PRL 110, 147201 (2013)

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PHYSICAL REVIEW LETTERS

Comparison of Quantum and Classical Relaxation in Spin Dynamics R. Wieser Institut fu¨r Angewandte Physik, Universita¨t Hamburg, D-20355 Hamburg, Germany (Received 17 October 2012; published 2 April 2013) The classical Landau-Lifshitz equation with a damping term has been derived from the time evolution of a quantum mechanical wave function under the assumption of a non-Hermitian Hamilton operator. Further, the trajectory of a classical spin (S) has been compared with the expectation value of the spin ^ A good agreement between classical and quantum mechanical trajectories can be found for operator (S). Hamiltonians linear in S^ or S, respectively. Quadratic or higher order terms in the Hamiltonian result in a disagreement. DOI: 10.1103/PhysRevLett.110.147201

PACS numbers: 75.78.n, 75.10.Jm, 75.10.Hk

The Landau-Lifshitz equation [1] is one of the most often used equations in physics. This equation is of importance not only in micromagnetism [2] or for the spin dynamics at the atomic level [3], but also in disciplines like astronomy [4], biology [5], chemistry [6], and medicine [7]. In micromagnetism, it describes the motion of a magnetic moment in a local magnetic field. The equation of motion can be augmented easily by additional interactions that can be incorporated into an effective field or, e.g., by temperature effects. Moreover, there are many similar, equivalent, or alternative approaches, namely, the Bloch equation [8], the Ishimori equation [9], and the LandauLifshitz-Bloch equation [10]. All these approaches are capable of describing magnetization dynamics, starting from a single atomic spin up to several micrometers. Originally, the Landau-Lifshitz equation was introduced as a pure phenomenological equation [1]. Later, it has been shown that the precessional term can be derived by quantum mechanics [11], but the damping term in the LandauLifshitz equation remained phenomenological until Gilbert proposed to use the Lagrange formalism with the classical Rayleigh damping instead of the original Landau-Lifshitz damping to improve the equation, resulting in the LandauLifshitz-Gilbert equation [12]. In this Letter, I will describe an alternative and thereby close the lack of knowledge: A simple derivation of the Landau-Lifshitz equation with damping starting from the quantum mechanical time evolution will be given. Such a derivation provides a deeper understanding of the underlying mathematics and the connection between quantum mechanics and classical physics. The derivation starts with the quantum mechanical time evolution of the state j c ðtÞi:

 ^ t iH j c ð1Þ ðt þ tÞi  1^  j c ðtÞi: @

(2)

If we further assume that we have a non-Hermitian ^ , we get ^ þH Hamilton operator H  ^ þ t iH (3) h c ð1Þ ðt þ tÞj  h c ðtÞj 1^ þ @ and the norm n2 ¼ h c ð1Þ ðt þ tÞj c ð1Þ ðt þ tÞi  ^ þ t ^ t iH iH 1^  j c ðtÞi ¼ h c ðtÞj 1^ þ @ @ i ^ H ^ þ j c ðtÞi ¼ 1  r: (4) ¼ 1  th c ðtÞjH @ Now, we are now looking for a normalized wave function and make the ansatz j c ð1Þ ðt þ tÞi pffiffiffiffiffiffiffiffiffiffiffiffi : 1r Then, Eq. (2) can be rewritten as j c ðt þ tÞi ¼

(5)

j c ð1Þ ðt þ tÞi  j c ðtÞi i ^ ¼ H j c ðtÞi; (6) t @ and, with Eq. (5), pffiffiffiffiffiffiffiffiffiffiffiffi j c ðt þ tÞi 1  r  j c ðtÞi i ^ ¼ H j c ðtÞi: (7) t @ pffiffiffiffiffiffiffiffiffiffiffiffi Further, with the Taylor expansion 1  r  1  12 r, we get

(1)

j c ðt þ tÞi  j c ðtÞi 1 r i ^  j c ðt þ tÞi ¼  H j c ðtÞi; t 2 t @ (8)

Under the assumption of a small time step t, we can ^ þ t; tÞ  expand the time evolution operator Uðt ^ ð1^  iH t=@Þ þ Oðt2 Þ:

where r is given by Eq. (4). In the limit t ! 0, the differential quotient becomes a differential operator dt and j c ðt þ tÞi becomes j c ðtÞi. Finally, we get the following modified time-dependent Schro¨dinger equation:

^ þ t; tÞj c ðtÞi: j c ðt þ tÞi ¼ Uðt

0031-9007=13=110(14)=147201(4)

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Ó 2013 American Physical Society

PRL 110, 147201 (2013) i@

   ^ ^ þH H d  ^ þ c ðtÞ  j c ðtÞi ¼ H    dt 2

      c ðtÞ j c ðtÞi:    (9)

This formula is identical to the equation proposed by Mølmer et al. [13] for the calculation of Monte Carlo wave functions in quantum optics. ^ ¼H ^ þ¼H ^ H ^  i, ^ þ i^ ( 2 Rþ , ^ With H 0 ^ ^ Hermitian), and hi ¼ h c ðtÞjj c ðtÞi, Eq. (9) becomes i@

d ^ c ðtÞi: ^  i½^  hiÞj j c ðtÞi ¼ ðH dt

(10)

Gisin [14] has proposed a similar equation, however, with ^ ¼H ^  iH: ^ the use of H i@

d ^  i½H ^  hHiÞj ^ j c ðtÞi ¼ ðH c ðtÞi: dt

(11)

This special case of Eq. (10) is the quantum mechanical counterpart of the Landau-Lifshitz equation, as will be shown below. Equation (11) can be rewritten as i@

d ^  i½H; ^ j c ih c jÞj c i; j c i ¼ ðH dt

(12)

and for the corresponding transposed equation we can use ^ þ ¼ H. ^ Then, the transposed commu^T ¼ H the fact that H tator is given by ^ j c ih c jT ¼ ½H; ^ j c ih c j; ½H;

(13)

and therefore the corresponding transposed equation is i@

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PHYSICAL REVIEW LETTERS

d ^  i½H; ^ j c ih c jÞ: h c j ¼ h c jðH dt

(14)

(15)

In the Schro¨dinger picture, the time dependence of the ^ is invested in ^ and in the expectation value hSi ^ which is in the Schro¨dinger Heisenberg picture in S, picture time independent:   ^ dhSi d^ ^ ¼ i@Tr S ; (16) i@ dt dt therefore, we find under usage of the cyclic change under the trace ^ dhSi i ^ ^  ^ ^ ¼  h½S; Hi þ h½S; ½; ^ Hi: dt @ @

dS^ i ^ ^  ^ ^ ¼  ½S; H þ ½S; ½; ^ H: dt @ @

(18)

^ The problem is, there is still an additional ^ instead of S. For S ¼ 1=2, the density matrix is given by 1 ^ ¼ ð1^ þ hi ^ Þ: ^ (19) 2 The factor 1=2 is just for the normation because Tr^ ¼ 1. ^ therefore, this The unity matrix 1^ does commutate with H; term can be skipped and ^ is equal to the polarization hi ^ ^ with the Pauli matrix vector ^ ¼ ð^ x ; ^ y ; ^ z Þ. Here,  ,  2 fx; y; zg, are the Pauli matrices. For general S, the ^ S=S ^ polarization hi ^ ^ has to be replaced by hSi with the corresponding spin matrix vector S^ ¼ ðS^x ; S^y ; S^z Þ and N  S^ ¼ 2S n¼1 n ( 2 fx; y; zg) [16]: ^ 

^ S^ hSi : @S

(20)

^ ¼ S and Under the assumption of being in a pure state jhSij the further assumption that the z^ axis is the quantization ^ ¼ hS^z i ¼ @S, we get axis hSi hS^z iS^ ^ ¼ S: (21) @S Putting this in Eq. (18) gives the Heisenberg equation ^ 

Now, we are able to write down the corresponding von Neumann or quantum Liouville equation [15] of the density operator : ^ d^ d dj c i dh c j ¼ ðj c ih c jÞ ¼ hc j þ jc i dt dt dt dt i  ^  ½; ^ ¼ ½; ^ H ^ ½; ^ H: @ @

Please notice there is still a ^ included on the right-hand side of Eq. (17). To get the Heisenberg equation, we interpret the operators in the Heisenberg picture and skip the bras h c j and ^ ¼ h c jSj ^ c i). kets j c i on both sides of the equation (hSi The expectation values are identical in both pictures, and therefore we finally get

(17)

dS^ i ^ ^  ^ ^ ^ ¼  ½S; H þ ½S; ½S; H: dt @ @ In a previous publication [11], I have shown that

(22)

^ i ^ ^ @H ½S; H ¼ S^  þ Oð@Þ; (23) @ @S where the cross product and the gradient directly follow from the definition of the commutator [17] and the additional term occurs if the Hamilton operator is not linear in S^ n . The double commutator term on the right-hand side is more complicated. Here, we have to know that, within ^ H ^ ¼ the Clifford algebra, S^  S^ ¼ iS^ and ðS^  SÞ ^S  ðS^  HÞ ^ hold, which is not the case for normal vec^ ¼ ðH^ x ; H^ y ; H^ z Þ are tors. Here, S^ ¼ ðS^x ; S^y ; S^z Þ and H matrix vectors. Alternatively, we can use the following relation of the SO(3) Lie algebra: ^  x^ y^ ^y x^ ¼ ½x; ^ y^ ; x  y ¼ xy

(24)

where x and y are normal vectors and x^ and y^ are 3  3 skew-symmetric matrices

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x3

x2

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PHYSICAL REVIEW LETTERS x3 0 x1

x2

1

1

C x1 C A; (25) 0

with y and y^ accordingly. This relation can be proven with the aid of the Jacobi identity of the cross product. For details and additional depictions, see Ref. [18]. In the limit S ! 1 and @ ! 0, we get the classical Landau-Lifshitz equation

 is the gyromagnetic ratio coming from the relation between magnetic moment  and spin, S ¼ jj comes from the normalization,  is the damping constant, and Heff ¼ @H=@S is the effective field, with the classical Hamilton function H. To prove the agreement between the time dependent Schro¨dinger equation and the Landau-Lifshitz equation, we have performed numerical calculations. In the following, we use a simple single spin model. The description is just an example and can be extended to systems with ^ is given by N > 1. The corresponding Hamilton operator H ^ ¼ Dz ðS^z Þ2  S Bz S^z  S Bx ðtÞS^x : H

(27)

The first term of the Hamiltonian describes a uniaxial anisotropy, with the z axis as the easy axis. This term only makes sense if the spin is embedded in an environment of other atoms. A single atom itself shows no spin-orbit coupling and therefore has no uniaxial anisotropy. The second term represents a static external magnetic field in the þz direction. The last term is a time-dependent field pulse 2

Bx ðtÞ ¼ Bx0 e1=2½ðtt0 Þ=TW  ;

-1 0

200 300 time t (arb. units)

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mechanical expectation values (b) disagree. This is caused by the quadratic Hamilton operator due to the anisotropy ^ leads to S^  [11]. In this case, the commutator ½S^ n ; H ^ @H=@S plus an additional term of the order of @. This additional term vanishes in the classical limit S ! 1 and leads to the disagreement between quantum mechanical and classical trajectory. In the case of a linear Hamiltonian, ^ does not lead to an additional the commutator ½S^ n ; H correction, and both trajectories show a perfect agreement. Therefore, we can say that the second commutator in the damping term does not produce any corrections. (a)

(28)

with Gaussian shape to excite the spin. In an experimental setup using single atoms, such an excitation can be realized, e.g., by a current pulse coming from a scanning tunneling microscope (STM) tip. In the following, we investigate the two situations: either (i) Dz ¼ 0 and Bz  0 or (ii) Dz  0 and Bz ¼ 0. In case (i), all terms of the Hamiltonian ^ In case (ii), the Hamiltonian contains a are linear in S. quadratic term. In a previous publication [11], I have shown that, in case (i), under the assumption of a negligible damping ( ¼ 0), a good agreement between classical and quantum spin dynamics can be obtained. Case (ii) shows without relaxation a disagreement between classical and quantum spin dynamics due to the noncommutativity of the quadratic terms. Figure 1 shows the results of the calculation of a single spin with relaxation for case (i) (without anisotropy). Again, an excellent agreement between quantum mechanical expectation values hS^ i,  2 fx; y; zg, and the classical trajectories S of the Landau-Lifshitz equation is found. In case (ii) (Fig. 2), the classical trajectories (a) and quantum

100

FIG. 1 (color online). Magnetization as a function of time after a Gaussian field pulse: comparison of classical (bold lines) and quantum mechanical (thin lines) trajectories. Dz ¼ 0, S Bz ¼ 0:1, TW ¼ 0:02, t0 ¼ 10, S Bx0 ¼ 25:27, and  ¼ 0:2.

classical magnetization

(26)

0 -0.5

1 0.5 0

-0.5 -1 0

(b) spin expectation values

dS   ¼ S  Heff þ S  ðS  Heff Þ: dt S S

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magnetization

PRL 110, 147201 (2013) 0 1 0 0 x1 C B B x^ ¼ B x¼B A; @ x2 C @ x3

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200 300 time t (arb. units)

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1 0.5 0

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FIG. 2 (color online). Magnetization as a function of time after a Gaussian field pulse: (a) classical trajectory S , (b) quantum mechanical expectation values hS^ i,  2 fx; y; zg. Dz ¼ 0:1, S Bz ¼ 0, TW ¼ 0:02, t0 ¼ 10, S Bx0 ¼ 25:27, and  ¼ 0:2.

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PHYSICAL REVIEW LETTERS

(a) 0.3

magnetization

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0

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(b) 0.3

magnetization

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FIG. 3 (color online). Overdamped relaxation of magnetization after a Gaussian field pulse. S ,  2 fx; y; zg, corresponds to classical (bold lines) and quantum mechanical (thin lines) trajectories, respectively. (a) Dz ¼ 0 and S Bz ¼ 0:1, (b) Dz ¼ 0:1 and S Bz ¼ 0. TW ¼ 0:02, t0 ¼ 10, S Bx0 ¼ 25:27, and  ¼ 0:2.

In the case of the quadratic Hamiltonain with anisotropy, we have the additional term [11] (iDz =@) (S^xn , S^yn , 0), which ^ Therefore, the deviation between clascommutes with S. sical and quantum trajectories does not come from this term. The question is whether this correction comes from ^ only or whether the the precessional term ( i=@) ½S^ n ; H ^ ½S; ^ H ^ also leads to a correction. relaxation term (=@) ½S; To answer this question and to clarify the effect of the damping, we compare the trajectories in the overdamped limit (  1). Here, we assume that the damping dominates the dynamics and skip the precessional terms: The overdamped time dependent Schro¨dinger equation is given by   d  ^ ^ j c ðtÞi ¼ 0; þ ½H  hHi (29) dt @ and the overdamped Landau-Lifshitz equation is given by @S  ¼ S  ðS  Heff Þ: (30) @t S Figure 3 shows the trajectories of the overdamped relaxation process after a field pulse excitation. As expected in case (i) Dz ¼ 0, Bz  0, we see a perfect agreement between the quantum mechanical and the classical curves. In case (ii) Dz  0, Bz ¼ 0, we find the deviation which means that the second commutator also produces a correction which modifies the correction which comes from the

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precession term. A detailed description of the underlying physics is given in the Supplemental Material [19]. In summary, I have shown that it is possible to derive the Landau-Lifshitz equation from the quantum mechanical time evolution of a wave function. This derivation reveals the underlying mathematics and assumptions. During the derivation, we get different presentations in different physical pictures and descriptions. In quantum mechanics, we can find behavior which cannot be described by classical physics like quantum tunneling. In a previous publication [11], I have shown that quadratic or higher order Hamilton operators do not behave in a classical manner, meaning the Ehrenfest theorem does not hold in these cases. In this Letter, this concept has been used to proof the damping term. In the case of a linear Hamiltonian, we see a perfect agreement of classical physics and quantum mechanics, but for quadratic and higher order Hamiltonians a deviation appears, which comes from the damping term. Therefore, the described formalism gives us the possibility to compare the classical with the quantum spin dynamics. The author wants to thank N. Mikuszeit and S. Krause for helpful discussions. This work has been supported by the Deutsche Forschungsgemeinschaft (SFB 668 B3) and the Hamburg Cluster of Excellence NANOSPINTRONICS.

[1] D. L. Landau and E. M. Lifshitz, Phys. Z. Sowjetunion 8, 153 (1935). [2] W. F. Brown, Micromagnetics (Wiley, New York, 1963). [3] V. P. Antropov, M. I. Katsnelson, M. van Schilfgaarde, and B. N. Harmon, Phys. Rev. Lett. 75, 729 (1995). [4] G. Bo¨rner et al., Astron. Astrophys. 44, 417 (1975). [5] J. B. Bell, A. L. Garcia, and S. A. Williams, Phys. Rev. E 76, 016708 (2007). [6] A. Hucht, S. Buschmann, and P. Entel, Europhys. Lett. 77, 57003 (2007). [7] K. Witte et al., J. Spintr. Magn. Nanomater. 1, 40 (2012). [8] F. Bloch, Phys. Rev. 70, 460 (1946). [9] Y. Ishimori, Prog. Theor. Phys. 72, 33 (1984). [10] D. A. Garanin, Phys. Rev. B 55, 3050 (1997). [11] R. Wieser, Phys. Rev. B 84, 054411 (2011). [12] T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004). [13] K. Mølmer, Y. Castin, and J. Dalibard, J. Opt. Soc. Am. B 10, 524 (1993). [14] N. Gisin, Helv. Phys. Acta 54, 457 (1981). [15] D. A. Garanin, Adv. Chem. Phys. 147, 213 (2011). [16] U. Fano, Rev. Mod. Phys. 29, 74 (1957). [17] M. Lakshmanan, Phil. Trans. R. Soc. A 369, 1280 (2011). [18] C.-S. Liu, K.-C. Chen, and C.-S. Yeh, J. Mar. Sci. Tech. 17, 228 (2009). [19] See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.110.147201 for the underlying physics of the simulations which have been used to demonstrate that damped quantum and classic spin dynamics are identical as long as the Hamiltonian does not contain any nonlinear terms like the uniaxial anisotropy.

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Comparison of quantum and classical relaxation in spin dynamics.

The classical Landau-Lifshitz equation with a damping term has been derived from the time evolution of a quantum mechanical wave function under the as...
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