Diamond as a solid state quantum computer with a linear chain of nuclear spins system
aa r X i v : . [ qu a n t - ph ] O c t Diamond as a solid state quantumcomputer with a linear chain of nuclearspins system
G.V. L´opez ∗ Departamento de F´ısica de la Universidad de Guadalajara,Blvd. Marcelino Garc´ıa Barrag´an 1421, esq. Calzada Ol´ımpica,44430 Guadalajara, Jalisco, M´exicoPACS: 03.67.Hk, 03.67.Lx03.67.AcSeptember, 2013
Abstract
By removing a C atom from the tetrahedral configuration of the dia-mond, replace it by a C atom, and repeating this in a linear direction, it ispossible to have a linear chain of nuclear spins one half and to build a solidstate quantum computer. One qubit rotation and controlled-not (CNOT)quantum gates are obtained immediately from this configuration, and CNOTquantum gate is used to determined the design parameters of this quantumcomputer. ∗ [email protected] Introduction
So far, the idea of having a working quantum computer with enough number ofqubits (at least 1000) has faced two main problems: the decoherence [1]-[8] due theinteraction of the environment with the quantum system, and technological lim-itations (pick up signal from NMR quantum computer [9] and [10], laser controlcapability in ion trap quantum computer [11] and [12], physical build up for morethan two qubits like in photons cavities [13], atoms traps [14] and [15], Josephson’sjoint ions [16], Aronov-Bhom devices [17], diamond NV device [18], or high fieldand high field gradients in linear chain of paramagnetic atoms with spin one half[19]). In particular, the linear chain of paramagnetic atoms of spin one half becamea good mathematical model to make studies of quantum gates [20], quantum algo-rithms [21], and decoherence [22] which could be applied to other to other quantumcomputers. In this paper, one put together the ideas of using the diamond stablestructure and the linear chain of spin one half nucleus. To do this, on the tetrahe-dral C (with nuclear spin zero) configuration of the diamond main structure, oneremoves one C element of this configuration an replace it by a C (with nuclearspin one half) atom, and one repeats this replacement along a linear direction of thecrystal. By doing this replacement, one obtains a linear chain of atoms of nuclearspin one half which is protected from the environment by the crystal structure andthe electrons cloud. Therefore, one could have a quantum computer highly tolerantto environment interaction and maybe not so difficult to build it, from the techno-logical point of view. C - C diamond and spin-spin interaction The above idea is represented in Figure 1, where the C atoms are place on theposition of some C atoms. This replacement could be done using the same technicsused to construct the diamond NV structure [25], or using ion implantation technics[23] and neutralization of C in the diamond [24]. It is assumed in this paperthat this configuration can be built somehow. Now, as one can see, the importantinteraction on this configuration is the spin-spin interaction between the nucleus ofthe C atoms. This interaction is well known [26] and is given by U = µ o π ( m · x )( m · x ) − m · m | x | , (1)2igure 1: Diamond C - C .where the magnetic moment m i , i =1 , of C ′ s is related with the nuclear spin as m i = γ S i , (2)being γ the proton gyromagnetic ratio ( γ ≈ . × rad/T · s ). Without loosingthe main idea, it will be assumed here that C magnetic moment is due to proton.The variable x indicates the separation vector between two C nucleus, which hasmagnitude a = | x | ∼ − m . Aligning the chain of C nucleus along the x-axisof the reference system and assuming Ising interaction between C nucleus, thisenergy can be written as U = J ~ S z S z , (3)where the coupling constant J has been defined as J = µ o γ ~ πa . (4)3 Hamiltonian of the system
Consider a magnetic field of the form B ( x, t ) = ( b cos( ωt + ϕ ) , − b sin( ωt + ϕ ) , B ( x )) , (5)where b , ϕ , and ω are the magnitude, the phase, and the frequency of the trans-verse rf-field. The z-component of the magnetic field has a gradient on the x-axis,determined by the difference on Larmore’s frequencies of the C ′ s nuclear magneticmoments, (cid:18) ∆ B ∆ x (cid:19) = ∆ ωγ ∆ x . (6)The magnetic field at the location of the ith- C atom is B i ( t ) = B ( x i , t ), and theinteraction energy of the magnetic moments of the C atoms with the magneticfield is U = − N X i =1 m i · B i ( t ) , (7)where N is the number of C atoms aligned along the x-axis. This energy can bewritten as U = − N X j =1 ω j S zj − Ω2 N − X k =1 (cid:18) e iθ S − k + e − iθ S + k (cid:19) , (8)where ω j is the Larmore’s frequency of the ith- C , ω j = γB ( x j ) , (9)Ω is the Rabi’s frequency, Ω = γb, (10) S − j and S + j are the ascent and descent spin operators, S ± j = S xj ∓ iS yj , and θ hasbeen defined as θ = ωt + ϕ. (11)Let us consider first and second neighbor interactions among C nuclear spins, andassuming equidistant separation between any pair of spins, the Hamiltonian of the4ystem is H = − N X j =1 ω j S zj + J ~ N − X k =1 S zj S zj +1 + J ′ ~ N − X l =1 S zl S zl +2 − Ω2 N X j =1 (cid:18) e iθ S − j + e − iθ S + j (cid:19) , (12)where J is the coupling constant of first neighbor C atoms, and J ′ is the couplingconstant of second neighbor C atoms which must be about one order of magnitudelower than J . One can write this Hamiltonian as H = H + W ( t ), where H and W are defined as H = − N X j =1 ω j S zj + J ~ N − X k =1 S zj S zj +1 + J ′ ~ N − X l =1 S zl S zl +2 , (13)and W ( t ) = − Ω2 N X j =1 (cid:18) e iθ S − j + e − iθ S + j (cid:19) . (14)The operator H is diagonal on the basis {| ξ i = | ξ N . . . ξ i} ξ k =0 , of the Hilbert spaceof 2 N dimensionality. Its eigenvalues defines the spectrum of the system, E ξ = ~ ( − N X j =1 ( − ξ j ω j + J N − X k =1 ( − ξ k + ξ k +1 + J ′ N − X l =1 ( − ξ l + ξ l +2 ) . (15)Since J ′ < J ≪ ω j for j =1 ,...,N , this spectrum is not degenerated with E | ... i asthe energy of ground state, and E | ... i as the energy of the most exited state. Tocalculate the spectrum, one has used the following action of S zj operator S zj | ξ i = ~ − ξ j | ξ i . (16)The Schr¨odinger’s equation, i ~ ∂ | Ψ i ∂t = H | Ψ i , (17)is solved by proposing a solution of the form | Ψ i = X ξ C ξ ( t ) | ξ i , (18)5hich brings about the following system of first order differential equations on theinteraction representation i ~ ˙ a δ = X ξ a ξ e i ( E δ − E ξ ) t/ ~ W δ,ξ ( t ) , (19)where a δ and W δ,ξ are defined as a δ ( t ) = C δ ( t ) e − iE δ t/ ~ (20)and W δ,ξ ( t ) = h δ | W ( t ) | ξ i . (21)This is very well known procedure to solve time dependent Schr¨odinger’s equation,and the solution of Eq. (19) brings about he unitary evolution of the system (giventhe initial condition | Ψ o i ).Defining the evolution parameter τ through the change of variable t = ω o τ ( ω o =2 πM Hz ), the parameters ω j , Ω, J and J ′ are real numbers given in units of ω o .This evolution parameter will be used below in the analysis of the CNOT quantumgate. In order to get an operating quantum computer, one needs to show that, at least,one qubit rotation gate ( N = 1) and two qubits CNOT gate ( N = 2) or threequbits controlled-controlled-not (CCNOT) gate ( N = 3) can be constructed fromthis quantum system. Because this quantum system is homeomorphic [30] to thelinear chain of paramagnetic atoms with spin one half system [27], it is clear fromthe point of view of mathematical models that the above gates can be constructedwith this C - C diamond system. However, one needs to assign realistic workableparameters for the real design of a C - C diamond quantum computer. To do this,one studies in this section the behavior of a quantum CNOT gate as a function ofseveral parameters. One neglect one qubit rotation ( N = 1 , J = J ′ = 0) becauseit is obvious that one can get it through an arbitrary pulse on the rf-field with thefrequency given by the Larmore’s frequency of the qubit ( ω = ω ), for a single C atom in the diamond structure. In particular, the NOT quantum gate is obtainedusing a π -pulse duration ( τ = π/ Ω) with this frequency. Therefore, the study of the6NOT quantum gate is of the most interest ( N = 2 , J = 0 , J ′ = 0). Two qubitsdynamics is obtained from Eqs. (13), (14), and (19), resulting the equations i ˙ a = − Ω2 (cid:0) e − i ( ωt + ϕ +( E − E ) t/ ~ ) a + e − i ( ωt + ϕ +( E − E ) t/ ~ ) a (cid:1) (22) i ˙ a = − Ω2 (cid:0) e + i ( ωt + ϕ +( E − E ) t/ ~ ) a + e − i ( ωt + ϕ +( E − E ) t/ ~ ) a (cid:1) (23) i ˙ a = − Ω2 (cid:0) e + i ( ωt + ϕ +( E − E ) t/ ~ ) a + e − i ( ωt + ϕ +( E − E ) t/ ~ ) a (cid:1) (24) i ˙ a = − Ω2 (cid:0) e + i ( ωt + ϕ +( E − E ) t/ ~ ) a + e + i ( ωt + ϕ +( E − E ) t/ ~ ) a (cid:1) (25)where the complex variables a i for i =1 , , , correspond to the amplitude of probabilityto find the system on the states | i , | i , | i and | i . The energies E i for i =1 , , , are deduced from Eq. (15). Note that a (0) = C (0) and | a ( t ) | = | C ( t ) | (thesame for the other variables). CNOT quantum gate corresponds to the transition | i ↔ | i , and this one is gotten by selecting the rf-frequency as ω = ω − J/ . (26)Larmore’s frequencies are denoted by ω and ω , and ω is parametrized as ω = ω (1 + f ) , (27)where f measures the relative change of the frequencies of both qubits. The sepa-ration of the C nucleus, a , is parametrized as a = ξ · − m. (28)Figure 2 shows the CNOT quantum gate behavior with the initial conditions C (0) = C (0) = C (0) = 0 and C (0) = 1 during a π -pulse ( τ = π/ Ω) and with the pa-rameters B = 0 . T, ω = 21 . , J = 0 . , ξ = 1 , f = 0 . . (29)Figure 3 shows the fidelity parameter, F = |h Ψ ideal | Ψ real i| , (30)7t the end of the π -pulse, as a function of the Rabi’s frequency, where | Ψ real i is thestate obtained with the simulation, and | Ψ ideal i is the expected state ( | i ). Thesimulation was done for two different weak magnetic fields and for f = 0 .
01 (1), f = 0 .
05 (2), f = 0 . f = 0 . | i and | i ) to thedynamics of the system, which depends on Rabi’s frequency and they are explainedby the 2 πk -method [19]. As one can see from this picture , the CNOT gate is verywell produced either with B = 0 . T and f = 0 . B = 0 . T and f = 0 . J , and the fidelity F of the CNOT quantum gate as a function of the twoqubits separation (characterized by the parameter ξ , Eq. (28)), having f = 0 . ξ , mean-while the gradient and coupling constant have the strong variation deduce from Eq.(6) and Eq. (4). Considering the separation of the two C atoms about the thelength of the diamond unit cell, one can select ξ = 3, corresponding to a couplingconstant of J = 0 . B /a ≈ . × T /m .One needs to mention that in the case the alignment of the C atoms be along thez-axis (the same direction of the longitudinal magnetic field), the coupling constantdeduced from Eq. (1) would be given by − J , with J given by Eq. (4), and basicallythe results are the same as the presented here.8ccording to these results, one has now an idea of the value of the parameters forthe design of a quantum computer with the C - C diamond quantum system: (a)Separation between C atoms is a = 3 × − m which can be aligned along thex-axis, (b) coupling constant is J = 0 . π M Hz ), (c) longitudinal magneticfield is B = 0 . T , (d) gradient of this longitudinal magnetic field along the x-axisis ∆ B /a = 0 . × T /m , and (e) magnitude of the rf-magnetic field on the planex-y is b = 0 . T (Rabi’s frequency Ω = 0 . π M hz )).Although the gradient of the magnetic field might be a concern, the magnitudeof the longitudinal magnetic field is low enough to think that this gradient can beachieved. The scalability of the system is clear, the read out system could be basedon single spin measurement technics [28], and studies on decoherence remains to bedone on this system. This quantum computer resembles a solid state NMR system[29]. It was shown that by removing a C atom, replace it by a C atom in the tetra-hedral configuration of the diamond, and doing this process periodically in a lineardirection, one could get a linear chain of nuclear spins one half which can be work asa quantum computer. The interaction between C atoms is governed by the mag-netic dipole-dipole interaction, and the parameters of a possible quantum computerdesign were determined by studying the quantum CNOT gate with two qubits. Al-though there might be a concern about the gradient of the magnetic field along thelines of C atoms, it must not be so difficult to get this gradient since the magnitudeof this magnetic field is relatively low (0.5 T). In principle, it is possible to replacea C atom by any other spin one half atom. However, an unclose configurationof electrons in the lattice makes necessarily to take into account the interaction ofelectrons with this atom ( as it is the case of diamond NV configuration) whichmakes the analysis and the quantum computer much more complicated and sensi-tive to environment interaction. The misplacement of the C atom along the x-axisproduces different coupling constant in the interaction, but according to Figure 4,the fidelity of the CNOT quantum gate does not change, and one would expectthe same result for quantum algorithms. The displacement of C atoms off x-axischanges the coupling constant and the interaction itself, which has to be studied. Inaddition, it still remains to study the decoherence on quantum gates and quantumalgorithms of system. 9 eferences [1] H. -P. Breuer and F. Petruccione, ”The Theory of Open Quantum Systems,”Oxford University Press, 2006.[2] A.O. Caldeira and A.T. Legget, Physica A , , 587 (1983).[3] W.G. Unruh and W.H. Zurek, Phys. Rev. D Phys. Rev. D , 2843 (1992).[5] A. Venugopalan, Phys. Rev. A , 4307 (1997).[6] H.D. Zeh, Found. Phys. , 109 (1973).[7] J.P. Paz and W.H. Zurek, Proc. Les Houches, 111A , 409 (1997).[8] G. Lindblad,
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