The most outstanding experimental discovery in
recent times is Bose-Einstein condensation (BEC) in 1995 and
excellent reviews on BEC exist in the vast ocean of literature (Ketterle, Rev.
Mod. Phys. 74, (2002)1131). This has
triggered both theoretical and experimental works on this fascinating topic of
research. The reasons for these vigorous activities are: a) it gives an
opportunity to open a new window for a macroscopic view of quantum mechanics,
and b) it makes such studies most interesting in the field of matter-wave
relations. Experiments show nonlinear excitation within the wave that is known
as solitons, and the properties of such bosonic particles in a condensate could
manipulate them in periodic or other potential. Some evidence exists for the
formation of both dark and bright solitions in the condensate, but the concept
of bound state still remains somewhat illusive, despite a lot of activities on
nonlinear optical systems that are important for many devices. This could
create motivation to study bound state with a connection to BEC.
Further,
both lower and upper bounds are found out in relation to frequency and the
extent of condensation in the bound state. Although ferroelectrics data are
used for general theoretical study, it can also be extended to other relevant
systems, e.g. in magnon system with two-well Landau potential (Demokritov,
Nature, 433, (2006) 430). This effort is made through Klein-Gordon (K-G) model
which on perturbation gives rise to NLSE that is a variant of GPE, which is somehow
popular and commonly used for studying BEC. Landau-Ginzburg (L-G)
potential is used in our discrete Hamiltonian for nonlinear optical materials
like ferroelectrics (Phys. Rev. B 81, (2010) 064104) and metamaterials (Phys.
Rev. B 84, (2011) 155429) to probe localization for discrete breathers. Also, a
perturbation on our K-G model gives rise to NLSE (Phys. Rev. B 86, (2012)
184101) and showed both bright and dark solitons with ‘discrete energy levels’,
estimated via hypergeometric function.
The coherent molecular BEC was shown in
one-dimension by mean field theories of parametric nonlinearities that convert
two solitons to one (and vice-versa) for non-integrable equations (Drummond, Phys.
Rev E 54, (1996) 896). Molecular BEC was invoked for a reaction between molecules
to engineer condensates of heavier molecules where macroscopic occupation of
single molecular quantum state gives rise to the coherent bosonic stimulation
(Kokkelman, Phys Rev. A 63, (2001) R031601). BEC was also reported on lithium molecules in
an optical trap with spin mixture of fermionic Li atoms by measuring a
collective excitation mode (Denschlag, Science 302, (2003) 2101). This gives rise
to further motivation to theoretically explore BEC in the realm of bound state
that might help towards better understanding of switching phenomenon in a vast
area of devices including many nano devices involving lithium niobate. The
intention is to show that molecular BEC formation is possible in this system
that is described by frequency and temperature in the bound state.
The dark and bright solitons in lithium
niobate ferroelectric are already established in Klein-Gordon lattice. For
small oscillations, the modal dynamics is characterized as bound states,
revealed via associated Legendre polynomial. Bose Einstein condensation takes
place around bosonic particles having different wave functions within the bound
states in the temperature region T =
0 to Tc. The pairing and
interplay between the dark and bright solitons occur with their effect on the
condensation. The bound state disappears after a critical frequency (i.e. equivalent
to a critical temperature). The quasi-particles then become phonons in the
unbound states that propagate through the domains, when scattering length
increases.
Despite mathematical rigour, the physics
of the above work may not be palatable by many relevant Physics Journals.
However, a recent effective mass calculation shows the temperature of BEC
formation lies in the range of micro-Kelvin.
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