1. Dynamics and thermodynamics of quantum spins at low temperature Andrea Morello Kamerlingh Onnes Laboratory Leiden University UBC Physics & Astronomy TRIUMF

2. CaF2 CaF 2 : the “fruit fly” of spin systems CaF 2 : Non-magnetic insulator 19 F : simple cubic lattice of nuclear spins 1/2 with 100% natural abundance

3. CaF2 CaF 2 : the “fruit fly” of spin systems P.L. Kuhns et al., PRB 35, 4591 (1987) Still not understood... J.S. Waugh and C.P. Slichter, PRB 37, 4337 (1988) T. Room, PRB 40, 4201 (1989) J.S. Waugh and C.P. Slichter, PRB 40, 4203 (1989)

4. Single-molecule magnets Stoichiometric compounds based on macromolecules, each containing a core of magnetic ions surrounded by organic ligands, and assembled in an insulating crystalline structure e.g. Mn 12 12 Mn ions

5. Mn 12 Total spin = 10 The whole cluster behaves as a nanometer-size magnet. 4 Mn 4+ ions  s = 3/2 8 Mn 3+ ions  s = 2

6. Crystalline structure D. Gatteschi et al., Science 265, 1054 (1994) The clusters are assembled in a crystalline structure, with relatively small (dipolar) inter-cluster interactions  15 Å

7. Magnetic anisotropy The magnetic moment of the molecule is preferentially aligned along the z – axis. z H = -DS z 2

8. Magnetic anisotropy The magnetic moment of the molecule is preferentially aligned along the z – axis. z H = -DS z 2

9. Magnetic anisotropy The magnetic moment of the molecule is preferentially aligned along the z – axis. z H = -DS z 2

10. Magnetic anisotropy The magnetic moment of the molecule is preferentially aligned along the z – axis. z H = -DS z 2

11. Magnetic anisotropy The magnetic moment of the molecule is preferentially aligned along the z – axis. z H = -DS z 2

12. Magnetic anisotropy The magnetic moment of the molecule is preferentially aligned along the z – axis. z H = -DS z 2

13. Magnetic anisotropy Classically, it takes an energy  65 K to reverse the spin. z H = -DS z 2

14. Quantum tunneling of magnetization z Degenerate states H = -DS z 2 H = -DS z 2 + C(S + 4 + S - 4 )

15. Quantum tunneling of magnetization z Quantum mechanically, the spin of the molecule can be reversed by tunneling through the barrier L. Thomas et al., Nature 383, 145 (1996) H = -DS z 2 + C(S + 4 + S - 4 )

16. Macroscopic quantum superposition The actual eigenstates of the molecular spin are quantum superpositions of macroscopically different states   10 -11 - 10 -7 K

17. External field  z H = -DS z 2 + C(S + 4 + S - 4 ) - g  B S x B x The application of a perpendicular field allows to artificially introduce non-diagonal elements in the spin Hamiltonian environmental couplings coherence regime

18. Quantum coherence t h/  tunable over several orders of magnitude by application of a magnetic field Prototype of spin qubit with tunable operating frequency P.C.E. Stamp and I.S. Tupitsyn, PRB 69, 014401 (2004) A. Morello, P.C.E. Stamp and I.S. Tupitsyn, cond-mat/0605709 (2006)

19.  x y z |z -  |z +  |z -  HyHy  wave |A|A |S|S |z +  |z -   /2  0 -30 0 -10 20 30 -20 10  /2 3  /2  0 Energy (K)  22 Quantum coherence A. Morello, P.C.E. Stamp and I.S. Tupitsyn, cond-mat/0605709 (2006)

20. Decoherence rates A. Morello, P.C.E. Stamp and I.S. Tupitsyn, cond-mat/0605709 (2006) optimal coherent operation point at T = 50 mK Q  10 7

21. Nuclear spin bath Intrinsic source of decoherence N.V. Prokof’ev and P.C.E. Stamp, J. Low Temp. Phys. 104, 143 (1996)

22. Nuclear bias

28. The nuclear spin dynamics allows incoherent tunneling The electron spin tunneling triggers nuclear spin dynamics

29. Nuclear relaxation  electron spin fluctuations Energy At low temperature, the field produced by the electrons on the nuclei is quasi-static  NMR in zero external field The fluctuations of the electron spins induce nuclear relaxation  nuclei are local probes for (quantum?) fluctuations

30. Nuclear relaxation: inversion recovery

31. Quantum tunneling The nuclear spin relaxation is sensitive to quantum tunneling fluctuations A. Morello et al., PRL 93, 197202 (2004)

32. Precessional decoherence nuclear spin hyperfine field

33. ii e -  =  cos  i  e -   i 2 / 2 i  = # nuclear spins flipped by precession around the new local field Precessional decoherence   0 for 55 Mn nuclei

34. Topological decoherence = 1/2   i 2 i = # nuclear spins flipped by adiabatically following the new local field  i =  /2 ii 00  0 is the “bounce frequency”

35. Tunneling timescales  0 = frequency of the “small oscillations” on the bottom of the potential well t    T  1 s  10 -12 s  T = time between subsequent incoherent tunneling events ħ0ħ0  10 K = 200 GHz

36. = 1/2   i 2 i = # nuclear spins flipped by adiabatically following the new local field Topological decoherence  i =  /2 ii 00  0 is the “bounce frequency”  200 MHz 200 GHz = 10 - 3  0 The 55 Mn nuclei cannot adiabatically follow a tunneling event

37. Hyperfine-split manifolds The hyperfine fields before and after tunneling are antiparallel  The hyperfine-split manifolds on either sides of the barrier are simply mirrored with respect to the local nuclear polarization. M M - 1 M - 2 M - 3 - M - M + 1 - M + 2 - M + 1 - M M - 3 M - 2 M - 1 M N.V. Prokof’ev and P.C.E. Stamp, cond-mat/9511011 (1995) E0E0

38. Tunneling rate - unbiased case The most probable tunneling transition (without coflipping nuclei) is between states with zero nuclear polarization. M 2 1 0 - 1 - 2 - M n = 0 - M - 2 - 1 0 1 2 M  n -1 = n2n2 2  ħ   n =  n (,  ) since,   0  0 >>  n>0

39. Biased case  = e.g. dipolar field from neighboring clusters or external field M 2 1 0 - 1 - 2 - M - 2 - 1 0 1 2 M n = 0   0 -1 = 0202 2  ħ  exp(-  /  0 ) Tunneling  swapping dipolar and hyperfine energy

40. Tunneling rates  m -1 (  ) =  m 2 e -  ħ E 0m  m 2 / E 0m 2 e - |  m | /  0m m=10 m=9 m=8

41. Nuclear spin temperature The nuclear spins are in thermal equilibrium with the lattice spin temperature bath temperature

42. Dipolar magnetic ordering of cluster spins Mn 6 S = 12 High symmetry Small anisotropy Fast relaxation T c  0.16 K A. Morello et al., PRL 90, 017906 (2003) PRB 73, 134406 (2006)

43. Dipolar magnetic ordering of cluster spins Mn 4 S = 9/2 Lower symmetry Larger anisotropy “Fast enough” quantum relaxation M. Evangelisti et al., PRL 93, 117202 (2004) The electron spins can reach thermal equilibrium with the lattice by quantum relaxation

44. Isotope effect M. Evangelisti et al., PRL 95, 227206 (2005) Enrichment with I = 1/2 isotopes speeds up the quantum relaxation Fe 8 S = 10 Low symmetry Large anisotropy Isotopically substituted 57 Fe, I = 1/2  56 Fe, I = 0

45. Isotope effect Sample with proton spins substituted by deuterium  proton  deuterium = 6.5 W. Wernsdorfer et al., PRL 84, 2965 (2000)

46. Isotope effect in the nuclear relaxation The reduced tunneling rate is directly measured by the 55 Mn relaxation rate Sample with proton spins substituted by deuterium  proton  deuterium = 6.5

47. Landau-Zener tunneling C. Zener, Proc. R. Soc. London A 137, 696 (1932)  P  (d  / dt) -1 2ħ  2 P

48. Landau-Zener tunneling C. Zener, Proc. R. Soc. London A 137, 696 (1932)  P  (d  / dt) -1 2ħ  2 P 1 - P

49. Landau-Zener tunneling C. Zener, Proc. R. Soc. London A 137, 696 (1932)  P  (d  / dt) -1 2ħ  2 P

50. Landau-Zener tunneling C. Zener, Proc. R. Soc. London A 137, 696 (1932)  P  (d  / dt) -1 2ħ  2 P 1 - P

51. Landau-Zener tunneling P  (d  / dt) -1 2ħ  2 P P Can these probabilities be different? P = P ?

52. quantum dynamics probed by nuclear spins A wealth of detailed information Including:

53. A wealth of detailed information Including: quantum dynamics probed by nuclear spins

54. A wealth of detailed information quantum dynamics probed by nuclear spins dipolar ordering and thermal equilibrium Including:

55. Coherent  Incoherent t The physics behind incoherent quantum tunneling in nanomagnets is THE SAME that will determine their coherent dynamics Benchmark system for decoherence studies

56. Acknowledgements P.C.E. Stamp, I.S. Tupitsyn, W.N. Hardy, G.A. Sawatzky (UBC Vancouver) O.N. Bakharev, H.B. Brom, L.J. de Jongh (Kamerlingh Onnes lab - Leiden) Z. Salman, R.F. Kiefl(TRIUMF Vancouver) M. Evangelisti(INFM - Modena) R. Sessoli, D. Gatteschi, A. Caneschi(Firenze) G. Christou, M. Murugesu, D. Foguet(U of Florida - Gainesville) G. Aromi(Barcelona)