Dynamics and thermodynamics of quantum spins at low temperature Andrea Morello Kamerlingh Onnes...

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)