Abstract

By exchange of virtual microwave photon induced by a transmission line resonator, the nonlinear interaction between a nitrogen-vacancy-center ensemble (NVE) and a superconducting charge qubit is achieved in circuit quantum electrodynamics, where the nonlinear coupling results from the second order of the coupling between the magnetic field of the transmission line resonator and the charge qubit. In our case, the nonlinear coupling can be much enhanced by a factor of the total spin number in the NVE. As an application, we present a potentially practical scheme to realize the squeezing of the NVE using the nonlinear coupling, which is within reach of the currently available technology.

© 2015 Optical Society of America

1. Introduction

In quantum optics, quantum nonlinear effects are indispensable for investigating and controlling the dynamics of harmonic oscillators [1]. Especially, the giant nonlinearities would open up many potential applications, such as generating nonclassical field states [2], squeezing and parametric amplification, [1, 35], which have been regarded as important resources for universally processing quantum information with continuous variables [6]. Therefore, it is highly desirable to develop new quantum hybrid systems to realize strong nonlinear couplings [511], especially in the solid-state systems [515].

Recently, the strong coupling regime has been experimentally implemented between super-conducting qubit (SQ) and transmission line resonator (TLR) system [1618], and many important quantum information processes (QIPs) [19], including the coupling of Josephson qubits by the TLR as a quantum data bus [20] and the preparation of the TLR Fock states [18], have been demonstrated. Based on these experimental advances, there are various of interests in investigating nonlinear interactions in circuit QED systems [2125]. For example, a phase-preserving amplifier based on the nonlinear interaction was proposed and realized to improve its performance in the regime near the quantum limit [23]. By using a single artificial multilevel Cooper pair box molecule, one may achieve giant self-Kerr effect at microwave frequencies [12]. Additionally, there are promising efforts to realize squeezed states of microwaves by using three-wave mixing coupling in circuit QED [4, 5].

The nitrogen-vacancy defect center (NV) in diamond is of particular interest because it offers unprecedented tunability and scalability, which are easily integrated with other quantum systems [26, 27]. Moreover, it is optically controllable and has extremely long nuclear and electronic decoherence time [28]. Due to the enhanced magnetic-dipole coupling through the collective excitation of the spins in NV ensemble (NVE) involving large numbers of NV centers with N = 1011 ∼ 1013 [31], much effort has been devoted recently to combine the NVE with circuit quantum electrodynamics (QED) both experimentally and theoretically [3138]. There are many significant progresses in simulating quantum optics phenomena and implementing quantum information processing, such as quantum memories and continuous variable entanglement [3436, 38]. However, no nonlinear coupling in circuit QED has been considered yet in the combined NVE-circuit QED hybrid systems.

Here we focus on the weak quadratic coupling between the SQ and TLR, and realize a new kind of nonlinear interaction between spin ensemble and SQ in the hybrid system consisting of a NVE, a SQ and a TLR. By virtual excitation of microwave photons, which acts as a quantum bus, an effective nonlinear coupling between the NVE and the SQ is generated in the dispersive limit. More intriguingly, we consider the total spin number of NV centers as N = 1011 ∼ 1013 [31, 32], where the nonlinear NVE-SQ coupling actually benefits from the collective enhancement. Physically, as long as the ensemble has only few excited spins, the excitation behaves like a harmonic oscillator degree of freedom in the low-excitation regime. Therefore, a pair of bosons in the NVE is excited when the SQ decays from the excited state to the ground state and vice versa. On the other hand, nonlinear optical processes with the χ(2) terms can also be simulated successfully. The squeezed state of the NVE can be generated in this manner of the hybrid system. Comparing with the previous schemes with the nonlinear coupling of solid-state systems [4, 5], our scheme focuses on a new quantum system (involving NVE) with nonlinear coupling and our nonlinear strength can be enhanced using currently available technology. We will discuss all sources of decoherence and dissipation of the hybrid system in our following description.

2. The model

As schematically shown in Fig. 1, we consider a hybrid setup consisting of a NVE, a TLR and a SQ. The NVE contained in a single diamond crystal is placed at the antinodes of the magnetic field of the full-wave mode of the TLR with the geometric length L0, distributed inductor L, and capacitance C. Similarly, the SQ interacts with the TLR by the magnetic flux through the loop enclosed in the SQ. The state of the SQ can be separately controlled by the gate voltage Vg through a gate capacitance Cg. The combined NVE-TLR-SQ system we study is governed by the Hamiltonian

Htot=HNVE+HFQ+HTLR+HSTI+HNTI,
in which HNVE, HFQ and HTLR are the Hamiltonians of NVE, FQ and TLR, respectively. HSTI corresponds to the interaction between the SQ and TLR, and HNTI represents the coupling between the NVE and the TLR.

 figure: Fig. 1

Fig. 1 (a) Schematic of the hybrid system, where the NVE is placed in the antinode of the TLR coupled to the SQ through magnetic field induced by its quantized current. (b) The level structure of the NVE.

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The ground state of a NV is spin triplet with a zero-field splitting Dgs = 2.87 GHz between ms = 0 and the nearly degenerate sublevels ms = ± 1 in the absence of external magnetic field [39]. In our proposal, the logic states |g〉 and |e〉 of the NVE are denoted by the states |3A, ms = 0〉, and |3A, ms = −1〉, respectively. So the Hamiltonian of a single NVE reads (in units of ħ = 1),

HNVE=δSz,
where δDgsγeB0 is the energy gap between the ground state sublevels ms = 0 and ms = −1, with the external magnetic field B0 and γe the the gyromagnetic ratio of an electron. Sz=12j=1Nτjz is the collective spin operator for the NVE with the Pauli spin operators τz = |e〉〈e| − |g〉〈g|, τ+ = |e〉〈g| and τ = |g〉〈e|. By using the definition of the collective operators of the NVE, we can easily find that these operators satisfy the SU(2) angular momentum commutation relations [Sz, S±] = ±S± and [S+, S] = 2Sz, where the Casimir invariant is S2 = N(N/2 + 1)/2.

In the present system, a superconducting quantum interference devices (SQUID) is employed, in which the SQ is coupled to a large superconducting reservoir through two identical Josephson junctions with capacitance CJ and Josephson coupling energy EJ. The SQ with zero and one extra Cooper pair in the superconducting island represented by the charge states |0〉 and |1〉, respectively, can act as an effective two-level artificial atom. The SQUID configuration of the SQ allows one to apply external flux Φe to control the total effective Josephson coupling energy. In the charge representation, the Hamiltonian of the SQ reads [40]

HSQ=12EC(12ng)σzEJcos(πΦeΦ0)σx,
where EC=2e2CΣ is the single Cooper-pair charging energy of the SQ, and ng=CgVg(2e) is the amount of gate charge induced by the gate voltage Vg with CJ Jopsephson junction capacitance and Cg gate capacitance. CΣ = 2CJ + Cg is the total capacitance, Φ0 = π/e is the flux quantum. σz = |1〉〈1| − |0〉〈0|, σx = |1〉〈0| + |0〉〈1| are the usual Pauli operators. The state of the SQ can be separately controlled by the gate voltage Vg through a gate capacitance Cg.

The coupling between the SQ and the TLR comes from the quantized external flux through the total effective area S of the SQUID. The SQUID is also integrated at the antinode of the magnetic field in the transmission-line resonator, where the electric field is zero or can be neglected. The quantized flux induced by the quantized current Iq of the TLR is

Φq=μ0SIq2πd
with d being the distance between the SQ and the TLR. At the antinode, the current in the TLR reaches its maximal amplitude Iq=iΣkI0(k)(akak), where ak is the kth mode of the TLR, I0(k)=ωkLl [19] with l being the length of the TLR and ωk the frequency of the TLR. Considering that the SQ is placed at the point where the amplitude of the quantized current takes its maximal value, we can achieve the quantized flux as
Φq=ikϕk(akak),
ϕk=μ0SI0(k)2πd.
There is only one mode that couples to the SQ and the quantized flux felt by the SQUID becomes
Φq=iϕb(aa).
With the total biased flux, the effective Josephson coupling energy of the SQUID becomes
EJ(Φq)=EJcos(πΦe+ΦqΦ0).
Thus we could achieve the system combining the SQ and TLR by the Hamiltonian H1 = HSQ + HTLR + HSTI as
H1=ωaaa12EC(12ng)σzEjcos(πΦe+ΦqΦ0)σx.
It is obvious that the Josephson coupling energy in Eq. (9) is a nonlinear function of Φq. At some external biased points, for instance, Φe = mΦ0, with m an arbitrary integer, the Josephson coupling energy in Eq. (9) becomes (−1)m+1EJ cos(πΦq0). If we use the eigenenergy basis of the SQ Hamiltonian to simplify the Hamiltonian in Eq. (9) and expand the Josephson coupling energy to the second order in Φq0, Eq. (9) is simplified to (in units of ħ = 1),
H2=12Ωσ˜z+ωaaa+η(a2σ˜+a2σ˜+),
where the transition frequency of the SQ and the coupling coefficient between the TLR and the SQ in the nonlinear interaction regime are
Ω=Ec2(12ng)2+4EJ2,
η=(1)m+1EJπ2ϕb22Φ02cosθ,
respectively. a (a) is the annihilation (creation) operator of the full-wave mode of the TLR with the equilibrium frequency ωa. We have used in Eq. (10) the new basis
|0˜=cos(θ2)|0+sin(θ2)|1,
|1˜=sin(θ2)|0+cos(θ2)|1
with the mix angle θ = tan−1[(−1)m2EJ/(Ec(1 − 2ng))]. Therefore, the corresponding Pauli operators are σ̃z = |1̃〉〈1̃| − |0̃〉〈0̃|, σ̃+ = |1̃〉〈0̃| and σ̃ = |0̃〉〈1̃|. Equation (10) includes an important nonlinear interaction which could be applied for an outstanding task in quantum mechanics and quantum optics, namely generating squeezed states [4, 5, 10, 21, 22]. Here we notice that this nonlinear coupling is very weak,which is reported as η = 2π × 0.25 kHz [21].

On the other hand, all the spins in NVE interact symmetrically with a single mode of the electromagnetic field because the mode wavelength is larger than the spatial dimension of the NVE if the spin ensemble is placed near the antinode of TLR’s field. The dynamics of the magnetic-dipole coupling between the TLR and the NVE is governed by the corresponding Hamiltonian

HNTI=g(S+a+Sa),
which is a Jaynes-Cummings-type interaction with g the interaction strength between the NVE and the photon. Then the Hamiltonian of the whole system turns to be
HTot=12Ωσ˜z+ωaaa+δSz+η(a2σ˜+a2σ˜+)+g(S+a+Sa).
The last two terms of Eq. (16) shows two kinds of interactions, one of which is the quadratic coupling between the SQ and the TLR and the other is linear coupling between the NVE and the TLR. In following discussions, we mainly focus on the dispersive interaction with adiabatic elimination method by using a canonical transformation—the Fröhlich-Nakajima (FN) transformation [41]. In order to get to the effective nonlinear interaction between the NVE and SQ, the degrees of freedom of the TLR are eliminated. We will also show how this effective nonlinear interaction strength is enhanced by the factor of the total spin numbers in the NVE. As an application, we intend to use the nonlinear coupling to realize the squeezing of the NV, which is within reach of the currently available technology.

3. Enhanced strong nonlinear interaction

In terms of normalized collective operators S˜+=1Nj=1Nτjz, we notice that the collective spin operator + can create symmetric Dicke excitation states |nP, and we encode the qubit of the NVE as

|0P=|g1g2gN
|1P=S˜+|gNV=1Nk=1Nτ+k|g1g2gN.
Hence, in this form we end up again with a two-level spin system, where strong coupling can be achieved for sufficiently large N. Thus we can rewritten the Eq. (16) as
HTot=12Ωσ˜z+ωaaa+δS˜+S˜+η(a2σ˜+a2σ˜+)+Ng(S˜+a+S˜a).
In the dispersive regime, we consider the off-resonant case with the following conditions
ΔS=|ωaδ|Ng,
ΔN=|2ωaΩ|η,
which are also requisite by the usual second-order perturbation theory. The microwave photon is virtually excited under this situation. In terms of the FN transformation as H˜Tot=URHTotUR=exp(F)HTotexp(F), where F is the corresponding anti-Hermitian operator F for FN transformation adopts the following form
F=NgΔS(aS˜aS˜+)+ηΔN(a2σ˜a2σ˜+).
Further transformation using the Baker-Campbell-Hausdorff formula exp[F]HTotexp[F]=HTot+[HTot,F]+12![[HTot,F],F]+, expands the Hamiltonian Tot to the second order in Ng/ΔS and ηN. Since the cavity mode only supports the virtual excitation of photons, we can neglect the terms with microwave photons and achieve an effective Hamiltonian
H˜Tot12Ωσ˜z+(ω0+χ)S˜+S˜+iλeff(S˜+2σ˜S˜2σ˜+),
in which photon-induced dynamic energy shift of the SQ is neglected due to being a small quantity. Equation (23) shows clearly that, although no energy exchanges with the NVE, the microwave photon has still acted as a bus, generating long-range spin-spin interactions and the nonlinear NVE-SQ coupling. Here,
χ=Ng2ΔS
is the total photon-assisted long-range spin-spin interaction.
λeff=Ng2ηΔS(1ΔS+1ΔN)
is the nonlinear NVE-SQ coupling constant. This nonlinear coupling λeff (+2σ̃ + 2σ̃+) in the Hamiltonian Tot, to our knowledge, is new in quantum optics and condensed matter physics. For a single spin of NV, we know that τ+2=0 and τ2=0. Therefore, our achieved Hamiltonian only describes the coupling of the SQ to the collective excitation of the NVE.

When the SQ decays from the excited state to the ground state, two collective excitations of the NVE are achieved and vice versa. For large ensembles with few excitations our scheme is closely related to the bosonization procedure [42]. we find that for few excitations it is possible to identify + = c and = c with bosonic creation and annihilation operators. By using the Holstein-Primakoff (HP) transformation [43] as

S+Nc,SNc,SzccN2,
where the operator c (c) fulfills the approximate bosonic commutation relation [c, c] ≃ 1 in the low-excitation case, namely, in the limit of large N and low excitations, ccN. It means that the collective excitation governed by S± behaves like a harmonic oscillator degree of freedom. Thus, the Hamiltonian Tot can be rewritten as
H˜Tot12Ωσ˜z+δCcc+λeff(c2σ˜+c2σ˜),
where δC = ω0 + .

From the effective Hamiltonian H̃′Tot, one can find that this nonlinear coupling strength λeff is in direct proportion to the spin number N in ensemble. It implies that the coupling strength λeff can be enhanced greatly by increasing the value of N, as shown below. Physically, the effective Hamiltonian H̃′Tot describes an interaction between the SQ and the pair of bosons excited from the NVE. The interaction represents that a pair of bosons is excited when the SQ decays from the excited state to the ground state and vice versa. This is also the reason why the coupling strength is in direct proportion to the spin number N in ensemble, which is the nonlinear effect different from traditional linear coupling proportion to N in ensemble [3133]. This kind of nonlinear interaction might be useful for processing quantum information with continuous variables under strong coupling [6]. Moreover, the effective Hamiltonian H̃′Tot also shows that the nonlinear optical processes with the χ(2) term can be simulated successfully. Therefore, the quantum operations, such as squeezing, can be generated in our proposed hybrid systems. The details will be discussed in the next section.

To estimate the strength of the coupling between the NVE and the SQ, we consider currently available experimental parameters. Ω is allowed to be tunable in the range 2π × 103 ∼ 1010 Hz [19] and the energy gap of the NVE could be also adjusted by external magnetic field. In the nonlinear interaction regime, the coupling strength is η = 2π × 0.25 kHz [21]. On the other hand, a single diamond crystal (3 × 3 × 0.5 mm3) containing the NVE is glued on top of TLR, with the distance to the silicon substrate less than 0.5 μm to ensure a maximal spin-resonator coupling. With the increase of the spin number N, the effective SQ-NVE coupling strength will be enhanced rapidly. If the coupled number of the NV centers in NVE is 1012, the coupling between a single NV center and the TLR is 2π × 11 Hz and Ng/ΔS=η/ΔN=0.1. So we can estimate our nonlinear indirect interaction between the NVE and the SQ to be λeff=Ng2ηΔS(1ΔS+1ΔN)(Ng)2ΔS(ηΔN)=0.01Ng, and obtain λeff ≃ 2π × 0.1 MHz.

4. Squeezing

As discussed in the section III, the nonlinear optical processes with the χ(2) term in Eq. (27) means that the essential quantum operations such as squeezing and parametric amplification can be implemented in our proposed hybrid SQ-TLR-NVE system. If we choose Ω = 2δC and shift to the rotating frame defined by 12Ωσ˜z+δCcc, we obtain the following effective Hamiltonian

H˜Riλeff(c2σ˜+c2σ˜).
We now consider a spin-echo-like process in which we let the system evolve under the Hamiltonian in (28) for short periods of time Δt [5]. Between each of such time interval, we apply a quick π-pulse σ̃x to flip the state of the charge qubit, so that the evolution for two periods can be simplified as eiHsqt with Hsq = λeff (c2 + c2)σ̃x. If we initialize the charge qubit in the 〈σ̃x〉 = 1 state, and repeat this procedure for M times, the evolution operator on the state of the NVE becomes
S(κ)=iκ2(c2c2),
with κ = effΔt. This is a squeezing operator on the NVE with squeezing parameter κ. Assuming that the NVE initially in the state |0〉P and using the transformation S(κ)cS(κ) = ccoshκc sinhκ, we obtain the variances in the two quadratures X1 = (c + c)/2 and X2 = (cc)/2i,
ΔX1=X12X12=eκ2,
ΔX2=X22X22=eκ2.
As a result, the NVE evolves into the squeezed vacuum state
|κ=S(κ)|0=κ2(c2+c2)|0.
It shows that the operators or the spins are squeezed fully for any spin number.

In order to estimate the practical squeezing efficiency, we also have to consider the influence from the noise in this hybrid SQ-TLR-NVE system. In the following, we will study the influence from the phase fluctuation and the amplitude fluctuation on the squeezing process, respectively. We firstly consider the phase fluctuation with Θ being the fluctuations in the phase of the pumping field with the effective Rabi frequency Ωp.

From the Heisenberg equation [44], we can obtain the following stochastic multiplicative equation:

Ψ˙=(M0ΘM2)Ψ,
where
Ψ=[c+c+cc+c2(c+)2],M0=[02κ2κκ00κ00],M=[000010001].
With our specific initial condition 〈Ψ(0)〉 = {1, 0, 0}T, the Laplace-transform solution has the following exact solution:
c+c+cc+=2κ2πidzeztz(z+Θ)4κ2
=eΘt/2(Θsinh(ϰt/2)ϰ+cosh(ϰt/2))
and
c2=Ωκe2Θt2κ2Θ2/8+(4κ+5Θ)e(2κ32Θ)t16κ+4Θ+(4κ5Θ)e(2κ32Θ)t16κ4Θ,
where ϰ=Θ2+16κ2. From the above two equations, we obtain the following formulas for the variance of the Hermitian amplitudes with laser phase fluctuations:
ΔX1=eΘt/2[Θsinh(ϰt/2)+4ϰcosh(ϰt/2)]4ϰ+ξ
ΔX2=eΘt/2[Θsinh(ϰt/2)+4ϰcosh(ϰt/2)]4ϰξ,
where
ξ=2Θκe2Θt8κ2Θ2/2+(4κ5Θ)e(2κ32Θ)t32κ8Θ(4κ+5Θ)e(2κ32Θ)t32κ+8Θ.

In the limit Θ ≪ t−1 ≪ Ωp, where Ωp = κ/t, the variances in the two quadratures ΔX1 and ΔX2 are given by

ΔX1=(Θt2)e2Ωpt+e2Ωpt2,
ΔX2=eΩpt12Θt2.
Similarly, the influence of the amplitude fluctuation can also be calculated as
ΔX1=exp[2ΘtΩpt+2(eΘt1)]2,
ΔX2=exp[2Θt+Ωpt+2(eΘt1)]2,
where Θ is the fluctuations in the amplitude of the pumping field. We plot variances versus Ωpt under different cases in the Fig. 2. It is shown that there are minimums of ΔX1 and ΔX2 as shown in Fig. 2(a), which means that to observe the squeezing in a noisy environment, it is important to accurately tune the values of Ωpt. We also find that the phase fluctuation Θ affects the squeezed properties severely with increasing time t. On the other hand, in Fig. 2(b), the amplitude fluctuation Θ influences the squeezing efficiency nearly negligibly when Θ′/Ωp ≤ 0.2.

 figure: Fig. 2

Fig. 2 ((a) ΔX1 versus Ωpt under the different cases, where the soild red, dotted blue, dashed green, and dotdashed pink lines denote Θ/Ωp = 0, Θ/Ωp = 0.001, Θ/Ωp = 0.01, and Θ/Ωp = 0.1, respectively. The inset shows the ΔX2 versus Ωpt. (b) ΔX′1 versus Ωpt under the different cases, where the solid red, dotted blue, dashed green, and dotdashed pink lines denote Θp = 0, Θp = 0.1, Θp = 0.2, and Θp = 0.5, respectively. The inset shows the ΔX′2 versus Ωpt.

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Additionally, all sources of decoherence and dissipation of both the NVE and SQ should be considered. In current experiments, the electron spin relaxation time T1 of the NV ranges from 6 ms at room temperature [45] to sec at low temperature. In addition, the dephasing time T2 = 350 μs induced by the nearby nuclear-spin fluctuation has been reported [46]. But the latest experimental progress [47] with isotopically pure diamond sample has demonstrated a longer T2 with T2 = 2 ms. Current conservative experimental parameters of decoherence of the SQ show the relaxation rate and dephasing rate of the qubit as γq = 1 MHz and γϕ = 10 MHz [1618], respectively. On the other hand, the decoherence of the qubit brings insignificant affect on the squeezing efficiency due to the fact that the SQ colud always stay in the ground state with 〈σz〉 = 1 during the squeezing process [5, 22].

In the present hybrid system, the main obstacle on obtaining large squeezing efficiency comes from the decay rate of NVE [22]. Considering the typical decoherence, which is from the nuclear spins of 13C defects in the NVs, we can use isotopically purified 12C diamond through the purification technique [47]. The dipole-interaction between the redundant nitrogen spins and the NVs is another major decoherence. There are two different ways to reduce the influence. One is to improve the conversion rate which would reduce the linewidth of the NVs while maintaining the large collective coupling constants [31,38]. The other is to apply the external driving pulses to the electron spins on the redundant nitrogen atoms by using flip-flop processes, where the nitrogen spins are flipped by the spin-echo pulses on a time scale much faster than the flip-flop rate.

5. Conclusion

In summary, a hybrid system consisting of SQ, TLR, and NVE has been considered for achieving strong nonlinear interaction. Through the exchange of virtual microwave photons of the TLR, we have put forward a new nonlinear coupling between the NVE and the SQ. Our proposed hybrid system has described a nonlinear interaction between a pair of bosons and the SQ. In addition, we have also achieved nonlinear optical processes with the χ(2) term, based on the collective excitations of the NVE. Therefore, the squeezed states can be generated.

Acknowledgments

This work is supported by the National Fundamental Research Program (No. 2013CB921803), the National Natural Science Foundation of China under Grants (No. 11104326, No. 11274351, and No. 11274351).

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25. T. Ojanen and J. Salo, “Possible scheme for on-chip element for squeezed microwave generation,” Phys. Rev. B 75, 184508 (2007). [CrossRef]  

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

27. Y. M. Liu, “Virtual-photon-induced entanglement with two nitrogen-vacancy centers coupled to a high-Q silica microsphere cavity,” SCIENCE CHINA Physics, Mechanics Astronomy 56(11), 2138 (2013) [CrossRef]  

28. F. Jelezko, T. Gaebel, I. Popa, A. Gruber, and J. Wrachtrup, “Observation of coherent oscillations in a single electron spin,” Phys. Rev. Lett. 92, 076401 (2004). [CrossRef]   [PubMed]  

29. F. Jelezko, T. Gaebel, I. Popa, M. Domhan, A. Gruber, and J. Wrachtrup, “Observation of coherent oscillation of a single nuclear spin and realization of a two-qubit conditional quantum gate,” Phys. Rev. Lett. 93, 130501 (2004). [CrossRef]   [PubMed]  

30. F. Shi, X. Rong, N. Xu, Y. Wang, J. Wu, B. Chong, X. Peng, J. Kniepert, R. S. Schoenfeld, W. Harneit, M. Feng, and J. Du, “Room-temperature implementation of the Deutsch-Jozsa algorithm with a single electronic spin in diamond,” Phys. Rev. Lett. 105, 040504 (2010). [CrossRef]   [PubMed]  

31. 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]  

32. 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. 105210501 (2010). [CrossRef]  

33. X. Zhu, S. Saito, A. Kemp, K. Kakuyanagi, S. Karimoto, H. Nakano, W. J. Munro, Y. Tokura, M. S. Everitt, K. Nemoto, M. Kasu, N. Mizuochi, and K. Semba, “Coherent coupling of a superconducting flux qubit to an electron spin ensemble in diamond,” Nature 478, 221–224 (2011). [CrossRef]   [PubMed]  

34. W. L. Yang, Z. Q. Yin, Y. Hu, M. Feng, and J. F. Du, “High-fidelity quantum memory using nitrogen-vacancy center ensemble for hybrid quantum computation,” Phys. Rev. A 84, 010301 (2011). [CrossRef]  

35. W. L. Yang, Z. Q. Yin, Q. Chen, C. Y. Chen, and M. Feng, “Two-mode squeezing of distant nitrogen-vacancy-center ensembles by manipulating the reservoir,” Phys. Rev. A 85, 022324 (2012). [CrossRef]  

36. Z. L. Xiang, X. Y. Lü, 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]  

37. Xin-You Lü, Z. L. Xiang, W. Cui, J. Q. You, and F. Nori, “Quantum memory using a hybrid circuit with flux qubits and nitrogen-vacancy centers,” Phys. Rev. A 88, 012329 (2013). [CrossRef]  

38. Y. Kubo, C. Grezes, A. Dewes, T. Umeda, J. Isoya, H. Sumiya, N. Morishita, H. Abe, S. Onoda, T. Ohshima, V. Jacques, A. Dréau, J.-F. Roch, I. Diniz, A. Auffeves, D. Vion, D. Esteve, and P. Bertet, “Hybrid Quantum Circuit with a Superconducting Qubit Coupled to a Spin Ensemble,” Phys. Rev. Lett. 107, 220501 (2011). [CrossRef]   [PubMed]  

39. N. B. Manson, J. P. Harrison, and M. J. Sellars, “Nitrogen-vacancy center in diamond: Model of the electronic structure and associated dynamics,” Phys. Rev. B 74, 104303 (2006). [CrossRef]  

40. Y. Makhlin, G. Schön, and A. Shnirman, “Quantum-state engineering with Josephson-junction devices,” Rev. Mod. Phys. 73, 357 (2001). [CrossRef]  

41. H Fröhlich, “Theory of the superconducting state. I. The ground state at the absolute zero of temperature,” Phys. Rev. 79, 845 (1950) [CrossRef]  

42. K. Henschel, J. Majer, J. Schmiedmayer, and H. Ritsch, “Cavity QED with an ultracold ensemble on a chip: Prospects for strong magnetic coupling at finite temperatures,” Phys. Rev. A 82, 033810 (2010) [CrossRef]  

43. T. Holstein and H. Primakoff, “Field dependence of the intrinsic domain magnetization of a ferromagnet,” Phys. Rev. 58, 1098 (1940) [CrossRef]  

44. K. Wódkiewicz and M. S. Zubairy, “Effect of laser fluctuations on squeezed states in a degenerate parametric amplifier,” Phys. Rev. A 272003 (1983) [CrossRef]  

45. P. Neumman, N. Mizuochi, F. Rempp, P. Hemmer, H. Watanabe, S. Yamasaki, V. Jacques, T. Gaebel, F. Jelezko, and J. Wrachtrup, “Multipartite entanglement among single spins in diamond,” Science 320, 1326–1329 (2008) [CrossRef]  

46. T. Gaebel, M. Domhan, I. Popa, C. Wittmann, P. Neumann, F. Jelezko, J. R. Rabeau, N. Stavrias, A. D. Greentree, S. Prawer, J. Meijer, J. Twamley, P. R. Hemmer, and J. Wrachtrup, “Room-temperature coherent coupling of single spins in diamond,” Nat. Phys. 2, 408–413 (2006) [CrossRef]  

47. G. Balasubramanian, P. Neumann, D. Twitchen, M. Markham, R. Kolesov, N. Mizuochi, J. Isoya, J. Achard, J. Beck, J. Tissler, V. Jacques, P. R. Hemmer, F. Jelezko, and J. Wrachtrup, “Ultralong spin coherence time in isotopically engineered diamond,” Nat. Mater. 8, 383–387 (2009) [CrossRef]   [PubMed]  

References

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    [Crossref] [PubMed]
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    [Crossref]
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  14. M. J. Woolley, A. C. Doherty, G. J. Milburn, and K. C. Schwab, “Nanomechanical squeezing with detection via a microwave cavity,” Phys. Rev. A 78, 062303 (2008).
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  19. A. Blais, R. S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, “Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation,” Phys. Rev. A 69, 062320 (2004).
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  20. Mika A. Sillanpää, J. I. Park, and R. W. Simmonds, “Coherent quantum state storage and transfer between two phase qubits via a resonant cavity,” Nature 449, 438–442 (2007).
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  21. W. Y. Huo and G. L. Long, “Entanglement and squeezing in solid-state circuits,” New J. Phys. 10, 013026 (2008).
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  23. N. Bergeal, F. Schackert, M. Metcalfe, R. Vijay, V. E. Manucharyan, L. Frunzio, D. E. Prober, R. J. Schoelkopf, S. M. Girvin, and M. H. Devoret, “Preparation and measurement of three-qubit entanglement in a superconducting circuit,” Nature 467, 574–578 (2010).
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  24. A. M. Zagoskin, E. Il’ichev, M. W. McCutcheon, J. F. Young, and F. Nori, “Controlled generation of squeezed states of microwave radiation in a superconducting resonant circuit,” Phys. Rev. Lett. 101, 253602 (2008).
    [Crossref] [PubMed]
  25. T. Ojanen and J. Salo, “Possible scheme for on-chip element for squeezed microwave generation,” Phys. Rev. B 75, 184508 (2007).
    [Crossref]
  26. Z. L. Xiang, S. Ashhab, J. Q. You, and F. Nori, “Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems,” Rev. Mod. Phys. 85, 623 (2013).
    [Crossref]
  27. Y. M. Liu, “Virtual-photon-induced entanglement with two nitrogen-vacancy centers coupled to a high-Q silica microsphere cavity,” SCIENCE CHINA Physics, Mechanics Astronomy 56(11), 2138 (2013)
    [Crossref]
  28. F. Jelezko, T. Gaebel, I. Popa, A. Gruber, and J. Wrachtrup, “Observation of coherent oscillations in a single electron spin,” Phys. Rev. Lett. 92, 076401 (2004).
    [Crossref] [PubMed]
  29. F. Jelezko, T. Gaebel, I. Popa, M. Domhan, A. Gruber, and J. Wrachtrup, “Observation of coherent oscillation of a single nuclear spin and realization of a two-qubit conditional quantum gate,” Phys. Rev. Lett. 93, 130501 (2004).
    [Crossref] [PubMed]
  30. F. Shi, X. Rong, N. Xu, Y. Wang, J. Wu, B. Chong, X. Peng, J. Kniepert, R. S. Schoenfeld, W. Harneit, M. Feng, and J. Du, “Room-temperature implementation of the Deutsch-Jozsa algorithm with a single electronic spin in diamond,” Phys. Rev. Lett. 105, 040504 (2010).
    [Crossref] [PubMed]
  31. 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]
  32. 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. 105210501 (2010).
    [Crossref]
  33. X. Zhu, S. Saito, A. Kemp, K. Kakuyanagi, S. Karimoto, H. Nakano, W. J. Munro, Y. Tokura, M. S. Everitt, K. Nemoto, M. Kasu, N. Mizuochi, and K. Semba, “Coherent coupling of a superconducting flux qubit to an electron spin ensemble in diamond,” Nature 478, 221–224 (2011).
    [Crossref] [PubMed]
  34. W. L. Yang, Z. Q. Yin, Y. Hu, M. Feng, and J. F. Du, “High-fidelity quantum memory using nitrogen-vacancy center ensemble for hybrid quantum computation,” Phys. Rev. A 84, 010301 (2011).
    [Crossref]
  35. W. L. Yang, Z. Q. Yin, Q. Chen, C. Y. Chen, and M. Feng, “Two-mode squeezing of distant nitrogen-vacancy-center ensembles by manipulating the reservoir,” Phys. Rev. A 85, 022324 (2012).
    [Crossref]
  36. Z. L. Xiang, X. Y. Lü, 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]
  37. Xin-You Lü, Z. L. Xiang, W. Cui, J. Q. You, and F. Nori, “Quantum memory using a hybrid circuit with flux qubits and nitrogen-vacancy centers,” Phys. Rev. A 88, 012329 (2013).
    [Crossref]
  38. Y. Kubo, C. Grezes, A. Dewes, T. Umeda, J. Isoya, H. Sumiya, N. Morishita, H. Abe, S. Onoda, T. Ohshima, V. Jacques, A. Dréau, J.-F. Roch, I. Diniz, A. Auffeves, D. Vion, D. Esteve, and P. Bertet, “Hybrid Quantum Circuit with a Superconducting Qubit Coupled to a Spin Ensemble,” Phys. Rev. Lett. 107, 220501 (2011).
    [Crossref] [PubMed]
  39. N. B. Manson, J. P. Harrison, and M. J. Sellars, “Nitrogen-vacancy center in diamond: Model of the electronic structure and associated dynamics,” Phys. Rev. B 74, 104303 (2006).
    [Crossref]
  40. Y. Makhlin, G. Schön, and A. Shnirman, “Quantum-state engineering with Josephson-junction devices,” Rev. Mod. Phys. 73, 357 (2001).
    [Crossref]
  41. H Fröhlich, “Theory of the superconducting state. I. The ground state at the absolute zero of temperature,” Phys. Rev. 79, 845 (1950)
    [Crossref]
  42. K. Henschel, J. Majer, J. Schmiedmayer, and H. Ritsch, “Cavity QED with an ultracold ensemble on a chip: Prospects for strong magnetic coupling at finite temperatures,” Phys. Rev. A 82, 033810 (2010)
    [Crossref]
  43. T. Holstein and H. Primakoff, “Field dependence of the intrinsic domain magnetization of a ferromagnet,” Phys. Rev. 58, 1098 (1940)
    [Crossref]
  44. K. Wódkiewicz and M. S. Zubairy, “Effect of laser fluctuations on squeezed states in a degenerate parametric amplifier,” Phys. Rev. A 272003 (1983)
    [Crossref]
  45. P. Neumman, N. Mizuochi, F. Rempp, P. Hemmer, H. Watanabe, S. Yamasaki, V. Jacques, T. Gaebel, F. Jelezko, and J. Wrachtrup, “Multipartite entanglement among single spins in diamond,” Science 320, 1326–1329 (2008)
    [Crossref]
  46. T. Gaebel, M. Domhan, I. Popa, C. Wittmann, P. Neumann, F. Jelezko, J. R. Rabeau, N. Stavrias, A. D. Greentree, S. Prawer, J. Meijer, J. Twamley, P. R. Hemmer, and J. Wrachtrup, “Room-temperature coherent coupling of single spins in diamond,” Nat. Phys. 2, 408–413 (2006)
    [Crossref]
  47. G. Balasubramanian, P. Neumann, D. Twitchen, M. Markham, R. Kolesov, N. Mizuochi, J. Isoya, J. Achard, J. Beck, J. Tissler, V. Jacques, P. R. Hemmer, F. Jelezko, and J. Wrachtrup, “Ultralong spin coherence time in isotopically engineered diamond,” Nat. Mater. 8, 383–387 (2009)
    [Crossref] [PubMed]

2013 (4)

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

Y. M. Liu, “Virtual-photon-induced entanglement with two nitrogen-vacancy centers coupled to a high-Q silica microsphere cavity,” SCIENCE CHINA Physics, Mechanics Astronomy 56(11), 2138 (2013)
[Crossref]

Z. L. Xiang, X. Y. Lü, 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]

Xin-You Lü, Z. L. Xiang, W. Cui, J. Q. You, and F. Nori, “Quantum memory using a hybrid circuit with flux qubits and nitrogen-vacancy centers,” Phys. Rev. A 88, 012329 (2013).
[Crossref]

2012 (1)

W. L. Yang, Z. Q. Yin, Q. Chen, C. Y. Chen, and M. Feng, “Two-mode squeezing of distant nitrogen-vacancy-center ensembles by manipulating the reservoir,” Phys. Rev. A 85, 022324 (2012).
[Crossref]

2011 (5)

X. Zhu, S. Saito, A. Kemp, K. Kakuyanagi, S. Karimoto, H. Nakano, W. J. Munro, Y. Tokura, M. S. Everitt, K. Nemoto, M. Kasu, N. Mizuochi, and K. Semba, “Coherent coupling of a superconducting flux qubit to an electron spin ensemble in diamond,” Nature 478, 221–224 (2011).
[Crossref] [PubMed]

W. L. Yang, Z. Q. Yin, Y. Hu, M. Feng, and J. F. Du, “High-fidelity quantum memory using nitrogen-vacancy center ensemble for hybrid quantum computation,” Phys. Rev. A 84, 010301 (2011).
[Crossref]

Y. Kubo, C. Grezes, A. Dewes, T. Umeda, J. Isoya, H. Sumiya, N. Morishita, H. Abe, S. Onoda, T. Ohshima, V. Jacques, A. Dréau, J.-F. Roch, I. Diniz, A. Auffeves, D. Vion, D. Esteve, and P. Bertet, “Hybrid Quantum Circuit with a Superconducting Qubit Coupled to a Spin Ensemble,” Phys. Rev. Lett. 107, 220501 (2011).
[Crossref] [PubMed]

Y. Cao, W. Y. Huo, Q. Ai, and G. L. Long, “Theory of degenerate three-wave mixing using circuit QED in solid-state circuits,” Phys. Rev. A 84, 053846 (2011).
[Crossref]

F. Massel, T. T. Heikkilä, J. M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. J. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonator,” Nature 480, 351–354 (2011).
[Crossref] [PubMed]

2010 (8)

G. Chen, Y. P. Zhang, L. T. Xiao, J. Q. Liang, and S. T. Jia, “Strong nonlinear coupling between an ultracold atomic ensemble and a nanomechanical oscillator,” Opt. Express 18, 23016–23023 (2010).
[Crossref] [PubMed]

G. Chen, H. Zhang, Y. G. Yang, R. Wang, L. T. Xiao, and S. T. Jia, “Qubit-induced high-order nonlinear interaction of the polar molecules in a stripline cavity,” Phys. Rev. A 82, 013601 (2010).
[Crossref]

S. Kumar and D. P. DiVincenzo, “Exploiting Kerr cross nonlinearity in circuit quantum electrodynamics for nondemolition measurements,” Phys. Rev. B 82, 014512 (2010).
[Crossref]

N. Bergeal, F. Schackert, M. Metcalfe, R. Vijay, V. E. Manucharyan, L. Frunzio, D. E. Prober, R. J. Schoelkopf, S. M. Girvin, and M. H. Devoret, “Preparation and measurement of three-qubit entanglement in a superconducting circuit,” Nature 467, 574–578 (2010).
[Crossref]

F. Shi, X. Rong, N. Xu, Y. Wang, J. Wu, B. Chong, X. Peng, J. Kniepert, R. S. Schoenfeld, W. Harneit, M. Feng, and J. Du, “Room-temperature implementation of the Deutsch-Jozsa algorithm with a single electronic spin in diamond,” Phys. Rev. Lett. 105, 040504 (2010).
[Crossref] [PubMed]

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]

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. 105210501 (2010).
[Crossref]

K. Henschel, J. Majer, J. Schmiedmayer, and H. Ritsch, “Cavity QED with an ultracold ensemble on a chip: Prospects for strong magnetic coupling at finite temperatures,” Phys. Rev. A 82, 033810 (2010)
[Crossref]

2009 (3)

G. Balasubramanian, P. Neumann, D. Twitchen, M. Markham, R. Kolesov, N. Mizuochi, J. Isoya, J. Achard, J. Beck, J. Tissler, V. Jacques, P. R. Hemmer, F. Jelezko, and J. Wrachtrup, “Ultralong spin coherence time in isotopically engineered diamond,” Nat. Mater. 8, 383–387 (2009)
[Crossref] [PubMed]

K. Jacobs and A. J. Landahl, “Engineering giant nonlinearities in quantum nanosystems,” Phys. Rev. Lett. 103, 067201 (2009).
[Crossref] [PubMed]

S. Rebić, J. Twamley, and G. J. Milburn, “Giant Kerr nonlinearities in circuit quantum electrodynamics,” Phys. Rev. Lett. 103, 150503 (2009).
[Crossref]

2008 (4)

M. J. Woolley, A. C. Doherty, G. J. Milburn, and K. C. Schwab, “Nanomechanical squeezing with detection via a microwave cavity,” Phys. Rev. A 78, 062303 (2008).
[Crossref]

W. Y. Huo and G. L. Long, “Entanglement and squeezing in solid-state circuits,” New J. Phys. 10, 013026 (2008).
[Crossref]

A. M. Zagoskin, E. Il’ichev, M. W. McCutcheon, J. F. Young, and F. Nori, “Controlled generation of squeezed states of microwave radiation in a superconducting resonant circuit,” Phys. Rev. Lett. 101, 253602 (2008).
[Crossref] [PubMed]

P. Neumman, N. Mizuochi, F. Rempp, P. Hemmer, H. Watanabe, S. Yamasaki, V. Jacques, T. Gaebel, F. Jelezko, and J. Wrachtrup, “Multipartite entanglement among single spins in diamond,” Science 320, 1326–1329 (2008)
[Crossref]

2007 (4)

T. Ojanen and J. Salo, “Possible scheme for on-chip element for squeezed microwave generation,” Phys. Rev. B 75, 184508 (2007).
[Crossref]

Mika A. Sillanpää, J. I. Park, and R. W. Simmonds, “Coherent quantum state storage and transfer between two phase qubits via a resonant cavity,” Nature 449, 438–442 (2007).
[Crossref] [PubMed]

K. Jacobs, “Engineering quantum states of a nanoresonator via a simple auxiliary system,” Phys. Rev. Lett. 99, 117203 (2007).
[Crossref] [PubMed]

A. A. Houck, D. I. Schuster, J. M. Gambetta, J. A. Schreier, B. R. Johnson, J. M. Chow, L. Frunzio, J. Majer, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Generating single microwave photons in a circuit,” Nature 449, 328–331 (2007).
[Crossref] [PubMed]

2006 (4)

C. P. Sun, L. F. Wei, Y. X. Liu, and F. Nori, “Quantum transducers: Integrating transmission lines and nanomechanical resonators via charge qubits,” Phys. Rev. A 73, 022318 (2006).
[Crossref]

X. Zhou and A. Mizel, “Nonlinear coupling of nanomechanical resonators to Josephson quantum circuits,” Phys. Rev. Lett. 97, 267201 (2006).
[Crossref]

N. B. Manson, J. P. Harrison, and M. J. Sellars, “Nitrogen-vacancy center in diamond: Model of the electronic structure and associated dynamics,” Phys. Rev. B 74, 104303 (2006).
[Crossref]

T. Gaebel, M. Domhan, I. Popa, C. Wittmann, P. Neumann, F. Jelezko, J. R. Rabeau, N. Stavrias, A. D. Greentree, S. Prawer, J. Meijer, J. Twamley, P. R. Hemmer, and J. Wrachtrup, “Room-temperature coherent coupling of single spins in diamond,” Nat. Phys. 2, 408–413 (2006)
[Crossref]

2005 (2)

S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513 (2005).
[Crossref]

K. Moon and S. M. Girvin, “Theory of microwave parametric down-conversion and squeezing using circuit QED,” Phys. Rev. Lett. 95, 140504 (2005).
[Crossref] [PubMed]

2004 (5)

A. Blais, R. S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, “Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation,” Phys. Rev. A 69, 062320 (2004).
[Crossref]

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004).
[Crossref] [PubMed]

I. Chiorescu, P. Bertet, K. Semba, Y. Nakamura, C. J. P. M. Harmans, and J. E. Mooij, “Coherent dynamics of a flux qubit coupled to a harmonic oscillator,” Nature 431, 159–162 (2004).
[Crossref] [PubMed]

F. Jelezko, T. Gaebel, I. Popa, A. Gruber, and J. Wrachtrup, “Observation of coherent oscillations in a single electron spin,” Phys. Rev. Lett. 92, 076401 (2004).
[Crossref] [PubMed]

F. Jelezko, T. Gaebel, I. Popa, M. Domhan, A. Gruber, and J. Wrachtrup, “Observation of coherent oscillation of a single nuclear spin and realization of a two-qubit conditional quantum gate,” Phys. Rev. Lett. 93, 130501 (2004).
[Crossref] [PubMed]

2001 (1)

Y. Makhlin, G. Schön, and A. Shnirman, “Quantum-state engineering with Josephson-junction devices,” Rev. Mod. Phys. 73, 357 (2001).
[Crossref]

1997 (1)

A. Imamöglu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467 (1997).
[Crossref]

1986 (1)

B. Yurke and D. Stoler, “Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion,” Phys. Rev. Lett. 57, 13 (1986).
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1983 (1)

K. Wódkiewicz and M. S. Zubairy, “Effect of laser fluctuations on squeezed states in a degenerate parametric amplifier,” Phys. Rev. A 272003 (1983)
[Crossref]

1950 (1)

H Fröhlich, “Theory of the superconducting state. I. The ground state at the absolute zero of temperature,” Phys. Rev. 79, 845 (1950)
[Crossref]

1940 (1)

T. Holstein and H. Primakoff, “Field dependence of the intrinsic domain magnetization of a ferromagnet,” Phys. Rev. 58, 1098 (1940)
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Abe, H.

Y. Kubo, C. Grezes, A. Dewes, T. Umeda, J. Isoya, H. Sumiya, N. Morishita, H. Abe, S. Onoda, T. Ohshima, V. Jacques, A. Dréau, J.-F. Roch, I. Diniz, A. Auffeves, D. Vion, D. Esteve, and P. Bertet, “Hybrid Quantum Circuit with a Superconducting Qubit Coupled to a Spin Ensemble,” Phys. Rev. Lett. 107, 220501 (2011).
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Achard, J.

G. Balasubramanian, P. Neumann, D. Twitchen, M. Markham, R. Kolesov, N. Mizuochi, J. Isoya, J. Achard, J. Beck, J. Tissler, V. Jacques, P. R. Hemmer, F. Jelezko, and J. Wrachtrup, “Ultralong spin coherence time in isotopically engineered diamond,” Nat. Mater. 8, 383–387 (2009)
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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. 105210501 (2010).
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Ai, Q.

Y. Cao, W. Y. Huo, Q. Ai, and G. L. Long, “Theory of degenerate three-wave mixing using circuit QED in solid-state circuits,” Phys. Rev. A 84, 053846 (2011).
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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. Phys. 85, 623 (2013).
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Auffeves, A.

Y. Kubo, C. Grezes, A. Dewes, T. Umeda, J. Isoya, H. Sumiya, N. Morishita, H. Abe, S. Onoda, T. Ohshima, V. Jacques, A. Dréau, J.-F. Roch, I. Diniz, A. Auffeves, D. Vion, D. Esteve, and P. Bertet, “Hybrid Quantum Circuit with a Superconducting Qubit Coupled to a Spin Ensemble,” Phys. Rev. Lett. 107, 220501 (2011).
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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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Balasubramanian, G.

G. Balasubramanian, P. Neumann, D. Twitchen, M. Markham, R. Kolesov, N. Mizuochi, J. Isoya, J. Achard, J. Beck, J. Tissler, V. Jacques, P. R. Hemmer, F. Jelezko, and J. Wrachtrup, “Ultralong spin coherence time in isotopically engineered diamond,” Nat. Mater. 8, 383–387 (2009)
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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).
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Beck, J.

G. Balasubramanian, P. Neumann, D. Twitchen, M. Markham, R. Kolesov, N. Mizuochi, J. Isoya, J. Achard, J. Beck, J. Tissler, V. Jacques, P. R. Hemmer, F. Jelezko, and J. Wrachtrup, “Ultralong spin coherence time in isotopically engineered diamond,” Nat. Mater. 8, 383–387 (2009)
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Bergeal, N.

N. Bergeal, F. Schackert, M. Metcalfe, R. Vijay, V. E. Manucharyan, L. Frunzio, D. E. Prober, R. J. Schoelkopf, S. M. Girvin, and M. H. Devoret, “Preparation and measurement of three-qubit entanglement in a superconducting circuit,” Nature 467, 574–578 (2010).
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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).
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Bertet, P.

Y. Kubo, C. Grezes, A. Dewes, T. Umeda, J. Isoya, H. Sumiya, N. Morishita, H. Abe, S. Onoda, T. Ohshima, V. Jacques, A. Dréau, J.-F. Roch, I. Diniz, A. Auffeves, D. Vion, D. Esteve, and P. Bertet, “Hybrid Quantum Circuit with a Superconducting Qubit Coupled to a Spin Ensemble,” Phys. Rev. Lett. 107, 220501 (2011).
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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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I. Chiorescu, P. Bertet, K. Semba, Y. Nakamura, C. J. P. M. Harmans, and J. E. Mooij, “Coherent dynamics of a flux qubit coupled to a harmonic oscillator,” Nature 431, 159–162 (2004).
[Crossref] [PubMed]

Blais, A.

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004).
[Crossref] [PubMed]

A. Blais, R. S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, “Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation,” Phys. Rev. A 69, 062320 (2004).
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Braunstein, S. L.

S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513 (2005).
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Cao, Y.

Y. Cao, W. Y. Huo, Q. Ai, and G. L. Long, “Theory of degenerate three-wave mixing using circuit QED in solid-state circuits,” Phys. Rev. A 84, 053846 (2011).
[Crossref]

Chen, C. Y.

W. L. Yang, Z. Q. Yin, Q. Chen, C. Y. Chen, and M. Feng, “Two-mode squeezing of distant nitrogen-vacancy-center ensembles by manipulating the reservoir,” Phys. Rev. A 85, 022324 (2012).
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Chen, G.

G. Chen, Y. P. Zhang, L. T. Xiao, J. Q. Liang, and S. T. Jia, “Strong nonlinear coupling between an ultracold atomic ensemble and a nanomechanical oscillator,” Opt. Express 18, 23016–23023 (2010).
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G. Chen, H. Zhang, Y. G. Yang, R. Wang, L. T. Xiao, and S. T. Jia, “Qubit-induced high-order nonlinear interaction of the polar molecules in a stripline cavity,” Phys. Rev. A 82, 013601 (2010).
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Chen, Q.

W. L. Yang, Z. Q. Yin, Q. Chen, C. Y. Chen, and M. Feng, “Two-mode squeezing of distant nitrogen-vacancy-center ensembles by manipulating the reservoir,” Phys. Rev. A 85, 022324 (2012).
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Chiorescu, I.

I. Chiorescu, P. Bertet, K. Semba, Y. Nakamura, C. J. P. M. Harmans, and J. E. Mooij, “Coherent dynamics of a flux qubit coupled to a harmonic oscillator,” Nature 431, 159–162 (2004).
[Crossref] [PubMed]

Cho, S. U.

F. Massel, T. T. Heikkilä, J. M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. J. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonator,” Nature 480, 351–354 (2011).
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Chong, B.

F. Shi, X. Rong, N. Xu, Y. Wang, J. Wu, B. Chong, X. Peng, J. Kniepert, R. S. Schoenfeld, W. Harneit, M. Feng, and J. Du, “Room-temperature implementation of the Deutsch-Jozsa algorithm with a single electronic spin in diamond,” Phys. Rev. Lett. 105, 040504 (2010).
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Chow, J. M.

A. A. Houck, D. I. Schuster, J. M. Gambetta, J. A. Schreier, B. R. Johnson, J. M. Chow, L. Frunzio, J. Majer, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Generating single microwave photons in a circuit,” Nature 449, 328–331 (2007).
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Cui, W.

Xin-You Lü, Z. L. Xiang, W. Cui, J. Q. You, and F. Nori, “Quantum memory using a hybrid circuit with flux qubits and nitrogen-vacancy centers,” Phys. Rev. A 88, 012329 (2013).
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Deutsch, M.

A. Imamöglu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467 (1997).
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Devoret, M. H.

N. Bergeal, F. Schackert, M. Metcalfe, R. Vijay, V. E. Manucharyan, L. Frunzio, D. E. Prober, R. J. Schoelkopf, S. M. Girvin, and M. H. Devoret, “Preparation and measurement of three-qubit entanglement in a superconducting circuit,” Nature 467, 574–578 (2010).
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A. A. Houck, D. I. Schuster, J. M. Gambetta, J. A. Schreier, B. R. Johnson, J. M. Chow, L. Frunzio, J. Majer, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Generating single microwave photons in a circuit,” Nature 449, 328–331 (2007).
[Crossref] [PubMed]

Dewes, A.

Y. Kubo, C. Grezes, A. Dewes, T. Umeda, J. Isoya, H. Sumiya, N. Morishita, H. Abe, S. Onoda, T. Ohshima, V. Jacques, A. Dréau, J.-F. Roch, I. Diniz, A. Auffeves, D. Vion, D. Esteve, and P. Bertet, “Hybrid Quantum Circuit with a Superconducting Qubit Coupled to a Spin Ensemble,” Phys. Rev. Lett. 107, 220501 (2011).
[Crossref] [PubMed]

Diniz, I.

Y. Kubo, C. Grezes, A. Dewes, T. Umeda, J. Isoya, H. Sumiya, N. Morishita, H. Abe, S. Onoda, T. Ohshima, V. Jacques, A. Dréau, J.-F. Roch, I. Diniz, A. Auffeves, D. Vion, D. Esteve, and P. Bertet, “Hybrid Quantum Circuit with a Superconducting Qubit Coupled to a Spin Ensemble,” Phys. Rev. Lett. 107, 220501 (2011).
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DiVincenzo, D. P.

S. Kumar and D. P. DiVincenzo, “Exploiting Kerr cross nonlinearity in circuit quantum electrodynamics for nondemolition measurements,” Phys. Rev. B 82, 014512 (2010).
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Doherty, A. C.

M. J. Woolley, A. C. Doherty, G. J. Milburn, and K. C. Schwab, “Nanomechanical squeezing with detection via a microwave cavity,” Phys. Rev. A 78, 062303 (2008).
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Domhan, M.

T. Gaebel, M. Domhan, I. Popa, C. Wittmann, P. Neumann, F. Jelezko, J. R. Rabeau, N. Stavrias, A. D. Greentree, S. Prawer, J. Meijer, J. Twamley, P. R. Hemmer, and J. Wrachtrup, “Room-temperature coherent coupling of single spins in diamond,” Nat. Phys. 2, 408–413 (2006)
[Crossref]

F. Jelezko, T. Gaebel, I. Popa, M. Domhan, A. Gruber, and J. Wrachtrup, “Observation of coherent oscillation of a single nuclear spin and realization of a two-qubit conditional quantum gate,” Phys. Rev. Lett. 93, 130501 (2004).
[Crossref] [PubMed]

Dréau, A.

Y. Kubo, C. Grezes, A. Dewes, T. Umeda, J. Isoya, H. Sumiya, N. Morishita, H. Abe, S. Onoda, T. Ohshima, V. Jacques, A. Dréau, J.-F. Roch, I. Diniz, A. Auffeves, D. Vion, D. Esteve, and P. Bertet, “Hybrid Quantum Circuit with a Superconducting Qubit Coupled to a Spin Ensemble,” Phys. Rev. Lett. 107, 220501 (2011).
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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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Du, J.

F. Shi, X. Rong, N. Xu, Y. Wang, J. Wu, B. Chong, X. Peng, J. Kniepert, R. S. Schoenfeld, W. Harneit, M. Feng, and J. Du, “Room-temperature implementation of the Deutsch-Jozsa algorithm with a single electronic spin in diamond,” Phys. Rev. Lett. 105, 040504 (2010).
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Du, J. F.

W. L. Yang, Z. Q. Yin, Y. Hu, M. Feng, and J. F. Du, “High-fidelity quantum memory using nitrogen-vacancy center ensemble for hybrid quantum computation,” Phys. Rev. A 84, 010301 (2011).
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Esteve, D.

Y. Kubo, C. Grezes, A. Dewes, T. Umeda, J. Isoya, H. Sumiya, N. Morishita, H. Abe, S. Onoda, T. Ohshima, V. Jacques, A. Dréau, J.-F. Roch, I. Diniz, A. Auffeves, D. Vion, D. Esteve, and P. Bertet, “Hybrid Quantum Circuit with a Superconducting Qubit Coupled to a Spin Ensemble,” Phys. Rev. Lett. 107, 220501 (2011).
[Crossref] [PubMed]

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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Everitt, M. S.

X. Zhu, S. Saito, A. Kemp, K. Kakuyanagi, S. Karimoto, H. Nakano, W. J. Munro, Y. Tokura, M. S. Everitt, K. Nemoto, M. Kasu, N. Mizuochi, and K. Semba, “Coherent coupling of a superconducting flux qubit to an electron spin ensemble in diamond,” Nature 478, 221–224 (2011).
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Feng, M.

W. L. Yang, Z. Q. Yin, Q. Chen, C. Y. Chen, and M. Feng, “Two-mode squeezing of distant nitrogen-vacancy-center ensembles by manipulating the reservoir,” Phys. Rev. A 85, 022324 (2012).
[Crossref]

W. L. Yang, Z. Q. Yin, Y. Hu, M. Feng, and J. F. Du, “High-fidelity quantum memory using nitrogen-vacancy center ensemble for hybrid quantum computation,” Phys. Rev. A 84, 010301 (2011).
[Crossref]

F. Shi, X. Rong, N. Xu, Y. Wang, J. Wu, B. Chong, X. Peng, J. Kniepert, R. S. Schoenfeld, W. Harneit, M. Feng, and J. Du, “Room-temperature implementation of the Deutsch-Jozsa algorithm with a single electronic spin in diamond,” Phys. Rev. Lett. 105, 040504 (2010).
[Crossref] [PubMed]

Fröhlich, H

H Fröhlich, “Theory of the superconducting state. I. The ground state at the absolute zero of temperature,” Phys. Rev. 79, 845 (1950)
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Frunzio, L.

N. Bergeal, F. Schackert, M. Metcalfe, R. Vijay, V. E. Manucharyan, L. Frunzio, D. E. Prober, R. J. Schoelkopf, S. M. Girvin, and M. H. Devoret, “Preparation and measurement of three-qubit entanglement in a superconducting circuit,” Nature 467, 574–578 (2010).
[Crossref]

A. A. Houck, D. I. Schuster, J. M. Gambetta, J. A. Schreier, B. R. Johnson, J. M. Chow, L. Frunzio, J. Majer, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Generating single microwave photons in a circuit,” Nature 449, 328–331 (2007).
[Crossref] [PubMed]

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004).
[Crossref] [PubMed]

Gaebel, T.

P. Neumman, N. Mizuochi, F. Rempp, P. Hemmer, H. Watanabe, S. Yamasaki, V. Jacques, T. Gaebel, F. Jelezko, and J. Wrachtrup, “Multipartite entanglement among single spins in diamond,” Science 320, 1326–1329 (2008)
[Crossref]

T. Gaebel, M. Domhan, I. Popa, C. Wittmann, P. Neumann, F. Jelezko, J. R. Rabeau, N. Stavrias, A. D. Greentree, S. Prawer, J. Meijer, J. Twamley, P. R. Hemmer, and J. Wrachtrup, “Room-temperature coherent coupling of single spins in diamond,” Nat. Phys. 2, 408–413 (2006)
[Crossref]

F. Jelezko, T. Gaebel, I. Popa, M. Domhan, A. Gruber, and J. Wrachtrup, “Observation of coherent oscillation of a single nuclear spin and realization of a two-qubit conditional quantum gate,” Phys. Rev. Lett. 93, 130501 (2004).
[Crossref] [PubMed]

F. Jelezko, T. Gaebel, I. Popa, A. Gruber, and J. Wrachtrup, “Observation of coherent oscillations in a single electron spin,” Phys. Rev. Lett. 92, 076401 (2004).
[Crossref] [PubMed]

Gambetta, J. M.

A. A. Houck, D. I. Schuster, J. M. Gambetta, J. A. Schreier, B. R. Johnson, J. M. Chow, L. Frunzio, J. Majer, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Generating single microwave photons in a circuit,” Nature 449, 328–331 (2007).
[Crossref] [PubMed]

Girvin, S. M.

N. Bergeal, F. Schackert, M. Metcalfe, R. Vijay, V. E. Manucharyan, L. Frunzio, D. E. Prober, R. J. Schoelkopf, S. M. Girvin, and M. H. Devoret, “Preparation and measurement of three-qubit entanglement in a superconducting circuit,” Nature 467, 574–578 (2010).
[Crossref]

A. A. Houck, D. I. Schuster, J. M. Gambetta, J. A. Schreier, B. R. Johnson, J. M. Chow, L. Frunzio, J. Majer, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Generating single microwave photons in a circuit,” Nature 449, 328–331 (2007).
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K. Moon and S. M. Girvin, “Theory of microwave parametric down-conversion and squeezing using circuit QED,” Phys. Rev. Lett. 95, 140504 (2005).
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A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004).
[Crossref] [PubMed]

A. Blais, R. S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, “Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation,” Phys. Rev. A 69, 062320 (2004).
[Crossref]

Greentree, A. D.

T. Gaebel, M. Domhan, I. Popa, C. Wittmann, P. Neumann, F. Jelezko, J. R. Rabeau, N. Stavrias, A. D. Greentree, S. Prawer, J. Meijer, J. Twamley, P. R. Hemmer, and J. Wrachtrup, “Room-temperature coherent coupling of single spins in diamond,” Nat. Phys. 2, 408–413 (2006)
[Crossref]

Grezes, C.

Y. Kubo, C. Grezes, A. Dewes, T. Umeda, J. Isoya, H. Sumiya, N. Morishita, H. Abe, S. Onoda, T. Ohshima, V. Jacques, A. Dréau, J.-F. Roch, I. Diniz, A. Auffeves, D. Vion, D. Esteve, and P. Bertet, “Hybrid Quantum Circuit with a Superconducting Qubit Coupled to a Spin Ensemble,” Phys. Rev. Lett. 107, 220501 (2011).
[Crossref] [PubMed]

Gruber, A.

F. Jelezko, T. Gaebel, I. Popa, A. Gruber, and J. Wrachtrup, “Observation of coherent oscillations in a single electron spin,” Phys. Rev. Lett. 92, 076401 (2004).
[Crossref] [PubMed]

F. Jelezko, T. Gaebel, I. Popa, M. Domhan, A. Gruber, and J. Wrachtrup, “Observation of coherent oscillation of a single nuclear spin and realization of a two-qubit conditional quantum gate,” Phys. Rev. Lett. 93, 130501 (2004).
[Crossref] [PubMed]

Hakonen, P. J.

F. Massel, T. T. Heikkilä, J. M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. J. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonator,” Nature 480, 351–354 (2011).
[Crossref] [PubMed]

Harmans, C. J. P. M.

I. Chiorescu, P. Bertet, K. Semba, Y. Nakamura, C. J. P. M. Harmans, and J. E. Mooij, “Coherent dynamics of a flux qubit coupled to a harmonic oscillator,” Nature 431, 159–162 (2004).
[Crossref] [PubMed]

Harneit, W.

F. Shi, X. Rong, N. Xu, Y. Wang, J. Wu, B. Chong, X. Peng, J. Kniepert, R. S. Schoenfeld, W. Harneit, M. Feng, and J. Du, “Room-temperature implementation of the Deutsch-Jozsa algorithm with a single electronic spin in diamond,” Phys. Rev. Lett. 105, 040504 (2010).
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N. B. Manson, J. P. Harrison, and M. J. Sellars, “Nitrogen-vacancy center in diamond: Model of the electronic structure and associated dynamics,” Phys. Rev. B 74, 104303 (2006).
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Heikkilä, T. T.

F. Massel, T. T. Heikkilä, J. M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. J. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonator,” Nature 480, 351–354 (2011).
[Crossref] [PubMed]

Hemmer, P.

P. Neumman, N. Mizuochi, F. Rempp, P. Hemmer, H. Watanabe, S. Yamasaki, V. Jacques, T. Gaebel, F. Jelezko, and J. Wrachtrup, “Multipartite entanglement among single spins in diamond,” Science 320, 1326–1329 (2008)
[Crossref]

Hemmer, P. R.

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Y. M. Liu, “Virtual-photon-induced entanglement with two nitrogen-vacancy centers coupled to a high-Q silica microsphere cavity,” SCIENCE CHINA Physics, Mechanics Astronomy 56(11), 2138 (2013)
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Figures (2)

Fig. 1
Fig. 1 (a) Schematic of the hybrid system, where the NVE is placed in the antinode of the TLR coupled to the SQ through magnetic field induced by its quantized current. (b) The level structure of the NVE.
Fig. 2
Fig. 2 ((a) ΔX1 versus Ωpt under the different cases, where the soild red, dotted blue, dashed green, and dotdashed pink lines denote Θ/Ωp = 0, Θ/Ωp = 0.001, Θ/Ωp = 0.01, and Θ/Ωp = 0.1, respectively. The inset shows the ΔX2 versus Ωpt. (b) ΔX′1 versus Ωpt under the different cases, where the solid red, dotted blue, dashed green, and dotdashed pink lines denote Θp = 0, Θp = 0.1, Θp = 0.2, and Θp = 0.5, respectively. The inset shows the ΔX′2 versus Ωpt.

Equations (44)

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H tot = H NVE + H FQ + H TLR + H STI + H NTI ,
H NVE = δ S z ,
H SQ = 1 2 E C ( 1 2 n g ) σ z E J cos ( π Φ e Φ 0 ) σ x ,
Φ q = μ 0 S I q 2 π d
Φ q = i k ϕ k ( a k a k ) ,
ϕ k = μ 0 S I 0 ( k ) 2 π d .
Φ q = i ϕ b ( a a ) .
E J ( Φ q ) = E J cos ( π Φ e + Φ q Φ 0 ) .
H 1 = ω a a a 1 2 E C ( 1 2 n g ) σ z E j cos ( π Φ e + Φ q Φ 0 ) σ x .
H 2 = 1 2 Ω σ ˜ z + ω a a a + η ( a 2 σ ˜ + a 2 σ ˜ + ) ,
Ω = E c 2 ( 1 2 n g ) 2 + 4 E J 2 ,
η = ( 1 ) m + 1 E J π 2 ϕ b 2 2 Φ 0 2 cos θ ,
| 0 ˜ = cos ( θ 2 ) | 0 + sin ( θ 2 ) | 1 ,
| 1 ˜ = sin ( θ 2 ) | 0 + cos ( θ 2 ) | 1
H NTI = g ( S + a + S a ) ,
H Tot = 1 2 Ω σ ˜ z + ω a a a + δ S z + η ( a 2 σ ˜ + a 2 σ ˜ + ) + g ( S + a + S a ) .
| 0 P = | g 1 g 2 g N
| 1 P = S ˜ + | g N V = 1 N k = 1 N τ + k | g 1 g 2 g N .
H Tot = 1 2 Ω σ ˜ z + ω a a a + δ S ˜ + S ˜ + η ( a 2 σ ˜ + a 2 σ ˜ + ) + N g ( S ˜ + a + S ˜ a ) .
Δ S = | ω a δ | N g ,
Δ N = | 2 ω a Ω | η ,
F = N g Δ S ( a S ˜ a S ˜ + ) + η Δ N ( a 2 σ ˜ a 2 σ ˜ + ) .
H ˜ Tot 1 2 Ω σ ˜ z + ( ω 0 + χ ) S ˜ + S ˜ + i λ eff ( S ˜ + 2 σ ˜ S ˜ 2 σ ˜ + ) ,
χ = N g 2 Δ S
λ eff = N g 2 η Δ S ( 1 Δ S + 1 Δ N )
S + N c , S N c , S z c c N 2 ,
H ˜ Tot 1 2 Ω σ ˜ z + δ C c c + λ eff ( c 2 σ ˜ + c 2 σ ˜ ) ,
H ˜ R i λ eff ( c 2 σ ˜ + c 2 σ ˜ ) .
S ( κ ) = i κ 2 ( c 2 c 2 ) ,
Δ X 1 = X 1 2 X 1 2 = e κ 2 ,
Δ X 2 = X 2 2 X 2 2 = e κ 2 .
| κ = S ( κ ) | 0 = κ 2 ( c 2 + c 2 ) | 0 .
Ψ ˙ = ( M 0 Θ M 2 ) Ψ ,
Ψ = [ c + c + c c + c 2 ( c + ) 2 ] , M 0 = [ 0 2 κ 2 κ κ 0 0 κ 0 0 ] , M = [ 0 0 0 0 1 0 0 0 1 ] .
c + c + c c + = 2 κ 2 π i d z e z t z ( z + Θ ) 4 κ 2
= e Θ t / 2 ( Θ sinh ( ϰ t / 2 ) ϰ + cosh ( ϰ t / 2 ) )
c 2 = Ω κ e 2 Θ t 2 κ 2 Θ 2 / 8 + ( 4 κ + 5 Θ ) e ( 2 κ 3 2 Θ ) t 16 κ + 4 Θ + ( 4 κ 5 Θ ) e ( 2 κ 3 2 Θ ) t 16 κ 4 Θ ,
Δ X 1 = e Θ t / 2 [ Θ sinh ( ϰ t / 2 ) + 4 ϰ cosh ( ϰ t / 2 ) ] 4 ϰ + ξ
Δ X 2 = e Θ t / 2 [ Θ sinh ( ϰ t / 2 ) + 4 ϰ cosh ( ϰ t / 2 ) ] 4 ϰ ξ ,
ξ = 2 Θ κ e 2 Θ t 8 κ 2 Θ 2 / 2 + ( 4 κ 5 Θ ) e ( 2 κ 3 2 Θ ) t 32 κ 8 Θ ( 4 κ + 5 Θ ) e ( 2 κ 3 2 Θ ) t 32 κ + 8 Θ .
Δ X 1 = ( Θ t 2 ) e 2 Ω p t + e 2 Ω p t 2 ,
Δ X 2 = e Ω p t 1 2 Θ t 2 .
Δ X 1 = exp [ 2 Θ t Ω p t + 2 ( e Θ t 1 ) ] 2 ,
Δ X 2 = exp [ 2 Θ t + Ω p t + 2 ( e Θ t 1 ) ] 2 ,

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