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Dynamics and thermodynamicsDynamics and thermodynamicsof quantum spins at low temperatureof quantum spins at low temperature
Andrea MorelloAndrea MorelloKamerlingh Onnes
LaboratoryLeiden
University
UBCPhysics
&Astronomy
TRIUMF
CaF2CaF2CaFCaF22: the “fruit fly” of spin systems: the “fruit fly” of spin systems
CaF2: Non-magnetic insulator
19F : simple cubic lattice of nuclear spins 1/2with 100% natural abundance
CaF2CaF2CaFCaF22: the “fruit fly” of spin systems: 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)
Single-molecule magnetsSingle-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
MnMn1212
Total spin = 10
The whole cluster behaves as a nanometer-size magnet.
4 Mn4+ ions s = 3/2
8 Mn3+ ions s = 2
Crystalline structureCrystalline 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
Magnetic anisotropyMagnetic anisotropy
The magnetic moment of the molecule is preferentially aligned along the z – axis.
-70
-60
-50
-40
-30
-20
-10
0
-10 -5 0 5 10
Th-A T
QT
Sz
En
erg
y (
K)
zH = -DSz
2
-70
-60
-50
-40
-30
-20
-10
0
-10 -5 0 5 10
Th-A T
QT
Sz
En
erg
y (
K)
Magnetic anisotropyMagnetic anisotropy
The magnetic moment of the molecule is preferentially aligned along the z – axis.
zH = -DSz
2
-70
-60
-50
-40
-30
-20
-10
0
-10 -5 0 5 10
Th-A T
QT
Sz
En
erg
y (
K)
Magnetic anisotropyMagnetic anisotropy
The magnetic moment of the molecule is preferentially aligned along the z – axis.
zH = -DSz
2
-70
-60
-50
-40
-30
-20
-10
0
-10 -5 0 5 10
Th-A T
QT
Sz
En
erg
y (
K)
Magnetic anisotropyMagnetic anisotropy
The magnetic moment of the molecule is preferentially aligned along the z – axis.
zH = -DSz
2
-70
-60
-50
-40
-30
-20
-10
0
-10 -5 0 5 10
Th-A T
QT
Sz
En
erg
y (
K)
Magnetic anisotropyMagnetic anisotropy
The magnetic moment of the molecule is preferentially aligned along the z – axis.
zH = -DSz
2
-70
-60
-50
-40
-30
-20
-10
0
-10 -5 0 5 10
Th-A T
QT
Sz
En
erg
y (
K)
Magnetic anisotropyMagnetic anisotropy
The magnetic moment of the molecule is preferentially aligned along the z – axis.
zH = -DSz
2
-70
-60
-50
-40
-30
-20
-10
0
-10 -5 0 5 10
Th-A T
QT
Sz
En
erg
y (
K)
Magnetic anisotropyMagnetic anisotropy
Classically, it takes an energy Classically, it takes an energy 65 K to reverse 65 K to reverse the spin.the spin.
zH = -DSz
2
-70
-60
-50
-40
-30
-20
-10
0
-10 -5 0 5 10
Th-A T
QT
Sz
En
erg
y (
K)
Quantum tunneling of magnetizationQuantum tunneling of magnetization
z
Degenerate states
H = -DSz2H = -DSz2 + C(S+
4 + S-4)
-70
-60
-50
-40
-30
-20
-10
0
-10 -5 0 5 10
Th-A T
QT
Sz
En
erg
y (
K)
Quantum tunneling of magnetizationQuantum tunneling of magnetization
z
Quantum mechanically, the spin of the molecule can be Quantum mechanically, the spin of the molecule can be reversed by tunneling through the barrierreversed by tunneling through the barrier
L. Thomas et al., Nature 383, 145 (1996)
H = -DSz2 + C(S+
4 + S-4)
-70
-60
-50
-40
-30
-20
-10
0
-10 -5 0 5 10
Th-A T
QT
Sz
En
erg
y (
K)
Macroscopic quantum superpositionMacroscopic quantum superposition
The actual eigenstates of the molecular spin are quantum superpositions of macroscopically different states
10-11 - 10-7 K
External field External field zz
0 1 2 3 4 5 6 7 8 9 1010-12
1x10-10
1x10-8
1x10-6
1x10-4
1x10-2
1x100
1x102
(
K)
Bperp
(T)
H = -DSz2 + C(S+
4 + S-4) - gBSxBx
The application of a perpendicular field allows to artificially introduce non-diagonal elements in the spin Hamiltonian
environmentalcouplings
coherence regime
0 20 40
-1.0
-0.5
0.0
0.5
1.0
Y A
xis
Titl
e
X Axis Title
Quantum coherenceQuantum coherence
t
h/tunable over several orders of magnitude by application of a magnetic field
Prototype of spin qubit with tunable operating frequencyPrototype 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)
x
y
z
|z-
|z+
|z+
|z-
Hy
wave
|A
|S
|z+ |z-
/2
0
-30
0
-10
20
30
-20
10
/2
3/2
0
En
erg
y (
K)
2
Quantum coherenceQuantum coherence
A. Morello, P.C.E. Stamp and I.S. Tupitsyn, cond-mat/0605709 (2006)
Decoherence ratesDecoherence rates
A. Morello, P.C.E. Stamp and I.S. Tupitsyn, cond-mat/0605709 (2006)
1.5 2.0 2.5 3.0 3.5 4.01E-9
1E-8
1E-7
1E-6
1E-5
1E-4
1E-3
0.01
0.1
1
10
nuclear
phonon
dipolar 0.05 K 0.1 K 0.2 K 0.4 K
By (T)
optimal coherentoperation pointat T = 50 mK
Q 107
Nuclear spin bathNuclear spin bath
Intrinsic source of decoherenceIntrinsic source of decoherence
N.V. Prokof’ev and P.C.E. Stamp, J. Low Temp. Phys. 104, 143 (1996)
Nuclear biasNuclear bias
Nuclear biasNuclear bias
Nuclear biasNuclear bias
Nuclear biasNuclear bias
Nuclear biasNuclear bias
Nuclear biasNuclear bias
Nuclear biasNuclear bias
The nuclear spin dynamics The nuclear spin dynamics allows incoherent tunnelingallows incoherent tunneling
The electron spin tunneling The electron spin tunneling triggers nuclear spin triggers nuclear spin
dynamicsdynamics
Nuclear relaxation Nuclear relaxation electron spin electron spin
fluctuationsfluctuationsEn
erg
y
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
0
100
200
300
Tim
e af
ter
inve
rsio
n (s
)
9070503010
Time after refocus (s)
100
10
1
0.1
0.01
0.001
Inte
nsi
ty (
a.u
.)
Nuclear relaxation: inversion recoveryNuclear relaxation: inversion recovery
Quantum tunnelingQuantum tunneling
0.01 0.1 10.01
0.1
1
10
Nu
clea
r re
laxa
tio
n r
ate
(s-1
)
Temperature (K)
The nuclear spin relaxation is sensitive to quantum tunneling fluctuations
A. Morello et al., PRL 93, 197202 (2004)
-0.4 -0.2 0.0 0.2 0.4 0.60
5
10
15
20
25
Rel
axat
ion
rat
e (
10-3 s
-1)
Bz (T)
Precessional decoherencePrecessional decoherence
nuclear spin
hyperfine field
i
e - = cos i e - i
2 / 2
i
= # nuclear spins flipped by precession around the new local field
Precessional decoherencePrecessional decoherence
0 for 0 for 5555Mn nucleiMn nuclei
Topological decoherenceTopological decoherence
= 1/2 i2
i
= # nuclear spins flipped by adiabatically following the new local field
i = /2i
0
0 is the “bounce frequency”
Tunneling timescalesTunneling 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
-70
-60
-50
-40
-30
-20
-10
0
-10 -5 0 5 10
Th-A T
QT
Sz
En
erg
y (
K)
ħ0
10 K = 200 GHz
= 1/2 i2
i
= # nuclear spins flipped by adiabatically following the new local field
Topological decoherenceTopological decoherence
i = /2i
0
0 is the “bounce frequency”
200 MHz
200 GHz= 10 - 3
0
The 55Mn nuclei cannot adiabatically follow a tunneling event
-70
-60
-50
-40
-30
-20
-10
0
-10 -5 0 5 10
Th-A T
QT
Sz
En
erg
y (
K)
Hyperfine-split manifoldsHyperfine-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.
MM - 1M - 2M - 3
- M- M + 1- M + 2
- M + 2- M + 1- M
M - 3M - 2M - 1M
N.V. Prokof’ev and P.C.E. Stamp, cond-mat/9511011 (1995)
E0
Tunneling rate - unbiased caseTunneling 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
2 ħn = n(,)
since , 00 >> n>0
Biased caseBiased case
= e.g. dipolar field from neighboring clusters or external field
M 2 1 0- 1- 2- M
- M - 2 - 1 0 1 2 M
n = 0
0-1 =
02
2 ħexp(- / 0)
Tunneling Tunneling swapping dipolar and hyperfine energy swapping dipolar and hyperfine energy
-70
-60
-50
-40
-30
-20
-10
0
-10 -5 0 5 10
Th-A T
QT
Sz
En
erg
y (
K)
Tunneling ratesTunneling rates
0.0 0.5 1.0 1.5 2.00.01
0.1
1
10
100
Rel
axat
ion
rat
e (
s-1)
Temperature (K)
m-1 () =
m2 e-
ħ E0m
m2 / E0m
2
e - |m| / 0m
m=10
m=9
m=8
Nuclear spin temperatureNuclear spin temperature
The nuclear spins are in thermal equilibrium with the latticeThe nuclear spins are in thermal equilibrium with the lattice
0 1 2 3 4 5 60.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 3 6 9 12 150
50
100
150
Tnucl
K
wait
echo
pulse
(a)
Tem
per
atu
re (
K)
Time (h)
(b)
Tem
per
atu
re (
mK
)
Time (h)
spin temperature
bath temperature
Dipolar magnetic ordering of cluster spinsDipolar magnetic ordering of cluster spins
Mn6 S = 12High symmetry
Small anisotropyFast relaxation
0.1
1
nuclear spins
~ T - 2
c / R
crystal fieldanisotropy
Phonons
~ T 3
0.1 1 100
1
2
3
'
T (K)
Tc 0.16 K
A. Morello et al., PRL 90, 017906 (2003) PRB 73, 134406 (2006)
Dipolar magnetic ordering of cluster spinsDipolar magnetic ordering of cluster spins
Mn4 S = 9/2Lower symmetryLarger anisotropy
“Fast enough” quantum relaxation
M. Evangelisti et al., PRL 93, 117202 (2004)
The electron spins can reach thermal equilibrium with the latticeThe electron spins can reach thermal equilibrium with the latticeby quantum relaxationby quantum relaxation
Isotope effectIsotope effect
M. Evangelisti et al., PRL 95, 227206 (2005)
Enrichment with Enrichment with II = 1/2 isotopes speeds up the quantum relaxation = 1/2 isotopes speeds up the quantum relaxation
Fe8 S = 10Low symmetry
Large anisotropyIsotopically substituted57Fe, I = 1/2 56Fe, I = 0
Isotope effectIsotope effect
Sample with proton spins substituted by deuterium
protondeuterium
= 6.5
W. Wernsdorfer et al., PRL 84, 2965 (2000)
Isotope effect in the nuclear relaxationIsotope effect in the nuclear relaxation
The reduced tunneling rate is directly The reduced tunneling rate is directly measured by the measured by the 5555Mn relaxation rateMn relaxation rate
0.01 0.1 1 10 100 1000
-1.0
-0.5
0.0
0.5
1.0
W = 0.0035 s-1
W = 0.023 s-1
Ech
o in
ten
sity
Time (s)
Sample with proton spins substituted by deuterium
protondeuterium
= 6.5
Landau-Zener tunnelingLandau-Zener tunneling
C. Zener, Proc. R. Soc. London A 137, 696 (1932)
P (d / dt) -1
2ħ
2
P
Landau-Zener tunnelingLandau-Zener tunneling
C. Zener, Proc. R. Soc. London A 137, 696 (1932)
P (d / dt) -1
2ħ
2
P
1 - P
Landau-Zener tunnelingLandau-Zener tunneling
C. Zener, Proc. R. Soc. London A 137, 696 (1932)
P (d / dt) -1
2ħ
2
P
Landau-Zener tunnelingLandau-Zener tunneling
C. Zener, Proc. R. Soc. London A 137, 696 (1932)
P (d / dt) -1
2ħ
2
P
1 - P
Landau-Zener tunnelingLandau-Zener tunneling
P (d / dt) -1
2ħ
2
P
P
Can these probabilities be Can these probabilities be different?different?
P = P?
• quantum dynamics probed by nuclear spins
A wealth of detailed informationA wealth of detailed information
Including:
A wealth of detailed informationA wealth of detailed information
Including:
• quantum dynamics probed by nuclear spins
A wealth of detailed informationA wealth of detailed information
• quantum dynamics probed by nuclear spins
• dipolar ordering and thermal equilibrium
Including:
Coherent Coherent Incoherent Incoherent
0 20 40
-1.0
-0.5
0.0
0.5
1.0
Y Ax
is Ti
tle
X Axis Title
t
The physics behind incoherent quantum tunneling in nanomagnets isTHE SAME
that will determine their coherent dynamics
Benchmark system for decoherence studiesBenchmark system for decoherence studies
AcknowledgementsAcknowledgements
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)