Quantum state engineering with nitrogen-vacancy centers coupled to low-Q microresonator

Liu-Yong Cheng; Hong-Fu Wang; Shou Zhang; Kyu-Hwang Yeon

doi:10.1364/OE.21.005988

1. Introduction

Entanglement, as an unique physical resource in microscopic quantum systems, has a broad prospect of application in the current fiery quantum information and computing and testing the fundamental principles of quantum mechanics [1–5]. So preparation of entangled states in different quantum systems has been attracting great interest in recent years [3,6–10]. In previous studies, cavity quantum electrodynamics (QED) system with a atom confined in a quantized optical cavity is one of the most promising platform for quantum information processing (QIP) due to the strong coupling of qubits and short coherent interaction time [3, 11, 12]. However, the complexity of trapping single atoms precisely in microcavity is still at a groping stage. So a novel solid-state cavity QED system has been concerned, in which the nitrogen-vacancy (NV) centers in diamond are coupled to the microresonator with a quantized whispering-gallery mode (WGM) [13–17]. The NV center consists of nearest-neighbor pair of a nitrogen atom substituted for a carbon atom and a lattice vacancy in diamond. Recent researches confirm that electron spin states in NV centers have a long lifetime even at room temperature and can be manipulated by electromagnetic field or optical pulse [18, 19]. And by coupling to optical nanocavity, the zero phonon line (ZPL) emission from a single NV center can be significantly enhanced [20]. These unique characteristics make the NV center one of the most potential carrier of quantum information. Considering the scalability of the solid-state quantum devices, this microresonator-NV center system can serve as an efficient node for distributed quantum communication and computing. And connecting the nodes with optical fibre may pioneer a new route for the large scale quantum network.

In the last few years, many theoretical and experimental efforts have been devoted to quantum information and computing tasks using the NV centers coupled to the microresonator (cavity) with a quantized WGM [21–27]. But most of the previous schemes require high-quality (high-Q) cavity and strong coupling between the NV centers and WGM. In 2008, Dayan et al. expounded the single-photon input-output process from a microtoroidal resonator (MTR) with coupled optical fiber in experiment [13]. Based on the physical model in Ref. [13], Chen et al. proposed a potential scheme to deterministically entangle two separate NV centers fixed on the exterior surface of two MTRs [25]. On the other hand, the quantum non-demolition measurement of the electronic spin state of diamond NV centers has been proposed with high fidelity and low error rate [18, 28, 29].

Motivated by the former works, here we demonstrate that the fiber-MTR-NV centers composite device can generate not only two-NV centers entangled Bell states, but also two-photon polarization entangled states through MTR-assisted single-photon input-output process. And the former generation scheme can be extended to create three NV centers Greenberger-Horne-Zeilinger (GHZ) states. Then utilizing the prepared entanglement we achieve the quantum state transfer (QST) for distant NV centers or even between NV centers and photonic qubit. In our schemes, as the single photon input-output process for appropriate phase shift due to optical Faraday effect is the fundamental of implementing the entangled states and quantum state transfer, we analyze the effect of involved system parameters on the photon reflection characteristics. A few properties of the proposed schemes should be pointed out. First, the realized entanglement is prepared in the electronic spin ground states of the NV centers, which facilitates the entanglement store and future applications. Moreover, based on the MTR-NV center system as elementary unit, we can probably expand this quantum device for the large-scale quantum network with optical channels. In addition, the analysis of experimental feasibility shows that the schemes here only need the low-quality (low-Q) cavities to interact with the coupled NV centers, i.e., they are pretty applicable under the large cavity damping rate and weak coupling conditions.

2. The input-output process in MTR-NV center system

The configuration of the MTR-NV center system considered here is exhibited schematically in Fig. 1(a). An NV center, whose level structure is shown in Fig. 1(b) with the ground state of electronic spin triplet ³A and excited state of spin triplet ³E, is fixed on the surface of the MTR with WGM. For the spin triplet ³A, there is a zero-field splitting between the state |³A, m_s = 0〉 and the nearly degenerate states |³A, m_s = ±1〉, which is similar for the ³E with sub-levels |³E, m_s = 0〉 and |³E, m_s = ±1〉 as shown in Fig. 1(b) [30, 31]. To avoid the adverse effect from spontaneous emission, information is encoded in ground levels as |³A, m_s = +1〉 ≡ |+〉 and |³A, m_s = −1〉 ≡ |−〉. The excited state sub-level |³E, m_s = 0〉 is adopted as auxiliary level |e〉 [23, 25]. For the convenience of measuring the ground state information and driving suitable coupled transition, an external magnetic field is used to split the electronic spin ground state |+〉 and |−〉 as shown in Fig. 1(b). We consider the level transition between |−〉 (|+〉) and |e〉 is resonantly coupled to the left (right) circularly polarized photon |L〉 (|R〉) in MTR.

Fig. 1 Diagrammatic illustration of basic model of single photon input-output process. (a) An NV center is confined to a MTR with quantized WGM and a single photon pulse is introduced to interact with the NV center. (b) The electron energy level configuration of an NV center and the relevant transition coupling with the input polarized photon. The sophisticated level splitting of the excited states is not shown.

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The interaction between the MTR field and the NV center modelled as a Λ-type atom is governed by the basic Jaynes-Cummings (JC) model (h̄ = 1)

H = \sum_{j = R, L} [\frac{ω_{j 0}}{2} σ_{j z} + ω_{j C} a_{j}^{†} a_{j} + i g_{j} (a_{j} σ_{j +} - a_{j}^{†} σ_{j -})] + H_{R},

where a^† and a are the creation and annihilation operators of the MTR field; ω₀ and ω_C are the transition frequency of electronic energy levels and the frequency of the cavity field; σ_z, σ₊, and σ₋ are inversion, raising, and lowering operators of the NV center between the two corresponding transition levels, respectively. H_R represents the Hamiltonian involving the field and atomic reservoirs.

Then we introduce a single photon pulse with frequency ω_p to input the MTR cavity. Considering the low temperature reservoir and neglecting the vacuum input field, we can get the quantum Langevin equations for cavity field operator a and the NV center lowering operator σ₋[32–34]:

\frac{d a_{j}}{d t} = [i (ω_{p} - ω_{C}) - \frac{κ}{2}] a_{j} (t) - g σ_{j -} (t) - \sqrt{κ} a_{j, in} (t) + ζ,

\frac{d σ_{j -}}{d t} = [i (ω_{p} - ω_{0}) - \frac{γ}{2}] σ_{j -} (t) - g σ_{j, z} (t) a_{j} (t) + ξ,

by solving their Heisenberg equations of motion. Here κ and γ are the cavity decay rate and the NV center dipole decay rate, respectively. ζ and ξ are the noise operators related to reservoirs. Under the assumption of weak excitation limit, i.e., 〈σ_z〉 = −1, we can adiabatically eliminate the cavity mode and find the reflection coefficient for the photon pulse as [33]

r (ω_{p}) = \frac{a_{j, out} (t)}{a_{j, in} (t)} = \frac{[i (ω_{c} - ω_{p}) - \frac{κ}{2}] [i (ω_{0} - ω_{p}) + \frac{γ}{2}] + g^{2}}{[i (ω_{c} - ω_{p}) + \frac{κ}{2}] [i (ω_{0} - ω_{p}) + \frac{γ}{2}] + g^{2}},

where the output field

a_{j, out} (t) = a_{j, in} (t) + \sqrt{κ} a_{j} (t)

. Perhaps the input pulse encounters an NV center in detuning level, i.e. g = 0, then Eq. (4) changes to

r_{0} (ω_{p}) = \frac{i (ω_{c} - ω_{p}) - \frac{κ}{2}}{i (ω_{c} - ω_{p}) + \frac{κ}{2}} .

From Eqs. (4) and (5), it can be seen that if ω₀ = ω_C = ω_p, then

r (ω_{p}) = \frac{- κ γ + 4 g^{2}}{κ γ + 4 g^{2}}, r_{0} (ω_{p}) = - 1 .

It is this feature that enables a controlled phase flip gate on NV center and the incident photon pulse. More specifically, if a single polarized photon |L〉 is reflected from a MTR cavity after running into an NV center prepared in |−〉 or |+〉, it will experience a phase shift e^iϕ or e^iϕ⁰ from Eq. (4) or (5) owing to optical Faraday rotation, respectively. While if a single |R〉 polarized photon is reflected, it will experience a phase shift e^iϕ⁰ regardless of the electronic spin states because |+〉 state is decoupled to input pulse with frequency ω_p due to the large level splitting. Choosing ω₀ = ω_C = ω_p and

g \geq 5 \sqrt{κ γ}

, we approximately have ϕ = 0 and ϕ₀ = π from Eq. (6), which implies a controlled phase flip gate on the incident photon pulse and NV center after a π phase shifter on the photon output path,

\begin{array}{l} | R 〉 | + 〉 \to | R 〉 | + 〉, | R 〉 | - 〉 \to | R 〉 | - 〉, \\ | L 〉 | + 〉 \to | L 〉 | + 〉, | L 〉 | - 〉 \to - | L 〉 | - 〉 . \end{array}

3. Preparation of entanglement and quantum state transfer

With the optical transition selection rules of the electronic spin states shown in Fig. 1(b) and the controlled phase gate introduced in above section, we can generate several significative entangled states, such as the two NV centers entangled states achieved in Ref. [25]. In their schemes, the single-photon pulse initially prepared in a superposition of |R〉 and |L〉 interacts with the NV centers in superposition of |+〉 and |−〉 in two remote MTRs and two Bell states $| Φ^{+} 〉 = 1 / \sqrt{2} (| + + 〉 + | - - 〉)$ and $| Ψ^{+} 〉 = 1 / \sqrt{2} (| + - 〉 + | - + 〉)$ were accomplished deterministically after detection of the photon state. Interestingly, with a specific encoding form as logical qubits $| 0 〉 \equiv (| + 〉 + | - 〉) / \sqrt{2}$ and $| 1 〉 \equiv (| + 〉 - | - 〉) / \sqrt{2}$ , we here describe the generation scheme of three-qubit GHZ state of NV centers in three separated MTRs, as illustrated in Fig. 2(a). Consider all the NV centers are initially prepared in |0〉 states and introduce a single photon pulse with the initial state $(| R 〉 + | L 〉) / \sqrt{2}$ . On the basis of single photon input-output process in Eq. (6), the input photon passes through the three MTRs confining NV centers on the surface and a P₄₅ followed by a photon detector. Here P₄₅ is a 45° polarizer projecting the polarization to $(| R 〉 + | L 〉) / \sqrt{2}$ . Hence the whole evolution procedure between the input photon and three NV centers can be represented as

\begin{array}{l} \frac{| R 〉 + | L 〉}{\sqrt{2}} {| 000 〉}_{123} & \to & \frac{1}{\sqrt{2}} (| R 〉 {| 000 〉}_{123} + | L 〉 {| 111 〉}_{123}) \\ \to & \frac{1}{\sqrt{2}} (| R 〉 + | L 〉) ({| 000 〉}_{123} + {| 111 〉}_{123}) . \end{array}

So after the photon detector is triggered, the preparation of three-qubit GHZ state on separated NV centers has been accomplished. And it is evident that this scheme is straightforwardly scalable to generate N-qubit GHZ state with more MTRs and NV centers involved.

Fig. 2 Diagrammatic illustration of setups for entanglement generation and QST. (a)Generating GHZ state of NV centers in three separated MTRs via single photon pulse. (b)Generating Bell state |Φ⁻〉 using different polarization component of input photon. (c)Generating two-photon entangled state assisted by the NV center spin state. SPS: single-photon source; PBS: circular polarization beam splitter; HWP: half-wave plate; P₄₅: 45° polarizer; D: detector.

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Besides three-qubit GHZ state, for two NV centers in separated MTRs, we can directly get Bell state $| Φ^{-} 〉 = 1 / \sqrt{2} (| + + 〉 - | - - 〉)$ with the setup exhibited in Fig. 2(b). The innovation is that the input photon pulse is separated by a circular polarization beam splitter (C-PBS) into |L〉 component propagating in a transmission optical path and |R〉 component propagating in a reflection optical path. The |L〉 component interacts with the NV center 1 prepared in arbitrary single qubit state α₁|+〉 +β₁|−〉 and the |R〉 component passes thought a half-wave plate (HWP) before and after interacting with the NV center 2 prepared in α₂|+〉 + β₂|−〉. The HWP rotates the photon polarization as |R〉 ↔ |L〉. Then the |L〉 and |R〉 components from different paths are synthesized by the second C-PBS followed by a P₄₅ and a photon detector. Once the detector receives the photon signal, the two NV centers in separated MTRs collapse to |Φ⁻〉 state if we choose all coefficients meet $α_{i} = β_{i} = 1 / \sqrt{2}$ , i = 1, 2.

Notice that in the schemes above, we use the photon pulse as flying qubit to deterministically generate spin entangled states of NV centers. Next, we will show that photon polarization entanglement can be generated via electron spin of a NV center confined on the MTR surface [33]. The schematic and device description are shown in Fig. 2(c). We assume photons 1 and 2 with the same frequency are initially prepared in α₁|L〉+β₁|R〉 and α₂|L〉+β₂|R〉, respectively. They sequentially interact with the NV center in $(| + 〉 + | - 〉) / \sqrt{2}$ . According to the controlled phase flip gate given in Eq. (7), we will obtain a tripartite hybrid entangled state as

\begin{array}{l} \frac{1}{\sqrt{2}} (α_{1} α_{2} | R R + 〉 + α_{1} α_{2} | R R - 〉 + α_{1} β_{2} | R L + 〉 - α_{1} β_{2} | R L + 〉 \\ + α_{2} β_{1} | L R + 〉 - β_{1} α_{2} | L R - 〉 + β_{1} β_{2} | L L + 〉 + β_{1} β_{2} | L L - 〉) . \end{array}

Then we apply a Hadamard gate, which expresses as

| + 〉 \to (| + 〉 + | - 〉) / \sqrt{2}

and

| - 〉 \to (| + 〉 - | - 〉) / \sqrt{2}

, on the electron spin state by a π/2 microwave pulse. As a result, the entangled state in Eq. (9) becomes

\begin{array}{l} α_{1} α_{2} | R R + 〉 + α_{1} β_{2} | R L - 〉 + α_{2} β_{1} | L R - 〉 + β_{1} β_{2} | L L + 〉 \\ = | + 〉 (α_{1} α_{2} | R R 〉 + β_{1} β_{2} | L L 〉) + | - 〉 (α_{1} β_{2} | R L 〉 + α_{2} β_{1} | L R 〉) . \end{array}

This means that two-photon polarization entangled state can be accomplished in terms of the detection results on the spin state. The spin readout measurement can be achieved via resonant optical excitation as discussed in Refs. [18, 19, 23]. It is necessary to note that this photon polarization entanglement scheme is scalable to generate multi-photon entanglement, such as GHZ state and cluster state. However, we need to watch out for the time interval t between photons sequentially entering the MTR. Because if the practical operations can not meet the condition that t is much shorter than the electron spin relaxation time τ, it will be almost impossible for us to get high-fidelity photon entanglement. Fortunately, it has been confirmed that the electron spin relaxation time and dephasing time of NV centers in diamond are relatively long (ms) even at room temperature [19, 35, 36].

Transfer of quantum state information from one qubit to another is an important step for practical distributed quantum information storage and large-scale quantum communication networks. Next, we will discuss how to realize QST deterministically with generated entanglement between photon pulse and NV center. First, we propose a deterministic scheme to transfer states between two separate NV centers. Assume the state information to be transferred is prepared in NV center 1 as α₁|+〉₁ + β₁|−〉₁ and the receiving qubit is NV center 2 initially prepared in $({| + 〉}_{2} + {| - 〉}_{2}) / \sqrt{2}$ . The two NV centers are confined on the surface of two spatially separated MTRs, respectively. To achieve the QST, we introduce a single photon pulse in $(| R 〉 + | L 〉) / \sqrt{2}$ which in turn passes through the two MTRs. As discussed above, after the photon interacts with NV centers in the light of Eq. (7), we get

\frac{1}{2} [(| R 〉 + | L 〉) {(α_{1} | + + 〉 + β_{1} | - - 〉)}_{12} + (| R 〉 - | L 〉) {(α_{1} | + - 〉 + β_{1} | - + 〉)}_{12}] .

Then applying a π/2 microwave pulse, which works as a Hadamard gate, on the electronic spin state of NV center 1, Eq. (11) becomes

\begin{array}{l} \frac{1}{2 \sqrt{2}} {(| R 〉 + | L 〉) [{| + 〉}_{1} {(α_{1} | + 〉 + β_{1} | - 〉)}_{2} + {| - 〉}_{1} {(α_{1} | + 〉 - β_{1} | - 〉)}_{2}] \\ + (| R 〉 - | L 〉) [{| + 〉}_{1} {(α_{1} | - 〉 + β_{1} | + 〉)}_{2} + {| - 〉}_{1} {(α_{1} | - 〉 - β_{1} | + 〉)}_{2}]} . \end{array}

So according to the measurement results of the photon polarization and electron spin of NV center 1, we can deterministically regenerate the state α₁|+〉 +β₁|−〉 on the second NV center via appropriate local single-qubit operations.

Using similar principle, we can also accomplish the QST from photonic qubit to electron spin qubit, or vice versa. That is, a photon-spin interface will be possible with the entanglement between photon pulse and NV center generated first [34]. With the technological development of single photon and NV centers, these QST processes can be achieved effectively among different communication nodes. And it is of important significance for large-scale solid-state system quantum computing and communication network.

4. Analysis and discussion

In our schemes, the reflection coefficient of input photon pulse and the phase shift induced on output photon are playing crucial roles. If the controlled phase gate from the reflection phase shift in Eq. (7) can not be almost completely satisfied, the performance of our entanglement generation and QSTs may be inevitably reduced. Moreover, although the NV center in diamond is easy to be fixed in solid-state microcavity than atoms, its position precision may influence the coupling strength between the NV center and cavity field. So under the resonant condition ω₀ = ω_C = ω_p and Eq. (6), we here consider the effect of parameters variation on the reflection coefficient. In Fig. 3(a), we have plotted the reflectance against the ratio of coupling strength to dissipation factors [25]. It shows that the reflectance r(ω_p) ≈ 0.95 just requires the coupling strength $g \geq 3 \sqrt{κ γ}$ , which is not a difficult experimental requirements. The results presented in Ref. [16] give a set of effective parameters [g, κ, γ_total, γ_ZPL]/2π = [0.3, 26, 0.013, 0.0004] GHz for a hybrid diamond-GaP microdisk system with a relatively bad quality Q ∼ 10⁴. Here γ_ZPL is ZPL spontaneous emission rate of an NV center, which is mainly considered in this paper. Figures 3(b) and 3(c) show the reflectance as functions of g/κ and g/γ for different γ and κ. It is as expected that the reflection coefficient increases with the increase of the coupling strength g. However, r(ω_p) ≈ 0.95 can be obtained even when g < 0.1κ. That means the presented schemes do not essentially need strong coupling and high-Q cavity conditions. Figure 3(d) shows the reflectance against κ/γ when g is hundreds of times bigger than γ. It is apparent that the effect of the cavity damping increase on r(ω_p) is almost negligible when g > 300γ. As the proper parameter relationships involving in Fig. 3 are available in current technical conditions, such as in Ref. [16], the effective reflection coefficient and phase shift of photon pulse can be obtained in our schemes, which make the presented entanglement generation and QSTs could have high fidelity.

Fig. 3 The reflection coefficient as functions of parameters variation under the resonant condition ω₀ = ω_C = ω_p. (a) r(ω_p) against $g / \sqrt{κ γ}$ . (b) r(ω_p) against g/κ for γ = 6 × 10⁻⁵κ (red curve) and γ = 6 × 10⁻⁴κ (blue curve). The dotted lines in (a) and (b) indicate |r(ω_p)| = 0.95. (c) r(ω_p) against g/γ for κ = 500γ (red curve), κ = 1000γ (blue curve) and κ = 1500γ (green curve). (d) r(ω_p) against κ/γ for g = 300γ (red curve), g = 500γ (blue curve) and g = 800γ (green curve).

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For the actual experiment operations, the photon loss due to photon absorption and scattering, the mismatch between input pulse mode and WGM, and the inefficiency of the detector will bring ineffectiveness to our schemes. However, the successful generation or transfer is reflected in triggering the photon detector. So the photon loss will just affect the success efficiency, rather than the fidelity. It is fortunate that recent experimental technology can generate 300000 high quality single photons within 30 seconds [37], which ensures our scheme can be accomplished fast and near deterministically. And the weak excitation condition in input-output process can be satisfied by high quality single-photon sources as discussed in Ref. [38]. In addition, the current fabrication techniques can suppress the photon leakage and make it negligible compared with the main cavity decay into the input-output modes. Another characteristic of the system considered here is that the electron spin relaxation time and the dephasing time of NV centers are demonstrated to be ∼ms at room temperature and ∼s at low temperature in recent researches [19, 35, 36], which ensures enough operating time. However, with temperatures rising, the radiation spectrum of NV centers will have a large broadening and the ZPL emission will be debased. So it is essential to seek a solution to enhance the ZPL emission at higher temperature. One feasible method is coupling the NV centers to optical nanocavities and improving the ratio between the Q-factor and cavity mode volume V[20]. Afterwards, Koshino et al. have demonstrated a effective entanglement operation of homogeneously broadened NV qubits at about 30 K in the weak-coupling regime [39]. Moreover, some instructive room-temperature coherent manipulation of the spin of a single NV center to achieve quantum information and computation tasks have been proposed [40–42].

Although we consider here the well-known triplet ³A ↔³E optical transition in Fig. 1(a), more precise results show that the excited state triplet of the NV center is split by the inter-system interaction, e.g., spin-orbit interaction, into six spin states as {|A_1,2〉, |E_x,y〉, |E_1,2〉} with C₃_υ symmetry [43, 44]. However, Maze et al. subsequently presented a positive demonstration that the axial spin-orbit interaction could not mix the excited state due to the large gap and the non-axial spin-orbit interaction is anticipated to be small [45]. Thus we can select suitable transition-allowed energy levels and approximately consider the triplet ³A ↔³E transition. Moreover, the sizable splitting energy gap by the spin-orbit and spin-spin interactions is beneficial for encoding qubits and measuring the spin states of the NV centers.

Another point should be indicated here is that in order to facilitate manipulating and extending the MTR-NV center system, the NV center should be close to the surface of diamond, which may have effect on NV center properties [46]. And that the fibre transmitting input pulse is close to the resonator may influence the coupling between the resonator and NV center. These problems need to be further researched in the later.

5. Conclusion

In summary, we have analyzed the single photon input-output process from the MTR confining an NV center. The phase shift due to the Faraday rotation in this process can be used to deterministically generate the NV center Bell states, three NV centers GHZ states, as well as two-photon entangled states. With the generated spin-photon hybrid entanglement, a potential photon-spin interface could be realized to transfer state information from NV center to photon pulse, or vice versa. Strong coupling strength or high-Q cavity (resonator) is not requisite for our setups and the long coherence time of NV center electron state at higher temperature might relieves the cryogenic requirements, which greatly reduces the experimental difficulty. The final analysis shows that our schemes are feasible and may have good performance in the current experimental conditions, which is expected to be useful for future large-scale solid-state system quantum computing and communication network.

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant Nos. 61068001 and 11264042, China Postdoctoral Science Foundation under Grant No. 2012M520612 and the Talent Program of Yanbian University of China under Grant No. 950010001.

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Quantum state engineering with nitrogen-vacancy centers coupled to low-Q microresonator

Abstract

1. Introduction

2. The input-output process in MTR-NV center system

3. Preparation of entanglement and quantum state transfer

4. Analysis and discussion

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (3)

Equations (12)

Optics Express