Solitons in Bose-Einstein condensates
Discrete solitons in nonlinear optical systems
Breathers in condensed matter
Breathers in DNA
Protein folding mechanisms
The phenomenon of Bose–Einstein condensation is a quantum-phase transition originally
predicted by Bose and Einstein in 1924. In particular, it was shown that below a
critical transition temperature Tc, a finite fraction of particles of a boson gas (i.e. whose particles
obey the Bose statistics) condenses into the same quantum state, known as the Bose–Einstein
condensate (BEC). Although Bose–Einstein condensation is known to be a fundamental
phenomenon, connected, e.g. to superfluidity in liquid helium and superconductivity in metals,
BECs were experimentally realized 70 years after their theoretical prediction:
this major achievement took place in 1995, when different species of dilute alkali vapours
confined in a magnetic trap (MT) were cooled down to extremely low temperatures,
and has already been recognized through the 2001 Nobel prize in Physics. This first
unambiguous manifestation of a macroscopic quantum state in a many-body system sparked
an explosion of activity, as reflected by the publication of several thousand papers related to
BECs since then. Nowadays there exist more than fifty experimental BEC groups around the
world, while an enormous amount of theoreticalwork has followed and driven the experimental
efforts, with an impressive impact on many branches of physics.
From a theoretical standpoint, and for experimentally relevant conditions, the static and dynamical properties of a BEC can be described by means of an effective mean-field equation known as the Gross–Pitaevskii (GP) equation. This is a variant of the famous nonlinear Schrödinger (NLS) equation (incorporating an external potential used to confine the condensate), which is known to be a universal model describing the evolution of complex field envelopes in nonlinear dispersive media. As such, the NLS equation is a key model appearing in a variety of physical contexts, ranging from optics, to fluid dynamics and plasma physics, while it has also attracted much interest from a mathematical viewpoint. The relevance and importance of the NLS model is not limited to the case of conservative systems and the theory of solitons; in fact, the NLS equation is directly connected to dissipative universal models, such as the complex Ginzburg–Landau equation, which have been studied extensively in the context of pattern formation.
In the case of BECs, the nonlinearity in the GP (NLS) model is introduced by the interatomic interactions, accounted for through an effective mean field. Importantly, the mean-field approach, and the study of the GP equation, allows the prediction and description of important, and experimentally relevant, nonlinear effects and nonlinear waves, such as solitons and vortices. These, so-called, matter-wave solitons and vortices can be viewed as fundamental nonlinear excitations of BECs, and as such have attracted considerable attention. Importantly, they have also been observed in many elegant experiments using various relevant techniques. These include, among others, phase engineering of the condensates in order to create vortices or dark matter-wave solitons in them, the stirring (or rotation) of the condensates providing angular momentum creating vortices and vortex lattices, the change in scattering length (from repulsive to attractive via Feshbach resonances) to produce bright matter-wave solitons and soliton trains in attractive condensates or set into motion a repulsive BEC trapped in a periodic optical potential referred to as optical lattice to create gap matter-wave solitons. As far as vortices and vortex lattices are concerned, it should be noted that their description and connection to phenomena as rich and profound as superconductivity and superfluidity were one of the themes of the Nobel prize in Physics in 2003.
For many decades, optics has provided one of the traditional testbeds for investigations
of nonlinear wave propagation. For example, the (continuous) nonlinear
Schrödinger (NLS) equation provides a dispersive envelope wave model for describing
the electric field in optical fibers. In the presence of a spatially-discrete external
potential (such as a periodic potential), one can often reduce the continuous NLS to
the discrete NLS (DNLS).
An optical waveguide is a physical structure that guides electromagnetic waves in the optical spectrum. Early proposals in nonlinear optics suggested that light beams can trap themselves by creating their own waveguide through the nonlinear Kerr effect. Waveguides confine the diffraction, allowing spatial solitons to exist. In the late 1990s, Eisenberg et al. showed experimentally that a similar phenomenon (namely, discrete spatial solitons) can occur in a coupled array of identical waveguides. One injects low-intensity light into one waveguide (or a small number of neighboring ones); this causes an ever increasing number of waveguides to couple as it propagates, analogously to what occurs in continuous media. If the light has high intensity, the Kerr effect changes the refractive index of the input waveguides, effectively decoupling them from the rest of the array. That is, certain light distributions propagate with a fixed spatial profile in a limited number of waveguides.
Photorefractive crystals can be used to construct 2D periodic lattices via plane wave interference by employing a technique known as optical induction. This method has become a very important playground for investigations of nonlinear waves in optics. The theoretical prediction and subsequent experimental demonstration of 2D discrete optical solitons has led to the construction and analysis of entirely new families of discrete solitons. The extra dimension allows much more intricate nonlinear dynamics to occur than is possible in the 1D waveguides discussed above. Early experiments demonstrated novel self-trapping effects such as the excitation of odd and even nonlinear localized states. They also showed that photorefractive crystals can be used to produce index gratings that are more controllable than those in fabricated waveguide arrays.
Various researchers have since exploited the flexibility of photorefractive crystals to create interesting, robust 2D structures that have the potential to be used as carriers and/or conduits for data transmission and processing in the setting of all-optical communication schemes. In the future, 1D arrays in 2D environments might be used for multidimensional waveguide junctions, which has the potential to yield discrete soliton routing and network applications. An ever-larger array of structures has been predicted and experimentally obtained in lattices induced with a self-focusing nonlinearity.
Different experiments suggest that moving breathers could be responsible for dark tracks in mica muscovite, the sputtering of atoms after radiation with alpha particles or the removal of defects after ion irradiation, and the unusual low temperature of the reconstructive transformation of mica muscovite to lutetium disilicate. Preliminary results in simulations suggest that this explanation is possible, but it is necessary the simulation and study of more realistic models, with more particles, better parameters and taking into account more spatial dimensiones. All this will allow to determine for which materials and for which crystal structures moving and stationary breathers are possible and, therefore, will suggest new experiments. If this hypothesis is confirmed, there exist the possibility of technological applications as defect detection and cleaning, or the trapping or radionuclides.
DNA is the molecule that encodes the information that organisms need to live and reproduce
themselves. Fifty years after the discovery of its double helix structure  it is still fascinating
physicists, as well as biologists, who try to unveil its remarkable properties. But the structure
of the molecule is only a static picture and it is now understood that the dynamics of biological
molecules is also essential for their function. This is particularly true for DNA. The genetic
code, defined by the sequence of base pairs which form the plateaus of the double helix,
is hidden in the core of the helix. When a gene is transcribed the hydrogen bonds
that link the two bases in a base pair are broken in a region called the 'transcription bubble'
and the bases are exposed outside the helix for chemical reaction. This is possible because
those bonds that hold the bases together are weak. As a result, even in the absence of enzymes
involved in the reading or duplication of the code,DNAundergoes large amplitude fluctuations.
The lifetime of a base pair, i.e. the time during which it stays closed, is only of the order of a few
milliseconds. Experiments show that these fluctuations, known by biologists as the breathing
of DNA, are highly localized and may open a single base pair while the adjacent ones stay
Therefore, when it is viewed at the scale of a base pair, DNA appears as a rather regular lattice, which undergoes very large amplitude motions so that the nonlinear properties of the bonds that connect its elements cannot be ignored. DNA is an attractive system for nonlinear science because its properties can be probed very accurately by experiments that combine the methods of physics with biological tools. The experimental results impose constraints on the theoretical modelling and therefore help us in the design of an appropriate model. Although significant progress has been made since the first attempts to describe DNA with nonlinear models, there are however many points that stay open and raise interesting questions for nonlinear physics.
Physically, proteins are polymers whose units are the amino acids.
Although the way in which they function remains obscure, it is known that to perform their functions they must, first of all,
acquire a well defined average structure,
known as the native structure. A fundamental question for Biology, Medicine and the Pharmaceutical and Biotechnology industries,
known as the protein folding problem, is how a given sequence of amino acids, in cells, most of the times assumes the
The vibrational excites state (VES) hypothesis is the idea that protein folding and function involves a first step in which energy is stored in the form of vibrational excited states. Another way to describe it is to say that protein folding and function are not driven by thermal agitation, as happens according to Anfinsen's thermodynamic hypothesis and that instead, the drivers of protein folding and function are transient forces that arise from the non-radiative decay of vibrational excited states.
A biological role for vibrational excited states was first proposed by McClare in connection with a possible crisis in bioenergetics. McClare's proposal was taken up by Davydov who wanted to explain the conformational changes responsible for muscle contraction, where the trigger is the energy donating reaction of the hydrolysis of Adenosinetriphosphate (ATP). Davydov's assumption was that the first event after the hydrolysis of ATP is the storing of the energy released in the chemical reaction in a vibrational mode of the peptide group, known as amide I. In the Davydov model, the interaction of the amide I with the vibrations of hydrogen bonds leads to a localization of the amide I excitation in a few amino acids, much as the interaction of an electron with the phonon modes of a polarizable crystal leads to a polaron. In solid state physics the electron generates distortions (phonon modes) that in turn lead to a localized, lower energy state for the electron and in the Davydov model the amide I excitation generates a distortion in the neighbouring hydrogen bonds that in turn localizes the amide I excitation. In both cases, we say that we have a self-trapped state (of the amide I or of the electron). Both Davydov's analytical results and Scott and co-worker's simulations of more realistic discrete models indicate that the state constituted by an amide I excitation and its associated hydrogen bond distortions, a self-trapped state known in the literature as the Davydov soliton, is stable at low temperatures.
Biological systems live at finite temperature and thus it is always essential to find out how thermal fluctuations affect the states of both amide I vibrations and phonons. Monte Carlo studies of the Davydov/Scott model, both in the full quantum and in the mixed quantum-classical regimes, have already shown that the lattice distortion, at equilibrium, is indistinguishable in the two regimes. Our aim is to find out what happens in non-equilibrium conditions.
Suggestions for further reading
- Theory of Bose-Einstein condensation in trapped gases
F Dalfovo, S Giorgini, LP Pitaevskii and S Stringari
Reviews of Modern Physics 71 (1999) 463
- Nonlinear waves in Bose–Einstein condensates: physical relevance and mathematical techniques
R Carretero-González, DJ Frantzeskakis and PG Kevrekidis
Nonlinearity 21 (2008) R139.
- Discrete solitons in Optics
F Lederer et al
Physics Reports 463 (2008) 1
- Discrete breathers — Advances in theory and applications
S Flach and AV Gorbach
Physics Reports 467 (2008) 1
- Nonlinear dynamics and statistical physics of DNA
Nonlinearity 17 (2004) R1
- Modelling DNA at the mesoscale: a challenge for nonlinear science?
M Peyrard, S Cuesta-López and G James
Nonlinearity 21 (2008) T91
- Davydov's soliton
Physics Reports 217 (1992) 1