- Email: [email protected]
2 chains with an XY Hamiltonian
18 September 1998
Chemical Physics Letters 294 Ž1998. 297–304
From regular to erratic quantum dynamics in long spin 1r2 chains with an XY Hamiltonian ) E.B. Fel’dman 1, R. Bruschweiler , R.R. Ernst ¨
2
Laboratorium fur Switzerland ¨ Physikalische Chemie, ETH Zentrum, CH-8092 Zurich, ¨ Received 22 July 1998
Abstract The dynamics of spin polarization driven by the XY Hamiltonian of a linear spin 1r2 chain of arbitrary length N was studied analytically and numerically. For an initial state with polarization localized on a single spin, I j z , the dimension of the accessible Liouville subspace grows with N 2. The polarization propagates along the chain in the form of spin-wave packets that gradually split into sub-packets bouncing back and forth at the ends. The initially regular appearance turns at longer times into an erratic behavior due to the complex interference of the spin-wave packets. Even for large N and long times the density operator exhibits large-amplitude fluctuations about its long-term average, which is non-ergodic in the sense that it is dependent on the initial conditions of the dynamics. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction The understanding of spin evolution in extended many-spin quantum systems is important both from a fundamental and an applied point of view w1–3x. A comparison between experiment and theory for the evolution of large systems is usually hampered by the fact that for most Hamiltonians no analytical solution of the Schrodinger equation can be obtained. ¨ Numerical computer simulations, on the other hand, suffer from the exponentially increasing dimension of the Hilbert space with the number of spins N.
) Corresponding author. E-mail: [email protected] 1 On sabbatical leave from Institute of Chemical Physics, Russian Academy of Sciences, 142432 Chernogolovka, Moscowskaya oblast, Russia. 2 E-mail: [email protected]
Often they permit simulations of not more than about a dozen of spins. Alternatively, it is tempting to interpret the long-term behavior of large spin systems in terms of spin thermodynamics. Such an approach can however be treacherous as closed quantum systems can exhibit a significant degree of non-ergodic behavior irrespective of the number of spins w4–6x. In this Letter, we investigate the dynamics of a spin system under the influence of an XY Hamiltonian w7x. In contrast to the dipolar and the isotropic coupling Hamiltonians, the XY Hamiltonian can be analytically diagonalized for an arbitrarily large number of spins w8x. We have also recently demonstrated that an XY Hamiltonian can be generated experimentally in scalar coupled spin chains by taking advantage of liquid-state NMR multiple-pulse techniques w9x. For single-spin Zeeman polarization I j z as the initial condition, we derive in this Letter a
0009-2614r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 0 8 8 7 - 2
E.B. Fel’dman et al.r Chemical Physics Letters 294 (1998) 297–304
298
complete analytical solution of the Liouville–von Neumann equation and discuss the results based on numerical calculations. Some characteristic features of the solution are rationalized by analytical approximations.
The sum over the ‘index’ k in Eq. Ž2. and in forthcoming equations implies that the sum is carried out over n s 1, 2, . . . , N. The quantity k can be viewed as a wave vector that is related to the spinwave behavior of the system w7x. For an initial condition of the density operator s with a single spin Ij polarized,
2. Analytical solution of the Liouville–von Neumann equation for a spin chain with an XY Hamiltonian
s Ž 0. s Ij z ,
We consider a linear chain of N spins I s 1r2 in a magnetic field B0 with identical secular flip–flop Ž XY . interactions exclusively between nearest neighbors. The Hamiltonian H of this system can be written as H s HZ q HX Y
there is an exact solution to the Liouville–von Neumann equation s˙ s yiw H , s x:
s Ž t. s
is1
Nq1
where HZ is the Zeeman Hamiltonian with v 0 s yg B0 and the gyromagnetic ratio g . HX Y is the XY Hamiltonian with nearest-neighbor interactions. Ii a Ž a s x, y, z . are the cartesian spin operators, and D is the uniform XY coupling constant. The Hamiltonian H can be diagonalised exactly using the Jordan–Wigner transformation method w10x:
´ k b k† b k y
k
1 2
Nv 0 ,
´ k s D cos k q v 0 ,
Ž 2. where N
Ý gk Ž j . cj .
Ž 3.
js1
The operators c j are Fermion operators w10x: c j s Ž y2 .
jy1
I1 z I2 z . . . I z , jy1 Iy j
Ž 4.
gk Ž j . s
ž
X
yi Ž ´ k y ´ k X . t 1 2
,
Ž 7.
js1
was used. The density operator s Ž t . evolves in a Liouville subspace of dimension N Ž N q 1.r2 that is spanned by hermitian linear combinations of the Žnon-normalized. operators
Iyp I pq1, z . . .
q yX X X IpXy1, z Iq p , I p I pq1, z . . . I p y1, z I p
with 1 F p F pX F N .
4 Ž 8.
Evolution under the XY Hamiltonian generates for each pair of adjoint operators Bj , Bj† 4 of Eq. Ž8. either the sum Bj q Bj† or the difference iŽ Bj y Bj† . but not both. For example, for p s pX , only the q q y difference Iy p I p y I p I p s y2 I p z is realized, while y q q y the sum I p I p q Ip I p s 1 cannot be accessed from the trace-free initial condition of Eq. Ž6. by unitary evolution. The expectation values of the single-spin operators Ip z as a function of time can be expressed by ² Ip z : Ž t .
2
4
1r2
s
/
sin Ž kj . , Nq1 pn with k s , n s 1, 2, . . . , N . Nq1
Ý sin Ž kj . sin Ž lj . s d k l
² Ij z : Ž 0.
and the function g k Ž j . is defined by 2
Ý exp k, k
N
2
Ý Ii z q D Ý Ž Ii x Iiq1, x q Ii y Iiq1, y . , Ž 1.
bk s
Nq1
=sin Ž kj . sin Ž kX j . b k† b k X y
Ny1
is1
Hs Ý
2
where the well-known orthogonality relationship
N
sv0
Ž 6.
Ž N q 1.
2
Ý exp Ž yi ´ k t . sin Ž kj . sin Ž kp .
,
k
Ž 9. Ž 5.
where ² I p z :Ž t . s Tr s Ž t . I p z 4 . It directly follows that
E.B. Fel’dman et al.r Chemical Physics Letters 294 (1998) 297–304
² Ip z :Ž t .r² Ij z :Ž0. G 0. Thus, in contrast to the dipolar and the isotropic Hamiltonian, a sign change of the single-spin polarizations cannot occur w4,11x. Eq. Ž9. is symmetric in the initial spin Ij and the target spin Ip , implying that the transfers Ij ™ Ip and Ip ™ Ij are equally effective. For the multi-spin terms of Eq. Ž8. one obtains, using Eq. Ž7., qX : ² Iy X p I pq1, z . . . I p y1, z I p Ž t .
² Ij z : Ž 0. X
s
Ž y1.
pq p q1
X
2 py p q3
Ž N q 1.
2
f Ž p . f ) Ž pX . ,
Ž 10 .
where 1 F p - pX F N and f Ž p . s Ý exp Ž yi ´ k t . sin Ž kj . sin Ž kp . .
Ž 11 .
k
The complex conjugate of f Ž pX . is denoted by f ) Ž pX .. Eq. Ž10. is a real-valued expression for p q pX s even and a purely imaginary expression for X p q pX s odd. Hence, it follows ² Iq p I pq1, z . . . I p y1, z X pq p y yX :Ž . ² I p Ipq1, z . . . IpXy1, z Iq X :Ž t .. Ip t s Žy1. p
3. Numerical evaluation The general solution given in Eq. Ž9. was numerically evaluated for different initially polarized spins, I j z . In all calculations the value D s 2 p P 4444 sy1 is assumed, which corresponds to an XY dipole–dip o le c o u p lin g o f p ro to n sp in s D s yŽ m 0r4 p .g 2 Ž hr2 p .Ž1rr 3 . with a nearest neigh˚ bor distance r s 3 A. Fig. 1 shows the evolution of ² I1 z :Ž t .r² I1 z :Ž0. for a spin chain with N s 10000 spins where at t s 0 only spin I1 is polarized, s Ž0. s I1 z . The polarization on spin I1 , ² I1 z :Ž t ., decays within less than 1 ms to zero, as shown in Fig. 1A. It reappears and rises sharply for a first time after a time delay T s 716 ms Žsee inset of Fig. 1A.. It is followed by an oscillation with increasing frequency and decreasing amplitude, reflecting the dispersive behavior of the evolution Žsee Section 4.. The longer time behavior, given in Fig. 1B and C, shows recurrent ‘echoes’
299
at multiples of T. This periodic pattern reflects the propagation of a spin wave bouncing back and forth at the chain ends. The recurrence time T can be estimated by Ts
2 Nr
2 Nr f
ng
2N
Ž 12 .
s rD
D
with the group velocity ng f rD w12x. After several thousand milliseconds, the periodic echo train gradually disappears and an irregular behavior becomes evident. Even at very long times, as shown in Fig. 1D, no ‘equilibration’ is observed as the root-meansquare fluctuation amplitude remains comparable to the time-averaged polarization ² I1 z : Ž t . :
(Ž² I
1z
: Ž t . y ² I1 z : Ž t . . ² I1 z : Ž t .
2
f 1.4 .
Ž 13 .
The dynamics depends on the position of the initially polarized spin, on the position of the observer spin, and on the chain length. In the examples presented in Figs. 1 and 2, the observer spin was chosen to be the initially polarized spin. The characteristic behavior of spins not at the chain end is exemplified for a spin chain with N s 1000 spins and s Ž0. s I158, z in Fig. 2A and one with N s 10000 spins with s Ž0. s I1580, z in Fig. 2B. The evolution starts with two wave packets that propagate in opposite directions. For ² I158, z :Ž t .r² I158, z :Ž0., four different phases of evolution, labeled I–IV, are visible in Fig. 2A: the first three phases display complicated but nearly regular burst patterns while the last phase shows an erratic behavior that is reminiscent of ‘quantum chaos’. The first recurring burst arriving at time 158Tr1000 originates from reflection at spin I1. The reflection at spin I1000 gives rise to a further burst at 842Tr1000. The third burst at T s 71.6 ms is due to sequential reflections at both chain ends. It is followed by the erratic phase IV due to interference of the different wave packets. An increase of the chain length by a factor 10 ŽFig. 2B. leads to qualitatively the same behavior with the same types of phases with a ten-fold extended time scale. Spin dynamics under the XY Hamiltonian markedly differs from evolution under the dipolar
300 E.B. Fel’dman et al.r Chemical Physics Letters 294 (1998) 297–304 Fig. 1. Time-course of ² I1 z :Ž t .r² I1 z :Ž0. for a linear chain of N s10000 spins under the XY Hamiltonian with the initial condition s Ž0. s I1 z . ŽA. The short-time behavior. The inset shows the first recurrence of the spin-wave packets. ŽB. – ŽD. behavior for increasingly longer times. All calculations have been performed based on Eq. Ž9.. For the initial decay of Fig. 1A the simplified Eq. Ž18. is equivalent to Eq. Ž9..
E.B. Fel’dman et al.r Chemical Physics Letters 294 (1998) 297–304
Hamiltonian: While the XY Hamiltonian involves a Liouville subspace of dimension N Ž N q 1.r2, the dipolar Hamiltonian populates a Liouville subspace
301
of dimension 4 Ny 1, which rapidly leads to a smooth and ‘diffusion-like’ evolution of polarization exhibiting low-amplitude oscillations about the average
Fig. 2. ŽA. Time-course of ² I158, z :Ž t .r² I158, z :Ž0. in a 1000-spin chain under the XY Hamiltonian with the initial condition s Ž0. s I158, z . Four different phases I–IV are indicated at the top of the figure. ŽB. Time-course of ² I1580, z :Ž t .r² I1580, z :Ž0. in a 10000-spin chain with s Ž0. s I1580, z . The plots were calculated based on Eq. Ž9.. In both panels the curves start at t s 0 with amplitude 1.
E.B. Fel’dman et al.r Chemical Physics Letters 294 (1998) 297–304
302
density operator w1,4x. Dipolar evolution in linear chains also does not show pronounced echo recurrences of the kind observed in Fig. 1. Evolution under the isotropic Hamiltonian qualitatively resembles also the evolution under the dipolar Hamiltonian w9x.
the first reflection at one of the chain ends has occurred. Thus for j s 1, i.e. for initial polarization of spin I1 , the expression is useful for t-
N D
.
Ž 15 .
Combining Eq. Ž14. with Eq. Ž9. yields ² Im z : Ž t . ² Ij z : Ž 0.
4. Useful approximations for long chains
m
2 f Jjym Ž Dt . y 2 Ž y1. Jjym Ž Dt . 2 =J jqm Ž Dt . q Jjqm Ž Dt . ,
It is possible to analytically assess the fine-structure of the spin-wave evolution by replacing in Eq. Ž9. the summation by an integration:
where Jjy mŽ Dt . is the Bessel function of the first kind of order j y m. Application of the well-known identity of Bessel functions w13x, 2m
Ý exp Ž yi ´ k t . sin Ž kj . sin Ž km .
x
k
Ž 16 .
Jm Ž x . s Jmy1 Ž x . q Jmq1 Ž x . ,
Ž 17 .
yields for j s 1 the result N f p
p
H0 exp Ž yi ´
kt
. sin Ž kj . sin Ž km . d k .
Ž 14 .
This approximation is good for the evolution before
² Im z : Ž t . ² I1 z : Ž 0 .
4 m2 s
D2 t 2
Jm2 Ž Dt . .
Ž 18 .
It follows for an observer spin Im remote from the
Fig. 3. Time-course of ² I5000, z :Ž t .r² I1 z :Ž0. in a 10000-spin chain with s Ž0. s I1 z . The calculations were performed using the exact Eq. Ž9. and the alternative Eq. Ž18.. No difference between the resulting curves can be observed for the plotted time interval.
E.B. Fel’dman et al.r Chemical Physics Letters 294 (1998) 297–304
other chain end, m < N, that the envelope of the polarization ² Im z :Ž t . decays in time proportionally to ty3 . For t F 0.25 ms in Fig. 1A, Eq. Ž18. gives virtually the same result as Eq. Ž9.. Fig. 3 shows the decay of ² I5000, z :Ž t .r² I1 z :Ž0. of a 10000-spin chain for s Ž0. s I1 z . The approximation of Eq. Ž18. is superimposed on the exact result based on Eq. Ž9.. No deviation is visible, which shows that Eq. Ž18. represents a very good approximation of the exact behavior. The decay of ² I1 z :Ž t .r² I1 z :Ž0. at short times shown in Fig. 1A can be attributed to the evolution of a single wave packet. On the other hand for the appearance and decay of ² I5000, z :Ž t .r² I1 z :Ž0., the total polarization is split into many different sub-packets that travel with individual group velocities
nks
d´k dk
s yD sin k ,
Ž 19 .
where ´ k and k have been defined in Eqs. Ž2. and Ž5., respectively. The dispersion of group velocities leads to the emergence of a discrete number of wave packets manifested in the characteristic pattern of Fig. 3. For longer times Žnot shown., the spreading of the wave packets progresses, and new sub-packets emerge. Their interference leads to an increasingly erratic behavior similar to that in Figs. 1 and 2.
303
obtains for the average polarizations Žfor t 4 T . Žsee also Ref. w5x. ² Im z :
s
² Ij z : Ž 0.
Tr s Im z 4 Tr I j2z
4
1 s
1 q
Nq1
2 Ž N q 1.
Ž dm , j q dm , Nq1yj . . Ž 21 .
Thus the average polarizations of the initially polarized spin I j and of its symmetric partner INq1yj remain 50% higher than the ones of all other spins.
6. Conclusions The exact solution and computer calculations of the dynamics of a linear spin 1r2 chain under the XY Hamiltonian, investigated in this Letter, show that the evolution is confined to a Liouville subspace with a dimension that grows quadratically with the number spins. The dynamics of the system can be rationalized by the propagation of a growing number of spin-wave packets that are reflected at the chain ends. The interference of spin-wave packets leads to a transition from a regular to an erratic behavior. No ‘equilibrium’ is established even in the limit of large chains and long times.
5. Long-term averages Acknowledgements The density operator s Ž t . of Eq. Ž7. oscillates about the average state s , which is determined by
s s lim t™`
1
t
H s Ž t . dt t 0
2
s
1
Ý sin2 Ž kj . bk† bk y 2 , Nq1
Ž 20 .
k
where from Eq. Ž7. only the time-independent terms with k s kX remain. Since Tr s 2 Ž0.4 ) Tr s 2 4 , s corresponds to a hypothetical state that cannot be reached under unitary evolution. s is neither proportional to H of Eq. Ž1. nor is it independent of the initial condition I j z , implying that also the time-independent part of s has a non-ergodic character. One
We thank Christoph Scheurer for providing a FORTRAN program that efficiently evaluates Eq. Ž18.. We are grateful to Professor J.S. Waugh for sending a preprint of Ref. w5x. The work has been supported by the Swiss National Science Foundation and the Russian Foundation for Fundamental Investigations ŽGrant No. 98-03-33151..
References w1x N. Bloembergen, Physica 15 Ž1949. 386. w2x R. Tycko, G. Dabbagh, J. Am. Chem. Soc. 115 Ž1991. 3592. w3x P. Robyr, B.H. Meier, P. Fischer, R.R. Ernst, J. Am. Chem. Soc. 116 Ž1994. 5315.
304
E.B. Fel’dman et al.r Chemical Physics Letters 294 (1998) 297–304
w4x R. Bruschweiler, R.R. Ernst, Chem. Phys. Lett. 264 Ž1997. ¨ 393. w5x J.S. Waugh, Mol. Phys. Žto be published.. w6x E.B. Fel’dman, S. Lacelle, J. Chem. Phys. 108 Ž1998. 4709. w7x E.H. Lieb, T. Schultz, D.C. Mattis, Ann. Phys. ŽParis. 16 Ž1961. 407. w8x H.B. Cruz, L.L. Goncalves, J. Phys. C: Solid State Phys. 14 Ž1981. 2785.
w9x Z.L. Madi, B. Brutscher, T. Schulte-Herbruggen, R. ´ ¨ Bruschweiler, R.R. Ernst, Chem. Phys. Lett. 268 Ž1997. 300. ¨ w10x P. Jordan, E. Wigner, Z. Phys. 47 Ž1928. 631. w11x M. Rance, Chem. Phys. Lett. 154 Ž1989. 242. w12x H.M. Pastawski, G. Uzaj, P.R. Levstein, Chem. Phys. Lett. 261 Ž1996. 329. w13x M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Hemisphere, New York, NY, 1987.
Chemical Physics Letters 294 Ž1998. 297–304
From regular to erratic quantum dynamics in long spin 1r2 chains with an XY Hamiltonian ) E.B. Fel’dman 1, R. Bruschweiler , R.R. Ernst ¨
2
Laboratorium fur Switzerland ¨ Physikalische Chemie, ETH Zentrum, CH-8092 Zurich, ¨ Received 22 July 1998
Abstract The dynamics of spin polarization driven by the XY Hamiltonian of a linear spin 1r2 chain of arbitrary length N was studied analytically and numerically. For an initial state with polarization localized on a single spin, I j z , the dimension of the accessible Liouville subspace grows with N 2. The polarization propagates along the chain in the form of spin-wave packets that gradually split into sub-packets bouncing back and forth at the ends. The initially regular appearance turns at longer times into an erratic behavior due to the complex interference of the spin-wave packets. Even for large N and long times the density operator exhibits large-amplitude fluctuations about its long-term average, which is non-ergodic in the sense that it is dependent on the initial conditions of the dynamics. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction The understanding of spin evolution in extended many-spin quantum systems is important both from a fundamental and an applied point of view w1–3x. A comparison between experiment and theory for the evolution of large systems is usually hampered by the fact that for most Hamiltonians no analytical solution of the Schrodinger equation can be obtained. ¨ Numerical computer simulations, on the other hand, suffer from the exponentially increasing dimension of the Hilbert space with the number of spins N.
) Corresponding author. E-mail: [email protected] 1 On sabbatical leave from Institute of Chemical Physics, Russian Academy of Sciences, 142432 Chernogolovka, Moscowskaya oblast, Russia. 2 E-mail: [email protected]
Often they permit simulations of not more than about a dozen of spins. Alternatively, it is tempting to interpret the long-term behavior of large spin systems in terms of spin thermodynamics. Such an approach can however be treacherous as closed quantum systems can exhibit a significant degree of non-ergodic behavior irrespective of the number of spins w4–6x. In this Letter, we investigate the dynamics of a spin system under the influence of an XY Hamiltonian w7x. In contrast to the dipolar and the isotropic coupling Hamiltonians, the XY Hamiltonian can be analytically diagonalized for an arbitrarily large number of spins w8x. We have also recently demonstrated that an XY Hamiltonian can be generated experimentally in scalar coupled spin chains by taking advantage of liquid-state NMR multiple-pulse techniques w9x. For single-spin Zeeman polarization I j z as the initial condition, we derive in this Letter a
0009-2614r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 0 8 8 7 - 2
E.B. Fel’dman et al.r Chemical Physics Letters 294 (1998) 297–304
298
complete analytical solution of the Liouville–von Neumann equation and discuss the results based on numerical calculations. Some characteristic features of the solution are rationalized by analytical approximations.
The sum over the ‘index’ k in Eq. Ž2. and in forthcoming equations implies that the sum is carried out over n s 1, 2, . . . , N. The quantity k can be viewed as a wave vector that is related to the spinwave behavior of the system w7x. For an initial condition of the density operator s with a single spin Ij polarized,
2. Analytical solution of the Liouville–von Neumann equation for a spin chain with an XY Hamiltonian
s Ž 0. s Ij z ,
We consider a linear chain of N spins I s 1r2 in a magnetic field B0 with identical secular flip–flop Ž XY . interactions exclusively between nearest neighbors. The Hamiltonian H of this system can be written as H s HZ q HX Y
there is an exact solution to the Liouville–von Neumann equation s˙ s yiw H , s x:
s Ž t. s
is1
Nq1
where HZ is the Zeeman Hamiltonian with v 0 s yg B0 and the gyromagnetic ratio g . HX Y is the XY Hamiltonian with nearest-neighbor interactions. Ii a Ž a s x, y, z . are the cartesian spin operators, and D is the uniform XY coupling constant. The Hamiltonian H can be diagonalised exactly using the Jordan–Wigner transformation method w10x:
´ k b k† b k y
k
1 2
Nv 0 ,
´ k s D cos k q v 0 ,
Ž 2. where N
Ý gk Ž j . cj .
Ž 3.
js1
The operators c j are Fermion operators w10x: c j s Ž y2 .
jy1
I1 z I2 z . . . I z , jy1 Iy j
Ž 4.
gk Ž j . s
ž
X
yi Ž ´ k y ´ k X . t 1 2
,
Ž 7.
js1
was used. The density operator s Ž t . evolves in a Liouville subspace of dimension N Ž N q 1.r2 that is spanned by hermitian linear combinations of the Žnon-normalized. operators
Iyp I pq1, z . . .
q yX X X IpXy1, z Iq p , I p I pq1, z . . . I p y1, z I p
with 1 F p F pX F N .
4 Ž 8.
Evolution under the XY Hamiltonian generates for each pair of adjoint operators Bj , Bj† 4 of Eq. Ž8. either the sum Bj q Bj† or the difference iŽ Bj y Bj† . but not both. For example, for p s pX , only the q q y difference Iy p I p y I p I p s y2 I p z is realized, while y q q y the sum I p I p q Ip I p s 1 cannot be accessed from the trace-free initial condition of Eq. Ž6. by unitary evolution. The expectation values of the single-spin operators Ip z as a function of time can be expressed by ² Ip z : Ž t .
2
4
1r2
s
/
sin Ž kj . , Nq1 pn with k s , n s 1, 2, . . . , N . Nq1
Ý sin Ž kj . sin Ž lj . s d k l
² Ij z : Ž 0.
and the function g k Ž j . is defined by 2
Ý exp k, k
N
2
Ý Ii z q D Ý Ž Ii x Iiq1, x q Ii y Iiq1, y . , Ž 1.
bk s
Nq1
=sin Ž kj . sin Ž kX j . b k† b k X y
Ny1
is1
Hs Ý
2
where the well-known orthogonality relationship
N
sv0
Ž 6.
Ž N q 1.
2
Ý exp Ž yi ´ k t . sin Ž kj . sin Ž kp .
,
k
Ž 9. Ž 5.
where ² I p z :Ž t . s Tr s Ž t . I p z 4 . It directly follows that
E.B. Fel’dman et al.r Chemical Physics Letters 294 (1998) 297–304
² Ip z :Ž t .r² Ij z :Ž0. G 0. Thus, in contrast to the dipolar and the isotropic Hamiltonian, a sign change of the single-spin polarizations cannot occur w4,11x. Eq. Ž9. is symmetric in the initial spin Ij and the target spin Ip , implying that the transfers Ij ™ Ip and Ip ™ Ij are equally effective. For the multi-spin terms of Eq. Ž8. one obtains, using Eq. Ž7., qX : ² Iy X p I pq1, z . . . I p y1, z I p Ž t .
² Ij z : Ž 0. X
s
Ž y1.
pq p q1
X
2 py p q3
Ž N q 1.
2
f Ž p . f ) Ž pX . ,
Ž 10 .
where 1 F p - pX F N and f Ž p . s Ý exp Ž yi ´ k t . sin Ž kj . sin Ž kp . .
Ž 11 .
k
The complex conjugate of f Ž pX . is denoted by f ) Ž pX .. Eq. Ž10. is a real-valued expression for p q pX s even and a purely imaginary expression for X p q pX s odd. Hence, it follows ² Iq p I pq1, z . . . I p y1, z X pq p y yX :Ž . ² I p Ipq1, z . . . IpXy1, z Iq X :Ž t .. Ip t s Žy1. p
3. Numerical evaluation The general solution given in Eq. Ž9. was numerically evaluated for different initially polarized spins, I j z . In all calculations the value D s 2 p P 4444 sy1 is assumed, which corresponds to an XY dipole–dip o le c o u p lin g o f p ro to n sp in s D s yŽ m 0r4 p .g 2 Ž hr2 p .Ž1rr 3 . with a nearest neigh˚ bor distance r s 3 A. Fig. 1 shows the evolution of ² I1 z :Ž t .r² I1 z :Ž0. for a spin chain with N s 10000 spins where at t s 0 only spin I1 is polarized, s Ž0. s I1 z . The polarization on spin I1 , ² I1 z :Ž t ., decays within less than 1 ms to zero, as shown in Fig. 1A. It reappears and rises sharply for a first time after a time delay T s 716 ms Žsee inset of Fig. 1A.. It is followed by an oscillation with increasing frequency and decreasing amplitude, reflecting the dispersive behavior of the evolution Žsee Section 4.. The longer time behavior, given in Fig. 1B and C, shows recurrent ‘echoes’
299
at multiples of T. This periodic pattern reflects the propagation of a spin wave bouncing back and forth at the chain ends. The recurrence time T can be estimated by Ts
2 Nr
2 Nr f
ng
2N
Ž 12 .
s rD
D
with the group velocity ng f rD w12x. After several thousand milliseconds, the periodic echo train gradually disappears and an irregular behavior becomes evident. Even at very long times, as shown in Fig. 1D, no ‘equilibration’ is observed as the root-meansquare fluctuation amplitude remains comparable to the time-averaged polarization ² I1 z : Ž t . :
(Ž² I
1z
: Ž t . y ² I1 z : Ž t . . ² I1 z : Ž t .
2
f 1.4 .
Ž 13 .
The dynamics depends on the position of the initially polarized spin, on the position of the observer spin, and on the chain length. In the examples presented in Figs. 1 and 2, the observer spin was chosen to be the initially polarized spin. The characteristic behavior of spins not at the chain end is exemplified for a spin chain with N s 1000 spins and s Ž0. s I158, z in Fig. 2A and one with N s 10000 spins with s Ž0. s I1580, z in Fig. 2B. The evolution starts with two wave packets that propagate in opposite directions. For ² I158, z :Ž t .r² I158, z :Ž0., four different phases of evolution, labeled I–IV, are visible in Fig. 2A: the first three phases display complicated but nearly regular burst patterns while the last phase shows an erratic behavior that is reminiscent of ‘quantum chaos’. The first recurring burst arriving at time 158Tr1000 originates from reflection at spin I1. The reflection at spin I1000 gives rise to a further burst at 842Tr1000. The third burst at T s 71.6 ms is due to sequential reflections at both chain ends. It is followed by the erratic phase IV due to interference of the different wave packets. An increase of the chain length by a factor 10 ŽFig. 2B. leads to qualitatively the same behavior with the same types of phases with a ten-fold extended time scale. Spin dynamics under the XY Hamiltonian markedly differs from evolution under the dipolar
300 E.B. Fel’dman et al.r Chemical Physics Letters 294 (1998) 297–304 Fig. 1. Time-course of ² I1 z :Ž t .r² I1 z :Ž0. for a linear chain of N s10000 spins under the XY Hamiltonian with the initial condition s Ž0. s I1 z . ŽA. The short-time behavior. The inset shows the first recurrence of the spin-wave packets. ŽB. – ŽD. behavior for increasingly longer times. All calculations have been performed based on Eq. Ž9.. For the initial decay of Fig. 1A the simplified Eq. Ž18. is equivalent to Eq. Ž9..
E.B. Fel’dman et al.r Chemical Physics Letters 294 (1998) 297–304
Hamiltonian: While the XY Hamiltonian involves a Liouville subspace of dimension N Ž N q 1.r2, the dipolar Hamiltonian populates a Liouville subspace
301
of dimension 4 Ny 1, which rapidly leads to a smooth and ‘diffusion-like’ evolution of polarization exhibiting low-amplitude oscillations about the average
Fig. 2. ŽA. Time-course of ² I158, z :Ž t .r² I158, z :Ž0. in a 1000-spin chain under the XY Hamiltonian with the initial condition s Ž0. s I158, z . Four different phases I–IV are indicated at the top of the figure. ŽB. Time-course of ² I1580, z :Ž t .r² I1580, z :Ž0. in a 10000-spin chain with s Ž0. s I1580, z . The plots were calculated based on Eq. Ž9.. In both panels the curves start at t s 0 with amplitude 1.
E.B. Fel’dman et al.r Chemical Physics Letters 294 (1998) 297–304
302
density operator w1,4x. Dipolar evolution in linear chains also does not show pronounced echo recurrences of the kind observed in Fig. 1. Evolution under the isotropic Hamiltonian qualitatively resembles also the evolution under the dipolar Hamiltonian w9x.
the first reflection at one of the chain ends has occurred. Thus for j s 1, i.e. for initial polarization of spin I1 , the expression is useful for t-
N D
.
Ž 15 .
Combining Eq. Ž14. with Eq. Ž9. yields ² Im z : Ž t . ² Ij z : Ž 0.
4. Useful approximations for long chains
m
2 f Jjym Ž Dt . y 2 Ž y1. Jjym Ž Dt . 2 =J jqm Ž Dt . q Jjqm Ž Dt . ,
It is possible to analytically assess the fine-structure of the spin-wave evolution by replacing in Eq. Ž9. the summation by an integration:
where Jjy mŽ Dt . is the Bessel function of the first kind of order j y m. Application of the well-known identity of Bessel functions w13x, 2m
Ý exp Ž yi ´ k t . sin Ž kj . sin Ž km .
x
k
Ž 16 .
Jm Ž x . s Jmy1 Ž x . q Jmq1 Ž x . ,
Ž 17 .
yields for j s 1 the result N f p
p
H0 exp Ž yi ´
kt
. sin Ž kj . sin Ž km . d k .
Ž 14 .
This approximation is good for the evolution before
² Im z : Ž t . ² I1 z : Ž 0 .
4 m2 s
D2 t 2
Jm2 Ž Dt . .
Ž 18 .
It follows for an observer spin Im remote from the
Fig. 3. Time-course of ² I5000, z :Ž t .r² I1 z :Ž0. in a 10000-spin chain with s Ž0. s I1 z . The calculations were performed using the exact Eq. Ž9. and the alternative Eq. Ž18.. No difference between the resulting curves can be observed for the plotted time interval.
E.B. Fel’dman et al.r Chemical Physics Letters 294 (1998) 297–304
other chain end, m < N, that the envelope of the polarization ² Im z :Ž t . decays in time proportionally to ty3 . For t F 0.25 ms in Fig. 1A, Eq. Ž18. gives virtually the same result as Eq. Ž9.. Fig. 3 shows the decay of ² I5000, z :Ž t .r² I1 z :Ž0. of a 10000-spin chain for s Ž0. s I1 z . The approximation of Eq. Ž18. is superimposed on the exact result based on Eq. Ž9.. No deviation is visible, which shows that Eq. Ž18. represents a very good approximation of the exact behavior. The decay of ² I1 z :Ž t .r² I1 z :Ž0. at short times shown in Fig. 1A can be attributed to the evolution of a single wave packet. On the other hand for the appearance and decay of ² I5000, z :Ž t .r² I1 z :Ž0., the total polarization is split into many different sub-packets that travel with individual group velocities
nks
d´k dk
s yD sin k ,
Ž 19 .
where ´ k and k have been defined in Eqs. Ž2. and Ž5., respectively. The dispersion of group velocities leads to the emergence of a discrete number of wave packets manifested in the characteristic pattern of Fig. 3. For longer times Žnot shown., the spreading of the wave packets progresses, and new sub-packets emerge. Their interference leads to an increasingly erratic behavior similar to that in Figs. 1 and 2.
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obtains for the average polarizations Žfor t 4 T . Žsee also Ref. w5x. ² Im z :
s
² Ij z : Ž 0.
Tr s Im z 4 Tr I j2z
4
1 s
1 q
Nq1
2 Ž N q 1.
Ž dm , j q dm , Nq1yj . . Ž 21 .
Thus the average polarizations of the initially polarized spin I j and of its symmetric partner INq1yj remain 50% higher than the ones of all other spins.
6. Conclusions The exact solution and computer calculations of the dynamics of a linear spin 1r2 chain under the XY Hamiltonian, investigated in this Letter, show that the evolution is confined to a Liouville subspace with a dimension that grows quadratically with the number spins. The dynamics of the system can be rationalized by the propagation of a growing number of spin-wave packets that are reflected at the chain ends. The interference of spin-wave packets leads to a transition from a regular to an erratic behavior. No ‘equilibrium’ is established even in the limit of large chains and long times.
5. Long-term averages Acknowledgements The density operator s Ž t . of Eq. Ž7. oscillates about the average state s , which is determined by
s s lim t™`
1
t
H s Ž t . dt t 0
2
s
1
Ý sin2 Ž kj . bk† bk y 2 , Nq1
Ž 20 .
k
where from Eq. Ž7. only the time-independent terms with k s kX remain. Since Tr s 2 Ž0.4 ) Tr s 2 4 , s corresponds to a hypothetical state that cannot be reached under unitary evolution. s is neither proportional to H of Eq. Ž1. nor is it independent of the initial condition I j z , implying that also the time-independent part of s has a non-ergodic character. One
We thank Christoph Scheurer for providing a FORTRAN program that efficiently evaluates Eq. Ž18.. We are grateful to Professor J.S. Waugh for sending a preprint of Ref. w5x. The work has been supported by the Swiss National Science Foundation and the Russian Foundation for Fundamental Investigations ŽGrant No. 98-03-33151..
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w9x Z.L. Madi, B. Brutscher, T. Schulte-Herbruggen, R. ´ ¨ Bruschweiler, R.R. Ernst, Chem. Phys. Lett. 268 Ž1997. 300. ¨ w10x P. Jordan, E. Wigner, Z. Phys. 47 Ž1928. 631. w11x M. Rance, Chem. Phys. Lett. 154 Ž1989. 242. w12x H.M. Pastawski, G. Uzaj, P.R. Levstein, Chem. Phys. Lett. 261 Ž1996. 329. w13x M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Hemisphere, New York, NY, 1987.