QUANTUM COMPUTING WITH MOLECULAR MAGNETS: SPIN-PHONON RELAXATION AND THE PROBLEM OF DECOHERENCE

Transcription

1 QUANTUM COMPUTING WITH MOLECULAR MAGNET: PIN-PHONON RELAXATION AND THE PROBLEM OF DECOHERENCE Alex Tarantul, Boris Tsukerblat Chemistry Department, Ben-Gurion University of the Negev, Beer-heva, Israel In this article we consider spin-phonon relaxation processes in two molecular magnets for which the long living rabi oscillations were observed. This study is aimed to reveal the decoherence mechanisms which are important in vie of possible application of molecular magnets as qubits in quantum computing. Introduction Molecular nanomagnets [] became an object of active research not only because of their fascinating quantum-mechanical properties like magnetic tunneling and bistability but also due to their promising potential to be implemented in the novel molecular electronic and spintronic devices as well as for the quantum computing []. The main idea of quantum computing is that a quantum bit (qubit) constitutes neither pure 0 nor pure value (as in the classical bit) but some quantum superposition 0 of two (or even more) pure quantum states. The content of quantum information stored in the qubit is therefore a set of the mixture coefficients,,.. []. The problem however is that any wanted superposition of the pure states exists only limited period of time until it is destroyed by various decoherence processes whose origin is an interaction of a qubit with outer world. o one of the main requirements for quantum computer is to have its decoherence time sufficiently long so that computation operations may be done before the qubit s content gets destroyed []. Will molecular nanomagnet provide a decoherence time long enough to serve as a qubit?. While the positive answers had been given in [5] the breakthrough happened in 008 when the long-living ( 8 s ) coherent Rabi oscillations have been observed [6] in the well-known (see review [7] and refs. therein) vanadium cluster known as V 5 (Fig. ). Molecular 65

2 nanomagnets of the low-spin and high-spin types are quite different in many points including the decoherence issue. Fig.. Ball-and-stick representation of of the cluster anion [V IV 5As 6 O (H O)] 6 emphasizing the V triangle Fig.. The ball and stick representation of the Fe - complex (see ection ). 66

3 The high-spin Fe clusters (Fig. ) became the second example of molecular magnet in which the coherent oscillations had been observed [8]. This article summarize some preliminary results regarding these two molecular nanomagnets.. The Fe family: exchange interactions The star-like Fe, hereafter Fe (different from the butterfly clusters [9]) marks a family of ingle Molecule Magnets (MM) in which three Fe III are located in the corners and connected to the fourth central ion in such a way that the whole system possesses the approximate trigonal symmetry C (Fig. ). While the first member of this family has been studied for more then ten years ago [0], up to now there are at least five species which are collected in Table. Table Magnetic clusters of the Fe family, stoichiometry and references for detailed information hort toichiometry References name Fe - Fe (acac) (Brmp) 6 [8,,5] Fe - Fe (acac) 6 (tmp) [,5] Fe - Fe (dpm) 6 (OCH ) 6 [0-] Fe - Fe (dpm) 6 in [0] Fe -5 Fe {RN(CH CH O) } 6 [6] In these compounds the major exchange interaction is a strong antiferromagnetic isotropic coupling between the central and corner spins so the total spin in the ground state is 5. These high-spin molecular magnets also demonstrate the typical for the MMs zero field splitting (ZF) and a barrier-type for spin reorientation. During last decade Fe clusters became an object of intensive research. The inelastic neutron scattering (IN) measurements within the ground manifold as well as between the 5 ground state and excited multiplets have been reported in Refs. [-]. A comprehensive study of the Fe - and Fe - complexes is given in refs [,5]. This study includes the 67

4 synthesis information as well as the analysis of the magnetic measurements such as DC/AC susceptibilities, frequency-domain magnetic resonance (FDMR) and continuous-wave electron paramagnetic resonance (CW EPR, W band) spectra. The studies [5] of the anisotropic properties of Fe - and Fe - in frozen solutions show how he ZF parameter D depends on the cluster s solvent environment. The coherent Rabi oscillations in Fe - have been detected [8]. This article provides a detailed investigation of the decoherence mechanisms and relaxation times in Fe -. Among other studies of the Fe family one should mention the work [7] studying the magnetooptical properties of Fe in different polymeric environments, the density-functional studies in the Ref. [8] and the energy-spectrum study in the Ref. [9]. Ref. [0] reports that while anchored to the gold surface the Fe molecules continue to exhibit their intrinsic magnetic hysteresis and it may be a first step toward the MM-based memory devices. Practical implementation of the molecular magnets may become a close reality, hence the aim of this work is an investigation of the role played by spin-lattice relaxation mechanisms in the overall picture of decoherence in the Fe clusters. In the star-like Fe clusters the isotropic Heisenberg-Dirac- Van Vleck (HDVV) superexchange Hamiltonian of C symmetry has a general form H I O ij J i j J i j (),, ij,, Here indexes,, label the corner ions while refers to the ion in a center of the star. The J parameter relates to the corner-to-center exchange pathway and is that species in the Fe family have a large parameter () J between the corner ions. While all J for at least one member - the Fe - complex the ferromagnetic corner-to-corner exchange has been experimentally demonstrated []. The () corresponding J however is much weaker then superexchange i alongside the rays of a star; all up-to-date information about the HDVV 68

5 parameters in the Fe family is to be given in the []. Fig. shows the energy levels of the isotropic exchange model with J 8.5cm (of the Fe - complex) and J 0. The first excited state with is located about cm above the 5 ground state (Fig., inset) and all levels cover the range of 700 cm. The numbers on the right to the energy level show how many intermedia te spins, states possess this energy; one sees that in most cases it is more then one. This excessive degeneracy is legitimate in the spin-frustrated systems like V 5 (see review [] and references therein) where it reflects a real situation in which some state may have more then one possible spin alignment or correspond to the orbital degeneracy in the system. In fact, it was shown [,] that accidental degeneracy in spin-frustrated systems is actually related to the orbital degeneracy. But when appears in the nonfrustrated complexes like Fe, extra degeneracy indicates that the HDVV model should be supplemented by the high order interactions. ome deviation from Fig.. HDVV energy levels of the Fe - complex the strict trigonal symmetry might have remove, at least partially, that degeneracy. In fact, most of references report this deviation and it has been estimated, although ambiguously, using the IN data (Ref. []). It 69

6 is also known [,] that in the model of high-spin systems the higher-order scalar effective Hamiltonians like biquadratic exchange H s s can also play a role. However there are no BQ ij ij i j experimental data about the biquadratic exchange in the Fe. The ground level is not degenerate and in the coupling scheme, is 5,5,5, 5. s,, All Fe species exhibit significant measure of anisotropy which is described by the axial and transversal local anisotropy Hamiltonians expressed in terms of the single ion spins: axial s i si H LA di sz i i and trans H s s LA i i x i y i where i,.. enumerates the ions. It was shown [] that within the orbitaly non-degenerate and well-isolated HDVV multiplet the action of the local anisotropy operators is approximately described by the fullspin Hamiltonian H ANI B Oˆ B Oˆ B Oˆ B Oˆ B Oˆ () m where Ô n are tevens operator equivalents [5]. One should note that 0 B is identical to D if D is a parameter of the conventional ZF axial operator H ZF DZ and B is nothing but doubled parameter E in the transverse anisotropic second order operator trans H ZF E X Y. Quite comprehensive experimental data on the anisotropic parameters (within the ground multiplet) for some Fe compounds is to be given in []. Anisotropic effects split the HDVV eigenstates. Fig. shows the well-known double well emerging within the ground state multiplet 5 as a result of second-order ZF 70

7 described by the Hamitonian given by M DM axial H ZF. The energy levels in the wells are and calculated with D appropriate for Fe -. If trans transverse anisotropy H ZF is neglected, spin projection M is a good quantum number labeling the states in the wells. The overall splitting of the ground state is about 9 cm and the gap between the lowest sublevel M 5 and the first excited sublevel M is cm. Fig.. Double-well ZF inside the ground state of the Fe - complex.. The Fe family: spin-lattice relaxation times The detailed approach for the calculation of spin-lattice relaxation times in Fe is to be given in []. This approach is resulted in the Equation () D 8 b E D () 5 Vˆ () coth kt for the time of the direct (one-phonon) relaxation from the level M to level M 5 (see Fig. ) where 7

8 6 J R R 6 J R ij ij ij 0 R RR 0 (ij=,,) is the spin-vibronic parameter (estimated in []) of the HDVV exchange along the center-to-corners pathways in a star-like molecule; R 0 is an equilibrium center-to-corner distance. Other parameters involved in the Eq. () are: crystal density ρ and sound velocity, averaged squared Van Vleck coefficient b corresponding to the vibrational mode ς [], V ˆ is a spin operator corresponding to the same mode [] and is an energy gap between the levels where the relaxation is being discussed. There is no data regarding the transverse ZF parameter E for the Fe - complex, but one can find the ratio E D 0 from Fe - for which the transverse parameter is known []. With all values 7 and estimations mentioned above one gets the result. 0 s for the one-phonon relaxation time in the transition 5 at T K. Of course, the same result is valid for the second potential well where spin projections have opposite sign. pin-lattice relaxation times were measured for the Fe - molecules in toluene with the aid of pulsed-epr technique 8 with no stationary magnetic field B 0. The Rabi oscillations occur between the ground state sublevels M 5 and, and the value obtained is about 6 0 s which is about five times slower then that in our estimation. This shows that the presented approach in conjunction with the model of the spin-phonon coupling in Fe cluster gives a good estimation for the spin-phonon relaxation rates. A relatively small discrepancy is not surprising due to the use of approximate key parameters (spin-vibronic coupling parameter, sound velocity). The temperature dependence and experimental data [8] show distinctively clear dependence, much stronger then that given by the direct (one-phonon) processes (Eq. ()). The conclusion so is that the two-phonon (either Orbach or Raman) processes play the main role in this system. The direct transitions are insufficient to build the two-phonon trajectories; there is a need for some additional kind of direct transitions in which M is odd D 7

9 number. These transition may come from the interaction of spins with the oscillating transversal magnetic field B of the EPR pulse itself. Another important point is that the mechanism of decomposition of molecular vibrations into the whole-lattice vibrations using the Van Vleck coefficients comes from the solid state while in the experiment [8] the Fe were placed in the frozen solution. This raises a need for a comprehensive theory of spin - thermal bath relaxation mechanisms in the amorphous non-crystal environment.. Exchange interactions in V 5 cluster The model of spin triangle for the low lying spin excitations in the V 5 system suggested in ref. [6,7] includes the isotropic Heisenberg- Dirac-Van Vleck (HDVV) exchange interaction and the antisymmetric (A) exchange proposed by Dzyaloshinskii [8] and Moriya [9] (see also refs. [,]) as an origin of spin canting in magnetic materials. The last interaction was shown [,,-6] to be especially important for the spin frustrated system possessing triangular structure. It was shown that the spin of hexagons are paired due to relatively strong antiferromagnetic interactions while the coupling inside the triangle is relatively small. That is why the model of an effective spin triangle of vanadium ions ( ) belonging to the central triangle provides an adequate description of the whole system at low temperatures. This triangular vanadium magnetic layer in the V 5 structure is shaded in the Fig.. Three spins of the central triangle are coupled through the antiferromagnetic isotropic exchange. The full Hamiltonian of the system looks as: D H H 0 H A J ij i j () The eigen-values of H 0 with the antiferromagnetic ( J 0 ) coupling include two accidentally degenerate spin doublets (ground manifold) and excited spin quadruplet separated by the gap J. According to the overall point symmetry D the vector constants D ij ( ij,, numerate the sides) of the A exchange have, in general, three independent components: along and perpendicular to the side (in plane of the triangle) and perpendicular to the plane component whose i, j 7

10 absolute values are Hamiltonian A A D l, D t and D n respectively. Consequently the H ( normal ) and H can be divided into two parts, H ( in-plane ) which are defined as: H A D n Z Z Z A l X X Y (5) H D X Y D t Y X Y X (6) Y where axes X and Y of the global coordinates system are in the plane while the axis Z is perpendicular to the plane. These two parts of the A exchange are explicitly separated as they play quite different physical roles. The normal part of A exchange splits the ground doublets into two Kramers doublets and gives rise to a strong (first order) magnetic anisotropy while the in-plane part is responsible for the doublet-quadruplet mixing. This mixing leads to the second order zero-field splitting of the level (that is D l Dt 8J ). It was shown [] that D l and D are combined into an effective parameter D t D t D l so the exchange model is fully specified by the three parameters J, D n and D [,5].. pin-lattice relaxation times in V 5 cluster The one-phonon (direct) relaxation times in the V 5 were calculated [6,7] within the same approach as for the Fe. However since in the one-phonon transitions the density of the phonon states at the resonance frequencies is small, an important rule play the twophonon relaxation processes. These are the Raman processes and the Orbach-Aminov type processes [9-]. Due to the fact that the excited levels corresponding to the unpaired spins of the hexagons in V 5 are higher then the Debye energy, only the levels of the central vanadium triangle contribute significantly to the second order relaxation. That is why one can expect that only the resonant Orbach-Aminov type relaxation is actual for the V 5 system at the relatively low temperatures A 7

11 corresponding to the actual EPR measurements. The Raman processes become important only at high temperatures. In ref. [] the equation was given which allows to calculate the Orbach-Aminov relaxation times in a multilevel system in terms of the one-phonon relaxation R probabilities. We adopt here this equation assuming that m and m are the labels for the initial and final states while j numerates the intermediate levels and denotes the one-phonon transition probabilities : wm j m j jm (7) R D j w w w jm w w m j jm w jm Eq. (7) contains contributions to the overall relaxation time arising from the direct one-phonon processes (first term) that have been evaluated earlier as well as the contributions from the two-phonon resonance processes. The estimated one-phonon and Orbach-Aminov relaxation times for the five most intense EPR transitions (see [7] and refs therein) at the frequency 0. cm 9 GHz in parallel field and at different temperatures are given in the Table. At K for both the one-phonon and Orbach-Aminov processes the relaxation times are of the order of 0 s within the manifold while for the excited manifold the direct processes show very long relaxation times of the order of tens of seconds (note, that this estimation in ref [] contains an error). In this manifold Orbach-Aminov processes are much faster ( R ~ 0 s ). At higher temperatures Orbach-Aminov mechanism becomes the leading one in the ground manifold as well while the onephonon processes are less significant. At these temperatures in both manifolds the Orbach-Aminov relaxation times are of the order of 0 s. The relaxation times so far estimated are two orders longer then the coherence time of s measured in the Rabi oscillation 75

12 experiment []. These estimations lead to the conclusion that the decoherence in this system can be probably attributed to the dipolar and hyperfine interactions. Table Relaxation times in the direct and Orbach-Aminov transitions at the frequency of 9 GHz and at different temperatures. ymbol * is related to almost forbidden one-phonon transitions ( D is of the order of tens of sec.) T = K T = 5 K m m τ D, s τ R, s m m τ D, s τ R, s * * * * * * Conclusion To put the results into the perspective the following remarks are to be done. Once again, the second (after V 5 ) molecular magnet where coherent Rabi oscillations have been observed is the Fe - complex (ection ). pin-lattice relaxation time T has been measured for this system and was found to be of the order of 0 s. Due to its strong temperature dependence, T in Fe was attributed to the two-phonon processes, either Raman or Orbach-Aminov types. ince spin-lattice relaxation time has not been measured for V 5, and in this view it would be interesting to compare the results for two systems so far mentioned. One can rudely estimate that the isotropic spin-phonon coupling parameters in V 5 and Fe relate to each other as :0, as the isotropic constants J 0 relate ( 0.85 cm in V 5, 8. cm in Fe ). pinlattice relaxation times then should relate to each other as squared 6 76

13 reciprocal values and one may expect the order of 0 s for T in V 5. However, since the ground state spins also relate as :0 ( in V 5, 5 in Fe ), relaxation rates in V 5 may turn out to be significantly longer then even 0 s and reach the values estimated in this work. Vanadium cluster is known for its very small spin-vibronic interaction. However, in the trinuclear low-spin systems similar to V 5 but with strong spin-vibronic coupling (like copper cluster [,]) the phonon assisted relaxation is expected to be much faster, especially when the main decoherence mechanism, spin-to nuclei and dipole-dipole interactions, are substantially suppressed. According to ref. [6] the coherence time may be increased up to 0 s when the main decoherence mechanisms are suppressed by the chemical means. The approach so far developed can be applied to more complicated spinfrustrated systems, like high-nuclearity magnetic polyoxometalates [5]. Acknowledgments We thank the Israel cience Foundation for the financial support (IF grant no. 68/09). References. D. Gatteschi, R. essoli, J. Villain. Molecular Nanomagnets. Oxford University Press. Oxford M. N. Leuenberger, D. Loss. Nature 00. V. 0. P M. Nielsen, I. L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press. Cambridge D. Loss, D. P. DiVincenzo. Phys. Rev. A 998 V. 57. P A. Ardavan, O. Rival, J. J. L. Morton,. Blundell, A. M. Tyryshkin, G. A. Timco, R. E. P. Winpenny. Phys. Rev. Lett V. 98. P. 0570() 6.. Bertaina,. Gambarelli, T. Mitra, B. Tsukerblat, A. Müller, B. Barbara. Nature 008. V. 5. P B. Tsukerblat, A. Tarantul. The nanoscopic V5 cluster: a unique magnetic polyoxometalate. In Molecular Cluster Magnets. Ed. Winpenny, R. E. P., World cientific. ingapore. 0. P

14 8. C. chlegel, J. van lageren, M. Manoli, E. K. Brechin, M. Dressel. Direct observation of quantum coherence in singlemolecule magnets// Physical review letters 008. V. 0. (), P A. L. Barra, A. Caneschi, A. Cornia, F. F. de Biani, D. Gatteschi, C. angregorio, R. essoli, L. orace. ingle-molecule magnet behavior of a tetranuclear iron(iii) complex. The origin of slow magnetic relaxation in iron(iii) clusters// Journal of the American Chemical ociety 999. V.. P Carretta, P. antini, G. Amoretti, T. Guidi, R. Caciuffo, A. Candini, A. Cornia, D. Gatteschi, M. Plazanet, J. A. tride. Intraand inter-multiplet magnetic excitations in a tetrairon(iii) molecular cluster// Physical Review B 00. V. 70. P.... Gomez-Coca, T. Cauchy, E. Ruiz. Extended Fe() butterfly complexes: theoretical analysis of magnetic properties and magnetostructural maps// Dalton Transactions V. 9. (0), P T. Guidi, G. Amoretti, R. Caciuffo,. Carretta, A. Cornia, C. D. Frost, E. Liviotti. Inter-multiplet transitions in the Fe- magnetic cluster// Journal of Magnetism and Magnetic Materials 00. V. 7. P. E777-E778.. G. Amoretti, R. Caciuffo,. Carretta, A. Cornia, D. Gatteschi, J. Kulda, E. Liviotti. Intra- and inter-multiplet neutron transitions in an Fe- magnetic cluster// Applied Physics a-materials cience & Processing 00. V. 7. P C. chlegel, J. van lageren, M. Manoli, E. K. Brechin, M. Dressel. ynthesis, structures and magnetic properties of two novel tetranuclear iron(iii) single-molecule magnets: Enhanced energy barriers in solution// Polyhedron 009. V. 8. (9-0), P R. L. Novak, F. Pineider, C. D. Fernandez, L. Gorini, L. Bogani, C. Danieli, L. Cavigli, A. Cornia, R..essoli. Magneto-optical studies on the molecular cluster Fe in different polymeric environments// Inorganica Chimica Acta 008. V. 6. (-5), P R. W. aalfrank, A. cheurer, I. Bernt, F. W. Heinemann, A. V. Postnikov, V. chunemann, A. X. Trautwein, M.. Alam, H. Rupp, P. Muller. The {Fe(III)[Fe(III)(L())()]()} star-type single- 78

15 molecule magnet// Dalton Transactions 006. V.. P C. chlegel, E. Burzuri, F. Luis, F. Moro, M. Manoli, E. K. Brechin, M. Murrie, J. van lageren. Magnetic Properties of Two New Fe() ingle-molecule Magnets in the olid tate and in Frozen olution// Chemistry-a European Journal 00. V. 6. P J.Ribas-Arino, T. Baruah, M. R. Pederson. Density-functional study of two Fe--based single- molecule magnets// Journal of Chemical Physics 005. V.. P T. J. Burns, J. Oitmaa. Energy spectrum of the Fe- molecular magnet// Europhysics Letters 00. V. 6. P M. Mannini, F. Pineider, P. ainctavit, C. Danieli, E. Otero, C. ciancalepore, A. M. Talarico, M. A. Arrio, A. Cornia, D. Gatteschi, R. essoli. Magnetic memory of a single-molecule quantum magnet wired to a gold surface// Nature Materials 009. V. 8. P A. Tarantul, B. Tsukerblat. pin-phonon relaxation in high-spin molecular nanomagnet Fe// in preparation.. B.. Tsukerblat, M. I. Belinskii, V. E. Fainzilberg. Magnetochemistry and pectroscopy of Transition Metal Exchange Clusters. In Book. Magnetochemistry and pectroscopy of Transition Metal Exchange Clusters. Ed. Harwood Acad. Pub. New York Vol. 9. P B.. Tsukerblat, M. I. Belinskii. Magnetochemistry and Radiospectroscopy of Exchange Clusters (Russian). tiinza. Kishinev C. A. Bates, R. F. Jasper. Relaxation of Exchange Coupled Triads in Ruby// Journal of Physics Part C olid tate Physics 97. V.. P.. 5. M. T. Hutchings. olid tate Physics. Academic. New York A. L. Barra, D. Gatteschi, L. Pardi, A. Muller, J. Doring. Magnetic-Properties of High-Nuclearity pin Clusters - - Oxovanadium(Iv) and 5-Oxovanadium(Iv) Clusters// Journal of the American Chemical ociety 99. V.. P D. Gatteschi, L. Pardi, A.-l. Barra,Müller A. Mol. Eng. 99. V.. P I. E. Dzyaloshinskii. Zh.Exp.Teor.Fiz V.. P

16 9. T. Moriya. Phys. Rev. 96. V. 0. P I. Chiorescu, W. Wernsdorfer, A. Muller, H. Bogge, B. Barbara. Butterfly hysteresis loop and dissipative spilt reversal in the = /, V-5 molecular complex// Physical Review Letters 000. V. 8. P Bertaina,. Gambarelli, T. Mitra, B. Tsukerblat, A. Müller,B. Barbara. Nature 008. V. 5. P I. Chiorescu, W. Wernsdorer,M. A.,H. Bögge,B. Barbara. J. Magn. Magn. Mater V.., P B. Tsukerblat, A. Tarantul, A. Muller. Crossover of the magnetic levels and adiabatic magnetization of the mesoscopic cluster V 5 // Physics Letters A 006. V. 5. P A. Tarantul, B. Tsukerblat, A. Muller. tatic magnetization of V-5 cluster at ultra-low temperatures: Precise estimation of antisymmetric exchange// Inorganic Chemistry 007. V. 6. P A. Tarantul, B. Tsukerblat, A. Muller. High-field magnetization of V-5 cluster at ultra-low temperatures: Importance of antisymmetric exchange and its precise estimation// Chemical Physics Letters 006. V. 8. P A. Tarantul, B. Tsukerblat. Inorg. Chim. Acta 00. V. 6. P A. Tarantul, B. Tsukerblat. Direct and two-phonon Orbach- Aminov type spin-latticerelaxation in molecular magnet V 5 // J. of Phys.: Conference eries. 0. V.. P B. Tsukerblat, A. Tarantul, A. Muller. Low temperature EPR spectra of the mesoscopic cluster V 5 :The role of antisymmetric exchange// Journal of Chemical Physics 006. V. 5. P R. Orbach. Proc. R. oc. 96. V. a5(9). P L. K. Aminov. ov. Phys. -JETP. 96. V. 5. P L. K. Aminov. Zh. Exp. Teor. Phys. 97. V. 67. P L. K. Aminov, B. Z. Malkin Dynamics and Kinetics of electronic and vibronic excitations in the paramagnetic crystals,. Kazan tate University. Kazan M. I. Belinsky. Inorg. Chem V. 7. P M. I. Belinsky. Inorg. Chem V. 7. P P. Kögerler, B. Tsukerblat, M. A. Dalton Transactions. 00. V. 9. P