1

1

)(21

)(2

)(

)(

B

B

BM

BM

eq

-1

-0.5

0

0.5

1

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

0 s0.1 ms0.5 ms1 ms2 ms3 ms

M/M

s

µ0H (T)

0.04 K11 GHz

0.001 T/s

period: 10 ms

Absorption of microwaves

Max ~ 5 s-1

W. Wernsdorfer et al , EPL (2003)

Gaussian absorption lines

Important broadening by nuclear spins Loss of coherence

R ~ b ~ 30 kHz2~ ~ 0.2 GHz

Rabi oscillations, require larger b.

N = BMax/2 = B2/ ~20

Precession ~ 20 turns

tbBbB

bP 2

1222

22

2

)(2

1sin

)()(

)(

)()(4

2LL Bfb

Photon assisted tunneling in a SMM (Fe8) Absorption of circular polarized microwaves

-10 -5 0 5 10

En

erg

y

quantum number m

²M = +1

tunneling

²M = -1

H = 0

-1 -0.5 0 0.5 1-40

-30

-20

-10

0

En

erg

y (

K)

µ0Hz (T)

²M = ±1

-10

-9

-8

-7

10

9

8

7

Absorption of circular polarized microwaves(115 GHz)

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

00.091

M/M

s

µ0H (T)

0.007 T/s

P/P 0 =

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

00.119

M/M

s

µ0H (T)

0.007 T/s

P/P 0 =

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

00.131

M/M

s

µ0H (T)

0.007 T/s

P/P 0 =

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

00.151

M/M

s

µ0H (T)

0.007 T/s

P/P 0 =

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

00.167

M/M

s

µ0H (T)

0.007 T/s

P/P 0 =

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

00.190

M/M

s

µ0H (T)

0.007 T/s

P/P 0 =

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

00.207

M/M

s

µ0H (T)

0.007 T/s

P/P 0 =

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

00.237

M/M

s

µ0H (T)

0.007 T/s

P/P 0 =

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

00.256

M/M

s

µ0H (T)

0.007 T/s

P/P 0 =

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

00.292

M/M

s

µ0H (T)

0.007 T/s

P/P 0 =

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

00.320

M/M

s

µ0H (T)

0.007 T/s

P/P 0 =

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

00.366

M/M

s

µ0H (T)

0.007 T/s

P/P 0 =

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

00.458

M/M

s

µ0H (T)

0.007 T/s

P/P 0 =

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

00.568

M/M

s

µ0H (T)

0.007 T/s

P/P 0 =

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

00.693

M/M

s

µ0H (T)

0.007 T/s

P/P 0 =

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

00.841

M/M

s

µ0H (T)

0.007 T/s

P/P 0 =

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

01

M/M

s

µ0H (T)

0.007 T/s

P/P 0 =

Sorace et al, PRB 2003

Photon induced tunnel probabilityPassisted = P - n±10P±10

10-7

10-6

10-5

10-4

10-3

10-2

10-1

0.001 0.01 0.1

P_EPRB

B

PEP

R

(au)

n = 1n = 0

Ts

0.8

n=0

n=1

Ts= T0 + ћmws /Cs

Sorace et al, PRB (2003)

Environmental effects

Central molecule spinMn12, Fe8

Spin-bathEnvironmental spins

Enhance tunnelingMesoscopic spins

Decoherence

Phonon-bath

Spin-phonons transitionBottleneck (TB>>T1)

Electromagnetic radiation bath

Spin-photons transitions(incoherent)

Free carriersStrong decoherenceRKKY interactions

Kondo, Heavy fermions

Central ionic spin Rare-earths

Strong hyperfine interactions

Coherent dynamicsTowards new spin-qubits

V15

Rare-earths ions

A new direction

Tunneling of the angular momentum J ofHo3+ ions in Y0.998Ho0.002LiF4

Example of a metallic matrix: Ho3+ ions in Y0.999Ho0.001Ru2Si2

Mesocopic nanomagnetism

Resonant microwave absorption : towards spin qubits

A new direction:

Tunneling of the angular momentum of rare-earths ions

A quasi- infinite number of systems for the study of mesoscopic quantum dynamics:

- different CF and 4f symmetries - different concentrations - insulating, metallic, semi-conducting …

Ho3+ in Y0.998Ho0.002LiF4

Tetragonal symmetry (Ho in S4); (J = L+S = 8; gJ=5/4)

Dipolar interactions ~ mT << levels separation

-6 -4 -2 0 2 4 6-200

-150

-100

-50

0

50

100

150

-9 -6 -3 0 3 6 9-240

-200

-160

-120

-80

-40b)a) E (K)

<Jz>

E (K)

0H

z (T)

R. Giraud, W. Wernsdorfer, D. Mailly, A. Tkachuk, and B. Barbara, PRL, 87, 057203-1 (2001)

B20 = 0.606 K, B40 = -3.253 mK, B44 =- 42.92 mK, B60 =-8.41mK, B64 =- 817.3mK Sh. Gifeisman et al, Opt. Spect. (USSR) 44, 68 (1978);

N.I. Agladze et al, PRL, 66, 477 (1991)

Barrier short-cuts

Energy barrier ( ~ 10 K)

Strong mixing

Singlet excited state

Doublet ground-state

Large 1 (Orbach

process)

CF levels and energy barrier of Ho3+ in Y0.998Ho0.002LiF4

46

46

44

44

06

06

04

04

02

02 OBOBOBOBOBHCF

Hysteresis loop of Ho3+ ions in YLiF4

-1

-0,5

0

0,5

1

-3 -2 -1 0 1 2 3

1.5K

1.6K

1.9K

2.4K

M/M

S

BL (T)

Comparison with Mn12-ac

dH/dt=0.55 mT/s

-80 -40 0 40 80 120

-1,0

-0,5

0,0

0,5

1,0

200 mK 150 mK 50 mK

M/M

S

0H

z (mT)

-20 0 20 40 60 800

100

200

300

n=0n=3

n=1

n=-1

n=2

dH/dt > 0

1/ 0

dm

/dH

z (1/

T)

Many steps !

L.Thomas, F. Lionti, R. Ballou, R. Sessoli, R. Giraud, W. Wernsdorfer, D. Mailly, A.Tkachuk,

D. Gatteschi,and B. Barbara, Nature, 1996. and B. Barbara, PRL, 2001

Steps at Bn = 450.n (mT) Steps at Bn = 23.n (mT)

Tunneling of Mn12-ac Molecules Tunneling of Ho3+ ion

… Nuclear spins…

Ising CF Ground-state + Hyperfine Interactions

H = HCF-Z + A{JzIz + (J+ I- + J- I+ )/2}

-80 -40 0 40 80 120

-1,0

-0,5

0,0

0,5

1,0

200 mK 150 mK 50 mK

M/M

S

0H

z (mT)

-20 0 20 40 60 800

100

200

300

n=0n=3

n=1

n=-1

n=2

dH/dt > 0

1/ 0

dm/d

Hz (

1/T)

-200 -150 -100 -50 0 50 100 150 200

-180,0

-179,5

-179,0

-178,5

I = 7/2

E (

K)

0H

z (mT)

-7/2

7/2

7/2

5/2

3/2

-7/2

Co-Tunneling of electronic and nuclear momenta: Electro-nuclear entanglement

The ground-state doublet 2(2 x 7/2 + 1) = 16 states

-5/2

5/2

gJBHn = n.A/2 A = 38.6 mK

Avoided Level Crossings between |, Iz and |+, Iz’ if I= (Iz -Iz

’ )/2= odd

-75 -50 -25 0 25 50 75-1.0

-0.5

0.0

0.5

1.0

T = 30 mKv = 0.6 mT/s

HT=190 mT

HT=170 mT

HT=150 mT

HT=130 mT

HT=110 mT

HT=90 mT

HT=70 mT

HT=50 mT

HT=30 mT

HT=10 mT

M/M

S

0H

z (mT)

dB/dt ~ 1 mT/s

Acceleration of quantum dynamicsin a transverse field

…. slow sweeping field: meas >> bott > 1

Near thermodynamical equilibrium at the cryostat temperature…

-1

-0.5

0

0.5

1

-0.08 -0.04 0 0.04 0.08

0.136 mT/s0.068 mT/s0.034 mT/s0.017 mT/s

M/M

s

µ0H (T)

0.04 K

n=1

n=2

Case of a metallic matrix: Ho3+ ions in Y0.999Ho0.001Ru2Si2

n=0

These steps come from tunneling transitions of J+I of single Ho3+ ions,In a sea of free electrons.

Y0.998Ho0.002LiF4

Ho0.001Y0.999Ru2Si2

-1

-0.5

0

0.5

1

-0.08 -0.04 0 0.04 0.08

0.136 mT/s0.068 mT/s0.034 mT/s0.017 mT/s

M/M

s

µ0H (T)

0.04 K

-80 -60 -40 -20 0 20 40 60 80-1,0

-0,5

0,0

0,5

1,0

-80 -60 -40 -20 0 20 40 60 80

-180,0

-179,5

v = 0.11 mT/s

b)

M/M

S

0H

z (mT)

a)

E (

K)

0H

z (mT)

The resonances fields of Ho3+

ions, in YLiF4 and

YCu2Si2 are the same

Y1-HoRu2Si2 ~ 0.1%

Same resonance

fields

Many body tunneling events

mediated by RKKY interactions ?

Multiparticle Kondo ?Screening ?

(See Stamp and Prokofiev, 1997)

Effect of a transverse field: Step 2 merges with the continuous one

-80 -60 -40 -20 0 20-1.0

-0.5

0.0

0.5

1.0v = 0.14 mT/s

n = 2

n = 1

HT = 0

HT = 10 mT

T = 40 mK

M/M

S

0H

z (mT)

Ising CF Ground-state + Hyperfine Interactions

H = HCF-Z + A{JzIz + (J+ I- + J- I+ )/2}

-80 -40 0 40 80 120

-1,0

-0,5

0,0

0,5

1,0

200 mK 150 mK 50 mK

M/M

S

0H

z (mT)

-20 0 20 40 60 800

100

200

300

n=0n=3

n=1

n=-1

n=2

dH/dt > 0

1/ 0

dm/d

Hz (

1/T)

-200 -150 -100 -50 0 50 100 150 200

-180,0

-179,5

-179,0

-178,5

I = 7/2

E (

K)

0H

z (mT)

-7/2

7/2

7/2

5/2

3/2

-7/2

Co-Tunneling of electronic and nuclear momenta: Electro-nuclear entanglement

The ground-state doublet 2(2 x 7/2 + 1) = 16 states

-5/2

5/2

gJBHn = n.A/2 A = 38.6 mK

Avoided Level Crossings between |, Iz and |+, Iz’ if I= (Iz -Iz

’ )/2= odd

-200 -150 -100 -50 0 50 100 150 200

-180,0

-179,5

-179,0

-178,5

I = 7/2

E (

K)

0H

z (mT)

50 mK0.3 T/s

120 160 200 240

0

4

8

-150 -75 0 75 150 225

0

20

40

60

-300 -200 -100 0 100 200 300-1,0

-0,5

0,0

0,5

1,0

-8 -6 -4 -2 0 2 4 6 8 10-180

-120

-60

0

60

120

180

240

n = 6

n = 7n = 8

n = 9

b)

dH/dt<0

n=1

n=0

1/ 0

dm

/dH

z (1/

T)

0H

z (mT)

a)

M/M

S

0H

z (mT)

integer n half integer n

linear fit

0H

n = n x 23 mT

0H

n (

mT

)

n

Giraud et al, PRL 87, 057203 1 (2001)

Additional steps at fields: Hn = (23/2).n (mT)single Ho3+ tunneling being at avoided level crossings at Hn = 23.n (mT)

50 mK 200 mK0.3 T/s

Simultaneous tunneling of Ho3+ pairs (4-bodies entanglement)Two Ho3+ Hamiltonian avoided level crossings at Hn = (23/2).n

Fast measurements: meas ~ bott > 1 >> s

Single-ion level structure En = nE geffBHn/2

Tunneling: gJBHnn’ = (n’-n)A/2

Co-tunneling: gJBHnn’=(n’-n+1/2)A/2

Two-ions Level structureCo-tunnelingBiais tunnelingDiffusive tunneling

-2000 -1000 0 1000 2000

-180.0

-179.5

-179.0

-178.5

-2000 -1000 0 1000 2000

-360

-359

-358

-357

0 100 200 300 400 500

-360.0

-359.6

n=-9b)

a)

n=-8 n=3/2

. . .

. . .

mI=+5/2

mI=+7/2

mI=+5/2

mI=+7/2

I = 7/2E

ner

gy

(K)

Hz (Oe)

87654

32

1

0

En

erg

y (K

)

Hz (Oe)

n = 0

Hbias

n = 2n = 3/2n = 1/2

n = 1

En

erg

y (K

)

Hz (Oe)

Toy model of two coupled effective spins, with gz /gx >> 1

H/J = ijSi

zSjz +

ij(Si

+Sj- + Sj

+Si-)/2 + ij (Si

+Sj+ + Sj

-Si-)

with

= (Jx + Jy)/4J = (Jx - Jy)/4J

This is why dipolar interactions induce co-tunneling

Co-tunnelingDiffusive tunneling

Single-ion level structure En = nE geffBHn/2

Tunneling: gJBHnn’ = (n’-n)A/2

Co-tunneling: gJBHnn’=(n’-n+1/2)A/2

Two-ions Level structureCo-tunnelingBiais tunnelingDiffusive tunneling

-2000 -1000 0 1000 2000

-180.0

-179.5

-179.0

-178.5

-2000 -1000 0 1000 2000

-360

-359

-358

-357

0 100 200 300 400 500

-360.0

-359.6

n=-9b)

a)

n=-8 n=3/2

. . .

. . .

mI=+5/2

mI=+7/2

mI=+5/2

mI=+7/2

I = 7/2E

ner

gy

(K)

Hz (Oe)

87654

32

1

0

En

erg

y (K

)

Hz (Oe)

n = 0

Hbias

n = 2n = 3/2n = 1/2

n = 1

En

erg

y (K

)

Hz (Oe)

Higher temperatures: cross-spin relaxation through excited singlets

R. Giraud et al PRL, 2003 and JMMM (also ICM’2003, Rome).

S. Bertaina, B. Barbara, R. Giraud, B. Malkin, M. Vanyunin, A. Takchuk, PRB submitted.

-Single-ion tunneling (LT: spins-bath and phonons-bath )

- Co-tunneling (LT: spins-bath, HT: phonons-bath )

Extension to N >2 multi-tunneling

gJBHn(N) = nA/2N n-D

Multi-molecule resonant tunneling at gBHn(N) = nD/2N n-D

Case of strong coupling (J>>D): S =S1+S2+…+ SN gBHn(N)=nD …Wrong!

Reason: D decreases when S increases.

Multi-tunneling should fill the space between single spins tunneling

Spin-glass regimeProfile of (Hz/A)

-200 0 200 400 600 8000

2

4

6

8

10 ' T=1.75 K

LiYF4:Ho (0.11%), 1200 Hz

-200 0 200 400 600 800

0

2

4

T=1.75 K

''

-200 0 200 400 600 8000

2

4

6

8

10

T=2.5 K

-200 0 200 400 600 800

0

2

4

T=2.5 K

-200 0 200 400 600 8000

2

4

6

8

10

T=3 K

-200 0 200 400 600 800

0

2

4

T=3 K

-200 0 200 400 600 8000

2

4

6

8

10

T=3.5 K

Magnetic field (Oe)

-200 0 200 400 600 800

0

2

4

T=3.5 K

Magnetic field (Oe)

Numerical fits (Malkin, Vanyunin et al, PRB submitted)

Why D decreases when S increases:

Take N spins with anisotropy energies: En= DnSn2

Assume they are coupled with J >> Dn to form a SMM:

The total energy ET =∑DnSn2 = DT ST

2 DT = ∑DnSn2 / (∑Sn)2 << Dn

Dn=D and Sn=S DT = D/N gBHn(N)=n(D/N) n-D, as for Weak C.

…

1 10 100 1000

Quantum worldClassical world

Mn4

Mn12 Mn84Mn30

Technological applications : Magnetic recording on nm scale Quantum information, Molecular electronic and spintronics,Biomedical applications…

…..

Incredible impact on molecular and supra-molecular chemistry.

Larger and larger molecules DS2

AFTER Mn12-ac…

Co cluster

Assume:

DT

N0

2,8 3,2 3,6 4,0 4,4 4,8 5,2 5,6 6,0 6,4 6,8 7,2 7,640

60

80

100

120

140

160

180

200

220

240

Magnetic field (kOe)

J

frequency 100 GHz

0,00 0,05 0,10 0,15 0,20 0,25 0,30-250

-225

-200

-175

-0,5

0,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

m=0

Ener

gy (G

Hz)

Magnetic field B (T)

m=2

Hyperfine sublevels of Ho3+ ion in LiYF4

Direct check of hyperfine sublevels from EPR In Ho:YLiF4 (Malkin group)

G. Shakurov et al, Appl. Magn. Res. 2005

250 GHz

0,00 0,05 0,10 0,15 0,20 0,25 0,30-250

-225

-200

-175

-0,5

0,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

m=0

Ener

gy (G

Hz)

Magnetic field B (T)

m=2

Hyperfine sublevels of Ho3+ ion in LiYF4

…but too small transition amplitude …

0 200 400 600 800 1000 1200 1400 1600

-4000

-2000

0

2000

4000

dI/d

H (

u.a

.)

Champ magnétique (Oe)

LiYF4 - Ho:0.001%

0 200 400 600 800 1000 1200 1400 1600-3000

-2000

-1000

0

1000

2000

dI/d

H (

u.a

.)

Champ magnétique (Oe)

CaWO4 - Ho:0.05%

RPE continue de Ho3+ (9.5 GHz)

CaWO4 :

Same Structure as YLiF4Almost no nuclear spins

0,00 0,05 0,10 0,15

210.80

210.55

210.35

210.90

Magnetic field B (T)

210.25

(GHz)

An example of the direct observation of the anticrossing of hyperfine sublevels (m=2)

in the EPR spectra (G. Shakurov, B. Malkin, B.Barbara. Appl. Magn. Res. 2005 )

7

8

20 22 24 26205,2

205,5

205,8

206,1

206,4

206,7

209,7

210,0

210,3

210,6

210,9

Iz=-3/2, 1/2

w

w

w

Tra

nsi

tion

fre

qu

en

cy (

GH

z)

s

s

s

s

w

Iz=-1/2

68 70 72

Iz=-5/2, -1/2

Iz=-3/2

w

w

w

w

Magnetic field (mT)

s

s

s

s

116 118 120

Iz=-7/2, -3/2

Iz=-5/2

m=0

w

w

w

ws

s

s

s

m=2

The anticrossings detected in the EPR spectra in LiYF4 (0.1% Ho)

0,00 0,05 0,10 0,15 0,20 0,25 0,30-250

-225

-200

-175

-0,5

0,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

m=0

Ener

gy (G

Hz)

Magnetic field B (T)

m=2

Hyperfine sublevels of Ho3+ ion in LiYF4

…but too smal transition amplitude …

0 200 400 600 800 1000 1200 1400 1600

-4000

-2000

0

2000

4000

dI/d

H (

u.a

.)

Champ magnétique (Oe)

LiYF4 - Ho:0.001%

0 200 400 600 800 1000 1200 1400 1600-3000

-2000

-1000

0

1000

2000

dI/d

H (

u.a

.)

Champ magnétique (Oe)

CaWO4 - Ho:0.05%

Continuous EPR on Ho3+ (9.5 GHz)

CaWO4 :

Structure isomorphe à LiYF4Amost no nuclear spins

CONCLUSION

NanoparticlesThe Micro-SQUID technique : unique tool for single particles measurements

(from micron to nanometer scales)

Classical spins dynamics

Molecular magnetsQuantum Tunneling and quantum dynamics of large spins

Effects of environmental degrees of freedom (spin-bath)

Very short coherent time in molecular magnets (in « normal » conditions)

Rare-Earth in insulating and metalic matrixesEvidence for tunneling of the total angular momentum J

Crucial role of hyperfine interactions

Multi-tunneling effects

Coherent quantum dynamics and new type of spin-qubits