dynamics of the nuclear spin bath in molecular nanomagnets: a test for decoherence andrea morello...
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Dynamics of the nuclear spin bath Dynamics of the nuclear spin bath in molecular nanomagnets: a test for decoherencein molecular nanomagnets: a test for decoherence
Andrea MorelloAndrea MorelloKamerlingh Onnes
LaboratoryLeiden
University
UBCPhysics
&Astronomy
TRIUMF
Single-molecule magnetsSingle-molecule magnetsSchrSchrödinger … what?ödinger … what?
How can we quantify the “macroscopicity” of a quantum state ?
?
Extensive differenceExtensive 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 / μB), …
Can be very big! Think of a particle in a double slit
= d / Å
A.J. Leggett, J. Phys.: Condens. Matter 14, R415 (2002)
DisconnectivityDisconnectivity
D = number of particles that “behave differently” in the two branches of the superposition
e.g = a 1N + 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
DisconnectivityDisconnectivity
What about a superconductor?
BCS = (ri , ri+1)
Josephson effect: (r1 , r2) = a L(r1 , r2) + b R(r1 , r2) D = 2 !!
e.g. Cooper pair box
Y. Nakamura et al., Nature 398, 786 (1999)
High-disconnectivity examplesHigh-disconnectivity examples
Molecular interference: e.g. C60 D = 1080
M. Arndt et al., Nature 401, 680 (1999)I. Chiorescu et al., Science 299, 1869 (2003)
Flux qubit D 106
Single-molecule magnetsSingle-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. Mn12
12 Mn ions
Single-molecule magnetsSingle-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 magnetsSingle-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 Future directions: magnetic molecules on
surfacessurfaces
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-assemblyon gold
Magnetic islandsin polycarbonate
M. Cavallini et al., Angew. Chem. 44, 888 (2005)
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 = -DSz2
-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 = -DSz2
-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 = -DSz2
-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 = -DSz2
-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 = -DSz2
-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 = -DSz2
-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 = -DSz2
-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 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 perperndicular 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)
How macroscopic ?How macroscopic ?
4 x 3 = 12
Disconnectivity = 44
8 x 4 = 32+Sz = +10
20 B
Sz = -10
- 20 B
-
Extensive difference = 40
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 can stimulate the quantum can stimulate the quantum
tunnelingtunneling
N.V. Prokof’ev and P.C.E. Stamp, J. Low Temp. Phys. 104, 143 (1996)
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
220 240 260 280 300 320 340 360 3800
2
4
6
8
10
hyperfine fields: 21.8 T 26.2 T 34.5 T
Mn3+Mn3+
Mn4+
Inte
nsi
ty (
a.u
.)
Frequency (MHz)
5555Mn NMR spectra in zero applied fieldMn NMR spectra in zero applied field
Inuclear = 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, 104401 (2001)T. Kubo et al., PRB 65, 224425 (2002)
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
Thermal activationThermal activation
1.0 1.2 1.4 1.6 1.8 2.00.1
1
10
100
Nu
clea
r re
laxa
tio
n r
ate
(s-1
)
Temperature (K)
-70
-60
-50
-40
-30
-20
-10
0
-10 -5 0 5 10
Th-A T
QT
Sz
En
erg
y (
K)
-70
-60
-50
-40
-30
-20
-10
0
-10 -5 0 5 10
Th-A T
QT
Sz
En
erg
y (
K)
Thermal activationThermal activation
1.0 1.2 1.4 1.6 1.8 2.00.1
1
10
100
Nu
clea
r re
laxa
tio
n r
ate
(s-1
)
Temperature (K)
-70
-60
-50
-40
-30
-20
-10
0
-10 -5 0 5 10
Th-A T
QT
Sz
En
erg
y (
K)
Quantum tunnelingQuantum tunneling
0.01 0.1 10.01
0.1
1
10
100
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)
T1 30 s
External field External field BBzz zz
By applying an external longitudinal field Bz, the resonance condition for tunneling is destroyed
-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)
T = 20 mKdemagnetized
Fast-relaxing moleculesFast-relaxing molecules
Every real sample contains minority species with a flipped Jahn-Teller axis
Smaller anisotropy barrier (35 instead of 65 K) fast
normalW. Wernsdorfer et al., Europhys. Lett. 47, 254 (1999) Z. Sun et al., Chem. Comm., 1973 (1999)
Faster tunneling rateFaster tunneling rate
Intercluster nuclear couplingIntercluster nuclear coupling
0 5 10 15 20 250.01
0.1
1
Ech
o in
ten
sity
(a.
u.)
Time (ms)
Intercluster nuclear couplingIntercluster nuclear coupling
0 5 10 15 20 250.01
0.1
1
Ech
o in
ten
sity
(a.
u.)
Time (ms)
T2 10 ms
Intercluster nuclear couplingIntercluster nuclear coupling
0 5 10 15 20 250.01
0.1
1
Ech
o in
ten
sity
(a.
u.)
Time (ms)
0 5 10 15 20 250.01
0.1
1
Ech
o in
ten
sity
(a.
u.)
Time (ms)
Intercluster nuclear couplingIntercluster nuclear coupling
ratio 2
Nuclei in different cluster are mutually coupled Nuclei in different cluster are mutually coupled spin diffusion spin diffusion
A. Morello et al., PRL 93, 197202 (2004)
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
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)
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) cond-mat/0509261 (2005)
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
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
Are these probabilities still the Are these probabilities still the same when the bias is a quantum same when the bias is a quantum
excitation?excitation?
P = P?
muon spin relaxationmuon spin relaxation
beam
sample
backwarddetector
forwarddetector
implantation
relaxation
decay= 2.2 s
external field(optional)
asymmetry =B - F
B + F polarization
SR in MnSR in Mn1212
0.01 0.1 1 10 1001E-3
0.01
0.1
1
10
100
B = 0.7 T
B = 0
mu
on
rel
axat
ion
rat
e (s
-1)
Temperature (K)
Mn12-acwith fast-relaxing molecules
0.01 0.1 1 10 1001E-3
0.01
0.1
1
10
100
B = 0.7 T
B = 0
Temperature (K)
Mn12-tBuwithout fast-relaxing molecules
?
Inexplicably fast relaxation down to T << 1 KInexplicably fast relaxation down to T << 1 K
Z. Salman, A. Morello et al., unpublished
SR in MnSR in Mn44 dimers dimers
Mn4 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 MnSR in Mn44 dimers dimers
1 10 1000.1
1
10
B = 0
B = 0.2 T
mu
on
rel
axat
ion
rat
e (s
-1)
Temperature (K)
Z. Salman, A. Morello et al., unpublished
• 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
• nuclear spin diffusion
Including:
A wealth of detailed informationA wealth of detailed information
• quantum dynamics probed by nuclear spins
• nuclear spin diffusion
• dipolar ordering and thermal equilibrium
Including:
A wealth of detailed informationA 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 informationA 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 informationA 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 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 , 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 setupUltra-low temperature setup
(a) Coaxial cable connected to the NMR spectrometeer and pulse generator.
(b) 3He 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 3He phase. (l) Double-wall Kapton tail. (m) Forced downwards flow of 3He in the dilute
phase. (n) Sample with NMR coil.(o) Openings in the inner Kapton tube to allow the
return of the 3He flow. (p) Vacuum plug.
8Li
H1cos(ωt )
H0backward
forward
ISAC ISAC -NMR facility at TRIUMF-NMR facility at TRIUMF
18895 18900 18905 18910-0.30
-0.25
-0.20
-0.15
-0.10
Asym
metr
y
Frequency (kHz)
0 2 4 6 8 10-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
Asym
metr
y
Time (s)
8Li 8Be + e- + e
G.D. Morris et al., PRL 93, 157601 (2004)
Preliminary Results - MnPreliminary Results - Mn1212 on silicon on silicon
silicon substrate
DipolarFields
10 100
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
28 keV
Lin
ewid
th (
kHz)
Temperature (K)
28 keV
G.G. Condorelli et al., Angew. Chem. Int. Ed. 43, 4081 (2004)
10 100
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
28 keV
Lin
ewid
th (
kHz)
Temperature (K)
Preliminary Results - MnPreliminary Results - Mn1212 on silicon on silicon
1 10
2
4
6
8
Lin
ewid
th (
kHz)
Implantation Energy (keV)
T = 5 KT = 3 K
silicon substrate
DipolarFields
28 keV
10 100
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
28 keV 1 keV
Lin
ewid
th (
kHz)
Temperature (K)
1 keV
Z. Salman et al., unpublished