## 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, 3–5], 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 [5–11], especially in the solid-state systems [5–15].

Recently, the strong coupling regime has been experimentally implemented between super-conducting qubit (SQ) and transmission line resonator (TLR) system [16–18], 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 [21–25]. 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* = 10^{11} ∼ 10^{13} [31], much effort has been devoted recently to combine the NVE with circuit quantum electrodynamics (QED) both experimentally and theoretically [31–38]. There are many significant progresses in simulating quantum optics phenomena and implementing quantum information processing, such as quantum memories and continuous variable entanglement [34–36, 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* = 10^{11} ∼ 10^{13} [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 *L*_{0}, 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 *V _{g}* through a gate capacitance

*C*. The combined NVE-TLR-SQ system we study is governed by the Hamiltonian

_{g}*H*,

_{NVE}*H*and

_{FQ}*H*are the Hamiltonians of NVE, FQ and TLR, respectively.

_{TLR}*H*corresponds to the interaction between the SQ and TLR, and

_{STI}*H*represents the coupling between the NVE and the TLR.

_{NTI}The ground state of a NV is spin triplet with a zero-field splitting *D _{gs}* = 2.87 GHz between

*m*= 0 and the nearly degenerate sublevels

_{s}*m*= ± 1 in the absence of external magnetic field [39]. In our proposal, the logic states |

_{s}*g*〉 and |

*e*〉 of the NVE are denoted by the states |

^{3}

*A*,

*m*= 0〉, and |

_{s}^{3}

*A*,

*m*= −1〉, respectively. So the Hamiltonian of a single NVE reads (in units of

_{s}*ħ*= 1),

*δ*≈

*D*−

_{gs}*γ*

_{e}B_{0}is the energy gap between the ground state sublevels

*m*= 0 and

_{s}*m*= −1, with the external magnetic field

_{s}*B*

_{0}and

*γ*the the gyromagnetic ratio of an electron. ${S}_{z}=\frac{1}{2}{\sum}_{j=1}^{N}{\tau}_{{j}_{z}}$ is the collective spin operator for the NVE with the Pauli spin operators

_{e}*τ*= |

_{z}*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 [

*S*,

_{z}*S*

_{±}] = ±

*S*

_{±}and [

*S*

_{+},

*S*

_{−}] = 2

*S*, where the Casimir invariant is

_{z}*S*

^{2}=

*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 *C _{J}* and Josephson coupling energy

*E*. 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 Φ

_{J}*to control the total effective Josephson coupling energy. In the charge representation, the Hamiltonian of the SQ reads [40]*

_{e}*V*with

_{g}*C*Jopsephson junction capacitance and

_{J}*C*gate capacitance.

_{g}*C*

_{Σ}= 2

*C*+

_{J}*C*is the total capacitance, Φ

_{g}_{0}=

*π/e*is the flux quantum.

*σ*= |1〉〈1| − |0〉〈0|,

_{z}*σ*= |1〉〈0| + |0〉〈1| are the usual Pauli operators. The state of the SQ can be separately controlled by the gate voltage

_{x}*V*through a gate capacitance

_{g}*C*.

_{g}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 *I _{q}* of the TLR is

*d*being the distance between the SQ and the TLR. At the antinode, the current in the TLR reaches its maximal amplitude ${I}_{q}=-i{\mathrm{\Sigma}}_{k}{I}_{0}^{(k)}\left({a}_{k}^{\u2020}-{a}_{k}\right)$, where

*a*is the

_{k}*k*th mode of the TLR, ${I}_{0}^{(k)}=\sqrt{\frac{\hslash {\omega}_{k}}{Ll}}$ [19] with

*l*being the length of the TLR and

*ω*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 There is only one mode that couples to the SQ and the quantized flux felt by the SQUID becomes With the total biased flux, the effective Josephson coupling energy of the SQUID becomes

_{k}*H*

_{1}=

*H*+

_{SQ}*H*+

_{TLR}*H*as

_{STI}*. At some external biased points, for instance, Φ*

_{q}*=*

_{e}*m*Φ

_{0}, with

*m*an arbitrary integer, the Josephson coupling energy in Eq. (9) becomes (−1)

^{m+1}

*E*cos(

_{J}*π*Φ

*/Φ*

_{q}_{0}). 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 Φ

*/Φ*

_{q}_{0}, Eq. (9) is simplified to (in units of

*ħ*= 1),

*a*(

*a*

^{†}) is the annihilation (creation) operator of the full-wave mode of the TLR with the equilibrium frequency

*ω*. We have used in Eq. (10) the new basis

_{a}*θ*= tan

^{−1}[(−1)

*2*

^{m}*E*/(

_{J}*E*(1 − 2

_{c}*n*))]. Therefore, the corresponding Pauli operators are

_{g}*σ̃*= |1̃〉〈1̃| − |0̃〉〈0̃|,

_{z}*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*= 2

*π*× 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

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

## 3. Enhanced strong nonlinear interaction

In terms of normalized collective operators
${\tilde{S}}_{+}=\frac{1}{\sqrt{N}}{\sum}_{j=1}^{N}{\tau}_{{j}_{z}}$, we notice that the collective spin operator *S̃*_{+} can create symmetric Dicke excitation states |*n*〉* _{P}*, and we encode the qubit of the NVE as

*F*for FN transformation adopts the following form

*H̃*to the second order in $\sqrt{N}g/{\mathrm{\Delta}}_{S}$ and

_{Tot}*η*/Δ

*. Since the cavity mode only supports the virtual excitation of photons, we can neglect the terms with microwave photons and achieve an effective Hamiltonian*

_{N}*λ*(

_{eff}*S̃*

_{+}

^{2}

*σ̃*

_{−}+

*S̃*

_{−}

^{2}

*σ̃*

_{+}) in the Hamiltonian

*H̃*, to our knowledge, is new in quantum optics and condensed matter physics. For a single spin of NV, we know that ${\tau}_{+}^{2}=0$ and ${\tau}_{-}^{2}=0$. Therefore, our achieved Hamiltonian only describes the coupling of the SQ to the collective excitation of the NVE.

_{Tot}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 *S̃*_{+} = *c*^{†} and *S̃*_{−} = *c* with bosonic creation and annihilation operators. By using the Holstein-Primakoff (HP) transformation [43] as

*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,

*c*

^{†}

*c*≪

*N*. It means that the collective excitation governed by

*S*

_{±}behaves like a harmonic oscillator degree of freedom. Thus, the Hamiltonian

*H̃*can be rewritten as

_{Tot}*δ*=

_{C}*ω*

_{0}+

*Nχ*.

From the effective Hamiltonian *H̃′ _{Tot}*, one can find that this nonlinear coupling strength

*λ*is in direct proportion to the spin number

_{eff}*N*in ensemble. It implies that the coupling strength

*λ*can be enhanced greatly by increasing the value of

_{eff}*N*, as shown below. Physically, the effective Hamiltonian

*H̃′*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

_{Tot}*N*in ensemble, which is the nonlinear effect different from traditional linear coupling proportion to $\sqrt{N}$ in ensemble [31–33]. This kind of nonlinear interaction might be useful for processing quantum information with continuous variables under strong coupling [6]. Moreover, the effective Hamiltonian

*H̃′*also shows that the nonlinear optical processes with the

_{Tot}*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 2*π* × 10^{3} ∼ 10^{10} 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* = 2*π* × 0.25 kHz [21]. On the other hand, a single diamond crystal (3 × 3 × 0.5 mm^{3}) containing the NVE is glued on top of TLR, with the distance to the silicon substrate less than 0.5 *μ*m 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 10^{12}, the coupling between a single NV center and the TLR is 2*π* × 11 Hz and
$\sqrt{N}g/{\mathrm{\Delta}}_{S}=\eta /{\mathrm{\Delta}}_{N}=0.1$. So we can estimate our nonlinear indirect interaction between the NVE and the SQ to be
${\lambda}_{\mathit{eff}}=\frac{N{g}^{2}\eta}{{\mathrm{\Delta}}_{S}}\left(\frac{1}{{\mathrm{\Delta}}_{S}}+\frac{1}{{\mathrm{\Delta}}_{N}}\right)\simeq \frac{{\left(\sqrt{N}g\right)}^{2}}{{\mathrm{\Delta}}_{S}}\left(\frac{\eta}{{\mathrm{\Delta}}_{N}}\right)=0.01\sqrt{N}g$, and obtain *λ _{eff}* ≃ 2

*π*× 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 = 2*δ _{C}* and shift to the rotating frame defined by
$\frac{1}{2}\mathrm{\Omega}{\tilde{\sigma}}_{z}+{\delta}_{C}{c}^{\u2020}c$, we obtain the following effective Hamiltonian

*t*[5]. Between each of such time interval, we apply a quick

*π*-pulse

*σ̃*to flip the state of the charge qubit, so that the evolution for two periods can be simplified as

_{x}*e*

^{−iHsqt}with

*H*=

_{sq}*λ*(

_{eff}*c*

^{2}+

*c*

^{†2})

*σ̃*. If we initialize the charge qubit in the 〈

_{x}*σ̃*〉 = 1 state, and repeat this procedure for

_{x}*M*times, the evolution operator on the state of the NVE becomes with

*κ*=

*Mλ*Δ

_{eff}*t*. This is a squeezing operator on the NVE with squeezing parameter

*κ*. Assuming that the NVE initially in the state |0〉

*and using the transformation*

_{P}*S*

^{†}(

*κ*)

*cS*(

*κ*) =

*c*cosh

*κ*−

*c*

^{†}sinh

*κ*, we obtain the variances in the two quadratures

*X*

_{1}= (

*c*+

*c*

^{†})/2 and

*X*

_{2}= (

*c*−

*c*

^{†})/2

*i*,

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 Ω* _{p}*.

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

*, the Laplace-transform solution has the following exact solution:*

^{T}In the limit Θ ≪ *t*^{−1} ≪ Ω* _{p}*, where Ω

*=*

_{p}*κ/t*, the variances in the two quadratures Δ

*X*

_{1}and Δ

*X*

_{2}are given by

*′*is the fluctuations in the amplitude of the pumping field. We plot variances versus Ω

*under different cases in the Fig. 2. It is shown that there are minimums of Δ*

_{p}t*X*

_{1}and Δ

*X*

_{2}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 Ω

*. We also find that the phase fluctuation Θ affects the squeezed properties severely with increasing time*

_{p}t*t*. On the other hand, in Fig. 2(b), the amplitude fluctuation Θ

*′*influences the squeezing efficiency nearly negligibly when Θ

*′/*Ω

*≤ 0.2.*

_{p}Additionally, all sources of decoherence and dissipation of both the NVE and SQ should be considered. In current experiments, the electron spin relaxation time *T*_{1} of the NV ranges from 6 ms at room temperature [45] to *sec* at low temperature. In addition, the dephasing time *T*_{2} = 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 *T*_{2} with *T*_{2} = 2 ms. Current conservative experimental parameters of decoherence of the SQ show the relaxation rate and dephasing rate of the qubit as *γ _{q}* = 1 MHz and

*γ*= 10 MHz [16–18], 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 〈

_{ϕ}*σ*〉 = 1 during the squeezing process [5, 22].

_{z}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 ^{13}*C* defects in the NVs, we can use isotopically purified ^{12}*C* 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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