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Abstract
In this paper, we investigate the spin squeezing in a hybrid quantum system consisting of a Silicon-Vacancy (SiV) center ensemble coupled to a diamond acoustic waveguide via the strain interaction. Two sets of non-overlapping driving fields, each contains two time-dependent microwave fields, are applied to this hybrid system. By modulating these fields, the one-axis twist (OAT) interaction and two-axis two-spin (TATS) interaction can be independently realized. In the latter case the squeezing parameter scales to spin number as $\xi _R^2\sim 1.61N^{-0.64}$ with the consideration of dissipation, which is very close to the Heisenberg limit. Furthermore, this hybrid system allows for the study of spin squeezing generated by the simultaneous presence of OAT and TATS interactions, which reveals sensitivity to the parity of the number of spins Ntot, whether it is even or odd. Our scheme enriches the approach for generating Heisenberg-limited spin squeezing in spin-phonon hybrid systems and offers the possibility for future applications in quantum information processing.
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1. Introduction
The spin-squeezed state [1–3], one of the typical quantum many-body entangled states [4,5], is a critical resource in quantum information processing [6–10] and quantum metrology [11–15]. Such a state can be used to overcome the shot-noise limit [15–19] and to study many-body entanglement [20–25]. The ability to efficiently generate spin-squeezed states is the first step in unlocking its potential applications [26–28]. There are various approaches to produce spin squeezing, such as transferring the squeezing from squeezed light to spin ensembles [29–32], quantum nondemolition (QND) measurements of collective spins [33–36], and utilizing the OAT and two-axis twist (TAT) squeezing interactions [1,3,37–45]. The OAT interaction is able to deterministically generate spin squeezing and has been experimentally realized in different systems, such as the atomic [46–48] and trapped ion systems [49], with the squeezing degree scales as $\propto N^{-2/3}$ [1,3]. As for the TAT interaction, it can provide a squeezing degree scaling as $\propto N^{-1}$ in an ideal case [1,3,39,40,50]. Thus, many efforts have been devoted to engineering the TAT interaction and generating highly squeezed spin states in current state-of-the-art systems [3,27,28,37,39,40,45,50–56].
The quantum hybrid systems based on color centers have been extensively developed in quantum information processing benefiting from their longer coherence time and better controllability [57–68]. The SiV center, a type of color centers, has the fourfold-degenerate ground state, which enables various interactions between corresponding electronic levels [69–76]. Moreover, due to the strong coupling between SiV centers and acoustic modes, the entanglement of SiV centers can be generated mediated by phonons [73,77,78]. Such a strong coupling can also be employed to achieve better spin squeezing in spin-phonon hybrid systems based on SiV centers [62,73,77,79–81].
In this work, we propose a scheme for generating spin-squeezed states in a hybrid system consisiting of an ensemble of SiV centers coupled to the acoustic mode of a diamond waveguide via the strain interaction [73,77]. Unlike the previous works [73,77], the SiV ensemble in this work is partitioned into two different segments resulting from two sets of non-overlapping microwave fields, as shown in Fig. 1(a). The strain-induced coupling enables effective spin-spin interactions mediated by virtual phonons, and then the OAT and TATS interactions can be induced independently, where the latter one allows for generating the Heisenberg-limited spin squeezing theoretically [1,3,38–40,50,54,82]. Furthermore, we investigate the spin-squeezed states generated by the mixed Hamiltonian of OAT and TATS interactions and show the sensitivity of these states to the even-odd spin particles [83–85], which holds potential for sensing applications. Considering practical dissipations in the system, the Wineland parameter $\xi _R^2$ has a trend as $\xi _R^2\sim 1.61N^{-0.64}$, which can be used to achieve a measurement precision close to the Heisenberg limit. Compared to other schemes that necessitate the use of squeezed field injection, complex pulse drive or parametric drive to generate better spin-squeezed states, our scheme requires only the appropriate modulation of microwave fields and allows better spin-squeezed states based on this spin-phonon hybrid system [86,87].
Fig. 1. (a) Sketch of an array of 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 the inset. In this system, there are two different segments $S_1$ and $S_2$ in the SiV-center ensemble, which contains $N_1$ and $N_2$ SiV centers, respectively, resulted from the different set of driving fields. (b) The level structure of the electronic ground state of the SiV center. The 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.
Our paper is organized as follows. In Sec. 2, we introduce the theoretical model of a hybrid quantum system consisting of two SiV-center segments embedded in a quasi 1D acoustic waveguide. Section 3 shows the time evolution of squeezing parameters $\xi _S^2$ and $\xi _R^2$ in the case of the OAT, TATS and mixed OAT-TATS Hamiltonians. In Sec. 4, we discuss the experimental feasibility of this scheme and analyze the influence caused by the experimental dissipation of this hybrid system. Finally, we make a summary in Sec. 5.
2. Model
As depicted in Fig. 1(a), we consider a spin-phonon hybrid system, where two sets of $N_1$ and $N_2$ SiV centers in segment $S_1$ and segment $S_2$, respectively, are coupled to an acoustic mode of a 1D diamond waveguide via the strain-induced interaction. This interaction arises from the change of Coulomb energy of the electronic states due to the displacement of atoms forming the defect. First, we consider the SiV centers in segment $S_1$ which are driven by two time-dependent microwave fields $\Omega _1(t)$ and $\Omega _2(t)$, and this system can be described by the Hamiltonian [73,78]
where $H_{\rm SiV_{S_1}}$ and $H_{\rm ph}$ are the Hamiltonians of SiV centers in segment $S_1$ and the acoustic mode, respectively, and $H_{\rm strain_{S_1}}$ denotes the strain-induced coupling between the orbital degree of the SiV center in segment $S_1$ and the common acoustic mode of the waveguide, as shown in Fig. 1(b).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, with 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 Jahn-Teller (JT) effect along $\vec x(\vec y)$ direction [69,73]. Two time-dependent microwave fields $\Omega _{1,2}(t)$ with frequencies $\omega _{1,2}$ will induce transitions between states $\vert 1\rangle \leftrightarrow \vert 4\rangle$ and $\vert 2\rangle \leftrightarrow \vert 3\rangle$, as shown in Fig. 1(b). Consequently, the dynamics of SiV centers can be described by the Hamiltonian $(\hbar =1)$ [73,78]
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. The quantized Hamiltonian of acoustic modes can be written as
where $a_{n,k}$ is the annihilation operator of one acoustic mode.Within the framework of linear elasticity theory, the strain-induced coupling Hamiltonian between SiV centers in segment $S_1$ and acoustic modes in the waveguide is given by [69,73]
where $j_+=j_-^{\dagger }=\vert 3\rangle _j\langle 1\vert +\vert 4\rangle _j\langle 2\vert$ is the spin-conserving raising operator, and $g_{n,k}$ describes the coupling strength with the form where $d/2\pi \approx 1$ PHz is the strain sensitivity and the dimensionless function $\xi _{n,k}(y,z)$ denotes specific strain distribution [72,73]. For a small quasi 1D diamond acoustic waveguide, the length, width, and thickness can be chosen as $L\sim 10-100~\mu$m, $w,d \lesssim 200$ nm$^2$, respectively, and it has the group velocity $\upsilon \sim 1\times 10^4~\rm {m/s}$, thus the strain-induced coupling strength is $g=2\pi \times (4\sim 14)$ MHz [72,73,77].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 mode as a single standing wave with $\omega _n\approx 2\pi \times 46$ GHz for simplicity. Then, the Hamiltonian Eq. (1) can be written as
Performing a unitary transformation with respect to $U=e^{-iH_0t}$, where $H_0=\sum _j[\left ({{\omega _n}-{\omega _2}}\right )\vert 2_j\rangle$ $\langle 2\vert +\omega _n\vert 3_j\rangle \langle 3\vert +{\omega _1}\vert 4_j\rangle \langle 4 \vert +{\omega _{n}}{a^{\dagger }}a]$, the Hamiltonian in the interaction picture reads
Next we consider the total hybrid system with two segments $S_{1,2}$, which are connected by a common acoustic mode. The effective Hamiltonian of this whole hybrid system can be written as [91]
As shown in Eq. (9), Eq. (11), by properly adjusting the microwave fields, the effective detunings can be set as $\varepsilon _1=-\varepsilon _2=\Delta _s$, which implies that the two parts of the SiV centers are physically different. The effective coupling strength between SiV-center segments and the acoustic mode can also be tuned through modulating the driven fields. In addition, we set $w_1=w_2=0$.
Assuming that the setup works at 100mK temperature, thus the phonon number of acoustic mode is close to 0, i.e. $\left \langle a^{\dagger } a\right \rangle \sim 0$. Moreover, with the condition $\Delta _s\gg \lambda _{1,2}g_n,\Lambda _{1,2}g_n$ is satisfied, applying the canonical transformation $H \rightarrow e^{-S}H_{eff}e^S$ with [90,92]
Finally, we could obtain an effective projected Hamiltonian in the spin-ensemble subspace [92,93] as following
The validity of this Hamiltonian may be verified by comparing the dynamical processes under the Hamiltonians Eq. (6) and Eq. (13). As shown in Fig. 2, we plot the time evolution of populations of corresponding levels by solving the quantum master equation based on the Hamiltonians Eq. (6) and Eq. (13). For example, four SiV centers are employed in the numerical calculation. We can see that the dynamical evolution of states $\vert 1\rangle$ and $\vert 2\rangle$ based on two Hamiltonians are almost identical in a short time, with only a small difference in amplitude. In addition, the phonon number and the populations of states $\vert 3\rangle$ and $\vert 4\rangle$ are almost zero. Consequently, we use the effective Hamiltonian to study the squeezing properties in the following sections.
Fig. 2. Time evolution of the populations of SiV-center levels. (a) Population evolution of the level $\vert 1\rangle$. (b) Population evolution of the level $\vert 2\rangle$. The red and green lines in (a) and (b) represent the results based on Eq. (6) and Eq. (13), respectively. (c) Population evolution of the level $\vert 3\rangle$ and level $\vert 4\rangle$. (d) Time evolution of phonon numbers.
The terms in line 2 of Eq. (13) represent the OAT interaction, while the term in line 3 indicates the TATS interaction [2,3,29]. Thus, by tunning the driving fields, one could realize an OAT Hamiltonian along the z axis, a TATS Hamilatonian, and a mixed Hamiltonian containing the OAT and TATS interactions, respectively. In addition, the dynamical evolution of the system can be described by the quantum master equation
3. Spin squeezing
In this section, we quantify the degree of spin-squeezed states by calculating two commonly used squeezing parameters, the Ueda-Kitagawa parameter $\xi _S^2$ and the Wineland parameter $\xi _R^2$. $\xi _S^2$ is closely linked to quantum correlations (entanglement), whereas $\xi _R^2$ finds wide applications in quantum metrology. These two parameters are defined as follows [2,3,29,94]
3.1 OAT interaction Hamiltonian
When the terms in Eq. (13) are set as $\lambda _1\Lambda _2-\lambda _2\Lambda _1=0$ and $\lambda _1^2\neq \Lambda _1^2,\lambda _2^2\neq \Lambda _2^2$ through tunning the amplitudes and frequencies of the driving fields, we can obtain the following OAT Hamiltonian along the z axis,
Figure 3 shows the time evolution of the squeezing parameters $\xi _S^2$ and $\xi _R^2$. The red line, black line, and green dotted line represent the squeezing parameters of $N_1=N_2=N=20,30,50$, respectively. Particularly, due to the inequality of OAT interactions $G_{\rm OAT1}=0.3\varGamma, G_{\rm OAT2}=-0.15\varGamma$, the two SiV-center segments each generate different spin squeezing. Thus the plots of whole spin squeezing in the hybrid system as shown in Fig. 3 look different from common appearance of spin squeezing generated by the OAT interaction in a single ensemble, but their maximal squeezing degrees are same with equal spin numbers, only the optimal times are different. The hybrid system evolves from a spin coherent state distributed on the x-axis, in which state the values of both parameters $\xi _S^2$ and $\xi _R^2$ are 1, as depicted in the Fig. 3 at time 0. As the system begins to evolve, these two parameters become smaller than 1, indicating that the spin squeezing has been generated in this hybrid system. As shown in Fig. 3, the resulting spin-squeezed states have the maximal squeezing when $\varGamma t\sim 60$, and these minimum values decrease with increasing the particle number. The corresponding minimum squeezing parameters are $\xi _S^2\approx 0.18,0.11,0.08$ and $\xi _R^2\approx 0.22,0.15,0.13$ with the case of $N=20,30,50$, respectively. In addition, we find that $\xi _S^2<\xi _R^2$ for the same spin numbers, which is consistent with the results as mentioned in Ref. [3].
Fig. 3. The effects of spin number to the spin squeezing generated by the OAT interaction. (a) Time evolution of the squeezing parameter $\xi _S^2$ with $N_1=N_2=N=20,30,50$. (b) Time evolution of the squeezing parameter $\xi _R^2$ with $N_1=N_2=N=20,30,50$.
3.2 TATS interaction Hamiltonian
Similar to the case of OAT interaction, we can also set $\lambda _1\Lambda _2-\lambda _2\Lambda _1\neq 0$ and $\lambda _1^2=\Lambda _1^2,\lambda _2^2=\Lambda _2^2$ by tunning the amplitudes and frequencies of the driving fields. Then, the Hamiltonian with a TATS interaction could be obtained from Eq. (13) as follows
Figure 4 depicts the time evolution of squeezing parameters $\xi _S^2$ and $\xi _R^2$ with different spin numbers in the case of TATS interaction Hamiltonian. The red, blue, green lines in this figure represent the squeezing parameters $\xi _S^2$ and $\xi _R^2$ of $N_1=N_2=N=20,30,50$, respectively. We can see that the minimum values of $\xi _S^2$ and $\xi _R^2$ have decreased significantly compared to the OAT interaction case in Fig. 4 with the same spin number, specifically, $\xi _S^2\approx 0.03,0.021,0.013$ and $\xi _R^2\approx 0.113,0.079,0.049$ with the case of $N=20,30,50$, respectively. Similarly, $\xi _S^2<\xi _R^2$ for the same spin numbers. Figure 4 shows that both squeezing parameters would reach smaller values faster with the increasing spin number. Moreover, the spin squeezing generated by this TATS interaction Hamiltonian can approach the Heisenberg limit $1/N$ for large spin number in the ideal case [3].
Fig. 4. The effects of spin number to the spin squeezing generated by the TATS interaction. (a) Time evolution of the squeezing parameter $\xi _S^2$ with $N_1=N_2=N=20,30,50$. (b) Time evolution of the squeezing parameter $\xi _R^2$ with $N_1=N_2=N=20,30,50$.
3.3 Mixed Hamiltonian of OAT and TATS interaction
With appropriate tuning of the microwave driving fields, it is also possible to obtain a mixed Hamiltonian that comprises both OAT and TATS interactions from Eq. (13),
We also plotted the time evolution of the squeezing parameters $\xi _S^2$ and $\xi _R^2$ for different spin numbers in Fig. 5, and the red, blue, green lines in this figure represent the squeezing parameters $\xi _S^2$ and $\xi _R^2$ of $N_1=N_2=N=20,30,50$, respectively. In the case of mixed Hamiltonian of the OAT and TATS interaction, the minimum values of corresponding squeezing parameters are $\xi _S^2\approx 0.08,0.063,0.047$ and $\xi _R^2\approx 0.146,0.109,0.075$ with the case of $N=20,30,50$, respectively, which are smaller than the case of OAT interaction induced spin squeezing, but also slightly larger than the ideal TATS case. From the Fig. 5, we can also see that the time for the system to reach the optimal squeezing is significantly smaller than the OAT (Fig. 3) and TATS (Fig. 4) cases.
Fig. 5. The effects of spin number to the spin squeezing generated by the mixed Hamiltonian. (a) Time evolution of the squeezing parameter $\xi _S^2$ with $N_1=N_2=N=20,30,50$. (b) Time evolution of the squeezing parameter $\xi _R^2$ with $N_1=N_2=N=20,30,50$.
In particular, in the mixed OAT-TATS interaction case, we find that the spin squeezing effect differs significantly depending on whether the total number of spins is odd or even. This property may be utilized to detect changes of the number, $N_{tot}$, of coupled spins at the single-particle level. Figure 6 shows the time evolution of the Ueda-Kitagawa parameters $\xi _S^2$ with $N_{tot}=40$ and $N_{tot}=39$ ($|N_1-N_2|=1$). Notably, during the first instance of spin squeezing, the parameters $\xi _S^2$ of $N_{tot}=40$ and $N_{tot}=39$ are almost identical. However, as the hybrid evolves from the spin coherent state to spin-squeezed state for the second time, the spin squeezing in the $N_{tot}=39$ case is significantly poorer compared to the the $N_{tot}=40$ case, as shown in Fig. 6(a). When the number of total spins is odd, there will be a difference in the parity of spin numbers between the two segments, resulting in the overall dynamics of spin squeezing, the combination of two parts with different periods and parities. Therefore, like destructive and constructive interference, the squeezing parameters $\xi _S^2$ with odd total spins will display the maximum squeezed value that alternates between large and small in odd and even periods. In contrast, in the case with even total spins, such alternations in the maximum value of spin squeezing are absent. Figure 6(b) illustrates that how this odd-even sensitivity could be used for sensing. We plot the value of $\left \langle J_X^2 \right \rangle$ with different total spin numbers $N_{tot}=40,39,38,37,36$. Here, the spin numbers $N_1=N_2$ when $N_{tot}$ is even, while $|N_1-N_2|=1$ with $N_{tot}$ being odd. As shown in Fig. 6(b), if the spins left or decoupled from the waveguide one by one, the corresponding values of $\left \langle J_X^2 \right \rangle$ would become smaller and smaller. In this sense, we are able to detect the change of spins by measuring $\left \langle J_X^2 \right \rangle$.
Fig. 6. The effects of even and odd total spin numbers to the spin squeezing generated by the mixed Hamiltonian. (a) Time evolution of the squeezing parameter $\xi _S^2$ with $N_{tot}=40$ and $N_{tot}=39$. (b) An example of how the even-odd sensitivity of the spin squeezing could be used for sensing. Time evolution of the value of $\left \langle J_X^2 \right \rangle$ with $N_{tot}=40,39,38,37,36$.
4. Experimental feasibility
In this section, we discuss the relevant parameters used in numerical simulations to assess the practical feasibility of this schemement. First, we take the value of the strain-induced coupling strength between the acoustic mode in the diamond waveguide and SiV centers to $g=2\pi \times 5\rm {MHz}$. To embed the SiV centers into the 1D diamond waveguide, we can utilize ion implantation techniques based on state-of-the-art nanofabrication techniques [74]. The ground state splitting of SiV centers is $\Delta \approx$ 46GHz, and the transitions between states $\left | 1\right \rangle \leftrightarrow \left | 4\right \rangle$ and $\left | 2\right \rangle \leftrightarrow \left | 3\right \rangle$ can be induced by the microwave driving fields or via an equivalent optical Raman process, which has already been experimentally realized [73,95,96]. At 100 mK, the coherence time of SiV centers could be extended to $T_s\sim$10 ms by using dynamical decoupling techniques [70,97–99], corresponding the spin dephasing rate of a single SiV center is about $\gamma _{d}\sim 100$Hz. The driving fields adopted here are $\Omega _1,\Omega _2,\Omega _3,\Omega _4\sim 2\pi \times (30\sim 50)\rm {MHz}$ with $\delta _1,\delta _2,\delta _3,\delta _4\sim 2\pi \times (300\sim 500)\rm {MHz}$, respectively. It should be noted that we have not taken into account the effect of dissipation in numerical simulations of the squeezing in the previous section. A quality factor of $Q\approx \times 10^5$ for the mechanical phonon modes of the small-sized diamond waveguide has been demonstrated [100,101], which leads to a mechanical dissipation value of $\varGamma \sim 2\pi \times 500~{\rm kHz}$. Consequently, the effective collective decay rate induced by mechanical dissipation in Eq. (14) has a value as $\varGamma _{eff}\sim 2\pi \times 50\rm {Hz}$ in the TATS interaction case, which is of the same order of magnitude as the spin dephasing rate. Here, we modify the master equation Eq. (14) by including the spin dephasing term as follows
In Fig. 7, we plot the optimal squeezing parameter $(1/\xi _R^2)_{max}$ versus the total number of spins $N_{tot}$ in TATS interaction case, taking into account the collective decoherence induced by mechanical dissipation of the acoustic mode and the dephasing rate of SiV centers. Using the numriacl results obtained from Eq. (19), we fit a curve and obtain the trend of the optimal squeezing parameters $(\xi _R^2)_{max}$ with respect to the number of spins as $\xi _R^2\sim 1.61N^{-0.64}$, which is better than the corresponding results of recent works [39,43,84,102–104]. Moreover, the spin squeezing generated in our model could also be further improved by engineering the environment [43,84]. As such, our scheme can generate highly squeezed-spin states under currently available experimental conditions in the hybrid system base on SiV centers.
Fig. 7. The optimal squeezing parameter $(1/\xi _R^2)_{max}$ versus the total spin numbers $N_{tot}$ with the consideration of experimental disspation in TATS interaction case. The black dots is the value of $(1/\xi _R^2)_{max}$ with corresponding spin numbers, and the red line represents the curve fitting of numerical results.
5. Conclusion
In summary, we have designed a hybrid quantum system, consisting of an ensemble of SiV centers coupled to the acoustic mode of a diamond waveguide via the strain-induced coupling. The system is partitioned into two segments with different sets of microwave driving fields, and by ajusting the frequencies and amplitudes of fields, we can achieve the OAT interaction, the TATS interaction and mixed Hamiltonian with both OAT and TATS interactions. The scheme can still work when the numbers of SiV centers in the two segments differ, despite a reduction in the squeezing effect. In the ideal TATS scenario with large numbers of spins, the two spin-squeezing parameters $\xi _R^2$ and $\xi _S^2$ scale with total spin numbers as $\xi _S^2,\xi _R^2\sim N^{-1}$, reaching the Heisenberg limit. In the mixed interaction case, our hybrid system can generate the optimal spin squeezing more rapidly, and these spin-squeezed states is sensitive to the parity of the total number of spins. Moreover, we have provided a possible method for measuring the change of spin numbers at the single particle level. Considering the realistic dissipation, $\xi _R^2$ scales with the total number of spins as $\xi _R^2\sim 1.61N^{-0.64}$, demonstrating its potential for application in quantum metrology. Consequently, our scheme can work well under experimental conditions and extend the applications of the SiV-based hybrid quantum systems in quantum information processing and quantum metrology.
Funding
National Natural Science Foundation of China (Grant No. 12375025, Grant No. 11874432); Fundamental Research Funds for the Central Universities (Grant No. 22qntd3001); National Key Research and Development Program of China (Grant No. 2019YFA0308200).
Acknowledgment
Zhen-qiang Ren thank Yuan Zhou for valuable discussions. Part of the simulations are coded in Python using the QuTiP library [105,106]
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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