Dynamics of the nuclear spin bath in molecular nanomagnets: a test for decoherence Andrea Morello Kamerlingh Onnes Laboratory Leiden University UBC Physics & Astronomy TRIUMF

Single-molecule magnets Schrödinger … what? How can we quantify the “macroscopicity” of a quantum state ? ?

Extensive difference  i = difference between the i-th extensive property in the two branches of the superposition, expressed in typical units for atomic scale objects  = max {  1,  2, …,  N } e.g. (distance / Å), (magnetic moment / μ B ), …  Can be very big! Think of a particle in a double slit  = d / Å A.J. Leggett, J. Phys.: Condens. Matter 14, R415 (2002)

Disconnectivity D = number of particles that “behave differently” in the two branches of the superposition e.g  = a  1 N + b  2 N  D = N  This is often quite small… A.J. Leggett, Supp. Prog. Theor. Phys. 69, 80 (1980) A.J. Leggett, J. Phys.: Condens. Matter 14, R415 (2002) D = 1

Disconnectivity What about a superconductor?  BCS =   (r i, r i+1 ) Josephson effect:  (r 1, r 2 ) = a  L (r 1, r 2 ) + b  R (r 1, r 2 )  D = 2 !! e.g. Cooper pair box Y. Nakamura et al., Nature 398, 786 (1999)

High-disconnectivity examples Molecular interference: e.g. C 60  D = 1080 M. Arndt et al., Nature 401, 680 (1999) I. Chiorescu et al., Science 299, 1869 (2003) Flux qubit  D  10 6

Single-molecule magnets Molecular 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 Mn ions

Single-molecule magnets D. Gatteschi et al., Science 265, 1054 (1994) The whole cluster behaves as a nanometer-size magnet. Total spin S = 10

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

Future directions: magnetic molecules on surfaces Patterning on silicon M. Cavallini et al., Nano Lett. 3, 1527 (2003) Isolated molecules on gold L. Zobbi et al., Chem. Comm. 1604, (2005) A. Nait-abdi et al., J. App. Phys. 95, 7345 (2004) Monolayer self-assembly on gold Magnetic islands in polycarbonate M. Cavallini et al., Angew. Chem. 44, 888 (2005)

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

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

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

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

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

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

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

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

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 S - 4 )

Macroscopic quantum superposition  The actual eigenstates of the molecular spin are quantum superpositions of macroscopically different states  K

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

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, (2004)

How macroscopic ? 4 x 3 = 12 Disconnectivity = 44 8 x 4 = 32 + S z =  B S z =  B - Extensive difference = 40

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

Nuclear bias

The nuclear spin dynamics can stimulate the quantum tunneling N.V. Prokof’ev and P.C.E. Stamp, J. Low Temp. Phys. 104, 143 (1996)

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

55 Mn NMR spectra in zero applied field I nuclear = 5/2 3 NMR lines corresponding to the 3 inequivalent Mn sites central frequencies: 231, 277, 365 MHz hyperfine field at the nuclear site parallel to the anisotropy axis for the electron spin Y. Furukawa et al., PRB 64, (2001) T. Kubo et al., PRB 65, (2002)

Nuclear relaxation: inversion recovery

Thermal activation

Quantum tunneling The nuclear spin relaxation is sensitive to quantum tunneling fluctuations A. Morello et al., PRL 93, (2004) T 1  30 s

External field B z  z By applying an external longitudinal field B z, the resonance condition for tunneling is destroyed T = 20 mK demagnetized

Fast-relaxing molecules Every real sample contains minority species with a flipped Jahn-Teller axis  Smaller anisotropy barrier (35 instead of 65 K) fast normal W. Wernsdorfer et al., Europhys. Lett. 47, 254 (1999) Z. Sun et al., Chem. Comm., 1973 (1999) Faster tunneling rate

Intercluster nuclear coupling

T 2  10 ms

Intercluster nuclear coupling

ratio   2 Nuclei in different cluster are mutually coupled  spin diffusion A. Morello et al., PRL 93, (2004)

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

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

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

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, (2003) cond-mat/ (2005)

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, (2004) The electron spins can reach thermal equilibrium with the lattice by quantum relaxation

Isotope effect M. Evangelisti et al., PRL 95, (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

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

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

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

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

Landau-Zener tunneling P  (d  / dt) -1 2ħ  2 P P Are these probabilities still the same when the bias is a quantum excitation? P = P ?

muon spin relaxation beam sample backward detector forward detector implantation relaxation decay   = 2.2  s external field (optional) asymmetry =  B - F  B + F  polarization

 SR in Mn 12 Mn 12 -ac with fast-relaxing molecules Mn 12 - t Bu without fast-relaxing molecules ? Inexplicably fast relaxation down to T << 1 K Z. Salman, A. Morello et al., unpublished

 SR in Mn 4 dimers Mn 4 molecular cores, S = 9/2 Antiferromagnetic superexchange interaction Exchange bias  no tunneling in zero field W. Wernsdorfer et al., Nature 416, 406 (2002)

 SR in Mn 4 dimers Z. Salman, A. Morello et al., unpublished

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

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

A wealth of detailed information quantum dynamics probed by nuclear spins nuclear spin diffusion Including:

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

A wealth of detailed information quantum dynamics probed by nuclear spins nuclear spin diffusion dipolar ordering and thermal equilibrium fast dynamics induced by local polarized probes Including:

A wealth of detailed information Including: quantum dynamics probed by nuclear spins nuclear spin diffusion dipolar ordering and thermal equilibrium fast dynamics induced by local polarized probes

A wealth of detailed information Including: quantum dynamics probed by nuclear spins nuclear spin diffusion dipolar ordering and thermal equilibrium fast dynamics induced by local polarized probes

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

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, K.H. Chow, R.I. Miller, W.A. MacFarlane (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)

Ultra-low temperature setup (a) Coaxial cable connected to the NMR spectrometeer and pulse generator. (b) 3 He distillator ("still") with cold plate. (c) Rotating shafts for the tunable capacitors, accessible from the top of the refrigerator. (d) NMR matching capacitor. (e) NMR tuning capacitor. (f) Upper heat exchanger. (g) 80 mK pot. (h) -cable, cut to one wavelength at  280 MHz. (i) Lower heat exchanger. (j) Araldite mixing chamber. (k) Pure 3 He phase. (l) Double-wall Kapton tail. (m) Forced downwards flow of 3 He in the dilute phase. (n) Sample with NMR coil. (o) Openings in the inner Kapton tube to allow the return of the 3 He flow. (p) Vacuum plug.

8 Li H 1 cos(ωt ) H0H0 backward forward ISAC  -NMR facility at TRIUMF 8 Li 8 Be + e - + e G.D. Morris et al., PRL 93, (2004)

Preliminary Results - Mn 12 on silicon silicon substrate Dipolar Fields 28 keV G.G. Condorelli et al., Angew. Chem. Int. Ed. 43, 4081 (2004)

Preliminary Results - Mn 12 on silicon silicon substrate Dipolar Fields 28 keV1 keV Z. Salman et al., unpublished