Virtual-photon-induced entanglement with two nitrogen-vacancy centers coupled to a high-q silica microsphere cavity

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1 . Article. SCIENCE CHINA Physics, Mechanics & Astronomy November 2013 Vol. 56 No. 11: doi: /s Virtual-photon-induced entanglement with two nitrogen-vacancy centers coupled to a high-q silica microsphere cavity LIU YiMin * Department of Physics, Shaoguan University, Shaoguan , China Received January 27, 2013; accepted March 7, 2013; published online October 8, 2013 We propose a potentially practical scheme for efficient generation of entanglement with two nitrogen-vacancy centers (NVC) coupled to a whispering-gallery mode cavity. By virtue of the virtual-photon-excitation, the entanglement with two separate NVC can be produced in a deterministic way. The required operations are very close to the capabilities of current experimental techniques. The effects of decoherence induced by the cavity decay and the atomic spontaneous decay are also investigated. entanglement, nitrogen-vacancy center, cavity QED PACS number(s): a, Lx, Mi, Pq Citation: Liu Y M. Virtual-photon-induced entanglement with two nitrogen-vacancy centers coupled to a high-q silica microsphere cavity. Sci China-Phys Mech Astron, 2013, 56: , doi: /s As one of attractive candidates for solid-state for various quantum information processing (QIP) [1 8], the nitrogenvacancy center (NVC) [9,10] has attracted intensive attention. The electronic spins of NVC can be fastly polarized, coherently manipulated, and readout by optical means with long coherence time [11,12]. On the other hand, the nuclear spins, i.e., 13 C [13], 14 N [14] and 15 N [15] around the electron spin, have been demonstrated to be an excellent candidate for the quantum information storage, where the quantum state mapping between electronic and the nuclear spins can be realized by virtue of the hyperfine couplings. The progress also enable high-fidelity readout of quantum information from the electron spin [16]. However, all the above-mentioned experiments concentrated in the single NVC case. Based on the coupling between spin-dependent optical transitions and the spin states [17 19], Togan et al. [20] have experimentally demonstrated a reliable entanglement between the spinofannvcandanoptical photon, which is potentially practical for long-distance entanglement protocols and scalable QIP. Additionally, the en- *Corresponding author ( lym @163.com) tanglement of two of the separate NVCs has only been implemented experimentally using the magnetic-dipolar coupling method [21]. However, the idea is pretty hard to be applied to distant NVCs due to the negligible magnetic dipolar coupling strenghs. On the other hand, in the context of cavity quantum electrodynamics (QED), several protocols have employed the coupling between the NVCs and the quantized whisperinggallery mode (WGM) of high-q microsphere cavity to realized the controlled phase gating [18] and W states between NVCs [19], respectively. In this paper we propose a potentially practical scheme for efficient generation of entanglement with two separate NVC electron spins in the NVCmicrosphere cavity system, where the considerable enhancement of the zero phonon line (ZPL) [22] has been realized, and the Λ-type configuration of the optical transition in NVC [23] has been experimental demonstrated [20]. We will show below that entanglement with two NVCs can be generated efficiently by virtue of the Raman transition, where a smart encoding method of the qubits in different NVCs is used, as done in ref. [18]. Furthermore, we also analysize the nosie c Science China Press and Springer-Verlag Berlin Heidelberg 2013 phys.scichina.com

2 Liu Y M Sci China-Phys Mech Astron November (2013) Vol. 56 No effect caused by the cavity decay and the spontaneous emission decay on the concurrence between the two NVCs, by numerical simulation through calculating the Lindblod-form Master equation. We offer some brief remarks on this system before building our theoretical model. The NVCs we are studying are negatively charged with two unpaired electrons located at the vacancy, usually treated as electron spin-1. So the ground state is a spin triplet state labeled as (See the inset of Figure 1), where the levels m S = ±1 are degenerated. Additionally, there exsit a zero-field splitting with 2.87 GHz between state m S = ±1 andm S = 0 [24]. The excited state = ( E, m s =+1 + E +, m s = 1)/ 2 is one of the six excited states defined by group theory, where E ± are orbital states with angular momentum proection ±1 along the NV axis [20,25 27]. Note that the state is associated with a broadband photoluminescence emission with ZPL of ev, which allows optical detection of individual NVC using confocal microscopy. We begin by considering a combined system consisting of two separate NVCs and one WGM cavity, as shown in Figure 1. The reason we have considered fused-silica microsphere cavities instead of Fabry-Perot (FP) cavities is the strong coupling condition and physical scalability due to the lowest-order WGM corresponding to light traveling around the equator of the microsphere [28,29]. In the present model, each NVC can be treated as a Λ-type three-level structure through individual σ + circularly polarized laser pulse irradiation (Rabi frequencies Ω ) [23,30] along with the WGM field of the microsphere cavity. In our scheme, the states,,, m s = 1,and are denoted by the states 0, 1,and χ of the qubit, respectively, and the state is used as an auxiliary state h. The total Hamiltonian of this composite nanocrystalmicrosphere system under the rotating wave approximation (RWA), in the interaction picture, can be written in units of ( = 1) as [ H I = (g a + 0 χ +Ω χ 1 =1,2 + H.c.) +Δ χ χ ], where a + (a) is the creation (annihilation) operator for the WGM. Applying standard quantum optical techniques [31], under the large-detuning conditions Δ Ω, g, theexcited states χ are only virtually excited and adiabatically eliminated, which yields the effective Hamiltonian [32 34] H eff = [g Ω ( 1 0 a a + ) =1, g 2 a + a 0 0 ]/Δ, +Ω 2 where g Ω /Δ is the effective Rabi frequency. The effective Hamiltonian can be further reduced to H eff = =1,2 g [ 1 0 a a + ], where g = g Ω /Δ,if we use the additional lasers with appropriate frequencies to compensate the laser-induced level shifts in H eff [32,34]. Note that the resonant interactions only occur between the state 0 and the state 1, induced by the WGM field. So the auxiliary state h is not involved in the interaction with the WGM throughout our scheme [35]. Assuming that the system is initially in the state 1 1 h 2 0 c with 0 c ( 1 c )the vacuum (one-photon) state of the WGM field. In this case the second qubit does not interact with the WGM field. As a result, the system evolves as 1 1 h 2 0 c [cos( g 1 t) c isin( g 1 t) c ] h h 2 0 c, (1) when we choose the interaction time t = π/ g 1. Next, we consider another situation, that is, the system is initially in the state c, then the corresponding time evolution is c Ñ{[ g 2 1 cos( Gt) + g 2 2 ] c +[ g 2 1 g2 2 (cos( Gt) 1)] c isin( Gt) c } c (2) with the choice Gt = 2π, where G = g g2 2 and Ñ = 1/ G 2. We can satisfy eqs. (1) and (2) by g 2 = 3 g 1. Therefore we obtain the following operations as , 0 1 h h 2, (3) , 1 1 h h 2. If the logic state 1 ( 0) of the qubit 2 is represented by h ( 0) of the second NVC and consider the quantum information encoded in the subspace spanned by the states { 0 1, 1 1, 0 2, 1 2 }, we begin in a product state Ψ 0 = 1 2 ( ) ( ) 0 c. After the performance of the operations (eqs. (1) (3)), in the absence of noise, the final state becomes NV1 Microsphere cavity NV2 g Ω Figure 1 (Color online) The schematic setup of WGM microsphere system, where two identical NVCs in diamond nanocrystals are equidistantly attached around the equator of a single fused-silica microsphere cavity. The inset shows the level configuration of the th NVC, where Δ is the detuning, which meets corresponding two-photon resonance conditions. g is the coupling strength between NV center and WGM, and Ω is the coupling strength of the laser radiation to the NVC. is the zero-field splitting between the lowest energy sublevel and the sublevels, which are degenerate at zero magnetic field because of C 3v symmetry of NVC. We restrict our study to the down state and the up state m s = 1.

3 2140 Liu Y M Sci China-Phys Mech Astron November (2013) Vol. 56 No. 11 (a) g Ω (b) g Ω Figure 2 (Color online) (a) The qubit definition for the first NVC, where the bold lines for the states, and encode the logical qubits 1 and 0, respectively. (b) The qubit definition for the second NVC, where the bold lines for the states, m s = 1 and, encode the logical qubits 1 and 0, respectively. 1 Ψ f = 2 ( ) 0 c. (4) It implies that the two separate NVCs have evolved to a state of maximal entanglement. In practice, noise or decoherence places limits on the entanglement process discussed above. In our scheme, the noise rate includes three different types of noise, such as the radiative decay Γ 10 between 1 and 0,the radiative decay Γ ho between h and 0,andWGMfield decay rate κ. we can model the effect of decoherence by replacing Schrödinger equation with the Lindblad equation ρ = i[h,ρ] + κ(2aρa + a + aρ ρa + a) + D[ρ], (5) where D[ρ] = =1,2{Γ 10 (2σ 01 ρσ 10 σ 10 σ 01 ρ ρσ 10 σ 01 ) + Γ h0 (2σ 0h ρσ h0 σ h0 σ 0h ρ ρσ h0 σ 0h )}, andσ 01 = 0 1, σ 0h = 0 h. The effective spontaneous decay rate Γ10 of the excited state to 1 and 0 could be estimated as Γ 0 Ω g /Δ 2 [36,37] with Γ 0 the spontaneous decay rates of the state χ, where we have assumed that the spontaneous decay rate from χ to 1 is equal to the spontaneous decay rate from χ to 0. Based on ref. [38], the coupling strength between WGM and NVC could be g max /2π = 55 MHz, and the other experimental parameters can be adusted to be Δ = 2π 1 GHz and Ω max /2π = 100 MHz. So we have the operational time t 0 = π/ g 1 to be 0.91 μs ifwesimplyassume g 1 = g 2 = g max, Ω 1 =Ω max, and Ω 2 = 3Ω 1 2π 173 MHz, and the WGM field decay rate κ = ω 10 /Q = 2π 47 khz. Using the parameter values above, we have calculated in Figure 3 the fidelity of the entangled state Ψ f as a function of the noise rate Γ 10, Γ h0,andκ, respectively. It clearly indicates the high fidelity of our proposed entangled state Ψ f as long as κ, Γ 10,andΓ h0 are small enough with respect to the effective coupling rate. We can also study the influence of the noise rates on the concurrence [39] of the entangled state Ψ f. As a measure of entanglement in most previous investigations, concurrence [40] of two qubits is calculated by the following relation C AB = max ξ1 4 ξi ;0, (6) i=2 where the values arranged in a descending order {ξ 1,ξ 2,ξ 3, ξ 4 } are four non-negative eigenvalues of the oint density matrix ϱ = ρ AB (σ y A σy B )ρ AB (σy A σy B ), with ρ AB the density matrix of the system consisting of qubits A and B, andthe asterisk stands for the complex conugate. Figure 4 plots the concurrence of the final state Ψ f (eq. (4)) as a function of the three types of noise. With (κ, Γ 10, Γ h0 ) g 1, substantial entanglement between the two NVCs can still be retained. As shown in the Figure 4, the entanglement is the least tolerant to the decoherence between h and 0 and is the most tolerant to cavity decay. We survey the relevant experimental parameters. In general, WGM of microspheres have samll volume (V m 100 μm 3 ) [41], and ultrahigh Q factor [42], which offer predominant conditions for obtaining the strong coupling regime [43 49]. Recent experimental evidence for strong coupling between NVCs and the WGM of polystyrene WGM cavity [50] or silica WGM cavity [38] have been shown in the NVC-WGM system, respectively. On the other hand, In the Fidelity Noise rate Figure 3 (Color online) The fidelity of the entangled state Ψ f in the presece of noise effects, where the lines from top to bottom denote the case of only considering the cavity decay κ, the radiative decay Γ 10,andtheradiative decay Γ h0, respectively. Here we have set g 1 = g 2 = g = 1. The inset shows the fidelity of the entangled state Ψ f when all the three noise rates are considered. Concurrence Concurrence κ/g Γ/g 1 Figure 4 (Color online) The concurrence of entangled state Ψ f in the presense of the cavity decay κ (inset), the radiative decay Γ 10 (solid line), and the radiative decay Γ h0 (dashed line), respectively.

4 Liu Y M Sci China-Phys Mech Astron November (2013) Vol. 56 No realistic experiments, the NVC s electronic spin relaxation time T 1 can change from 6 ms [51] to seconds in the case of low temperature. In addition, ref. [52] has reported that the dephasing time T 2 induced by the nuclear-spin fluctuation inside the NVC can reach 350 μs, which implies that the influence from the intrinsic damping and dephasing of the NVC is possibly negligible in the present NVC-WGM system. In conclusion, we have proposed a scheme for efficient generation of entanglement with two separate NVCs coupled to a WGM cavity. By virtue of the virtual-photon-excitation and by smart qubit encoding, the entanglement with two separate NVCs can be produced in a deterministic way. The required operations are very close to the capabilities of current experimental techniques. The effect of decoherence induced by the cavity decay and the atomic spontaneous decay is also investigated. 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