Yuri P. Kalmykov, 1 William T. Coffey 2, and Serguey V. Titov 3

Size: px

Start display at page:

Download "Yuri P. Kalmykov, 1 William T. Coffey 2, and Serguey V. Titov 3"

Agnes Dorsey
6 years ago
Views:

Nonlinear magnetic relaxation of quantum superparamagnets with

Coffey and erguey V. Titov 3 1. Université de Perpignan Lab.

Trinity College Department of Electronic and Electrical Engineering

1 Nonlinear magnetic relaxation of quantum superparamagnets with arbitrary spin value : Phase-space method Yuri P. Kalmykov 1 William T. Coffey and erguey V. Titov 3 1. Université de Perpignan Lab. Mathématiques Physique et ystèmes Perpignan France. Trinity College Department of Electronic and Electrical Engineering Dublin Ireland 3. Russian Academy of ciences Institute of Radio Engineering and Electronics Moscow Russia 1

2 ˆ ( 1) C mn You may try if you want to understand how a classical vector is equal to a matrix 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 is the spin matrix operator m n 1 dim 1 1 ~ 0 μ? X X Z Z Y Y

3 Ruslan Leont'evich tratonovich was an outstanding physicist engineer probabilist. Professor tratonovich was born on May in Moscow Russia. He died on the 13th of January

4 Main objective is to discuss an universal (phase space) formulation of the spin dynamics for arbitrary value of i.e. a formulation applicable to both the quantum (0) and classical cases (>>>1) W. Wernsdorfer Adv. Chem. Phys

5 Dynamics of Uniaxial uperparamagnets (arbitrary ) Quantum Classical (>>1) V V V 1 0 m H Bˆ Aˆ ˆ Z Z ˆ i Hˆ ˆ ˆ ˆ Q t E. Chudnovsky D. Garanin J. Villain A. Würger J. L. García-Palacioset al. V Bcos A cos 1 W V N W sin W t sin L. Néel W. F. Brown A. Aharoni et al. 5

6 ummary Introduction 1. Wigner s phase space distributions. Wigner functions for a quantum oscillator 3. Wigner functions for spins 4. pins in phase space 5. Phase space distributions for spins 6. pins in an external field 7. Master and Langevin equations 8. Uniaxial superparamagnets 9. tochastic resonance 10. Conclusions 6

7 Quantum Mechanics W. Heisenberg ( ) Wave and Matrix Mechanics Approaches E. chrödinger ( ) Path Integral Approach R. Feynman ( ) 7

8 E. P. Wigner Phys. Rev (193) Wigner (193) formulation of quantum mechanics as quasi-probability distributions on phase space (xp) / ( ) ˆ ipy W x p x y x ye dy ˆ x x ) 1 - density matrix E. P. Wigner ( ) Evolution equation for W W p W 1 i i V x V x W t m x i p p Equation for the eigenvalues E 0 1 i p V x W EW m i x p 8

9 E. P. Wigner Phys. Rev (193) Wigner (193) formulation of quantum mechanics as quasi-probability distributions on phase space (xp) Â t Calculation of an observable Traditional approach Phace space approach Tr ˆ ˆ Aˆ t A A ˆ t Ax ( p )W ( x p ) dxdp Aˆ A A( x p) - Weyl symbol of the operator Â Thus observables can be calculated just as classical ones 9

n 0 n n n 1/4 0 m0 x / ( x) e H x m / Phase space Traditional approach: m x m x approach: 1 * ipy / n( ) n( /)

10 Wigner functions for a quantum oscillator n E 1 m n n! W. P. chleich Quantum Optics in Phase pace Wiley-VCH Berlin 001. n 0 n n n 1/4 0 m0 x / ( x) e H x m / Phase space Traditional approach: m x m x approach: 1 * ipy / n( ) n( /) n( /) W x p x y x y e dy n ( 1) p m 0x /( m0) Wn e L n p m 0x /( m 0) n 0 En : ( 1/) 0 n H x Hermite polynomials n : L x Laguerre polynomials n Evolution equation (Liouville equation) Wn p Wn V Wn 0 t m x x p 10

11 D. Kohen et al. Phase space approach to theories of quantum dissipation J. Chem. Phys (1997) Master equation for a quantum oscillator (weak coupling limit) W p W W m0 0 W m0 x pw coth t m x p m p kt p [G.. Agarwal Phys. Rev. A (1971)] imilar to the Fokker-Planck equation for a classical oscillator W p W W W m0 x pw mkt t m x p m p p Evolution equation for the density matrix t i m / a 0 1 e 1 H 0 e a a a a 0 11

12 pin operators 1/. ˆ 1 σˆ ˆσ Pauli matrices ˆ 10 1 ˆ 10 i ˆ 11 0 X Y Z 1 0 i pherical components of the spin operators for any ˆ ( 1) C mn m n 1 C Clebsch-Gordan coefficient m n 1 1

13 Phase space formulation of quantum mechanics for spins Z u Z u. R. L. tratonovich ov. Phys. JETP (1957) Transformation of the spin matrices Weyl symbols of the spin operators ˆ ˆ ˆ ˆ X Y Z ˆ ˆ ˆ 1 1 X Y Z u X X u Y Y tratonovichwigner transformation ( 1) ˆ ( 1) u u ux sin cos u Y sin sin u Z cos tratonovichwigner transformation ( 0) ˆ ( 1) u ˆ tratonovichwigner transformation ( 1) u J. M. Radcliffe (1971) F. A. Berezin (1975) G.. Agarwal ( ) J. C. Várilly and J. M. Gracia-Bondía (1989) C. Brif and A. Mann (1998) et al. 13

14 R. L. tratonovich ov. Phys. JETP (1957) Phase space formulation of quantum mechanics for spins (tratonovich 1956) representation (phase) space of the polar and azimuthal angles u X X Z u Z u u Y Y Phase space representation of the spin density matrix W ˆ Direct tratonovich Wigner transformation ( ) 1 4 wˆ Inverse tratonovichwigner transformation W Tr ˆ ˆ ˆ ˆ wˆ ˆ ˆ ˆ ˆ X Y Z X Y Z ( ) sindd u ( ux uy uz) (sin cos sinsin cos ) J. M. Radcliffe (1971) F. A. Berezin (1975) G.. Agarwal ( ) J. C. Várilly and J. M. Gracia-Bondía (1989) C. Brif and A. Mann (1998) et al

15 Generalized coherent state representation Kernel of the tratonovich Wigner transformation wˆ ( ) 1 tratonovich Wigner transformation : Explicit equation L 4 ( ) ˆ 1 ( ) ˆ w CL 0YLM ( ) T L M 1L0 ML J. M. Radcliffe (1971) F. A. Berezin (1975) G.. Agarwal ( ) J. C. Várilly and J. M. Gracia-Bondía (1989) C. Brif and A. Mann (1998) et al. 15

transformation ( 1) un 1 1 ( )cos H 1 Cubic ˆ cub H ˆ ˆ ˆ 4 4 4 X Y Z

16 Transformation of the spin Hamiltonians Traditional representation. Phase space representation Uniaxial Hˆ un 1 ˆ Z tratonovichwigner transformation ( 1) un 1 1 ( )cos H 1 Cubic ˆ cub H ˆ ˆ ˆ X Y Z tratonovichwigner transformation ( 1) cub H 1 3 ( 1) ( )( ) 4 4 sin sin sin 3 ( 3 1) / 4 16

witching field curves [A. Thiaville Phys. Rev. B 61 11 (000)] 5100.

)cos h h 3/ 3/ X Z 1 Cubic H cub 3 ( 3 1) / 4 1 3 4 ( 1)( )( ) sin

17 witching field curves [A. Thiaville Phys. Rev. B (000)] Quantum effects in toner-wohlfarth astroids H un Uniaxial 1 1 1( )cos h h 3/ 3/ X Z 1 Cubic H cub 3 ( 3 1) / ( 1)( )( ) sin sin sin 4 Yu. P. Kalmykov et al Phys. Rev. B 008 v. 77 No. 10 p

Example: pin in an external field H µ ˆ H Density matrix approach: ˆ Hˆ ˆ 0Z e Z ˆ ˆ 0 Tr B Z Z 1 1 1 x B

18 Example: pin in an external field H µ ˆ H Density matrix approach: ˆ Hˆ ˆ 0Z e Z ˆ ˆ 0 Tr B Z Z x B x coth x coth is the Brillouin function (Langevin function -> ) After W.E. Henry Phys. Rev (195) 18

19 H µ Y. Takahashi and F. hibata J. Phys. oc. Jpn (1975) Example: pin in an external field ˆ e Phase space approach: ˆ 0Z Z tratonovichwigner transformation ( 1) B x is the Brillouin function (Langevin function -> ) Classical limit ( ) 0 const ( 1) W ( ) Z cosh sinh cos W 0 ( 1) ( 1)cos W ( )sind Z B ( ) exp cos / Z cl Boltzmann distribution 19

20 H Density matrix approach: µ Phase space approach: tationary solution: Classical limit ( ) 0 const Y. Takahashi and F. hibata J. Phys. oc. Jpn (1975) pin in an external field: Master equation for the longitudinal relaxation ˆ i Hˆ ˆ ˆ ˆ H Q t Master Equation (Quantum Fokker-Planck like equation) W eq ˆ W () (1) D ( z) W D ( z) W t z z 0 0 ˆ 0Z ( ) Z cosh sinh cos The Fokker-Planck equation for rotational diffusion of a classical spin in an external field W 1 (1 z ) W W t z z N tationary solution: cos W ( ) e / Z eq cl 0

21 Quantum Langevin equation for a spin in a uniform field Z H 0 u Y 0 u = u H Dˆ h u Dˆ u AH q h 0 X h () t 0 i 1 i j i j h () t h ( t ) δ( t t) Classical limit ( ) u =uh 0 h u u H0 h Yu. P. Kalmykov W. T. Coffey and. V. Titov EPL (009). 1

22 H Linear Response: J. L. García-Palacios and D. Zueco J. Phys. A: Math. Gen. (006) Nonlinear Response: Yu. P. Kalmykov W. T. Coffey and. V. Titov Phys. Rev. E (007). pin in an external field continued µ ˆZ I eq ˆ () t Z ˆZ II eq cor / N 1: = 1 : = 3 3: = 4: 0 t d ˆ 1 () ˆ () ˆ Z t Z t Z 0 dt eq quantum : 0 / / N e ( 1) ( 1) 1 1 classical limit ( ) : / N ( 1)

23 Uniaxial superparamagnet in an external field H 0 H I H II ˆ () t I II ˆZ ˆZ eq eq Z Quantum ˆ ˆ Hˆ / / N Z Z 0 t d ˆ 1 () ˆ () ˆ Z t Z t Z 0 dt eq k1 II m ˆ sgn( mm ) II Z m b m II mk m II k 1 [ ( 1) kk ( 1)] k discrete sum D.A. Garanin J. L. García-Palacios D. Zueco et al. 0 t ( ) i Classical V ( ) cos cos 1 h Vz z e Vz e dzdz 1 z Vz 1 N e Vz e dzdz 1 z hef z 3/ 1 1 h 1 h ~ N 1h e 1 h e 1 h h / Continuous integral W.F. Brown A. Aharoni W.T. Coffey et al. 3

24 Uniaxial superparamagnet in an external field Hˆ ˆ / ˆ / Z Z d ˆ 1 () ˆ () ˆ Z t Z t Z 0 dt eq ( ) i = = 10 1: = 0 : = 4 3: = 8 N 1: = 3/ : = 4 3: = 10 4: = 0 5: = 80 6: Linear Response: J. L. García-Palacios and D. Zueco Phys. Rev. B (006) N Nonlinear Response: Yu. P. Kalmykov W. T. Coffey and. V. Titov Phys. Rev. B (010) classical limit

25 Uniaxial superparamagnet in an external field ( ) i ( ) 1 ~ N 1 i ef ' ()/ ''()/ 3 1: = 4 : = 6 3: = 8 4: = 10 J. L. García-Palacios D. Zueco et al. 4 1 = = N 1 Quantum ˆ ˆ Hˆ / / Z Z ef N ˆ Classical ˆ Z II Z ˆ ˆ ˆ Z Z II II V ( ) cos cos ef N cos II 1 cos cos II II 5

26 tochastic resonance Archetypal model: a one-dimensional overdamped bistable oscillator subjected to noise and excited by a weak periodic force E exite (t)=a cos t of frequency close to the Kramers escape rate from the well so that the noise induced hopping becomes synchronized with E exite (t) U kt U U E excit If the dynamic susceptibility () is known we may write the signal-to-noise ratio (NR) at = ( ) NR( T ) A. 4 kt ( ) tochastic Resonance has a bell-like shape of the curve NR(T) i.e. a maximum at certain temperature (noise) level. NR tochastic Resonance increase with increasing fluctuation intensity of the periodic signal and of the signal-to-noise ratio. L. Gammaitoni P. Hänggi P. Jung and F. Marchesoni Rev. Mod. Phys. 70 (1998) 3. T

Raikher V. tepanov A. Grigorenko P. Nikitin Phys. Rev. E 56 (1997) 6400.

27 V( ) n n kt Magnetic tochastic Resonance kt V = cos H(t) = Hcos t kt kt NR : = : = 4 3: = 10 4: = 0 5: = 40 N = 1 1/ Classical: Yu. Raikher V. tepanov A. Grigorenko P. Nikitin Phys. Rev. E 56 (1997) Quantum: Yu. P. Kalmykov. V. Titov W. T. Coffey Phys. Rev. B 81 (010) 17411;

Conclusions: The phase space formalism: provides a complementary method

that the powerful classical approaches (escape rate theory of

corrected dynamic susceptibilities reversal times hysteresis and

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 8

29 Phase space (Wigner) approach: Further reading. R. de Groot and L. G. uttorp Foundations of Electrodynamics (North-Holland Amsterdam 197). W. P. chleich Quantum Optics in Phase pace (Wiley-VCH Berlin 001). R. Puri Mathematical Methods of Quantum Optics (pringer Berlin 001). Quantum Mechanics in Phase pace edited by C. K. Zachos D. B. Fairlie and T. L. Curtright (World cientific ingapore 005). R. E. Wyatt Quantum Dynamics with Trajectories: Introduction to Quantum Hydrodynamics (pringer New York 005). 9

Magnetization reversal in the presence of thermal agitation and spin-transfer torques

Magnetization reversal in the presence of thermal agitation and spin-transfer torques Y.P. Kalmykov, W.T. Coffey, S.V. Titov, J.E. Wegrowe, D. Byrne Université de Perpignan Trinity College Dublin IREE