# Arbitrary Control of Entanglement between two nitrogen-vacancy center
ensembles coupling to superconducting circuit qubit

Wan-Jun Su<EMAIL_ADDRESS>Zhen-Biao Yang  Zhi-Rong Zhong  Department of
Physics, Fuzhou University, Fuzhou 350002, People’s Republic of China

###### Abstract

We propose a method to synthesize an arbitrary quantum state of two nitrogen-
vacancy center ensemble (NVEs) in circuit QED system. The key in this scheme
is requiring an independent state-selective qubit rotation. In light of
photon-number-dependent Stark shift, the circuit qubit can arbitrary selectly
interacts with one of the two fermionic NVEs spins by controling the Rabi
frequencies of driving fields. A series of driving microwave pulses are
applied to the circuit qubit, then, arbitrary quantum states of two NVEs can
be produced, such as NOON states, Multi-dimensional entangled states,
entangled coherence states. Numerical investigation demonstrates that the
present scheme is robust against decoherence caused by spontaneous emission
and photon loss. What’more, the scheme can be extended to more NVEs, which is
important for one-way quantum computation.

arbitrary control; entangled states; NV center

###### pacs:

03.65.Xp;42.50.Pq

##  I Introduction

Recently, the hybrid systems that combine the advantages of cavity QED
RaimondRMP01  or circuit QED MakhlinRMP01 ; WallraffNature04 ; BlaisPRA04 ;
MariantoniNP11 systems, have attracted much attention for quantum information
applications. Among these systems, the composite system consisting of a NVEs,
superconducting microstrip (SCM) cavity, and nonlinear element, such as a
transmon qubit, has emerged as one of excellent candidates for the
construction of solid-state quantum information processor YouNature11 ; APRL09
. In the low-excitation limit, the collective excitations of the NVEs behave
as the fermionic modes or the harmonic oscillators. Then, the collectively
enhanced magnetic dipole interaction will result in obtaining appreciable
coupling strength. Recent experiments have achieved strong coupling between
the NVEs and the superconducting cavity modes KuboPRL10 ; RPRL11 ;
SchusterPRL10 .

The question of creating arbitrary quantum state of a cavity field or atoms
(ion, NV centers) has been discussed in the literature LawPRL96 ; StrauchPRL10
; YangPRA12 ; ZhongOC10 ; SadowskiarXiv:1303.3757 . The method proposed by Law
and Eberly LawPRL96  has used a two-channel approach, in which a controllable
coupling strength between atom-cavity and a driving classical field with an
adjustable amplitude have been used. Based on Law and Eberly’s work, Strauch
et al StrauchPRL10  have presented a method to synthesize an arbitrary quantum
state of two superconducting resonators using a control qubit. They used the
photon-number-dependent Stark shift to achieve selective manipulations of the
quantum system. Yang et al YangPRA12 have presented a scheme to engineer a
two-mode squeezed state of effective fermionic modes realized by collective
excitations of two distant NVEs coupled to separated transmission line
resonators (TLRs), which are interconnected by a current-biased Josephson-
junction superconducting qubit. By making use of the engineered NVE-TLR
magnetic coupling with Raman transition between the ground sublevels of the
NVEs, they may manipulate the artificial reservoir by tuning the external
driving fields. Up to now, Fock states BertetPRL02 ; HofheinzNature08 ,
Schr$\ddot{o}$dinger cat state BrunePRL96 ; AuffevesPRL03 ; MeunierPRL05  and
entangled states RauschenbeutelSCi00  have been produced in cavity or circuit
quantum electrodynamics experiments.

Motivated by these works, we propose a scheme to engineer arbitrary entangled
states of two distant NVEs coupled to a SCM with a tunable circuit qubit. For
example, multi-dimensional entangled states, NOON state and entangled coherent
states of NVEs are produced. Treating the collective NVEs spin as a fermionic
mode, we can set up an effective Jaynes-Cummings model (JCM) SPRA08 . The main
idea of this scheme is that the frequency and intensity of the ”shift” pulse
can be modified to control the coupling between the NVEs and the circuit
qubit. So by performing a sequence of microwave pulses, the circuit qubit
alternately resonantly interacts with two NVEs. The present approach has the
following merits: (i) we can realize arbitrary control of entanglement between
two NVES by selective manipulation of the control qubit. (ii) The cavity field
would not be excited in the whole process, the interaction is a virtual-photon
process. In other words, our model works well in the bad-cavity limit, which
makes it more applicable to current laboratory techniques. (iii) More
importantly, our idea can be generalized to generate entangled states for two
or more NVEs, providing a potentially practical tool for large-scale one-way
quantum computation.

##  II Physical model and effective dynamics

Figure 1: (Color online)(a)The experimental setup for arbitrary controlling of
entanglement between two NV center ensembles coupling to superconducting
circuit qubit in a SCM cavity with $d\sim 10\mu m$ and $L\sim 1cm$. The length
of the orange lines indicates the strength of the cavity-mode electric field.
The blue lines that encircle the center conductor depict the magnetic field
lines at the locations where their strength is maximum. The presence of a
transmon qubit at an electric-field maximum ensures that the cavity has a
large nonlinearity. The NV centers ensemble of a diamond crystal trapped $\sim
10\mu m$ above the cavity structure. (b) Configurations of the transmon qubit
and the NV center ensemble level structure and relevant transitions.

The basic idea of this work is illustrated by the schematic setup shown in
Fig.1. Two NVEs (denoted 1 and 2) with different frequencies are coupled to a
superconducting circuit qubit in a SCM cavity. The circuit qubit is introduced
at an electric-field maximum of the superconducting cavity. In contrast, each
of the NVEs is separately placed at the corresponding locations where the
magnetic field is maximum. The SCM cavity with quality factor $\sim 10^{4}$
and frequency $\omega_{c}$, serves as a quantum bus, which is
electrostatically coupled to the qubit and magnetically coupled to the NVEs.
The NVEs consists of $\sim 10^{11}$ negatively changed NV color centers in a
diamond crystal. The ground state of NV center is a spin triplet, which
labeled as ${}^{3}A$. There is a zero-field splitting (2.88GHz) between the
state $\left|0\right\rangle$ ($m_{s}=0$) and $\left|\pm
1\right\rangle$($m_{s}=\pm 1$) for the spin-spin interaction LloydSCi93 . We
consider the ground state $|m_{s}=0\rangle=|0\rangle_{NV}$ and the excited
state $|m_{s}=\pm 1\rangle=|1\rangle_{NV}$, and the corresponding transition
energy is $\hbar\omega_{NV}$. The tunable qubit $Q$ is a transmon qubit, with
ground state $|g\rangle$ and excited state $|e\rangle$, which are separated by
level energy $\hbar\omega_{T}$. Meanwhile, the qubit is coupled to a nonlinear
resonator which is used to readout its state or apply a driving microwave, as
in related circuit QED experiments MalletNP09 . Through the nonlinear
resonator, the qubit is driven by a microwave fields with tunable Rabi
frequency $\Omega_{(t)}$ and frequency $\omega_{d}$. The NV center ensembles
and the qubit may be tuned in and out of resonance with each other by
controlling the microwave pulses.

In our model, the transmon qubit alternately interacts with one of the two
NVEs. So here, we fristly consider a combined system of a circuit qubit, a SCM
cavity and one NVEs, in the frame rotating with the cavity frequency
$\omega_{c}$, the corresponding Hamiltonian is given by ($\hbar=1$)

$\displaystyle H$ $\displaystyle=$
$\displaystyle[g_{c}\sigma_{eg}a_{c}e^{i\Delta_{T}t}+\Omega(t)\sigma_{eg}e^{i\Delta_{d}t},$
(2)
$\displaystyle+\sum_{i=1}^{N}g_{m}\sigma_{i}^{+}a_{c}e^{i\Delta_{m}t}]+H.c.,$

where $\sigma_{eg}=|e\rangle\langle g|$ and
$\sigma_{i}^{+}=|1\rangle_{NV}\langle 0|$ denotes the raising operator of the
transmon qubit (NV center). $a_{c}$ is the annihilation operator of the
superconducting cavity mode. $g_{c}$ $(g_{m})$ denotes the electric (magnetic)
dipole coupling strength of the transmon qubit (single NV center) to the
cavity mode. The detunings for the related transitions are
$\Delta_{T}=\omega_{T}-\omega_{c}$, $\Delta_{m}=\omega_{NV}-\omega_{c}$,
$\Delta_{d}=\omega_{T}-\omega_{d}$. Under the condition
$|\Delta_{T}|,|\Delta_{d}|\gg|g_{c}|,|\Omega_{(t)}|$, the classical field and
the cavity field only induce a Stark shift. In the case
$|\Delta_{m}|\gg|g_{m}\sqrt{N}|$, the dispersive interaction between NVEs and
cavity leads to a Stark shift and a dipole coupling for the NVEs. If the
cavity field initially is in the vacuum state, the photon number will remain
Zero during the whole process. The Hamiltonian can be rewritten as ZhengPRL00

$\displaystyle H$ $\displaystyle=$
$\displaystyle(\frac{2\Omega_{(t)}^{2}}{\Delta_{d}}+\frac{g_{c}^{2}}{\Delta_{T}})S_{z}+\sum_{i,j=1}^{N}\frac{g_{m}^{2}}{\Delta_{m}}\sigma_{i}^{+}\sigma_{j}^{-}$
(4)
$\displaystyle+\sum_{i=1}^{N}\frac{g_{m}g_{c}}{2}(\frac{1}{\Delta_{m}}+\frac{1}{\Delta_{T}})(\sigma_{i}^{+}\sigma_{ge}e^{-i\xi
t}+\sigma_{i}^{-}\sigma_{eg}e^{i\xi t}),$

where $S_{z}=(|e\rangle\langle e|-|g\rangle\langle g|)/2$ and we have
discarded the constant energy ${g_{c}^{2}}/(2\Delta_{T})(|e\rangle\langle
e|+|g\rangle\langle g|)$. Introducing a collective operator
$b^{+}=1/\sqrt{N}\sum_{i=1}^{N}\sigma_{i}^{+}$, in the low excition limit,
where almost all NV centers are in the ground state, the operator $b^{+}$ and
$b$ obey approximately fermionic commutation relations $[b,b^{+}]=1$. In this
case $\sum_{i,j=1}^{N}\sigma_{i}^{+}\sigma_{j}^{-}=Nb^{+}b$, so the NVEs spin
can be considered to be an effective fermionic mode. Assuming
$\Delta_{m}=\Delta_{T}=\Delta_{d}=\Delta$, and setting
$\omega_{z}(t)={2\Omega_{(t)}^{2}}/\Delta+{g_{c}^{2}}/\Delta$,
$\omega_{b}={g_{m}^{2}N}/\Delta$ and $g=\sqrt{N}g_{m}g_{c}/\Delta$, the Eq.(2)
can be reduced to an effective J-C model,

$\displaystyle H_{eff}$ $\displaystyle=$
$\displaystyle\omega_{z}(t)S_{z}+\omega_{b}b^{+}b+g(\sigma_{ge}b^{+}+\sigma_{eg}b).$
(5)

The above equation denotes the interaction between a two-level qubit and a
fermionic system.

Figure 2: (Color online) The populations of the basic states
$|e\rangle|0\rangle_{NVE}|0\rangle_{c}$ (P1) and
$|g\rangle|1\rangle_{NVE}|0\rangle_{c}$ (P2) governed by the full Hamiltonian
in Eq.(1) (red lines) and the effective Hamiltonian in Eq.(3) (blue lines).
Parameters: $g_{m}=0.1g$, $g_{c}=10g$, $\Omega=g$, $N=10^{4}$ (a)
$\Delta=100g$ (b) $\Delta=200g$.

In order to validate the feasibility of the above physical model, we perform a
direct numerical simulation the Schrödinger equation with the full Hamiltonian
in Eq. (1) and the effective Hamiltonian in Eq.(3). To satisfy the resonant
condition $\omega_{z}(t)=\omega_{b}$, setting the parameters $g_{m}=0.1g$,
$g_{c}=10g$, $\Omega=g$, $N=10^{4}$ and $\Delta=100g$, we can get
$\omega_{z}(t)\approx\omega_{b}=g$. In the following simulation, we calculate
the temporal evolution of the system with the initial state
$|e\rangle|0\rangle_{NVE}|0\rangle_{c}$. We plot the time-dependent
populations of the basic states $|e\rangle|0\rangle_{NVE}|0\rangle_{c}$ (P1)
and $|g\rangle|1\rangle_{NVE}|0\rangle_{c}$ (P2) governed by the full
Hamiltonian in Eq.(1) (red lines in Fig.2) and the effective Hamiltonian in
Eq.(3) (blue lines in Fig.2). We can see that the population dynamics the
effective and full exhibit excellent agreement when the the condition are
satisfied. While the deviations decrease at the cost of the long evolution
time. The larger the value of $\Delta$ is, the longer the evolution time is.
Eventually, the simulation result of full Hamiltonian is almost the same as
that of the effective Hamiltonian when $\Delta=200g$. Thus, the above
approximation for the Hamiltonian is reliable as long as $\Delta$ is large
enough.

Now, we consider the circuit qubit alternately interacts with two fermionic
modes (denoted $b_{1}$ and $b_{2}$). The Hamiltonian is given as

$\displaystyle H_{eff}$ $\displaystyle=$
$\displaystyle\omega_{z}(t)S_{z}+\omega_{b_{1}}{b_{1}}^{+}b_{1}+\omega_{b_{2}}{b_{2}}^{+}b_{2}+g_{1}(\sigma_{ge}{b_{1}}^{+}+\sigma_{eg}b_{1})+g_{2}(\sigma_{ge}{b_{2}}^{+}+\sigma_{eg}b_{2}).$
(6)

Choosing the microwave pulse to satisfy $\omega_{z}(t)=\omega_{b_{1}}$, the
control qubit is resonantly coupling with the NVEs fermionic mode 1, while the
bonsonic mode 2 is far off-resonant. In this case, the transition is given by

$\displaystyle|e\rangle|n_{1}\rangle|n_{2}\rangle$ $\displaystyle\rightarrow$
$\displaystyle
e^{-i\omega_{b_{1}}t}[\cos(g_{1}\sqrt{n_{1}+1}t)|e\rangle|n_{1}\rangle-i\sin(g_{1}\sqrt{n_{1}+1}t)|g\rangle|n_{1}+1\rangle]|n_{2}\rangle,$
(7) $\displaystyle|g\rangle|n_{1}+1\rangle|n_{2}\rangle$
$\displaystyle\rightarrow$ $\displaystyle
e^{-i\omega_{b_{1}}t}[\cos(g_{1}\sqrt{n_{1}+1}t)|g\rangle|n_{1}+1\rangle-i\sin(g_{1}\sqrt{n_{1}+1}t)|e\rangle|n_{1}\rangle]|n_{2}\rangle,$
(8)

where $|n\rangle$ denotes the Fock state for the collective NVEs spin. The
system will oscillate between $|e\rangle|n_{1}\rangle|n_{2}\rangle$ and the
state $|g\rangle|n_{1}+1\rangle|n_{1}\rangle$ at an angular frequency
$g_{1}\sqrt{n_{1}+1}$. The dynamics provides a way to realizing various
interesting phenomena in this hybrid system.

##  III arbitrary entangled states

A corresponding algorithm for the synthesis of an arbitrary quantum state of
two NVEs is described as

$\displaystyle|\psi\rangle=\sum_{n_{1}=0}^{N_{1}}\sum_{n_{2}=0}^{N_{2}}c_{n_{1},n_{2}}|n_{1},n_{2}\rangle.$
(9)

where $|n_{1},n_{2}\rangle$ is Fock states of collective fermionic mode of two
NVEs.

Now we use the physics model above to generate arbitrary entangled states for
two NVEs $|n_{1},n_{2}\rangle$. For definiteness, the system state denotes
$|q,n_{1},n_{2}\rangle$ (where the qubit state is q=e or g). The resonant
interaction of the qubit with each NVE leads to creating Fock states based on
the $H_{eff}$ in Eq.(3). These resonant interactions are efficient and fast,
so that the interaction time is so short to prevent the decoherence. However,
in order to generate an arbitrary quantum state of two NVEs, an independent
state-selective qubit rotation is required. How to address the qubit between
the resonant interactions is an important problem. To answer this question, a
driving microwave pulse (Rabi frequency $\Omega_{s}$) can be used to make the
photon-number-dependent Stark shift take effect FPRL11 ; AlexandrePRA04 . This
implies that in the dispersive regime, the transmon qubit will undergo Rabi
oscillations. To make this process happen, the frequency of the driving
microwave satisfies

$\displaystyle\omega_{s}=\omega_{z}(t)+\frac{{g_{1}}^{2}}{\delta_{1}}(2n_{b_{1}}+1)+\frac{{g_{2}}^{2}}{\delta_{2}}(2n_{b_{2}}+1),$
(10)

where $\delta_{1}=\omega_{z}(t)-\omega_{b_{1}}$ and
$\delta_{2}=\omega_{z}(t)-\omega_{b_{2}}$. We set
$\lambda={g_{1}}^{2}/\delta_{1}=-{g_{2}}^{2}/\delta_{2}$, which can always be
achieved for the transmon qubit with a tunable frequency $\omega_{z}(t)$ by
applying the ”shift” pulses $\omega_{d}$. To avoid nonresonant transitions, we
assume $|\Omega_{s}|<\lambda$ and $\omega_{b1}<\omega_{z}(t)<\omega_{b2}$. So
we can simplify the frequencies of the driving microwave (Stark-shifted Rabi
pulses)

$\displaystyle\omega_{n}=\omega_{z}(t)+2\lambda\Delta n,$ (11)

where $\Delta n=n_{b_{1}}-n_{b_{2}}$ is an integer. By choosing different
values of $\omega_{n}$, we can address each of the qubits between the resonant
interactions.

Figure 3: (Color online) Schematic set of operations to generate an arbitrary
state of two NV center ensembles. In this Fock-state diagram, the state
$|q,n_{1},n_{2}\rangle$ is represented by the node at location
($n_{1},n_{2}$),where q is control qubit state (q=e or g).Interactions lead to
couplings between these states, indicated by the arrows. Three key
interactions are used: $U_{1}$ transfers quanta between the qubit and NVEs 1
(solid lines), $U_{2}$ transfers quanta between the qubit and NVEs 2 (dashed
lines), and $R$ (curved arrows) rotates the qubit.

We now show how the interactions can be used to achieve the synthesis of
arbitrary joint state of two NVEs. Assuming that the control qubit is in the
ground state and the collective modes of NVEs are in the vacuum states, the
initial state of the system is

$\displaystyle|\psi_{0}\rangle=|g,0,0\rangle.$ (12)

Our goal is to force the system to evolve into a final state of the form

$\displaystyle|\psi_{(t)}\rangle=\sum_{n_{1}=0}^{N_{1}}\sum_{n_{2}=0}^{N_{2}}c_{n_{1},n_{2}}|g,n_{1},n_{2}\rangle.$
(13)

The time evolution operator of the system can be expressed as a product of
evolution operators associated with the time intervals, which is accomplished
by the following sequence of operations:

$\displaystyle
U(t)=[\prod_{j=1}^{N_{2}}(\prod_{k=0}^{N_{1}}U_{2,jk}R_{2,jk})]\prod_{j=1}^{N_{1}}U_{1,j}R_{1,j},$
(14)

where $U_{1,j}$ and $U_{2,j}$ describe the evolution due to resonantly
interaction between the NVEs and the qubit, corresponding to Eq.(5). When the
microwave $\Omega_{s}$ turns off, $U_{1,j}$ and $U_{2,j}$ do work. $R_{1,j}$
and $R_{2,j}$ describe the evolution due to the single-qubit rotations, which
use the Stark-shifted Rabi pulses.
$R|\psi_{(t)}\rangle=\sum_{n_{1}=0}^{N_{1}}\sum_{n_{2}=0}^{N_{2}}r_{q,n_{1},n_{2}}|g,n_{1},n_{2}\rangle$.
If the collective fermionic modes in different NVEs satisfy
$n_{1}-n_{2}=\Delta n$, the factors $r_{q,n_{1},n_{2}}$ are

$\displaystyle r_{e,n_{1},n_{2}}$ $\displaystyle=$ $\displaystyle
e^{-i\alpha}\cos(\Omega_{s}t)c_{e,n_{1},n_{2}}-ie^{-i\beta}\sin(\Omega_{s}t)c_{g,n_{1},n_{2}}$
(15) $\displaystyle r_{g,n_{1},n_{2}}$ $\displaystyle=$ $\displaystyle
e^{i\alpha}\cos(\Omega_{s}t)c_{g,n_{1},n_{2}}-ie^{i\beta}\sin(\Omega_{s}t)c_{e,n_{1},n_{2}}.$
(16)

To determine the precise sequence of operations for a given state
$|\psi_{(t)}\rangle$, one solves the equation of inverse evolution,

$\displaystyle|g,0,0\rangle$ $\displaystyle=$ $\displaystyle
U_{(t)}^{{\dagger}}|\psi_{(t)}\rangle$ (17) $\displaystyle=$
$\displaystyle\prod_{j=1}^{N_{1}}R_{1,j}^{{\dagger}}U_{1,j}^{{\dagger}}[\prod_{j=1}^{N_{2}}(\prod_{k=0}^{N_{1}}R_{2,jk}^{{\dagger}})U_{2,jk}^{{\dagger}}]|\psi_{(t)}\rangle.$

Each step of the sequence in the right side of Eq.(13) can remove fermionic
excition successively from the state $|\psi_{(t)}\rangle$ until all the
excition are exhausted. In the Fock-state diagram as shown in Fig.3,
$R_{2,jk}^{{\dagger}}U_{2,jk}^{{\dagger}}$ acts on $|\psi_{(t)}\rangle$, and
moves the system along the vertical paths. All populations in
$|g,n_{k},n_{j}\rangle$ and $|e,n_{k},n_{j-1}\rangle$ are transferred to
$|g,n_{k},n_{j-1}\rangle$. After all the excitions have been removed from the
columns in row $j$, the sequence repeats for $j-1$. The step will not stop
until the excition of NVES 2 fermionic mode is zeno, and then the system state
is $|g,n_{k},0\rangle$. Once there, the
$R_{1,j}^{{\dagger}}U_{1,j}^{{\dagger}}$ sequence moves population along the
horizontal paths to $|g,0,0\rangle$. By counting the number of operations in
$U$, we find that the general sequence requires $N_{1}$ $U_{1,j}^{{\dagger}}$
unitaries, $(N_{1}+1)N_{2}$ $U_{2,j}^{{\dagger}}$ unitaries, and about
$(N_{1}+1)(N_{2}+1)$ Rabi pulses. So the total interaction time is
approximately given by

$\displaystyle
t_{max}=(N_{1}+1)(N_{2}+1)\frac{\pi}{\Omega_{s}}+\sum_{j=1}^{N_{1}}\frac{\pi}{2\sqrt{j}g_{1}}+(N_{1}+1)\sum_{j=1}^{N_{2}}\frac{\pi}{2\sqrt{j}g_{2}}$
(18)

###  III.1 NOON states of NVEs

The maximally entangled $N-$photon states are described as

$\displaystyle|\psi\rangle=\frac{1}{\sqrt{N+1}}\sum_{k=0}^{N}|k,N-k\rangle,$
(19)

and the NOON states is in the from

$\displaystyle|\psi\rangle=\frac{1}{\sqrt{2}}(|N_{1},0\rangle+|0,N_{2}\rangle).$
(20)

States with the latter form have an advantage in its sensitivity for optical
interferometry over a coherent state and can achieve the Heisenberg limit of
$1/N$ in accuracy of phase measurement.

In order to generate NOON states of two NVEs, the NVEs collective modes are
initially prepared in vacuum states and we apply a Rabi pulse resonant with
the control qubit. When the time is chosen as $\Omega_{s}t=\pi/2$ , the qubit
is prepared in the state $|e\rangle$. Secondly, we turn off the Rabi pulse,
and then a controllable ”shift” pulse should be applied through the nonlinear
element. $\Omega$ and $\Delta_{d}$ are chosen to satisfy
$\omega_{z}(t)=\omega_{b1}$. The resonant interaction of NVEs 1 and qubit is
fast. Meanwhile, the detuning between the qubit and the NVEs 2 is large, so we
can consider the NVEs 2 has no affect during the interaction in this step.
After an interaction time $t_{1}$, the system evolves into the state

$\displaystyle|\psi(t_{1})\rangle=e^{-i\omega_{b1}t_{1}}(\cos(g_{1}t_{1})|e\rangle|0,0\rangle-i\sin(g_{1}t_{1})|g\rangle|1,0\rangle).$
(21)

Then we apply a Rabi pulse resonant to realize ”$\pi$” gate. After the Rabi
pulse has been turned off, a different ”shift” pulse is used, which brings a
resonantly coupling between the qubit and the NVEs 2, while the coupling of
the circuit qubit and the NVEs 1 is out. After an interaction time $\tau_{1}$,
the system evolves into the state

$\displaystyle|\psi(\tau_{1})\rangle$ $\displaystyle=$ $\displaystyle-
ie^{-i(\omega_{b1}t_{1}+\omega_{b2}\tau_{1})}[\cos(g_{1}t_{1})(\cos(g_{2}\tau_{1})|e\rangle|0,0\rangle$
(23)
$\displaystyle-i\sin(g_{2}\tau_{1})|g\rangle|0,1\rangle)-i\sin(g_{1}t_{1})|g\rangle|1,0\rangle].$

If the interaction time is chosen as $g_{2}\tau_{1}=\pi/2$, then the system is
in the state

$\displaystyle|\psi(\tau_{1})\rangle$ $\displaystyle=$ $\displaystyle
e^{(\omega_{b1}t_{1}+\omega_{b2}\tau_{1})}[-i\cos(g_{1}t_{1})|g\rangle|0,1\rangle+\sin(g_{1}t_{1})|g\rangle|1,0\rangle].$
(24)

Now applying a Rabi pulse resonant with the control qubit, with the time
$\Omega_{s}t=\pi/2$, to get $|g\rangle\rightarrow|e\rangle$. Then adding a
shift pulse, the qubit resonantly interacts with the NVEs 1. After an
interaction time $t_{2}$, the system evolves into the state

$\displaystyle|\psi(t_{2})\rangle$ $\displaystyle=$ $\displaystyle
e^{-i[\omega_{b1}(t_{1}+t_{2})+\omega_{b2}\tau_{1}]}[\cos(g_{1}t_{1})(\cos(g_{1}t_{2})|e\rangle|0,1\rangle$
(27)
$\displaystyle-i\sin(g_{1}t_{2})|g\rangle|1,1\rangle)+\sin(g_{1}t_{1})(\cos(\sqrt{2}g_{1}t_{2})|e\rangle|1,0\rangle$
$\displaystyle-i\sin(\sqrt{2}g_{1}t_{2})|g\rangle|2,0\rangle)].$

If the interaction time is chosen as
$\cos(\sqrt{2}g_{1}t_{2})=\sin(g_{1}t_{2})=0$, the system is on the state

$\displaystyle|\psi(t_{2})\rangle$ $\displaystyle=$ $\displaystyle
e^{-i[\omega_{b1}(t_{1}+t_{2})+\omega_{b2}\tau_{1}]}(\cos(g_{1}t_{1})|e\rangle|0,1\rangle-i\sin(g_{1}t_{1})|g\rangle|2,0\rangle).$
(28)

After applying a Rabi pulse resonant to realize ”$\pi$” gate, let the qubit
resonantly interacts with the NVEs 2 through another shift pulse. With an
interaction time $\tau_{2}$, which satisfies $\cos(\sqrt{2}g_{2}\tau_{2})=0$,
the system is in the state

$\displaystyle|\psi(\tau_{2})\rangle$ $\displaystyle=$ $\displaystyle
e^{-i[\omega_{b1}(t_{1}+t_{2})+\omega_{b2}(\tau_{1}+\tau_{2})]}[|g\rangle\otimes(\cos(g_{1}t_{1})|0,2\rangle+\sin(g_{1}t_{1})|2,0\rangle)],$
(29)

Repeating the process $N$ times, discarding the phase factor and choosing a
suitable interaction time each time
($\cos(\sqrt{N}g_{1}t_{N})=\sin(\sqrt{N-1}g_{1}t_{N})=0$ and
$\cos(\sqrt{N}g_{2}\tau_{N})=0$), the system state will collapse onto the
state

$\displaystyle|\psi(\tau_{N})\rangle=-\frac{1}{\sqrt{2}}[|g\rangle\otimes(|0,n\rangle+|n,0\rangle)],$
(30)

here we assume that $\cos(gt_{1})=\sin(gt_{1})=1/\sqrt{2}$ and discard the
phase factor. Then the collective NVEs spin are in the multi-particle NOON
state, and the control qubit is in the ground state. From the processes above,
NOON state can be produced by transferring amplitude along certain paths in
the Fock-state diagram in Fig.3. This sequence requires a linear number of
operations. The corresponding interaction time is given by

$\displaystyle
t_{NooN}=(N_{1}+N_{2}-\frac{1}{2})\frac{\pi}{\Omega_{s}}+\frac{\pi}{4g_{1}}+\sum_{j=2}^{N}\frac{\pi}{(2\sqrt{j})g_{1}}+\sum_{j=1}^{N}\frac{\pi}{2\sqrt{j}g_{2}},$
(31)

###  III.2 multi-dimensional entangled states of NVEs

The multi-dimensional entangled states (MDES) are described as

$\displaystyle|\psi\rangle=\frac{1}{\sqrt{N+1}}\sum_{k=0}^{N}|k,k\rangle,$
(32)

Compared with low-dimensional entanglement states, high-dimensional
entanglement states, i.e., entangled qudits (the dimension $d\geq 3$), has
been proved to be stronger in the violations of local realism
KaszlikowskiPRL00 ; CollinsPRL02 , and more resilience to error than two-
dimensional systems FujiwaraPRL03 . Besides, quantum cryptographic protocols
where qubits are replaced with qudits include higher information density
coding DurtPRA04  and realize the schemes fasterVaziriPRL02 . Bell
inequalities and time-energy degree of freedom for multipartite qudits have
been studied SonPRL06 ; RichartAPB12 . The application of qudits offers
interesting alternatives. For example, they allow the reduction of elementary
gates, and consequently of the number of physical information carriers.
Recently, some theoretical schemes have been proposed for implementing three-
dimensional entangled state of atoms in a cavity QED system DelgadoPLA07 ;
YePRA08 ; LinPRA07 . For instance, Song et al.songEPJD08  have proposed a
scheme to generate a maximally four-dimensional entangled state in two remote
cavities. In this scheme, by choosing suitable intensities and detunings,
atomic spontaneous radiation and photon leakage out of cavity and fiber are
efficiently suppressed. Shen et al. shenJPB11  also have proposed a scheme to
generate a four-dimensional entangled state between two atoms trapped in two
separate cavities via quantum Zeno dynamics.

Now, we will show how to produce high-dimensional entangled states of NVEs in
this physical model. The two NVEs collective modes are initially prepared in
vacuum states and the control qubit is initially prepared in the excited state
$|e\rangle$. The first step, let $H_{I}$ takes effect in Eq.(4). A
controllable ”shift” pulse is applied to satisfy the resonant interaction
between the control qubit and NVES 1, while the NVEs 2 is not affected during
the interaction for the large detuning. After an interaction time $t_{1}$, A
Rabi pulse is used to realize ”$\pi$”gate, and then we add another pulse to
let $R$ take effect, i.e. $|e\rangle\rightarrow|g\rangle$ and
$|g\rangle\rightarrow|e\rangle$. Then, a controllable ”shift” pulse is applied
to satisfy the resonant interaction between the control qubit and NVEs 2,
while the NVEs 1 is large detuning from the control qubit. After an
interaction time $\tau_{1}$, the system evolves into the state

$\displaystyle|\psi(t)\rangle$ $\displaystyle=$ $\displaystyle
e^{-i(\omega_{b1}t_{1}+\omega_{b2}\tau_{1})}[\cos(g_{1}t_{1})|g,0,0\rangle$
(34)
$\displaystyle-i\sin(g_{1}t_{1})(\cos(g_{2}\tau_{1})|e,1,0\rangle-i\sin(g_{2}\tau_{1})|g,1,1\rangle)].$

Choosing the interaction time $t_{1}$ and $\tau_{1}$, which satisfy
$\cos(g_{1}t_{1})=1/\sqrt{2}$ and $\cos(g_{2}\tau_{1})=0$, and discarding the
phase factor, the system state can be reduced to

$\displaystyle|\psi(t)\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}[|g\rangle\otimes(|0,0\rangle+|11\rangle)].$
(35)

The control qubit is in the ground state and the two NVES are in a two-
dimensional entangled state.

The second step, we use a Rabi pulse to realize a single qubit rotation, i.e.
$|g\rangle\rightarrow|e\rangle$. Implementing the process above once again,
after an interaction time $t_{2}$ and $\tau_{2}$, the system evolves into the
state

$\displaystyle|\psi(t)\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}e^{-i(\omega_{b1}t_{2}+\omega_{b2}\tau_{2})}[\cos(g_{1}t_{2})|g,0,0\rangle$
(39)
$\displaystyle-i\sin(g_{1}t_{2})(\cos(g_{2}\tau_{2})|e,1,0\rangle-i\sin(g_{2}\tau_{2})|g,1,1\rangle)$
$\displaystyle+\cos(\sqrt{2}g_{1}t_{2})(\cos(g_{2}\tau_{2})|g,1,1\rangle-i\sin(g_{2}\tau_{2})|e,1,0\rangle)$
$\displaystyle-i\sin(\sqrt{2}g_{1}t_{2})(\cos(\sqrt{2}g_{2}\tau_{2})|e,2,1\rangle-i\sin(\sqrt{2}g_{2}\tau_{2})|g,2,2\rangle)].$

Choosing the interaction time $t_{2}$ and $\tau_{2}$, which satisfy
$\sin(\sqrt{2}g_{1}t_{2})=\cos(g_{1}t_{2})$,
$\sin(g_{1}t_{2})\cos(g_{2}\tau_{2})+\cos(\sqrt{2}g_{1}t_{2})\sin(g_{2}\tau_{2})=0$
and $\cos(\sqrt{2}g_{2}\tau_{2})=0$, and discarding the phase factor, the
system state can be reduced to

$\displaystyle|\psi(t)\rangle$ $\displaystyle=$
$\displaystyle|g\rangle\otimes(c_{00}|0,0\rangle+c_{11}|1,1\rangle+c_{22}|2,2\rangle),$
(40)

where the control qubit is in the ground state and the two NVES are in the
three-dimensional entangled states. Repeating the process $N$ times,
discarding the phase factor and choosing a suitable interaction time each
time, the NVEs states will collapse onto the N-dimensional entangled states.
The corresponding interaction time is given by

$\displaystyle
t_{MDES}=(N_{1}+N_{2}-\frac{1}{2})\frac{\pi}{\Omega_{s}}+\sum_{j=1}^{N}\frac{\pi}{(2\sqrt{j}+2)g_{1}}+\sum_{j=1}^{N}\frac{\pi}{2\sqrt{j}g_{2}},$
(41)

###  III.3 entangled coherent states of NVEs

We consider a transmon qubit coupled to the collective excition modes of two
NVEs with different frequencies. In the strong coupling regime, $g\gg\kappa$,
where dissipation can be neglected. We can define the Hamiltonian for the
system

$\displaystyle H$ $\displaystyle=$ $\displaystyle H_{0}+H_{int},$
$\displaystyle H_{0}$ $\displaystyle=$
$\displaystyle\delta_{1}b_{1}^{+}b_{1}+\delta_{2}b_{2}^{+}b_{2}+\Omega_{s}(t)\sigma_{eg}+\Omega_{s}^{*}(t)\sigma_{ge},$
$\displaystyle H_{int}$ $\displaystyle=$ $\displaystyle
g_{1}(\sigma_{ge}b_{1}^{+}+\sigma_{eg}b_{1})+g_{2}(\sigma_{ge}b_{2}^{+}+\sigma_{eg}b_{2})$
(42)

In the interaction picture, the Hamiltonian changes to

$\displaystyle H_{int}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(|+\rangle\langle+|-|-\rangle\langle-|+e^{i2\Omega_{s}t}|+\rangle\langle-|+e^{-i2\Omega_{s}t}|-\rangle\langle+|)$
(43)
$\displaystyle\times(g_{1}b_{1}e^{-i\delta_{1}t}+g_{2}b_{2}e^{-i\delta_{2}t})+H.c.$

Where the dressed states
$\left|\pm\right\rangle=(\left|g\right\rangle\pm\left|e\right\rangle)/\sqrt{2}$
are the eigenstates of $\sigma_{x}=\sigma_{eg}+\sigma_{ge}$ with the
eigenvalues $\pm 1$, respectively. In the strong driving limit
$\Omega_{s}>>\\{\delta,g\\}$, we can realize a rotating-wave approximation and
neglect the terms that oscillate fast. The effective Hamiltonian
$H_{eff}^{{}^{\prime}}$ can be written asSolanoPRL03

$\displaystyle
H_{eff}^{{}^{\prime}}=\frac{\sigma_{x}}{2}(g_{1}b_{1}e^{-i\delta_{1}t}+g_{2}b_{2}e^{-i\delta_{2}t})+H.c.$
(44)

If the NVEs collective modes is prepared in vacuum state and the qubit is
prepared in the ground state $|g\rangle$. The initial state of the system is
$|g\rangle|0\rangle|0\rangle=(|+\rangle+|-\rangle)|0\rangle|0\rangle/\sqrt{2}$,
the system at time $t$ will be

$\displaystyle|\psi(t)\rangle=\frac{1}{\sqrt{2}}(|+\rangle|\alpha\rangle|\beta\rangle+|-\rangle|-\alpha\rangle|-\beta\rangle)$
(45)

with $\alpha=g_{1}(e^{i\delta_{1}t}-1)/2\delta_{1}$,
$\beta=g_{2}(e^{i\delta_{2}t}-1)/2\delta_{2}$. Taking a measurement of the
control qubit will produce even or odd coherent state of two NVEs.

##  IV Discussion

Now we investigate the fidelity of the arbitrary state of two NVEs. Here we
take the collective fermionic of NVEs $(n_{1}=n_{2}=3)$ for example. The
fidelity of the prepared states is given by

$\displaystyle F_{total}=\prod_{n_{1}}\prod_{n_{2}}F_{n_{1},n_{2}},$ (46)

and

$\displaystyle
F_{n_{1},n_{2}}=\left\langle\Psi_{(t)}|\rho|\Psi_{(t)}\right\rangle,$ (47)

where $\left|\Psi_{(t)}\right\rangle$ is the final state described by the
Schrödinger equation
$i(d\left|\Psi_{(t)}\right\rangle/dt)=H\left|\Psi_{(t)}\right\rangle$. Here
$H$ is the full Hamiltonian governed by Eq.(1). Setting the Rabi frequency of
microwave $\Omega=g$ and the detuning $\Delta=100g$, we plot the fidelity as
functions of frequency of microwave and the detuning between cavity and NVEs,
as is shown in Fig.4. One can find that the fidelity keeps to be high values
$(\geq 0.993)$ when the scaled frequency $\Omega/g$ is evaluated within the
domain $[1,5]$, as shown in Fig4.(a). The result shows the optimal fidelity of
entangled states is almost unaffected when the fluctuation of the Rabi
frequency of the classical field becomes large, which will reduce the
difficulty in the experiment. The fidelity is larger than 0.90 when
$\Delta/g\geq 60$, and the fidelity keeps to be high values $(\geq 0.992)$
when the scaled $\Delta/g\geq 100$, as is shown in Fig4.(b). We see that, in a
specific range, the larger the detuning is, the higher the fidelity is.
However, the large detuning prolongs the evolution time, which will lead to
the worse impacts of decoherence. It would be interesting to perform a
numerical analysis taking into account the the effect spontaneous emission of
the NV centers or superconducting circuit qubit, and cavity losses. The master
equation of the whole system can be expressed by

$\displaystyle\dot{\rho}$ $\displaystyle=$
$\displaystyle-i\left[H,\rho\right]+\frac{\kappa}{2}(2a\rho
a^{+}-aa^{+}\rho-\rho{a}^{+}a)$ (48)
$\displaystyle+\sum_{j=1}^{N}\frac{\gamma}{2}(2{\sigma}_{01}^{j}\rho{\sigma}_{10}^{j}-{\sigma}_{01}^{j}{\sigma}_{10}^{j}\rho-\rho{\sigma}_{10}^{j}{\sigma}_{01}^{j})$
$\displaystyle+\frac{\gamma}{2}(2{\sigma}_{ge}\rho{\sigma}_{eg}-{\sigma}_{eg}{\sigma}_{ge}\rho-\rho{\sigma}_{eg}{\sigma}_{ge}),$

where $\kappa$ denotes the effective decay rate of the cavity. For simplicity,
we assume $\gamma_{eg}=\gamma_{10}=\gamma$, where $\gamma$ represents the
branching ration of the spontaneous decay from level $\left|e\right\rangle$ to
$\left|g\right\rangle$ in control qubit or $\left|1\right\rangle$ to
$\left|0\right\rangle$ in NVEs. By solving the master equation numerically, we
obtain the relation of the fidelity versus the scaled ratio $\gamma/g$ and
$\kappa/g$ with $\Delta/g=100$ in Fig.5. We see that the spontaneous emission
dominates the reduction of fidelity, while the decay rates of the cavity
influence the fidelity slightly, which can be understood by the virtual
excitation of the cavity field mode. When set $\Delta/g=200$, the fidelity
will be larger, while the time needed will be longer.

Figure 4: (Color online) The fidelity of an arbitrary state of two NVEs
($n_{1}=n_{2}=3$) versus the experiment parameters, (a) the scaled Rabi
frequency $1\leq\Omega/g\leq 5$, $\Delta=100g$ (b) the scaled detuning
$40\leq\Delta/g\leq 200$, $\Omega=g$. The other parameters are $g_{m}=0.1g$,
$g_{c}=10g$, $N=10^{4}$. Figure 5: (Color online) The fidelity of an arbitrary
state of two NVEs ($n_{1}=n_{2}=3$) versus the scaled cavity decay rate
$\kappa/g$ and atomic spontaneous emission rate $\gamma/g$. $\Delta/g=100$,
$\Omega=g$ and the other parameters are $g_{m}=0.1g$, $g_{c}=10g$, $N=10^{4}$.

In order to bridge the difference in frequency between qubit and NVEs, the SCM
cavity frequency $\omega_{c}$ can be tuned on a nanosecond time scale by
applying current pulses through an on chip flux line, inducing a magnetic flux
through a SQUID embedded in cavity PalaciosJLTP08 . $\omega_{c}$ is varied in
order to transfer coherently quantum information between qubit and NVEs.
Single qubit rotation and flux pulses placing NVEs and cavity in and out
resonance have been realized in Hybrid quantum circuit system KuboPRL11 .
Qubit state readout is performed by measuring the phase of a microwave pulse
reflected on the nonlinear resonator, which depends on the qubit state; the
probability $P_{e}$ to find the qubit in $|e\rangle$ is then determined by
repeating $\sim 10^{4}$ times the same experimental sequence. In this scheme,
we have ignored the phase factors that arise when adding the different shift
pulse. These phase can be corrected by including brief pauses between the Rabi
and shift pulsesLNature09 , and these do not significantly affect the time
evolution during the whole process.

To satisfy the requirement of the physical model, the relevant cavity QED
parameters $[g,g_{m},g_{c},\Omega,\Delta]/2\pi=[0.15,0.015,1.5,0.15,15]$ GHz
KuboPRL11 . By tuning the transmon qubit or the cavity to produce a large
detuning of the NVEs transitions or transmon qubit from the cavity, the
decoherence of the NVEs-qubit system will be reduced by $\sim 10^{6}$APRL09 .
The total parparing time of the arbitrary state of two NVEs is associated with
the coupling strength $g_{i}$ in according to the Eq.(14). Because of
$g=\sqrt{N}g_{m}g_{c}/\Delta$, we emphasize that the growth of the number of
NV centers in each spin ensemble could greatly reduce the operation time. In
this scheme, we assume $|\Omega_{s}|<\lambda$ to avoid nonresonant
transitions, where $\lambda={g}^{2}/(\omega_{z}(t)-\omega_{b})$. According to
the parameters $\lambda=50g$ and $|\Omega_{s}|=5g$, the generation of an
arbitrary state of two NVEs with $n_{1}=n_{2}=3$ will take only $200ns$. The
time compares quite favorably to the coherence time of qubit, which is now
consistently greater than $0.5\mu s$LNature09 . What’more, The implement time
is also shorter than the coherence time of NVEMizuochiPRB09 . So it is
possible to efficiently manipulate and measure the entangled states of NVEs in
experiment.

##  V conclusion

In summary, we have shown how to realize the J-C Model with the collective
NVEs fermionic mode. The model provides a possibility for engineering
arbitrary entangled states of two distant NVEs coupled to a superconducting
cavity with a tunable quantum qubit. The coupling between the qubit and the
NVEs is induced by the nonresonant cavity mode, which is always in the vacuum
state. Thus the evolution of the system is insensitive to cavity decay. With
the help of shift microwave pulse, the qubit alternately resonantly interacts
with the two NVEs. The idea can also be used for preparation of NOON states,
N-dimensional entangled states and coherent states of two NVEs. The scheme can
be applied to more NVEs and opens up a way to implement quantum information
processing with NVEs-circuit cavity system.

##  VI acknowledge

This work is supported from the National Natural Science Foundation of China
under Grant No. 11405031, Research Program of Fujian Education Department
under Grant No. JA14044.

## References

  * (1) J. M. Raimond, M. Brune, and S. Haroche, Rev. Mod. Phys. 73, 565 (2001).
  * (2) Y.Makhlin,G. Schon, and A. Shnirman, Rev. Mod. Phys. 73, 357 (2001);
  * (3) A. Wallraff, D. I. Schuster, A. Blais, et. al., Nature (London) 431, 162 (2004).
  * (4) A. Blais, R. S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. A 69, 062320 (2004).
  * (5) M. Mariantoni, H. Wang, R. C. Bialczak, et.al., Nat. Phys.7, 287 (2011).
  * (6) J. Q.You and F. Nori, Nature 474 589(2011).
  * (7) A.Imamo$\breve{g}$lu, Phys. Rev. Lett., 102,083602(2009).
  * (8) Y. Kubo, F. R. Ong, P. Bertet, et.al., Phys. Rev. Lett. 105, 140502 (2010).
  * (9) R. Ams$\ddot{u}$ss, Ch. Koller, et. al., Phys. Rev. Lett. 107, 060502 (2011).
  * (10) D. I. Schuster, A. P. Sears, E. Ginossar, et. al., Phys. Rev. Lett. 105, 140501 (2010).
  * (11) C. K. Law and J. H. Eberly, Phys. Rev.Lett. 76, 1055 (1996).
  * (12) Frederick W. Strauch, Kurt Jacobs, and Raymond W. Simmonds, Phys. Rev. Lett. 105, 050501 (2010).
  * (13) W. L. Yang, Z. Q. Yin, Q. Chen, et. al. Phys. Rev. A 85 022324(2012).
  * (14) Zhi Rong Zhong, Optics communications 283 189(2010).
  * (15) Przemyslaw Sadowski, arXiv:1303.3757
  * (16) P. Bertet et al., Phys. Rev. Lett. 88, 143601 (2002).
  * (17) M. Hofheinz et al., Nature (London) 454, 310 (2008).
  * (18) M. Brune et al., Phys. Rev. Lett. 77, 4887 (1996).
  * (19) A. Auffeves et al., Phys. Rev. Lett. 91, 230405(2003).
  * (20) T. Meunier et al., Phys. Rev. Lett. 94, 010401 (2005).
  * (21) A. Rauschenbeutel et al., Science 288, 2024 (2000).
  * (22) S B Zheng, Phys. Rev. A 77 045802(2008)
  * (23) S. Lloyd, Science, 261,1569 (1993).
  * (24) F. Mallet et al., Nature Phys. 5, 791 (2009).
  * (25) S B Zheng and G C Guo, Phys. Rev. Lett., 85,2392(2000)
  * (26) F. R. Ong, M. Boissonneault, F. Mallet, et al, Phys. Rev. Lett., 106,167002(2011)
  * (27) Alexandre Blais et al., Phys. Rev. A 69,062320(2004)
  * (28) D. Kaszlikowski, P. Gnacinski, M. Zukowski, W. Mik- laszewski, and A. Zeilinger, Phys. Rev. Lett. 85, 4418(2000).
  * (29) D. Collins, N. Gisin, N. Linden, S. Massar, and S. Popescu, Phys. Rev. Lett. 88, 040404 (2002).
  * (30) M. Fujiwara,M. Takeoka, J.Mizuno, and M. Sasaki, Phys. Rev. Lett. 90,167906 (2003).
  * (31) T. Durt, D. Kaszlikowski, J. -L. Chen, and L. C. Kwek, Phys. Rev. A 69, 032313 (2004).
  * (32) Vaziri A, Weihs G and Zeilinger A, Phys. Rev. Lett. 89, 240401 (2002).
  * (33) Son W, Lee J Y and Kim M S, Phys. Rev. Lett. 96, 060406 (2006).
  * (34) Richart D, Fischer Y, Weinfurter H, Appl Phys B 106, 543 (2012).
  * (35) A. Delgado, C. Saavedra, and J. C. Retamal, Phys. Lett. A 370, 22 (2007).
  * (36) S.Y. Ye, Z. R. Zhong, and S. B. Zheng, Phys. Rev. A 77, 014303(2008).
  * (37) G. W. Lin, M. Y. Ye, L B Chen, et al, Phys. Rev. A 76,014308 (2007).
  * (38) Song J, Xia Y, Song H S and Liu B, Eur. Phys. J. D 50, 91 (2008).
  * (39) Shen L T, Wu H Z and Chen R X, J.Phys. B: At. Mol. Opt. Phys. 44,205503 (2011).
  * (40) E. Solano, G. S. Agarwal and H. Walther.Phys. Rev. Lett., 90, 027903 (2003) .
  * (41) A. Palacios-Laloy et al., J. Low Temp. Phys. 151, 1034 (2008).
  * (42) Y. Kubo, C. Grezes, A. Dewes, et al, Phys. Rev. Lett. 107,220501(2011).
  * (43) L.Di Carlo et al.,Nature(London)460,240(2009).
  * (44) N. Mizuochi, P. Neumann, F. Rempp,_et al._ , Phys. Rev. B 80,041201 (2009) .