1. Introduction

Ensembles of nitrogen-vacancy (NV) centers in diamond provide an advantageous solid-state platform for quantum information technologies [1–4]. Quantum information can be stored in such systems over a longer period of time due to the extraordinarily long coherence time of spin ensembles [1,3,4], which can reach 1 s even at relatively high temperatures [5], offering unique opportunities for quantum memories [6]. Since the spins in NV centers can be coherently coupled to both optical and microwave fields [7–9], NV ensembles (NVEs) are readily integrable into field-controlled hybrid physical systems designed for specific quantum-science applications [7–9]. Recent achievements in this field include hybrid systems integrating NV centers in diamond with superconducting flux qubits and resonators as a promising solution for quantum information technologies and as a platform for experimental tests of the fundamental concepts of quantum mechanics [7,10–13].

As one of their central advantages, NVEs can help enhance coupling with superconducting coplanar waveguide resonators (CPWRs) relative to single NV centers. While for a single NV center, a typical NV–CPWR coupling constant is on the order of a few hertz [12,14], well below the CPWR dissipation rate, the coupling of an NVE can be collectively enhanced, as the coupling strength scales as $\sqrt{N_0}$ with the number of spins N₀ in an NVE [14–16]. This opens the ways to overcome the limitations on quantum information processing inherent in single NV centers.

Here, we consider a hybrid quantum system containing three NVEs coupled to a common CPWR. In such a system, collectively enhanced magnetic coupling between the NVEs and the CPWR can give rise to a quantum entanglement in NVEs once those have been initially prepared in their ground state. We analyze a tripartite entanglement among NVEs in such a system and demonstrate that this entanglement can be meaningfully characterized in terms of suitably defined entanglement witnesses [17–20]. This analysis suggests that systems of this class can provide a practical platform for the generation of multipartite entangled states as a powerful resource for large-scale quantum information processing and quantum networking.

2. Entanglement generation

The superconducting coplanar waveguide resonator consisting of the length L, the capacitance C_c, and the inductance F_c contains two nearby lateral ground planes and a narrow center conductor, which can be described by the harmonic oscillator operators, and thus, its Hamiltonian can be written as H_C = ħω_ca^†a, where a (a^†) is the annihilation (creation) operator of the microwave mode of the resonator, and its corresponding eigenfrequency is determined by ${\omega }_c=2\pi /\sqrt{C_c{F}_c}$ [21, 22]. The distributions of current and voltage inside the CPWR can be described as $I_{\mathit{qpw}}=i\sqrt{{\omega }_c/F_c}(a^{†}-a)\text{sin}(2\pi /Lx)$ and $V_{\mathit{qpw}}=\sqrt{{\omega }_c/C_c}(a^{†}+a)\text{cos}(2\pi /Lx)$.

The ground state of an NV center is a spin triplet with zero-magnetic-field splitting D_gs/2π = 2.87 GHz between the m_s = 0 sublevel and the degenerate m_s = ±1 sublevels. An external magnetic field B_ext applied along the [111] direction of the diamond crystal lattice (parallel to the resonator plane) removes the degeneracy of |m_s = ±1〉 sublevels by inducing Zeeman splitting and tunes the transition energies. In the study we only use one m_s = 0 to m_s = −1 transition of each crystal. Thus, the spin operators of the jth NV center are defined as σ_zj = | − 1_j〈〉−1_j | − |0_j〈〉0_j|, σ_+j = | − 1_j〈〉0_j| and σ_−j = |0_j〈〉−1_j|. For an NVE consisting of N₀ NV centers, collective spin operators can then be written as $S_{\tau }={\sum }_{k=1}^{N_0}{\sigma }_{\tau k}(\tau =z,\pm )$.

In the regime of weak excitation, the spin operators of an NVE with large N₀ can be mapped onto bosonic operators via Holstein–Primakoff (HP) transformation [23], ${\sum }_{k=1}^{N_0}{\sigma }_{+k}^j=b_j^{†}\sqrt{N_0-b_j^{†}{b}_j}\backsimeq \sqrt{N_0}{b}_j^{†}$,${\sum }_{k=1}^{N_0}{\sigma }_{-k}^j=b_j\sqrt{N_0-b_j^{†}{b}_j}\backsimeq \sqrt{N_0}{b}_j$, and ${\sum }_{k=1}^{N_0}{\sigma }_{zk}^j=2b_j^{†}{b}_j-N_0$, where j = 1, 2 for the first and the second NVEs, respectively, and [b_j, $b_j^{†}$] = 1. The coupling of an ensemble of N₀ spins is thus enhanced by a factor of $\sqrt{N_0}$ compared to the coupling of a single spin [15,16].

The total Hamiltonian of the NVE–CPWR hybrid system considered here can now be written (in units of ħ = 1) as

We are going to show now that, starting from this Hamiltonian, we can create an entangled state shared among the NVE memories. To this end, we prepare the CPWR (up to a normalization factor) in the macroscopic cat even (odd) state |α〉_c ± | − α〉_c, where the plus (minus) sign is for the even (odd) cat state [242526] and |α〉_c denotes the coherent state of the microwave mode. The ith NVE is prepared in its ground state |0〉. The initial state of the entire system is thus written as

The time-dependent wave function |ψ_±(t)〉 of the whole NVE–CPWR system is found by applying the time evolution operator U(t) = e^−iHt/ħ to |ψ_±(0)〉. For an NVE–CPWR system prepared in the initial state of Eq. (2), this gives

Generally, both the NVEs and the CPWR may become entangled as a result of this time evolution. However, we are concerned here with the entanglement that builds up in the NVEs. To quantify this entanglement, we trace out the CPWR subspace to isolate the reduced density operator related to the NVEs, ρ_NV±(t) = Tr_CPWR(|ψ_±(t)〉〈ψ_±(t)|). This leads to

3. Witnessing the entanglement among NVEs

With an appropriate encoding of the logical bases, the above density matrix can be mapped onto a three-qubit state [27]. With this goal in mind, we choose the orthogonal basis of |0〉_i ≡ |β_i(t)〉 and ${|\mathbf{1}〉}_i\equiv \left(|-{\beta }_i(t)〉-p_i(t)|{\beta }_i(t)〉\right)/\sqrt{1-p_1^2(t)}$, where p_i(t) = e^{−2(g_i/G)2|α|2sin2(Gt)} [28].

The coherent states |± β_i(t)〉 can then be expressed as |β_i(t)〉 ≡ |0〉_i, $|-{\beta }_i(t)〉\equiv p_i(t){|\mathbf{0}〉}_i+\sqrt{1-p_i^2(t)}{|\mathbf{1}〉}_i$. To identify the entanglement in this tripartite state, it is instructive to consider the entanglement witness of a three-qubit GHZ state [29,30], $|\mathit{GHZ}〉=\left(|000〉+|111〉\right)/\sqrt{2}$.

We first consider the case of an even coherent input state N₊(|α〉_c + |− α〉_c) with the density matrix ρ_NV+(t). The relevant entanglement witness is then given by [29,30] ${\widehat{W}}_{\mathit{GHZ}}=\frac{1}{2}I\otimes I\otimes I-|\mathit{GHZ}〉〈\mathit{GHZ}|$.

Importantly, the entanglement witness is represented by a Hermitian operator and is, therefore, at least in principle, measurable. The witness defined above connects to the genuine three-partite entanglement when Tr(Ŵ_GHZ ρ_NV+(t)) < 0. The expectation value of the witness operator Ŵ_GHZ with respect to ρ_NV+(t), encoded into a three-qubit density matrix, is

If one of the NVEs is not coupled to the rest of the system, we get a biseperable state where the two remaining NVEs may become entangled. Specifically, if the first NVE is uncoupled from the rest of the system (g₁ = 0), we have p₁(t) = 1. The expectation value of the entanglement witness then reduces to $\mathit{EW}(p_1(t)=1)=\left[1-p_2^2(t)p_3^2(t)\right]/(4+4e^{-2{\left|\alpha \right|}^2})$.

We now see that the considered entanglement witness can become negative only when the entanglement is distributed among all the three NVEs. Figure 1 displays the dynamics of the expectation values of the entanglement witnesses for different coherence parameters. When the coupling strengths of all the NVEs are the same, g₁ = g₂ = g₃ [Fig. 1(a)], we find p₁(t) = p₂(t) = p₃(t) = p(t). In this case, EW simplifies to

 

Fig. 1 Time evolution of EW for an NVE with an even-cat input state for (a) g₁ = g₂ = g₃ and (b) g₂ = 0.5g₁ and g₃ = 1.5g₁.

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Accordingly, an entangled state can be detected when Gt is close to (2k + 1)π/2, so that EW displays a periodic behavior as a function of Gt. For large α, EW may get close to −1/2, indicating a large overlap between the NVE state and the three-qubit GHZ state. Coupling strengths of individual NVEs do not effect the periodic behavior of EW, which is fully controlled by the overall coupling G. EW reaches its minima at Gt = (2k + 1)π/2, when all the photons are shared between the NVEs, leaving the resonator completely depleted. In a system where the coupling strengths are not equal, the overlap between the NVE states and the GHZ state is smaller. As can be seen from Fig. 1(b), when the coherence parameter α is very small, EW is positive, and the detection of entangled state fails. On the other hand, when α is very large (Fig. 1), entanglement can be detected only within a very narrow range of Gt near Gt = (2k + 1)π/2. For α → ∞, we find EW → −1/2 when Gt = (2k + 1)π/2; EW → 0 otherwise.

Next, we consider the case of an odd-cat input state of the form N₋(|α〉_c − | − α〉_c), with the density matrix of the final state given by ρ_NV−(t). In this case, we use another maximally entangled GHZ state as a reference. To define this state, we apply a local operation to the GHZ state, $({\sigma }_z\otimes I\otimes I)|\mathit{GHZ}〉=\frac{1}{\sqrt{2}}\left(|000〉-|111〉\right)$.

The corresponding entanglement witness for the odd-cat input state is [20,29,30]

The expectation values of the entanglement witness are again periodic functions of Gt (Fig. 2). However, we see that EW have larger peak values. In this case when p₁(t) = p₂(t) = p₃(t) = p(t), EW simplifies to

 

Fig. 2 Time evolution of EW for an NVE with an odd-cat input state for (a) g₁ = g₂ = g₃ and (b) g₂ = 0.5g₁ and g₃ = 1.5g₁.

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Thus, entanglement can be detected when $3p^2(t)-2\left[q(t)-p^3(t)\right]\sqrt{1-p^2(t)}<0$. Comparing this inequality with a similar condition obtained for even-cat input states, we see that, entanglement detection in the case of odd-cat input state imposes more stringent requirements on p(t). When α is small, entanglement detection through the entanglement witness fails.

As is readily seen in Fig. 2, minimum EW values are achieved at Gt = (2k + 1)π/2, when all the photons are shared between the NVEs, leaving the resonator completely depleted of photons. When the couplings between individual NVEs are not identical, the overlap of the NVE state and the GHZ state is smaller. As can be seen from Fig. 2(b) when the coherence parameter α is small, EW becomes positive, and entanglement detection fails. When α is very large, entanglement can be detected only within a very narrow range of Gt near (2k + 1)π/2. For α → ∞, we have EW → −1/2 when Gt = (2k + 1)π/2; EW → 0 otherwise.

To quantify the overlap between the considered class of entangled states and the maximally entangled GHZ state, we consider the fidelity of the considered entangled states with respect to the ideal GHZ states. Applying a standard definition of the fidelity [31] $F\left(\rho ,\sigma \right)=Tr{\left(\sqrt{\sqrt{\rho }\sigma \sqrt{\rho }}\right)}^2$ to the density matrices ρ and σ of the considered entangled states and the ideal GHZ states, we find

In the special case of p₁(t) = p₂(t) = p₃(t) = p(t), F_± simplifies to

For both even- and odd-cat input states, the fidelity is a periodic function of dimensionless time Gt (Fig. 3), reaching its maximum value at Gt = (2k + 1)π/2. As the coherence parameter increases, the NVE state gets closer to the maximally entangled state, with the fidelity approaching 1 for moderate values of the coherence parameter. This indicates that NVEs are approaching maximally entangled GHZ states. At t = 0, the fidelity is exactly 1/2, as expected, since all the NVEs are in their ground states.

 

Fig. 3 Time dependence of the fidelity (a) F₊ and (b) F₋ with g₁ = g₂ = g₃.

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4. Effect of decoherence

In any realistic experimental implementation, entanglement detection is inevitably subject to decoherence. To examine decoherence effects, we consider an experimental scheme where the resonator is tuned to the zero-field splitting frequency of NV centers with a suitable choise of electric circuit parameters. Specifically, to tune the resonator frequency to D_gs/2π = 2.87 GHz, the CPWR should be designed to have an inductance F_c = 60.7 nH and a capacitance C_c = 2 pF [21].

For a single NV center placed 1 μm above the center, the coupling strength induced by the CPWR conductor is g₀/2π ≈ 12 Hz. This is well below the decoherence rate of the resonator, κ/2π ≈ 1 MHz [32]. However, for ≈ 10¹² NV centers coupled to a superconducting resonator, a collective coupling, g/2π ≈ 10 MHz, exceeding the coherence rate of both the CPWR and diamonds can be achieved [14, 32], enabling the desired strong coupling regime. With these experimental parameters, the first maxima of entanglement are observed at t ≈ 14 ns, i.e., well within the lifetime of microwave photons in a superconducting resonator (≈ 1 ms [34]). To let the NVE–CPWR coupling reach its maximum, the NVEs should be positioned symmetrically at the points where the magnetic field of the resonator reaches its maximum.

The coherence time of NV centers in diamond is extraordinarily long, especially at lower temperatures [5, 33]. Quantum control techniques based on dynamically decoupling pulse sequences enable a further suppression of NV spin decoherence. CPWRs are operated at temperatures of a few millikelvin. Decoherence due to NV centers can be efficiently suppressed at these temperatures, leaving decoherence related to the CPWR the dominant source of decoherence. We include these effects in our model through the quantum jump approach [21], leading to a modified Hamiltonian

The dynamics of the CPWR modes is now giverned by

The initial state defined by Eq. (2) evolves to the state

In Figs. 4(a) and 4(b), we plot the EW dynamics for even- and odd-cat input states in the case when the coupling strengths of all the NVEs are the same, g₁ = g₂ = g₃ and λ = 0.1. For both classes of input states, EW is seen to decrease as a function of time due to the dissipation of microwave photons from the resonator. For small coherence parameters, the damping can make entanglement detection completely impossible. This limitation is especially serious in the case of odd-cat input states. For large coherence parameters, however, entanglement can still be efficiently detected via the entanglement witnesses. Specifically, for the first peak in entanglement dynamics, observed at Gt = π/2, the effect of damping is still weak in the case of a high-quality resonator, allowing a robust generation of strongly entangled macroscopic quasi-GHZ states. At later moments of times, damping inevitably becomes a significant factor, leading to a loss of photons from the resonator. These photons can be detected with a suitable photon-counting scheme. No photon count in such a scheme would indicate that all the entanglement is shared among the NV memories.

 

Fig. 4 Time evolution of EW for an NVE with decoherence for (a, b) g₁ = g₂ = g₃ and (c, d) g₁ = 1, g₂ = 0.5g₃ = 1.5: (a, c) even-cat input state and (b, d) odd-cat input state.

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Figures 4(c) and 4(d) display the EW dynamics in a system where the individual NVE–resonator coupling strengths g_i are not equal to each other (we set g₁ = 1, g₂ = 0.5, g₃ = 1.5). For large α, the behavior of EW is similar to the dynamics of EW in a system with equal g_i (Figs. 4(a), 4(b)). However, since a system with unequal g_i features a weaker NVE entanglement, the entanglement witnesses fail to detect the entanglement in such a system already for α ≃ 1 (Figs. 4(c), 4(d)).

Figure 5 illustrates the influence of decoherence on the fidelity of the generated NVE state calculated with respect to the maximally entangled GHZ state. Similar to the case of no decoherence (Fig. 3), the fidelity reaches its maximum at Gt = (2k + 1)π/2. However, unlike the case of no decoherence, the fidelity never reaches 1 at these points, decreasing from one maximum to another. The rate of this decoherence-induced fidelity decay depends on the coherence parameter. Specifically, for α = 2, the fidelity is still larger than 0.94 at Gt = π/2 for both even- and odd-cat inputs, deceasing at later moments of time.

 

Fig. 5 Effect of decoherence on the time evolution of the fidelity (a) F₊ and (b) F₋ with g₁ = g₂ = g₃.

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5. Conclusion

A hybrid quantum system consisting of three NV centers coupled to a common superconducting coplanar waveguide resonator is shown to enable the generation of tripartite macroscopic entangled states. The degree of entanglement of such states can be controlled by varying the initial state. The density matrix of the NVEs can be encoded to recast a three-qubit system state, which can be characterized in terms of entanglement witnesses in relation to the GHZ states. We have identified the parameter space within which the generated entangled states can have an arbitrarily large overlap with the GHZ states, indicating an enhanced entanglement in the system.