1

Yuri P. Kalmykov,1 William T. Coffey2, and Serguey V. Titov3

Nonlinear magnetic relaxation of quantumsuperparamagnets with arbitrary spin value S:

Phase-space method

1.Université de Perpignan, Lab. Mathématiques, Physique et Systèmes, Perpignan, France

2.Trinity College, Department of Electronic and Electrical Engineering, Dublin, Ireland

3.Russian Academy of Sciences, Institute of Radio Engineering and Electronics, Moscow, Russia

2

“You may try, if you want, to understand how a classical vector is equal to a matrix S, and maybe you will discover something – but don’t break your head on it. …”

R. Feynman The Feynman Lectures on Physics, Tome III, Quantum Mechanics

is the magnetic dipole vector, S is the spin matrix operator

dim 2 1 2 1 ,~ 0,

S SS

S

? S μ ,, ,1,

ˆ ( 1) S mS nm n

S S S C

Y

Y

Z

X

Z

X

3

Ruslan Leont'evich Stratonovich was an outstanding physicist, engineer, probabilist. Professor Stratonovich was born on May 31,

1930 in Moscow, Russia. He died on the 13th of January, 1997

http://en.wikipedia.org/wiki/Ruslan_L._Stratonovich

4

Main objective is to discuss an universal (phase space) formulation of the spin dynamics for arbitrary

value of S, i.e., a formulation applicable to both the quantum

(S20) and classical cases (S>>>1)

W. Wernsdorfer, Adv. Chem. Phys.,

2001

5

Dynamics of Uniaxial Superparamagnets (arbitrary S)

2ˆ ˆˆS Z ZH BS AS

2cos cosV B A

ˆ ˆˆ ˆ ˆ,Si H Q

t

Quantum Classical (S>>1)

m 0

V

V1

V2

E. Chudnovsky, D. Garanin, J. Villain, A. Würger, J. L. García-Palacios,et al.

L. Néel, W. F. Brown, A. Aharoni, et al.

12 sinsinN

W VW Wt

6

Introduction1. Wigner’s phase space distributions 2. Wigner functions for a quantum oscillator3. Wigner functions for spins4. Spins in phase space5. Phase space distributions for spins6. Spins in an external field7. Master and Langevin equations 8. Uniaxial superparamagnets9. Stochastic resonance 10. Conclusions

Summary

7

E. Schrödinger (1887-1961)

Wave and Matrix Mechanics Approaches

Quantum Mechanics

W. Heisenberg (1901-1976)

Path Integral Approach

R. Feynman(1918-1988)

8

E. P. Wigner (1902-1995)

Wigner (1932) formulation of quantum mechanics as quasi-probability distributions on phase space (x,p)

/1 11( )2

,2

ˆ ,2

ipyx y x yW x p e dy

1 2ˆ , )x x - density matrix

E. P. Wigner, Phys. Rev. 40, 749 (1932)

Evolution equation for W

1 02 2

W p W i iV x V x Wt m x i p p

Equation for the eigenvalues E21

2 2 2ip V x W EW

m i x p

9

Wigner (1932) formulation of quantum mechanics as quasi-probability distributions on phase space (x,p)

ˆ ˆTr ˆA t A

ˆ ( , )A AA x p - Weyl symbol of the operator

E. P. Wigner, Phys. Rev. 40, 749 (1932)

Thus observables can be calculated just as classical ones

Calculation of an observable A t

A

, ( , )ˆ ( )W x pA t dA x p x dp

Traditional approach

Phace space approach

10

Wigner functions for a quantum oscillator

0n n nW W Wp Vt m x x p

20

1 / 4/ 20

01( ) /

2 !m x

n nn

mx e H x m

n

* /1( , ) ( / 2) ( / 2)2

ipyn n nW x p x y x y e dy

0 ( 1/ 2)nE n

2 2 2 2

0 0/( ) 2 2 2 20 0

( 1) 2 /( )n

p m x mn nW e L p m x m

W. P. Schleich, Quantum Optics in Phase Space, Wiley-VCH, Berlin, 2001.

Phase space approach:

Evolution equation (Liouville equation)

2 2 2 20

22 2n

n n nm xE

m x

:nH x Hermite polynomials

:nL x Laguerre polynomials

Traditional approach:

11

Master equation for a quantum oscillator(weak coupling limit)

2 0 00 coth

2 2mW p W W Wm x pW

t m x p m p kT p

[G. S. Agarwal, Phys. Rev. A 4, 739 (1971)]

Similar to the Fokker-Planck equation for a classical oscillator

D. Kohen et al., Phase space approach to theories of quantum dissipation, J. Chem. Phys. 107, 5236 (1997)

20

W p W W Wm x pW mkTt m x p m p p

0

0

0 , 1 , ,2

/ 2 , ,1

i H e a a a at m

m a a a ae

Evolution equation for the density matrix

12

.1 / 2S

Spin operators

0 1 0 1 01 1 1ˆ ˆ ˆ, ,1 0 0 0 12 2 2X Y Z

iS S S

i

,

,

1ˆ ˆ2

S σ σ Pauli matrices

Spherical components of the spin operators for any S

,, ,1,

ˆ ( 1) S mS nm n

S S S C

,, ,1

S mS nC Clebsch-Gordan coefficient

13

.

uY

u

Y

Z

uX

uZ

X

Phase space formulation of quantum mechanics for spins

( 1)ˆ ( 1)

Stratonovich Wigner transformation

S

S u

sin cossin sin

cos

X

Y

Z

uuu

u

Transformation of the spin matrices

ˆ ˆ ˆ,ˆ ,X Y ZS S SS

( 0)

( 1)

ˆ ( 1)

ˆ

Stratonovich Wigner transformation

Stratonovich Wigner transformation

S S

S

S u

S u

Weyl symbols of the spin operators

R. L. Stratonovich, Sov. Phys. JETP, 4, 891 (1957)

J. M. Radcliffe (1971), F. A. Berezin (1975), G. S. Agarwal (1981,1994), J. C. Várilly and J. M. Gracia-Bondía (1989), C. Brif and A. Mann (1998) et al.

ˆ ˆ ˆ, , 2 1 2 1X Y ZS S S S S

14

Phase space formulation of quantum mechanics for spins(Stratonovich, 1956)

representation (phase) space of the polar and

azimuthal angles

,

,

ˆ ˆ ˆ ˆT ˆr , ,

2 1 ˆ sin

( , )

( ,4

)ˆ

Direct Stratonovich Wigner transformation

Inverse Stratonovich Wigner transformation

X Y ZS S S w

S w d

W

dW

J. M. Radcliffe (1971), F. A. Berezin (1975), G. S. Agarwal (1981,1994), J. C. Várilly and J. M. Gracia-Bondía (1989), C. Brif and A. Mann (1998) et al.

1,0

uY

u

Y

Z

uX

uZ

X

R. L. Stratonovich, Sov. Phys. JETP, 4, 891 (1957)

ˆ ˆ ˆ,ˆ ,X Y ZS S S

Phase space representation of the

spin density matrix

( , , )(sin cos , sin sin ,cos )

X Y Zu u u

u

15

Generalized coherent state representation

J. M. Radcliffe (1971), F. A. Berezin (1975), G. S. Agarwal (1981,1994), J. C. Várilly and J. M. Gracia-Bondía (1989), C. Brif and A. Mann (1998) et al.

1ˆ ( , ) , , , ,w S S

2,

1 , , 00

( ), , ,

4ˆ ( , )2 1

ˆ( , )S L

S SS S L

L M LL M

SL Mw C T

SY

Kernel of the Stratonovich Wigner transformation

:Stratonovich Wigner transformation Explicit equation

, ,S

16

.

Transformation of the spin Hamiltonians

( 1)

2 4 4 4ˆ ˆ ˆˆ2

Stratonovich Wignertrans

cubX Y Z

formation

SH S S S

( 1)

12ˆˆ

Stratonovich Wignertransfo

Z

run

mat on

S

i

SH

111 2

2( )cos2

unS

SH S S

32

2 31

2

2

2 4

2

(2 3 1) / 4

( 1)

sin 2 sin si

( )(

24

n

)

cubSH S S S

S S S S

Uniaxial

Cubic

Traditional representation Phase space representation

17

.

Switching field curves [A. Thiaville, Phys. Rev. B 61, 12221 (2000)]

2 111 2( )cos

2unS

SH S S

32

2 4 22 312 2

(2 3 1) / 4

( 1)( )( ) sin 2 sin sin 24

cubSH S S S

S S S S

Uniaxial

Cubic

5 02, ,10S

Quantum effects inStoner-Wohlfarth astroids

Yu. P. Kalmykov et al, Phys. Rev. B 2008, v. 77, No. 10, p. 104418

3/2 3/2 1X Zh h

18

Example: Spin in an external field

0ˆˆTrZ Z SS S SB S

0ˆˆ

S ZH S

is the Brillouin function (Langevin function, S -> )

ˆˆ SHSe Z

Density matrixapproach:

After W.E. Henry , Phys. Rev. 88, 559 (1952)

2 1 2 1 1coth coth2 2 2 2SS S xB x x

S S S S

H

µ

19

Example: Spin in an external field

Classical limit (S , )

SB x is the Brillouin function (Langevin function, S -> )

0ˆ ( 1)

ˆ Z

Stratonovich Wignertransfo

SS

rmation

e Z

2( 1) 1 1 10 02 2( ) cosh sinh cos

SSW Z

102 ( ) exp cos / clS S ZW

( 1)12 0

0

( 1)cos ( )sinZ

S

S S W d

SB S

S

Y. Takahashi and F. Shibata, J. Phys. Soc. Jpn. 38, 656 (1975)

0S const

Phase space approach:

______________________________

Boltzmann distribution

H

µ

20

Spin in an external field: Master equation for the longitudinal relaxation

Classical limit (S , )

0ˆˆ

S ZH S

Y. Takahashi and F. Shibata, J. Phys. Soc. Jpn. 38, 656 (1975)

0S const

______________________________Phase spaceapproach:

Density matrixapproach:

______________________________

ˆ ˆˆ ˆ ˆ,Si H Q

t

(2) (1)( ) ( )W D z W D z Wt z z

The Fokker-Planck equation for rotational diffusion of a classical spin in an external field

Master Equation (Quantum Fokker-Planck like equation)

cos( ) /eq clW e Z

21 1 10 02 2( ) cosh sinh cos

Seq SW Z

21 (1 )2 N

W z W Wt z z

Stationary solution:

Stationary solution:

H

µ

21

Quantum Langevin equation for a spin in a uniform field

Classical limit (S )1,

( ) 0

( ) ( ) 2 ( )i

i j i j

h t

h t h t δ t t

u

Y

Z

X

H0

Yu. P. Kalmykov, W. T. Coffey, and S. V. Titov, EPL (2009).

0 0 = u u H h u u H h

0

0

ˆ

ˆ

= D

D A

u u H h

u u hqH

22

Spin in an external field continuedNonlinear Response: Yu. P. Kalmykov, W. T. Coffey, and S. V. Titov, Phys. Rev. E (2007).

,

,

,

,

H

µ

1ˆ ˆ ˆ( ) ( ) 0Z Z Z eq

d S t S t Sdt

IIˆZ eq

S

ˆ ( )ZS t

0 t

IˆZ eq

S

23

4

1

1: S = 12: S = 33: S = 4: S

cor /

N

0S

1/

quantum :

/ ( 1)

( 1)

SN S e

1

classical limit ( ) :

/( 1)

N

S

Linear Response: J. L. García-Palacios and D. Zueco, J. Phys. A: Math. Gen. (2006)

23

Uniaxial superparamagnet in an external field

,

,

,

,

D.A. Garanin, J. L. García-Palacios, D. Zueco, et al.

2 2ˆ ˆˆ / /S Z ZH S S S S

ClassicalQuantum

1

II II

II

II1

ˆ sgn( )2

[ ( 1) ( 1)]

S k

Z m b mSm k m SN

k S k

m S m m

S S k k

1

21 11

11 1

2

114

1ef

h zV zV z

V zN V z

h z

e e dz dzz

e e dz dzz

2( ) cos cosV

W.F. Brown, A. Aharoni, W.T. Coffey, et al.

2 23/2 11 1

2~ 1 1 ,1

/ 2

h hN h e h e

hh

( ) 1 , 11 i

0 t

H0

HII HI

IIˆZ eq

S

ˆ ( )ZS t

0 t

IˆZ eq

S

1ˆ ˆ ˆ( ) ( ) 0Z Z Z eq

d S t S t Sdt

discrete sumContinuous integral

24

Uniaxial superparamagnet in an external field

Nonlinear Response: Yu. P. Kalmykov, W. T. Coffey, and S. V. Titov, Phys. Rev. B (2010).

,

,

,

,

Linear Response: J. L. García-Palacios and D. Zueco, Phys. Rev. B (2006)

2 2ˆ ˆˆ / /S Z ZH S S S S

0 5 10 15

= 5

N 1: S = 3/2

2: S = 43: S = 104: S = 205: S = 806: S

15

0 10 20100

101

102

103

104

N

= 10

S

1: = 0 2: = 4 3: = 8

1

2

3 clas

sica

l lim

it

( ) 1 , 11 i

1ˆ ˆ ˆ( ) ( ) 0Z Z Z eq

d S t S t Sdt

25

Uniaxial superparamagnet in an external field

,

,

,

,

J. L. García-Palacios, D. Zueco, et al.

2 2ˆ ˆˆ / /S Z ZH S S S S

Classical

Quantum

2( ) cos cosV

( ) 1 , 11 i

( ) 1~ 1N

efi

3

2

4

= 2 = 10

' (

)/

1: S = 42: S = 63: S = 84: S = 10

1

3

2 1

4

N

''(

)/

22

II II2 2

II

ˆ ˆ2

ˆ ˆ ˆZ Z

ef NZ Z

S S

S S

S

22IIII

2

II

cos cos2

1 cosef N

Stochastic resonance Archetypal model: a one-dimensional overdamped bistable oscillator

subjected to noise and excited by a weak periodic force Eexite (t)=A cos tof frequency close to the Kramers escape rate from the well so that the noise

induced hopping becomes synchronized with Eexite (t)

Stochastic Resonance has a bell-like shape of the curve SNR(T) ,i.e., a maximum at certain temperature (noise) level.

Stochastic Resonance increase, with increasing fluctuation intensity, of the periodic signal and of the signal-to-noise ratio.

U

U kT

excitU E

If the dynamic susceptibility () is known we may write the signal-to-noise ratio (SNR) at =

22( )

( ) .4 ( )

SNR T AkT

T

SNR

L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys. 70 (1998) 223.

Magnetic Stochastic Resonance

Classical: Yu. Raikher, V. Stepanov, A. Grigorenko, P. Nikitin, Phys. Rev. E 56 (1997) 6400.Quantum: Yu. P. Kalmykov, S. V. Titov, W. T. Coffey, Phys. Rev. B 81 (2010) 172411;

5

4

3

2

1N = 1

1: S = 2 2: S = 43: S = 104: S = 205: S = 40 S

SNR

1/

V( )

n

–n

kTkT

kTkT V = –cos2 H(t) = Hcos t

28

Conclusions: The phase space formalism:

• provides a complementary method of study of static and dynamic properties of spin systems

• indicates that the powerful classical approaches (escape rate theory of multidimensional systems, methods of solution of classical diffusion equation, etc.) may be directly carried over to the quantum domain yielding quantum corrected dynamic susceptibilities, reversal times, hysteresis and switching curves, etc.

• may also be extended to describe the macroscopic quantum tunneling in spin systems such as magnetic nanoclusters and molecular magnets

29

Phase space (Wigner) approach: Further reading

S. R. de Groot and L. G. Suttorp, Foundations of Electrodynamics(North-Holland, Amsterdam, 1972).

W. P. Schleich, Quantum Optics in Phase Space (Wiley-VCH, Berlin, 2001).

R. Puri, Mathematical Methods of Quantum Optics (Springer, Berlin, 2001).

Quantum Mechanics in Phase Space, edited by C. K. Zachos, D. B. Fairlie, and T. L. Curtright (World Scientific, Singapore, 2005).

R. E. Wyatt, Quantum Dynamics with Trajectories: Introduction to Quantum Hydrodynamics (Springer, New York, 2005).