We present an exact asymptotic solution for electron transition amplitudes in
an infinite linear chain driven by an external homogeneous time-dependent
electric field. This solution extends the Landau-Zener theory for the case of
infinite number of states in discrete spectrum. In addition to transition
amplitudes we calculate an effective diffusion constant.Comment: 3 figure

A. Garg

A. Onipko

C. Kergueris

C. Zener

C.E. Carroll

D.S. Crothers

D.W. Hone

F.T. Hioe

G.H. Wannier

H. Fukuyama

H. Ness

L.D. Landau

M. Holthaus

N. A. Sinitsyn

S. Brundobler

S.G. Devison

V. L. Pokrovsky

V.N. Ostrovsky

W. Wernsdorfer

W.V. Houston

Yu.N. Demkov

English

arXiv

Journals published by the American Physical Society can be found at http://journals.aps.org/We present an exact asymptotic solution for electron transition amplitudes in an infinite linear chain driven by an external time-dependent electric field. This solution extends the Landau-Zener theory for the case of an infinite number of states in the discrete spectrum. In addition to the transition amplitudes we calculate the effective diffusion constant

Pokrovsky, Valery L.

Sinitsyn, NA.

Texas A&M University

PHYSICAL REVIEW B, VOLUME 65, 153105Landau-Zener transitions in a linear chain
V. L. Pokrovsky1,2 and N. A. Sinitsyn1
1Department of Physics, Texas A&M University, College Station, Texas 77843-4242
2Landau Institute of Theoretical Physics, Chernogolovka, Moscow Region 142432, Russia
~Received 19 September 2001; published 4 April 2002!
We present an exact asymptotic solution for electron transition amplitudes in an infinite linear chain driven
by an external time-dependent electric field. This solution extends the Landau-Zener theory for the case of an
infinite number of states in the discrete spectrum. In addition to the transition amplitudes we calculate the
effective diffusion constant.
DOI: 10.1103/PhysRevB.65.153105 PACS number~s!: 72.10.2dLandau-Zener ~LZ! theory1,2 treats a quantum system
placed in a slowly varying external field. If such a system
was prepared in a state of its discrete spectrum, it adiabati-
cally follows this state until its time-dependent energy level
crosses another one. Near the crossing point the adiabaticity
can be violated and the system can escape from the state it
occupied initially to another one. Landau and Zener found
the transition probability for two-level crossing. The crossing
of more than two levels at the same time is generally an
unlikely coincidence. However, in some systems such a mul-
tilevel crossing may occur systematically, due to the high
symmetry of the underlying Hamiltonian. The transition ma-
trix for special cases of multilevel crossing was studied in
Refs. 3–8. Presently only a few exact results for multilevel
crossing are known. One of them relates to a multiplet of
atomic electronic states with a total spin S or total rotational
moment J larger than 1/2 in a varying external magnetic
field.3,4 The Zeeman splitting between 2S11 or 2J11 lev-
els regularly vanishes at nodes of the magnetic field. Another
exactly solvable model displaying multilevel crossing is the
so-called bow-tie model,5 whose physical interpretation is
not obvious.
Since its creation in 1932, LZ theory has had numerous
applications. They include molecular predissociation,10 slow
atomic and molecular collisions,11 and electron transfer in
biomolecules.12 Recently Wernsdorfer et al.13,14 employed
the LZ theory to describe consistently the steplike shape of
the hysteresis loop in special molecules with large magnetic
moments called nanomagnets. Using the LZ probability for-
mula these authors were able to find the extremely small
tunnel splitting of the classic degenerate ground states and
even to reveal oscillations of this value in an external mag-
netic field. This beautiful experiment, together with its clever
treatment, is a triumph of quantum mechanics and, in par-
ticular, LZ theory.
The problem considered in this article is closely related to
another application of LZ theory: electronic transfer in
donor-acceptor complexes.9 In this process of biological and
chemical importance, an electron tunnels between initial and
final positions through a long chain of identical sites. There
are two limiting cases for such a process. In the first case
there is no coherence between two sequential tunneling pro-
cesses connecting nearest-neighbor sites. In this case the
probability of tunneling through several sites is very small in
comparison to that for one-site tunneling. This limiting case0163-1829/2002/65~15!/153105~4!/$20.00 65 1531was studied earlier.9 We consider the opposite limiting case
in which the sequential tunneling processes are highly coher-
ent and tunneling through many sites becomes available.
If the coherence between LZ transitions is lost, the prob-
lem is reduced to the multiplication and addition of prob-
abilities, each described by a proper LZ expression. The
price we must pay for incorporating the coherence between
different transitions is a strong reduction of the class of quan-
tum systems considered. The number of crossing levels in
such systems must be infinite. The hopping amplitudes from
a site to its neighbors must be all identical. Physically it
describes the quantum electron transfer between a donor and
acceptor separated by a long polymer strand ~molecular
bridge!. The bridge can be considered as a linear array of
identical sites. Such one-dimensional atomic-scale
wires were intensely studied, both experimentally and
theoretically.15–17 Our results can be also applied
to transitions among electron states in semiconductor
superlattices.18,19
We study the tunneling of a particle in such systems
driven by a time-dependent homogeneous external field. An
important assumption is that all molecular fragments in the
chain are identical. An electric field splits the energy levels at
different sites of the chain and suppresses the transitions,
which occur within a narrow intervals about times when the
electric field becomes zero. Since the tunneling is a fast pro-
cess, we disregard the oscillatory relaxation originating from
phonons and other elementary excitations.
Let denote un& a state located at the nth site of the chain.
We assume that these states form a complete orthonormal set
~Wannier basis!. In terms of this set the electron Hamiltonian
reads
Hˆ 5 (
n51
N
~gun&^n11u1c.c.!1F~ t !nun&^nu, ~1!
F~ t !5eE~ t !a ,
where E(t) is the electric field, e is the electron charge, a is
the distance between sites, and g is the coupling constant
~hopping amplitude!. A series of exact solutions for the time-
dependent Shro¨dinger equation with the Hamiltonian ~1! for
N5‘ , known as drifting plane waves, was found long©2002 The American Physical Society05-1
BRIEF REPORTS PHYSICAL REVIEW B 65 153105ago.19–23 Below we solve the same problem for fixed initial
conditions, thus resolving the multistate LZ problem.
The states un& are conventionally called the diabatic
states. They are the eigenstates of the diagonal part of this
Hamiltonian H05F(t)nun&^nu. The eigenstates of the total
Hamiltonian ~1! depending on t or F(t) as parameters are
called adiabatic states. Until uF(t)u@g , the diabatic levels
are close to the adiabatic ones and the transitions between
levels are suppressed. This is the adiabatic regime. The adia-
baticity is violated in the vicinity of the electric field nodes
determined by the inequality uF(t)u<g , where all transitions
proceed. By level crossing we mean that the diabatic levels
cross and the exact eigenvalues of the Hamiltonian ~1! never
cross, in accordance with the Wigner-Neumann theorem. It is
convenient to place the time origin t50 directly at the node
of F(T). Since only a narrow vicinity of the node is substan-
tial for transitions, the exact dependence of the field on time
can be reasonably approximated by a linear one: F(t)
’F˙ (0)t . At zero electric field E and free boundary condi-
tions the Hamiltonian ~1! can be diagonalized analytically.
Its spectrum is
« j52g cos~p j /N !, j51, . . . ,N . ~2!
For nonzero fields we have found the adiabatic eigenval-
ues numerically. The result is shown in Fig. 1 for a finite
chain with 15 sites. For comparison the diabatic levels are
depicted in the same figure.
We proceed to solve the time-dependent Schro¨dinger
equation with the Hamiltonian ~1!. Its matrix representation
reads
Hnm5nF˙ ~0 !tdnm1g~dm ,n111dm ,n21!. ~3!
For an infinite chain (N→‘), Eq. ~3! is valid for all n and m.
After a proper rescaling of time the Hamiltonian ~3! becomes
dimensionless:
Hmn5ntdmn1g~dm ,n111dm ,n21!. ~4!
FIG. 1. Dependence of eigenvalues of the Hamiltonian ~1! on E.
Solid lines show adiabatic energies as a function of E; dashed lines
depict diabatic energies. While the diabatic levels intersect in one
point; the adiabatic levels do not intersect.15310It depends on only one dimensionless number g5g/AF˙ (0),
which is the Landau-Zener parameter. Let the time-
dependent state vector be ua ,t&5(ncn(t)un& . Then the sys-
tem of equations for the amplitudes cn(t) reads
ic˙ n5ntcn1g~cn211cn11!. ~5!
The transition matrix element Tn ,n8 should be identified with
the t→1‘ asymptote of an amplitude cn(t) for a solution
obeying the initial condition ucm(t)u25dm ,n8 at t→2‘ .
Since all cm(t) except of cn8(t) are zero at t→2‘ , the initial
condition can be more explicitly written as
cm~ t→2‘!5dm ,n8exp~2in8t2/2!. ~6!
We multiply the asymptotic values of cn(t) by exp@int2/2# to
remove strongly oscillating phase factors from Tn ,n8 .
Now introduce an auxiliary function u(w ,t)
5(n52‘
‘ cn(t)einw. The system ~5! is equivalent to the fol-
lowing equation in partial derivatives for u(w ,t):
]u
]t
1t
]u
]w
12igu cos w50. ~7!
The initial condition ~6! is equivalent to the initial condition
u(w ,t→2‘)→exp@in8(2t2/21w)# . Given the solution
u(w ,t), the amplitudes cn(t) can be found by the inverse
Fourier transformation cn(t)5(1/2p)*02pu(w ,t)e2inwdw .
The solution of Eq. ~7! that obeys proper boundary condi-
tions is
u~w ,t !5expH 2iF 2gE
2‘
t
cosS w2 t22 1 t822 D dt8
1n8S w2 t22 D G J . ~8!
Putting t51‘ in the solution ~8! and taking the inverse
Fourier transform, we arrive at following asymptotic values:
cn~ t !’expS 2 int22 1i ~n82n !p4 D J un2n8u~2A2pg !. ~9!
Thus, the scattering amplitudes in terms of modified states,
with the fast phase factor exp(2int2/2) incorporated, are
Tn ,n8[^nuTun8&5e
i(n82n)p/4J un2n8u~2A2pg !, ~10!
where the operator T is expressed in terms of the evolution
operator U(t ,t8) for the Hamiltonian ~4! in the interaction
representation:
T5 lim
t→‘ ,t8→2‘
expS iE
t8
0
H0~t!dt D U~ t ,t8!
3expS iE
0
t
H0~t!dt D . ~11!
The matrix elements Tn ,n8 display an infinite number of os-
cillations with the LZ parameter g. However, for large5-2
BRIEF REPORTS PHYSICAL REVIEW B 65 153105un2n8u the oscillations start with g.un2n8u. These oscilla-
tions can be observed experimentally by varying the field
sweep rate E˙ (0). For small values of g the amplitudes are
small and quickly decrease with growing un2n8u. In Fig. 2
we depict transition probabilities for several levels n closest
to the initial one n8 versus the Landau-Zener parameter g.
Figure 3 shows the dependence of the transition amplitude
on un2n8u at a fixed value of g.
For large g@un2n8u the asymptotic values of the ampli-
tudes ~10! are
^nuTun8&;
ei(n82n)p/4
~A2p3g !1/2
cosS 2A2pg2 ~n2n8!p2 2 p4 D .
~12!
It is instructive to compare this result with other exactly
solvable generalized Landau-Zener models. Most of them
refer to systems with a finite number of states N. In the limit
g@N the transition probabilities behave like an exponent
exp@2C(n,n8)g2#, where the C(n ,n8) do not depend on g. In
contrast, the result ~12! displays a power law with oscilla-
tions instead of an exponential dependence on g for large g.
This is the manifestation of quantum interference of different
Feynman trajectories, which are discrete in the chain. A step
FIG. 2. Transition probabilities to sites closest to an initially
filled one as functions of g: ~a! un2n8u50, ~b! un2n8u51, ~c!
un2n8u52, and ~d! un2n8u53.
FIG. 3. Transition probabilities vs n2n8 at a fixed g51.6. Solid
line is a guide for the eye.15310in a trajectory has average length g ~see below!. Such a step
cannot be realized in a system with a finite number of states
if g@N .
The mean square displacement at one crossing event is
^~n2n8!2&5 (
n52‘
‘
~n2n8!2uJ un2n8u~2A2pg !u
254pg2.
~13!
If the external field is periodic in time and the coherence
between crossing events is lost,24 the electron performs a
random walk; i.e., it diffuses. Assume the field to oscillate
harmonically as F(t)5F0sin(vt). At the nodes tk5pk/v
(k is an integer! all diabatic levels cross together. The
squared Landau-Zener parameter is g25g2/F0v . The diffu-
sion coefficient is D52a2^(n2n8)2&/T , where T52p/v is
the period of oscillations and the factor of 2 accounts for two
crossing events per period. Collecting these results and Eq.
~13!, we find
D5
4ag2
F0
. ~14!
This result does not depend on the frequency of the external
field.
The theory can be extended to a more general Hamil-
tonian incorporating hopping between any two sites, but con-
serving translational invariance:
H5(
m ,n
Hm ,num&^nu,
Hmn5ntdmn1g um2nu , g2k5gk* . ~15!
For simplicity we present below the result for real hopping
amplitudes gk5g2k :
^nuTun8&5
ei(n82n)p/4
2p E0
2p
exp@2i2A2p f ~w ,g j!
1i~n82n !w#dw , ~16!
where f (w ,g j)5(k(gk /Ak)cos kw.
The model ~1! can be generalized also to incorporate in-
ternal degrees of freedom of identical chain fragments. In
this case the local states are described by amplitudes an ,a
with two indices. The first index n denotes the position and
the second index a labels the inner states. The Schro¨dinger
equation for the amplitudes then reads
ia˙ n ,a5~nt1ea1dat !an ,a1(
b
ga ,b~an11,b1an21,b!,
~17!
where the indexes a ,b51, . . . ,Nint run over the internal
states of the molecular wire segment. Changing to variables
an ,a5bn ,ae2int
2/2 we eliminate the term proportional to t in
Eq. ~17!. Introducing a new function ua(w)
5(n52‘
‘ bn ,aeinw, we reduce the infinite system ~17! to a
finite set of Nint ordinary differential equations:5-3
BRIEF REPORTS PHYSICAL REVIEW B 65 153105iu˙ a5~ea1dat !ua12 cos~ t21w!(
b
ga ,bub , ~18!
in which w plays the role of a parameter. The initial condi-
tions are bn ,a(t→2‘)5exp(2idat2/2)dn ,n8 and ua(w ,t→
2‘)5exp@(2idat2/2)1in8w# . Thus, the variable ~param-
eter! w enters not only in the system ~18!, but also in the
initial conditions. This system must be solved for all values
of parameter w in the interval (0,2p). The inverse Fourier
transformation yields the evolution operator just as for the
case Nint51. An analytical solution of the system ~18! is
possible for some special choices of the parameters «a , da ,
and ga ,b . For example, two identical coupled chains corre-
spond to Nint52. Then the indices a ,b take on the values
1,2. The simplest solvable choice of parameters is «150,
«25«; d150, d25d; g1,15g2,25g; g1,25g2,15g8. An ex-
act solution of this model can be reduced to ones solved in
this article together with the solvable two-level LZ model.
In conclusion, we have generalized the LZ theory to an
infinite number of crossing levels. Physically it describes an
electron on an infinite chain subject to a time-dependent
electric field. The high symmetry of the problem allows us to
find not only the asymptotics, but also the intermediate val-
ues of the amplitudes. Our solution is valid for an infinite
chain with translational symmetry. We demonstrated it for
the case of a simple primitive cell, but it can be generalized
and in some cases exactly solved even if the primitive cell
contains more than one site.
Finite-size effects do not permit us to apply our result ~10!
directly to the transition amplitude from the first to the last
site of the chain even if it contains many sites. However, our
calculation of the diffusion coefficient ~14! is valid since
diffusion presumably proceeds far from the ends of the
chain. Certainly, we assume that coherence is lost during the
time interval between two sequential crossing events.15310Our solution demonstrates a phenomenon that is probably
common for most systems with multilevel crossing: oscilla-
tions of the transition probabilities as a function of the LZ
parameter and site position ~distance between diabatic lev-
els!. However, their asymptotic values for large values of the
LZ parameter differ from those for other solvable multistate
LZ models with a finite number of states. We expect that in a
general situation with N@1 crossing levels, the transition
probabilities will behave similarly to those found in this
work for 1!g!N , provided that the initially occupied states
are far enough from the diabatic spectrum boundaries.
Finally we discuss the relationship between our problem
and a typical problem for semiconductor superlattices.18,19
The latter is associated with Anderson localization. The di-
abatic levels at sites are randomly distributed. In one and two
dimensions all sites are localized. If the width of the energy
distribution D is much less than the tunneling amplitude g ,
the localization length in one dimension is ag/D . To enhance
the tunneling through a chain it is reasonable to apply a
time-dependent electric field. The electric field is substantial
if Fg/D>D where F is a typical value of F(t). Our approxi-
mation is valid if the inequality is strong: Fg/D@D . Tunnel-
ing transitions in the field proceed during an interval of time
t defined by relation F(t);g . The value F(t) can be ac-
cepted for F. We see that the strong inequality g@D guar-
antees the existence of the strong-field limit in which the
randomness of levels can be ignored. This requirement does
not impose any limitations on the LZ parameter g.
We thank W. Saslow for a critical reading of the manu-
script. This work was supported by the NSF under Grant No.
DMR 0072115 and by the DOE under Grant No. DE-FG03-
96ER45598. One of us ~V.P.! acknowledges the support of
the Humboldt Foundation.1 L.D. Landau, Phys. Z. Sowjetunion 2, 46 ~1932!.
2 C. Zener, Proc. R. Soc. London, Ser. A 137, 696 ~1932!.
3 F.T. Hioe, J. Opt. Soc. Am. B 4, 1327 ~1987!.
4 C.E. Carroll and F.T. Hioe, J. Phys. A 19, 1151 ~1986!.
5 V.N. Ostrovsky and H. Nakamura, J. Phys. A 30, 6939 ~1997!.
6 Yu.N. Demkov and V.I. Osherov, Zh. Exp. Teor. Fiz. 53, 1589
~1967! @Sov. Phys. JETP 26, 916 ~1968!#.
7 Yu.N. Demkov and V.N. Ostrovsky, J. Phys. B 28, 403 ~1995!.
8 S. Brundobler and V. Elser, J. Phys. A 26, 1211 ~1993!.
9 V. May and O. Kuhn, Charge and Energy Transfer Dynamics in
Molecular Systems ~Wiley—VCH-Verlag, Berlin, 2000!.
10 L.D. Landau and E.M. Lifshitz, Quantum Mechanics ~Pergamon,
Oxford, 1967!.
11 D.S. Crothers and J.G. Huges, J. Phys. B 10, L557 ~1977!.
12 A. Garg, N.J. Onuchi, and V. Ambegaokar, J. Chem. Phys. 83,
4491 ~1985!.
13 W. Wernsdorfer and R. Sessoli, Science 284, 133 ~1999!.
14 W. Wernsdorfer, T. Ohm, C. Sangregorio, R. Sessoli, D. Mailly,and C. Paulsen, Phys. Rev. Lett. 82, 3903 ~1999!.
15 C. Kergueris et al., Phys. Rev. B 59, 12 505 ~1999!.
16 H. Ness and A.J. Fisher, Appl. Surf. Sci. 162-163, 613 ~2000!.
17 A. Onipko, Yu. Klymenko, L. Malysheva, and S. Stafstrom, Solid
State Commun. 108, 555 ~1998!.
18 M. Holthaus, G.H. Ristow, and D.W. Hone, Phys. Rev. Lett. 75,
3914 ~1995!.
19 D.W. Hone and X. Zhao, Phys. Rev. B 53, 4834 ~1996!.
20 W.V. Houston, Phys. Rev. 57, 184 ~1940!.
21 G.H. Wannier, Phys. Rev. 117, 432 ~1960!.
22 S.G. Devison, R.A. Miskovic, F.O. Goodman, A.T. Amos, and
B.L. Burrows, J. Phys.: Condens. Matter 9, 6371 ~1997!.
23 H. Fukuyama, R.A. Ban, and H.C. Fogedby, Phys. Rev. B 8, 5579
~1973!.
24 Do not confuse the decoherence between different crossings
events with the decoherence between tunneling on different dis-
tances during one crossing.5-4


Landau-Zener transitions in a linear chain

Crossref

Texas A&amp;M Repository

PHYSICAL REVIEW B, VOLUME 65, 153105Landau-Zener transitions in a linear chainV. L. Pokrovsky1,2 and N. A. Sinitsyn11Department of Physics, Texas A&M University, College Station, Texas 77843-42422Landau Institute of Theoretical Physics, Chernogolovka, Moscow Region 142432, Russia~Received 19 September 2001; published 4 April 2002!We present an exact asymptotic solution for electron transition amplitudes in an infinite linear chain drivenby an external time-dependent electric field. This solution extends the Landau-Zener theory for the case of aninfinite number of states in the discrete spectrum. In addition to the transition amplitudes we calculate theeffective diffusion constant.DOI: 10.1103/PhysRevB.65.153105 PACS number~s!: 72.10.2dLandau-Zener ~LZ! theory1,2 treats a quantum systemplaced in a slowly varying external field. If such a systemwas prepared in a state of its discrete spectrum, it adiabati-cally follows this state until its time-dependent energy levelcrosses another one. Near the crossing point the adiabaticitycan be violated and the system can escape from the state itoccupied initially to another one. Landau and Zener foundthe transition probability for two-level crossing. The crossingof more than two levels at the same time is generally anunlikely coincidence. However, in some systems such a mul-tilevel crossing may occur systematically, due to the highsymmetry of the underlying Hamiltonian. The transition ma-trix for special cases of multilevel crossing was studied inRefs. 3–8. Presently only a few exact results for multilevelcrossing are known. One of them relates to a multiplet ofatomic electronic states with a total spin S or total rotationalmoment J larger than 1/2 in a varying external magneticfield.3,4 The Zeeman splitting between 2S11 or 2J11 lev-els regularly vanishes at nodes of the magnetic field. Anotherexactly solvable model displaying multilevel crossing is theso-called bow-tie model,5 whose physical interpretation isnot obvious.Since its creation in 1932, LZ theory has had numerousapplications. They include molecular predissociation,10 slowatomic and molecular collisions,11 and electron transfer inbiomolecules.12 Recently Wernsdorfer et al.13,14 employedthe LZ theory to describe consistently the steplike shape ofthe hysteresis loop in special molecules with large magneticmoments called nanomagnets. Using the LZ probability for-mula these authors were able to find the extremely smalltunnel splitting of the classic degenerate ground states andeven to reveal oscillations of this value in an external mag-netic field. This beautiful experiment, together with its clevertreatment, is a triumph of quantum mechanics and, in par-ticular, LZ theory.The problem considered in this article is closely related toanother application of LZ theory: electronic transfer indonor-acceptor complexes.9 In this process of biological andchemical importance, an electron tunnels between initial andfinal positions through a long chain of identical sites. Thereare two limiting cases for such a process. In the first casethere is no coherence between two sequential tunneling pro-cesses connecting nearest-neighbor sites. In this case theprobability of tunneling through several sites is very small incomparison to that for one-site tunneling. This limiting case0163-1829/2002/65~15!/153105~4!/$20.00 65 1531was studied earlier.9 We consider the opposite limiting casein which the sequential tunneling processes are highly coher-ent and tunneling through many sites becomes available.If the coherence between LZ transitions is lost, the prob-lem is reduced to the multiplication and addition of prob-abilities, each described by a proper LZ expression. Theprice we must pay for incorporating the coherence betweendifferent transitions is a strong reduction of the class of quan-tum systems considered. The number of crossing levels insuch systems must be infinite. The hopping amplitudes froma site to its neighbors must be all identical. Physically itdescribes the quantum electron transfer between a donor andacceptor separated by a long polymer strand ~molecularbridge!. The bridge can be considered as a linear array ofidentical sites. Such one-dimensional atomic-scalewires were intensely studied, both experimentally andtheoretically.15–17 Our results can be also appliedto transitions among electron states in semiconductorsuperlattices.18,19We study the tunneling of a particle in such systemsdriven by a time-dependent homogeneous external field. Animportant assumption is that all molecular fragments in thechain are identical. An electric field splits the energy levels atdifferent sites of the chain and suppresses the transitions,which occur within a narrow intervals about times when theelectric field becomes zero. Since the tunneling is a fast pro-cess, we disregard the oscillatory relaxation originating fromphonons and other elementary excitations.Let denote un& a state located at the nth site of the chain.We assume that these states form a complete orthonormal set~Wannier basis!. In terms of this set the electron HamiltonianreadsHˆ 5 (n51N~gun&^n11u1c.c.!1F~ t !nun&^nu, ~1!F~ t !5eE~ t !a ,where E(t) is the electric field, e is the electron charge, a isthe distance between sites, and g is the coupling constant~hopping amplitude!. A series of exact solutions for the time-dependent Shro¨dinger equation with the Hamiltonian ~1! forN5‘ , known as drifting plane waves, was found long©2002 The American Physical Society05-1BRIEF REPORTS PHYSICAL REVIEW B 65 153105ago.19–23 Below we solve the same problem for fixed initialconditions, thus resolving the multistate LZ problem.The states un& are conventionally called the diabaticstates. They are the eigenstates of the diagonal part of thisHamiltonian H05F(t)nun&^nu. The eigenstates of the totalHamiltonian ~1! depending on t or F(t) as parameters arecalled adiabatic states. Until uF(t)u@g , the diabatic levelsare close to the adiabatic ones and the transitions betweenlevels are suppressed. This is the adiabatic regime. The adia-baticity is violated in the vicinity of the electric field nodesdetermined by the inequality uF(t)u<g , where all transitionsproceed. By level crossing we mean that the diabatic levelscross and the exact eigenvalues of the Hamiltonian ~1! nevercross, in accordance with the Wigner-Neumann theorem. It isconvenient to place the time origin t50 directly at the nodeof F(T). Since only a narrow vicinity of the node is substan-tial for transitions, the exact dependence of the field on timecan be reasonably approximated by a linear one: F(t)’F˙ (0)t . At zero electric field E and free boundary condi-tions the Hamiltonian ~1! can be diagonalized analytically.Its spectrum is« j52g cos~p j /N !, j51, . . . ,N . ~2!For nonzero fields we have found the adiabatic eigenval-ues numerically. The result is shown in Fig. 1 for a finitechain with 15 sites. For comparison the diabatic levels aredepicted in the same figure.We proceed to solve the time-dependent Schro¨dingerequation with the Hamiltonian ~1!. Its matrix representationreadsHnm5nF˙ ~0 !tdnm1g~dm ,n111dm ,n21!. ~3!For an infinite chain (N→‘), Eq. ~3! is valid for all n and m.After a proper rescaling of time the Hamiltonian ~3! becomesdimensionless:Hmn5ntdmn1g~dm ,n111dm ,n21!. ~4!FIG. 1. Dependence of eigenvalues of the Hamiltonian ~1! on E.Solid lines show adiabatic energies as a function of E; dashed linesdepict diabatic energies. While the diabatic levels intersect in onepoint; the adiabatic levels do not intersect.15310It depends on only one dimensionless number g5g/AF˙ (0),which is the Landau-Zener parameter. Let the time-dependent state vector be ua ,t&5(ncn(t)un& . Then the sys-tem of equations for the amplitudes cn(t) readsic˙ n5ntcn1g~cn211cn11!. ~5!The transition matrix element Tn ,n8 should be identified withthe t→1‘ asymptote of an amplitude cn(t) for a solutionobeying the initial condition ucm(t)u25dm ,n8 at t→2‘ .Since all cm(t) except of cn8(t) are zero at t→2‘ , the initialcondition can be more explicitly written ascm~ t→2‘!5dm ,n8exp~2in8t2/2!. ~6!We multiply the asymptotic values of cn(t) by exp@int2/2# toremove strongly oscillating phase factors from Tn ,n8 .Now introduce an auxiliary function u(w ,t)5(n52‘‘ cn(t)einw. The system ~5! is equivalent to the fol-lowing equation in partial derivatives for u(w ,t):]u]t1t]u]w12igu cos w50. ~7!The initial condition ~6! is equivalent to the initial conditionu(w ,t→2‘)→exp@in8(2t2/21w)# . Given the solutionu(w ,t), the amplitudes cn(t) can be found by the inverseFourier transformation cn(t)5(1/2p)*02pu(w ,t)e2inwdw .The solution of Eq. ~7! that obeys proper boundary condi-tions isu~w ,t !5expH 2iF 2gE2‘tcosS w2 t22 1 t822 D dt81n8S w2 t22 D G J . ~8!Putting t51‘ in the solution ~8! and taking the inverseFourier transform, we arrive at following asymptotic values:cn~ t !’expS 2 int22 1i ~n82n !p4 D J un2n8u~2A2pg !. ~9!Thus, the scattering amplitudes in terms of modified states,with the fast phase factor exp(2int2/2) incorporated, areTn ,n8[^nuTun8&5ei(n82n)p/4J un2n8u~2A2pg !, ~10!where the operator T is expressed in terms of the evolutionoperator U(t ,t8) for the Hamiltonian ~4! in the interactionrepresentation:T5 limt→‘ ,t8→2‘expS iEt80H0~t!dt D U~ t ,t8!3expS iE0tH0~t!dt D . ~11!The matrix elements Tn ,n8 display an infinite number of os-cillations with the LZ parameter g. However, for large5-2BRIEF REPORTS PHYSICAL REVIEW B 65 153105un2n8u the oscillations start with g.un2n8u. These oscilla-tions can be observed experimentally by varying the fieldsweep rate E˙ (0). For small values of g the amplitudes aresmall and quickly decrease with growing un2n8u. In Fig. 2we depict transition probabilities for several levels n closestto the initial one n8 versus the Landau-Zener parameter g.Figure 3 shows the dependence of the transition amplitudeon un2n8u at a fixed value of g.For large g@un2n8u the asymptotic values of the ampli-tudes ~10! are^nuTun8&;ei(n82n)p/4~A2p3g !1/2cosS 2A2pg2 ~n2n8!p2 2 p4 D .~12!It is instructive to compare this result with other exactlysolvable generalized Landau-Zener models. Most of themrefer to systems with a finite number of states N. In the limitg@N the transition probabilities behave like an exponentexp@2C(n,n8)g2#, where the C(n ,n8) do not depend on g. Incontrast, the result ~12! displays a power law with oscilla-tions instead of an exponential dependence on g for large g.This is the manifestation of quantum interference of differentFeynman trajectories, which are discrete in the chain. A stepFIG. 2. Transition probabilities to sites closest to an initiallyfilled one as functions of g: ~a! un2n8u50, ~b! un2n8u51, ~c!un2n8u52, and ~d! un2n8u53.FIG. 3. Transition probabilities vs n2n8 at a fixed g51.6. Solidline is a guide for the eye.15310in a trajectory has average length g ~see below!. Such a stepcannot be realized in a system with a finite number of statesif g@N .The mean square displacement at one crossing event is^~n2n8!2&5 (n52‘‘~n2n8!2uJ un2n8u~2A2pg !u254pg2.~13!If the external field is periodic in time and the coherencebetween crossing events is lost,24 the electron performs arandom walk; i.e., it diffuses. Assume the field to oscillateharmonically as F(t)5F0sin(vt). At the nodes tk5pk/v(k is an integer! all diabatic levels cross together. Thesquared Landau-Zener parameter is g25g2/F0v . The diffu-sion coefficient is D52a2^(n2n8)2&/T , where T52p/v isthe period of oscillations and the factor of 2 accounts for twocrossing events per period. Collecting these results and Eq.~13!, we findD54ag2F0. ~14!This result does not depend on the frequency of the externalfield.The theory can be extended to a more general Hamil-tonian incorporating hopping between any two sites, but con-serving translational invariance:H5(m ,nHm ,num&^nu,Hmn5ntdmn1g um2nu , g2k5gk* . ~15!For simplicity we present below the result for real hoppingamplitudes gk5g2k :^nuTun8&5ei(n82n)p/42p E02pexp@2i2A2p f ~w ,g j!1i~n82n !w#dw , ~16!where f (w ,g j)5(k(gk /Ak)cos kw.The model ~1! can be generalized also to incorporate in-ternal degrees of freedom of identical chain fragments. Inthis case the local states are described by amplitudes an ,awith two indices. The first index n denotes the position andthe second index a labels the inner states. The Schro¨dingerequation for the amplitudes then readsia˙ n ,a5~nt1ea1dat !an ,a1(bga ,b~an11,b1an21,b!,~17!where the indexes a ,b51, . . . ,Nint run over the internalstates of the molecular wire segment. Changing to variablesan ,a5bn ,ae2int2/2 we eliminate the term proportional to t inEq. ~17!. Introducing a new function ua(w)5(n52‘‘ bn ,aeinw, we reduce the infinite system ~17! to afinite set of Nint ordinary differential equations:5-3BRIEF REPORTS PHYSICAL REVIEW B 65 153105iu˙ a5~ea1dat !ua12 cos~ t21w!(bga ,bub , ~18!in which w plays the role of a parameter. The initial condi-tions are bn ,a(t→2‘)5exp(2idat2/2)dn ,n8 and ua(w ,t→2‘)5exp@(2idat2/2)1in8w# . Thus, the variable ~param-eter! w enters not only in the system ~18!, but also in theinitial conditions. This system must be solved for all valuesof parameter w in the interval (0,2p). The inverse Fouriertransformation yields the evolution operator just as for thecase Nint51. An analytical solution of the system ~18! ispossible for some special choices of the parameters «a , da ,and ga ,b . For example, two identical coupled chains corre-spond to Nint52. Then the indices a ,b take on the values1,2. The simplest solvable choice of parameters is «150,«25«; d150, d25d; g1,15g2,25g; g1,25g2,15g8. An ex-act solution of this model can be reduced to ones solved inthis article together with the solvable two-level LZ model.In conclusion, we have generalized the LZ theory to aninfinite number of crossing levels. Physically it describes anelectron on an infinite chain subject to a time-dependentelectric field. The high symmetry of the problem allows us tofind not only the asymptotics, but also the intermediate val-ues of the amplitudes. Our solution is valid for an infinitechain with translational symmetry. We demonstrated it forthe case of a simple primitive cell, but it can be generalizedand in some cases exactly solved even if the primitive cellcontains more than one site.Finite-size effects do not permit us to apply our result ~10!directly to the transition amplitude from the first to the lastsite of the chain even if it contains many sites. However, ourcalculation of the diffusion coefficient ~14! is valid sincediffusion presumably proceeds far from the ends of thechain. Certainly, we assume that coherence is lost during thetime interval between two sequential crossing events.15310Our solution demonstrates a phenomenon that is probablycommon for most systems with multilevel crossing: oscilla-tions of the transition probabilities as a function of the LZparameter and site position ~distance between diabatic lev-els!. However, their asymptotic values for large values of theLZ parameter differ from those for other solvable multistateLZ models with a finite number of states. We expect that in ageneral situation with N@1 crossing levels, the transitionprobabilities will behave similarly to those found in thiswork for 1!g!N , provided that the initially occupied statesare far enough from the diabatic spectrum boundaries.Finally we discuss the relationship between our problemand a typical problem for semiconductor superlattices.18,19The latter is associated with Anderson localization. The di-abatic levels at sites are randomly distributed. In one and twodimensions all sites are localized. If the width of the energydistribution D is much less than the tunneling amplitude g ,the localization length in one dimension is ag/D . To enhancethe tunneling through a chain it is reasonable to apply atime-dependent electric field. The electric field is substantialif Fg/D>D where F is a typical value of F(t). Our approxi-mation is valid if the inequality is strong: Fg/D@D . Tunnel-ing transitions in the field proceed during an interval of timet defined by relation F(t);g . The value F(t) can be ac-cepted for F. We see that the strong inequality g@D guar-antees the existence of the strong-field limit in which therandomness of levels can be ignored. This requirement doesnot impose any limitations on the LZ parameter g.We thank W. Saslow for a critical reading of the manu-script. This work was supported by the NSF under Grant No.DMR 0072115 and by the DOE under Grant No. DE-FG03-96ER45598. One of us ~V.P.! acknowledges the support ofthe Humboldt Foundation.1 L.D. Landau, Phys. Z. Sowjetunion 2, 46 ~1932!.2 C. Zener, Proc. R. Soc. London, Ser. A 137, 696 ~1932!.3 F.T. Hioe, J. Opt. Soc. Am. B 4, 1327 ~1987!.4 C.E. Carroll and F.T. Hioe, J. Phys. A 19, 1151 ~1986!.5 V.N. Ostrovsky and H. Nakamura, J. Phys. A 30, 6939 ~1997!.6 Yu.N. Demkov and V.I. Osherov, Zh. Exp. Teor. Fiz. 53, 1589~1967! @Sov. Phys. JETP 26, 916 ~1968!#.7 Yu.N. Demkov and V.N. Ostrovsky, J. Phys. B 28, 403 ~1995!.8 S. Brundobler and V. Elser, J. Phys. A 26, 1211 ~1993!.9 V. May and O. Kuhn, Charge and Energy Transfer Dynamics inMolecular Systems ~Wiley—VCH-Verlag, Berlin, 2000!.10 L.D. Landau and E.M. Lifshitz, Quantum Mechanics ~Pergamon,Oxford, 1967!.11 D.S. Crothers and J.G. Huges, J. Phys. B 10, L557 ~1977!.12 A. Garg, N.J. Onuchi, and V. Ambegaokar, J. Chem. Phys. 83,4491 ~1985!.13 W. Wernsdorfer and R. Sessoli, Science 284, 133 ~1999!.14 W. Wernsdorfer, T. Ohm, C. Sangregorio, R. Sessoli, D. Mailly,and C. Paulsen, Phys. Rev. Lett. 82, 3903 ~1999!.15 C. Kergueris et al., Phys. Rev. B 59, 12 505 ~1999!.16 H. Ness and A.J. Fisher, Appl. Surf. Sci. 162-163, 613 ~2000!.17 A. Onipko, Yu. Klymenko, L. Malysheva, and S. Stafstrom, SolidState Commun. 108, 555 ~1998!.18 M. Holthaus, G.H. Ristow, and D.W. Hone, Phys. Rev. Lett. 75,3914 ~1995!.19 D.W. Hone and X. Zhao, Phys. Rev. B 53, 4834 ~1996!.20 W.V. Houston, Phys. Rev. 57, 184 ~1940!.21 G.H. Wannier, Phys. Rev. 117, 432 ~1960!.22 S.G. Devison, R.A. Miskovic, F.O. Goodman, A.T. Amos, andB.L. Burrows, J. Phys.: Condens. Matter 9, 6371 ~1997!.23 H. Fukuyama, R.A. Ban, and H.C. Fogedby, Phys. Rev. B 8, 5579~1973!.24 Do not confuse the decoherence between different crossingsevents with the decoherence between tunneling on different dis-tances during one crossing.5-4

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Landau-Zener transitions in a linear chain

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Landau-Zener transitions in a linear chain

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