Interplay of classical and quantum spin dynamics Текст научной статьи по специальности «Физика»

ISSN 2072-5981

Volume 14, 2012 No. 2, 12201 - 9 pages

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Editors-in-Chief Jean Jeener (Universite Libre de Bruxelles, Brussels) Boris Kochelaev (KFU, Kazan) Raymond Orbach (University of California, Riverside)

Executive Editor

Yurii Proshin (KFU, Kazan) Editor@ ksu. ru

Editors

Vadim Atsarkin (Institute of Radio Engineering and Electronics, Moscow) Detlef Brinkmann (University of Zürich, Zürich) Yurij Bunkov (CNRS, Grenoble) John Drumheller (Montana State University, Bozeman) Mikhail Eremin (KFU, Kazan) Yoshio Kitaoka (Osaka University,

Osaka)

Boris Malkin (KFU, Kazan) Haruhiko Suzuki (Kanazawa University, Kanazava) Murat Tagirov (KFU, Kazan)

*

In Kazan University the Electron Paramagnetic Resonance (EPR) was discovered by Zavoisky E.K. in 1944.

Interplay of classical and quantum spin dynamics*

F.S. Dzheparov

Institute for Theoretical and Experimental Physics,

B. Cheremushkinskaya, 25, Moscow 117218, Russia

E-mail: [email protected]

(Received November 12, 2012; accepted November 20, 2012)

Quantum and classical spin dynamics have more similarity than difference in theoretical part. Many processes can be described within general formalism where the type of dynamics became important at final steps only. The lecture is illustrated by consideration of operator perturbation theory and multispin resonance transitions.

PACS: 45.10.Hj, 76.60.-k, 76.30.-v, 75.75.Jn

Keywords: spin dynamics, perturbation theory, master equation, magnetic nanoparticles, nuclear relaxation, electron relaxation, combination frequency resonance

1. Introduction

Classical spins behavior becomes of interest in connection with modern studies of magnetic resonance in magnetic nanoparticles (see for example [1-4] and references therein).

Traditional consideration of quantum spin transitions in NMR and EPR [5, 6] was very different from analysis of ferromagnetic resonance [7], where motion of huge classical moment of total sample was used. The difference produces difficulties in comparison of corresponding results, especially for complex multi-spin and "multi-quantum" transitions.

Modern theory of multi-spin transitions starts, probably, with treating of two-spin transitions in the book [8]. Most extensive consideration was fulfilled in Ref. [9], where several versions of the theory were constructed, and intensities of many transitions were calculated and compared with precision results of beta-NMR spectroscopy [10, 11]. The theory can be characterized as a quantum mechanical unitary operator perturbation theory, constructed in commutator form. It has many common properties with treating of multi-pulse narrowing methods of solid-state NMR [12, 13]. Main aim of the lecture consists in concentrated description of the theory in application to multi-spin transitions, which is equally applicable both to quantum and classical spin systems. The description produces shortest way to separation of quantum and classical effects in results of measurements.

It should be stressed, that construction of quantum mechanical unitary operator perturbation theory started in first half of 20-th century. Probably it was initiated both internal requirements of quantum mechanics and analogies with canonical perturbation theory of classical mechanics (see Ref. [14] as an introduction for example). The influence of canonical classical mechanics was very strong from the beginning of the quantum theory. For example, quantum theory introduces new space of states instead of classical trajectories, but definition of operators in this space is strictly connected with Hamiltonian classical mechanics in Cartesian coordinates and prescribes a substitution of canonical momentum • • • ^ p = 3Z(q,q) / dq (not kinematical one mq !) by the operator p = -id/dq as a quantization rule. Here • •

L(q,q) is a Lagrangian as a function of Cartesian coordinate q and velocity q. We will use presumably units with h = 1 and c = 1. From mathematical point of view the quantum mechanical states form a Hilbert space, it will be referred to as Schrödinger space here in order to separate from Liouville space, discussed below.

* This short review is prepared on base of invited lecture at XV International Youth Scientific School "Actual problems of magnetic resonance and its application", Kazan, 22 - 26 October 2012

Separation of canonical and kinematical momentums gives a possibility to take into account magnetic field, because (in simplest case) the Hamiltonian

H(p, q) = qp - L =1 m ^q j + U(q) = ^(p + eA(q)f + u(q).

Here A(q) is a vector-potential, e is electrical charge, and U(q) is scalar potential energy. With more general definition we have canonical momentum as pM = dS (q) / dqM, where

S (q) = j0qp^q - (p, q) dt = -¡"pMdqM

is an action as a function of final 4-dimentional coordinate q^ = (t, q) and integration is carried out

along real trajectory. This definition produces the same space components p of the 4-momentum pM,

and new component p0 = -H(p,q). Therefore, applying the same quantization rule to canonical momentum p0 = p0 = -H(p, q) we have two different definitions for corresponding momentum

d

operator: p0 = -i— and p0 = -H(p,q). For compatibility we should require their coincidence in

dt

action on realizable state | \y), that produce the main equation of the quantum theory - Schrödinger

equation i—\y) = H(p,q)|i^). This short excursion demonstrates exclusive importance of canonical

classical mechanics for foundations of the quantum theory.

Unitary transformations form one of the most important sections of quantum mechanics. Nowadays workers know and understand it, as a rule, much better than canonical transformations in classical theory. Similar education effect is known for a long time; it was indicated, for example, in the "Introduction" in Ref. [15]. Therefore our consideration will be based on unitary transformations in Liouville space, which can be considered as necessary and usual extension of standard Schrödinger quantum mechanical space or natural space for classical dynamics, based on Liouville equation for distribution functions. In quantum mechanics Liouville space is formed by density matrices or by usual operators of Schrödinger space.

2. Perturbation theory. General outlines

Absolute majority of interesting theoretical problems have no exact solutions. Perturbation theory produces a possibility to receive approximate (and verifiable in an experiment) result starting from exactly solvable simplified problem. From my point of view the best perturbation theory for spin dynamics consists of two very different parts. First part is directed on simplification of the Hamiltonian up to form, suitable for application in derivation of a master equation, which produces

description of evolution of observables. The master equation derivation forms second part of the

perturbation theory. This strategy has long history, and its elements can be found, for example, in Refs. [8] and [16].

Evolution both quantum and classic systems is governed by Liouville equation

^p = -iLp, (1)

dt

where p is density matrix or distribution function in quantum and classical theory respectively. In quantum case the Liouville operator L (Liouvillian) is defined as

Lp = [ H, p] = H p-pH, (2)

A

T i TT i dH dp dH dp T-pd

Lp = -i {H,p\ = -i-------------------------=-i > >

I dp dq dq dp I 1=l a=1

dH dp dH dp

Kdpa dq« dq« dp*

ß

Pi, qß

= ^jk^aß • Heisenberg equations of

Here N is a number of considered particles, while d is space dimension.

Eqs. (1)-(3) unify main equations of motion of quantum and classical theory. A tendency to such unification existed from the beginning of quantum mechanics. Partially one of form of main quantization postulate consists in substitution of Poisson bracket of canonical variables by the operator

commutator according the same rule {pj , qk}=^jk^aß -

motion have the same similarity to Hamilton equations.

From operator point of view the Liouvillian L is a Hermite operator in a new (relative to Schrödinger space) Hilbert space (Liouville space), where density matrices and other quantum mechanical operators works as vectors, while in classical theory Liouville space is formed by distribution functions and other functions in the same way. The Liouville operator is hermitian relative to scalar production (a, b ) = Tr(a+b) or (a, b ) = J dpdq • a+b in quantum and classical theory

respectively.

Formal Nakajima-Zwanzig derivation of master equation starts from separation of a small, but important part p1 of p, which is sufficient for calculation of observables. In simplest case this operation is introduced by a time-independent projection operator P and

pi = Pp, P2 = P. (4)

Multiplying Eq. (1) on P we have

where

and, evidently,

dt pi =dfPp = -iPL (P + P )p = -i (PLPpi + PLP p2), (5)

p2 = Pp, P = 1-P, (P )2 = P, (6)

d — — — — —

- p2 = -iPL (P + P )p = -i (PLPp, + PLP p2). (7)

Solving Eq. (7) with initial condition p2(t = 0) = 0 and substituting the solution into Eq. (5) we receive a master equation

dtp = -/Qp1 - (t,T)Pi (T) (8)

with definitions of frequency matrix Q and memory kernel M(t,r):

Q = PLP, M(t,r) = PL(t)P •T exp (-i Jtdr1PL (t1 )P)-PL(r)P .

(9)

Here standard chronological exponent (Texp) is introduced.

These formal operations are well known and equally applicable both for quantum and classical theory. Real calculations require reasonable approximations for memory kernel which can depend on type of dynamics and we will not discuss them later.

We will concentrate our attention on the first part of perturbation theory consisting of simplification of the Hamiltonian. We will consider the Hamiltonian of the form

H =X ’ H++= H-m . (10)

As a rule the representation (10) is a consequence of application of so called representation of interaction, when strong but exactly solvable part of the evolution is excluded from the equation of motion, and all terms in (10) have the same order of value, while many of frequencies am are large.

We come to this representation, for example, in discussion of resonances at frequencies, obeying the condition ka = maI. Here k and m are integer, and aI and a are Larmor frequency and frequency of alternating field. The transitions can take place, for example, in homo-spin system formed by spins I, placed in strong static magnetic field (directed along z-axis with value aI) and orthogonal it radio-frequency field with value a1I (we use frequency units for magnetic fields) in presence of dipole-dipole interactions. The terms Hm at that represent so called Van Vleck’ alphabet [5, 7] (with corresponding frequencies {am} = 0,±aI, ±2aI ) and rf-interaction (with frequencies ±(aI -a)) in the system, rotating with the frequency aI around the static field. Other example is produced by nuclear or electron spins in presence of quadrupole interaction

HQ =a, [(nl)2-1I(I + o) (H)

and static magnetic field, if aQ aI. The Hamiltonian (11) produces the same set of primary frequencies am as homo-spin dipole interaction. More complex examples together with solutions can be found in Refs. [9-11].

The aim of considered perturbation theory consists in such transformation of the Liouville equation, which conserve Hamilton form (2) of the main equation (with a new Hamiltonian) and suppress fast oscillating terms of the Hamiltonian. Slow oscillating terms should be treated via master equation.

Very important property of the theory consists in the fact, that new (transformed) Hamiltonian is constructed from powers of commutations (or Poisson brackets) that automatically produce no volume divergences.

Other important property is specific for spin dynamics, where spin variables are included only instead of full set of coordinates and momenta, and main quantum mechanical (QM) commutators are in exact agreement with Poisson brackets of classical mechanics (CM) again:

[i;ji] = iS^l (12)

Here Ia is a component of j-th spin, and [a,b] means the commutator of operators a and b in quantum mechanics, or [a, b] = -i {a, b} in classical theory. A summation is meant in (12) over index Y on right side, which is absent on left side. With these notations Eqs. (1)-(3) can be written as

d

—p = -iLp = -i [ H, p] = -iH yp (13)

both in quantum and in classical theory, and the operation [a,b] will be referred as commutator in

both theories, if it will not require additional refinement. Last relation in (13) can be written as

L = Hx, and it indicates that operator L in Liouville space (superoperator) is formed by H via commutator that is rather special form for superoperators.

3. Unitary operator perturbation theory

We can separate the Hamiltonian (10) (and corresponding Liouvillian) into fast (F) and slow (S) oscillating parts:

H (t) = H„ (t) + H„ (t), H„ (t) =-£^<a ‘"JH- , HF (t) = Z|.,M ‘“JH-.

i\am |>Q

(14)

The boundary frequency Q should be defined later from the requirement of self-consistency of calculations.

We can introduce a new variable p(1) (t) via unitary transformation in Liouville space

P(t ) = U (t )p(1)(t ) = exp (-iSx (t ) )p(1)(t ), Sx (t ) = £,

1

' i®„.

-e^Hl.

(15)

In action on typical states, for example on Ia or IaIf, the superoperator Hym produces finite result even for infinite systems, when number of spins N ^<x>. Therefore

Sx (t) ~ Hp / Q ~ £ (16)

can be considered as small value proportional to small parameter s .

It is evident that 3SX (t)/ dt = HyP (t), and equation of motion for p(1)(t) does not contain fast oscillating terms in main order in s:

d

More exactly

It is evident, that

-P(1) = -iHSp(1) + O(s). dt

-p(1) = -iZ(1)p(1), L(1) = U+LU + iU+ U. dtH H

U+LU p(1) = U

h • (u P- )„ ]=[(U * h )0,

P

(1)

(17)

(18)

(19)

Here a symbol (U p(1) ^ (or (U+H ^) means, that action of superoperators U (or U +) is concentrated within the bracket, and the result is a vector in Liouville space.

Differentiation of the exponential operator U(t) is a standard action (see for example [9] or Appendix in [17]) with a result

dt

Therefore

-es = i

i1 daeiaS Sy Jo

ii(1-a)S x

(20)

UJ+ Up(1) = i dae,aS Sx e

-ia p(1) = i da eaSx HxP e-'aSXp(r)

= i J0 d“e“x[ hp , ( e-“xp“’ )01 = i J0 d«\( ea HP )0

(1)

= 1

reS -1 iSx

H

, p

(1)

/0

(21)

We see that new Liouvillian L(1) = H(1)x is formed of new Hamiltonian according to relations (18), (19) and (21) as

iSx i

L(1)p(1) = H(1)xp(1) =[H(1),p(1)], H(1) = eiSH-------— HF . (22)

The transformation from Eqs. (13) and (14) to (18) and (22) defines an iteration method, which can be continued later. At every step of iterations fast oscillating term of the Hamiltonian is suppressed, and

new slow oscillating terms are created. After n steps the fast oscillating term has an order Hp) ~ s2 as in super-convergent classical Kolmogorov-Arnol’d-Moser theory. But, for typical conditions, relaxation speed wp,n), produced by HFn), is not smaller than the speed wF = wFn=0), calculated directly from HF ~ s0 [8, 9]. There is no contradiction here, because, as a rule, |ln wnF |~ 1/ s, while the

iterations display the power accuracy ~ (s2 ) for wFn) only [9].

Main result for applications is concentrated in effective Hamiltonian of slow motions Hf presented by slow part of the new Hamiltonian, because it contains new terms relative to initial HS . Two iterations produce accuracy up to s3 and [9]

Hf = Hf + O(s4) = ((1 -1Sx2 -±Sx3)Hs + (^Sx-1Sx2 -±Sx3)Hf)s

S (23)

- i(|_(( ( Hs + -i (p ))0p, S(Hs + i Hp ))0p ])s + O (s4).

»re A = £mAm exp(iamt)/ (iam) f°r A = EmAm exp(iamt). <he symbol (~)0 is mtroduced in (19),

and subscripts S and F indicate separation of slow and fast oscillating parts, as in (14). Main attention of many workers was attracted to situations, where instead of separation of slow part of the Hamiltonian the time averaging can be used. Corresponding result with accuracy up to s2 was received in Refs. [18, 19], it was repeated for spin dynamics in [13], and the accuracy was refined up to s5 in Ref. [20].

4. An example

Let us consider a resonance at the frequency a = 2aj in homo-spin system, produced by dipole interactions as an example. The Hamiltonian can be written as

H = Hz + HD + Hf (t). (24)

Here Hz = jjIz is Zeeman interaction, HD is dipole-dipole Hamiltonian and

Hf (t) = ia„ (I+e-at + I_e'at) (25)

represents the action of resonance alternating field. Main influence on resonance at the frequency

a = 2aj is produced by so called C-term of HD :

Hdc = 2 E k (cJk (i;i+ + I+IZ)+H .C.)=HC + h+c ,

3 r2*2

c,t =-T^sin(2&]k)e'^, \_Hz,H+c~] = ±aIH±c,

(26)

+k 4 rjk

where polar angles 3jk and </>jk define direction of the interspin vector rjk.

Effective Hamiltonian of slow motion will include secular part of dipole interactions HD0 and, according to (22), a resonance term

F.S. Dzheparov

Hres (t) = 2 [ S ,Hp ]S, Hp = HC ea + HC e-at + 2ji (i+ea a + I_e-iai a),

Aa,t -ia,t

S = Hp = HC e----------HC ea--------------------- (I+ ei(aia -1 e^1a).

iaj iaj 2i(at -a)' !

After simplifying

Hes (t) = ya ) eKa a E j1+ + H C. = H+ (t) + H- (t),

4aI (a-al) +

h+ (t) - a(2aia e JkcJki+n, h-(t)=(h+ (t))+.

(28)

In the simplest theory (neglecting the influence of dipole order) important part of the density matrix can be chosen as a function of Iz only and master equation (8) after standard transformations became a form

d

p1 =-J

Hess (t T) = eiHD 0tHreS (r) e

St p = -Jo” dT\-Hres (t,t) , LHres (t-T,t-T) , p1(t) ]], (29)

Correspondingly

■^(¡,) = J, (1:. fl) = -j„" *( i-. [ Hres (t • t), [ Hres (t-T, t-t) , p(t) ]])

= -2

£ dre(2ai a ([H+ (t,0) ,H-] ,p ).

(30)

Here scalar production in Liouville space is applied and the relation [Hz, H± ] = ±2aIH± is taken into account.

One of most representative parameter of resonance at combination frequency a = kaj is its forbidding factor A (a = kaI). To define it we can introduce a measurable parameter

. d 4n/[ H+ ,h;]\

Wg (a= 2ai) = -limtf da^~ln (II) =------------------------------------------------------7^-0, (31)

J dt i1-)o

which presents intensity of the resonance. Here (F)0 =(F,p1(t = 0)) for any F = F+. The argument a = 2aj on the left side in (30) indicates type of the resonance, while frequency integration in second term is fulfilled near the resonance frequency 2aI and does not include resonances of other types, which are supposed as well separable. The limit t ^ 0 in (31) implies small t»T2, according to applicability of Eq. (29). Similar parameter for Larmor resonance is

W0 (a = aj ) = na12I. (32)

The forbidding factor is

A (a = 2aI ) = W0 (a = 2aI) / W0 (a = a:). (33)

We see that it depends on equilibrium properties of the system. Of course results will be very different for large classical spins and for small quantum spins with the same gyromagnetic ratio. Direct calculation produces

Interplay of classical and quantum spin dynamics

W0(a = 2a,) = 8^] N+1)-^)2)- (34)

J

It is evident that W0 (a = aI) does not depend on spin value at all. Contrary that W0 (a = 2aI) has strong corresponding dependence, and "extra"-quantum case I = 1/2 has no dependence on initial state, while classic limit has strong corresponding dependence.

5. Conclusion

The analysis indicates that description of classical and quantum spin dynamics can be carried out in general way, where difference between these theories go into action at last steps in calculations of such values as forbidding factor or resonance form function for example. These exist strong indications, that dipole resonance form function for classical and quantum theories are very close as well, see for example [21].

Acknowledgments

I am grateful to Prof. V.A. Atsarkin for stimulating discussions. The work was supported by RFBR (project # 11-02-00880).

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