Dynamics of a particle in a perfect chain with one nonlinear impurity and in
a perfect nonlinear chain under the action of dc field is studied numerically.
The nonlinearity appears due to the coupling of the electronic motion to
optical oscillators which are treated in adiabatic approximation.
  We study for both the low and high values of field strength. Three different
range of nonlinearity is obtained where the dynamics is different. In low and
intermediate range of nonlinearity, it reduces the localization. In fact in the
intermediate range subdiffusive behavior in the perfect nonlinear chain is
obtained for a long time. In all the cases a critical value of nonlinear
strength exists where self-trapping transition takes place. This critical value
depends on the system and the field strength. Beyond the self-trapping
transition nonlinearity enhances the localization.Comment: 9 pages, Revtex, 6 ps figures include

A. M. Jayannavar

B. C. Gupta

D. H. Dunlap

E. E. Mendez

F. Bloch

H. N. Nazareno

L. Esaki

M. I. Molina

M. Johansson

P. K. Datta

R. Tsu

V. M. Kenkre

English

arXiv

Dynamics of a particle in a perfect chain with one nonlinear impurity and in a perfect nonlinear chain under the action of dc field is studied numerically. The nonlinearity appears due to the coupling of the electronic motion to optical oscillators that are treated in the adiabatic approximation. We study both the cases of low and high values of field strength. Three different ranges of nonlinearity are obtained each of which has a different dynamics. In the low and intermediate ranges of nonlinearity, the localization effects are reduced. In fact, in the intermediate range case subdiffusive behavior in the perfect nonlinear chain is obtained for a long time. In all cases a critical value of nonlinear strength exists where a self-trapping transition takes place. This critical value depends on the system and the field strength. Beyond the self-trapping transition, nonlinearity enhances the localization

Datta, P. K.

Jayannavar, A. M.

Publications of the IAS Fellows

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Effect of nonlinearity on the dynamics of a particle in dc
field-induced systems
P. K. Datta†a and A. M. Jayannavar‡b
†Mehta Research Institute of Mathematics and Mathematical Physics, Allahabad- 211 019, India
‡Institute of Physics, Bhubaneswar-751 005, India
Abstract
Dynamics of a particle in a perfect chain with one nonlinear impurity and in a
perfect nonlinear chain under the action of dc field is studied numerically. The
nonlinearity appears due to the coupling of the electronic motion to optical
oscillators which are treated in adiabatic approximation. We study for both
the low and high values of field strength. Three different range of nonlinearity
is obtained where the dynamics is different. In low and intermediate range
of nonlinearity, it reduces the localization. In fact in the intermediate range
subdiffusive behavior in the perfect nonlinear chain is obtained for a long
time. In all the cases a critical value of nonlinear strength exists where self-
trapping transition takes place. This critical value depends on the system and
the field strength. Beyond the self-trapping transition nonlinearity enhances
the localization.
PACS numbers : 52.35.Nx, 63.20.Ls, 72.20.Jv
Typeset using REVTEX
1
The application of dc field on a charge particle in a perfect periodic lattice would cause the
oscillatory movement of the particle with frequency ω = eEa/h¯. Here e is the charge of the
particle, a is the lattice spacing and E is the strength of the electric field. This oscillation
is called Bloch oscillation [1]. The rapid oscillation of the charge particle may act as a
fast emitter of electromagnetic radiation. However, the detection in bulk sample is almost
impossible due to the fact that scattering time is much smaller than the period of Bloch
oscillator. Esaki and Tsu [2] first studied the Bloch oscillation in superlattice structure (SL)
which have long periods that makes the possible application for obtaining electromagnetic
radiation in terahertz range. Recently these phenomena have been confirmed by various
laboratory experiments [3]. A simple theoretical approach which captures the underlying
physics is to use tight-binding models. Recently, the dynamical localization of a carrier has
been studied in a one-dimensional perfect lattice with one linear impurity [4]. The resonance
occurs for a particular value of the applied field and the strength of the impurity. This is
observed only when the electric field is very high [5]. In this case the two on-site energies
coincide which enhances the hopping of the carriers between the two degenerate sites. On
the other hand, for weak field the degenerate sites are far apart from each other and as a
result there is almost complete localization at the impurity site.
In this work we numerically study the dynamics of a charged particle in a linear chain
with a single nonlinear impurity and in a perfect nonlinear chain under the action of dc
electric field. The general equation of motion for the system is given as
ih¯
dcm
dt
= V (cm+1 + cm−1)− (ǫm + χm|cm|
2 + eamE)cm. (1)
Here cm(t) is the probability amplitude of the particle at site m at time t, V is the nearest-
neighbor transfer matrix element, ǫm and χm are on-site energy and nonlinearity strength
of the m-th site, respectively. Here, we assume e = a = h¯ = V = 1. The nonlinearity arises
due to the coupling of quasiparticles with optical oscillators in adiabatic approximation [6].
When the electric field E is zero the equation is called one-dimensional discrete nonlinear
Schro¨dinger equation (DNLS) [6–10]. The interesting property of DNLS equation is that
2
self-trapping transition occurs if the nonlinearity strength exceeds a critical value. The mean-
square displacement (MSD) for perfect nonlinear systems [7] as well as random nonlinear
systems [8,9] goes as ∼ t2. It has been found that the time-averaged site-probability is a
good candidate to study the dynamics of the system [9,10]. So, we mainly study here the
time-averaged probability of different sites of the systems discussed above for low as well as
high electric field. The time-averaged probability at m-th site is defined as
〈Pm〉 = lim
T→∞
1
T
∫ T
0
|cm(t)|
2 (2)
with |cm(0)|
2 = δm,0. We solve the coupled nonlinear differential equations numerically using
4-th order Runge-Kutte method. To avoid the boundary effect we use self-expanding lattice
[11]. For time-averaging we have taken T = 2000. The accuracy of numerical investigation
is checked through the total probability. To examine the long time behavior of the system
we also study the particle propagation and MSD. The MSD is defined as
< m2 >=
∞∑
m=−∞
m2|cm(t)|
2 (3)
with the same initial as above.
We first study the dynamical properties of a dc field-induced perfect chain containing
only one nonlinear impurity at the zeroth site. The equation of motion is same as Eq. 1
with ǫm = 0 for all m and χm = χδm,0. The nonlinearity alters the effective strength of the
zeroth site dynamically as it depends on the occupation probability at that site. In a single
linear impurity problem the degeneracy between the zeroth site and mth site occurs when
ǫ0 = mE (where ǫ0 is the defect strength at zeroth site) [4,5]. But here, the degeneracy
(χ|c0(t)|
2 = mE) may appear dynamically i.e. the strength of the zeroth site and any other
site become equal at different instants of time. This is due to the oscillatory behavior of
|c0(t)|
2 with time t. For low field (e.g. E = 0.5) the time-averaged probability of a few
sites as a function of nonlinearity strength χ is shown in Fig. 1. In this study we can
define three regions of χ where the behavior changes significantly: (1) for small values of
nonlinearity strength (χ < 3) where we find smooth behavior of time-averaged probability
3
of the sites under study; (2) intermediate values of χ, (3 < χ < 4.8) where we obtain small
fluctuations and (3) high values of χ(> 4.8) where the particle is confined at the zeroth
site with maximum probability. In region (1) for very small values of χ we do not find
any significant change. However, if we increase the value of χ further particle gets confined
towards the right side (m > 0) of zeroth site with maximum probability (see Fig. 2). Due
to the presence of nonlinearity at zeroth site, it becomes dynamically degenerate with the
sites right to the zeroth site. On the other hand, the energy difference between the zeroth
site and its left neighbor (m = −1) is in general greater than the applied electric field.
Consequently, the particle prefers to move towards m > 0. The MSD oscillates and its
amplitude also changes with time. In region 2, the fluctuations occur due to the chaotic
behavior of site-probabilities [10]. It should be noted that the values of 〈Pm〉 with m ≥ 0
under study are almost equal. This implies particle localizes within a region uniformly. The
region of localization depends on the dynamical degeneracy of the furthest site with the
zeroth site (i.e. the site which is degenerate with zeroth site at time t = 0). This behavior is
a sharp contrast to the linear impurity problem [4]. In linear impurity problem the particle
gets localized at the defect site more and more with increasing the defect strength. In
the fluctuation regime the value of MSD is even larger than that of perfect field-induced
systems. Thus, the nonlinearity reduces field-induced localization. Beyond the fluctuations
regime we find a self-trapping transition at χcr ∼ 4.8 where the maximum probability is
gathered at the initial excitation site (i.e. zeroth site). Just below χcr, as χ increases the
time spending by the particle at zeroth site with maximum probability also increases. This
indicates the self-trapping transition at χcr. However, after χcr if we increase the value of
χ the time-averaged probability of the zeroth site increases and of other sites it decreases.
The value of MSD decreases even than that of field-induced perfect system. In the absence
of electric field the self-trapping transition occurs at χcr ∼ 3.2 [9,10]. Thus, the value of χcr
increases due to the presence of electric field. The particle has a tendency to move to the
dynamical degenerate sites. Consequently, to get the self-trapping transition larger value of
χ is required.
4
In Fig. 3 we have plotted the time-averaged probability of different sites as a function
χ for high electric field (e.g. E = 5). Here, 〈P0〉 and 〈P1〉 initially decreases and increases
respectively with increasing χ. Beyond χ ∼ 7.7 we find the fluctuations in the time-averaged
probabilities. The self-trapping transition occurs at χcr ∼ 12.1. Without any nonlinearity
the particle gets localized at the zeroth site with maximum probability. As we increase
the value of nonlinearity the amplitude of the occupation probability of zeroth and its
right neighbor site (m = 1) increases. Thus the time-averaged probability of these two sites
decreases and increases respectively with increasing χ. In the fluctuation regime particle first
gets localized within the sites m = 0 and 1 with maximum probability. Here, |c0(t)|
2 ≤ 1
and it has oscillatory behavior. So, to obtain the degeneracy (χ|c0(t)|
2 = E) between zeroth
and its right neighbor site at many instants of time, the value of χ should be larger than E.
This is exactly obtained. As we increase χ the site m = 2 also starts becoming degenerate
with zeroth site at different instants of time. Thus 〈P2〉 gradually increases with increasing
χ and at some values of χ we find 〈P0〉, 〈P1〉 and 〈P2〉 are almost equal. This means that the
particle gets localized within the three sites m = 0, 1 and 2. To find the long time behavior
we also study the particle propagation. The behavior is consistent with that of time-averaged
probability. It should be noted that in single linear impurity problem the resonance occurs
between the two sites [4]. However, in the study of MSD we find the nonlinearity suppresses
the field-induced localization as like low field case. Beyond χcr we find increasing of 〈P0〉 and
decreasing of the time-averaged probability of all other sites with increasing χ. This means
that the particle gets localized at the initial excitation site with maximum probability. The
study of MSD shows the enhancement of the localization. It should be noted that the critical
value of χ for self-trapping transition increases with increasing the strength of dc field. As
we increase the field strength larger values of χ are required to get the degeneracy between
the sites at many instants of time. The particle prefers to move to the degenerate sites.
Consequently, larger χ values are required to get the self-trapping transition.
We now discuss the behavior of quasiparticles in field-induced perfect nonlinear chain.
The equation of motion is same as Eq. 1 with χm = χ and ǫm = 0 for all m. The strength of
5
all the sites are altered dynamically depending on the probability of the corresponding sites.
We first study for low field strength (e.g. E = 0.5). The time-averaged probability of a few
sites as a function of χ is shown in Fig. 4. For small values of nonlinearity the time-averaged
probability of the sites under study shows smooth behavior. We find that the dynamical
localization gets destroyed even for small values of χ. However, particle gets localized within
a few sites. For further increase of χ fluctuations, albeit small, in time-averaged probability
of the sites under study occur. The fluctuation regime starts from χ ∼ 2. In this regime MSD
shows subdiffusive behavior for a long time with many oscillations as shown in Fig. 5(a).
The dynamic alteration of the site energies makes the localization weak. The oscillating
behavior in MSD indicates the strong localization due to the applied electric field. Near
the self-trapping transition we find particle initially gets localized at the initial excitation
site for some time. Then it gets diffused to other sites. As we move towards the critical
value of χ the particle gets trapped at the zeroth site for longer and longer time. This also
reflects in the study of time-averaged probability. Two sharp falls in 〈P0〉 occur near χcr.
Thus we get the self-trapping transition at χcr ∼ 5.7. It should be noted that the value of
χcr of field-induced perfect nonlinear system is larger than that of single nonlinear problem
(compare Fig. 1 and Fig. 4). In the former case the localization is weaker than the later
one and consequently, to get the self-trapping transition more χ value is needed. However,
above χcr nonlinearity enhances the field-induced localization.
In Fig. 6 we have plotted the time-averaged probability of a few sites of perfect nonlinear
chain for high dc field (e.g. E = 5). For small values of χ they show smooth behavior. In this
region the asymmetric particle propagation and oscillatory behavior in the amplitude of MSD
is also obtained. For further increase of χ, the fluctuations in time-averaged probabilities
occur. Due to the alteration of the site energies in this region (6.9 < χ < 12.68) particle
moves away from the site which is degenerate with the zeroth site at t = 0. Here also
subdiffusive behavior in MSD is obtained for a long time as shown in Fig. 5(b). As the
energy difference between the sites (due to the electric field) is large, the particle gets
localized within a few sites with maximum probability. Thus the value of MSD is much
6
smaller than that of low field case. However, after self-trapping transition (χcr ∼ 12.68) we
find the particle gets localized at the initial excitation site and consequently the value of
MSD is smaller than that of perfect field-induced systems. Thus if χ exceeds the critical
value where the self-trapping transition occurs nonlinearity makes the localization stronger.
In conclusion, our numerical study of the dynamics of the particle in a perfect chain with
one nonlinear impurity and in a perfect nonlinear chain under the action of low as well as high
electric field indicates the existence of three regions of χ where the behavior is different. The
values of χ are different for different systems and field strengths. In all the cases we found a
sharp self-trapping transition of 〈P0〉 at χcr. The critical value of χ depends on the system as
well as field strength. When χ < χcr the nonlinearity reduces the field-induced localization.
Even we obtained subdiffusive behavior in field-induced perfect nonlinear systems. However,
above the critical value of χ nonlinearity enhances the localization.
7
REFERENCES
a Electronic-mail : pkd@mri.ernet.in
b Electronic-mail : jayan@iop.ren.nic.in
[1] F. Bloch, Z. Phys. 52, 555 (1928).
[2] L. Esaki and R. Tsu, IBM J. Res. DEv. 14, 61 (1970); R. Tsu and L. Esaki, Appl. Phys.
Lett. 19, 245 (1971).
[3] E. E. Mendez and G. Bastard, Phys. Today 46 (6), 334 (1993).
[4] H. N. Nazareno, C. A. A. da Silva and P. E. de Brito, Phys. Rev. B 50, 4503 (1994).
[5] B. C. Gupta and P. A. Sreeram,cond-mat/9705003.
[6] V. M. Kenkre and D. K. Campbell, Phys. Rev. B 34 4959 (1986)).
[7] M. Johansson, M. Ho¨rnquist and R. Riklund, Phys. Rev. B 52, 231 (1995).
[8] M. I. Molina and G. P. Tsironis, Phys. Rev. Lett. 73, 464 (1994).
[9] P. K. Datta and K. Kundu, Phys. Rev. B 53, 14929 (1996).
[10] M. I. Molina and G. P. Tsironis, Int. J. Mod. Phys. B 9, 1899 (1995).
[11] D. H. Dunlap, K. Kundu and P. Phillips, Phys. Rev. B 40, 10999 (1989)).
8
FIGURES
FIG. 1. Plot of time-averaged probability < Pm > as a function of χ for different values of m
in a perfect lattice containing a nonlinear impurity. The value of E is 0.5.
FIG. 2. Electronic probability propagation profile as a function of time (t) for χ = 2.5 and
E = 0.5 in the same system as in Fig. 1.
FIG. 3. Same as Fig. 1 but E = 5.
FIG. 4. Time-averaged probability < Pm > as a function χ for different values of m in a perfect
nonlinear chain. Here E = 0.5.
FIG. 5. log-log plot of MSD as a function of time t of a perfect nonlinear system. (a) The solid
curve corresponds to E = 0.5 and χ = 3.5. (b) The dotted curve corresponds to E = 5 and χ = 11.
FIG. 6. Same as Fig. 4 but E = 5.
9
0.2
0.4
0.6
0.8
0 1 2 3 4 5 6 7 8
hP
m
i

FIG. 1
m = 0
m = 1
m = 2
m = 3
m = -1
m = -3
FIG. 2
10
20
30
40
50
-12
-9
-6
-3 0 3 6 9 12
TIME
SITES
0.2
0.4
0.6
0.8
0 2 4 6 8 10 12 14 16
hP
m
i

FIG. 3
m = 0
m = 1
m = 2
00.2
0.4
0.6
0.8
0 1 2 3 4 5 6 7 8
hP
m
i

FIG. 4
m = 0
m = 3
m = 6
m = -3
m = -6
01
2
3
4
5
6
7
4 5 6 7 8 9 10 11 12
MSD
TIME
FIG. 5
00.2
0.4
0.6
0.8
0 2 4 6 8 10 12 14 16
hP
m
i

FIG. 6
m = 0
m = 1
m = 2
m = 3





























m = -1


Effect of nonlinearity on the dynamics of a particle in field-induced systems

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Effect of nonlinearity on the dynamics of a particle in dc field-induced
  systems

Effect of nonlinearity on the dynamics of a particle in dc field-induced systems

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