_-,\uparrow \rangle$, where $\vert e_\pm \rangle =(\vert e_x\rangle \pm i\vert e_y\rangle )/\sqrt 2$ are eigenstates of the orbital-angular-momentum operator $L_z$. The corresponding energy splitting $\Delta =[\lambda _g^2+\left (\Upsilon _x^2+\Upsilon _y^2\right )]^{1/2}\approx 2\pi \times 46$ GHz, where $\lambda _g=2\pi \times 45$ GHz is the spin-orbit coupling strength, and $\Upsilon _{x(y)}$ describes the strength of the JT effect along $\vec x(\vec y)$ direction. In contrast to previous works in the same system [8,60,62,64–66], here we apply two time-dependent microwave fields $\Omega _{1,2}(t)$ with frequencies $\omega _{1,2}$, which induce transitions between states $\vert 1\rangle \leftrightarrow \vert 4\rangle$ and $\vert 2\rangle \leftrightarrow \vert 3\rangle$ on the SiV center, as shown in Fig. 1(b). Consequently, the dynamics of SiV centers can be described by the Hamiltonian [8,65]
Now we consider acoustic modes in the quasi 1D diamond waveguide. The length, width, and thickness of the waveguide are $L$, $w$, $d$, respectively, as shown in Fig. 1(a), satisfying $L\gg w,d$ meanwhile. Within the frequency range of interest, the acoustic modes can be modeled as elastic waves with a displacement field $\vec u(\vec r,t)$ obeying the equation of motion for a linear, isotropic medium. Then under the periodic boundary condition, the quantized displacement field has the following form
Considering that the acoustic modes are well separated from frequency ($\Delta \omega _n\geq 2\pi \times 50$ MHz) in the waveguide with small size, we could treat the mechanical modes as a single standing wave with $\omega _n\approx 2\pi \times 46$ GHz for simplicity. Meanwhile, for the convenience of the following derivation, we could assume that a single SiV center is located at the position $x_j=0$, and the distance between neighboring centers is set to $\Delta x = x_j-x_{j-1} =n \lambda$, with $\lambda$ being the wavelength of the corresponding acoustic mode and n is an integer. Finally, we obtain the total Hamiltonian of this hybrid system
3. Generation of Dicke states
In this section, we discuss the generation of symmetric Dicke state, which is an excellent multipartite entangled state as its entanglement is maximally persistent and robust under particle losses [70–72,75,76]. Here we consider generating two non-trivial symmetric Dicke states $\vert D_3^{(1)}\rangle =(\vert 211\rangle +\vert 121\rangle +\vert 112\rangle )/\sqrt 3$ and $\vert D_3^{(2)}\rangle =(\vert 221\rangle +\vert 212\rangle +\vert 122\rangle )/\sqrt 3$ of three SiV centers through MS transformation [69], as shown in Fig. 2. The MS transformation can decompose two coupled sets of degenerate levels into a coupled nondegenerate two-state system and a set of uncoupled dark states. This method can furthermore be extended to three and more degenerate levels [68].
Fig. 2. Generation of two symmetric Dicke states. (a) The energy-level pattern of a string of three SiV centers with an initial state $\vert \alpha \rangle \vert 1\rangle _p$ globally driven by microwave fields, and $\vert \alpha \rangle =\vert 111\rangle$, the initial state could absorb a phonon to jump to one of the states $\vert \beta \rangle \vert 0\rangle _p$, $\vert \beta _1\rangle \vert 0\rangle _p$, $\vert \beta _2\rangle \vert 0\rangle _p$. After MS transformation, the energy-level pattern is reduced to a two-level subsystem, with $\vert \beta '\rangle =\vert D_3^{(1)}\rangle$ and two uncoupled dark states, $\vert \beta _1'\rangle \vert 0\rangle _p$, $\vert \beta _2'\rangle \vert 0\rangle _p$. (b) The energy-level pattern of a string of three SiV centers whose initial state is $\vert \alpha \rangle \vert 2\rangle _p$ globally driven by microwave fields similarly. Through MS transformation similarly, we could obtain corresponding Dicke states in an equivalent coupled three-level subsystem, i.e. $\vert \eta '\rangle =\vert D_3^{(1)}\rangle$ and $\vert \gamma '\rangle =\vert D_3^{(2)}\rangle$. See text for more details.
We first consider the preparation of the Dicke state $\vert D_3^{(1)}\rangle$. The acoustic mode of the waveguide and the electron-spin states of SiV center are initially prepared in the one-phonon state $\vert 1\rangle _p$ and the ground state $\vert 111\rangle$ [77–79], respectively. Through MS transformation, the original system could be factorized into a two-level subsystem and two independent uncoupled dark state subsystems, as shown in Fig. 2(a). Interestingly, the state $\vert \beta '\rangle =(\vert 211\rangle +\vert 121\rangle +\vert 112\rangle )/\sqrt 3$ is just the Dicke state $\vert D_3^{(1)}\rangle$ that we need, while the two uncoupled states $\vert \beta _1'\rangle =(\vert 211\rangle -2\vert 121\rangle +\vert 112\rangle )/\sqrt 6$, $\vert \beta _2'\rangle =(\vert 211\rangle -\vert 112\rangle )/\sqrt 2$ are not necessary here. An effective Hamiltonian describes the dynamical process of the two-level subsystem can be written as
Then we prepare the state $\vert D_3^{(2)}\rangle$. Different from the case of $\vert D_3^{(1)}\rangle$, the initial state of the hybrid system is prepared to a two-excitation state as $\vert 111\rangle \vert 2\rangle _p$, and we could therefore construct a three-state subspace to generate the corresponding Dicke state $\vert D_3^{(2)}\rangle$. After performing MS transformation similarly, we can obtain an equivalent three-level subsystem $\vert \alpha \rangle \vert 2\rangle _p\leftrightarrow \vert \eta '\rangle \vert 1\rangle _p\leftrightarrow \vert \gamma '\rangle \vert 0\rangle _p$, where $\vert \alpha \rangle =\vert 111\rangle$, $\vert \eta '\rangle =\vert D_3^{(1)}\rangle$, $\vert \gamma '\rangle =\vert D_3^{(2)}\rangle$, as shown in Fig. 2(b), there are also two decoupled two-level system in the new MS basis, $\vert \eta '_1\rangle =(\vert 211\rangle -2\vert 121\rangle +\vert 112\rangle )/\sqrt 6$, $\vert \gamma _1'\rangle =(\vert 221\rangle -2\vert 212\rangle +\vert 122\rangle )/\sqrt 6$, $\vert \eta _2'\rangle =(\vert 211\rangle -\vert 112\rangle )/\sqrt 2$, $\vert \gamma _3'\rangle =(\vert 221\rangle -\vert 122\rangle )/\sqrt 2$ are not in our consideration. The dynamical process of this three-level system is governed by the effective Hamiltonian
4. Numerical results and disentanglement process
In this section, we numerically simulate the population conversion between the initial state and corresponding Dicke states with experimentally feasible parameters. The population conversion can be used to describe the generation process of these two three-particle symmetric Dicke states [81,82]. We use two microwave fields which vary linearly with time in the scheme, the strengths of microwave fields are $\Omega _1=\Omega _2=2\pi \times 50~{\rm MHz}$, and corresponding detunings are $\delta _1=2\pi \times 500 ~{\rm MHz},\delta _2=2\pi \times 400~{\rm MHz}$, respectively. The strain-induced coupling strength between SiV centers and the acoustic mode used here is $g=2\pi \times 10$ MHz. Moreover, when the system began to evolve, the effective coupling strengths in Eq. (12) and Eq. (13) are $\epsilon =2\pi \times 270~{\rm MHz}$ and $\lambda =2\pi \times 30~{\rm MHz}$, respectively.
Figure 3 shows the populations of initial states and Dicke states $\vert D_3^{(1)}\rangle,\vert D_3^{(2) }\rangle$ during the time evolution. Both the probability of realizing two nontrivial three-particle symmetric Dicke states are $\rm {P_{\vert D_3^{(1)}\rangle,\vert D_3^{(2)}\rangle }}>0.99$. The conversion time of population is about 40 $\mu$s, while the coherence time of SiV centers could be extended to $T_s\sim$10 ms by using dynamical decoupling techniques [83–86], which is much longer than the conversion time of population.
Fig. 3. Population conversion of states versus the time evolution. (a) Population evolution of the initial state $\vert \alpha \rangle$ (green) and the Dicke state $\vert \beta '\rangle$ (red). (b) Population evolution of the initial state $\vert \alpha \rangle$ (blue), the Dicke states $\vert \eta '\rangle$ (green) and $\vert \gamma '\rangle$ (red).
Furthermore, we can optimize the generation process of Dicke states by adjusting the microwave fields. When increasing the strength of microwave fields monotonically, effective coupling between initial state and corresponding Dicke states is also increased, as a result, the rate of population conversion could become faster. Figure 4 shows the numerical curve fitting results of the population transfer versus different driven strengths, we could see the changing of the population-conversion time under different microwave fields driving. For example, the time of the populations transferred to $\vert D_3^{(1)}\rangle$ state from initial state is shortened about 50 $\mu$s when the driven strength increased to 50 ${\rm MHz}$ from 30 MHz. Similarly, in the case of $\vert D_3^{(2)}\rangle$ state generation, the corresponding conversion time is also shortened about 40 $\mu$s versus same microwave fields changing with Fig. 4 (a). Note that, in recent studies, it is found that the adiabatic passage can be further shortened by using the shortcut method [87]. In Fig. 4 (b), the populations of the steady-states after the transfer are slightly larger than that of the original states due to errors introduced by the numerical simulations and curve fitting.
Fig. 4. Curve fitting of numerical results of the population transfer versus different driven strengths. (a) and (b) are the cases for the generations of $\vert D_3^{(1)}\rangle$ state and $\vert D_3^{(2)}\rangle$ state, respectively. In this figure, dotted lines denote the populations of corresponding Dicke states $\vert D_3^{(1)}\rangle$ and $\vert D_3^{(2)}\rangle$, and solid lines represent the populations of initial states. In (b), dashed lines denote the populations of $\vert D_3^{(1)}\rangle$ state. And the black, red, green lines correspond the microwave fields strengths 30 MHz, 40 MHz, 50 MHz, respectively.
Here is another interesting phenomenon when the term $\nu -\delta _2$ is not zero in Eq. (11) via adjusting the microwave fields, the system may evolve into the target Dicke state $D_3^{(2)}$, stay at it for a while, and then reverse to its initial state, which is disentangled. This phenomenon could be interpreted by occurring avoid-crossing adiabatic passages twice. In addition, the disentanglement procedure could also be useful in future quantum applications. For instance, one could disentangle states of the SiV centers simply by optically reinitializing them or utilize this procedure to cross the obstacle of incomplete discharge in quantum batteries due to entanglement between spins (emitters) [88,89]. We numerically calculate the eigenstates of ${H_{\rm eff}^{(2)}}$ during the time evolution as shown in Fig. 5(a), there are two avoid crossings between initial state and Dicke state $D_3^{(2)}$. As a result, the populations of Dicke state $D_3^{(2)}$ will convert into the ground state through the second avoid-crossing region. In addition, the maintenance time of stable entangled Dicke states is related to the magnitude of the term $\nu -\delta _2$. When we appropriately choose the value of $\nu -\delta _2$, the system will stay at the target Dicke state for a longer time, as shown in Fig. 5(b) and (c), which provides more possibilities for further quantum operations. Finally, based on this hybrid quantum system, we have realized the generation and disentanglement of symmetric Dicke states only in the presence of two simple linear microwave fields compared to other complicated pulse-driven schemes.
Fig. 5. Adiabatic evolution of the three-level subsystem. (a) is the corresponding level energies versus time evolution, solid lines and dashed lines denote level energies with and without coupling respectively. (b) and (c) are the populations conversion of three-level subsystem when the term $\nu -\delta _2=$100 MHz and $\nu -\delta _2=$200 MHz, respectively.
So far, dissipations of this hybrid system have not been considered. The main decoherence sources associated with our implementation are dephasing of the electron spins and intrinsic dissipation of the waveguide. The mechanical dissipation is
where $\omega$ is the frequency of acoustic mode and $Q$ is the corresponding quality factor. In the practical situations, a mechanical quality factor $Q\approx 7\times 10^5$ for the mechanical phonon modes of the small size diamond waveguide has been demonstrated [90,91]. Moreover, at cryogenic temperatures, the mechanical Q-factor of diamond resonators has reached the same order of $10^6$ [92]. As a result, the mechanical dissipation has a value is that $\varGamma =10\sim 100~{\rm kHz}$. At 100 mK temperature, the dephasing rate of a single SiV center is $\gamma _{d}\sim 100$ Hz [29,83,84], which is much smaller than the mechanical dissipation $\varGamma$. Thus, we may neglect the spin dephasing in the calculation of fidelity. Finally, the effect dissipation can be described by the master equation: where $k=1,2$ corresponding Eq. (12), Eq. (13), and $\rho$ is the density operator of the effective two-level (three-level) subsystem. The lindblad form $L\rho$ can be written asThen, we can numerically calculate the fidelity of corresponding Dicke states to express the influence of system dissipation $\varGamma$. Since the Dicke state is the maximally entangled multipartite state and the fidelity describes the "closeness" of the generated state and the ideal Dicke state, the entanglement degree of generated state could also be equivalently characterized by the corresponding fidelity.
In Fig. (6), we plot the corresponding fidelity of prepared Dicke states with same parameters as in Fig. (3) versus the dissipation $\varGamma$ induced by mechanical mode of the waveguide at 100 mK temperature. The dissipation ranges from 1 kHz to 100 kHz. Both the state $\vert D_3^{(1)}\rangle$ and $\vert D_3^{(2)}\rangle$ keep a high fidelity with the increasement of $\varGamma$. The fidelity of $\vert D_3^{(2)}\rangle$ is larger than 0.92 with $\varGamma =50$ kHz, while the state $\vert D_3^{(1)}\rangle$ has a fidelity larger than 0.94 even with $\varGamma =100$ kHz.
Fig. 6. Fidelity of corresponding Dicke state versus the mechanical dissipation in the waveguide. The solid and dashed lines represent the fidelity of $\vert D_3^{(1)}\rangle$ and $\vert D_3^{(2)}\rangle$, respectively.
5. Conclusion
In summary, we construct a hybrid quantum-phonon system, where the SiV centers are coupled with acoustic modes in a quasi 1D diamond acoustic waveguide via strain-induced coupling. The SiV centers can be embedded in the 1D diamond waveguide through ion implantation techniques based on state-of-the-art nanofabrication techniques [93,94]. In SiV centers, transitions between states $\vert 1\rangle \leftrightarrow \vert 4\rangle$ and $\vert 2\rangle \leftrightarrow \vert 3\rangle$ can be induced by the microwave driving fields or via an equivalent optical Raman process [8], which has already been experimentally realized. We assume that the setup works at 100mK temperature, and the phonon number of acoustic mode is close to 0, which means that the acoustic mode is near its ground state. Thus the stain coupling between the SiV center and the acoustic mode allows a coherent transfer of quantum states between the spin and the acoustic modes. And the combination of optical pumping and readout techniques for spin states provides the basic ingredients for generating and detecting of various Fock states of the acoustic mode. [77–79]. Based on this hybrid quantum system, the symmetric Dicke states are deterministically generated through Morris-Shore transformation. The generation process of Dicke states could be optimized only by tuning two linear microwave fields. We also realize the disentanglement of prepared Dicke states, which could be interpreted by occurring avoid-crossing adiabatic passages twice. In addition, considering the dissipation in the hybrid system, both the generated Dicke states have high fidelity. Consequently, this scheme can work well under experimental conditions and expand applications of the SiV-based hybrid quantum systems in quantum information processing.
Funding
National Natural Science Foundation of China (Grant No. 11874432); National Key Research and Development Program of China (Grant No. 2019YFA0308200); Fundamental Research Funds for the Central Universities (Grant No. 22qntd3001).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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