Entanglement of systems of dipolar coupled nuclear spins at the adiabatic demagnetization
aa r X i v : . [ c ond - m a t . o t h e r] D ec Entanglement of systems of dipolar coupled nuclearspins at the adiabatic demagnetization
S. I. Doronin , E. B. Fel’dman , M. M. Kucherov ,A. N. Pyrkov Institute of Problems of Chemical Physics of Russian Academy of Sciences,Chernogolovka, Moscow Region, Russia, 142432 Siberian Federal University, Krasnoyarsk, 660074, RussiaE-mail: [email protected]
Abstract.
We consider the adiabatic demagnetization in the rotating reference frame(ADRF) of a system of dipolar coupled nuclear spins s = 1 / ntanglement of systems of nuclear spins at the adiabatic demagnetization
1. Introduction
Entangled states are very important for quantum computing, teleportation,cryptography [1]. Although entanglement is a profound concept of quantum informationtheory its experimental realization in many-body systems is an unsolved problem up tonow. Meanwhile the existing criterions of the existence of entanglement [2, 3] allow us toinvestigate entangled states in conventional NMR experiments [4]. It is well known thatentanglement emerges in systems of nuclear spins at microkelvin temperatures [5, 6].Such temperatures can be achieved with the adiabatic demagnetization in the rotatingreference frame (ADRF) [7].In the present article we consider the emergence of entanglement in the courseof the ADRF in the system of nuclear spins coupled by the dipole-dipole interaction(DDI). We perform numerical calculations for a chain consisting of nine spins and aplane nine-spin cluster in order to prove the emergence of entanglement of its differentsubsystems. We use the Wootters criterion [2] for the investigation of the spin pairentanglement. Entanglement of subsystems with larger numbers of spins is investigatedwith the positive partition transposition (PPT) criterion [3]. Entanglement emerges atapproximately the same temperature of the order of microkelvins for all subsystem sizes.However, the temperature depends on the space dimension of the system.
2. The density matrix of a spin system at the ADRF
We consider a system of N nuclear spins s = 1 / w ( t ) , which is perpendicular to the permanent magnetic field. The Hamiltonian, H lab , of the system in the laboratory frame can be written as follows H lab = w I z + H dz + 2 w I x cos (cid:20)Z t w ( t ′ ) dt ′ (cid:21) , (1)where w is the Larmor frequency, w is the amplitude of the rf field (in frequency units), I nα is the projection of the angular momentum operator of the n-th spin ( n = 1 , . . . N )on the α axis ( α = x, y, z ), I α = P Nn =1 I nα , and H dz is the secular part of the DDIHamiltonian [4] which can be written as H dz = X i 3. The reduced density matrix of a spin pair at the ADRF and theWootters criterion In order to obtain the reduced density matrix of an arbitrary pair of spins i and j we usethe approach developed in [5, 6]. The density matrix, ρ eq ( t ) , of (8) can be representedas [5, 6] ρ = X ξ ,ξ ,...,ξ N =0 α ξ ξ ...ξ N ...N x ξ ⊗ . . . ⊗ x ξ N N , (12)where N is a number of spins in the system, ξ k ( k = 1 , , . . . , N ) is one of thevalues { , , , } , x k = I k is the unit matrix of the dimension 2 × , x k = I kx ,x k = I ky , x k = I kz , and α ξ ξ ...ξ N ...N is a numerical coefficient. Averaging the densitymatrix of (12) over all spins except spins i and j and taking into account thattr { x ξ k k } = 0 ( k = 1 , , . . . , N ; ξ k = 1 , , 3) we arrive at the following expression forthe reduced density matrix, ρ ( ij ) eq ( t ) , of the i th and j th spins ρ ( ij ) eq ( t ) = X ξ i ,ξ j =0 α ξ i ξ j ij x ξ i i ⊗ x ξ j j , (13)where α ξ i ξ j ij = 2 N − tr { ρx ξ i i x ξ j j } tr { ( x ξ i i ) ( x ξ j j ) } . (14)The coefficients, α ξ i ξ j ij , of (14) can be calculated numerically. Then the reduced densitymatrix of the pair of spins i and j is determined completely. In order to apply theWootters criterion [2] one should find the ”spin-flipped” density matrix, ˜ ρ ( ij ) eq ( t ) , whichis ˜ ρ ( ij ) eq ( t ) = ( σ y ⊗ σ y )[ ρ ( ij ) eq ( t )] ∗ ( σ y ⊗ σ y ) (15)where the asterisk denotes complex conjugation in the standard basis {| i , | i , | i , | i} and the Pauli matrix σ y = − ii ! . The calculation of the density matrix, ˜ ρ ( ij ) eq ( t ) , and the diagonalization of the matrix product ρ ( ij ) eq ( t ) ˜ ρ ( ij ) eq ( t ) are performed numerically.The concurrence of the two–spin system with the density matrix ρ ( ij ) eq ( t ) is equal to [2] C = max { , λ − λ − λ − λ − λ } , λ = max { λ , λ , λ , λ } (16)where λ , λ , λ , and λ are the square roots of the eigenvalues of the product ρ ( ij ) eq ( t ) ˜ ρ ( ij ) eq ( t ) . 4. The entangled state of a spin subsystem with its environment at theADRF The spin pair entangled states are a simple type of entanglement which can bedescribed with the Wootters criterion [2] completely. Meanwhile entanglement of bigger ntanglement of systems of nuclear spins at the adiabatic demagnetization 5. Numerical analysis of entanglement in a nine-spin chain at the ADRF The results of the numerical investigation of the spin pair entanglement for a linearchain consisting of nine spins coupled by the DDI at the ADRF are represented infigure 1. We started the ADRF with the offset ∆ = 10 D where D is the DDI Figure 1. The concurrence, C, versus the dimensionless inverse temperature, βD , for spins 1 and 2 of the nine-spin chain according to the Wootters criterion. The dashline shows the same result obtained with the PPT criterion; w /D = 2. ntanglement of systems of nuclear spins at the adiabatic demagnetization n = (10 − n ) D , n = 0 , , . . . , . The corresponding values of the inversetemperatures, β n = ~ /kT n ( n = 0 , , . . . , , are found from (9) for the dimensionlessentropy S/k = 0 . . Figure 1 shows the concurrence of spins 1 and 2 of the nine-spin chainas a function of the dimensionless parameter βD . At comparatively high temperaturesthe concurrence is equal to zero (see figure 1) and the spin system is in a separablestate. When the temperature is getting sufficiently low in the course of the ADRFthe concurrence is sharply increasing and entanglement emerges. The entangled stateappears at βD ≈ . T ≈ . µK when D = 2 π s − . Notice that ordered states of nuclear spins were observed in a CaF single crystal atmicrokelvin temperatures [7]. Figure 2. The double negativity versus the dimensionless inverse temperature, βD , for spin 1 of the nine-spin chain and the other spins; w /D = 2. The dash line in figure 1 demonstrates that the result for the spin pair entanglementobtained with the PPT criterion coincide practically with the ones obtained with theWootters criterion. Figure 2 shows the double negativity versus the dimensionlessparameter βD for the first spin of the chain (the first subsystem) and the other spinsof the chain (the second subsystem) in the course of the ADRF when the dimensionlessentropy S/k = 0 . . One can conclude that entanglement emerges at βD ≈ . . Thisresult is close to the one for the spin pair entanglement. Figure 3 shows the doublenegativity versus the parameter βD for the first three spins of the chain (the firstsubsystem) and the other six spins of the chain (the second subsystem) at the sameconditions. Here entanglement is getting sufficiently large at βD ≈ . . In fact we havefound that entanglement of different subsystem emerges approximately at the sametemperature. ntanglement of systems of nuclear spins at the adiabatic demagnetization Figure 3. The double negativity versus the dimensionless inverse temperature, βD , for the first three spins of the nine-spin chain and the other spins; w /D = 2. Figure 4. The square cluster of nine spins. 6. Entanglement in the square cluster of nine spins The suggested method allows us to study entanglement in the systems of arbitrary spacedimensions. As an example, we consider the square cluster consisting of nine spins (seefigure 4). The dipolar coupling constant of spins j and k (the numbers of spins are ntanglement of systems of nuclear spins at the adiabatic demagnetization D jk = γ ~ r jk (1 − θ jk ) , (17)where γ is gyromagnetic ratio, r jk is the distance between spins j, k and θ jk is the anglebetween the vector, ~r jk , and the external magnetic field, ~H . The simple analysis yieldscos θ jk = 9( { ( j − / } − { ( k − / } ) ([( j − / − [( k − / + 9( { ( j − / } − { ( k − / } ) , (18)and r jk = a p ([( j − / − [( k − / + 9( { ( j − / } − { ( k − / } ) , (19)where a is the distance between the nearest neighbors in the cluster, [ q ] is the integerpart of q and { q } is the fractional part of q . Figure 5. The double negativity versus the dimensionless inverse temperature, βD , for spins 1, 2, 4 (first subsystem) and spins 5, 6, 8, 9 (second subsystem), in the squarecluster; w /a = 2. Figure 5 and figure 6 show that the entangled states emerge at βD = 1 . ntanglement of systems of nuclear spins at the adiabatic demagnetization Figure 6. The double negativity versus the dimensionless inverse temperature, βD , for spins 1, 2 (first subsystem) and spins 3, 6, 9 (second subsystem), at w /a = 2 inthe square cluster of nine spins. 7. Conclusion We investigated numerically entanglement in the chain of nuclear spins and in the squarecluster with the DDI in the course of the ADRF using a special computer program. Weshowed that the entangled states emerge at microkelvin temperatures for typical DDIcoupling constants. Two criterions [2, 3] of entanglement yield the same results for thespin pair entangled states. Entanglement of different subsystems emerges approximatelyat the same temperature and the pairwise entanglement can be used as an indicator ofentanglement of bigger subsystems. It is also worth to notice that we take into accountthe DDI of the remote spins in contrast to the works [5, 6] where the nearest-neighborinteractions were only considered. The performed calculations show that there are nothe entangled states of remote spins both in one-dimensional and two-dimensional cases.Entanglement emerges only if subsystems are in a direct contact. Entanglement ofdifferent subsystems at microkelvin temperatures suggest possible applications of linearspin chains in quantum information processing.We thank Professor D. E. Feldman and Professor V. A. Atsarkin for stimulatingdiscussions and E. I. Kuznetsova for assistance in our work. This work was supportedby the Russian Foundation for Basic Research (grant 07-07-00048). ntanglement of systems of nuclear spins at the adiabatic demagnetization References [1] Nielsen M A and Chuang I L, 2000 Quantum Computation and Quantum Information (CambridgeUniversity Press)[2] Wootters W K, 1998 Phys. Rev. Lett. , 2245[3] Peres A, 1996 Phys. Rev. Lett. , 1413[4] Goldman M, 1970 Spin Temperature and Nuclear Magnetic Resonance in Solids (Clarendon,Oxford)[5] Doronin S I, Pyrkov A N and Fel’dman E B, 2007 JETP Letters , 519[6] Doronin S I, Pyrkov A N and Fel’dman E B, 2007 Journal of Experimental and Theoretical Physics , 953[7] Abragam A and Goldman M, 1982 Nuclear Magnetism: Order and Disorder (Oxford UniversityPress, Oxford, England)[8] Vidal G, Werner R F, 2002 Phys. Rev. A , 032314[9] Doronin S I, Fel’dman E B, Guinzbourg I Ya and Maximov I I, 2001 Chem. Phys. Lett. , 144[10] Doronin S I, Fel’dman E B and Lacelle S, 2002 J. Chem. Phys.117