Cross-entangling electronic and nuclear spins of distant nitrogen-vacancy centers in noisy environments by means of quantum microwave radiation
Viviana Gómez Azuero, Ferney Rodríguez Dueñas, Luis Quiroga Puello
CCross-entangling electronic and nuclear spins of distant nitrogen-vacancy centers innoisy environments by means of quantum microwave radiation
A. Viviana G´omez, ∗ Ferney J. Rodr´ıguez, and Luis Quiroga Departamento de F´ısica, Universidad de Los Andes, A.A.4976, Bogot´a D.C., Colombia (Dated: July 27, 2018)Nitrogen-vacancy (NV) defect centers in diamond are strong candidates to generate entangledstates in solid-state environments even at room temperature. Quantum correlations in spatiallyseparated NV systems, for distances between NVs ranging from a few nanometers to a few kilometers,have been recently reported. In the present work we consider the entanglement transfer from two-mode microwave squeezed (entangled) photons, which are in resonance with the two lowest NVelectron spin states, to initially unentangled NV centers. We first demonstrate that the entanglementtransfer process from quantum microwaves to isolated NV electron spins is feasible. We then proceedto extend the previous results to more realistic scenarios where C nuclear spin baths surroundingeach NV are included, quantifying the entanglement transfer efficiency and robustness under theeffects of dephasing/dissipation noisy nuclear baths. Finally, we address the issue of assessing thepossibility of entanglement transfer from the squeezed microwave light to two remote nuclear spinsclosely linked to different NV centers.
PACS numbers: 03.67.Mn,03.65.Ud,03.67.Lx
I. INTRODUCTION
Recently a great deal of interest has arisen in quan-tum systems operating in the microwave sector of theelectromagnetic spectrum since they provide new oppor-tunities for exploring fundamental aspects of quantumphysics as well as possible applications in the field ofquantum information and computation. Important stepsin profiting microwave active quantum architectures in-clude superconducting (SC) circuits [1–5] and the manip-ulation of nuclear and electronic spins in solids [6–11]. Apromising idea pursued by several groups is to combinedifferent matter subsystems in a hybrid quantum sys-tem to take advantage of the scalability, flexibility andlarge coupling to microwave fields of some of them, forinstance SC circuits, and to exploit large coherence timesof other subsystems, such as solid-state spin systems, forstoring quantum information in stable quantum registers[12, 13]. From this perspective, nuclear spins prove moresuitable than electronic spins. However, the direct con-trol of spatially distant nuclear spins is challenging due tothe weak coupling between themselves. Thus, the searchfor nuclear long-range entangling mechanisms which al-low for opportunities to overcome those limitations areof great interest.An excellent platform for undertaking that search isprovided by nitrogen-vacancy (NV) centers in diamond.A single NV center is a well characterized defect in dia-mond consisting of a substitutional nitrogen atom nextto a carbon vacancy in an adjacent lattice site [14]. Ithas been demonstrated the selective addressing and con-trolling of a single NV, even at room temperature, and ∗ [email protected]; Present address: Civil EngineerDepartment, Universidad Mariana, San Juan de Pasto, Colom-bia. how their constituent electronic and nuclear spins can beeffectively manipulated and potentially coupled together[15, 16]. On the other hand, the dipolar and hyperfine in-teractions between the electronic and nuclear spins in NVcenters have been extensively studied. Individual controland readout of nuclear spin qubits coupled to the elec-tronic spin has been demonstrated [17]. Besides that,the control of two nuclear spins on an individual basis,generates entanglement of two C nuclear spins at thefirst coordination shell of the vacancy [18] and mediatethe entanglement between multiple photons [19].Numerous quantum information protocols with NVcenters have been previously discussed in the literature.The quantum dynamics of distant C nuclear spins hasbeen probed using a weak coupling with the electronicspin in NV centers [20]. Furthermore, the initializa-tion of electron and nuclear spin qubits [21], the transferof quantum states [21, 22] and the generation of con-trolled quantum gate between distant nitrogen nuclearspins [23] represent a step forward to build a quantumrepeater network for long distances. An important is-sue in the field of quantum information is the generationof entangled states in a scalable way. The combinationof radiation excitation from different wavelength sectorsof the electromagnetic spectrum (optical, microwave andradio-frequency) has allowed to engineer protocols forreaching entanglement between electron spins in two sep-arate NVs [24], the electron spin of a single NV and itsneighbor nitrogen nucleus [25] or the NV electron anda closely placed C nucleus [18]. Moreover, other pro-posals show protocols to generate spin-photon entangledstates between the ground state spin of a single NV centerand the polarization of an emitted optical photon [26],heralded entanglement between solid-state qubits usingoptical photons [27] and entanglement between NV elec-tron spins separated up to 1 . a r X i v : . [ c ond - m a t . o t h e r] J u l In the present work we present a theoretical proposalbased on NV defect centers in diamond to reach entangle-ment between distant electron and/or nuclear spins me-diated by a quantum (squeezed) microwave field (QMF)as provided by a two-mode Josephson mixer [29], seeFig. 1. The NV center has an electronic spin S = 1mostly localized at the defect bond. However, about 11%of its electron spin density is distributed over the near-est neighbor carbon atoms and as a result substantialhyperfine and dipolar couplings with neighboring carbonnuclear spins ( C ) are sizeable [18]. On the other hand,a diluted network of spin-1 / C -nuclei forms a meso-scopic spin bath for a NV center. Under these conditions,we demonstrate that it is feasible the transfer of entan-glement from the QMF to a pair of distant NVs (bothelectronic and nuclear spins) in such a noisy solid-stateenvironment. First, we propose to entangle the electronicspins with a third party o mediator: If the electronic spinsare strongly coupled to their nearest nuclear spins, thehyperfine interaction between them allows an effectiveentanglement transfer to the nuclear spins.Previous related works have proposed the use of NVcenters as hybrid quantum systems [30–33] in which elec-tron spins provide high fidelity control and readout whilenuclear spins, with ultra-long coherence times, supportrobust quantum registers. Also, the entanglement trans-fer from continuous variables to discrete spin systemshas been considered from different approaches [34–37].By contrast with most of previous studies, our presentapproach not only propose the entanglement generationbetween NV electronic spins but, most importantly, italso predicts the entanglement transfer to distant nuclearspins in noisy spin environments.The paper is organized as follows: In Sect. II we ad-dress the entanglement transfer from a two-mode entan-gled QMF to the electronic NV-spins in noisy environ-ments associated with nuclear spin baths. In Sect. III weextend previous results to the coupled electron-nuclearNV-spins by discussing three different scenarios: two dis-tant NV electron spins, two nuclear spins and one non-local electron-nucleus spin pair. A relevant result of thisanalysis is the identification of regimes for which max-imum entanglement is obtained in noisy environments.In Sect. IV, we report numerical results for the time de-pendent entanglement generation and the identificationof optimal parameters for maximum entanglement trans-fer under nuclear spin bath effects. Finally, in Sect. V wedraw our conclusions and discuss some possible outlooks. II. QMF POWER ENTANGLING OVER TWODISTANT ELECTRON-ELECTRON SPINS INNOISY NVS
The physics contained in the full system displayedin Fig. 1 is quite rich and it is therefore instructive toconsider a limiting case before analyzing the full cross-entangling processes in the composite multi-bath envi-
A B N V N V D i a m o n d s u b s t r a t e D i amond s ub str a t e JPA (a)(b)
JPA
FIG. 1. a) Two distant single NV centers, each one embed-ded in its own nuclear spin bath, in different branches of aparametric Josephson amplifier producing highly entangledmicrowave photons. (b) Schematics of (a) where e i , ν i denotethe electronic and nuclear spins of the individual NV centerin branch i ( i = A, B ). ronment. In the following, we derive and discuss sepa-rately results for the uncoupled electron-nucleus NV sys-tem, for short e i - ν i system, i = A, B , because of its highrelevance for the existing theoretical and experimentalliterature.Thus, we start by considering the simplest sce-nario where we disregard the effects of the closest nuclearspin (see Fig. 1-(b)): a two-arm device where in eachpath, A and B , we place a single NV-electron driven byan entangled QMF in presence of a diluted C nuclearnoisy bath. In each path a microwave cavity enhancesthe NV-microwave field coupling strength. The subsys-tems labeled by A and B are assumed to be identical.We assume that a magnetic field is applied along the z axis, leading to a Zeeman splitting between the electronicsub-levels with spin z-component m s = ± m s = 0, m s = − / C spin bath on theentanglement transfer process.Thus, for the uncoupled e i − ν i system the two-armwhole Hamiltonian isˆ H = (cid:88) j = A,B (cid:104) ω j σ z,j + Ω j ˆ a † j ˆ a j + g j (cid:16) ˆ a † j ˆ σ − j + ˆ a j ˆ σ + j (cid:17) + ˆ H EB,j + ˆ H B,j (cid:105) . (1)The first three terms in Eq.(1) correspond to the usualJaynes-Cummings (JC) Hamiltonian, where ω j and Ω j denote the electronic spin splitting and microwave cav-ity frequency, respectively and g j describe the electron-cavity coupling in arm j . The ˆ σ j,z operator representsthe Pauli spin matrix for the selected two-level NV tran-sition, while ˆ a † j , ˆ a j are the creation and annihilation op-erators for the QMF mode in arm j . The nuclear bathcouples to the NV-electron spin through the termˆ H EB,j = ˆ σ z,j N j (cid:88) k =1 [ A ( (cid:126)r k )ˆ τ k,z + B ( (cid:126)r k ) (ˆ τ k,x cos φ k + ˆ τ k,y sin φ k )] , (2)where ˆ τ k,x and ˆ τ k,y denote the Pauli spin operators forthe nuclear spin. The unit vector joining the electron andthe k-th nuclear spin (cid:126)r k = ( r k , θ k , φ k ) is characterized bythe polar angle θ k and azimuthal angle φ k and N j is thenumber of C nuclear spin in the j -th diamond lattice.The large difference between electron and nuclear Zee-man energies leads to ignore flip-flop terms involving ˆ σ x and ˆ σ y operators. The coupling strengths in Eq.(2) are A ( (cid:126)r k ) = − µ π γ NV γ C r k (cid:2) ( θ k ) − (cid:3) , (3)and B ( (cid:126)r k ) = − µ π γ NV γ C r k θ k )sin( θ k ) , (4)where γ NV ( γ C ) denotes the gyromagnetic ratio of theNV electron (nuclear) spin and r k is the distance betweenthe NV and the k-th nucleus in the diluted spin bath.The local nuclear spin bath Hamiltonian, ˆ H B,j , is givenby: ˆ H B,j = ˆ H N,j + ˆ H DD,j , (5)where ˆ H N,j = N j (cid:88) k =1 ω k τ k,z , (6)and ˆ H DD,j = (cid:88) i In the previous section we discussed a simple situa-tion where only the electronic spin of each NV centerhave been considered. We are now able to go beyondthat simple scenario. More realistically, each NV centeris composed of an electronic spin, e j , coupled via a hy-perfine interaction to a nearest neighbor nuclear spin ν j ( j = A, B ) which can be that of the substitutional nitro-gen atom itself − N or a C atom in the first-shell,see Fig. 1-(b). The Hamiltonian for this system includingthe nuclear bath within the mean field approximation isˆ H = ˆ H A + ˆ H B = (cid:88) j = A,B (cid:110) ω j σ j,z + Ω j ˆ a † j ˆ a j + G j ( t ) (cid:16) ˆ a † j ˆ σ − j + ˆ a j ˆ σ + j (cid:17) + ˆ σ z,j [ A ( (cid:126)r j )ˆ τ j,z + B ( (cid:126)r j ) (ˆ τ j,x cos φ j + ˆ τ j,y sin φ j )] } , (20) where the unit vector joining the electron-nuclear spinpair in the j -th NV is given by (cid:126)r j = ( r j , θ j , φ j ) with thepolar angle θ j and azimuthal angle φ j , respectively. Ex-pressions for coefficients A ( (cid:126)r j ) and B ( (cid:126)r j ) are the sameas those quoted in Eqs. (3)- (4). Let us now proceed toanalyze the spin pair system’s entanglement dynamics.Since subsystems A and B are independent, their respec-tive Hamiltonian operators commute ˆ H = ˆ H A + ˆ H B with (cid:104) ˆ H A , ˆ H B (cid:105) = 0. The Hamiltonian in Eq.(20) commuteswith the total excitation number operatorˆ N = ˆ N A + ˆ N B , (21)ˆ N j = ˆ a † j ˆ a j + (cid:18) ˆ σ j,z + 12 (cid:19) + (cid:18) ˆ τ j,z + 12 (cid:19) ; j = A, B. (22)Consequently, for each subsystem a sub-space with awell defined number of excitations presents a closed dy-namics which proceeds independently from other sub-spaces with different excitation number. Let | n, e σ , ν τ (cid:105) j denotes a general state for the subsystem j with n =0 , , , , ... photons, the electron in one of the states | e σ (cid:105) j = | e g (cid:105) j , | e e (cid:105) j with ˆ σ j,z | e g (cid:105) j = − | e g (cid:105) j , ˆ σ j,z | e e (cid:105) j = | e e (cid:105) j and the nucleus in state | ν τ (cid:105) j = | ν g (cid:105) j , | ν e (cid:105) j withˆ τ j,z | ν g (cid:105) j = − | ν g (cid:105) j , ˆ τ j,z | ν e (cid:105) j = | ν e (cid:105) j . Thus, the fullHilbert space for each subsystem can be partitionedinto independent sub-spaces in the following way: aone-dimensional subspace corresponding to the state | , e g , ν g (cid:105) j with N j = 0 excitations; a single three dimen-sional sub-space, with N j = 1 , spanned by the vectors | , (cid:105) j = | , e e , ν g (cid:105) j , (23) | , (cid:105) j = | , e g , ν g (cid:105) j , (24) | , (cid:105) j = | , e g , ν e (cid:105) j , (25)and finally an infinite number of four dimensional sub-spaces with N j ≥ n ≥ N j = n j + 1) spanned by vectors | N, (cid:105) j = | n, e e , ν g (cid:105) j , (26) | N, (cid:105) j = | n + 1 , e g , ν g (cid:105) j , (27) | N, (cid:105) j = | n − , e e , ν e (cid:105) j , (28) | N, (cid:105) j = | n, e g , ν e (cid:105) j . (29)We assume an unentangled initial state of the form | ψ (0) (cid:105) = | r (cid:105) ⊗ | e g , ν g (cid:105) A ⊗ | e g , ν g (cid:105) B , where the initialstate for the microwave radiation has the same form asin Eq.(12). At later times the system’s state becomes | ψ ( t ) (cid:105) = ˆ U A,B ( t ) r c ∞ (cid:88) n =0 r nt | n, e g , ν g (cid:105) A ⊗ | n, e g , ν g (cid:105) B = r c ∞ (cid:88) n =0 r nt (cid:104) ˆ U A ( t ) | n, e g , ν g (cid:105) A (cid:105) ⊗ (cid:104) ˆ U B ( t ) | n, e g , ν g (cid:105) B (cid:105) , (30)The evolution operator is ˆ U A,B ( t ) = ˆ U A ( t ) ⊗ ˆ U B ( t ) be-cause we consider independent subsystems. The totalevolution operator ˆ U A,B is determined by the system’sHamiltonian given by Eq. (20). The state in Eq. (30),can be expandend in terms of a set of time dependentcoefficients and the base states Eqs. (23)-(29) | ψ ( t ) (cid:105) = (cid:88) j = A,B (cid:34) C ,j | , e g , ν g (cid:105) j + (cid:88) k =1 C ,k ( t ) | , k (cid:105) j + ∞ (cid:88) N =2 4 (cid:88) k =1 C N,k ( t ) | N, k (cid:105) j (cid:35) . (31)Due to the inclusion of the hyperfine interaction betweenthe e j − ν j spins we can not obtain analytical expressionsfor the density matrix that characterize the dynamicalevolution of the spin system ρ ( t ) = | ψ ( t ) (cid:105) (cid:104) ψ ( t ) | , there-fore we have calculated numerically the density matrixfor the system ¯ ρ ( t ) = | ψ ( t ) (cid:105) (cid:104) ψ ( t ) | and then the reduceddensity operator ¯ ρ Q ( t ) (16 × 16 matrix) tracing over thestates of the field¯ ρ Q ( t ) = ∞ (cid:88) p =0 ∞ (cid:88) q =0 (cid:104) p, q | ¯ ρ ( t ) | p, q (cid:105) , (32)where p and q represent the photon states number inthe two branches and the bar in ¯ ρ Q ( t ) and ¯ ρ ( t ) denotesaverages over the stochastic term affecting the spin-cavitycoupling term.Before starting to use this formalism, we have com-pared the numerical results in the case where the hyper-fine interaction between the e j − ν j spins is zero, withthe analytical expressions obtained in sec. II. First, wefound numerically the term (cid:68) | C , ( t ) | (cid:69) , where C , ( t ) isone of the time dependent coefficient in Eq. (31), thenwe compare this solution with the analytical expression (cid:68) | C , ( t ) | (cid:69) = 12 (1 − cos (2 gt ) R ( t, , b A , τ A )) . (33)The above expression was obtained using the analyticalresult for the density matrix presented in appendix A.(Eq. A1). In Fig. 2(a) we present the obtained results. The next step, was compare the analytical and numericaldensity matrix elements, in Fig. 2(b) we show the resultfor ρ ( t ), and similar results were obtained for the otherdensity matrix entries. Finally, we evaluate the concur-rence between two electronic spins e A − e B with the an-alytical and numerical techniques, the results are shownin Fig. 2(c). The Fig. 2(a)-(b)-(c), were realized withnoise conditions b A = b B = 0 . g , gτ A = gτ B = 0 . n = 1000 numerical realizations. This results allow de-termine the number of realizations where the numericalresults converge with the analytical solutions. Now weare ready to use this formalism, following the proceduredescribed before, to evaluate the photon induced spinquantum correlations. In the stochastic simulation wehave considered 10 realizations for assuring numericalconvergence in the calculation of these averages. A. Non-local electron-electron ( e A − e B )entanglement In this section we calculate the entanglement be-tween e A − e B spins under noise conditions, includingthe hyperfine interaction between the nuclear spinsassociated to each NV center ( ν A and ν B ). In thissituation Eqs. (19) are not valid, therefore we requireto return to Eqs. (32) and calculate the reduced den-sity matrix. Due to the inclusion of the nuclear spininteraction we cannot anymore evaluate analytically the C o n c u rr e n c e (a)(b)(c) FIG. 2. Comparison between the analytical and numericalsolution in the case where the hyperfine interaction betweenthe e j − ν j ( j = A, B ) spins is zero. The red line correspondsto the analytical solution, the blue line (points) the numericalresults. Noise conditions were included with b A = b B = 0 . g , gτ A = gτ B = 0 . n = 10 numerical realizations.(a) (cid:10) | C , ( t ) | (cid:11) coefficient associated to two electronic spins as afunction of time (in g units) with the corresponding error bars.(b) Density matrix entrie ρ ( t ) of two electronic spins as afunction of time (in g units), with r = 0 . 5. (c) Concurrencebetween two electronic spins as a function of time (in g units),for r = 0 . density matrix entries, therefore we require to evaluatethem numerically with an appropriate average overmany realizations of the noise effects. In a four-spinbase, ordered as {| e g , ν g (cid:105) , | e g , ν e (cid:105) , | e e , ν g (cid:105) , | e e , ν e (cid:105)} A ⊗{| e g , ν g (cid:105) , | e g , ν e (cid:105) , | e e , ν g (cid:105) , | e e , ν e (cid:105)} B , we obtain a 16 × {| e A,g , e B,g (cid:105) , | e A,g , e B,e (cid:105) , | e A,e , e B,g (cid:105) , | e A,e , e B,e (cid:105)} ,reads as e ( t ) = e , ( t ) 0 0 e , ( t )0 e , ( t ) 0 00 0 e , ( t ) 0 e , ( t ) 0 0 e , ( t ) , (34) with e , ( t ) = ρ , ( t ) + ρ , ( t ) + ρ , ( t ) + ρ , ( t ) ,e , ( t ) = ρ , ( t ) + ρ , ( t ) + ρ , ( t ) + ρ , ( t ) ,e , ( t ) = ρ , ( t ) + ρ , ( t ) + ρ , ( t ) + ρ , ( t ) ,e , ( t ) = ρ , ( t ) + ρ , ( t ) + ρ , ( t ) + ρ , ( t ) , (35) e , ( t ) = e ∗ , ( t )= ρ , ( t ) + ρ , ( t ) + ρ , ( t ) + ρ , ( t ) . (36)Numerical results for different QMF and noise parame-ters will be discussed in Sect. IV. B. Non-local electron-nuclear ( e A − ν B )entanglement Let us now consider the QMF entangling power overan electron-nuclear spin pair in distant NVs under theeffects of separate C spin baths. In a base ordered as {| e A,g , ν B,g (cid:105) , | e A,g , ν B,e (cid:105) , | e A,e , ν B,g (cid:105) , | e A,e , ν B,e (cid:105)} the A electron B nucleus reduced density matrix reads as q ( t ) = q , ( t ) 0 0 q , ( t )0 q , ( t ) 0 00 0 q , ( t ) 0 q , ( t ) 0 0 q , ( t ) , (37)with q , ( t ) = ρ , ( t ) + ρ , ( t ) + ρ , ( t ) + ρ , ( t ) ,q , ( t ) = ρ , ( t ) + ρ , ( t ) + ρ , ( t ) + ρ , ( t ) ,q , ( t ) = ρ , ( t ) + ρ , ( t ) + ρ , ( t ) + ρ , ( t ) ,q , ( t ) = ρ , ( t ) + ρ , ( t ) + ρ , ( t ) + ρ , ( t ) , (38) q , ( t ) = q ∗ , ( t )= ρ , ( t ) + ρ , ( t ) + ρ , ( t ) + ρ , ( t ) . (39)Specific form for the density matrix elements are pre-sented in the appendix B. In order to analyze the entan-glement transfer from the QMF to the e A − ν B system andin particular investigate in detail its dependence on thenoise sources we have evaluated numerically Eqs. (38)-(39) with averages over the noise realizations. C. Non-local nuclear-nuclear ( ν A − ν B )entanglement In the previous section we show the mechanism to gen-erate entangled states between electronic spins with acorrelated field. Now we investigate the most intrigu-ing possibility of a controlled entanglement generationin a nuclear spin pair in separate NV centers in thediamond lattice. Due to the weak coupling betweenthe correlated field and the nuclear spins, we will usethe hyperfine interaction between the electronic and nu-clear spins as a mediator of the correlation or quan-tum bus, this kind of mechanism has been proposedin past to connect a finite number of nuclear spins I = 1 / {| ν A,g , ν B,g (cid:105) , | ν A,g , ν B,e (cid:105) , | ν A,e , ν B,g (cid:105) , | ν A,e , ν B,e (cid:105)} thenucleus-nucleus density matrix reads as ν ( t ) = ν , ( t ) 0 0 ν , ( t )0 ν , ( t ) 0 00 0 ν , ( t ) 0 ν , ( t ) 0 0 ν , ( t ) , (40)with ν , ( t ) = ρ , ( t ) + ρ , ( t ) + ρ , ( t ) + ρ , ( t ) ,ν , ( t ) = ρ , ( t ) + ρ , ( t ) + ρ , ( t ) + ρ , ( t ) ,ν , ( t ) = ρ , ( t ) + ρ , ( t ) + ρ , ( t ) + ρ , ( t ) ,ν , ( t ) = ρ , ( t ) + ρ , ( t ) + ρ , ( t ) + ρ , ( t ) , (41) ν , ( t ) = ν ∗ , ( t )= ρ , ( t ) + ρ , ( t ) + ρ , ( t ) + ρ , ( t ) , (42)in the Appendix B we show the expressions for Eqs. (41)-(42). We have evaluated numerically the expressionsEqs. (41)-(42) for determining each of the density ma-trix entries. IV. RESULTS AND DISCUSSION Up to now, we have described the general theoreti-cal formalism necessary for addressing the entanglementtransfer from two-mode microwave squeezed radiation toa bipartite system composed of electronic and/or nuclearspins of spatially separated NV centers. Before going tothe discussion of our results, it is important to assess thepoint concerning realistic numbers for the NV-microwavecoupling strength to which we turn now our attention bybriefly reviewing different proposed setups. Direct mag-netic coupling between an ensemble of NVs and trans-mission line resonators (TLR) has been experimentallyachieved in the linear or Gaussian regime [12, 45], con-firming additionally the scaling of the collective couplingstrength with the square root of the number of emitters.The reported value for the collective coupling constantbetween an ensemble of 10 NV centers and the TLRcan attain values up to g col / π ≈ M Hz . Furthermore,the possibility of reaching strong coupling between indi-vidual NV electronic spins and TLR, g/ π ≈ . M Hz , has been analyzed for the case of an interconnectingquantum system such as a nanomechanical resonator[46]. Moreover, a closely related method extended thosepossibilities for reaching strong coupling between a sin-gle NV electronic spin and a TLR [47]. In addition,related works have proposed a direct coupling betweenNVs and superconducting flux qubits with a coupling of g/ π ≈ M Hz for a NV diamond located at the centerof the superconducting small loop [31], and the trans-fer of single excitations between the NV ensemble with aflux qubit has also been presented in [48]. Finally, thestrong coupling between NV qubits and superconduct-ing resonators has made possible the transfer of quan-tum states between them, under conditions of a couplingstrength on the order of g/ π ≈ M Hz as discussed in[49]. We stress that the plots we describe below are givenin terms of dimensionless quantities (for instance gt fordimensionless time, among others). So that a feature inthe entanglement evolution seen at dimensionless gt = 1means approximately occurring at a time t ∼ − − µ s, well within the experimental reach of most of the pre-viously quoted works. Thus, our general results may betestable under realistic experimental conditions.In this section we provide additional analysis of theentanglement transfer in the three bipartite systems pre-sented before: electron-electron, electron-nucleus andnucleus-nucleus. In particular we investigate in detailits dependence on the noise sources. The experimentalvalues considered in our calculations are: a driven mi-crowave frequency Ω = Ω resonant with the electronspin frequencies ω = ω = (3 / × g .As shown in Fig. 3, we start using the formalism pre-sented in Sect. II where the hyperfine coupling betweenthe e i − ν j ( i, j = A, B ) spins A ( (cid:126)r ) = B ( (cid:126)r ) = 0 andwe calculate analytically the concurrence and quantumdiscord between the e A − e B spins. We have plotted twocases: in Fig. 3(a) we evidence the effective entanglementfor the e A − e B spins as function of the squeezing param-eter r and time with symmetric conditions for the twobranches, g ,A = g ,B = g . Furthermore, as the coher-ent dynamics of the NV centers is strongly influenced bythe coupling with neighboring spins ( C spin bath) thenoise effect in entanglement transfer simulated with theparameters b and τ is shown in Fig. 3(b). Comparingthe results between isolated spins Fig. 3(a) and the re-alistic situation of the spin bath Fig. 3(b) we observe awide region of strong entanglement even with the noisyconditions. As a consequence of the spins bath we note adecrease in the concurrence but principally for large r val-ues. An appreciable entanglement is obtained for r ≤ . G E =cosh [ r ] = 2 . e A -e B concurrence no spin bath e A -e B concurrence with nuclear spin bathe A -e B quantum discord no spin bath Concurrence and Quantum Discord values e A -e B quantum discord with nuclear spin bath FIG. 3. e A − e B concurrence (panels (a) and (b)) andquantum discord (panels (c) and (d)) as a function of theQMF squeezing parameter r and dimensionless time gt . Nonuclear spin bath effects in panels (a) and (c) while nu-clear spin effects are displayed in panels (b) and (d) with gτ A = gτ B = 0 . 5. In all plots the static QMF-spin couplingstrength is g ,A = g ,B = g . glement, we have calculated the quantum discord [51]as a function of r and time gt between e A − e B spinsFig. 3(c) without spin bath and in a noise environmentFig. 3(d). Comparing the results between concurrenceand quantum discord we can evidence similar behaviors,however the quantum discord persist a longer times whileconcurrence fall to zero and vanish in the same period oftime. A more detailed comparison between concurrenceand quantum discord is presented in Fig. 4 where we haveselected r = 0 . 87 values from Fig. 3 (dashed lines).Next, we have included the effect of the hyperfinecoupling and considered A ( (cid:126)r ) = B ( (cid:126)r ) ≈ g . We haveevaluated numerically the expressions Eqs. (35)-(36),Eqs. (38)-(39) and Eqs. (41)-(42) which include the av-erages over the coefficients that determine the densitymatrix entries. In the simulation we have considered 10 stochastic realizations for determining the averages overnuclear noises.For isolated spin systems the time dependent concur-rence is presented in Fig. 5(a), Fig. 5(c) and Fig. 5(e):for e A − e B , e A − ν B and ν A − ν B , respectively. Thebath effect in the entanglement transfer is illustrated inFig. 5(b), Fig. 5(d) and Fig. 5(f) where the noise param-eters are b = 0 . g and gτ = 0 . 5. For the electronic spinsthe maximum entanglement is achieved for small squeez- QD C on c u rr e n ce gt FIG. 4. e A − e B concurrence (dashed lines) and quantum dis-cord (continuous lines) with symmetric conditions as a func-tion of gt for a selected QMF squeezing parameter r = 0 . b = 0 . g and gτ = 0 . ing value r even including the hyperfine interaction withthe proximal nuclear spin. The bath inclusion changesslightly the optimal region to obtain entanglement butsmall r values are again needed. The dynamics for nu-clear spins or the combination of electronic and nuclearspins allows to characterize the strength of the entan-glement in terms of the squeezing microwave parame-ter. The results obtained show that for these systemsthe amount of squeezing in the microwaves required toproduce entanglement is greater compared with the elec-tron pair situation. Beside we can observe that the batheffect is greater in the entanglement between electronicspins, this effect is evidenced more clearly in Fig. 6 wherewe have selected r = 0 . 87 of Fig. 5(dashed red lines) andevaluate the concurrence as a function of time. The bluecontinuous line represents the e A − e B entanglement whilethe medium dashed red line the e A − ν B and the smalldashed black line ν A − ν B .It is worth noting that a longtime interest has existedfor reaching cross entanglement between different spinspecies, in special electron-nucleus entanglement, due tothe fact of its non-trivial consequences for quantum com-puting devices. In the field of NMR based quantum infor-mation processing, malonic acid molecular single crystalswere used to demonstrate that the entanglement betweendisparate spins (electronic spin resonance in GHz whilethe nuclear spin resonance is in the frequency domain ofMHz) is not only achievable but detectable [52]. Onthe other hand, magic number transitions in few electronquantum dots have been proposed for affecting and de-tecting the entanglement between the electron spins anda single nuclear spin, providing reliable quantum gateoperations [53]. We stress that results discussed in thissection bring an alternative path for reaching such cross0 e A -e B no spin bath e A -e B with spin bath v A -e B no spin bath v A -e B with spin bath v A - v B no spin bath v A - v B with spin bath Concurrence values FIG. 5. Concurrence of different spin pairs in separate NVsas a function of the QMF squeezing parameter r and dimen-sionless time gt . Panels (a) and (b) denote e A − e B , panels (c)and (d) represent e B − ν A (or equivalently e A − ν B ), panels (e)and (f) correspond to ν A − ν B . In all plots the static QMF-spin coupling strength is fixed to g ,A = g ,B = g . No nuclearbath effects yield to results in (a), (c) and (e). Nuclear batheffects with b A = b B = 0 . g and gτ A = gτ B = 0 . entangling, with the added possibility of affecting spa-tially separated different spin species.Finally, the effect of the hyperfine coupling betweenthe e i − ν j spin is illustrated in Fig. 7 where we com-pare the exact analytical solution for the concurrencebetween two electronic spins e A − e B (dashed line) with-out hyperfine interaction with the numerical solution for I = 0 . g , I = g and I = 2 g , where we have considered A ( (cid:126)r ) = B ( (cid:126)r ) = I . In Fig. 7(a) no spin bath included and C on c u rr e n ce gt FIG. 6. Concurrence for distant NV spins as a function ofdimensionless time gt , symmetric case g ,A = g ,B = g , fora selected QMF squeezing parameter r = 0 . 87 (marked bydashed red lines in Fig. 5-(b),(d),(f)). The continuous blueline represents the e A − e B spin pair, the dashed red line e A − ν B (or ( e B − ν A )) and the black dashed line ν A − ν B . in Fig. 7(b) symmetric noise conditions were includedwith b A = b B = 0 . g and gτ A = gτ B = 0 . 5. The resultsevidence that if we reduce the hyperfine coupling betweenthe e j − ν j spins the numerical solutions go identical tothe analytical results, validating the above results.Now, we want highlight two elements of the presentedresults: first, we note that the nuclear entanglement per-sists for longer times compared with the electron en-tanglement even under noise environments. Second, inFigs. 3-5 we observe very definite frequencies for the en-tanglement evolution in each system: e j − e j , ν j − ν j , e j − ν j . Therefore, we calculate the Fourier transform ofthe concurrence in order to determine relevant frequen-cies in the system’s entanglement dynamics. In Fig. 8 weshow the results for the Fourier transform of the concur-rence between e A − e B (Fig. 8(a)) and ν A − ν B (Fig. 8(b))as a function of the frequency ω in g units, with sym-metric conditions g ,A = g ,B = g and no spin bath.The squeezing parameter r was fixed as r = 0 . 87 be-cause we note that the central frequency in the Fouriertransform does not change with the squeezing of the mi-crowaves. Besides that, in the nuclear entanglement wehave a greater spectrum of relevant frequencies comparedwith the e A − e B entanglement where the frequency ap-pears as a more defined peak. Finally, we present how tochange the position of the peaks in the frequency scale( ω p ) of the Fourier transform for e A − e B (Fig. 9(a))and ν A − ν B (Fig. 9(b)) by varying the hyperfine cou-pling A ( (cid:126)r ) = B ( (cid:126)r ) = I . The results show a high de-pendence with the hyperfine coupling, and additionalthey recover the expected result for the uncoupled case A ( (cid:126)r ) = B ( (cid:126)r ) = 0 where ω p = 21 C on c u rr e n ce gt C on c u rr e n ce gt (a)(b) FIG. 7. e A − e B concurrence as a function of dimensionlesstime gt , symmetric case g ,A = g ,B = g and QMF squeez-ing parameter r = 0 . 87, for selected values of the hyperfineinteraction between e j − ν j in the local j -th NV. Solid linesrepresent the numerical solution: the small blue line corre-sponds to I = 0 . g , the red medium line is for I = 1 g whilethe black large line for I = 2 g . The dashed line representsthe exact analytical solution for the case where no hyperfineinteraction. (a) No nuclear spin baths. (b) Nuclear spin bathswith symmetric noise parameters b = 0 . g and gτ = 0 . V. CONCLUSIONS In summary, we have derived analytical expressionsfor the density matrix describing the dynamics of distantelectronic spins interacting with a two mode squeezedstate in a noise environment. We have characterized thedynamical entanglement in terms of the concurrence forthe two spins approximating the effect of the bath, with aclassical theory, as a Ornstein-Uhlenbeck process. Fromour analytical and numerical results, we conclude that asqueezed microwave field produced by a parametric am-plifier can be efficiently employed to induce entanglementin initially uncorrelated spin systems even when embed-ded in a noisy environment. We performed numericalsimulations with the same initial states by varying the - - 10 0 10 200.000.050.100.150.200.250.30 ω / g F [ C ] - - 10 0 10 200510152025 ω / g F [ C ] (a)(b) FIG. 8. Fourier transform of the concurrence C as a functionof dimensionless frequency ω/g , symmetric case g ,A = g ,B ,for r = 0 . 87 and no spin bath was included. Panel (a) denote e A − e B , panel (b) represent ν A − ν B . QMF and noise parameters, and obtained qualitativelysimilar results.The proposed scheme allows to evidence as the inclu-sion of noise environments change the optimal r values toobtain maximum entanglement. In a realistic scenario,we have included the hyperfine interaction between theproximal N spin and the electronic spin. In this sit-uation the analytical expressions are not valid then anumerical solution was realized.We extend our calculations to nuclear spins andelectron-nucleus entanglement. Our result probes thateven for nuclear spins which no interact directly with theentangled microwave field is possible an effective trans-fer of correlations mediated by the hyperfine electron-nuclear interaction. Besides for the nuclear systems theentanglement persist in spin baths environments thatproduce decoherence. While maximum entanglement isreached for small squeezing values for the electronic spinshighly entangled states for the microwaves is required toentangle nuclear spins in a spin bath.2 (b) (a) / g ω p / g / g ω p / g FIG. 9. Peak position of frequency in the Fourier transform ofconcurrence between spin pairs as a function of the hyperfineinteraction A ( (cid:126)r ) = B ( (cid:126)r ) = I . (a) e A − e B spins. (b) ν A − ν B spins A shifting in the squeezing value for obtain maximumentanglement was shown for the electronic spins in pres-ence of a spin bath, while for nuclear spins this value isconstant. Moreover this scheme show the required val-ues of squeezing in the studied systems and the limitingvalues for get entangled states in a spin bath.Finally, we show that other quantum correlations be-sides entanglement persist even in noise environmentsand the effect of the spin bath is small on other cor-relations beyond entanglement. ACKNOWLEDGMENTS A.V.G., F.J.R. and L.Q. acknowledge financial sup-port from Facultad de Ciencias at UniAndes-2015 project”Transfer of correlations from non-classically correlatedreservoirs to solid state systems” and project ”Quan-tum control of non-equilibrium hybrid systems-Part II”,UniAndes-2015. Appendix A: Entanglement dynamics formalism forNV electronic spins It is well known that the Hamiltonian Eq.(9) commuteswith the operator associated to the total number of ex- citations ˆ N = (cid:80) j = A,B (cid:104) ˆ a j † ˆ a j + (cid:16) ˆ σ z,j +12 (cid:17)(cid:105) . From thissymmetry it follows that the full spin-QMF Hilbert spacecan be separated in invariant sub-spaces of dimension 2for each arm ˆ H j = (cid:88) n ⊕ ˆ H n,j , (A1)each sub-space spanned by orthonormal bases with n j excitations {| ( n j − 1) + (cid:105) , | ( n j − −(cid:105)} expressed as: | ( n j − 1) + (cid:105) = cos (cid:16) α n,j (cid:17) | n j − , e e (cid:105) + sin (cid:16) α n,j (cid:17) | n j , e g (cid:105) , | ( n j − −(cid:105) = − cos (cid:16) α n,j (cid:17) | n j − , e e (cid:105) + sin (cid:16) α n,j (cid:17) | n j , e g (cid:105) , (A2)with tan( α n,j ) = g ,j √ n j δ j , the detuning is given by δ j = ω j − Ω j , and | e g (cid:105) , | e e (cid:105) , represent the ground and ex-cited states for the electronic spin. This latter symmetrycan also be exploited by associating a su(2)-Lie algebrawithin each invariant sub-space with n total excitationsas ˆ J x,j = 12 (cid:113) ˆ N j (cid:16) ˆ a † j σ − j + ˆ a j σ + j (cid:17) , (A3)ˆ J y,j = i (cid:113) ˆ N j (cid:16) ˆ a † j σ − j − ˆ a j σ + j (cid:17) , (A4)ˆ J z,j = 12 σ z,j . (A5)Therefore, the Hamiltonian for the sub-space with N j excitations can be written asˆ H ( t ) = (cid:88) j = A,B Ω j ˆ N j + δ j ˆ J z,j + 2 √ n j g j ( t ) ˆ J x,j − Ω j . (A6)In the interaction picture the Hamiltonian in Eq.(A6)describes an effective spin in a time-dependent magneticfield ˆ H j ( t ) = (cid:88) j = A,B (cid:126) ˆ J j · (cid:126)B j ( t ) , (A7)with (cid:126)B j ( t ) = (cid:18) (cid:113) ˆ N j g j ( t ) , , δ j (cid:19) . (A8)From now on we restrict to the resonance case δ j = 0yielding to a time-dependent field in the x -direction. Un-der this latter assumption the Hamiltonian commuteswith itself at different times, leading to an exactly solv-able evolution operatorˆ U j ( t ) = e i θ j,n ( t ) ˆ J x,j , (A9)3with θ j,n ( t ) = √ n j (cid:90) t dt j g ( t j ) . (A10)Note specially that at resonance | ( n j − , + (cid:105) = 1 √ | ( n j − , e e (cid:105) + | n j , e g (cid:105) ] , | ( n j − , −(cid:105) = 1 √ − | ( n j − , e e (cid:105) + | n j , e g (cid:105) ] . (A11)Within the sub-space with N j excitations it holds thatˆ J x,j | ( n j − 1) + (cid:105) = 12 | ( n j − 1) + (cid:105) , (A12) ˆ J x,j | ( n j − −(cid:105) = − | ( n j − −(cid:105) . (A13)Eq.(A9) acting on the initial state for the system | ψ ( t = 0) (cid:105) = | r (cid:105) | e g (cid:105) A | e g (cid:105) B (with r given by the Eq.(12)),and properties in Eqs.(A12)-(A13), allow us to easily ob-tain the NV-cavity quantum state at time t as | ψ ( t ) (cid:105) = ˆ U A,B ( t ) r c ∞ (cid:88) n =0 r nt | n, g (cid:105) A ⊗ | n, g (cid:105) B , (A14)where we have written r c = 1 / cosh(r) and r t =tanh(r).In order to proceed further, individual terms in Eq.(A14)can be developed asˆ U j ( t ) | n, g (cid:105) j = r c ∞ (cid:88) n =0 r nt √ (cid:104) e i √ n j θ j,n ( t ) | ( n j − 1) + (cid:105) + e − i √ n j θ j,n | ( n j − −(cid:105) (cid:105) j , (A15)with θ j,n ( t ) given by Eq.(A10). Appendix B: Two-NV full density matrix Here we summarize some important intermediate stepsto reach the analytical expression for the reduced two NV density matrix. The density operator at time t becomes4 r c ¯ ρ ( t ) = | e g , (cid:105) AA (cid:104) e g , | ⊗ | e g , (cid:105) BB (cid:104) e g , | ++ 12 ∞ (cid:88) n =1 r nt (cid:104)(cid:68) e − iθ A,n ( t ) (cid:69) | e g , (cid:105) AA (cid:104) ( n − 1) + | + (cid:68) e iθ A,n ( t ) (cid:69) | e g , (cid:105) AA (cid:104) ( n − −| (cid:105) ⊗⊗ (cid:104)(cid:68) e − iθ B,n ( t ) (cid:69) | e g , (cid:105) BB (cid:104) ( n − 1) + | + (cid:68) e iθ B,n ( t ) (cid:69) | e g , (cid:105) BB (cid:104) ( n − −| (cid:105) ++ 12 ∞ (cid:88) n =1 r nt (cid:104)(cid:68) e iθ A,n ( t ) (cid:69) | ( n − 1) + (cid:105) AA (cid:104) e g , | + (cid:68) e − iθ A,n ( t ) (cid:69) | ( n − −(cid:105) AA (cid:104) e g , | (cid:105) ⊗⊗ (cid:104)(cid:68) e iθ B,n ( t ) (cid:69) | ( n − 1) + (cid:105) BB (cid:104) e g , | + (cid:68) e − iθ B,n ( t ) (cid:69) | ( n − −(cid:105) BB (cid:104) e g , | (cid:105) +14 ∞ (cid:88) n =1 ∞ (cid:88) m =1 r n + mt (cid:104)(cid:68) e i ( θ A,n ( t ) − θ A,m ( t )) (cid:69) | ( n − 1) + (cid:105) AA (cid:104) ( m − , + | ++ (cid:68) e i ( θ A,n ( t )+ θ A,m ( t )) (cid:69) | ( n − 1) + (cid:105) AA (cid:104) ( m − , −| ++ (cid:68) e − i ( θ A,n ( t )+ θ A,m ( t )) (cid:69) | ( n − −(cid:105) AA (cid:104) ( m − , + | ++ (cid:68) e − i ( θ A,n ( t ) − θ A,m ( t )) (cid:69) | ( n − −(cid:105) AA (cid:104) ( m − , −| (cid:105) ⊗ (cid:104)(cid:68) e i ( θ B,n ( t ) − θ B,m ( t )) (cid:69) | ( n − 1) + (cid:105) BB (cid:104) ( m − , + | ++ (cid:68) e i ( θ B,n ( t )+ θ B,m ( t )) (cid:69) | ( n − 1) + (cid:105) BB (cid:104) ( m − , −| ++ (cid:68) e − i ( θ B,n ( t )+ θ B,m ( t )) (cid:69) | ( n − −(cid:105) BB (cid:104) ( m − , + | ++ (cid:68) e − i ( θ B,n ( t ) − θ B,m ( t )) (cid:69) | ( n − −(cid:105) BB (cid:104) ( m − , −| (cid:105) . (B1)The following expressions are valuable for that pur- pose: ∞ (cid:88) p =0 (cid:104) p | | ( n − 1) + (cid:105) (cid:104) ( m − 1) + | | p (cid:105) = 12 [ δ n,m ( | e g (cid:105) (cid:104) e g | + | e e (cid:105) (cid:104) e e | ) + δ n − ,m | e e (cid:105) (cid:104) e g | + δ n,m − | e g (cid:105) (cid:104) e e | ] ∞ (cid:88) p =0 (cid:104) p | | ( n − 1) + (cid:105) (cid:104) ( m − −| | p (cid:105) = 12 [ δ n,m ( | e g (cid:105) (cid:104) e g | − | e e (cid:105) (cid:104) e e | ) + δ n − ,m | e e (cid:105) (cid:104) e g | − δ n,m − | e g (cid:105) (cid:104) e e | ] ∞ (cid:88) p =0 (cid:104) p | | ( n − −(cid:105) (cid:104) ( m − 1) + | | p (cid:105) = 12 [ δ n,m ( | e g (cid:105) (cid:104) e g | − | e e (cid:105) (cid:104) e e | ) − δ n − ,m | e e (cid:105) (cid:104) e g | + δ n,m − | e g (cid:105) (cid:104) e e | ] ∞ (cid:88) p =0 (cid:104) p | | ( n − −(cid:105) (cid:104) ( m − −| | p (cid:105) = 12 [ δ n,m ( | e g (cid:105) (cid:104) e g | + | e e (cid:105) (cid:104) e e | ) − δ n − ,m | e e (cid:105) (cid:104) e g | + δ n,m − | e g (cid:105) (cid:104) e e | ] . (B2) Appendix C: Density matrix for electronic NV spinswith constant spin-cavity couplings In order to calculate the density matrix elements in thecase where the spin-cavity coupling is constant, we haveevaluated the expression Eq.(14) with g ,A = g ,B = g ,in this limit we have ρ , ( t ) = 1 r c ∞ (cid:88) n =0 r nt cos (cid:0) √ ng t (cid:1) , (C1) ρ , ( t ) = ρ , ( t ) = 1 r c ∞ (cid:88) n =0 r nt sin (cid:0) √ ng t (cid:1) cos (cid:0) √ ng t (cid:1) , (C2) ρ , ( t ) = 1 r c ∞ (cid:88) n =0 r nt sin (cid:0) √ ng t (cid:1) , (C3)5and the non-diagonal term becomes ρ , ( t ) = − r c ∞ (cid:88) n =0 r n +1 t sin (cid:0) √ n + 1 g t (cid:1) cos (cid:0) √ ng t (cid:1) . (C4) Appendix D: Time dependent coefficients In this section we provide the elements of the reduced16 × 16 density matrix for the electronic and nuclear spins in a noisy environment. In the main text we present sim-plified analytical expression for two electronic spins whenno hyperfine coupling with the proximal nuclear spin.However a numerical solution is needed if we include thisinteraction and the spin C bath. We start defining thesystems’s state at time t as | ψ ( t ) (cid:105) = ˆ U A,B ( t ) ∞ (cid:88) n =0 α n | n, e g , ν g (cid:105) A ⊗ | n, e g , ν g (cid:105) B (D1)= ∞ (cid:88) n =0 α n (cid:104) ˆ U A ( t ) | n, e g , ν g (cid:105) A (cid:105) ⊗ (cid:104) ˆ U B ( t ) | n, e g , ν g (cid:105) B (cid:105) = α | n, e g , ν g (cid:105) A ⊗ | n, e g , ν g (cid:105) B + α (cid:34) (cid:88) i =1 C ,i ( t ) | , i (cid:105) (cid:35) A ⊗ (cid:34) (cid:88) i =1 C ,j ( t ) | , j (cid:105) (cid:35) B + ∞ (cid:88) N =2 α N (cid:34) (cid:88) i =1 C N,i ( t ) | N, i (cid:105) (cid:35) A ⊗ (cid:88) j =1 C N,j ( t ) | N, j (cid:105) B where | α N | = tanh ( r ) N cosh ( r ) . (D2)The terms C ,i ( t ), C ,j ( t ) (where i and j can take values1 , , 3) are the coefficients at time t in the expansion forthe state in the sub-space with N = 1 excitations in thebranches A and B , respectively. The coefficients C N,i ( t )and C N,j ( t ) (where i and j in this case can take values1 , , , 4) allow determine the state at time t in the four dimensional subspaces with N ≥ 2. Now, we can proceedto evaluate the density matrix as ˆ ρ ( t ) = | ψ ( t ) (cid:105) (cid:104) ψ ( t ) | andthe reduced density matrix tracing over the state of thefield ¯ ρ Q ( t ) = ∞ (cid:88) p =0 ∞ (cid:88) q =0 (cid:104) p, q | ¯ ρ ( t ) | p, q (cid:105) , (D3)with p and q the photon number in the two branches. Thebar in ¯ ρ Q ( t ) and ¯ ρ ( t ) represent stochastic terms due tothe noise spin bath. The diagonal elements obtained forthe density matrix are given by ρ , = | α | + | α | (cid:68) | C , ( t ) | (cid:69) A (cid:68) | C , ( t ) | (cid:69) B + ∞ (cid:88) N =2 | α N | (cid:68) | C N, ( t ) | (cid:69) A (cid:68) | C N, ( t ) | (cid:69) B (D4) ρ , = | α | (cid:68) | C , ( t ) | (cid:69) A (cid:68) | C , ( t ) | (cid:69) B + ∞ (cid:88) N =2 | α N | (cid:68) | C N, ( t ) | (cid:69) A (cid:68) | C N, ( t ) | (cid:69) B (D5)6 ρ , = | α | (cid:68) | C , ( t ) | (cid:69) A (cid:68) | C , ( t ) | (cid:69) B + ∞ (cid:88) N =2 | α N | (cid:68) | C N, ( t ) | (cid:69) A (cid:68) | C N, ( t ) | (cid:69) B (D6) ρ , = ∞ (cid:88) N =2 | α N | (cid:68) | C N, ( t ) | (cid:69) A (cid:68) | C N, ( t ) | (cid:69) B ρ , = | α | (cid:68) | C , ( t ) | (cid:69) A (cid:68) | C , ( t ) | (cid:69) B + ∞ (cid:88) N =2 | α N | (cid:68) | C N, ( t ) | (cid:69) A (cid:68) | C N, ( t ) | (cid:69) B ρ , = | α | (cid:68) | C , ( t ) | (cid:69) A (cid:68) | C , ( t ) | (cid:69) B + ∞ (cid:88) N =2 | α N | (cid:68) | C N, ( t ) | (cid:69) A (cid:68) | C N, ( t ) | (cid:69) B ρ , = | α | (cid:68) | C , ( t ) | (cid:69) A (cid:68) | C , ( t ) | (cid:69) B + ∞ (cid:88) N =2 | α N | (cid:68) | C N, ( t ) | (cid:69) A (cid:68) | C N, ( t ) | (cid:69) B ρ , = ∞ (cid:88) N =2 | α N | (cid:68) | C N, ( t ) | (cid:69) A (cid:68) | C N, ( t ) | (cid:69) B ρ , = | α | (cid:68) | C , ( t ) | (cid:69) A (cid:68) | C , ( t ) | (cid:69) B + ∞ (cid:88) N =2 | α N | (cid:68) | C N, ( t ) | (cid:69) A (cid:68) | C N, ( t ) | (cid:69) B ρ , = | α | (cid:68) | C , ( t ) | (cid:69) A (cid:68) | C , ( t ) | (cid:69) B + ∞ (cid:88) N =2 | α N | (cid:68) | C N, ( t ) | (cid:69) A (cid:68) | C N, ( t ) | (cid:69) B ρ , = | α | (cid:68) | C , ( t ) | (cid:69) A (cid:68) | C , ( t ) | (cid:69) B + ∞ (cid:88) N =2 | α N | (cid:68) | C N, ( t ) | (cid:69) A (cid:68) | C N, ( t ) | (cid:69) B ρ , = ∞ (cid:88) N =2 | α N | (cid:68) | C N, ( t ) | (cid:69) A (cid:68) | C N, ( t ) | (cid:69) B ρ , = ∞ (cid:88) N =2 | α N | (cid:68) | C N, ( t ) | (cid:69) A (cid:68) | C N, ( t ) | (cid:69) B ρ , = ∞ (cid:88) N =2 | α N | (cid:68) | C N, ( t ) | (cid:69) A (cid:68) | C N, ( t ) | (cid:69) B ρ , = ∞ (cid:88) N =2 | α N | (cid:68) | C N, ( t ) | (cid:69) A (cid:68) | C N, ( t ) | (cid:69) B ρ , = ∞ (cid:88) N =2 | α N | (cid:68) | C N, ( t ) | (cid:69) A (cid:68) | C N, ( t ) | (cid:69) B (D7)where N vary between 0 and the photon number state inthe field. In our simulation we have considered N = 85and we have probed the essential conditions for a den-sity matrix. The results yield to T r { ¯ ρ Q ( t ) } = 1 as itshould be. The stochastic realizations in the coefficients (cid:104) ... (cid:105) take into account many realizations in the systemswhen we include the noise parameters. In our calcula-tion we have evaluated the average taking approximately10000 realizations in the coefficients average. Symmet-ric conditions have been considered in the two branches.Non-diagonal elements in the density matrix are:7 ρ , = ρ ∗ , = α α ∗ (cid:104) C , ( t ) (cid:105) A (cid:104) C , ( t ) ∗ (cid:105) B + ∞ (cid:88) N =1 α N α ∗ N +1 (cid:10) C N, ( t ) C ∗ N +1 , ( t ) (cid:11) A (cid:10) C N, ( t ) C ∗ N +1 , ( t ) (cid:11) B ρ , = ρ ∗ , = α α ∗ (cid:104) C , ( t ) (cid:105) A (cid:104) C , ( t ) ∗ (cid:105) B + ∞ (cid:88) N =1 α N α ∗ N +1 (cid:10) C N, ( t ) C ∗ N +1 , ( t ) (cid:11) A (cid:10) C N, ( t ) C ∗ N +1 , ( t ) (cid:11) B ρ , = ρ ∗ , = α α ∗ (cid:104) C , ( t ) (cid:105) A (cid:104) C , ( t ) ∗ (cid:105) B + ∞ (cid:88) N =1 α N α ∗ N +1 (cid:10) C N, ( t ) C ∗ N +1 , ( t ) (cid:11) A (cid:10) C N, ( t ) C ∗ N +1 , ( t ) (cid:11) B ρ , = ρ ∗ , = α α ∗ (cid:104) C , ( t ) (cid:105) A (cid:104) C , ( t ) ∗ (cid:105) B + ∞ (cid:88) N =1 α N α ∗ N +1 (cid:10) C N, ( t ) C ∗ N +1 , ( t ) (cid:11) A (cid:10) C N, ( t ) C ∗ N +1 , ( t ) (cid:11) B ρ , = ρ ∗ , = ∞ (cid:88) N =0 α N α ∗ N +2 (cid:10) C N, ( t ) C ∗ N +2 , ( t ) (cid:11) A (cid:10) C N, ( t ) C ∗ N +2 , ( t ) (cid:11) B ρ , = ρ ∗ , = | α | (cid:68) | C , ( t ) | (cid:69) A (cid:104) C , ( t ) C , ( t ) ∗ (cid:105) B + ∞ (cid:88) N =2 | α N | (cid:68) | C N, ( t ) | (cid:69) A (cid:10) C N, ( t ) C ∗ N, ( t ) (cid:11) B ρ , = ρ ∗ , = α α ∗ (cid:104) C , ( t ) C , ( t ) ∗ (cid:105) A (cid:104) C , ( t ) C , ( t ) ∗ (cid:105) B + ∞ (cid:88) N =2 α N α ∗ N +1 (cid:10) C N, ( t ) C ∗ N +1 , ( t ) (cid:11) A (cid:10) C N, ( t ) C ∗ N +1 , ( t ) (cid:11) B ρ , = ρ ∗ , = α α ∗ (cid:104) C , ( t ) C , ( t ) ∗ (cid:105) A (cid:104) C , ( t ) C , ( t ) ∗ (cid:105) B + ∞ (cid:88) N =2 α N α ∗ N +1 (cid:10) C N, ( t ) C ∗ N +1 , ( t ) (cid:11) A (cid:10) C N, ( t ) C ∗ N +1 , ( t ) (cid:11) B ρ , = ρ ∗ , = ∞ (cid:88) N =1 α N α ∗ N +1 (cid:10) C N, ( t ) C ∗ N +1 , ( t ) (cid:11) A (cid:10) C N, ( t ) C ∗ N +1 , ( t ) (cid:11) B ρ , = ρ ∗ , = ∞ (cid:88) N =1 α N α ∗ N +1 (cid:10) C N, ( t ) C ∗ N +1 , ( t ) (cid:11) A (cid:10) C N, ( t ) C ∗ N +1 , ( t ) (cid:11) B ρ , = ρ ∗ , = | α | (cid:104) C , ( t ) C , ( t ) ∗ (cid:105) A (cid:68) | C , ( t ) | (cid:69) B + ∞ (cid:88) N =2 | α N | (cid:10) C N, ( t ) C ∗ N, ( t ) (cid:11) A (cid:68) | C N, ( t ) | (cid:69) B ρ , = ρ ∗ , = α α ∗ (cid:104) C , ( t ) C , ( t ) ∗ (cid:105) A (cid:104) C , ( t ) C , ( t ) ∗ (cid:105) B + ∞ (cid:88) N =2 α N α ∗ N +1 (cid:10) C N, ( t ) C ∗ N +1 , ( t ) (cid:11) A (cid:10) C N, ( t ) C ∗ N +1 , ( t ) (cid:11) B ρ , = ρ ∗ , = α α ∗ (cid:104) C , ( t ) C , ( t ) ∗ (cid:105) A (cid:104) C , ( t ) C , ( t ) ∗ (cid:105) B + ∞ (cid:88) N =2 α N α ∗ N +1 (cid:10) C N, ( t ) C ∗ N +1 , ( t ) (cid:11) A (cid:10) C N, ( t ) C ∗ N +1 , ( t ) (cid:11) B ρ , = ρ ∗ , = | α | (cid:68) | C , ( t ) | (cid:69) A (cid:104) C , ( t ) C , ( t ) ∗ (cid:105) B + ∞ (cid:88) N =2 | α N | (cid:68) | C N, ( t ) | (cid:69) A (cid:10) C N, ( t ) C ∗ N, ( t ) (cid:11) B ρ , = ρ ∗ , = ∞ (cid:88) N =2 | α N | (cid:10) C N, ( t ) C ∗ N, ( t ) (cid:11) A (cid:68) | C N, ( t ) | (cid:69) B ρ , = ρ ∗ , = ∞ (cid:88) N =2 | α N | (cid:10) C N, ( t ) C ∗ N, ( t ) (cid:11) A (cid:10) C N, ( t ) C ∗ N, ( t ) (cid:11) B ρ , = ρ ∗ , = α α ∗ (cid:10) C , ( t ) C ∗ , ( t ) (cid:11) A (cid:10) C , ( t ) C ∗ , ( t ) (cid:11) B + ∞ (cid:88) N =2 α N α ∗ N +1 (cid:10) C N, ( t ) C ∗ N +1 , ( t ) (cid:11) A (cid:10) C N, ( t ) C ∗ N +1 , ( t ) (cid:11) B ρ , = ρ ∗ , = | α | (cid:10) C , ( t ) C ∗ , ( t ) (cid:11) A (cid:10) C , ( t ) C ∗ , ( t ) (cid:11) B + ∞ (cid:88) N =2 | α N | (cid:10) C N, ( t ) C ∗ N, ( t ) (cid:11) A (cid:10) C N, ( t ) C ∗ N, ( t ) (cid:11) B ρ , = ρ ∗ , = | α | (cid:10) C , ( t ) C ∗ , ( t ) (cid:11) A (cid:68) | C , ( t ) | (cid:69) B + ∞ (cid:88) N =2 | α N | (cid:10) C N, ( t ) C ∗ N, ( t ) (cid:11) A (cid:68) | C N, ( t ) | (cid:69) B ρ , = ρ ∗ , = α α ∗ (cid:10) C , ( t ) C ∗ , ( t ) (cid:11) A (cid:10) C , ( t ) C ∗ , ( t ) (cid:11) B + ∞ (cid:88) N =2 α N α ∗ N +1 (cid:10) C N, ( t ) C ∗ N +1 , ( t ) (cid:11) A (cid:10) C N, ( t ) C ∗ N +1 , ( t ) (cid:11) B ρ , = ρ ∗ , = ∞ (cid:88) N =2 | α N | (cid:10) C N, ( t ) C ∗ N, ( t ) (cid:11) A (cid:68) | C N, ( t ) | (cid:69) B ρ , = ρ ∗ , = ∞ (cid:88) N =1 α N α ∗ N +1 (cid:10) C N, ( t ) C ∗ N +1 , ( t ) (cid:11) A (cid:10) C N, ( t ) C ∗ N +1 , ( t ) (cid:11) B ρ , = ρ ∗ , = ∞ (cid:88) N =1 α N α ∗ N +1 (cid:10) C N, ( t ) C ∗ N +1 , ( t ) (cid:11) A (cid:10) C N, ( t ) C ∗ N +1 , ( t ) (cid:11) B ρ , = ρ ∗ , = | α | (cid:10) C , ( t ) C ∗ , ( t ) (cid:11) A (cid:10) C , ( t ) C ∗ , ( t ) (cid:11) B + ∞ (cid:88) N =2 | α N | (cid:10) C N, ( t ) C ∗ N, ( t ) (cid:11) A (cid:10) C N, ( t ) C ∗ N, ( t ) (cid:11) B ρ , = ρ ∗ , = α α ∗ (cid:10) C , ( t ) C ∗ , ( t ) (cid:11) A (cid:10) C , ( t ) C ∗ , ( t ) (cid:11) B + ∞ (cid:88) N =2 α N α ∗ N +1 (cid:10) C N, ( t ) C ∗ N +1 , ( t ) (cid:11) A (cid:10) C N, ( t ) C ∗ N +1 , ( t ) (cid:11) B ρ , = ρ ∗ , = ∞ (cid:88) N =1 α N α ∗ N +1 (cid:10) C N, ( t ) C ∗ N +1 , ( t ) (cid:11) A 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