Abstract

A hybrid quantum device consisting of three ensembles of nitrogen-vacancy centers (NVEs) whose spins are collectively coupled to a superconducting coplanar waveguide resonator is shown to enable the generation of controllable tripartite macroscopic entangled states. The density matrix of such NVEs can be encoded to recast a three-qubit system state, which can be characterized in terms of the entanglement witnesses in relation to the Greenberger-Horne-Zeilinger (GHZ) states. We identify the parameter space within which the generated entangled states can have an arbitrarily large overlap with GHZ states, indicating an enhanced entanglement in the system.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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 N0 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 Cc, and the inductance Fc 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 HC = ħωcaa, where a (a) is the annihilation (creation) operator of the microwave mode of the resonator, and its corresponding eigenfrequency is determined by ωc=2π/CcFc [21, 22]. The distributions of current and voltage inside the CPWR can be described as Iqpw=iωc/Fc(aa)sin(2π/Lx) and Vqpw=ωc/Cc(a+a)cos(2π/Lx).

The ground state of an NV center is a spin triplet with zero-magnetic-field splitting Dgs/2π = 2.87 GHz between the ms = 0 sublevel and the degenerate ms = ±1 sublevels. An external magnetic field Bext applied along the [111] direction of the diamond crystal lattice (parallel to the resonator plane) removes the degeneracy of |ms = ±1〉 sublevels by inducing Zeeman splitting and tunes the transition energies. In the study we only use one ms = 0 to ms = −1 transition of each crystal. Thus, the spin operators of the jth NV center are defined as σzj = | − 1j〈〉−1j | − |0j〈〉0j|, σ+j = | − 1j〈〉0j| and σj = |0j〈〉−1j|. For an NVE consisting of N0 NV centers, collective spin operators can then be written as Sτ=k=1N0στk(τ=z,±).

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], k=1N0σ+kj=bjN0bjbjN0bj,k=1N0σkj=bjN0bjbjN0bj, and k=1N0σzkj=2bjbjN0, where j = 1, 2 for the first and the second NVEs, respectively, and [bj, bj] = 1. The coupling of an ensemble of N0 spins is thus enhanced by a factor of N0 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

H=ωcaa+j=13ωNVbjbj+j=13gj(bja+abj),
where gj is the coupling strength for the ith NVE and the CPWR. We assume that the NVEs are frequency-tuned to be at resonance with the CPWR ωc = ω = ωNV.

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 |αc ± | − αc, where the plus (minus) sign is for the even (odd) cat state [242526] and |αc 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

|ψ±(0)=12±2e|α|2/2(|αc±|αc)|0|0|0.

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

|ψ±(t)=|α(t)c|β1(t)|β2(t)|β3(t)±|α(t)c|β1(t)|β2(t)|β3(t),
where α(t) = eiωtα cos(Gt), βi(t)=ieiωtαgisin(Gt)G, and G=g12+g22+g32is the effective coupling constant.

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, ρNV±(t) = TrCPWR(|ψ±(t)〉〈ψ±(t)|). This leads to

ρNV±(t)=N±2[|β1(t)β1(t)||β2(t)β2(t)||β3(t)β3(t)|+|β1(t)β1(t)||β2(t)β2(t)||β3(t)β3(t)|±q(t)||β1(t)β1(t)||β2(t)β2(t)||β3(t)β3(t)|±q(t)|β1(t)β1(t)||β2(t)β2(t)||β3(t)β3(t)|],
where q(t) = e−2|α|2cos2 (Gt) and N±=[2±2e2|α|2]12 is the normalization factor.

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 |0i ≡ |βi(t)〉 and |1i(|βi(t)pi(t)|βi(t))/1p12(t), where pi(t) = e−2(gi/G)2|α|2sin2(Gt) [28].

The coherent states |± βi(t)〉 can then be expressed as |βi(t)〉 ≡ |0i, |βi(t)pi(t)|0i+1pi2(t)|1i. To identify the entanglement in this tripartite state, it is instructive to consider the entanglement witness of a three-qubit GHZ state [29,30], |GHZ=(|000+|111)/2.

We first consider the case of an even coherent input state N+(|αc + |− αc) with the density matrix ρNV+(t). The relevant entanglement witness is then given by [29,30] W^GHZ=12III|GHZGHZ|.

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(ŴGHZ ρNV+(t)) < 0. The expectation value of the witness operator ŴGHZ with respect to ρNV+(t), encoded into a three-qubit density matrix, is

EW=Tr(W^GHZρNV+(t))=14+4e2|α|2[p12(t)+p22(t)+p32(t)p12(t)p22(t)p12(t)p32(t)p22(t)p32(t)2[q(t)+p1(t)p2(t)p3(t)]1p12(t)1p22(t)1p32(t)].

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 EW(p1(t)=1)=[1p22(t)p32(t)]/(4+4e2|α|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

EW(p(t))=[1p2(t)]4+4e2|α|2[3p2(t)2[q(t)+p3(t)]1p2(t)].
Thus, entanglement can be detected when 3p2(t)2[q(t)+p3(t)]1p2(t)<0.

 figure: Fig. 1

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(|αc − | − αc), with the density matrix of the final state given by ρNV(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, (σzII)|GHZ=12(|000|111).

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

W^GHZ=12III(σzII)|GHZGHZ|(σzII)
The expectation value of the witness operator Ŵ′GHZ with respect to ρNV(t), after encoding to a three-qubit density matrix, is calculated as
EW=Tr(W^GHZρNV(t))=144e2|α|2[p12(t)+p22(t)+p32(t)p12(t)p22(t)p12(t)p32(t)p22(t)p32(t)2[q(t)p1(t)p2(t)p3(t)]1p12(t)1p22(t)1p32(t)].

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

EW(p(t))=[1p2(t)]44e2|α|2[3p2(t)2[q(t)p3(t)]1p2(t)].

 figure: Fig. 2

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 3p2(t)2[q(t)p3(t)]1p2(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(ρ,σ)=Tr(ρσρ)2 to the density matrices ρ and σ of the considered entangled states and the ideal GHZ states, we find

F±=14±4q(t)p1(t)p2(t)p3(t)[1+p12(t)p22(t)p32(t)±2q(t)p1(t)p2(t)p3(t)+(1p12(t))(1p22(t))×(1p32(t))+2[q(t)±p1(t)p2(t)p3(t)]1p12(t)1p22(t)1p32(t)],
where the fidelity F+ and F measures the closeness of the state with the density matrix ρNV+ to the maximally entangled GHZ state |GHZ 〉 and F characterizes the closeness of the state with ρNV to the GHZ state (σzII)|GHZ 〉.

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

F±=14±4q(t)p1(t)p2(t)p3(t)[1+p6(t)±2q(t)p3(t)+2(q(t)±p3(t))(1p2(t))3/2+(1p2(t))3].

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.

 figure: Fig. 3

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 Dgs/2π = 2.87 GHz, the CPWR should be designed to have an inductance Fc = 60.7 nH and a capacitance Cc = 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

HT=Hiκ2aa,
where H is the Hamiltonian as defined by Eq. (1).

The dynamics of the CPWR modes is now giverned by

a(t)=ei(ωiκ/4)t[cos(G2(κ/4)2t)a(0)+iκ/4G2(κ/4)2sin(G2(κ/4)2t)a(0)i1G2(κ/4)2×sin(G2(κ/4)2t)(g1b1(0)+g2b2(0)+g3b3(0))],

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

|ψ±(t)=|α(t)c|β1(t)|β2(t)|β3(t)
±|α(t)c|β1(t)|β2(t)|β3(t).
Here,
α(t)=ei(ωiκ/4)tα[cos(Gζt)+i(λ/ζ)sin(Gζt)],
βi(t)=iei(ωiκ/4)tαgiGζsin(Gζt)
are the time-dependent coherence parameters, λ = κ/4G, and ζ=1(κ/4G)2.

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.

 figure: Fig. 4

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 gi 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 gi (Figs. 4(a), 4(b)). However, since a system with unequal gi 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.

 figure: Fig. 5

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.

Funding

Welch Foundation (Grant No. A-1801-20180324), ONR (Award No. 00014-16-1-2578), Government of Russian Federation (project no. 14.Z50.31.0040, Feb. 17, 2017), Russian Foundation for Basic Research (projects nos. 16-02-00843, 16-32-60164, and 18-52-00025), and Russian Science Foundation (project no. 17-12-01533).

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31. D. Petz, Quantum Information Theory and Quantum Statistics (Springer–VerlagBerlin Heidelberg, 2008).

32. Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J. -F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, “Strong Coupling of a Spin Ensemble to a Superconducting Resonator,” Phys. Rev. Lett. 105, 140502 (2010). [CrossRef]  

33. A. Jarmola, V. M. Acosta, K. Jensen, S. Chemerisov, and D. Budker, “Temperature- and Magnetic-Field-Dependent Longitudinal Spin Relaxation in Nitrogen-Vacancy Ensembles in Diamond,” Phys. Rev. Lett. 108, 197601 (2012). [CrossRef]   [PubMed]  

34. M. Reagor, W. Pfaff, C. Axline, R. W. Heeres, N. Ofek, K. Sliwa, E. Holland, C. Wang, J. Blumoff, K. Chou, M. J. Hatridge, L. Frunzio, M. H. Devoret, L. Jiang, and R. J. Schoelkopf, “Quantum memory with millisecond coherence in circuit QED,” Phys. Rev. B 94, 014506 (2016). [CrossRef]  

References

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  1. L. Childress, M. V. G. Dutt, J. M. Taylor, A. S. Zibrov, F. Jelezko, J. Wrachtrup, P. R. Hemmer, and M. D. Lukin, “Coherent dynamics of coupled electron and nuclear spin qubits in diamond,” Science 314, 281–285 (2006).
    [Crossref] [PubMed]
  2. J. R. Weber, W. F. Koehl, J. B. Varley, A. Janotti, B. B. Buckley, C. G. Van de Walle, and D. D. Awschalom, “Quantum computing with defects,” PNAS 07, 8513–8518 (2010).
    [Crossref]
  3. W. Song, Z. Yin, W. Yang, X. Zhu, F. Zhou, and M. Feng, “One-step generation of multipartite entanglement among nitrogen-vacancy center ensembles,” Sci. Rep. 5, 7755 (2015).
    [Crossref] [PubMed]
  4. W. B. Gao, A. Imamoglu, H. Bernien, and R. Hanson, “Coherent manipulation, measurement and entanglement of individual solid-state spins using optical fields,” Nature Photonics 9, 363–373 (2015).
    [Crossref]
  5. N. Bar-Gill, L. M. Pham, A. Jarmola, D. Budker, and R. L. Walsworth, “Solid-state electronic spin coherence time approaching one second,” Nat. Commun. 4, 1743 (2013).
    [Crossref] [PubMed]
  6. T. Liu, Q. -P. Su, S. -J. Xiong, J. -M. Liu, C. -P. Yang, and F. Nori, “Generation of a macroscopic entangled coherent state using quantum memories in circuit QED,” Sci. Rep. 6, 32004 (2016).
    [Crossref] [PubMed]
  7. Z. -L. Xiang, S. Ashhab, J. Q. You, and F. Nori, “Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems,” Rev. Mod. Physics,  85, 623–653 (2013).
    [Crossref]
  8. E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. V. G. Dutt, A. S. Sørensen, P. R. Hemmer, A. S. Zibrov, and M. D. Lukin, “Quantum entanglement between an optical photon and a solid-state spin qubit,” Nature 466, 730–734 (2010).
    [Crossref] [PubMed]
  9. W. L. Yang, Y. Hu, Z. Q. Yin, Z. J. Deng, and M. Feng, “Entanglement of nitrogen-vacancy-center ensembles using transmission line resonators and a superconducting phase qubit,” Phys. Rev. A 83, 022302 (2011).
    [Crossref]
  10. D. Marcos, M. Wubs, J. M. Taylor, R. Aguado, M. D. Lukin, and A. S. Sørensen, “Coupling nitrogen-vacancy centers in diamond to superconducting flux qubits,” Phys. Rev. Lett. 105, 210501 (2010).
    [Crossref]
  11. Z. -L. Xiang, X. -Y. Lu, T. F. Li, J. Q. You, and F. Nori, “Hybrid quantum circuit consisting of a superconducting flux qubit coupled to a spin ensemble and a transmission-line resonator,” Phys. Rev. B 87, 144516 (2013).
    [Crossref]
  12. R. Amsüss, C. Koller, T. Nöbauer, S. Putz, S. Rotter, K. Sandner, S. Schneider, M. Schramböck, G. Steinhauser, H. Ritsch, J. Schmiedmayer, and J. Majer, “Cavity QED with Magnetically Coupled Collective Spin States,” Phys. Rev. Lett. 107, 060502 (2011).
    [Crossref] [PubMed]
  13. Y. Maleki and A. M. Zheltikov, “Generating maximally-path-entangled number states in two spin ensembles coupled to a superconducting flux qubit,” Phys. Rev. A 97, 012312 (2018).
    [Crossref]
  14. V. Ranjan, G. de Lange, R. Schutjens, T. Debelhoir, J. P. Groen, D. Szombati, D. J. Thoen, T. M. Klapwijk, R. Hanson, and L. DiCarlo, “Probing Dynamics of an Electron-Spin Ensemble via a Superconducting Resonator,” Phys. Rev. Lett. 110, 067004 (2013).
    [Crossref] [PubMed]
  15. M. G. Raizen, R. J. Thompson, R. J. Brecha, H. J. Kimble, and H.J. Carmichael, “Normal-mode splitting and linewidth averaging for two-state atoms in an optical cavity,” Phys. Rev. Lett. 63, 240 (1989).
    [Crossref] [PubMed]
  16. S. Saito, X. Zhu, R. Amsüss, Y. Matsuzaki, K. Kakuyanagi, T. Shimo-Oka, N. Mizuochi, K. Nemoto, W. J. Munro, and K. Semba, “Towards realizing a quantum memory for a superconducting qubit: Storage and retrieval of quantum states,” Phys. Rev. Lett. 111, 107008 (2013).
    [Crossref]
  17. M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of mixed states: necessary and sufficient conditions,” Phys. Lett. A 223, 1 (1996).
    [Crossref]
  18. N. Ganguly, S. Adhikari, A. S. Majumdar, and J. Chatterjee, “Entanglement Witness Operator for Quantum Teleportation”, Phys. Rev. Lett. 107, 270501 (2011).
    [Crossref]
  19. M. Barbieri, F. D. Martini, G. D. Nepi, P. Mataloni, G. M. D’Ariano, and C. Macchiavello,” Detection of Entanglement with Polarized Photons: Experimental Realization of an Entanglement Witness,” Phys. Rev. Lett. 91, 227901 (2003).
    [Crossref] [PubMed]
  20. N. Behzadi, B. Ahansaz, and S. Shojaei, “Genuine entanglement among coherent excitonic states of three quantum dots located individually in separated coupled QED cavities,” Eur. Phys. J. D 67, 5 (2013).
    [Crossref]
  21. Yimin Liu, Jiabin You, and Qizhe Hou, “Entanglement dynamics of Nitrogen-vacancy centers spin ensembles coupled to a superconducting resonator,” Sci. Rep. 6, 21775 (2016).
    [Crossref] [PubMed]
  22. T. Astner, S. Nevlacsil, N. Peterschofsky, A. Angerer, S. Rotter, S. Putz, J. Schmiedmayer, and J. Majer, “Coherent Coupling of Remote Spin Ensembles via a Cavity Bus”, Phys. Rev. Lett. 118, 140502 (2017).
    [Crossref] [PubMed]
  23. K. Hammerer, A. S. Søensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. 82, 1041 (2010).
    [Crossref]
  24. Y. -C. Ou and H. Fan, “Monogamy inequality in terms of negativity for three-qubit states,” Phys. Rev. A 75062308 (2007).
    [Crossref]
  25. W. K. Wootters, “Entanglement of Formation of an Arbitrary State of Two Qubits,” Phys. Rev. Lett. 80, 2245 (1998).
    [Crossref]
  26. B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L. Frunzio, S. M. Girvin, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “Deterministically encoding quantum information using 100-photon schrödinger cat states,” Science 342, 607–610 (2013).
    [Crossref]
  27. Y. Maleki, “Generation and entanglement of multi–dimensional multi–mode coherent fields in cavity QED,” Quantum Inf. Process. 15, 4537–4562 (2016).
    [Crossref]
  28. Y. Maleki and A. Maleki, “Entangled multimode spin coherent states of trapped ions,” JOSA B 35, 1211–1217 (2018).
    [Crossref]
  29. A. Acin, D. Brus, M. Lewenstein, and A. Sanpera, “Classification of Mixed Three-Qubit States,” Phys. Rev. Lett. 87, 040401 (2001).
    [Crossref] [PubMed]
  30. M. Bourennance, M. Eibl, C. Kurtsiefer, S. Gaertner, H. Weinfurter, O. Guhne, P. Hyllus, D. Brus, M. Lewenstein, and A. Sanpera, “Experimental Detection of Multipartite Entanglement using Witness Operators,” Phys. Rev. Lett. 92, 087902 (2004).
    [Crossref]
  31. D. Petz, Quantum Information Theory and Quantum Statistics (Springer–VerlagBerlin Heidelberg, 2008).
  32. Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J. -F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, “Strong Coupling of a Spin Ensemble to a Superconducting Resonator,” Phys. Rev. Lett. 105, 140502 (2010).
    [Crossref]
  33. A. Jarmola, V. M. Acosta, K. Jensen, S. Chemerisov, and D. Budker, “Temperature- and Magnetic-Field-Dependent Longitudinal Spin Relaxation in Nitrogen-Vacancy Ensembles in Diamond,” Phys. Rev. Lett. 108, 197601 (2012).
    [Crossref] [PubMed]
  34. M. Reagor, W. Pfaff, C. Axline, R. W. Heeres, N. Ofek, K. Sliwa, E. Holland, C. Wang, J. Blumoff, K. Chou, M. J. Hatridge, L. Frunzio, M. H. Devoret, L. Jiang, and R. J. Schoelkopf, “Quantum memory with millisecond coherence in circuit QED,” Phys. Rev. B 94, 014506 (2016).
    [Crossref]

2018 (2)

Y. Maleki and A. M. Zheltikov, “Generating maximally-path-entangled number states in two spin ensembles coupled to a superconducting flux qubit,” Phys. Rev. A 97, 012312 (2018).
[Crossref]

Y. Maleki and A. Maleki, “Entangled multimode spin coherent states of trapped ions,” JOSA B 35, 1211–1217 (2018).
[Crossref]

2017 (1)

T. Astner, S. Nevlacsil, N. Peterschofsky, A. Angerer, S. Rotter, S. Putz, J. Schmiedmayer, and J. Majer, “Coherent Coupling of Remote Spin Ensembles via a Cavity Bus”, Phys. Rev. Lett. 118, 140502 (2017).
[Crossref] [PubMed]

2016 (4)

Yimin Liu, Jiabin You, and Qizhe Hou, “Entanglement dynamics of Nitrogen-vacancy centers spin ensembles coupled to a superconducting resonator,” Sci. Rep. 6, 21775 (2016).
[Crossref] [PubMed]

Y. Maleki, “Generation and entanglement of multi–dimensional multi–mode coherent fields in cavity QED,” Quantum Inf. Process. 15, 4537–4562 (2016).
[Crossref]

M. Reagor, W. Pfaff, C. Axline, R. W. Heeres, N. Ofek, K. Sliwa, E. Holland, C. Wang, J. Blumoff, K. Chou, M. J. Hatridge, L. Frunzio, M. H. Devoret, L. Jiang, and R. J. Schoelkopf, “Quantum memory with millisecond coherence in circuit QED,” Phys. Rev. B 94, 014506 (2016).
[Crossref]

T. Liu, Q. -P. Su, S. -J. Xiong, J. -M. Liu, C. -P. Yang, and F. Nori, “Generation of a macroscopic entangled coherent state using quantum memories in circuit QED,” Sci. Rep. 6, 32004 (2016).
[Crossref] [PubMed]

2015 (2)

W. Song, Z. Yin, W. Yang, X. Zhu, F. Zhou, and M. Feng, “One-step generation of multipartite entanglement among nitrogen-vacancy center ensembles,” Sci. Rep. 5, 7755 (2015).
[Crossref] [PubMed]

W. B. Gao, A. Imamoglu, H. Bernien, and R. Hanson, “Coherent manipulation, measurement and entanglement of individual solid-state spins using optical fields,” Nature Photonics 9, 363–373 (2015).
[Crossref]

2013 (7)

N. Bar-Gill, L. M. Pham, A. Jarmola, D. Budker, and R. L. Walsworth, “Solid-state electronic spin coherence time approaching one second,” Nat. Commun. 4, 1743 (2013).
[Crossref] [PubMed]

Z. -L. Xiang, S. Ashhab, J. Q. You, and F. Nori, “Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems,” Rev. Mod. Physics,  85, 623–653 (2013).
[Crossref]

V. Ranjan, G. de Lange, R. Schutjens, T. Debelhoir, J. P. Groen, D. Szombati, D. J. Thoen, T. M. Klapwijk, R. Hanson, and L. DiCarlo, “Probing Dynamics of an Electron-Spin Ensemble via a Superconducting Resonator,” Phys. Rev. Lett. 110, 067004 (2013).
[Crossref] [PubMed]

Z. -L. Xiang, X. -Y. Lu, T. F. Li, J. Q. You, and F. Nori, “Hybrid quantum circuit consisting of a superconducting flux qubit coupled to a spin ensemble and a transmission-line resonator,” Phys. Rev. B 87, 144516 (2013).
[Crossref]

S. Saito, X. Zhu, R. Amsüss, Y. Matsuzaki, K. Kakuyanagi, T. Shimo-Oka, N. Mizuochi, K. Nemoto, W. J. Munro, and K. Semba, “Towards realizing a quantum memory for a superconducting qubit: Storage and retrieval of quantum states,” Phys. Rev. Lett. 111, 107008 (2013).
[Crossref]

B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L. Frunzio, S. M. Girvin, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “Deterministically encoding quantum information using 100-photon schrödinger cat states,” Science 342, 607–610 (2013).
[Crossref]

N. Behzadi, B. Ahansaz, and S. Shojaei, “Genuine entanglement among coherent excitonic states of three quantum dots located individually in separated coupled QED cavities,” Eur. Phys. J. D 67, 5 (2013).
[Crossref]

2012 (1)

A. Jarmola, V. M. Acosta, K. Jensen, S. Chemerisov, and D. Budker, “Temperature- and Magnetic-Field-Dependent Longitudinal Spin Relaxation in Nitrogen-Vacancy Ensembles in Diamond,” Phys. Rev. Lett. 108, 197601 (2012).
[Crossref] [PubMed]

2011 (3)

N. Ganguly, S. Adhikari, A. S. Majumdar, and J. Chatterjee, “Entanglement Witness Operator for Quantum Teleportation”, Phys. Rev. Lett. 107, 270501 (2011).
[Crossref]

R. Amsüss, C. Koller, T. Nöbauer, S. Putz, S. Rotter, K. Sandner, S. Schneider, M. Schramböck, G. Steinhauser, H. Ritsch, J. Schmiedmayer, and J. Majer, “Cavity QED with Magnetically Coupled Collective Spin States,” Phys. Rev. Lett. 107, 060502 (2011).
[Crossref] [PubMed]

W. L. Yang, Y. Hu, Z. Q. Yin, Z. J. Deng, and M. Feng, “Entanglement of nitrogen-vacancy-center ensembles using transmission line resonators and a superconducting phase qubit,” Phys. Rev. A 83, 022302 (2011).
[Crossref]

2010 (5)

D. Marcos, M. Wubs, J. M. Taylor, R. Aguado, M. D. Lukin, and A. S. Sørensen, “Coupling nitrogen-vacancy centers in diamond to superconducting flux qubits,” Phys. Rev. Lett. 105, 210501 (2010).
[Crossref]

E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. V. G. Dutt, A. S. Sørensen, P. R. Hemmer, A. S. Zibrov, and M. D. Lukin, “Quantum entanglement between an optical photon and a solid-state spin qubit,” Nature 466, 730–734 (2010).
[Crossref] [PubMed]

J. R. Weber, W. F. Koehl, J. B. Varley, A. Janotti, B. B. Buckley, C. G. Van de Walle, and D. D. Awschalom, “Quantum computing with defects,” PNAS 07, 8513–8518 (2010).
[Crossref]

Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J. -F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, “Strong Coupling of a Spin Ensemble to a Superconducting Resonator,” Phys. Rev. Lett. 105, 140502 (2010).
[Crossref]

K. Hammerer, A. S. Søensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. 82, 1041 (2010).
[Crossref]

2007 (1)

Y. -C. Ou and H. Fan, “Monogamy inequality in terms of negativity for three-qubit states,” Phys. Rev. A 75062308 (2007).
[Crossref]

2006 (1)

L. Childress, M. V. G. Dutt, J. M. Taylor, A. S. Zibrov, F. Jelezko, J. Wrachtrup, P. R. Hemmer, and M. D. Lukin, “Coherent dynamics of coupled electron and nuclear spin qubits in diamond,” Science 314, 281–285 (2006).
[Crossref] [PubMed]

2004 (1)

M. Bourennance, M. Eibl, C. Kurtsiefer, S. Gaertner, H. Weinfurter, O. Guhne, P. Hyllus, D. Brus, M. Lewenstein, and A. Sanpera, “Experimental Detection of Multipartite Entanglement using Witness Operators,” Phys. Rev. Lett. 92, 087902 (2004).
[Crossref]

2003 (1)

M. Barbieri, F. D. Martini, G. D. Nepi, P. Mataloni, G. M. D’Ariano, and C. Macchiavello,” Detection of Entanglement with Polarized Photons: Experimental Realization of an Entanglement Witness,” Phys. Rev. Lett. 91, 227901 (2003).
[Crossref] [PubMed]

2001 (1)

A. Acin, D. Brus, M. Lewenstein, and A. Sanpera, “Classification of Mixed Three-Qubit States,” Phys. Rev. Lett. 87, 040401 (2001).
[Crossref] [PubMed]

1998 (1)

W. K. Wootters, “Entanglement of Formation of an Arbitrary State of Two Qubits,” Phys. Rev. Lett. 80, 2245 (1998).
[Crossref]

1996 (1)

M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of mixed states: necessary and sufficient conditions,” Phys. Lett. A 223, 1 (1996).
[Crossref]

1989 (1)

M. G. Raizen, R. J. Thompson, R. J. Brecha, H. J. Kimble, and H.J. Carmichael, “Normal-mode splitting and linewidth averaging for two-state atoms in an optical cavity,” Phys. Rev. Lett. 63, 240 (1989).
[Crossref] [PubMed]

Acin, A.

A. Acin, D. Brus, M. Lewenstein, and A. Sanpera, “Classification of Mixed Three-Qubit States,” Phys. Rev. Lett. 87, 040401 (2001).
[Crossref] [PubMed]

Acosta, V. M.

A. Jarmola, V. M. Acosta, K. Jensen, S. Chemerisov, and D. Budker, “Temperature- and Magnetic-Field-Dependent Longitudinal Spin Relaxation in Nitrogen-Vacancy Ensembles in Diamond,” Phys. Rev. Lett. 108, 197601 (2012).
[Crossref] [PubMed]

Adhikari, S.

N. Ganguly, S. Adhikari, A. S. Majumdar, and J. Chatterjee, “Entanglement Witness Operator for Quantum Teleportation”, Phys. Rev. Lett. 107, 270501 (2011).
[Crossref]

Aguado, R.

D. Marcos, M. Wubs, J. M. Taylor, R. Aguado, M. D. Lukin, and A. S. Sørensen, “Coupling nitrogen-vacancy centers in diamond to superconducting flux qubits,” Phys. Rev. Lett. 105, 210501 (2010).
[Crossref]

Ahansaz, B.

N. Behzadi, B. Ahansaz, and S. Shojaei, “Genuine entanglement among coherent excitonic states of three quantum dots located individually in separated coupled QED cavities,” Eur. Phys. J. D 67, 5 (2013).
[Crossref]

Amsüss, R.

S. Saito, X. Zhu, R. Amsüss, Y. Matsuzaki, K. Kakuyanagi, T. Shimo-Oka, N. Mizuochi, K. Nemoto, W. J. Munro, and K. Semba, “Towards realizing a quantum memory for a superconducting qubit: Storage and retrieval of quantum states,” Phys. Rev. Lett. 111, 107008 (2013).
[Crossref]

R. Amsüss, C. Koller, T. Nöbauer, S. Putz, S. Rotter, K. Sandner, S. Schneider, M. Schramböck, G. Steinhauser, H. Ritsch, J. Schmiedmayer, and J. Majer, “Cavity QED with Magnetically Coupled Collective Spin States,” Phys. Rev. Lett. 107, 060502 (2011).
[Crossref] [PubMed]

Angerer, A.

T. Astner, S. Nevlacsil, N. Peterschofsky, A. Angerer, S. Rotter, S. Putz, J. Schmiedmayer, and J. Majer, “Coherent Coupling of Remote Spin Ensembles via a Cavity Bus”, Phys. Rev. Lett. 118, 140502 (2017).
[Crossref] [PubMed]

Ashhab, S.

Z. -L. Xiang, S. Ashhab, J. Q. You, and F. Nori, “Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems,” Rev. Mod. Physics,  85, 623–653 (2013).
[Crossref]

Astner, T.

T. Astner, S. Nevlacsil, N. Peterschofsky, A. Angerer, S. Rotter, S. Putz, J. Schmiedmayer, and J. Majer, “Coherent Coupling of Remote Spin Ensembles via a Cavity Bus”, Phys. Rev. Lett. 118, 140502 (2017).
[Crossref] [PubMed]

Auffeves, A.

Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J. -F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, “Strong Coupling of a Spin Ensemble to a Superconducting Resonator,” Phys. Rev. Lett. 105, 140502 (2010).
[Crossref]

Awschalom, D. D.

J. R. Weber, W. F. Koehl, J. B. Varley, A. Janotti, B. B. Buckley, C. G. Van de Walle, and D. D. Awschalom, “Quantum computing with defects,” PNAS 07, 8513–8518 (2010).
[Crossref]

Axline, C.

M. Reagor, W. Pfaff, C. Axline, R. W. Heeres, N. Ofek, K. Sliwa, E. Holland, C. Wang, J. Blumoff, K. Chou, M. J. Hatridge, L. Frunzio, M. H. Devoret, L. Jiang, and R. J. Schoelkopf, “Quantum memory with millisecond coherence in circuit QED,” Phys. Rev. B 94, 014506 (2016).
[Crossref]

Barbieri, M.

M. Barbieri, F. D. Martini, G. D. Nepi, P. Mataloni, G. M. D’Ariano, and C. Macchiavello,” Detection of Entanglement with Polarized Photons: Experimental Realization of an Entanglement Witness,” Phys. Rev. Lett. 91, 227901 (2003).
[Crossref] [PubMed]

Bar-Gill, N.

N. Bar-Gill, L. M. Pham, A. Jarmola, D. Budker, and R. L. Walsworth, “Solid-state electronic spin coherence time approaching one second,” Nat. Commun. 4, 1743 (2013).
[Crossref] [PubMed]

Barthe, M. F.

Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J. -F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, “Strong Coupling of a Spin Ensemble to a Superconducting Resonator,” Phys. Rev. Lett. 105, 140502 (2010).
[Crossref]

Behzadi, N.

N. Behzadi, B. Ahansaz, and S. Shojaei, “Genuine entanglement among coherent excitonic states of three quantum dots located individually in separated coupled QED cavities,” Eur. Phys. J. D 67, 5 (2013).
[Crossref]

Bergonzo, P.

Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J. -F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, “Strong Coupling of a Spin Ensemble to a Superconducting Resonator,” Phys. Rev. Lett. 105, 140502 (2010).
[Crossref]

Bernien, H.

W. B. Gao, A. Imamoglu, H. Bernien, and R. Hanson, “Coherent manipulation, measurement and entanglement of individual solid-state spins using optical fields,” Nature Photonics 9, 363–373 (2015).
[Crossref]

Bertet, P.

Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J. -F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, “Strong Coupling of a Spin Ensemble to a Superconducting Resonator,” Phys. Rev. Lett. 105, 140502 (2010).
[Crossref]

Blumoff, J.

M. Reagor, W. Pfaff, C. Axline, R. W. Heeres, N. Ofek, K. Sliwa, E. Holland, C. Wang, J. Blumoff, K. Chou, M. J. Hatridge, L. Frunzio, M. H. Devoret, L. Jiang, and R. J. Schoelkopf, “Quantum memory with millisecond coherence in circuit QED,” Phys. Rev. B 94, 014506 (2016).
[Crossref]

Bourennance, M.

M. Bourennance, M. Eibl, C. Kurtsiefer, S. Gaertner, H. Weinfurter, O. Guhne, P. Hyllus, D. Brus, M. Lewenstein, and A. Sanpera, “Experimental Detection of Multipartite Entanglement using Witness Operators,” Phys. Rev. Lett. 92, 087902 (2004).
[Crossref]

Brecha, R. J.

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N. Bar-Gill, L. M. Pham, A. Jarmola, D. Budker, and R. L. Walsworth, “Solid-state electronic spin coherence time approaching one second,” Nat. Commun. 4, 1743 (2013).
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M. Bourennance, M. Eibl, C. Kurtsiefer, S. Gaertner, H. Weinfurter, O. Guhne, P. Hyllus, D. Brus, M. Lewenstein, and A. Sanpera, “Experimental Detection of Multipartite Entanglement using Witness Operators,” Phys. Rev. Lett. 92, 087902 (2004).
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M. Reagor, W. Pfaff, C. Axline, R. W. Heeres, N. Ofek, K. Sliwa, E. Holland, C. Wang, J. Blumoff, K. Chou, M. J. Hatridge, L. Frunzio, M. H. Devoret, L. Jiang, and R. J. Schoelkopf, “Quantum memory with millisecond coherence in circuit QED,” Phys. Rev. B 94, 014506 (2016).
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V. Ranjan, G. de Lange, R. Schutjens, T. Debelhoir, J. P. Groen, D. Szombati, D. J. Thoen, T. M. Klapwijk, R. Hanson, and L. DiCarlo, “Probing Dynamics of an Electron-Spin Ensemble via a Superconducting Resonator,” Phys. Rev. Lett. 110, 067004 (2013).
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V. Ranjan, G. de Lange, R. Schutjens, T. Debelhoir, J. P. Groen, D. Szombati, D. J. Thoen, T. M. Klapwijk, R. Hanson, and L. DiCarlo, “Probing Dynamics of an Electron-Spin Ensemble via a Superconducting Resonator,” Phys. Rev. Lett. 110, 067004 (2013).
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L. Childress, M. V. G. Dutt, J. M. Taylor, A. S. Zibrov, F. Jelezko, J. Wrachtrup, P. R. Hemmer, and M. D. Lukin, “Coherent dynamics of coupled electron and nuclear spin qubits in diamond,” Science 314, 281–285 (2006).
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M. Reagor, W. Pfaff, C. Axline, R. W. Heeres, N. Ofek, K. Sliwa, E. Holland, C. Wang, J. Blumoff, K. Chou, M. J. Hatridge, L. Frunzio, M. H. Devoret, L. Jiang, and R. J. Schoelkopf, “Quantum memory with millisecond coherence in circuit QED,” Phys. Rev. B 94, 014506 (2016).
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Yimin Liu, Jiabin You, and Qizhe Hou, “Entanglement dynamics of Nitrogen-vacancy centers spin ensembles coupled to a superconducting resonator,” Sci. Rep. 6, 21775 (2016).
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W. L. Yang, Y. Hu, Z. Q. Yin, Z. J. Deng, and M. Feng, “Entanglement of nitrogen-vacancy-center ensembles using transmission line resonators and a superconducting phase qubit,” Phys. Rev. A 83, 022302 (2011).
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M. Bourennance, M. Eibl, C. Kurtsiefer, S. Gaertner, H. Weinfurter, O. Guhne, P. Hyllus, D. Brus, M. Lewenstein, and A. Sanpera, “Experimental Detection of Multipartite Entanglement using Witness Operators,” Phys. Rev. Lett. 92, 087902 (2004).
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W. B. Gao, A. Imamoglu, H. Bernien, and R. Hanson, “Coherent manipulation, measurement and entanglement of individual solid-state spins using optical fields,” Nature Photonics 9, 363–373 (2015).
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J. R. Weber, W. F. Koehl, J. B. Varley, A. Janotti, B. B. Buckley, C. G. Van de Walle, and D. D. Awschalom, “Quantum computing with defects,” PNAS 07, 8513–8518 (2010).
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N. Bar-Gill, L. M. Pham, A. Jarmola, D. Budker, and R. L. Walsworth, “Solid-state electronic spin coherence time approaching one second,” Nat. Commun. 4, 1743 (2013).
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Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J. -F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, “Strong Coupling of a Spin Ensemble to a Superconducting Resonator,” Phys. Rev. Lett. 105, 140502 (2010).
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L. Childress, M. V. G. Dutt, J. M. Taylor, A. S. Zibrov, F. Jelezko, J. Wrachtrup, P. R. Hemmer, and M. D. Lukin, “Coherent dynamics of coupled electron and nuclear spin qubits in diamond,” Science 314, 281–285 (2006).
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A. Jarmola, V. M. Acosta, K. Jensen, S. Chemerisov, and D. Budker, “Temperature- and Magnetic-Field-Dependent Longitudinal Spin Relaxation in Nitrogen-Vacancy Ensembles in Diamond,” Phys. Rev. Lett. 108, 197601 (2012).
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M. Reagor, W. Pfaff, C. Axline, R. W. Heeres, N. Ofek, K. Sliwa, E. Holland, C. Wang, J. Blumoff, K. Chou, M. J. Hatridge, L. Frunzio, M. H. Devoret, L. Jiang, and R. J. Schoelkopf, “Quantum memory with millisecond coherence in circuit QED,” Phys. Rev. B 94, 014506 (2016).
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E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. V. G. Dutt, A. S. Sørensen, P. R. Hemmer, A. S. Zibrov, and M. D. Lukin, “Quantum entanglement between an optical photon and a solid-state spin qubit,” Nature 466, 730–734 (2010).
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B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L. Frunzio, S. M. Girvin, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “Deterministically encoding quantum information using 100-photon schrödinger cat states,” Science 342, 607–610 (2013).
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V. Ranjan, G. de Lange, R. Schutjens, T. Debelhoir, J. P. Groen, D. Szombati, D. J. Thoen, T. M. Klapwijk, R. Hanson, and L. DiCarlo, “Probing Dynamics of an Electron-Spin Ensemble via a Superconducting Resonator,” Phys. Rev. Lett. 110, 067004 (2013).
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M. Bourennance, M. Eibl, C. Kurtsiefer, S. Gaertner, H. Weinfurter, O. Guhne, P. Hyllus, D. Brus, M. Lewenstein, and A. Sanpera, “Experimental Detection of Multipartite Entanglement using Witness Operators,” Phys. Rev. Lett. 92, 087902 (2004).
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B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L. Frunzio, S. M. Girvin, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “Deterministically encoding quantum information using 100-photon schrödinger cat states,” Science 342, 607–610 (2013).
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M. Bourennance, M. Eibl, C. Kurtsiefer, S. Gaertner, H. Weinfurter, O. Guhne, P. Hyllus, D. Brus, M. Lewenstein, and A. Sanpera, “Experimental Detection of Multipartite Entanglement using Witness Operators,” Phys. Rev. Lett. 92, 087902 (2004).
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Yimin Liu, Jiabin You, and Qizhe Hou, “Entanglement dynamics of Nitrogen-vacancy centers spin ensembles coupled to a superconducting resonator,” Sci. Rep. 6, 21775 (2016).
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[Crossref] [PubMed]

L. Childress, M. V. G. Dutt, J. M. Taylor, A. S. Zibrov, F. Jelezko, J. Wrachtrup, P. R. Hemmer, and M. D. Lukin, “Coherent dynamics of coupled electron and nuclear spin qubits in diamond,” Science 314, 281–285 (2006).
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M. Barbieri, F. D. Martini, G. D. Nepi, P. Mataloni, G. M. D’Ariano, and C. Macchiavello,” Detection of Entanglement with Polarized Photons: Experimental Realization of an Entanglement Witness,” Phys. Rev. Lett. 91, 227901 (2003).
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T. Astner, S. Nevlacsil, N. Peterschofsky, A. Angerer, S. Rotter, S. Putz, J. Schmiedmayer, and J. Majer, “Coherent Coupling of Remote Spin Ensembles via a Cavity Bus”, Phys. Rev. Lett. 118, 140502 (2017).
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N. Ganguly, S. Adhikari, A. S. Majumdar, and J. Chatterjee, “Entanglement Witness Operator for Quantum Teleportation”, Phys. Rev. Lett. 107, 270501 (2011).
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Figures (5)

Fig. 1
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.
Fig. 2
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.
Fig. 3
Fig. 3 Time dependence of the fidelity (a) F+ and (b) F with g1 = g2 = g3.
Fig. 4
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.
Fig. 5
Fig. 5 Effect of decoherence on the time evolution of the fidelity (a) F+ and (b) F with g1 = g2 = g3.

Equations (17)

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H = ω c a a + j = 1 3 ω NV b j b j + j = 1 3 g j ( b j a + a b j ) ,
| ψ ± ( 0 ) = 1 2 ± 2 e | α | 2 / 2 ( | α c ± | α c ) | 0 | 0 | 0 .
| ψ ± ( t ) = | α ( t ) c | β 1 ( t ) | β 2 ( t ) | β 3 ( t ) ± | α ( t ) c | β 1 ( t ) | β 2 ( t ) | β 3 ( t ) ,
ρ NV ± ( t ) = N ± 2 [ | β 1 ( t ) β 1 ( t ) | | β 2 ( t ) β 2 ( t ) | | β 3 ( t ) β 3 ( t ) | + | β 1 ( t ) β 1 ( t ) | | β 2 ( t ) β 2 ( t ) | | β 3 ( t ) β 3 ( t ) | ± q ( t ) | | β 1 ( t ) β 1 ( t ) | | β 2 ( t ) β 2 ( t ) | | β 3 ( t ) β 3 ( t ) | ± q ( t ) | β 1 ( t ) β 1 ( t ) | | β 2 ( t ) β 2 ( t ) | | β 3 ( t ) β 3 ( t ) | ] ,
EW = Tr ( W ^ GHZ ρ NV + ( t ) ) = 1 4 + 4 e 2 | α | 2 [ p 1 2 ( t ) + p 2 2 ( t ) + p 3 2 ( t ) p 1 2 ( t ) p 2 2 ( t ) p 1 2 ( t ) p 3 2 ( t ) p 2 2 ( t ) p 3 2 ( t ) 2 [ q ( t ) + p 1 ( t ) p 2 ( t ) p 3 ( t ) ] 1 p 1 2 ( t ) 1 p 2 2 ( t ) 1 p 3 2 ( t ) ] .
EW ( p ( t ) ) = [ 1 p 2 ( t ) ] 4 + 4 e 2 | α | 2 [ 3 p 2 ( t ) 2 [ q ( t ) + p 3 ( t ) ] 1 p 2 ( t ) ] .
W ^ GHZ = 1 2 I I I ( σ z I I ) | GHZ GHZ | ( σ z I I )
EW = Tr ( W ^ GHZ ρ NV ( t ) ) = 1 4 4 e 2 | α | 2 [ p 1 2 ( t ) + p 2 2 ( t ) + p 3 2 ( t ) p 1 2 ( t ) p 2 2 ( t ) p 1 2 ( t ) p 3 2 ( t ) p 2 2 ( t ) p 3 2 ( t ) 2 [ q ( t ) p 1 ( t ) p 2 ( t ) p 3 ( t ) ] 1 p 1 2 ( t ) 1 p 2 2 ( t ) 1 p 3 2 ( t ) ] .
EW ( p ( t ) ) = [ 1 p 2 ( t ) ] 4 4 e 2 | α | 2 [ 3 p 2 ( t ) 2 [ q ( t ) p 3 ( t ) ] 1 p 2 ( t ) ] .
F ± = 1 4 ± 4 q ( t ) p 1 ( t ) p 2 ( t ) p 3 ( t ) [ 1 + p 1 2 ( t ) p 2 2 ( t ) p 3 2 ( t ) ± 2 q ( t ) p 1 ( t ) p 2 ( t ) p 3 ( t ) + ( 1 p 1 2 ( t ) ) ( 1 p 2 2 ( t ) ) × ( 1 p 3 2 ( t ) ) + 2 [ q ( t ) ± p 1 ( t ) p 2 ( t ) p 3 ( t ) ] 1 p 1 2 ( t ) 1 p 2 2 ( t ) 1 p 3 2 ( t ) ] ,
F ± = 1 4 ± 4 q ( t ) p 1 ( t ) p 2 ( t ) p 3 ( t ) [ 1 + p 6 ( t ) ± 2 q ( t ) p 3 ( t ) + 2 ( q ( t ) ± p 3 ( t ) ) ( 1 p 2 ( t ) ) 3 / 2 + ( 1 p 2 ( t ) ) 3 ] .
H T = H i κ 2 a a ,
a ( t ) = e i ( ω i κ / 4 ) t [ cos ( G 2 ( κ / 4 ) 2 t ) a ( 0 ) + i κ / 4 G 2 ( κ / 4 ) 2 sin ( G 2 ( κ / 4 ) 2 t ) a ( 0 ) i 1 G 2 ( κ / 4 ) 2 × sin ( G 2 ( κ / 4 ) 2 t ) ( g 1 b 1 ( 0 ) + g 2 b 2 ( 0 ) + g 3 b 3 ( 0 ) ) ] ,
| ψ ± ( t ) = | α ( t ) c | β 1 ( t ) | β 2 ( t ) | β 3 ( t )
± | α ( t ) c | β 1 ( t ) | β 2 ( t ) | β 3 ( t ) .
α ( t ) = e i ( ω i κ / 4 ) t α [ cos ( G ζ t ) + i ( λ / ζ ) sin ( G ζ t ) ] ,
β i ( t ) = i e i ( ω i κ / 4 ) t α g i G ζ sin ( G ζ t )

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