Dynamics and thermodynamics of quantum spins at low temperature

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Dynamics and thermodynamicsof quantum spins at
low temperature
Andrea Morello
Kamerlingh Onnes Laboratory
UBC PhysicsAstronomy
TRIUMF
Leiden University
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CaF2
CaF2 the fruit fly of spin systems
CaF2 Non-magnetic insulator 19F simple cubic
lattice of nuclear spins 1/2 with 100 natural
abundance
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CaF2
CaF2 the fruit fly of spin systems
P.L. Kuhns et al., PRB 35, 4591 (1987)
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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. Mn12 12 Mn ions
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Mn12
4 Mn4 ions ? s 3/2
8 Mn3 ions ? s 2
Total spin 10 The whole cluster behaves as a
nanometer-size magnet.
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Crystalline structure
? 15 Å
The clusters are assembled in a crystalline
structure, with relatively small (dipolar)
inter-cluster interactions
D. Gatteschi et al., Science 265, 1054 (1994)
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Magnetic anisotropy
z
H -DSz2
The magnetic moment of the molecule is
preferentially aligned along the z axis.
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Magnetic anisotropy
z
H -DSz2
The magnetic moment of the molecule is
preferentially aligned along the z axis.
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Magnetic anisotropy
z
H -DSz2
The magnetic moment of the molecule is
preferentially aligned along the z axis.
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Magnetic anisotropy
z
H -DSz2
The magnetic moment of the molecule is
preferentially aligned along the z axis.
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Magnetic anisotropy
z
H -DSz2
The magnetic moment of the molecule is
preferentially aligned along the z axis.
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Magnetic anisotropy
z
H -DSz2
The magnetic moment of the molecule is
preferentially aligned along the z axis.
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Magnetic anisotropy
z
H -DSz2
Classically, it takes an energy ? 65 K to reverse
the spin.
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Quantum tunneling of magnetization
z
H -DSz2
H -DSz2 C(S4 S-4)
Degenerate states
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Quantum tunneling of magnetization
z
H -DSz2 C(S4 S-4)
Quantum mechanically, the spin of the molecule
can be reversed by tunneling through the barrier
L. Thomas et al., Nature 383, 145 (1996)
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Macroscopic quantum superposition
? 10-11 - 10-7 K
The actual eigenstates of the molecular spin are
quantum superpositions of macroscopically
different states
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External field ? z
H -DSz2 C(S4 S-4) - g?BSxBx
The application of a perpendicular field allows
to artificially introduce non-diagonal elements
in the spin Hamiltonian
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Quantum coherence
t
h/?
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)
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Quantum coherence
A. Morello, P.C.E. Stamp and I.S. Tupitsyn,
cond-mat/0605709 (2006)
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Decoherence rates
optimal coherent operation point at T 50 mK
Q ? 107
A. Morello, P.C.E. Stamp and I.S. Tupitsyn,
cond-mat/0605709 (2006)
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Nuclear spin bath
Intrinsic source of decoherence
N.V. Prokofev and P.C.E. Stamp, J. Low Temp.
Phys. 104, 143 (1996)
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Nuclear bias
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Nuclear bias
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Nuclear bias
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Nuclear bias
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Nuclear bias
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Nuclear bias
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Nuclear bias
The nuclear spin dynamics allows incoherent
tunneling
The electron spin tunneling triggers nuclear spin
dynamics
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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
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Nuclear relaxation inversion recovery
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Quantum tunneling
The nuclear spin relaxation is sensitive to
quantum tunneling fluctuations
A. Morello et al., PRL 93, 197202 (2004)
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Precessional decoherence
hyperfine field
nuclear spin
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Precessional decoherence
? ? 0 for 55Mn nuclei
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Topological decoherence
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Tunneling timescales
h?0
? 10 K 200 GHz
?0 frequency of the small oscillations on the
bottom of the potential well
? 10-12 s
?T ? 1 s
???
t
???
?T time between subsequent incoherent tunneling
events
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Topological decoherence
? nuclear spins flipped by adiabatically
following the new local field
? 0
?0 is the bounce frequency
The 55Mn nuclei cannot adiabatically follow a
tunneling event
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Hyperfine-split manifolds
M M - 1 M - 2 M - 3
- M - M 1 - M 2
E0
M - 3 M - 2 M - 1 M
- M 2 - M 1 - M
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.
N.V. Prokofev and P.C.E. Stamp, cond-mat/9511011
(1995)
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Tunneling rate - unbiased case
since ?,? ? 0 ?0 gtgt ?ngt0
?n ?n(?,?)
The most probable tunneling transition (without
coflipping nuclei) is between states with zero
nuclear polarization.
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Biased case
? e.g. dipolar field from neighboring
clusters or external field
?
Tunneling ? swapping dipolar and hyperfine energy
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Tunneling rates
m8
m9
m10
?m2 / E0m2
?m2 e-
?m / ?0m
e -
?m-1 (?)
?? h E0m
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Nuclear spin temperature
The nuclear spins are in thermal equilibrium with
the lattice
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Dipolar magnetic ordering of cluster spins
Tc ? 0.16 K
Mn6 S 12 High symmetry Small anisotropy Fast
relaxation
A. Morello et al., PRL 90, 017906 (2003)
PRB 73, 134406 (2006)
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Dipolar magnetic ordering of cluster spins
Mn4 S 9/2 Lower symmetry Larger
anisotropy Fast enough quantum relaxation
The electron spins can reach thermal equilibrium
with the lattice by quantum relaxation
M. Evangelisti et al., PRL 93, 117202 (2004)
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Isotope effect
Fe8 S 10 Low symmetry Large
anisotropy Isotopically substituted 57Fe, I 1/2
? 56Fe, I 0
Enrichment with I 1/2 isotopes speeds up the
quantum relaxation
M. Evangelisti et al., PRL 95, 227206 (2005)
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Isotope effect
Sample with proton spins substituted by
deuterium ?proton
6.5
?deuterium
W. Wernsdorfer et al., PRL 84, 2965 (2000)
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Isotope effect in the nuclear relaxation
Sample with proton spins substituted by
deuterium ?proton
6.5
?deuterium
The reduced tunneling rate is directly measured
by the 55Mn relaxation rate
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Landau-Zener tunneling
?
C. Zener, Proc. R. Soc. London A 137, 696 (1932)
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Landau-Zener tunneling
P
?
C. Zener, Proc. R. Soc. London A 137, 696 (1932)
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Landau-Zener tunneling
?
C. Zener, Proc. R. Soc. London A 137, 696 (1932)
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Landau-Zener tunneling
P
?
C. Zener, Proc. R. Soc. London A 137, 696 (1932)
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Landau-Zener tunneling
P
P
?
P P
Can these probabilities be different?
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A wealth of detailed information
Including

• quantum dynamics probed by nuclear spins

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A wealth of detailed information
Including

• quantum dynamics probed by nuclear spins

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A wealth of detailed information
Including

• quantum dynamics probed by nuclear spins
• dipolar ordering and thermal equilibrium

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Coherent ? Incoherent
t
Benchmark system for decoherence studies
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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)