# Generation and complete nondestructive analysis of hyperentanglement assisted by nitrogen-vacancy centers in resonators

## Abstract

We present two efficient schemes for the deterministic generation and the complete nondestructive analysis of hyperentangled Bell states in both the polarization and spatial-mode degrees of freedom (DOFs) of two-photon systems, assisted by the nitrogen-vacancy (NV) centers in diamonds coupled to microtoroidal resonators as a result of cavity quantum electrodynamics (QED). With the input-output process of photons, two-photon polarization-spatial hyperentangled Bell states can be generated in a deterministic way and their complete nondestructive analysis can be achieved. These schemes can be generalized to generate and analyze hyperentangled Greenberger-Horne-Zeilinger states of multi-photon systems as well. Compared with previous works, these two schemes relax the difficulty of their implementation in experiment as it is not difficult to obtain the phase shift in single-sided NV-cavity systems. Moreover, our schemes do not require that the transmission for the uncoupled cavity is balanceable with the reflectance for the coupled cavity. Our calculations show that these schemes can reach a high fidelity and efficiency with current technology, which may be a benefit to long-distance high-capacity quantum communication with two DOFs of photon systems.

###### pacs:

03.67.Bg, 03.67.Hk, 42.50.Pq## I Introduction

Recently, hyperentanglement, which is defined as the entanglement in
multiple degrees of freedom (DOFs) of a quantum system
(1); (2); (3), has attracted much attention as it has
some important applications in quantum information processing. It
can speedup quantum computation (e.g., hyper-parallel photonic
quantum computation (4); (5)) and
it can be used to assist the complete Bell-state analysis and
entanglement purification. In 2003, Walborn *et al.*
(6) presented a simple linear-optical scheme for the
complete Bell-state analysis of photons with hyperentanglement. In
2006, Schuck *et al.* (7) demonstrated the complete
deterministic analysis of polarization Bell states with only linear
optics assisted by polarization-time-bin hyperentanglement. In 2007,
Barbieri *et al.* (8) demonstrated the complete
and deterministic discrimination of polarization Bell states
assisted by momentum entanglement. In 2009, Wilde and Uskov
(9) proposed a quantum-error-correcting-code scheme
assisted by linear-optical hyperentanglement. In 2010, some
deterministic entanglement purification protocols (EPPs)
(10); (11); (12) were proposed with the
polarization-spatial hyperentanglement of photon systems, which can
solve the troublesome problem that a large amount of quantum
resources would be sacrificed in the conventional EPPs, and they are
very useful in practical quantum repeaters. In 2012, Wang, Song, and
Long (13) proposed an important quantum repeater
protocol with polarization-spatial hyperentanglement.

The hyperentanglement of photon systems can also be used to largely
increase the channel capacity of quantum communication. For example,
Barreiro *et al.* (14) beat the channel capacity limit
of photonic superdense coding with
polarization-orbital-angular-momentum hyperentanglement in linear
optics in 2008. In 2010, Sheng, Deng, and Long (15) gave the
first scheme for the complete hyperentangled-Bell-state analysis
(HBSA) for quantum communication, and they designed the pioneering
model for the quantum teleportation with two DOFs of photon pairs,
resorting to cross-Kerr nonlinearity. In 2011, Pisenti *et al*.
(16) pointed out the limitations for manipulation and
measurement of entangled systems with inherently linear unentangling
devices. In 2012, Ren *et al.* (17) proposed another
interesting scheme for the complete HBSA for photon systems by using
the giant nonlinear optics in quantum dot-cavity systems and they
presented the entanglement swapping with photonic
polarization-spatial hyperentanglement. In 2012, Wang, Lu, and Long
(18) introduced an interesting scheme for the complete HBSA
for photon systems by the giant circular birefringence induced by
double-sided quantum-dot-cavity systems. In 2013, Ren, Du, and Deng
(19) proposed the parameter-splitting method to extract the
polarization-spatial maximally hyerentangled photons when the
coefficients of the initial partially hyperentangled states are
known, and this fascinating method can be achieved with the maximum
success probability by performing the protocol only once, resorting
to linear-optical elements only. They (19) also gave the
first hyperentanglement concentration protocol (hyper-ECP) for
unknown polarization-spatial less-hyperentangled states with
linear-optical elements only (19). Ren and Deng
(20) presented the first hyperentanglement purification
protocol (hyper-EPP) for two-photon systems in polarization-spatial
hyperentangled states, and it is very useful in the high-capacity
quantum repeaters with hyperentanglement. In 2014, Ren, Du, and
Deng (21) gave a two-step hyper-EPP for
polarization-spatial hyperentangled states with the
quantum-state-joining method, and it has a far higher efficiency. In
the same time, Ren and Long (22) proposed a
general hyper-ECP for photon systems assisted by quantum dot spins
inside optical microcavities. Recently, Li and Ghose presented an
interesting hyper-ECP resorting to linear optics
(23) and another efficient hyper-ECP for the
multipartite hyperentangled state via the cross-Kerr nonlinearity
(24).

Another attractive candidate for solid-state quantum information
processing is the nitrogen-vacancy (NV) center in a diamond, owing
to its long decoherence time even at room temperature
(25), and its spin can be initialized and readout via a
highly stable optical transition (26); (27). By using the
coherent manipulation of an electron spin and nearby individual
nuclear spins, Dutt *et al.* (28) demonstrated a
controllable quantum register in NV centers in 2007.
Decoherence-protected quantum gates for a hybrid solid-state
register (29) was also experimentally demonstrated on a
single NV center. As this system allows for high-fidelity
polarization and detection of single electron and nuclear spin
states even under ambient conditions (30); (31); (27); (32), the
multipartite entanglement among single spins in diamond was
demonstrated by Neumann *et al.* (33) in 2008. In
2010, Togan *et al.* (34) realized the quantum
entanglement generation of an optical photon and an NV center.
Photon Fock states on-demand can be implemented in a low-temperature
solid-state quantum system with an NV center in a nano-diamond
coupled to a nearby high-Q optical cavity (35).

Recently, a combination of NV centers and microcavities, a
promising solid-state cavity quantum electrodynamics (QED) system,
has gained widespread attention
(36); (37); (38); (41); (42); (39); (40); (43); (44). One of the
microcavities is called microtoroidal resonator (MTR) with a
quantized whispering-gallery mode (WGM) and required to be of a high
Q factor and a small mode volume (45); (46). However, when MTR
couples to a fiber, its Q factor is surely degraded (38). The
single-photon input-output process from a MTR in experiment also has
been demonstrated (38). In 2009, the quantum nondemolition
measurement on a single spin of an NV center has been proposed with
a low error rate (39) and it was experimentally demonstrated
through Faraday rotation (40) in 2010. In 2011, Chen *et
al.* (42) proposed an efficient scheme to entangle separate NV
centers by coupling to MTRs. In 2013, Wei and Deng (43)
proposed some interesting schemes for compact quantum gates on
electron-spin qubits assisted by diamond NV centers inside cavities.

In this paper, we present two efficient schemes to generate deterministically hyperentangled states, i.e., hyperentangled Bell states and hyperentangled Greenberger-Horne-Zeilinger (GHZ) states, in which photons are entangled in both the polarization and spatial-mode DOFs, assisted by the NV centers in diamonds coupled to MTRs and the input-output process of photons as a result of cavity QED. We also propose a scheme to distinguish completely the 16 polarization-spatial hyperentangled Bell states, and it works in a nondestructive way. After analyzing the hyperentangled Bell states, the photon systems can be used for other tasks in quantum information processing. Compared with previous works (15); (17); (18), these two schemes relax the difficulty of their implementation in experiment as it is not difficult to obtain the phase shift in single-sided NV-cavity systems. Moreover, they do not require that the transmission for the uncoupled cavity is balanceable with the reflectance for the coupled cavity. Our calculations show that these schemes can work with a high fidelity and efficiency with current experimental techniques, which may be beneficial to long-distance high-capacity quantum communication, such as quantum teleportation, quantum dense coding, and quantum superdense coding with two DOFs of photon systems.

This article is organized as follows. In Sec. II, we will introduce the diamond-NV-center system and its single-photon input-output process. The generation of hyperentangled Bell states and hyperentangled GHZ states, and the complete nondestructive HBSA are presented in Secs. III and IV. A discussion and a summary are given in Sec. V.

## Ii A nitrogen-vacancy center in microtoroidal resonator

As shown in Fig. 1(a), an NV center, composed of a substitutional nitrogen atom and an adjacent vacancy in diamond lattice, is coupled to an MTR with a WGM. The NV center is negatively charged with two unpaired electrons located at the vacancy, and the energy-level structure of the NV center coupling to the cavity mode is shown in Fig. 1(b). The ground state is a spin triplet with the splitting at GHz between the levels and owing to spin-spin interactions. The specifically excited state, which is one of the six eigenstates of the full Hamitonian including spin-orbit and spin-spin interactions in the absence of any perturbation, such as by an external magnetic field or crystal strain, is labeled as (34) , where and are the orbital states with the angular momentum projections and along the NV axis, respectively. The optical transition is allowed between the ground state and the excited state owing to the total angular momentum conservation (34); (47).

An NV center can be modeled as a -type three-level structure with the ground state and , and the excited state is . The transitions and in the NV center are resonantly coupled to the right (R) and the left (L) circularly polarized photons with the identical transition frequency, respectively. Considering the structure in Fig.1, which can be modeled as a single-sided cavity, one can write down the Heisenberg equations of motion for this system as follows:

(1) |

where , , and are the frequencies of the photon, cavity mode, and the atomic-level transition, respectively. g is the coupling strength between the NV center and the cavity mode. and are the decay rates of the NV center and the cavity field, respectively. is the side leakage rate of the cavity. and are the input and the output field operators.

In the weak excitation approximation, the reflection coefficient in the steady state can be obtained,

For , the reflection coefficient is

(3) |

From Eqs. (II) and (3), one can see that if ,

(4) |

If the NV center is in the initial state () and a single polarized photon () is input, the photon will experience a phase shift owing to the Faraday rotation. However, if the initial state of the NV center is (), the input photon with () polarization will get a phase shift . In the resonant condition , when and , we approximately have and from Eq. (II). The change of the input photon can be summarized as follows (43):

(5) |

## Iii Photonic hyperentanglement generation

We first describe how to generate two-photon polarization-spatial
hyperentangled Bell states assisted by NV centers coupled to MTRs as
a result of cavity QED, and then extend this approach for the
generation of three-photon polarization-spatial hyperentangled GHZ states.

### iii.1 Generation of two-photon hyperentangled Bell states

A two-photon hyperentangled Bell state in both the polarization and the spatial-mode DOFs can be expressed as

(6) |

Here, and denote the right-circular polarization and the left-circular polarization of photons, respectively. () and () are the different spatial modes for photon (b). The subscripts and denote the polarization and the spatial-mode DOFs, respectively. and represent the two photons in the hyperentangled state. The four Bell states in the polarization DOF can be expressed as

(7) |

and those in the spatial-mode DOF can be written as

(8) |

The principle of our scheme for the two-photon polarization-spatial hyperentangled Bell states generation (HBSG) assisted by NV centers coupled to MTRs as a result of cavity QED is shown in Fig.2. Here SW is an optical switch and BS represents a beam splitter which can accomplish the following transformation in the spatial-mode DOF of the photons,

(9) |

Suppose that two NV centers NV and NV are initialed to the superposition states , and the two photons and with the same frequency are prepared in the same initial state . Photons and are successively sent into the device shown in Fig.2. A time interval exists between two photons, and should be less than the spin coherence time . When photon passes through the cavity, the optical switch (SW) is switched to await photon . After passing through the two NV centers, the two photons and can be entangled with the electron spins in the NV centers in the two cavities. The corresponding transformations on the states can be described as follows:

(10) | |||||

Here .

From Eq. (10), one can see that the injecting photons and pass through PBS in sequence. Each of them is split into two wave-packets. The part of photon () in the state is transmitted to path (), while the part in the state is reflected by PBS to path () and subsequently it interacts with the first NV center (named NV). The wave-packets from the spatial modes and ( and ) are mixed at the beam splitter (BS). HWP is used to keep the polarization state of the photon unchanged. After BSs, the states of the two-photon systems are divided into two groups, according to the state of NV when it is measured with the basis . So do the outcomes of the measurement on NV. The relationship between the measurement outcomes of these two NV centers and the polarization-spatial hyperentangled Bell states of the two photons is shown in Table 1.

NV | NV | hyperentangled Bell states |
---|---|---|

From Table 1, one can see that if NV is in the state and NV is in the state , the two-photon system is in the hyperentangled Bell state . When the two NV centers are in the states and , respectively, the two-photon system is in the hyperentangled Bell state . The hyperentangled Bell states and also correspond to different combinations of the states of the two NV centers. Therefore, one can generate the polarization-spatial hyperentangled Bell states of the two-photon system by measuring the states of the two NV centers. By applying a Hadamard operation on the electron-spin state, its states and can be rotated to and , respectively. The measurement on spin readout can be achieved with the resonant optical excitation (48). Other polarization-spatial hyperentangled Bell states can be obtained in a similar way or by resorting to the single-qubit operations and the acquired hyperentangled Bell state.

### iii.2 Generation of three-photon hyperentangled GHZ states

Similar to the case for two-photon polarization-spatial hyperentangled Bell states, we denote a three-photon polarization-spatial hyperentangled GHZ state as

Here , , and represent the three photons in the hyperentangled state. The eight three-photon GHZ states in the polarization DOF can be expressed as

(12) |

and the eight GHZ states in the spatial-mode DOF are

(13) |

The principle of our scheme for the generation of a three-photon polarization-spatial hyperentangled GHZ state is shown in Fig.3. Considering all the three NV centers are in the initial state , and the flying photons , , and are in the superposition state . A brief description of our scheme for hyperentangled-GHZ-state generation can be written as follows.

Each of the three photons , , and is split into two wave-packets by PBS. The photon in the state is injected into the pathes , , and , while the photon in the state is sent into the pathes , , and . The photons in the pathes and interact with NV, and those in the pathes and interact with NV. BSs mix the spatial modes and , and , and and . The states of the three photons are divided into four groups, according to the states of NV and NV. Under the condition that NV is imported, the hyperentangled states of the photons can be determined by measuring the states of the NV centers. The evolution of the whole system can be described as

(14) | |||||

Here represents the total operation by PBS, NV, NV, BS, and NV in sequence. . The relation between the outcomes of the measurements on the three NV centers and the obtained final polarization-spatial hyperentangled GHZ states is shown in Table 2.

NV | NV | NV | hyperentangled GHZ states |
---|---|---|---|

Table 2 shows that if the three NV centers are in the states , , and , respectively, the three-photon system is in the polarization-spatial hyperentangled GHZ state . When the NV centers are in the state , , and , the final hyperentangled GHZ state of the three photons is . Our scheme can in principle generate the eight deterministic hyperentangled three-photon GHZ states by measuring the states of the NV centers.

## Iv Complete nondestructive polarization-spatial hyperentangled-Bell-state analysis

The principle of our scheme for distinguishing the 16 hyperentangled Bell states in both the polarization and the spatial-mode DOFs of entangled photon pairs is shown in Fig. 4. Here, we use four NV centers coupled to four MTRs and a few linear-optical elements to achieve the complete nondestructive analysis of hyperentangled Bell states. Considering that two NV centers are both initialized in the superposition state , and a hyperentangled photon pair is in one of the 16 hyperentangled Bell states. As shown in Fig. 4, one can let photon pass through the cavities first and then photon . SW is used for the arrival of photon after photon passes through the cavity. is the time interval between photon and photon which is smaller than the spin coherence time of the NV center.

According to Eq.(5), after the photons pass through NV and NV, the evolution of the system composed of the two photons and the two NV centers is shown in Table 3. One can see that the 16 polarization-spatial hyperentangled states can be divided into four groups. If both the spins of NV and NV are changed to be , the photon pair is in one of the four hyperentangled Bell states and . If both NV and NV stay in the initial state , the photon pair is in one of the four hyperentangled Bell states and . Otherwise, when NV and NV are in the state of (or ), the photon pair is in one of the four hyperentangled Bell states and (or and ).

Initial states | Final states |
---|---|

Subsequently, one can let the photons pass through the BS and QWP which are used to transform the phase information discrimination to parity information discrimination both for the polarization DOF and the spatial-mode DOF. According to Table 3, after the photons pass through BS and QWP, the initial hyperentangled Bell state and for the same group will become

(15) |

From Table 3, one can see that the final hyperentangled Bell states of two-photon systems in the same group will be changed into four different groups. By letting the photons pass through NV and NV which has the same syndetic connection with NV and NV, one can distinguish those hyperentangled Bell states. The hyperentangled Bell states in the other three groups will have the same conditions. After passing through all the elements shown in Fig. 4, the final states of the two photons become those ones shown in Table 4.

Initial states | Final states | Initial states | Final states |
---|---|---|---|