Abstract
We describe a onestep, deterministic and scalable scheme for creating macroscopic arbitrary entangled coherent states (ECSs) of separate nitrogenvacancy center ensembles (NVEs) that couple to a superconducting flux qubit. We discuss how to generate the entangled states between the flux qubit and two NVEs by the resonant driving. Then the ECSs of the NVEs can be obtained by projecting the flux qubit, and the entanglement detection can be realized by transferring the quantum state from the NVEs to the flux qubit. Our numerical simulation shows that even under current experimental parameters the concurrence of the ECSs can approach unity. We emphasize that this method is straightforwardly extendable to the case of many NVEs.
Introduction
As one of the promising building blocks for solidstate quantum information processing (QIP)^{1,2,3,4,5,6}, the nitrogenvacancy (NV) center consisting of a substitutional nitrogen atom and an adjacent vacancy in diamond can be optically controlled and fast microwave manipulated with long coherence time even at room temperature^{7,8}. Additionally, benefiting from the enhancement of the magneticdipole coupling by a factor through the collective excitation, the strong coupling regime has been reached in the hybrid systems consisting of NV center ensemble (NVE) and superconducting circuit^{9,10,11,12,13,14,15,16,17}, which have recently attracted much attention. Several potential applications based on the collective excitation of the NVEs are investigated both theoretically and experimentally, such as quantum memories^{18,19,20,21}, continuous variable entanglement^{22,23}, quantum state transfer^{24} and quantum simulation^{25}.
The practical quantum information processing requires generation of multipartite entanglement among NVEs, which is usually quite complicated and timeconsuming. For example, to generate entanglement between two NVEs, we may employ a sequence of single and twoqubit operations, such as SWAP gates. In this way, either the gate number or the operation time increases much more quickly as the number of the NVEs increases. Therefore, to efficiently generate multipartite entanglement among NVEs, we need to find a faster method with less operation time and gates. To this end, we present in this article a scalable and tunable framework for generating macroscopic arbitrary entangled coherent state (ECS)^{26,27,28,29,30,31,32} among many separate NVEs in a deterministic way. The key point of our proposal is that the flux qubit plays the role of data bus and the collective excitations of the NVEs behave as bosonic modes or harmonic oscillators in the lowexcitation limit. Through the collective magnetic coupling and the in situ tunability of the flux qubit, we show that a macroscopic ECS of NVEs can be generated with high success possibilities even under the decoherence of the flux qubit and NVEs. More importantly, we consider the magnetic field applied along a direction [100] of the NV centers^{18} to suppress the broadening of the ODMR spectral lines due to excitations in four crystalline orientations. This can largely reduce the overheads in gate operations with the increment of the number of NVEs. So our scheme not only allows an efficient way to accessing the largescale continuous variable quantum computing in manybody systems, but also serves as a critical building block towards scalable architectures based on solidstate QIP. It should be noted that the ECS owns unique quantum characteristics^{33,34,35} (e.g., robustness to singleparticle decoherence) and can be used for various applications beside quantum information processing^{36}, ranging from the fundamental quantum information theory^{37} and the implementation of quantum communication^{38,39} to quantum metrology^{40}. The quantum error correction may also be easily implemented with the logical qubit defined by the superposition of the coherent states^{41,42}.
The remainder of this paper is organized as follows. We present our system architecture in details in Sec. II, and the preparation and readout of the ECS of NVEs in Sec. III. Sec. IV focuses on the influence from the decoherence effects of the system. Discussion is presented in Sec. V. Finally, we conclude our findings in Sec. VI.
Results
System and Model
As schematically shown in Fig. 1, we consider two diamond NVEs glued near the boundary of a big superconducting gaptunable flux qubit, which consists of four Josephson junctions characterized by Josephson energies E_{J} and αE_{J}, respectively. Near each of the NVEs, there are small flux qubits employed for detection, which are initially tuned to be uncoupled with both the NVEs and the big flux qubit. When the biasing of the main loop is close to onehalf of a flux quantum Φ_{0} = h/2e, the gaptunable flux qubit can be treated as an effective twolevel system described by the Hamiltonian (in units of ħ = 1 throughout the paper) ^{18,43}, where σ_{z} and σ_{x} are usual Pauli operators in the basis of flux eigenstates 0〉_{f}(clockwise persistent current) and 1〉_{f}(counterclockwise persistent current). Here, the energy splitting of the qubit is given by ħε(Φ_{ext}) = 2I_{p}(Φ_{ext}−Φ_{0}/2) with I_{p} the persistent current in the flux qubit. is the energy of the tunnel splitting, which induces flip of the qubit states^{44,45}. Note that ε(Φ_{ext}) and can be controlled independently by the external magnetic flux threading the main loop and α loop, respectively, through two extra microwave lines for a finite time period, e.g., of the order of nanosecond^{46,47,48}. This enables us to couple the flux qubit to the NVE at the optimal point where the flux qubit has its longest coherence time. In our case we tune ε(Φ_{ext}) to be zero, and tune to be equal to NVEs' transition frequency. In this way, the flux qubit basis {0〉_{f}, 1〉_{f}} is defined in the eigenstates of the Pauli operator σ_{x}. Additionally, a resonantly driven microwave on the flux qubit with Rabi frequency Ω_{d} is also required to control the dynamics of the system.
For the spintriplet ground state of a NV center with a zerofield splitting D_{gs} = 2.87 GHz between m_{S} = 0 and the nearly degenerate sublevels m_{S} = ±1〉^{49}, we apply an external magnetic field along the crystalline direction [100] of the NV centers to split the degenerate sublevels m_{S} = ±1〉, which results in a twolevel system denoted by g〉 = ^{3}A, m_{s} = 0〉 and e〉 = ^{3}A, m_{s} = −1〉, respectively^{16}. So the Hamiltonian of the NVEs is given by where the last term represents the Zeeman splitting under the magnetic field B_{z} (i.e., the part of the external magnetic field along the crystalline directions). E is the straininduced splitting, g_{e} is the groundstate Lander factor, µ_{B} is the Bohr magneton, and N_{j} is the number of the NV centers involved in the jth spin ensemble. To have a good twolevel approximation for the NV centers, the diamond crystal is assumed to be bonded on top of the flux qubit chip with its [001] surface facing the chip^{18}. , , and are spin1 Pauli operators for the ith NV center in the jth NVE. Furthermore, we consider the case where the straininduced finestructure splitting is negligible compared to the Zeeman splitting, i.e., ^{9,10}, and thereby the second term in H_{NV E} can be neglected. Considering the subspace spanned by the states m_{S} = 0〉 and m_{S} = −1〉, we have the total Hamiltonian for the whole system as^{17} where ω_{j} = D_{gs} − g_{e}µ_{B}B_{z}, the last term is the interaction between the NVE and the flux qubit, and B^{f} is the magnetic field (along the direction [001] of the NV centers) produced by the flux qubit. , and are the Pauli operators for the ith NV center in the jth spin ensemble. Actually, the additional magnetic field B^{f} is attributed to the superposition state of the clockwise and counterclockwise persistent currents. Ω_{d} (ω_{f}) is the Rabi frequency (frequency) of the additional microwave resonantly driving on the flux qubit.
Using the HolsteinPrimakoff (HP) transformation^{50}, we map the spin operators to the bosonic operators as follows: , and , where the operators b_{j} and obey the standard bosonic commutator in the weak excitation limit . Here we denote the Fock state basis of the bosonic mode b_{j} as , for j = 1, 2. The coherent state of the mode b_{j} is defined as αα〉_{j} = b_{j}α〉_{j}. Then effective Hamiltonian on the basis of the HP transformations above becomes where is the effective coupling strength between the NVE and the flux qubit. In the case of and , the Hamiltonian in the basis of the eigenstates of the flux qubit takes the form , where the rotatingwave approximation (RWA) is adopted to neglect fast oscillating terms. The ladder operators σ_{±} are defined as σ_{±} = (σ_{z} ± iσ_{y})/2.
Preparation and readout of the ECS
In what follows, we describe how to generate deterministically the ECS of two spin ensembles in our setup by following three steps:
(i) The system is initially prepared in the state ψ(0)〉 = 0〉_{f} 0〉_{1} 0〉_{2}. Assuming that the splitting frequency of the flux qubit and the frequency ω_{f} of the driving field are set to be equal to the NVE collective excitation frequency ω_{j}, the effective Hamiltonian in the rotating frame becomes . At this step, Ω_{d}(t) in Eq. (3) is set to be zero.
(ii) Turning on the drive Ω_{d}(t) from zero to Ω_{d} () on the time scale of nanoseconds, we can switch from the flux qubit basis {0〉_{f}, 1〉_{f}} to the dressed state basis {+〉_{f}, −〉_{f}}, with , and the evolution operator takes the form after neglecting the fast oscillating terms, with , and is the rotating unitary operator. This operation U(t) displaces the collective bosonic mode of NVE by the amount of ±iG_{j}t/2 conditional on the flux qubit states 0〉_{f} or 1〉_{f}, and this complex amplitude iG_{j}t/2 is proportional to the evolution time. As a result, the system state ψ(0)〉 evolves into with α_{1} = −iG_{1}t/2 and α_{2} = −iG_{2}t/2. For simplicity we set the phase factor Ω_{d}t/2 = nπ (), which can be realized by precisely controlling the operation time. One can see the generated entanglement among the flux qubit and the two NVEs.
(iii) To generate the ECS of NVEs, we need to measure the state of the flux qubit on 0〉_{f} and 1〉_{f}. This task can be performed by applying a pulse sequence on a dc SQUID attached to the flux qubit (not shown in Fig. 1), where the voltage state of the SQUID is very sensitive to the faint change of the flux, and depends on the switching probability of the energy eigenstates of the flux qubit^{16}. So the two NVEs will be projected into the ECS ψ_{−}〉 = α_{1}〉_{1} α_{2}〉_{2} − −α_{1}〉_{1} −α_{2}〉_{2} or ψ_{+}〉 = α_{1}〉_{1} α_{2}〉_{2} + −α_{1}〉_{1} −α_{2}〉_{2}, corresponding to the readout from the state 1〉_{f} or 0〉_{f} with respect to the flux qubit, respectively. This implies that each ECS is created with the success possibility of 100% if the projection measurement is perfect.
The detection of the ECS is not a trivial task. In order to measure the ECS, we need to transfer the states from the NVEs to two additional small flux qubits, each of which is attached on a NVE, as shown in Fig. 1. So the task of entanglement detection can be performed by the direct measurement on the states of flux qubits. Here the qubit states in NVE are defined as , and with j = 1, 2, respectively^{36}. It is easy to verify that and are orthogonal to each other once α_{j}> 0. In the limit of , the coherent qubit states can be approximated as a superposition of Fock states and . Therefore, in the case of , we can transfer the state from NVEs to flux qubits in the Fock state basis using the SWAP gate between the jth NVE and the jth flux qubit.
For a more general case of α_{j}^{2}> 1, the usual way to the state transfer from the jth NVE to the jth flux qubit, , can be performed by a sequence of singlequbit gates and the operation^{36} where the operator plays the role of flipping the flux qubit, achievable by first applying a singlequbit rotation and then by performing the operation on the flux qubit. Alternatively, the operations above can be accomplished by the following method. Suppose that the effective Hamiltonian between the NVEs and the small flux qubit is similar to Eq. (3) but without the microwave drive Ω_{d}. So the corresponding operation is , where , and Δ_{j} is the detuning between the transition frequencies of the jth small flux qubit and the jth NVE. Additionally, the operation with ε_{j} = π/(2α_{j}) is actually the conditional displacement of the bosonic mode if and only if the flux qubit is in the excited state 1〉_{fj}. Such conditional displacements have been achieved by Eq. (3) with a strong microwave drive. Hence the operation U_{s} should be achievable with currently experimental technology, and then the entanglement between NVEs can be detected by measuring the states of small flux qubits.
We should note that the detection in our scheme requires the coupling strength of each NV center to the large flux qubit to be proportional to that to the small one, which guarantees the collective mode detected by the small flux qubit to be identical to that already prepared in the large flux qubit. Although it looks like an additional technical difficulty, the operations are experimentally doable. For example, the NVEs can be placed to the positions where the magnetic fields generated by both the small and the large flux qubits are nearly homogenous due to careful design of the qubits' circuit patterns^{51}.
Discussion
To visualize the decoherence effect on the evolution of concurrence between the NVEs, we have plotted the timedependent C_{12}(t) in Fig. 2. Due to the presence of decoherence, the concurrence increases first and then decreases gradually to zero. So to carry out our scheme more efficiently, we have to suppress these imperfect factors as much as we can. On the other hand, the concurrence keeps to be of very high values (≥ 0.95) under the decoherence as long as the time is evaluated within the domain [2/G, 2.5/G], and the maximal entanglement () can be obtained if the operation time . We emphasize that a stronger decoherence only slightly reduces the maximal value of the concurrence, as shown in the bottom panel of Fig. 2.
In Fig. 3, we have exactly calculated the evolution of the system, where the fast oscillating terms are involved. We may compare the generated ECS ψ′(t)〉 in the nonRWA case with the ECS ψ(t)〉 generated by the effective unitary operator . Here, the fidelity is defined as F(t) = 〈ψ′(t)ψ(t)〉. As shown in Fig. 3, the fidelity F(t) decreases in time evolution, which is larger than 0.98 only for t_{0} ≤ 2/G and Ω/G = 30. Therefore, effective unitary operator U(t) is valid within this regime, and the generated ECS has the amplitude α = β = it_{0}G/2 = 1 at time t_{0}, which is large enough to define the orthogonal qubit. In practice, if we choose G = 25 MHz, the microwave drive is Ω = 750 MHz, which is within the reach of experimental feasibility^{55}.
The condition on HP transformation for the NVEs requires that the total number of the excitations be much smaller than the number of the NV centers in each NVE. In our case, the average photon number in the two NVEs can be calculated as where ξ = e^{− γt} + e^{γt} − 2qP_{1}P_{2} cos(Ω_{d}t). Considering the case of G_{1} ≠ G_{2}, we set G_{2} = (1 + Δ)G_{1}. As shown in Fig. 4, we plot the total photon number N = N_{1} + N_{2} as a function of the parameter Δ and time. One can find that the value of N is smaller than 20. So the small excitation number ensures a reasonable HP transformation in our scheme.
We address some remarks for experimental implementation of our scheme. In our proposal, the flux qubit is strongly driven and the qubit basis is changed to dressed states +〉 and −〉. This is equivalent to the continuous dynamical decoupling^{21}. Based on recent experiments^{16}, the NVEflux qubit coupling strength has reached up to 70 MHz. In our proposal, if we set G_{1} = G_{2} = 50 MHz and , the operation time for obtaining the ECS of NVEs with high values of concurrence is about 40 ns, which is much shorter than the coherence times of the NV centers and the flux qubit. In addition, the decoherence of the NVE increases with the density of the NV centers. Thus, it is required to suppress the decoherence of the NVE by the spinecho pulses and/or by improving the conversion rate from nitrogen to NV which decrease the redundant nitrogen spins and thereby reduce the linewidth of the NVEs.
As a final remark, we emphasize that the setup plotted in Fig. 1 is straightforwardly extendable to the case with many NVEs involved, which would be essential to scalable QIP. For a more specific description, we have designed in Fig. 5 a novel tunable architecture that allows superconducting flux qubits to provide flexibility in arrangement of NVEs. In such a way, all spin ensembles are arranged in two or three spatial dimensions, where the flux qubit serves as a quantum bus for the noninteracting spin ensembles. The total Hamiltonian is given by with N the total number of NVEs. Eq. (6) can take an effective form in the dressed state basis by performing the similar transformations to the twoNVE case. As a result, all the NVEs will be projected into the ECS with the normalization factor Y, and α_{j} = −iG_{j}t/2 corresponding to the readout from the states 0〉_{f} and 1〉_{f} with respect to the flux qubit, respectively. We note that optical pulses are required to individually address the NVEs in the spatial scale. The present method would be very efficient for generating a continuousvariable entanglement among separate spin ensembles, which is a necessary step on the path towards a scalable solidstate quantum computing.
In summary, we have proposed a scheme of onestep creation of arbitrary macroscopic ECSs among many separate NVEs through the collective magnetic coupling to the flux qubit. We have shown that the ECSs of the spin ensembles can be achieved with high success possibilities even under the influence of decoherence from the flux qubit and NVEs. We have realized the worry about the large loop of the flux qubit which may cause more noise. However, we consider that the gaptunable flux qubit employed in our scheme can work at the optimal point, strongly suppressing the dephasing time. On the other hand, we have recently fabricated some large loop gaptunable flux qubits experimentally with the coherent time on the order of one microsecond, in which no significant drop was observed when the flux qubit loop size is increased from 4 × 5 µm^{2} to 20 × 20 µm^{2}. Considering the fact that we can use small pieces of NV sample with the size about 1 × 1 µm^{2}, we are confident of the feasibility of a multiqubit entanglement using a flux qubit with the loop size of 400 µm^{2}. Therefore we argue that the proposal is practical under the present experimental parameters, and should be helpful for largescale QIP in solidstate quantum systems, particularly for continuousvariable quantum computing and quantum communication.
Method
Modeling of decoherence effects
From now on we analyze the influence from decoherence of the flux qubit and the NVEs. Below we may focus on the decoherence effect in the step (ii) using the master equation where γ (Γ) is the decay rate of the flux qubit (NVE). For simplicity, we have assumed Γ_{1} = Γ_{2} = Γ. Using the superoperator technique^{52} and Hausdorff similarity transformation^{53}, we deduce the following differential equations with respect to the density matrix of the flux qubit, Then we solve the differential equations above with the initial state and obtain where is a factor reflecting the competition between the spinboson coupling and the dissipation.
After the step (iii), the final state of the NVEs, with the readout of the flux qubit being 0〉_{f}, turns to be where S = e^{−γt} + e^{γt} + 2qP_{1}P_{2} cos(Ω_{d}t) is the normalization constant with P_{1} = 〈α −α〉 and P_{2} = 〈β −β〉. Note that we can obtain the similar result if the readout of the flux qubit is 1〉_{f}.
We employ below the concept of concurrence for bipartite entangled nonorthogonal states^{54} to measure entanglement between the NVEs. To this end, we first make a suitable transformation on Eq. (11) from the nonorthogonal form of Eq. (11) into an orthogonal form by rebuilding two orthogonal and normalized states as the basis states of the twodimensional Hilbert space, i.e., we define 0〉_{1} = α〉_{1}, 1〉_{1} = (−α〉_{1} − P_{1} α〉_{1})/M_{1} with for NVE1, and 0〉_{2} = β〉_{2}, 1〉_{2} = (−β〉_{2} − P_{2}β〉_{2})/M_{2} with for NVE2. Then the reduced density operator (Eq. (11)) in the new basis states is rewritten as So we have the concurrence for the ECS in the form
References
 1.
Childress, L. et al. Coherent dynamics of coupled electron and nuclear spin qubits in diamond. Science 314, 281 (2006).
 2.
Dutt, M. V. G. et al. Quantum register based on individual electronic and nuclear spin qubits in diamond. Science 316, 1312 (2007).
 3.
Jiang, L. et al. Repetitive readout of a single electronic spin via quantum logic with nuclear spin ancillae. Science 326, 267 (2009).
 4.
Neumann, P. et al. Singleshot readout of a single nuclear spin. Science 329, 542 (2010).
 5.
Neumman, P. et al. Quantum register based on coupled electron spins in a roomtemperature solid. Nature Phys. 6, 249 (2010).
 6.
Togan, E. et al. Quantum entanglement between an optical photon and a solidstate spin qubit. Nature 466, 730 (2010).
 7.
Balasubramanian, G. Ultralong spin coherence time in isotopically engineered diamond. Nat. Materials 8, 383 (2009).
 8.
Mizuochi, N. Coherence of single spins coupled to a nuclear spin bath of varying density. Phys. Rev. B 80, 041201 (2009).
 9.
Xiang, Z. L., Ashhab, S., You, J. Q. & Nori, F. Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems. Rev. Mod. Phys. 85, 623 (2013).
 10.
Xiang, Z. L., Lü, X. Y., Li, T. F., You, J. Q. & Nori, F. Hybrid quantum circuit consisting of a superconducting flux qubit coupled to a spin ensemble and a transmissionline resonator. Phys. Rev. B 87, 144516 (2013).
 11.
Kubo, Y. et al. Strong coupling of a spin ensemble to a superconducting resonator. Phys. Rev. Lett. 105, 140502 (2010).
 12.
Schuster, D. I. et al. Highcooperativity coupling of electronspin ensembles to superconducting cavities. Phys. Rev. Lett. 105, 140501 (2010).
 13.
Amsüss, R. et al. Cavity QED with magnetically coupled collective spin states. Phys. Rev. Lett. 107, 060502 (2011).
 14.
Kubo, Y. et al. Hybrid quantum circuit with a superconducting qubit coupled to a spin ensemble. Phys. Rev. Lett. 107, 220501 (2011).
 15.
Kubo, Y. et al. Storage and retrieval of a microwave field in a spin ensemble. Phys. Rev. A 85, 012333 (2012).
 16.
Zhu, X. et al. Coherent coupling of a superconducting flux qubit to an electron spin ensemble in diamond. Nature 478, 221 (2011).
 17.
Marcos, D. et al. Coupling nitrogenvacancy centers in diamond to superconducting flux qubits. Phys. Rev. Lett. 105, 210501 (2010).
 18.
Saito, S. et al. Towards realizing a quantum memory for a superconducting qubit: Storage and retrieval of quantum states. Phys. Rev. Lett. 111, 107008 (2013).
 19.
Julsgaard, B., Grezes, C., Bertet, P. & Mølmer, K. Quantum memory for microwave photons in an inhomogeneously broadened spin ensemble. Phys. Rev. Lett. 110, 250503 (2013).
 20.
Yang, W. L., Yin, Z. Q., Hu, Y., Feng, M. & Du, J. F. Highfidelity quantum memory using nitrogenvacancy center ensemble for hybrid quantum computation. Phys. Rev. A 84, 010301(R) (2011).
 21.
Cai, J., Jelezko, F., Katz, N., Retzker, A. & Plenio, M. B. Longlived driven solidstate quantum memory. New. J. Phys. 14, 093030 (2012).
 22.
Yang, W. L., Yin, Z. Q., Chen, Q., Chen, C. Y. & Feng, M. Twomode squeezing of distant nitrogenvacancycenter ensembles by manipulating the reservoir. Phys. Rev. A 85, 022324 (2012).
 23.
Ma, S. L., Li, P. B., Fang, A. P., Gao, S. Y. & Li, F. L. Dissipationassisted generation of steadystate singlemode squeezing of collective excitations in a solidstate spin ensemble. Phys. Rev. A 88, 013837 (2013).
 24.
Chen, Q., Yang, W. L. & Feng, M. Controllable quantum state transfer and entanglement generation between distant nitrogenvacancycenter ensembles coupled to superconducting flux qubits. Phys. Rev. A 86, 022327 (2012).
 25.
Yang, W. L. et al. Quantum simulation of an artificial Abelian gauge field using nitrogenvacancycenter ensembles coupled to superconducting resonators. Phys. Rev. A 86, 012307 (2012).
 26.
Sanders, B. C. Entangled coherent states. Phys. Rev. A 45, 6811 (1992).
 27.
Zhang, W. M., Feng, D. H. & Gilmore, R. Coherent states: Theory and some applications. Rev. Mod. Phys. 62, 867 (1990).
 28.
Chen, M. Y., Tu, M. W. Y. & Zhang, W. M. Entangling two superconducting LC coherent modes via a superconducting flux qubit. Phys. Rev. B 80, 214538 (2009).
 29.
Chen, Q., Yang, W. L. & Feng, M. Generation of macroscopic entangled coherent states for distant ensembles of polar molecules via effective coupling to a superconducting charge qubit. Phys. Rev. A 86, 045801 (2012).
 30.
Zhou, L. & Xiong, H. A macroscopical entangled coherent state generator in a V configuration atom system. J. Phys. B: At. Mol. Opt. Phys. 41, 025501 (2008).
 31.
Jia, L. J., Yang, Z. B., Wu, H. Z. & Zheng, S. B. Generation of cluster states for cavity fields. Chin. Phys. B 17, 4207 (2008).
 32.
Munro, W. J., Milburn, G. J. & Sanders, B. C. Entangled coherentstate qubits in an ion trap. Phys. Rev. A 62, 052108 (2000).
 33.
Lukin, M. D., Yelin, S. F. & Fleischhauer, M. Entanglement of atomic ensembles by trapping correlated photon states. Phys. Rev. Lett. 84, 4232 (2000).
 34.
Braunstein, S. L. & Kimble, H. J. Teleportation of continuous quantum variables. Phys. Rev. Lett. 80, 869 (1998).
 35.
Nielsen, M. A. & Chuang, I. L. Quantum computing and Quantum Information (Cambridge University Press, Cambridge, UK, 2000).
 36.
Munro, W. J., Milburn, G. J. & Sanders, B. C. Entangled coherentstate qubits in an ion trap. Phys. Rev. A 62, 052108 (2000).
 37.
Jeong, H., Paternostro, M. & Ralph, T. C. Failure of local realism revealed by extremelycoarsegrained measurements. Phys. Rev. Lett. 102, 060403 (2009).
 38.
Park, K. & Jeong, H. Entangled coherent states versus entangled photon pairs for practical quantuminformation processing. Phys. Rev. A 82, 062325 (2010).
 39.
Azuma, K. & Kato, G. Optimal entanglement manipulation via coherentstate transmission. Phys. Rev. A 85, 060303(R) (2012).
 40.
Joo, J., Munro, W. J. & Spiller, T. P. Quantum metrology with entangled coherent states. Phys. Rev. Lett. 107, 083601 (2011).
 41.
Leghtas, Z., Kirchmair, G., Vlastakis, B., Schoelkopf, R. J., Devoret, M. H. & Mirrahimi, M. Hardwareefficient autonomous quantum memory protection. Phys. Rev. Lett. 111, 120501 (2013).
 42.
Vlastakis, B. et al. Deterministically encoding quantum information using 100photon Schrödinger cat states. Science 342, 607 (2013).
 43.
Stern, M. et al. Flux qubits with long coherence times for hybrid quantum circuits. Phys. Rev. Lett. 113, 123601 (2014).
 44.
Chiorescu, I., Bertet, P., Semba, K., Nakamura, Y., Harmans, C. J. P. M. & Mooij, J. E. Coherent dynamics of a flux qubit coupled to a harmonic oscillator. Nature 431, 159 (2004).
 45.
Qiu, Y., Xiong, W., Tian, L. & You, J. Q. Coupling spin ensembles via superconducting flux qubits. Phys. Rev. A 89, 042321 (2014).
 46.
Zhu, X. B., Kemp, A., Saito, S. & Semba, K. Coherent operation of a gaptunable flux qubit. Appl. Phys. Lett. 97, 102503 (2010).
 47.
Paauw, F. G., Fedorov, A., Harmans, C. J. P. M. & Mooij, J. E. Tuning the gap of a superconducting flux qubit. Phys. Rev. Lett. 102, 090501 (2009).
 48.
Fedorov, A., Feofanov, A. K., Macha, P., Díaz, P. F., Harmans, C. J. P. M. & Mooij, J. E. Strong coupling of a quantum oscillator to a flux qubit at its symmetry point. Phys. Rev. Lett. 105, 060503 (2010).
 49.
Manson, N. B., Harrison, J. P. & Sellars, M. J. Nitrogenvacancy center in diamond: Model of the electronic structure and associated dynamics. Phys. Rev. B 74, 104303 (2006).
 50.
Hammerer, K., Sørensen, A. S. & Polzik, E. S. Quantum interface between light and atomic ensembles. Rev. Mod. Phys. 82, 1041 (2010).
 51.
Zhu, X. B. in preparation.
 52.
Faria, J. G. P. D. & Nemes, M. C. Dissipative dynamics of the JaynesCummings model in the dispersive approximation: Analytical results. Phys. Rev. A 59, 3918 (1999).
 53.
Witschel, W. Ordered products of exponential operators by similarity transformations. Int. J. Quantum Chem. 20, 1233 (1981).
 54.
Wang, X. G. Bipartite entangled nonorthogonal states. J. Phys. A 35, 165 (2002).
 55.
Yoshihara, F., Nakamura, Y., Yan, F., Gustavsson, S., Bylander, J., Oliver, W. D. & Tsai, J. S. Flux qubit noise spectroscopy using Rabi oscillations under strong driving conditions. Phys. Rev. B 89, 020503(R) (2014).
Acknowledgements
The authors thank JunHong An and Naheed Akhtar for enlightening discussions and for their serious reading of the manuscript before submission. This work is supported partially by the National Fundamental Research Program of China under Grant No. 2012CB922102 and No. 2013CB921803, by the NNSF of China under Grants No. 11274351, No. 11274352, No. 11104326. ZQY was supported by the NBRPC (973 Program) 2011CBA00300 (2011CBA00302), NNSFC 11105136, 61361136003, and 61033001.
Author information
Affiliations
State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China
 Wanlu Song
 , Wanli Yang
 , Fei Zhou
 & Mang Feng
University of the Chinese Academy of Sciences, Beijing 100049, China
 Wanlu Song
Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, China
 Zhangqi Yin
The Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
 Xiaobo Zhu
Collaborative Innovation Center of Quantum Matter, Beijing 100190, China
 Xiaobo Zhu
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W.L.Y., Z.Q.Y. and X.B.Z. conceive the idea. W.L.S., Z.Q.Y. and W.L.Y. carry out the research with input from X.B.Z. and F.Z. W.L.S., Z.Q.Y., W.L.Y., X.B.Z., F.Z. and M.F. discuss the results. W.L.S., W.L.Y. and Z.Q.Y. write the manuscript with comments and refinements from X.B.Z. and M.F.
Competing interests
The authors declare no competing financial interests.
Corresponding authors
Correspondence to Wanli Yang or Xiaobo Zhu or Mang Feng.
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Further reading

Macroscopic tripartite entanglement of nitrogenvacancy centers in diamond coupled to a superconducting resonator
Journal of the Optical Society of America B (2019)

Quantum information processing with nitrogen–vacancy centers in diamond
Chinese Physics B (2018)

Witnessing quantum entanglement in ensembles of nitrogen–vacancy centers coupled to a superconducting resonator
Optics Express (2018)

Generating maximallypathentangled number states in two spin ensembles coupled to a superconducting flux qubit
Physical Review A (2018)

Tunable coupling of spin ensembles
Optics Letters (2018)
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