1.  From a single molecule to an ensemble of molecules at T ~0 : Both tunneling rate and decoherence increase  LZ probability: P LZ = 1 – exp[-  (  /ħ) 2 /  c] ~  2 /c Spin-bath (Prokofiev and Stamp): P SB ~ (  2 /  0 )e -│  │/  0.n(E D ) >> P LZ  0 = hyperfine energy = tunnel window Large spins Mesoscopic tunneling (slow) Nuclear spins Observation possible Strong decoherence. H= - DS z 2 - BS z 4 - E(S + 2 + S - 2 ) - C(S + 4 + S - 4 ) - g  B S z H z

2. Barrier in zero field (symmetrical) H= - DS z 2 - BS z 4 - E(S + 2 + S - 2 ) - C(S + 4 + S - 4 ) H // -M New resonances at g  B H n = nD (B=0) Thermally activated tunneling Landau-Zener transition at avoided level crossing (single molecule) Tunneling probability: P=1 – exp[-  (  /ħ) 2 /  c] c = dH/dt  Coexistence of tunneling and hysteresis

3. Proposal of Morello, Stamp, Tupitsyn

4. Effect of a tilted field (Mn 12 -ac) J. Appl. Phys. (1997) Easy axis ө BBLBL BTBT

5. Transverse field with constant transverse field (Fe 8 ) H= - DS z 2 - BS z 4 - E(S + 2 + S - 2 ) - C(S + 4 + S - 4 ) - g  B S z H x - g  B S z H z D~ D~  ~ DS 2 (┴ / Il) 2S/p with ┴ << Il  2  (E/D) S  4  (CS 2 /D) S/2  1  (H x /DS) 2S (Parity)

6. Mn 12 -ac No effect of S = 9 A (small) parity effect on thermally activated tunneling (S=10) -(S-1) - S S-1 S -(S-1) -S S-2 S-1 S -(S-1) - S S-1 S -(S-1) -S S-1 S n= 0, 2… n=1, 3… JMMM (1999)  4  (E/D)  S/2 0 0

7. Large parity effect and quantum phase interference at low temperature (Fe 8 ) [Mn 12 ] -2e S = 10 W. Wernsdorfer et al, PRL (2005), Science (1999)  = 0°, n=0   =  °  cos  or  =  °  sin   g  B H x /[2E(E+D)] 1/2  (e.g. review Tupitsyn, BB)

8. Dephasing

9. How the system escapes from the quantum regime (Mn 12 -ac) Chiorescu et al, PRL, 83, 947 (1999) Data points and calculated linesLevel Scheme B n /n = D –B[(m-n) 2 +n 2 ]. Sharp or continuous transition

10. Crossover From Quantum to Classical Regime Activated Tunneling Measured ( ) and Calculated ( ) Resonance Fields Barbara et al, JMMM 140-144, 1891 (1995) and J. Phys. Jpn. 69, 383 (2000) Classical Thermal Activation T blocking Ground-state Tunneling T c-o (Mn 12 -ac) t ~  0 exp E(H)/kT B

11. Shorter timescales (ac susceptibility): Tunneling moves to higher temperatures

12. First relaxation curves (Mn 12 -ac)

13. Scaling of the Quantum Dynamics of Mn 12 -ac M/M s = f (t/  (H,T) ) Exponential to Square Root Relaxation N. Prokofiev and P. Stamp, PRL 80, 5794 (1998) L. Thomas et al, J. Low Temp. Phys. (1998); PRL (1999). Paulsen et al J. Low Temp (1998). t/  (T)

14. Sqrt(t) at in H // and H ┴ Calculated Energy Spectrum Measured relaxation Chiorescu et al, PRL (2000)

15. Resonance width and tunnel window Effects of magnetic couplings and hyperfine Interactions Chiorescu et al, PRL, 83, 947 (1999) Barbara et al, J. Phys. Jpn. 69, 383 (2000) Kent et al, EPL, 49, 521 (2000) 8-1 8-0 Inhomogeneous dipolar broadening and the electronic spin-bath Data points and calculated linesLevel Scheme Homogeneous broadening of the tunnel window by nuclear spins Wernsdorfer et al, PRL (1999 ) Prokofiev and Stamp (1998 ) Weak HF coupling: Broadens the tunnel window (x10 5 ) Strong decoherence

16. Environmental effects Central molecule spin Mn 12, Fe 8 Spin-bath Environmental spins Enhance tunneling Mesoscopic spins Decoherence Phonon-bath Spin-phonons transition Bottleneck (T B >>T 1 ) V 15

17. Spins bath Essential Important Phonons bath Depends on T Important

18. Time Reversal Symmetry  =0 (Kramers Theorem) Experimentally:  ~80 mK. D ~J  g /g ~ 50mK (Also hyperfine interactions ~20 mK) V 15 : a large molecule with collective spin ½ 15 spins ½ with AF coupled (D H =2 15 ) Müller, Döring, Angew. Chem. Intl. Engl., 27, 171 (1988) Diagonalization of the 15-Spin ½ Hamiltoninan H =  J ij SiSj (I. Tupitsyn) 200 calculated levels. The 8 levels lowest levels frustrated 3-spins ½ triangle Effective hamiltonian: H = |J |  (S1S2 + S2S3 + S3S4) – g  B B(S1 + S2 + S3) Measurements of M(H) and  (T) confirm this picture

19. Dissipative spin reversal in a two-level system ( T<0.1K) Effects of the phonon bath at low temperature Low sweeping rates / Strong coupling to the cryostat LZS transition at Finite Temperature (dissipative)  botl  1 >  meas Hysteresis (≠Orbach process). Measured Calculated Chiorescu et al, PRL 84, 3454 (2000) Abragam and Bleaney (Oxford, 1970) M(H): Irreversible Equilibrium (Reversible) M(H)=M s th{H/2kT}

20. Spin temperature: n 1 /n 2 =exp(  H /kT s ) n T = number of phonons with ћ  =   T s = T T s << T T s  T (n 1 /n 2 = constant) n Tph = n T n Tph increases rapidly hole in the phonons density n Tph ~ 0  0 Time-scales:  B >>  1 ( v = dB/dt)  B =(  /  H 2 )tanh 2 (  H /2kT)  < 0 In the presence of a barrier (large spins) Similar phonons emission: Recovery to the ground-state by Inelastic tunneling ?  ine  v 2   3 (1+n(  H ))

21. Now: fast sweeping rates / weak coupling to the cryostat Adiabatic LZS Spin Rotation is recovered (T s ~0, reversible but out of equilibrium) Fit to M = (1/2)(g  B ) 2 H/  2 +(g  B H) 2      80 mK Chiorescu et al PRB, 2003

22. Relaxation Experiments Inside  Outside   B << calculated value  B (B,T) ~ calculated value Nuclear spin-bath affects bottleneck Bottleneck only Fit of M(t) to the Bottleneck model   B (B,T)

23. Environmental effects Central molecule spin Mn 12, Fe 8 Spin-bath Environmental spins Enhance tunneling Mesoscopic spins Decoherence Phonon-bath Spin-phonons transition Bottleneck (T B >>T 1 ) Electromagnetic radiation bath Spin-photons transitions (incoherent) V 15