Concentration of entangled nitrogen-vacancy centers in decoherence free subspace

Chuan Wang; Tie-Jun Wang; Yong Zhang; Rong-zhen Jiao; Guang-sheng Jin

doi:10.1364/OE.22.001551

1. Introduction

Entanglement between solid state systems plays an important role in quantum information processing (QIP). Particularly, the strong coupling interaction between microcavities and various dipoles can be easy manipulated and exhibits scalability as quantum bits (qubits) [1]. These separate solid qubits can be used as quantum nodes to build a quantum information processor.

Over the past decades, QIP based on nitrogen-vacancy (N-V) centers has attracted growing interest. N-V centers are the diamond nanoparticles with one carbon been replaced by a nitrogen atom and another neighboring carbon been replaced by a vacuum. In 2007, Dutt et al. [2] proposed a controllable quantum register in N-V centers by using the coherent manipulation of an electron spin and nearby individual nuclear spins. Later, Robledo et al. [3] described a method for initialization and measurement on the N-V center qubit which achieves high-fidelity read-out of the electronic spin, and then project up to nearby nuclear spin qubits onto a well-defined state. Toyli et al. [4] fabricated N-V centers in diamond based on broad-beam nitrogen implantation through apertures in electron beam lithography resist which provides a promising platform for scalable QIP. Later in 2011, Hausmann et al. realized the optical diamond nanostructures containing a single-color center. Also Fuchs et al. [5] demonstrated a room-temperature quantum memory by using the nitrogen nucleus spin intrinsic to each N-V centre in diamond. They performed a 120ns high fidelity state transfer in this regime. And Maurer et al. [6] realized a room temperature quantum memory with the storage time exceeded 1 second which provides more applications of QIP. For the realization of N-V centers, Hausmann et al. [7] demonstrated the techniques of single color center implanting in nanostructure in which photon anti-bunching even at high pump powers can be observed.

Combining the N-V centers and microcavities, the interaction between the spin and the photon can be increased. Recently, many related theoretical and experimental works have been devoted to the system that combines the N-V center qubit with the high-Q cavity. In 2009, An et al. [8] demonstrated the quantum information processing protocol by using a single photon interacted with low-Q cavity. As the N-V centers possess long coherence time at room temperature [9], the N-V defect in diamond combines with the whispering-gallery mode (WGM) microcavity system has emerged as a promising solid-state candidate in the realization of QIP. Exploiting the N-V center and microresonator system, Yang et al. [10] theoretically proposed an entangled state generation protocol by using N-V centers coupled to WGM cavity. In 2011, Chen et al. [11] presented another entanglement generation protocol exploiting N-V centers via the input-output theory of the coupling system. Moreover, experimental works focus on the N-V centers and microresonators have been reported recently. Dayan et al. illustrated the dynamics of the output of the cavity field based on a single atom coupled with a microcavity resonator [12]. And entanglement between N-V center spins has been realized by Togan et al. [13]. As the N-V centers are useful solid qubits which could be rapidly and high fidelity readout of quantum information, they becomes a potentially practical resource for long-distance entanglement and scalable QIP. On the other hand, environment noise is a major obstacle in the realization of QIP which will reduce the fidelity of entanglement. Thus the preservation and concentration of entanglement between N-V centers attracts much attentions. Entanglement concentration (EC) is a potentially practical approach for the preservation and concentration of entanglement for solid state based QIP. In order to increase the entanglement fidelity of the quantum systems, the users often recur to EC which was first proposed by Bennett et al. in 1996 [14]. The users can distill a subset system in a maximally entangled state from a less-entangled pure state system. In 2003, Zhao [15] and Yamamoto [16] experimentally demonstrated the EC using linear optics independently. Over the past decades, the realization of EC has attracted much attention [17–19].

Decoherence free subspace(DFS) of the Hilbert space [20] is proposed to protect quantum information against the decoherence. The DFS is a set of states which is not affected at all by the interaction with the environment. In 2003, the DFS state to collective dephasing has been demonstrated with photons in Ref. [21]. For example, the channel collective dephasing noise in the two-dimensional Hilbert can be expressed as |0〉 → e^iθ₀ |0〉 and |1〉 → e^iθ₁ |1〉, as illustrated in Refs. [22,23], the DFS qubits can be defined by using the |0̃〉 and |1̃〉 are encoded as the logical qubit |0̃₁|1̃₂ and |1̃₁|0̃₂ which is immune to the collective dephasing. And there has been much related works on the DFS based QIP. In this paper, by combining the solid qubit candidate and the DFS encoding, we propose an efficient EC protocol to preserve maximally entanglement between N-V centers and microcavity systems. The key ingredient of this scheme is the parity-check process based on the solid DFS qubits. We first construct the parity-check process by exploiting the input-output process and then generalize it to the concentration protocol. We also discuss the experimental feasibilities and the efficiency of the EC protocol.

2. Single DFS qubit assisted entanglement concentration

2.1. The model of single photon input-output relation

Consider two three-level N-V centers trapped in a single mode optical resonant cavity as shown in Fig. 1, by using a and a^† for the annihilation and creation operators of the cavity mode with the frequency ω_c, the Hamiltonian of the composite system can be described as

\hat{H} = \sum_{j = 1, 2} [\frac{ω_{0}^{j} {\hat{σ}}_{z}^{j}}{2} + i g_{j} (\hat{a} {\hat{σ}}_{+}^{j} - {\hat{a}}^{†} {\hat{σ}}_{-}^{j})] + ω_{c} {\hat{a}}^{†} \hat{a},

where g_j denote the coupling strength between the jth N-V centers and microcavity. The operators σ_z and σ₊₍₋₎ represent the inversion and raising(lowering) operators of the N-V center with frequency ω₀.

Fig. 1 The energy level structure of the N-V center coupled to microcavity, where the lower levels are Zeeman sublevels of the ground state and the upper level is the excited one. Quantum information is encoded in the spin state |0〉 and |1〉. The energy level transition between |0〉 and |e〉 is resonated by the left(L) circularly polarized mode. And the energy level transition between |1〉 and |e〉 is resonated by the right(R) circularly polarized mode.

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By solving the Heisenberg equations of the cavity mode and the N-V center operators, we found the evolution operators of the cavity field and the atomic operators which can be described as [24]:

\frac{d \hat{a}}{d t} = - [i (ω_{c} - ω) + \frac{κ}{2} + \frac{κ_{s}}{2}] \hat{a} - \sum_{j} g_{j} {σ_{-}}^{j} - \sqrt{κ} {\hat{a}}_{in},

\frac{d σ_{-}^{j}}{d t} = - [i (ω_{0} - ω) + \frac{γ_{j}}{2}] σ_{-}^{j} - g_{j} \hat{a} σ_{-}^{i},

where the γ_j/2 denotes the decay rate of the jth N-V center. κ and κ_s are the cavity decay rate and the cavity leaky rate, respectively. ω, ω_c and ω₀ denote the frequencies of the input photon, cavity mode and the atomic level transition, respectively. For a simple solution, we assume that the coupling strength between the two N-V centers and the resonator are identical, as g₁ = g₂ = g and γ₁ = γ₂ = γ. The above Heisenberg equations of motion can be solved and the reflection coefficient of this system can be obtained:

r (ω) = \frac{i (ω_{c} - ω) - \frac{κ}{2} + 2 g^{2} / [i (ω_{0} - ω) + γ / 2]}{i (ω_{c} - ω) + \frac{κ}{2} + 2 g^{2} / [i (ω_{0} - ω) + γ / 2]},

here we omit the cavity loss as κ_s = 0. On the resonant condition of the system with ω_c = ω_p = ω₀, the reflection coefficient r for the uncoupled cavity system can be written as

r (ω) = \frac{2 g^{2} / γ - κ / 2}{2 g^{2} / γ + κ / 2} .

So the definition of the reflection coefficient on uncoupled condition shows unity reflectance |r₀(ω)| = 1 as g = 0.

In the next, we describe the mechanism of the interaction in detail. At first, we consider the N-V center which is prepared in the state |0〉, the transition will be driven by the left circularly polarized photon in the state |L〉. The output photon pulse acquires a relative phase shift ϕ determined by the input-output relation: |Φ_out〉 = r(ω)|L〉 = e^iϕ |L〉. On the other hand, if the input photon is prepared in the right circularly polarized state |R〉, the output photon will evolve as |Φ_out〉 = r(ω)|R〉 = eⁱ^ψ|R〉. In all, the dynamics of the photon input-output process on the logic qubits can be described as:

\begin{array}{l} | 0 〉 | 0 〉 : | L 〉 \mapsto e^{i ϕ_{2}} | L 〉, | R 〉 \mapsto e^{i ϕ_{0}} | R 〉; \\ | 1 〉 | 1 〉 : | L 〉 \mapsto e^{i ϕ_{0}} | L 〉, | R 〉 \mapsto e^{i ϕ_{2}} | R 〉 . \end{array}

The relative phase shifts are defined by the reflection coefficient r(ω) during the input-output process which can be set as: ϕ₀ = π/2 and ϕ₂ = −π/2 in our further application. Recently, in Ref. [25], Chen et al. combines the idea of input-output process and the DFS encoding theory, and presents a high quality QIP protocol. And in Ref. [26], Liu et al. a hybrid controlled phase flip gate based on the inputCoutput process of the DFS qubit and microcavity system.

2.2. Single DFS qubit assisted entanglement concentration

Suppose the two N-V centers are encoded in the state |0̃〉 = |0〉₁|1〉₂ and |1̃〉 = |1〉₁|0〉₂. The generated entangled state in DFS requires local operations σ_x on one atom to flip one spin direction of the logical qubit, and the entangled state can be described as $({| \tilde{0} 〉}_{1} {| \tilde{0} 〉}_{2} \pm {| \tilde{1} 〉}_{2} {| \tilde{1} 〉}_{2}) / \sqrt{2}$ and $({| \tilde{0} 〉}_{1} {| \tilde{1} 〉}_{2} \pm {| \tilde{1} 〉}_{1} {| \tilde{0} 〉}_{2}) / \sqrt{2}$ . Here the DFS qubits expanded Bell state is introduced to against the detrimental effects from decoherence. However, the probability amplitude of the superposed state will be effected and made the maximally entanglement to nonmaximally entangled state (α|0̃〉₁|0̃〉₂ + β|1̃〉₁|1̃〉₂), here the probability amplitude of the state obeys |α|² + |β|² = 1.

Here following the left part of the setup shown in Fig. 2, if we choose the input pulse in the state $| ↺ 〉 = (| L 〉 + | R 〉) / \sqrt{2}$ or in the state $| ↻ 〉 = (| L 〉 - | R 〉) / \sqrt{2}$ , we found that during the input-output process of the Cavity-1, the relative phase shift is added on the input pulses as:

\begin{array}{l} | \tilde{0} 〉 | \tilde{0} 〉 | ↺ 〉 \mapsto | \tilde{0} 〉 | \tilde{0} 〉 | ↻ 〉; | \tilde{1} 〉 | \tilde{1} 〉 | ↺ 〉 \mapsto e^{i π} | \tilde{1} 〉 | \tilde{1} 〉 | ↻ 〉; \\ | \tilde{0} 〉 | \tilde{1} 〉 | ↺ 〉 \mapsto | \tilde{0} 〉 | \tilde{1} 〉 | ↺ 〉; | \tilde{1} 〉 | \tilde{0} 〉 | ↺ 〉 \mapsto | \tilde{1} 〉 | \tilde{0} 〉 | ↺ 〉 . \end{array}

If the two spins are in the odd parity, the left circularly polarized photon remains unchanged, otherwise, the left circularly polarized photon will change to the right circularly polarized state corresponding to the even parity of the spins. After the parity check process, the parity information of the DFS qubits can be readout by measuring the state of the output photon.

Fig. 2 The schematic diagram shown the principle of nonlocally entanglement concentration by using single photon input-output process.

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In the following, we described the protocol in details. The solid qubits are distributed on Alice’s and Bob’s side, respectively. The environment noise will effect the maximally entangled state and change to the less-entangled state as α|0̃〉_A|0̃〉_B + β|1̃〉_A|1̃〉_B. One party, say Alice, prepares a single logic qubit in the state α|0̃〉_a + β|1̃〉_a. At first, a local operation σ_x on Alice’s side is needed to flip the single qubit state in the DFS. Exploiting the setup shown in Fig. 2, the evolution of the system can be described as:

\begin{array}{l} (α {| \tilde{0} 〉}_{A} {| \tilde{0} 〉}_{B} + β {| \tilde{1} 〉}_{A} {| \tilde{1} 〉}_{B}) \otimes (α {| \tilde{0} 〉}_{a} + β {| \tilde{1} 〉}_{a}) \otimes \frac{(| L 〉 + | R 〉)}{\sqrt{2}} \\ \to α β (| \tilde{0} 〉 | \tilde{0} 〉 | \tilde{1} 〉 + | \tilde{1} 〉 | \tilde{1} 〉 | \tilde{0} 〉) \otimes \frac{e^{2 i ϕ_{2}} | L 〉 + e^{2 i ϕ_{0}} | R 〉}{\sqrt{2}} \\ + (α^{2} | \tilde{0} 〉 | \tilde{0} 〉 | \tilde{0} 〉 + β^{2} | \tilde{1} 〉 | \tilde{1} 〉 | \tilde{1} 〉) \otimes \frac{e^{i (ϕ_{2} + ϕ_{0})} (| L 〉 + | R 〉)}{\sqrt{2}} . \end{array}

It is obvious that the relative phase of the photon state can be distinguished. By performing single particle measurement on the ancillary photon, the state of the composite N-V centers system collapse to c₁(α²|0̃〉|0̃〉|0̃〉 + β²|1̃〉|1̃〉|1̃〉) and

(| \tilde{0} 〉 | \tilde{0} 〉 | \tilde{1} 〉 + | \tilde{1} 〉 | \tilde{1} 〉 | \tilde{0} 〉) / \sqrt{2}

corresponds to the photon state

(| L 〉 + | R 〉) / \sqrt{2}

and

(| L 〉 - | R 〉) / \sqrt{2}

, respectively. The final step is to perform single spin measurement on the ancillary N-V center in the basis

(| \tilde{0} 〉 \pm | \tilde{1} 〉) / \sqrt{2}

. The yield of the EC process is defined as 2|αβ|² that the state has been concentrated to the maximally entanglement.

For the remaining part, the state can be described as

| ϕ_{2} 〉 = \frac{1}{\sqrt{{| α |}^{2} + {| β |}^{2}}} (α^{2} | \tilde{0} 〉 | \tilde{0} 〉 \pm β^{2} | \tilde{1} 〉 | \tilde{1} 〉) .

Then we further iterate the concentration process. Alice prepares another single logic qubit in the state |ψ₂〉 = α²|1̃〉 + β²|0̃〉. By iterating the process exploiting the setup shown in Fig. 2, the composite system state |ϕ₂〉|ψ₂〉 can be concentrated to the maximally entangled state with the yield 2Z|αβ|⁴. Here Z represents the normalized constant. Moreover, the system without been concentrated are remaining in the state

| ϕ_{3} 〉 = (α^{4} | \tilde{0} 〉 | \tilde{0} 〉 \pm β^{4} | \tilde{1} 〉 | \tilde{1} 〉) / \sqrt{{| α |}^{4} + {| β |}^{4}}

. And the yield of the third round concentration is

y_{3} = \frac{2 {| α β |}^{4}}{{| α |}^{4} + {| β |}^{4}}

after the third round iteration.

We can conclude that the total yield of the ECP can be represented as

Y = \sum_{n} \frac{2 | α^{2^{n}} β^{2^{n}} |}{{| α |}^{2^{n}} + {| β |}^{2^{n}}},

here n denotes the iteration times. In Fig. 3, we numerically simulated the yield of the protocol versus the initial coefficient α in ideal conditions. By comparing the yield with different iteration times, the total yield of the protocol is efficiently improved. As the iteration times increase to three, the yield values increase from 0.4998 to 0.812 if the probability amplitude α = 0.7.

Fig. 3 The yield of the ECP after different iteration times in ideal cases without considering the system decoherence. Here we simulated the yield of the protocol by one time concentration (dotted line), two times (dashed line) and three times iteration (solid line).

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2.3. Experiment feasibilities

The key ingredients of our protocol can be described as follows: the system that consists two N-V centers and microcavity, and the efficient generation of entangled states in N-V centers. Firstly, the multi-particle and microcavity coupled system had been realized in experiment recently. As demonstrated by Zhu et al. [27], individual nanoparticles which are 30 nm in radius can be coupled to a microtoroid nanoresonator by using a nozzle for nanoparticle deposition. The process can be iterated several times to deposit more than one particle. Also in 2006, Park et al. [28] experimentally observed the mode splitting in which N-V centers in diamond nanocrys-tals are coupled to whispering gallery modes in a silica microsphere with the Q factor larger than 10⁸. Secondly, Bernien et al. [29] observed the two-photon quantum interference from separate N-V centers in experiment. The visibility of the two photons is 66%. And they also suggested that the coincidence rates may be enhanced by embedding N-V centers into optical cavities which enables the remote N-V centers to be entangled.

As N-V centers have long coherence electron spin relaxation time [9], it is well suited for QIP. The electronic spins in N-V centers are easy for initialization, manipulation and measurement. The important building block of our scheme is the strong coupling between the two N-V centers and the optical microcavity. In realistic experiment, optical microcavity with high-Q factor can be fabricated in various systems, such as microsphere and microtoroid resonators [30, 31]. Moreover, the coupling strength on the order of hundreds of megahertz between the N-V centers and microcavity has been realized. Strong coupling between a single N-V center and a gallium phosphide microdisc [32] has been reported. The N-V centers are experimentally coupled with chip-based microcavity with Q > 25, 000 and the coupling strength between a single microdisk photon and the N-V center zero photon line (ZPL) is g/2π = 0.3GHz. The total spontaneous emission rates of the N-V center is γ/2π = 0.013GHz. As illustrated in Ref. [33], the coupling of the zero-phonon line of a single N-V center to a photonic crystal microcavity is realized. The spontaneous emission into the ZPL is enhanced by a factor of ∼ 70, which corresponds to more than 70% of the photons being emitted into the cavity mode.

3. Discussion and summary

Consider the imperfection of the system, the efficiency of the protocol relies on the realistic experimental parameters. As shown in Fig. 4, we numerically simulated the efficiencies of the EC protocol under realistic experiment parameters by using three times iteration. We can conclude that the success probability is larger than 80% on condition that the coherent coupling strength 2g/κ = 0.8 in the dephasing error mode. In realistic experiment, suppose the coefficient $α = 1 / \sqrt{3}$ , the failure rate caused by coupling strength and the cavity decay is about 2%. And photon loss reduces the success rate by 5%. The dark count of the single-photon detector reduces the efficiency by 10⁻⁴. We can simply estimate the success probability of the protocol to be p = 2 × (1 – 2%) × (1 – 10⁻⁴) × 95% × 4/9 = 82.7%. On condition that there are dephasing process, the success probability reduces to (1 + cos²δθ)p/2 where δθ represents the relative phase shift induced by the noise on the orthogonal states. Moreover, we can improve the success probabilities of our protocol by repeating our operations.

Fig. 4 The yield of the ECP versus the coupling strength. Here we simulated the yield of the protocol by three times iteration, where γ/κ = 0.01.

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In addition, there are many ingredients which may also reduce the success probability of the proposed protocol, such as the spin decoherence and the photon loss during the cavity reflection, scattering and fiber absorption. Obviously, the EC process is repeated until the click of the photon detector. So the photon loss will not affect the yield of the protocol, but only the time of success. The main effect on the yield is the spin decoherence of the N-V center which is characterized by spin relaxation time T₁ and dephasing time T₂. In Ref. [34], it is reported that the spin relaxation time varies depend on the temperature. The results show that the time T₁ approaches to microsecond at room temperature, and the longest T₁ observed was on the order of minutes at 10 K for the sample. Also in Ref. [35], the electron spin relaxation time T₁ of N-V centers in diamond scales from microseconds to seconds at low temperature. Moreover, the dephasing time T₂ of N-V center is about 2ms in a isotopically pure diamond [36] which provides our scheme enough time for gate operations.

In summary, we have demonstrated the possibility to concentrate maximally entanglement by DFS encoding logic qubits confined in low-Q cavities. The DFS qubits can be prepared in maximally entangled state and be preserved by using the scheme nonlocally. As the electron spin state in N-V centers is used for storing quantum information and the photons are used as the quantum bus to transmit information between remote nodes. The protocol is scalable as the N-V centers and microcavity resonator coupled system acts as a quantum node that are immune to dephasing, by simply performing projective measurement on the transmitted photons from different nodes, then the quantum network can be established which can be used for quantum repeaters and distributed quantum computation. Also the main building block of our DFS EC protocol is the parity check detectors. One can distinguish the parity information between the DFS qubits without measuring them which provides a novel way of parity non-demolition detectors. The detectors are useful for accomplishing some interesting QIP tasks with the N-V centers and low-Q microcavity systems. For example, the entanglement analysis between DFS qubits and quantum repeaters between the two quantum nodes, and so on.

Acknowledgments

This work is supported by the National Fundamental Research Program Grant No. 2010CB923202, the National Natural Science Foundation of China under Grant Nos. 61205117 and 11374042, Beijing Higher Education Young Elite Teacher Project No. YETP0456, and the Open Research Fund Program of the State Key Laboratory of Low-Dimensional Quantum Physics, Tsinghua University Grant No. KF201301.

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Concentration of entangled nitrogen-vacancy centers in decoherence free subspace

Abstract

1. Introduction

2. Single DFS qubit assisted entanglement concentration

2.1. The model of single photon input-output relation

2.2. Single DFS qubit assisted entanglement concentration

2.3. Experiment feasibilities

3. Discussion and summary

Acknowledgments

References and links

Cited By

Figures (4)

Equations (10)

Optics Express