dynamics of the nuclear spin bath in molecular nanomagnets: a test for decoherence andrea morello...

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