Heisenberg-limited spin squeezing in a hybrid system with Silicon-Vacancy centers

In this paper, we investigate 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 $\xi_R^2\sim1.61N^{-0.64}$ 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 $N_{tot}$, 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.

In this work, we propose a scheme for generating spinsqueezed 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.This SiV ensemble is partitioned into two different segments resulting from two sets of non-overlapping microwave fields.The strain-induced coupling enables effective spin-spin interactions mediated by virtual phonons, then the OAT and TATS interactions can be induced independently, where the latter one can realize Heisenberg-limited spin squeezing [1][2][3].Furthermore, we investigate the spinsqueezed states generated by the mixed Hamiltonian of OAT and TATS interactions and show the sensitivity of these states to the even-odd spin particles, which holds potential for sensing applications.Considering practical dissipations in the system, the squeezing parameter ξ 2 R has a trend as ξ 2 R ∼ 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.
Our paper is organized as follows.In Sec.II, we introduce the theoretical model of a hybrid quantum system consisting of two SiV-center segments embedded in a quasi 1D acoustic waveguide.Section III shows the time evolution of squeezing parameters ξ 2 S and ξ 2 R in the case of the OAT, TATS and mixed OAT-TATS Hamiltonians.In Sec.IV, 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.V. 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 Ω 1 (t) and Ω 2 (t), and this system can be described by the Hamiltonian [63,65] where H SiVS 1 and H ph are the Hamiltonians of SiV centers in segment S 1 and the acoustic mode, respectively, and H strainS 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 ∆ = [λ 2 g + Υ 2 x + Υ 2 y ] 1/2 ≈ 2π × 46 GHz, where λ g = 2π × 45 GHz is the spin-orbit coupling strength, and Υ x(y) describes the strength of the Jahn-Teller (JT) effect along ⃗ x(⃗ y) direction [59,63].Two time-dependent microwave fields Ω 1,2 (t) with frequencies ω 1,2 will induce transitions between states |1⟩ ↔ |4⟩ and |2⟩ ↔ |3⟩, as shown in Fig. 1(b).Consequently, the dynamics of SiV centers can be described by the Hamiltonian(ℏ = 1) [63,65] where ω B = γ s B 0 denotes the energy-level splitting induced by the Zeeman effect, and we set ω B ≈ 2π ×5 GHz here.γ s is the spin gyromagnetic ratio, and j labels the j-th SiV center in segment S 1 .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 ≫ 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.
Considering that the acoustic modes are well separated from frequency (∆ω n ≥ 2π × 50 MHz) in the waveguide with small size, we could treat the mechanical mode as a single standing wave with ω n ≈ 2π × 46 GHz for simplicity.Then, the Hamiltonian Eq. ( 1) can be written as (6) Performing a unitary transformation with respect to U = e −iH0t , where the Hamiltonian in the interaction picture reads where ν, δ 1 , δ 2 are the corresponding detunings between the frequencies ω n,k , ω 1 , ω 2 and eigenfrequencies of states |3⟩, |4⟩, as shown in Fig. 1(b), and , we may further eliminate the higher energy levels |3⟩ and |4⟩ via Froehlich-Nakajima transformation [69][70][71].Finally, we obtain an equivalent two-level Hamiltonian where the parameters ε 1 , λ 1 , Λ 1 in Eq. ( 8) have the forms as following 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 where the operators j=1 |1⟩ j ⟨2| are the collective spin operators of the SiV centers, subscripts 1 and 2 denote the two parts and N 1 , N 2 are the total spin numbers of corresponding SiV-center segments.Moreover, the parameters ε 2 , λ 2 , Λ 2 in Eq. ( 10) have the forms as following As shown in Eq. ( 9), Eq. ( 11), by properly adjusting the microwave fields, the effective detunings can be set as ε 1 = −ε 2 = ∆ s , which implies that the two parts of the SiV centers are physically different.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. a † a ∼ 0.Moreover, with the condition ∆ s ≫ λ 1,2 g n , Λ 1,2 g n is satisfied, applying the canonical transformation H → e −S H ef f e S with [71,72] Finally, we could obtain an effective projected Hamiltonian in the spin-ensemble subspace [72,73] as following Here the terms in lines 2 and 3 represent the OAT interaction, while the term in line 4 indicates the TATS interaction [2,3,23].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 where n th is the average thermal phonon number, and indicates the collective spin relaxation induced by mechanical dissipation Γ m of the corresponding acoustic mode.

III. SPIN SQUEEZING
In this section, we quantify the degree of spin-squeezed states by calculating two most frequently used squeezing where (∆J− → n ⊥ ) 2 min is the minimum variance in a direction which perpendicular to the mean spin direction, and − → J = ⟨J 2 x ⟩ + J 2 y + ⟨J 2 z ⟩ denotes the magnitude of the mean spin.N tot = N 1 + N 2 is the total number of SiV centers in the waveguide, and for the sake of simplicity, we assume that N 1 ≃ N 2 .

A. OAT interaction Hamiltonian
When the terms in Eq. ( 13) are set as through tunning the amplitudes and frequencies of the driving fields, we can obtain the following OAT Hamiltonian along the z axis, where 2 )/∆ s describe the OAT interaction strengths of corresponding spin ensembles, and z + J z , the last two terms in Eq. ( 13) indicate a standard OAT interaction Hamiltonian [3].
Figure 2 shows the time evolution of the squeezing parameters ξ 2 S and ξ 2 R .The red line, black line, and green dotted line represent the squeezing parameters of N 1 = N 2 = N = 20, 30, 50, respectively.The hybrid system evolves from a spin coherent state distributed on the x-axis, in which state the values of both parameters ξ 2 S and ξ 2 R are 1, as depicted in the figure 2 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 figure 2, the generated spin-squeezed states reaches their optimal squeezing at time t ∼ 20us, with minimum values are ξ 2 S ≈ 0.18, 0.11, 0.08 and ξ 2 R ≈ 0.22, 0.15, 0.13 with the case of N = 20, 30, 50, respectively.In addition, we find that ξ 2 S < ξ 2 R for the same spin numbers, which is consistent with the results as mentioned in Ref. [3].

B. TATS interaction Hamiltonian
Similar to the case of OAT interaction, we can also set 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 where G T AT S = (λ 1 Λ 2 − λ 2 Λ 1 )/∆ s indicates the TATS interaction strength, and ∆ s1,s2 = ∆ s + 2λ 2 1,2 /∆ s .Figure 3 depicts the time evolution of squeezing parameters ξ 2 S and ξ 2 R with different spin numbers in the case of TATS interaction Hamiltonian.The red, blue, green lines in this figure represent the squeezing parameters ξ 2 S and ξ 2 R of N 1 = N 2 = N = 20, 30, 50, respectively.We can see that the minimum values of ξ 2 S and ξ 2 R have decreased significantly compared to the OAT interaction case in fig. 3 with the same spin number, specifically, ξ 2 S ≈ 0.03, 0.021, 0.013 and ξ 2 R ≈ 0.113, 0.079, 0.049 with the case of N = 20, 30, 50, respectively.Similarly, ξ 2 S < ξ 2 R for the same spin numbers.Figure 3 shows that both squeezing parameters would reach their minimum values more quickly with the increment of the spin numbers.Moreover, we can see that the spin-squeezed state generated by this TATS interaction Hamiltonian could approach the Heisenberg limit 1/N for large spin numbers, which is not possible in the OAT case [3].

C. 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), where G mix represents the mixed interaction strength, and ∆ s1,s2 = ∆ s + 2min λ 2 1,2 , Λ 2 1,2 /∆ s .We also plotted the time evolution of the squeezing parameters ξ 2 S and ξ 2 R for different spin numbers in Fig. 4, and the black, red, green lines in this figure represent the squeezing parameters ξ 2 S and ξ 2 R 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 ξ 2 S ≈ 0.08, 0.063, 0.047 and ξ 2 R ≈ 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. 4, we can also see that the time for the system to reach the optimal squeezing is significantly smaller than the OAT (Fig. 2) and TATS (Fig. 3) cases.
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 5 shows the time evolution of squeezing parameters ξ 2 S with N tot = 40 and N tot = 39.Notably, during the first instance of spin squeezing, the parameters ξ 2 S 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. 5(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 ξ 2 S 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 5(b) illustrates that how this odd-even sensitivity could be used for sensing.We plot the value of J 2 X with different total spin numbers N tot = 40, 39, 38, 37, 36.When the spins leaving or decoupling from the waveguide one by one, the corresponding values of J 2 X would become smaller and smaller, as shown in Fig. 5(b).

IV. 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 = 2π × 5MHz.To embed the SiV centers into the 1D diamond waveguide, we can utilize ion implantation techniques based on state-of-the-art nanofabrication techniques [75].The ground state splitting of SiV centers is ∆ ≈ 46GHz, and the transitions between states |1⟩ ↔ |4⟩ and |2⟩ ↔ |3⟩ can be induced by the microwave driving fields or via an equivalent optical Raman process, which has already been experimentally realized [63,76,77].At 100 mK, the spin dephasing rate of a FIG. 6.The optimal squeezing parameter (1/ξ 2 R )max versus the total spin numbers Ntot with the consideration of experimental disspation in TATS interaction case.The black dots is the value of (1/ξ 2 R )max with corresponding spin numbers, and the red line represents the curve fitting of numerical results.
single SiV center is about γ d ∼ 100Hz, corresponding to a coherent time is T s ∼10ms [47,60,78].The driving fields adopted here are Ω 1 , Ω 2 , Ω 3 , Ω 4 ∼ 2π × (30 ∼ 50)MHz with δ 1 , δ 2 , δ 3 , δ 4 ∼ 2π × (300 ∼ 500)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 ≈ ×10 5 for the mechanical phonon modes of the small-sized diamond waveguide has been demonstrated [79,80], which leads to a mechanical dissipation value of Γ ∼ 2π × 500 kHz.Consequently, the effective collective decay rate induced by mechanical dissipation in Eq. ( 14) has a value as Γ ef f ∼ 2π × 50Hz 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. 6, we plot the optimal squeezing parameter (1/ξ 2 R ) max versus the total number of spins N tot in TATS interaction case, taking into account the collective deco-herence 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 (ξ 2 R ) max with respect to the number of spins as ξ 2 R ∼ 1.61N −0.64 .As such, our scheme can generate highly squeezed-spin states under currently available experimental conditions in the hybrid system base on SiV centers.

V. 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 spinsqueezing parameters ξ 2 R and ξ 2 S scale with total spin numbers as ξ 2 S , ξ 2 R ∼ 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, ξ 2 R scales with the total number of spins as ξ 2 R ∼ 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.

FIG. 1 .
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 S1 and S2 in the SiV-center ensemble, which contains N1 and N2 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 timedependent microwave driving fields induce the transitions between levels |1⟩ ↔ |4⟩ and |2⟩ ↔ |3⟩, while the transitions between levels |1⟩ ↔ |3⟩ and |2⟩ ↔ |4⟩ are caused by the strain-induced coupling.

FIG. 5 .
FIG. 5.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 ξ 2 S with Ntot = 40 and Ntot = 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 J 2 X with Ntot = 40, 39, 38, 37, 36.