## Research Overview

The Carr Theoretical Physics Group works primarily in quantum many body theory, nonlinear dynamics,
and artificial solid state materials, although our interests
extend into many fields of physics and both science and mathematics as a whole.
For a brief description of some specific areas we are working in at the moment,
please see the summaries below. For a list of people and projects,
see the group page.

**Quantum Phase Transitions** are phase transitions which are not caused
by a temperature change, as in a solid to liquid transition, but by an external parameter in the system. They
are most clearly observed at very low temperatures, where quantum mechanical effects are often strongest. Shown at
the left is an interference experiment at the University of Munich in Germany on ultracold bosonic
atoms trapped in laser standing waves. The standing waves
make an egg crate, and the atoms sit in the wells of the crate. In each panel, the peaks of the crate have
been made successively higher. Then the lasers have been turned off and the whole system has been allowed
to expand, resulting in a matter-wave interference pattern. The interference peaks gradually wash out from
the top left to lower right panels. This is a realization of the phase-number uncertainty relationship.
When the peaks of the egg crate are low, the atoms tunnel from well to well, the phase is maximally certain, and one has a
superfluid. When the peaks are high, the atoms are prevented from tunneling, the number
of atoms in each well is maximally certain,
one has what is called a Mott insulator, and the relative phase between wells in unknown, which destroys
the interference pattern. For an introduction to quantum phase transitions, see
Subir Sachdev's book.

**Ultracold Molecules** have recently been brought to the edge of quantum degeneracy
at JILA at the University of Colorado and
at the University of Innsbruck. Ultracold molecules in optical lattices
represent a designer solid state system which can be shaped to any crystal structure and can be used
to explore condensed matter physics in a well-controlled system. These molecules can have temperatures
smaller than one millionth of one Kelvin. Molecules have internal
rotational, vibrational, and other degrees of freedom which give them an unprecedented
level of versatility for applications in quantum computing, quantum information processing,
and quantum simulations. Moreover, cold and ultracold molecules
bring a new idea of chemistry into play in which, for example, chemical reactions can be Bose-enhanced.
The picture to the left shows heteronuclear diatomic molecules with an electric field to create a dipole moment.
This leads to long range interactions, an exciting new possibility to realize a strongly interacting ultracold system.
The pale purple arrows represent laser beams used to create the oscillatory standing wave, or optical lattice potential depicted.

**Graphene nanoengineering** seeks to manipulate a
newly created two-dimensional solid state material, graphene, to obtain novel electronic, mechanical,
magnetic, thermal, and chemical properties. We are
used to processor speed in computers doubling every two years (Moore's law). However,
we are out of room in silicon to write smaller features. In order to increase computing speed
we need new materials and new ideas. Graphene is an excellent candidate because of its high mobility at room temperature,
among other reasons. There is a lot of excitement about graphene on many fronts, from fundamental theory to applications.
The picture at the left shows a graphene metacrystal, which is made by laying down a pattern of defects on a 2D carbon surface.
Another such pattern leads to a new allotrope of carbon in two dimensions, which we
call dimerite.
Examples of different allotropes of carbon
in three dimensions are graphite (pencil lead) and diamond. Graphene is a 2D allotrope of carbon which takes the
form of a honeycomb lattice, just like a beehive. Dimerite is based on converting 4 hexagons into a pair of pentagons
and a pair of septagons by introducing an Inverse Stone-Thrower-Wales
defect in a regular pattern. A review on the electronic properties of graphene can be found here.

**Spin wave fractals** refer to fractal patterns
in excitations of the local magnetic moment of thin ferromagnetic films. Fractals
are self-similar structures which repeat at progressively finer scales. For instance,
in the fern at the left, the pattern of each frond is repeated in each of its sections.
Each section has a subsection which in turn repeats the original pattern, and so forth.
Fractals occur frequently in Nature. The
Magnetic Materials
and Applied Magnetics Laboratory at
Colorado State University has recently discovered fractals and chaos in magnetic feedback
rings. We believe a simple theory based on a modified nonlinear Schrodinger
equation can explain the results. The nonlinear Schrodinger
equation has very special mathematical properties,
including an infinite number of conservation laws. Spin waves in ferromagnetic feedback rings
have been used to generate a whole host of intriguing nonlinear phenomena, including
bright and dark soliton trains.

**Bose Einstein Condensates** (BEC's) were predicted in 1924 and realized in 1995
at JILA at the University of Colorado and
at the Massachusetts Institute of Technology, for which the
investigators jointly received the 2001 Nobel prize, as well as at Rice University, where the observation of
a BEC remained
controversial until 1997.
The picture at the left shows the original
JILA experiment. As the temperature decreases, a macroscopic number of Rubidium atoms, in this case about a million, "Bose
condense" into the same state. This means they all have the same energy and momentum. Since they are held
in a harmonic trap, entering the same momentum state means they also condense in space -- thus the formation
of the peak in the rightmost image. All particles in the Universe are divided into two categories: bosons and fermions.
Bosons prefer to be in the same state -- this is the essential principle behind a laser, where all the photons,
which are also bosons, "lase" in a beam of a single color. BEC's are extremely versatile and
have generated an extraordinary amount of research activity all around the world over the last fifteen years. Besides
their beauty in terms of fundamental physics, they have applications in precision measurement, interferometry,
and atom lasers.
For a great review from the quantum many body perspective by a Nobel prize
winning theorist (2003), see
Tony Leggett's article in Review of Modern Physics.

**Macroscopic Quantum Tunneling** is the study of
what quantum mechanical tunneling means for a many-body wavefunction. The essential idea
of tunneling is that the probability distribution of a particle can extend through
a potential barrier: put an atom in a closed container, and everyone once in a while
it pops out. This is in severe contrast to something like a marble in a jar. One of the
fundamental questions in quantum mechanics is how the microscopic behavior of things like
atoms becomes the macroscopic behavior of things like marbles. The picture at the left shows a potential
barrier in blue. The red curve shows a many body wavefunction, in this case the mean field
of a Bose-Einstein condensate. The green line sketches the path atoms take as they tunnel
through the barrier. This is a bit
like testing gravitational attraction at different length scales: does gravity really work the same
on the scale of astronomical units as it does on the scale of micrometers? Here we are asking if
quantum mechanics works in the same way for a million atoms in synch as it does for a single one in isolation.

**Emergent phenomena** refers to complex properties of a system
whose constituents are simple. For example, consciousness could be considered an emergent phenomenon
in a collection of neurons; the great red spot of Jupiter is an emergent property of its atmosphere.
Shown at the left are matter wave solitons, a specific example of an emergent property in
a Bose-Einstein condensate. Solitons are localized waves which do not disperse.
They have a number of beautiful mathematical properties, and are to nonlinear equations
what plane or sinusoidal waves are to linear ones. The image shows an experiment at
Rice University in which
a train of solitons is created from a Bose-Einstein condensate
condensate with attractive interactions. Each of the peaks acts like an independent particle which runs
up and down the blue strip. The attractive interactions exactly and robustly balance the dispersion, or quantum
pressure, which would otherwise cause the solitons to spread out like a wavepacket. Solitons appear
in many places in Nature, including fiber optics, plasmas, DNA, and as tsunamis in the ocean. They have been
observed in many contexts in Bose-Einstein condensates. Solitons are fundamental to the definition of
superfluidity in one dimension in the same way that their higher dimensional cousins, vortices, are
key to defining superfluidity in two and three dimensions. A superfluid is a special quantum
mechanical state which flows with zero friction. Solitons are sufficiently important to be
directly in the title of a major scientific journal,
Chaos, Solitons, and Fractals. Solitons, vortices, and other emergent phenomena described
by nonlinear wave equations are
an important subject of study in our group.

**Macroscopic superposition states** bring quantum mechanics up from the single particle atomic or subatomic
level to the macroscopic world we live in. One of the more radical predictions of quantum mechanics is that a particle can be both here
and there simultaneously. We have very good evidence that this is true, in the form of interference patterns and other measurable
predictions. But is this radical idea true at macrosopic scales? The picture at the left shows a cat in a box. The apparatus next to
the cat represents a radioactive device which hinges on the decay of a single unstable nucleus. If the nucleus decays, a vial of poison
is broken and the cat dies (no actual cats were harmed in this thought experiment). The decay process is quantum mechanical,
so that the nucleus is in a superposition of two states, decayed and "un-decayed", so to speak.
Before the observor opens the box, is the cat
alive or dead? Quantum mechanics says that the cat is both alive and dead. Can we actually set up such an experiment
in physics, to realize these completely
counter-intuitive results? Creating macroscopic superposition states is an extension of quantum information processing, for
which there are many candidates. Particularly promising possibilities are Bose-Einstein
condensates in a double well and superconducting current loops.