Article
Quantum breathers in a nonlinear Klein Gordon lattice
Service de Recherches de Métallurgie Physique, CEASaclay/DEN/DMN 91191GifsurYvette Cedex, France
Physica D Nonlinear Phenomena (Impact Factor: 1.64). 04/2006; 216(1):191199. DOI: 10.1016/j.physd.2005.12.019 Source: arXiv
Fulltext
Available from: Laurent Proville, Jul 15, 2014arXiv:quantph/0507033v1 4 Jul 2005
Quantum breathers in a nonlinear Klein Gordon lattice
Laurent Proville
∗
Service de Recherches de M´etallurgie Physique,
CEASaclay/DEN/DMN 91191GifsurYvette Cedex, France
(Dated: February 1, 2008)
Abstract
The quantum modes of a nonlinear Klein Gordon lattice have been computed numerically [L.
Proville, Phys. Rev. B 71, 104306 (2005)]. The onsite nonlinearity has been found to lead to
a phonon pairing and consequently some ph on on bound states. In the present paper, the time
dependent Wannier transform of these states is shown to exhibit a breatherlike behavior, i.e., it is
spatially localized and timeperiodic. The typical time the lattice may sustain such breather states
is studied as a function of the trapped energy and the intersite lattice coupling.
PACS numbers: 63 .20.Ry, 03.65.Ge, 11.10.Lm, 63.20.Dj
1
Page 1
I. NONLINEAR LATTICE MODES
The discrete breather solutions are currently a matter of intensive research (see
Refs.
1,2,3,4,5,6,7,8,9,10,11,12,13,14
). The distinctive property of those lattice modes is to gather
the spatial localization and the time periodicity so they lead to a energy trapping and thus
a delay in the equipartition
5
. As a general consequence of anharmonicity, the emergence
of breathers may be recognized as a paradigm of physics since it occurs at diﬀerent scales
in various contexts, e.g., in macroscopic networks as a chain of coupled pendulums, in mi
croscopic Josephson arrays
15
as well a s in molecules
16
, polymers
17
and crystals as the PtCl
ethylene diamine chlorat e
18
.
The nonlinear excitations in materials have been studied for several decades. In the late
ﬁfties
19
, the possible existence of a twophonon bound state was pointed out in the infrared
(IR) spectroscopy of H
2
solid. The vibrational and rotational nonlinear excitations in the H
2
crystal have been thoroughly investigated both experimentally
20
and theoretically
21
. About
the same period
22
, the spectrum anomalies of the crystalline a cetanilide (ACN) was revealed
and later interpreted with diﬀerent theories (see Refs.
23,24,25
and for a historical survey see
Ref.
26
). Early in the sixties
27
, in the HCl solid, the anharmonicity of the ﬁrst overtone of
hydrogen vibration has been measured by IR adsorption. It has been interpreted as a two
phonon bound state, namely a biphonon
28
, in regard of the earlier theoretical work of V.M.
Agranovich
29,30
. The tr iphonon has a lso been identiﬁed in the spectrum of HCl
28
. In sev
enties, similar phonon bo und states have been recognized in several molecular crystals such
as CO
2
, N
2
O and OCS
31
, as well as in water ice
32
by measuring the anharmonic IR absorp
tions. F. Bogani achieved some convincing simulations of these anharmonic sp ectra
33
by
using the technics of renormalized perturbation theory. For the last decade, the nonlinearity
has emerged in several other materials:
 The inelastic neutron scattering (INS) has revealed the phonon bound states in the
metal hydrides as PdH
34
or TiH and Z rH
35
.
 The INS has also permitted to infer proton dynamics in the molecular crystals as
polyglycine
36
and 4methylpyridine
37
. In the latter, the bound states of the methyl
group rotational modes proved to last several days (see Refs.
37,38
and Ref.
26
for a
survey of the theory).
2
Page 2
 The stretch overtone of carbon monoxide adsorbed on Ru(100) has been found
to exhibit a strong a nharmonicity at low surface coverage
40,41
. Several theoretical
approaches
40,41,42
have been attempt to analyze the IR spectroscopy on Ru(100):CO.
The previous list is probably not complete but it is suﬃcient to emphasize that the nonlinear
excitations have been worked out in many diﬀerent materials, whether it is a molecular
crystal
33
, a hydrogenbonded crystal
28
or a metal hydride
34
. Furthermore the nonlinearity
may occur in one
37
, two
40
and threedimensional
33
systems. In most of the above cited
examples, the phonon dispersion may be evaluated as smaller than 10% of the fundamental
optical excitation and the anharmonicity proves to reach less than 5%. The latter estimation
holds for the ﬁrst overtone whereas for higher orders the strength of the anharmonicity may
increase as it is the case in HCl solid
28
or stabilize as in PdH
34
. The phonon b ound states
are the siblings of breathers as they all stem from anharmonicity (see Refs.
14,26
and Refs.
therein). The quantum breather may be viewed as a Wannier transform, applied to the
phonon bound states that participate to a same energy band. Presently, the purpose of our
work is to study this idea within the nonlinear KG model.
The accurate computation of nonlinear modes, whether it is in a classical lattice or in
a quantum one requires the use o f numerics. Recently
43
, we proposed a numerical method
that permits to compute the nonlinear quantum modes in a Klein Gordon lattice (KG) for
diﬀerent type of nonlinearity. In the present paper, we use those developments to study the
Wannier transform
14
of the la tt ice eigenmodes that exhibits a quasiparticle spectrum, i.e.,
a narrow energy band. The time dependant Wannier transform of these states is found to
exhibit a breatherlike behavior, i.e., it is spatially localized and timeperiodic. The life time
of such breather states is studied as a function of their energy. We f ound that the higher
the energy spike is, the longer it remains localized. That study has been carried out f or
diﬀerent model parameters, including the case where the phonon dispersion is larger than
the anharmonicity.
After a brief intro duction of the nonlinear KG lattice model, our computing method is
spelled out in Sec. II. In Sec. III and Sec. IV, we present and discuss our results on phonon
bound states and breathers, respectively. Some perspectives are given in Sec. V.
3
Page 3
FIG. 1: The plot of energy spectrum of a 1D chain, composed of N = 13 atoms for A
4
= 0.2,
versus the dimensionless coupling C. The eigenvalues are plotted as empty circles excepted the
phonon bound states energies, plotted as fu ll black circles. The tags indicate the order of phonon
bound states.
II. LATTICE MODEL AND NUMERICAL METHOD
The energy of a lattice made of identical particles is expressed as a Hamiltonian operato r:
H =
X
l
[
p
2
l
2m
+ V (x
l
) +
X
j=<l>
W (x
l
− x
j
)]. (1)
where x
l
and p
l
are displacement and momentum of the particle at site l, in a onedimensional
lattice. Such a lattice may prove relevant to model the quasionedimensional networks of
quantum particles in ZrH or in PtCl. The quantum particle of mass m evolves in a onsite
potential V , being coupled to its nearest neighbors, j =< l > by the interaction W . The
onsite potential V is developed to the fourth order whereas W is modelled by a quadratic
term:
V (x
l
) = a
2
x
2
l
+ a
3
x
3
l
+ a
4
x
4
l
W (x
l
− x
j
) = −c(x
l
− x
j
)
2
. (2)
Higher o r der terms could have been added with no diﬃculty for our theory. It is possi
ble to ﬁxe the coeﬃcients o f V within a ﬁrst principle calculation as done for PdH
44
and
conﬁrmed by the analyze of the INS spectrum
34
. For simplicity, we choose to ﬁxe a
3
= 0.
Introducing the dimensionless operators P
l
= p
l
/
√
m~Ω, X
l
= x
l
p
mΩ/~ and the f r equency
Ω =
p
2(a
2
− 2.c)/m, the Hamiltonian is rewritten as follows:
H = ~Ω
X
l
P
2
l
2
+
X
2
l
2
+ A
4
X
4
l
+
C
2
X
l
X
j=<l>
X
j
(3)
where the dimensionless coeﬃcients are A
4
= a
4
~
m
2
Ω
3
and C =
4c
mΩ
2
. The ﬁrst step of o ur
method is concerned with the exact diagonalization of the Hamiltonian where no interaction
couples displacements. The procedure has been detailed in Ref.
43
. Arranging the onsite
eigenvalues in increasing order, the αth eigenstate is denoted φ
α,i
and its eigenvalue is γ(α).
In case of a negligible intersite coupling, the H eigenstates can be written as some Bloch
4
Page 4
FIG. 2: Energy spectrum of a 1D chain which model parameters are A
4
= 0.2 and C = 0.05.
The chain is composed of N = 13 sites. Four en ergy regions have been reported: (a) phonons,
(b) biphonons, (c) triphonons and (d) quadriphonons. The eigenenergies are p lotted as empty
symbols and the phonon bound states energies have been signalized by full sy mbols.
(c) (d)
waves as follows:
B
[Π
i
α
i
]
(q) =
1
p
A
[Π
i
α
i
]
X
j
e
−iq.j
Π
i
φ
α
i
,i+j
(4)
where A
[Π
i
α
i
]
ensures the normalization. The label [Π
i
α
i
] identiﬁes a single onsite state
product Π
i
φ
α
i
,i
among the diﬀerent products that may be derived from the present one by
translation. The set of states {B
[Π
i
α
i
]
(q)}
q,N
cut
, including the uniform state Π
i
φ
0,i
at q = 0,
form a truncated basis where N
cut
ﬁxes the upper boundary on the onsite excitations:
P
i
α
i
≤ N
cut
. In case of a nonzero coupling, the states Eq.(4) may be thought as some
Hartree approximation of the t r ue eigenstates. The perturbation theory might be applied
to the intersite coupling so as to estimate the eigenspectrum. However, we have chosen
to carry out a computation as accurate as possible. Thus the Bloch wave basis is used to
expand t he Hamiltonian in. As the waves with diﬀerent q, are not hybridized by H, the
Hamiltonian can be expanded separately for each q. It can be achieved analytically whereas
the diagonalization of the resulting matrix has been realized numerically with a standard
method, from a numerical library
45
. The accuracy of our calculations has been tested both
in a anharmonic
43
and harmonic
46
chain. For these two comparisons, a very good ag r eement
has been found in the twophonon energy region and lower.
We denote by ψ
λ
(q) and E
λ
(q) the H eigenstates and the corresponding eigenenergies,
respectively. The subscript λ ﬁxes the correspondence between a eigenstate and its eigen
energy. Our numerical t echnics allows us t o compute the scalar product V
λ,[Π
i
α
i
]
(q) between
ψ
λ
(q) and B
[Π
i
α
i
]
(q). Among the Bloch waves B
[Π
i
α
i
]
(q), we note those bearing a single on
site excitation of order α
j
> 0, all the other lattice sites l, being such as α
l
= 0. For those
states the label [Π
i
α
i
] reduces to α. In case of V
λ,α
(q) > 0.5, we choose to distinguish t he
eigenstate ψ
λ
by setting λ = α. It simply means that the Bloch wave B
α
(q) has a dominant
contribution into ψ
λ
(q). At C = 0, one notes tha t V
α,α
(q) = 1. As it may be exp ected, the
scalar product V
α,α
(q) decreases as C increases but its variation is smooth as found in Fig. 1
5
Page 5
FIG. 3: Energy spectrum of a 1D chain which model parameters are A
4
= 0.2 and C = 0.3. The
chain is composed of either N = 13 sites. Two energy regions have been reported: (a) phonons
and (b) quadriphonons. The eigenenergies are plotted as empty symbols and the phonon bound
states energies have been signalized by full symbols.
(a) (b)
(the results shown in this ﬁgure are examined thoroughly in the following). The eigenstates
ψ
α
(q) correspond to the α phonon bound states. That terminology may be rightly thought
as ambiguous since a binding energy usually r efers to a g roundstate rat her than to some
excited states. However it is convenient as the excitation order α appears in the name. This
order corresponds, indeed, to the energy level of the anharmonic onsite potential.
III. PHONON BOUND STATES
In lattices, treated in Ref.
43
, the sites number was N = 33 for a basis cutoﬀ N
cut
= 4,
which proves suﬃcient for the study of the twophonon energy r egion. Here, we would like
to extend our study to the case of a four phonon bound state (quadriphonon). We thus
increased N
cut
but the number of Bloch waves, involved in our basis for N = 33, would
overload our computer’s memory, so we had to work with smaller lattices. For N = 13 and
N
cut
= 6, the rank of our basis reaches 6564 which can be managed within a reasonable
time. We worked also with a even smaller lattice, N = 7 which allows us to increase again
N
cut
as large a s N
cut
= 9. That case serves us as a reference in order to test the precision o f
our computations on the larger lattice.
Varying C from the anticontinuous limit
4
, i.e., C = 0 we plotted in Fig. 1 the eigenspec
trum as a function of C. Every circle symbol represents a single eigenvalue in the half ﬁrst
Brillouin zone. The eigenvalues that correspond to the eigenstates ψ
α
(q) (described in Sec.II)
have been plotted as full circles in Fig. 1, instead of empty ones for the other eigenstates.
As f ar as we increased the coupling C (see comment in Ref.
47
), in Fig.1, f or a given order α
and a ﬁxed wave vector q, we found a unique eigenstate that veriﬁes V
λ,α
(q) > 0.5. This is
the numerical proof t hat the nonlinear excitations may be continued from C = 0 to larger
coupling. This involves that the solutions ψ
α
(q) conserve some features similar to the Bloch
waves B
α
(q). Such a behavior could have been expected
43
while the energy gaps of the zero
6
Page 6
coupling spectrum remain. The hybridization between bound and unbound phonon states
is, indeed, thought to be weak in that case. The point that was very unexpected is that
even though C is larg e enough for gaps to close (between the triphonon and the surround
ing unbound phonon bands, for instance) we found a dominant contribution of B
α
(q) into
ψ
α
(q). Moreover, for parameters in Fig .1 this property does not depend on the order of the
excitation α. It holds for phonons as for higher order phonon bound states. What diﬀers,
however, for the latter is their band width which increases with C much smo other. In Figs.2
(ad), at a ﬁxed coupling, the eigenspectrum is plotted for diﬀerent energy regions versus
the wave vector. The same symbols a s in Fig.1 are used. We note that the larger the energy
is, the narrower the band of t he phonon bound states is. Indeed, the phonon band width is
about 0.06, whereas for the biphonon it is less than 0.01, for triphonon it is around 10
−3
and
quadriphonon the band width falls to 10
−5
, in our energy unit. Although we approach only
the very ﬁrst energy excitations, up to the fourth o rder, we may reasonably extrapolate our
results t o higher energies. We then expect that the band width of the phonon bound states
becomes exponentially narrower as the eigenenergy increases. In Figs.3 (ab), the coupling
parameter is such as the energy ga ps close at high energy. We note that even though the
energy spectrum exhibits no gap, we ﬁnd some eigenstates ψ
λ
(q) that verify V
λ,α
> 0.5 for
α = 4. In that case, the binding energy of the so called α phonon bound states vanishes.
However a strong compo nent of B
α
(q) takes part in ψ
α
(q). For that reason, we propose to
dub the ψ
α
(q) eigenstates as nonlinear α phonons to emphasize that these states diﬀer from
the linear superposition of phonons, as well as to stress their quantized feature. In Fig.3
(b), the band width of the nonlinear fo ur phonons is around 0.03 instead of 10
−5
in Fig.2
(d). Although the width of that band increases substantially with t he coupling, it is yet one
order of magnitude below the phonon band width which is roughly 0.45. The exponential
decrease noted at low coupling seems to be no longer valid at larger coupling. This point
deserves a thorough study that we propose to report in a future work. In Figs.2 (d) and 3
(b), the band of the quadriphonon does not exhibit the anomaly which appears when N
cut
is diminished and that consists in a breaking of the band continuity. As far as C < 0.3, our
numerical approach seems to be reliable to treat the ﬁrst phonon bound states. When t he
nonlinear parameter A
4
is small, i.e., of the order of 10
−2
in our dimensionless model, the
scalar product V
λ,α
falls below 1/2 for a suﬃciently large C which depends on α. The larger
α is, t he larger the transition coupling C
α
is. Moreover the C
α
is found to depend on t he
7
Page 7
wave vector q. At the edge of the lattice Brillouin zone, C
α
is larger t han in the center. In
the limit where A
4
equals zero, the strictly harmonic eigenstates verify V
λ,α
= 0 as soon as
C is switched on. In t hat particular case, C
α
= 0 for all α but for non zero A
4
, the C
α
are
larger than zero, even f or α = 1 which corresponds to the single phonon.
In Ref.
46
, the author attempted t o compare his theoretical computations, similar to F ig .1,
to some experimental measures in H
2
20
solid and Ru(100):CO
40
. Although such a exercise
was based on qualitative considerations, it is worthy to complete these comparison by not
ing that some spectral bands are due to the linear superpo sition of a biphonon and a single
phonon (see Fig.1 in the present paper and Fig. 3 in Ref.
46
). The signature of these states
has been measured in the IR spectrum of HCl solid
28
. Following a theoretical approach
proposed earlier
33
, C. Gellini et al. carried out t he computation
28
of renormalized G r een
functions to interpret the HCl sp ectrum. Although such a theory would be inadequate for
strong intersite coupling, for molecular crystals whose the molecule’s bond anharmonicity
dominates the intermolecular coupling, as crystalline CO
2
or HCl for instance, the renormal
ized Green functions seems relevant to capture the main physical properties. A convincing
demonstration has been g iven by Bogani in Ref.
48
where a precise ﬁt of the IR adsorp
tion spectrum has been a chieved in several molecular crystals. The earlier work of V.M.
Agranovich introduced initially the concept of phonon bound states and more speciﬁcally of
biphonon f or interpreting some experiments where anharmonic modes had been measured
(see Ref.
29
for infrared and Ref.
49
for neutron spectra). The lattice model of Agranovich
involves several energy terms that can be described brieﬂy as f ollows. The elementary ex
citation is proport io na l to the onsite product of BoseEinstein operators a
+
i
a
i
, while the
tunneling between neighboring sites i and j is modelled by a hopping term Ca
+
j
a
i
. The
onsite Hubbard interaction between the boson pairs simulates the lattice anharmonicity by
adding locally the energy op erator U(a
+
i
)
2
a
2
i
. Some other terms can be incorporated in the
model to modify, for instance, the biphonon tunneling
30
or the triphonon energy
35
. These
energy contributions are parameterized by independent coeﬃcients, e.g., C, U. For instance,
the Hubbard model for bo son has been used to interpret the INS in metal hydrides
35
. Here
the model parameters have been adjusted to exhibit t he same energy resonances as the INS
spectrum. It is possible to achieve a similar work within the KG model, as shown in Ref.
50
.
The Hubbard model for boson involves to neglect the energy terms that do not conserve
the t otal boson number, despite the fact that these terms stem from the potential energy
8
Page 8
FIG. 4: Energy spectrum of a single anharmonic oscillator versus the eigenvalue rank f or diﬀerent
parameters: (a) A
4
= 0.2 and (b) A
4
= 0.4. The semiclassical calculation (empty square symbols,
dashed line) is compared to the Hamiltonian diagonalization (full circle symbols, solid line) onto
the truncated Einstein basis (see Ref.
43
) The Y axis unit is ~Ω.
of atoms and molecules. The consequence of such an approximation is exempliﬁed in com
puting the phonon dispersion law. For a onedimensional lattice, the boson Hubbard lattice
would exhibit a phonon branch of the form (1 + Ccos(q)), whereas the form
p
1 + 2Ccos(q)
would be expected fro m the harmonic approximation with similar parameters. The latter
case corresponds to the exact diagona lization of the linear KG Hamiltonian which includes
only the quadratic potential energy of atoms. Since the two formula above diverge as C
increases, the Hubbard model f or boson proves inappropriate to treat the normal modes at
strong coupling. It might however be relevant to work out the high order nonlinear modes as
proposed in Ref.
51
(see also the contribution of G.P. Tsironis in the present volume). Then
the skipping of the nonconservative boson terms might ﬁnd some substantiation in the fact
that we fo und nonlinear narrow bands in continuous spectra as in Fig.1. A comparison be
tween the boson Hubbard and our KG model would b e very interesting to tentatively infer
the properties of the high order phonon b ound states.
IV. QUANTUM BREATHERS
One introduces t he time dependent Wannier state W
α
(t, n), which is constructed from
a combination of the α phonon bound states ψ
α
(q). We recognize these eigenstates among
others ψ
λ
(q) by the fact that they verify V
λ,α
(q) > 0.5, for a ﬁxed α. This deﬁnition permits
us to build a Wannier state even though the energy spectrum has no gap. Then the Wannier
transform is written as follows:
W
α
(t, k) >=
1
√
N
X
q
e
−i(q×k+E
α
(q )Ωt)
ψ
α
(q) > . (5)
The subscript k indicates the lattice site where is centered the Wannier transform. In Fig.1,
we found that the band of the α phonon bound states contains a single state per wave vector
so that the sum over q in the Wannier transform is complete. In case of a small intersite
coupling, the Bloch wave B
α
(q) is a good approximate of the α phonon bound state with
9
Page 9
the same wave vector. This may be thought as a Hartree approximation. To a ﬁrst order in
C, we found
43
that the E
α
(q) dependence on q is negligible provided that α > 1 and V is a
single well po t ential. Then the Wannier state W
α
(t, k) > can be rewritten as:
W
α
(t, k) >= e
−i(E
α
Ωt)
φ
α,k
Π
l6=k
φ
0,i
. (6)
Such a state is localized and time periodic so it may be considered as the quantum counter
part of the breather solutions for the classical nonlinear discrete KG lattice. These classical
breather solutions have two impor tant features that are ﬁrst their spatial localization and
second their time periodicity with a frequency and its overtones that are out of the linear
classical phonon branch
2
. Our proposition could be veriﬁed by comparing the energies of
a localized time periodic Wannier state and the semiclassical quantization of the classi
cal breather orbits, in same lattice. In the simple case of zero intersite coupling, such a
comparison has been carried out in Ref.
43
and Fig.4, for diﬀerent onsite potentials. The
remarkable agreement allows to expect that our proposition on breather quantum counter
part holds for la r ger values of C. To enforce our arguments, we dwell upon Fig.1. To a ﬁxed
α > 1, the energy of the corresponding Wannier state, given by the bracket of H, equals the
mean energy
˜
E
α
, deﬁned as the sum of E
α
(q)/N over the ﬁrst Brillouin zone. According
to Fig.1, that mean energy does not vary much with C. Indeed, provided that α > 1 and
C < 0.3 (the upper boundary on C to obtain a satisfactory precision), the bisecting line o f
the phonon bound states band is roughly parallel to the X axis. Consequently, the energy
of the Wannier state of order α is comparable to the same quantity computed at C = 0.
In turn, t he lat ter approaches very well the semiclassical quantization (see Fig.4) so the
energy of the Wannier state and the one of semiclassical breather orbits do not diﬀer in a
signiﬁcant manner provided that C remains weak. It seems reasonable in the following to
call α breather what is indeed the Wannier state of order α. It would be worth carrying
out the semiclassical quantization at non zero coupling in order to evaluate to what extend
our expectations might be conﬁrm. We think it should not contradict o ur arguments unless
the classical breather becomes unstable, i.e., its fr equency or one of its overtones fa ll in the
spectrum of the classical normal modes.
We now study the dynamics of a α breather as a function of the order α and for a non
zero coupling parameter. To that purpose, we integrate the time evolution of the onsite
10
Page 10
FIG. 5: The time evolution of kinetic energy of 17 atoms in onedimensional KG chain, for a
Wannier state mad e of phonon. The model parameters are A
4
= 0.2, A
3
= 0 and C = 0.05. The
lattice sites are reported on the Y axis while the X axis bears the time scale. The time unit is th e
inverse of Ω.
time
site
FIG. 6: Proﬁle of the 3Dplot described in Fig.5 for diﬀerent breathers made of: (a) phonon, (b)
biphonon, (c) triphonon and (d) quadriphonon. The parameters are same as in Fig.5. The time is
reported on the X axis. The Y axis unit is ~Ω).
(c) (d)
kinetic energy P
2
j
/2. The expectation of this operator is given by the bracket:
< W
α
(t, k)
P
2
j
2
W
α
(t, k) >=
1
N
X
q,q
′
< ψ
α
(q
′
)
P
2
j
2
ψ
α
(q) > ×e
−i((q−q
′
)×k+ (E
α
(q )−E
α
(q
′
))Ωt)
. (7)
For α = 1, the Wannier state Eq.(5) is constructed from phonons. In the 3D plot (see
Fig.5) of the kinetic energy time evolution, one notes that the energy is initially localized
and quickly spreads over the lat tice. In Fig.6 (a) , the proﬁle of the 3D plot shows tha t
after 40 time unit, the energy is no longer localized. The same proﬁle plot for α = 2, in
Fig.6 (b) shows that the quantum breather made of biphonons may last 10 times longer
than for phonons. With the Wannier transform of triphonons, the life time of the localized
excitation is again raised by one o r der of magnitude (see Fig.6 (c)). The life time of the
fourth quantum breather (i.e., the Wannier transform of the quadriphonons) overpasses
the ﬁrst case in Fig.6 (a), by three orders. Conclusively, we found that the nonlinear KG
lattice may sustain a high energy spike for longer than 10
3
times the typical relaxation of
low energy excitations, imposed by phonons. This behavior is related to the dispersion of
the phonon bound states since the thinner the band is, the longer t he Wannier tr ansform
remains coherent. According to our results, the breather life time increases exponentially
with respect to α. As noted previously (see Sec.III), the band width decreases as the o rder of
phonon bound states increases, even though at high energy the sp ectrum becomes continuous
(see Fig .1). According to our results, this continuity does not involve a particular decay in
the breather life time. It is noteworthy that the decay of a quantum breather is athermal
as it stems from the decoherence of the phonon bound states.
11
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FIG. 7: The s ame as in Fig.6 but for C = 0.3 and diﬀerent sizes: (ab) N = 13 and (cde) N = 7.
The Wannier states are made of either (ac) phonon, (bd) quadriphonon or (e) pentaphonon. The
time is reported on th e X axis and the Y axis unit is ~Ω. In the insets, some distinct time intervals
have been magniﬁed.
(e)
As C increases, the α breather life t ime decreases, in agreement to the band width
enlargement shown in F ig.1. This can be worked out from the comparison o f Fig. 6 (a) and
Fig. 6 (d) for C = 0.05 to the left hand side insets in Fig. 7 (a) and Fig. 7 (b) for C = 0.3.
Each couple of ﬁgures concern the cases α = 1 and α = 4. Fo r α = 1, one sees that the
life time of the localized excitation decreases from 40 to 4 time units while for α = 4, the
breather life time decreases from 8×10
4
to 8×10
1
. The drop is sharper for the higher order.
However for a ﬁxed value of C, whatever this value is, the band width of phonon bound
states decreases as the energy increases so that the life time of the corresponding breather
increases too. In Fig . 7 (e), the life time o f the ﬁfth breather is still two order of magnitude
larger than for α = 1. We thus expect that for a suﬃciently high energy spike, the breathing
mode survives not iceably even though the intersite coupling is large.
For a ﬁxed order α, the breather life time does not depend on the lattice size, as found in
comparing Fig. 7 (a) to Fig. 7 (c) and Fig. 7 (b) to Fig . 7 (d). The Figs. 7 (ab) have been
obtained for a 13 sites lattice and the Figs. 7 (cd) for a 7 sites lattice. Another interesting
feature revealed by these results is the time recurrence of breather. Indeed, one notes that
a certain time after the energy spike has spread, the energy backs to its initial trapped
state, similar to t he original one (see right hand side insets in Figs. 7 (ae)). The breather
is then bear by few sites although it has not exactly the same amplitude as initially. The
retrapping process occurs fo r a time twice larger than the breather life time because of the
time inversion symmetry. According to our computations, there is no exact frequency fo r the
breather recurrence as no regular behavior may be depicted in Figs. 7 (ae). Though, we note
that the recurrence occurs sooner in a smaller lattice, which is demonstrated by comparison
either of Fig. 7 (a) to (c) or Fig. 7 (b) to (d). In a macroscopic crystal, t he recurrence
is thus expected never to take place. In contrast, the breather recurrence might occur in a
single molecule as benzen. To that respect, the breather recurrence might be worth studying
thoroughly. Eventually, the shortest time interval upon which the recurrence occurs after
12
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starting the breather dynamics seems to increase with α as shown by comparing Fig. 7 (c)
to Fig. 7 (d) or Fig. 7 ( d) to Fig . 7 (e).
V. CONCLUSION AND POSSIBLE DEVELOPMENTS
As a summary, we attempted to work out the breather modes in the quantum KG latt ice.
We provided a numerical method to estimate their life time and spatial expansion. At the
quantum scale, it proves that the breathers are closely related to what has been called earlier,
the phonon bound states that are anharmonic eigenmo des. It is, indeed, wellknown in
condensed matter physics that a narrow band excitations may be viewed as a quasiparticle
through a Wannier transform. We applied that theory to the phonon bound states and
showed that the lattice may sustain the corresponding breather for a time which increases
as the magnitude of the energy spike. At low intersite coupling, we found that the breather
life time increases exponentially with the trapped energy. This variation softens at larg er
coupling, mainly because of the hybridization between the phonon bound states and the
linear superpositions of lower energy modes. For seak of simplicity, we only treated a quartic
nonlinearity. We found nonlinear excitatio ns for all couplings we tested, i.e., up to C = 0.3
which corresponds to a dispersion that is larger than the anharmonicity. In t he classical
counterpart of our KG lattice, a similar result is obtained since the discrete breathers occur
at all coupling too because their frequency is higher than the normal modes band. However,
the cubic nonlinearity is known to modify signiﬁcantly this feature as the breather frequency
should be smaller than the normal modes. Consequently, for a given breather solution, that
is for a ﬁxed frequency, there is a coupling threshold above which the breather is no longer
stable. This transition occurs when t he breather frequency or one of its overtones fall into
the classical normal band
2
. A similar behavior is expected in the quantum case, which will
be studied in a future work.
The results we o bta ined in a KG lattice are rather encouraging for a possible future study
of the quantum acoustic lattices, as the FPU
52
chain. Even though we did not address
precisely that case, we found that in energy spectra where no gap occurs, the nonlinear
excitations may yet be distinguished and still exhibit a particlelike energy branch. Such
excitations ar e exp ected to emerge in the quantum F PU chain too, under the condition that
they corresponds to a suﬃciently large energy. Our numerical theory should be tractable
13
Page 13
on the o nedimensional FPU lattice with only few sites, even though the rank of our basis
might increase dramatically. Then, one could yet achieve the Hamiltonian diagonalization
with a iterative procedure as the Lanczos method.
Alongside the present work, we carried out the calculation of the dynamical structure
factor of the nonlinear KG lattice
50
. A simulation of the inelastic scattering has been
achieved so as to compare our theory to practical cases.
Acknowledgments
I gratefully acknowledge S. Aubry who introduced me to the theory of breathers, at coﬀee
breaks in Labo ratoire L´eon Brillouin (CEASaclay) .
∗
Electronic address: lproville@cea.fr
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 "Furthermore, Provile used a numerical method to study same properties of biphonons (twoboson bound states) in Klein Gordon model [29]. The result of Ref. [30] indicates that biphonons is the simplest quantum breathers. Some works have shown that biphonons can also exist in the FermiPastaUlam atomic lattice model [31][32][33][34] . "
[Show abstract] [Hide abstract] ABSTRACT: Twodiscrete breathers are the bound states of two localized modes that can appear in classical nonlinear lattices. I investigate the quantum signature of twodiscrete breathers in the system of ultracold bosonic atoms in optical lattices, which is modeled as Bose–Hubbard model containing n bosons. When the number of bosons is small, I find numerically quantum twobreathers by making use of numerical diagonalization and perturbation theory. For the cases of a large number of bosons, I can successfully construct quantum twobreather states in the Hartree approximation. 
 "In addition, we obtained the energy level formula of the system for quantum breather states, which suggests the energy of quantum breathers is quantized. It is worth noting that quantum breathers that we have found have obvious quantum properties, which are different from twophonon bound states or biphonons567. We believe that our work may be useful for understanding localization phenomenon in some quantum systems. "
[Show abstract] [Hide abstract] ABSTRACT: In this paper, quantum solitons in the Fermi–Pasta–Ulam (FPU) model are investigated analytically. By using the canonical transform method and numberconserving approximation, we obtain the normal form of the phononconserving quantized Hamiltonian. In order to convert the quantized Hamiltonian into the coordinate space, we employ the inverse Fourier transform. With the help of the Hartree approximate and the semidiscrete multiplescale method, the nonlinear Schrödinger (NLS) equation is derived. The results show that quantum solitons may exist in the FPU model. Moreover, it is found that moving quantum solitons become quantum intrinsic localized modes under certain condition. In addition, we obtain the energy level of quantum solitons, which indicates that the energy of such quantum solitons is quantized. 
 "Inspection of the spatial dependence of the various components of the n = 2 excitation operator revealed the localized nature of the excitation [40] . Proville has addressed the spatial and temporal correspondence between the classical and quantum breathers of the nonlinear KleinGordon lattice [41], by forming a Wannier wavepacket as a linear superposition of energy eigenstates which exhibited both the localized and oscillatory nature of the quantum excitations. In this paper, we shall investigate the lowest members of the hierarchy of quantized breathers in the α and β FermiPastaUlam problem within the ladder approximation. "
[Show abstract] [Hide abstract] ABSTRACT: We have calculated the lowest energy quantized breather excitations of both the β and the α FermiPastaUlam monoatomic lattices and the diatomic β lattice within the ladder approximation. While the classical breather excitations form continua, the quantized breather excitations form a discrete hierarchy labeled by a quantum number n. Although the number of phonons is not conserved, the breather excitations correspond to multiple bound states of phonons. The n=2 breather spectra are composed of resonances in the twophonon continuum and of discrete branches of infinitely longlived excitations. The nonlinear attributes of these excitations become more pronounced at elevated temperatures. The calculated n=2 breather and the resonance of the monoatomic β lattice hybridize and exchange identity at the zone boundary and are in reasonable agreement with the results of previous calculations using the numberconserving approximation. However, by contrast, the breather spectrum of the α monoatomic lattice couples resonantly with the singlephonon spectrum and cannot be calculated within a numberconserving approximation. Furthermore, we show that for sufficiently strong nonlinearity, the α lattice breathers can be observed directly through the singlephonon inelastic neutronscattering spectrum. As the temperature is increased, the singlephonon dispersion relation for the α lattice becomes progressively softer as the lattice instability is approached. For the diatomic β lattice, it is found that there are three distinct branches of n=2 breather dispersion relations, which are associated with three distinct twophonon continua. The twophonon excitations form three distinct continua: One continuum corresponds to the motion of two independent acoustic phonons, another to the motion of two independent optic phonons, and the last continuum is formed by propagation of two phonons that are one of each character. Each breather dispersion relation is split off the top from of its associated continuum and remains within the forbidden gaps between the continua. The energy splittings from the top of the continua rapidly increase, and the dispersions rapidly decrease with the decreasing energy widths of the associated continua. This finding is in agreement with recent observations of sharp branches of nonlinear vibrational modes in NaI through inelastic neutronscattering measurements. Furthermore, since the band widths of the various continua successively narrow as the magnitude of their characteristic excitation energies increase, the finding is also in agreement the theoretical prediction that breather excitations in discrete lattices should be localized in the classical limit.