Braids in Classical Dynamics
Christopher Moore
PRL 70, 3675--3679 (1993)
URL: http://link.aps.org/abstract/PRL/v70/p3675
This was a fascinating paper! I found it in the references of the paper on choreographic orbits. I never learned classical mechanics like this ...
In this paper, Moore analyzes orbits of N-body systems in 2+1 dimensions. He considers two-body potentials of the form
V(i,j) = A m(i) m(j) r(i,j)^X
as a function of the exponent X.
His approach to finding allowed orbits is novel. He suggests a topological classification. Basically, Moore starts out with a braid with N strands. (Every orbit in a plane defines a braid in 2+1 dimensions, and mathematicians know how to determine which braids are topologically equivalent.) Next, he defines an action for N-particle orbits whose minima are solutions of Newton's second law. He shows that starting with the braid and relaxing it toward a minimum of the action can result in only three things:
1) A particle flies off to infinity.
2) Two particles collide.
3) The braid relaxes to a topologically equivalent orbit that satisfies F = ma.
Armed with this tool, he examines the types of braids that yield solutions for a particular exponent X in the potential. First, he shows that escape is impossible if X < 2. That in itself is in interesting result. Next, he shows that for X <= -2, relaxation never leads to a collision. Thus, every braid is an allowed orbit for a potential that falls off at least as fast as 1/R^2.
The X=2 case is interesting because the system is integrable, and all particles orbit the center of mass with the same period. Moore shows that the only allowed braids are those in which any two particles have a winding number of +1 or -1 (as long as they don't collide at the origin). These are called harmonic braids.
After this, he reports numerical investigations of other braid types. He identifies solutions to solutions to the 3-body problem I didn't know about last week: the figure eight discussed in the choreographic orbits paper, and a braid I'll call the cross-circle. The table on page 3 is really interesting.
The application of topology to the 3-body problem is interesting and impressive. I'll have to get a copy of Arnol'd's book and see if I can learn a few of these methods.
Friday, May 25, 2007
Electric NMR
Electric Dipole Echoes in Rydberg Atoms
S. Yoshida, et al.
PRL 98, 203004 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e203004
This group from Rice has demonstrated the electric analog of spin echoes.
In NMR, spins precess around an applied magnetic field. Due to local variations in the magnetic field, not all spins precess at the same rate. However, one can apply a pulse that flips every spin in the sample, and all the spins basically reverse their motion. If the spins start precessing at t=0 and the flipping pulse (a pi-pulse) is applied at t=T, then at t=2T, all the spins will be right back where they started, even though they were precessing at different rates! The large magnetic moment of the sample at t=2T is called a spin echo. Effects like collisions, diffusion, thermal excitations, and other interactions prevent the system from returning to its exact initial configuration, but the echo can be detected after rather long delays.
So what does this have to do with electric dipoles? I learned that to first order in the applied field, the equation of motion for the electric dipole also describes precession. That's not exactly true --- the thing that actually precesses is a pseudospin comprised of the orbital angular momentum and something called the Runge-Lenz vector, which is proportional to the electric dipole moment. The difference of two pseudospins gives the Runge-Lenz vector. The authors say this is true of a classical dipole as well as the quantum theory.
One major difference between NMR and the experiment described here is that electric dipoles oscillate much more rapidly than magnetic dipoles. To make the relevant time scale as long as possible, the use Rydberg atoms --- potassium atoms in the n=350 level.
When they performed the experiment, the authors observed a marked increase of the survival probability when a flipping pulse was applied, in contrast with the exponential decay observed without the flipping pulse.
Another difficulty of this experiment versus NMR is the effect of the environment. Couplings to and interactions with spurious electric fields in the environment lead to much more rapid decoherence in the Rydberg gas than in a typical NMR sample. The authors point out that this is not necessarily a bad thing: dipole echoes could prove to be a useful tool in studying decoherence.
S. Yoshida, et al.
PRL 98, 203004 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e203004
This group from Rice has demonstrated the electric analog of spin echoes.
In NMR, spins precess around an applied magnetic field. Due to local variations in the magnetic field, not all spins precess at the same rate. However, one can apply a pulse that flips every spin in the sample, and all the spins basically reverse their motion. If the spins start precessing at t=0 and the flipping pulse (a pi-pulse) is applied at t=T, then at t=2T, all the spins will be right back where they started, even though they were precessing at different rates! The large magnetic moment of the sample at t=2T is called a spin echo. Effects like collisions, diffusion, thermal excitations, and other interactions prevent the system from returning to its exact initial configuration, but the echo can be detected after rather long delays.
So what does this have to do with electric dipoles? I learned that to first order in the applied field, the equation of motion for the electric dipole also describes precession. That's not exactly true --- the thing that actually precesses is a pseudospin comprised of the orbital angular momentum and something called the Runge-Lenz vector, which is proportional to the electric dipole moment. The difference of two pseudospins gives the Runge-Lenz vector. The authors say this is true of a classical dipole as well as the quantum theory.
One major difference between NMR and the experiment described here is that electric dipoles oscillate much more rapidly than magnetic dipoles. To make the relevant time scale as long as possible, the use Rydberg atoms --- potassium atoms in the n=350 level.
When they performed the experiment, the authors observed a marked increase of the survival probability when a flipping pulse was applied, in contrast with the exponential decay observed without the flipping pulse.
Another difficulty of this experiment versus NMR is the effect of the environment. Couplings to and interactions with spurious electric fields in the environment lead to much more rapid decoherence in the Rydberg gas than in a typical NMR sample. The authors point out that this is not necessarily a bad thing: dipole echoes could prove to be a useful tool in studying decoherence.
Wednesday, May 23, 2007
Experimental Model of Graphene
Interlayer Interaction and Electronic Screening in Multilayer Graphene Investigated with Angle-Resolved Photoemission Spectroscopy
T. Ohta, et al.
PRL 98, 206802 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e206802
This group from Berkeley has used the Advanced Light Source to generate a phenomenological tight-binding model for graphite samples with 1 to 4 layers. They used angle-resolved photoemission to construct the energy bands of their samples, then tuned the parameters of their tight-binding model to reproduce the observed band structure.
There were several interesting findings:
1) The splitting of the pi-electron bands in a 4-layer sample is nearly identical to that in bulk graphite. However, the hopping integral is larger in the layered sample and the screening length is smaller.
2) The charge density in the samples was nearly constant. It did not depend on the number of layers or the way in which they were stacked.
3) In layered samples, the oscillations of photoemission intensity as a function of momentum perpendicular to the beam are similar to those observed in quantized thin film states.
4) Single graphene sheets show no oscillations of this nature. The authors point out that there is more to this than the fact that graphene is only a single layer of carbon atoms. It indicates that there is virtually no interaction with the substrate. Graphene really is like an isolated 2D system.
5) The on-site Coulomb interactions differ from layer to layer. I have not seen this effect included in simple models of graphene bilayers. Then again, it's been a while since I read one of those papers.
T. Ohta, et al.
PRL 98, 206802 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e206802
This group from Berkeley has used the Advanced Light Source to generate a phenomenological tight-binding model for graphite samples with 1 to 4 layers. They used angle-resolved photoemission to construct the energy bands of their samples, then tuned the parameters of their tight-binding model to reproduce the observed band structure.
There were several interesting findings:
1) The splitting of the pi-electron bands in a 4-layer sample is nearly identical to that in bulk graphite. However, the hopping integral is larger in the layered sample and the screening length is smaller.
2) The charge density in the samples was nearly constant. It did not depend on the number of layers or the way in which they were stacked.
3) In layered samples, the oscillations of photoemission intensity as a function of momentum perpendicular to the beam are similar to those observed in quantized thin film states.
4) Single graphene sheets show no oscillations of this nature. The authors point out that there is more to this than the fact that graphene is only a single layer of carbon atoms. It indicates that there is virtually no interaction with the substrate. Graphene really is like an isolated 2D system.
5) The on-site Coulomb interactions differ from layer to layer. I have not seen this effect included in simple models of graphene bilayers. Then again, it's been a while since I read one of those papers.
General Relativity and Dance Steps
Choreographic Solution to the General Relativistic Three Body Problem
T. Imai, T. Chiba, and H. Asada
PRL 98, 201102 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e201102
This group from Hirosaki University in Japan has found a periodic solution of the 3-body problem for Newtonian gravity plus first order corrections from general relativity. Though the paper is not well-written and the background is not sufficient for understanding the group's work, I found the result interesting enough to read through all four pages.
I know a little about the three-body problem. My main impression is that there is no good way to separate coordinates and find a general solution. It admits both regular and chaotic solutions.
The authors provided some interesting historical background. Euler found one of the first solutions to the 3-body problem, with all three masses in a line. Though the distance between masses may change, the ratio of the distances does not. Euler found his solution in 1765. Seven years later, Laplace found a second, highly symmetric solution in which the three masses are at the corners of an equilateral triangle. According to the authors, these are the only solutions in which each particle's orbit is an ellipse.
Later, Poincare showed that not all solutions can be obtained analytically. In 1993, Christopher Moore found a solution in which all three particles move along a figure eight --- a very interesting solution to the problem! Moore was looking at general potentials in 2+1 dimensions and the types of braids the three bodies would weave in spacetime.
This figure eight, along with the solutions of Euler and Laplace, are examples of a more general class of solutions called choreographic orbits. In such an orbit, each of the three particles moves in a single closed orbit. It's as if all three particles are performing a dance routine. Some of Moore's braids are quite complicated indeed!
One of the best known effects of general relativity is the precession of Mercury's perhelion. Mercury's orbit around the sun is an ellipse with rotating axes. Therefore, it is not periodic (at least on short time scales). Do the effects of general relativity make choreographic orbits impossible?
The authors show that the first order corrections to Newtonian gravity do admit a figure eight solution, so at least some choreographic orbits are permitted. The authors had to choose the initial conditions carefully to obtain a periodic orbit.
In the conclusion, the authors point out that the higher order effects of general relativity might not have solutions. It seems to me that once gravitational waves are included, no periodic solutions will be possible. If a system is radiating away its energy, it can never return to its initial condition, thus the orbits of its constituent particles are not closed. I don't believe any 3-body system is symmetric enough that it would not emit gravitational wave.
T. Imai, T. Chiba, and H. Asada
PRL 98, 201102 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e201102
This group from Hirosaki University in Japan has found a periodic solution of the 3-body problem for Newtonian gravity plus first order corrections from general relativity. Though the paper is not well-written and the background is not sufficient for understanding the group's work, I found the result interesting enough to read through all four pages.
I know a little about the three-body problem. My main impression is that there is no good way to separate coordinates and find a general solution. It admits both regular and chaotic solutions.
The authors provided some interesting historical background. Euler found one of the first solutions to the 3-body problem, with all three masses in a line. Though the distance between masses may change, the ratio of the distances does not. Euler found his solution in 1765. Seven years later, Laplace found a second, highly symmetric solution in which the three masses are at the corners of an equilateral triangle. According to the authors, these are the only solutions in which each particle's orbit is an ellipse.
Later, Poincare showed that not all solutions can be obtained analytically. In 1993, Christopher Moore found a solution in which all three particles move along a figure eight --- a very interesting solution to the problem! Moore was looking at general potentials in 2+1 dimensions and the types of braids the three bodies would weave in spacetime.
This figure eight, along with the solutions of Euler and Laplace, are examples of a more general class of solutions called choreographic orbits. In such an orbit, each of the three particles moves in a single closed orbit. It's as if all three particles are performing a dance routine. Some of Moore's braids are quite complicated indeed!
One of the best known effects of general relativity is the precession of Mercury's perhelion. Mercury's orbit around the sun is an ellipse with rotating axes. Therefore, it is not periodic (at least on short time scales). Do the effects of general relativity make choreographic orbits impossible?
The authors show that the first order corrections to Newtonian gravity do admit a figure eight solution, so at least some choreographic orbits are permitted. The authors had to choose the initial conditions carefully to obtain a periodic orbit.
In the conclusion, the authors point out that the higher order effects of general relativity might not have solutions. It seems to me that once gravitational waves are included, no periodic solutions will be possible. If a system is radiating away its energy, it can never return to its initial condition, thus the orbits of its constituent particles are not closed. I don't believe any 3-body system is symmetric enough that it would not emit gravitational wave.
Monday, May 21, 2007
Shining Light through a Wall with Axions
Transparency of the Sun to Gamma Rays Due to Axionlike Particles
Malcolm Fairbairn, Timur Rashba, and Sergey Troitsky
PRL 98, 201801 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e201801
Interesting stuff! The paper centers around the following logical deduction:
If axions allow light to pass through walls, then axions would allow gamma rays to pass through the sun.
• What is an axion?
An axion is a hypothetical pseudoscalar particle. (It behaves like a scalar function, except it changes sign under spatial reflections.) Axions cn couple to photons in a magnetic field. They are expected --- if they exist --- tobe light and weakly coupled.
Although the coupling between axions and photons is weak, it would have observable consequences. In a magnetic field, some light would be converted to axions, and interaction between left and right circularly polarized light would be mediated by axions. In QED, there is no coupling between the two opposite polarizations.
• Have axions been observed?
Maybe. An experiment called PVLAS detected a shift in the polarization of a laser passing through a strong magnetic field. The results are consistent with an axion mass of 1 meV and an inverse coupling of 100 TeV. However, an experiment called CAST claims to have ruled out a coupling that strong --- even though 100 TeV is a rather weak, unless you're a string theorist!
• How might one look for axions?
One method is that employed by the people at PVLAS: to search for polarization shifts in a strong magnetic field. However, the coupling of axions and photons offers a second, more exotic search technique: look for the transmission of light through an opaque object.
A strong magnetic field on one side of the object would convert some photons to axions. The axions would be able to pass through the object while the photons would not. A second strong magnetic field on the other side of the barrier would convert some of the transmitted axions back into photons, so it would appear that some light passed through the object. In short, if axions exist, then two strong magnetic fields would allow you to shine your flashlight through a rock!
• What's the sun got to do with any of this?
The group of physicists who wrote this paper believe that astronomical observations could determine the existence of axions. They have constructed a simple analytic model and numerically investigated a more detailed model, both of which predict the magnetic fields of the sun should allow the transmission of gamma rays through the sun via the axion-photon conversion process described above. The sun is normally opaque to gamma rays, and it does not produce a significant amount of gamma rays, so there should be little background interference.
The requirement for this type of observation is that a distant gamma ray source be identified whose line of sight from earth (or a space-based gamma ray detector) is eclipsed by the sun. If there are no axions, or if the coupling is too weak, the gamma ray source will disappear behind the sun. If gamma rays are still detected when the sun is between the source and the detector, it will provide strong evidence for the existence of axions.
There are gamma ray detectors on earth. Unfortunately for the authors, only one gamma ray source has been studied whose line of sight is obstructed by the sun. The statistics on this single source are not good enough to rule either way on the existence of axions, however the predicted and observed fluxes agree rather well. The flux definitely did not drop to zero when the sun obstructed the source.
In coming years, astronomical observations like those discussed by the authors may resolve the contradictory findings of PVLAS and CAST. The authors also suggest that decommissioned magnets from particle accelerators may allow for observation of axion-assisted photon tunneling in earth-based labs. No matter where the experimental data comes from, our knowledge of axions is going to increase quite a lot in the coming years.
Malcolm Fairbairn, Timur Rashba, and Sergey Troitsky
PRL 98, 201801 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e201801
Interesting stuff! The paper centers around the following logical deduction:
If axions allow light to pass through walls, then axions would allow gamma rays to pass through the sun.
• What is an axion?
An axion is a hypothetical pseudoscalar particle. (It behaves like a scalar function, except it changes sign under spatial reflections.) Axions cn couple to photons in a magnetic field. They are expected --- if they exist --- tobe light and weakly coupled.
Although the coupling between axions and photons is weak, it would have observable consequences. In a magnetic field, some light would be converted to axions, and interaction between left and right circularly polarized light would be mediated by axions. In QED, there is no coupling between the two opposite polarizations.
• Have axions been observed?
Maybe. An experiment called PVLAS detected a shift in the polarization of a laser passing through a strong magnetic field. The results are consistent with an axion mass of 1 meV and an inverse coupling of 100 TeV. However, an experiment called CAST claims to have ruled out a coupling that strong --- even though 100 TeV is a rather weak, unless you're a string theorist!
• How might one look for axions?
One method is that employed by the people at PVLAS: to search for polarization shifts in a strong magnetic field. However, the coupling of axions and photons offers a second, more exotic search technique: look for the transmission of light through an opaque object.
A strong magnetic field on one side of the object would convert some photons to axions. The axions would be able to pass through the object while the photons would not. A second strong magnetic field on the other side of the barrier would convert some of the transmitted axions back into photons, so it would appear that some light passed through the object. In short, if axions exist, then two strong magnetic fields would allow you to shine your flashlight through a rock!
• What's the sun got to do with any of this?
The group of physicists who wrote this paper believe that astronomical observations could determine the existence of axions. They have constructed a simple analytic model and numerically investigated a more detailed model, both of which predict the magnetic fields of the sun should allow the transmission of gamma rays through the sun via the axion-photon conversion process described above. The sun is normally opaque to gamma rays, and it does not produce a significant amount of gamma rays, so there should be little background interference.
The requirement for this type of observation is that a distant gamma ray source be identified whose line of sight from earth (or a space-based gamma ray detector) is eclipsed by the sun. If there are no axions, or if the coupling is too weak, the gamma ray source will disappear behind the sun. If gamma rays are still detected when the sun is between the source and the detector, it will provide strong evidence for the existence of axions.
There are gamma ray detectors on earth. Unfortunately for the authors, only one gamma ray source has been studied whose line of sight is obstructed by the sun. The statistics on this single source are not good enough to rule either way on the existence of axions, however the predicted and observed fluxes agree rather well. The flux definitely did not drop to zero when the sun obstructed the source.
In coming years, astronomical observations like those discussed by the authors may resolve the contradictory findings of PVLAS and CAST. The authors also suggest that decommissioned magnets from particle accelerators may allow for observation of axion-assisted photon tunneling in earth-based labs. No matter where the experimental data comes from, our knowledge of axions is going to increase quite a lot in the coming years.
Thursday, May 17, 2007
Laser Cooling of Semiconductors
Surface Plasmon Assisted Laser Cooling of Solids
Jacob Khurgin
PRL 98, 177401 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e177401
Laser cooling of atoms is based on the Doppler effect. Laser frequencies are tuned below an atomic transition so that atoms moving toward the laser source will be more likely to absorb a photon than those moving away. The atom will then emit a photon in a random direction as it relaxes back down to its ground state, but the radiated photon will be of higher energy than the absorbed photon. As a result, the atom has radiated away some of its kinetic energy, so it has been cooled by a laser.
In solids the basic idea is similar. The solid absorbs one frequency of light and emits at a higher frequency, losing energy as a result. The mechanism is different, and it called anti-Stokes photoluminescence. Light is absorbed at a specfic frequency, then this excited state comes to thermal equilibrium with the system. Later, light is emitted at a higher frequency. The shift in frequency would be on the order of kT, where T is the temperature of the solid, and the system would gradually cool down.
Khurgin points out that there are several difficulties in the case of semiconductors. There are a lot of nonradiative decay channels, and the high index of refraction leads to low efficiencies --- i.e., even if you generate a photon of the right frequency, it's hard for it to get out of the semiconductor. One way to improve the efficiency is to make the density of absorbing states small and the density of emitting states large. When a photon is absorbed, the energy is more likely to be transferred to a higher-energy emitting state than to remain in the absorbing state for a time, then be re-emitted at the same frequency.
Khurgin's approach to the problem is to exploit surface plasmon polaritons. These occur at the interface between a dielectric and a metal at frequencies where the dielectric constants of the two media are equal in magnitude but opposite in sign. The density of plasmon states has a sharp resonance, which leads to an increase of spontaneous emission at the resonant frequency. (Apparently, Purcell worked this out back in 1946.)
Khurgin notes that the plasmon modes still have to couple to radiative modes before they give up their energy, so it might seem that nothing has been gained. However, he goes on to demonstrate that the plasmons can couple to the phonons of the metal. The metal will heat up, but the goal was never to cool the metal and the dielectric together --- only to cool the dielectric medium.
Based on this observation, Khurgin proposes placing a layer of silver on top of a gallium arsenide layer with a gap between them of a couple nanometers. The gap is a thermal insulator between the silver and gallium arsenide. The only coupling between the two systems are the plasmons. A laser will produce excitations in the gallium arsenide layer, and many of these will relax into the many available plasmon modes. The plasmon modes will couple to the phonon modes in silver, but not in gallium arsenide, so they will gradually transfer energy from the semiconductor to the metal. This four-step process leads to laser cooling of the semiconductor:
laser ---> semiconductor excitations ---> plasmons ---> phonons in metal
Khurgin estimates that the silver and gallium arsenide system could have a cooling efficiency of 2 percent or more.
Jacob Khurgin
PRL 98, 177401 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e177401
Laser cooling of atoms is based on the Doppler effect. Laser frequencies are tuned below an atomic transition so that atoms moving toward the laser source will be more likely to absorb a photon than those moving away. The atom will then emit a photon in a random direction as it relaxes back down to its ground state, but the radiated photon will be of higher energy than the absorbed photon. As a result, the atom has radiated away some of its kinetic energy, so it has been cooled by a laser.
In solids the basic idea is similar. The solid absorbs one frequency of light and emits at a higher frequency, losing energy as a result. The mechanism is different, and it called anti-Stokes photoluminescence. Light is absorbed at a specfic frequency, then this excited state comes to thermal equilibrium with the system. Later, light is emitted at a higher frequency. The shift in frequency would be on the order of kT, where T is the temperature of the solid, and the system would gradually cool down.
Khurgin points out that there are several difficulties in the case of semiconductors. There are a lot of nonradiative decay channels, and the high index of refraction leads to low efficiencies --- i.e., even if you generate a photon of the right frequency, it's hard for it to get out of the semiconductor. One way to improve the efficiency is to make the density of absorbing states small and the density of emitting states large. When a photon is absorbed, the energy is more likely to be transferred to a higher-energy emitting state than to remain in the absorbing state for a time, then be re-emitted at the same frequency.
Khurgin's approach to the problem is to exploit surface plasmon polaritons. These occur at the interface between a dielectric and a metal at frequencies where the dielectric constants of the two media are equal in magnitude but opposite in sign. The density of plasmon states has a sharp resonance, which leads to an increase of spontaneous emission at the resonant frequency. (Apparently, Purcell worked this out back in 1946.)
Khurgin notes that the plasmon modes still have to couple to radiative modes before they give up their energy, so it might seem that nothing has been gained. However, he goes on to demonstrate that the plasmons can couple to the phonons of the metal. The metal will heat up, but the goal was never to cool the metal and the dielectric together --- only to cool the dielectric medium.
Based on this observation, Khurgin proposes placing a layer of silver on top of a gallium arsenide layer with a gap between them of a couple nanometers. The gap is a thermal insulator between the silver and gallium arsenide. The only coupling between the two systems are the plasmons. A laser will produce excitations in the gallium arsenide layer, and many of these will relax into the many available plasmon modes. The plasmon modes will couple to the phonon modes in silver, but not in gallium arsenide, so they will gradually transfer energy from the semiconductor to the metal. This four-step process leads to laser cooling of the semiconductor:
laser ---> semiconductor excitations ---> plasmons ---> phonons in metal
Khurgin estimates that the silver and gallium arsenide system could have a cooling efficiency of 2 percent or more.
Light to Atom to Light Again
Reversible State Transfer between Light and a Single Trapped Atom
A.D. Boozer, A. Boca, R. Miller, T.E. Northup, and H.J. Kimble
PRL 98, 193601 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e193601
This group from Cal Tech claims the first demonstration of the transfer of a coherent state between a photon and an atom, then from the atom back to a photon. This is a necessity for a quantum network.
Much work in quantum computing has gone into developing and manipulating qbits. It's one thing to have a single working qbit in isolation, or a pair, or 16 of them. But if you want to develop a quantum computer with a large number of qbits, or if you want to transfer the output state of your computation somewhere else, what do you do? The authors suggest that coherent light would be able to transfer superpositions of quantum states over optical fibers. Transmitting light over optical fibers is not so hard. The difficult step is turning an atomic state into a photon, transmitting the photon, then turning the photon into the same atomic state somewhere else.
In this letter, the Cal Tech group demonstrates "the reversible mapping of a coherent optical field to and from the hyperfine ground states of a single trapped cesium atom."
The prototype for their experiment is a 3-level atom. The atom has two ground states |a> and |b>, and an excited state |e>. The atom is in an optical cavity that couples |b> and |e>, and an external field couples |a> and |e>. If the external field is turned on slowly, the state |a,n> is transformed into |b,n+1> --- i.e., there is a transition between atomic states and a single photon is generated in the cavity. Slowly turning the field off reverses the transition.
If the cavity is empty, the process can be used to generate a single photon. The atom is prepared in state |a,0> and the field is slowly turned on. The resulting state is |b,1>. This is interesting, but transitions between single atomic states are not the building blocks of quantum computing. Entanglement and coherent superpositions of states are the tools of the trade. The useful thing about the process just described is that is works on a superposition of states:
(A |a> + B |b> ) |0> <---> |b> ( A |0> + B |1> )
This represents the transfer of a superposition of atomic states to a superposition of photon states, and all that was required is the turning off of a classical field.
The authors never mention the word "entangled" when discussing their experiments. I don't know what the exact definition of entangled states is, but I recall something about the impossibility of writing such a state as a product of states. If this is true, then the relation above does not describe the transfer of an entangled state from atom to photon. Still, it's a good first step!
A.D. Boozer, A. Boca, R. Miller, T.E. Northup, and H.J. Kimble
PRL 98, 193601 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e193601
This group from Cal Tech claims the first demonstration of the transfer of a coherent state between a photon and an atom, then from the atom back to a photon. This is a necessity for a quantum network.
Much work in quantum computing has gone into developing and manipulating qbits. It's one thing to have a single working qbit in isolation, or a pair, or 16 of them. But if you want to develop a quantum computer with a large number of qbits, or if you want to transfer the output state of your computation somewhere else, what do you do? The authors suggest that coherent light would be able to transfer superpositions of quantum states over optical fibers. Transmitting light over optical fibers is not so hard. The difficult step is turning an atomic state into a photon, transmitting the photon, then turning the photon into the same atomic state somewhere else.
In this letter, the Cal Tech group demonstrates "the reversible mapping of a coherent optical field to and from the hyperfine ground states of a single trapped cesium atom."
The prototype for their experiment is a 3-level atom. The atom has two ground states |a> and |b>, and an excited state |e>. The atom is in an optical cavity that couples |b> and |e>, and an external field couples |a> and |e>. If the external field is turned on slowly, the state |a,n> is transformed into |b,n+1> --- i.e., there is a transition between atomic states and a single photon is generated in the cavity. Slowly turning the field off reverses the transition.
If the cavity is empty, the process can be used to generate a single photon. The atom is prepared in state |a,0> and the field is slowly turned on. The resulting state is |b,1>. This is interesting, but transitions between single atomic states are not the building blocks of quantum computing. Entanglement and coherent superpositions of states are the tools of the trade. The useful thing about the process just described is that is works on a superposition of states:
(A |a> + B |b> ) |0> <---> |b> ( A |0> + B |1> )
This represents the transfer of a superposition of atomic states to a superposition of photon states, and all that was required is the turning off of a classical field.
The authors never mention the word "entangled" when discussing their experiments. I don't know what the exact definition of entangled states is, but I recall something about the impossibility of writing such a state as a product of states. If this is true, then the relation above does not describe the transfer of an entangled state from atom to photon. Still, it's a good first step!
Tuesday, May 15, 2007
Phonon Effects in Graphene
Electron-Phonon Coupling Mechanism in Two-Dimensional Graphite and Single-Walled Carbon Nanotubes
G.G. Samsonidze, E.B. Barros, R. Saito, J.Jiang, G. and M.S. Dresselhaus
PRB 75, 155420 (2007)
URL: http://link.aps.org/abstract/PRB/v75/e155420
The authors have analyzed the effects of phonons on the fermi energies and wave vectors of graphene. Their analysis is based on the group theory of the wave vector and demonstrates both the power of the technique and my lack of understanding.
They observe two phenomena associated with the phonons. First, there is a Pierls instability, which means the phonon modes open up a frequency-dependent band gap. The second is a Kohn anomaly, which is electron screening of a particular phonon mode.
Section 2 and Appendix A are very useful, as they show the general procedure for introducing phonon effects into the tight-binding model. When the phonon mode breaks the symmetry of the lattice, the unit cell of graphene must be enlarged to become a supercell of six atoms. This leads to a 6 by 6 Hamiltonian instead of the more familiar 2 by 2 version. However, the larger Hamiltonian is what Wigner calls a supermatrix --- a matrix composed of smaller matrices. There are two diagonal 3 by 3 matrices for the on-site terms, H[AA] and H[BB]. The hopping terms are described by two 3 by 3 matrices, H[AB] and H[BA], with every entry equal to t (for k=0). The latter is surprising, as it suggests that every A site is connected to every B site --- a fact that is not obvious from the diagrams provided by the authors.
This Hamiltonian gives a six-band spectrum, with the middle four bands degenerate. The K-point phonon mode breaks the degeneracy from fourfold to twofold and opens a bandgap that depends on the phonon coupling strength and amplitude.
The appendix gives the corresponding Hamiltonian for phonons with wave vectors not at a highly symmetric point of the Brillioun zone. It is not as symmetric. Although the authors do not analyze the energy bands of the general tight-binding Hamiltonian with phonon interactions, I assume that a phonon that does not respect any symmetries of the underlying lattice would lift all the degeneracies. The only exception might be the Kramers degeneracy imposed by time-reversal symmetry.
One aspect of this work I don't understand is why the phonon modes considered by the authors are more important than others. Are they the modes of lowest energy? Do the phonon bands cross the electron bands at the fermi energy? It seems that they are simply the easiest to analyze, but that does not mean their physical effects are the most important.
G.G. Samsonidze, E.B. Barros, R. Saito, J.Jiang, G. and M.S. Dresselhaus
PRB 75, 155420 (2007)
URL: http://link.aps.org/abstract/PRB/v75/e155420
The authors have analyzed the effects of phonons on the fermi energies and wave vectors of graphene. Their analysis is based on the group theory of the wave vector and demonstrates both the power of the technique and my lack of understanding.
They observe two phenomena associated with the phonons. First, there is a Pierls instability, which means the phonon modes open up a frequency-dependent band gap. The second is a Kohn anomaly, which is electron screening of a particular phonon mode.
Section 2 and Appendix A are very useful, as they show the general procedure for introducing phonon effects into the tight-binding model. When the phonon mode breaks the symmetry of the lattice, the unit cell of graphene must be enlarged to become a supercell of six atoms. This leads to a 6 by 6 Hamiltonian instead of the more familiar 2 by 2 version. However, the larger Hamiltonian is what Wigner calls a supermatrix --- a matrix composed of smaller matrices. There are two diagonal 3 by 3 matrices for the on-site terms, H[AA] and H[BB]. The hopping terms are described by two 3 by 3 matrices, H[AB] and H[BA], with every entry equal to t (for k=0). The latter is surprising, as it suggests that every A site is connected to every B site --- a fact that is not obvious from the diagrams provided by the authors.
This Hamiltonian gives a six-band spectrum, with the middle four bands degenerate. The K-point phonon mode breaks the degeneracy from fourfold to twofold and opens a bandgap that depends on the phonon coupling strength and amplitude.
The appendix gives the corresponding Hamiltonian for phonons with wave vectors not at a highly symmetric point of the Brillioun zone. It is not as symmetric. Although the authors do not analyze the energy bands of the general tight-binding Hamiltonian with phonon interactions, I assume that a phonon that does not respect any symmetries of the underlying lattice would lift all the degeneracies. The only exception might be the Kramers degeneracy imposed by time-reversal symmetry.
One aspect of this work I don't understand is why the phonon modes considered by the authors are more important than others. Are they the modes of lowest energy? Do the phonon bands cross the electron bands at the fermi energy? It seems that they are simply the easiest to analyze, but that does not mean their physical effects are the most important.
Nonlinear Resistance in Nanotubes
Scaling of Resistance and Electron Mean Free Path of Single-Walled Carbon Nanotubes
M.S. Purewal, B.H. Hong, A. Ravi, B. Chandra, J. Hone, and Philip Kim
PRL 98, 186808 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e186808
This group from Columbia University has attached a series of electrodes to individual carbon nanotubes and measured the current-voltage characteristics of carbon nanotubes as a function of temperature and length between electrodes. This allowed them to analyze the scaling behavior of the resistance in single nanotubes.
Using the Landauer-Buttiker model of resitance, the group extracted the mean free path of electrons in the nanotube and studied this as a function of temperature. They find crossover behavior between two types of scaling. Below a critical temperature, the mean free path approaches a constant that differs from one nanotube to the next --- i.e., it depends on the diameter and chiral angle. Above the critical temperature, the authors find universal scaling: the mean free path is inversely proportional to the temperature.
The low temperature behavior is consistent with models based on impurity scattering while the high temperature behavior is consistent with electron-phonon scattering.
At extremely long length scales and low temperatures, the authors find that the resistance no longer scales linearly with the distance between the source and drain. The critical length at which nonlinear behavior starts to dominate is much larger than the electron mean free path, which suggests it is not the result of Anderson localization or a similar type of quantum interference. The phase coherence length is the same order of magnitude as the mean free path, and therefore much shorter than the critical length scale. The authors imply there is no satisfactory theoretical explanation of the observed behavior.
This was the first paper on nanotubes with a recommendation from the editors. It is well-written and reports an interesting result, so the endorsement is well-deserved.
M.S. Purewal, B.H. Hong, A. Ravi, B. Chandra, J. Hone, and Philip Kim
PRL 98, 186808 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e186808
This group from Columbia University has attached a series of electrodes to individual carbon nanotubes and measured the current-voltage characteristics of carbon nanotubes as a function of temperature and length between electrodes. This allowed them to analyze the scaling behavior of the resistance in single nanotubes.
Using the Landauer-Buttiker model of resitance, the group extracted the mean free path of electrons in the nanotube and studied this as a function of temperature. They find crossover behavior between two types of scaling. Below a critical temperature, the mean free path approaches a constant that differs from one nanotube to the next --- i.e., it depends on the diameter and chiral angle. Above the critical temperature, the authors find universal scaling: the mean free path is inversely proportional to the temperature.
The low temperature behavior is consistent with models based on impurity scattering while the high temperature behavior is consistent with electron-phonon scattering.
At extremely long length scales and low temperatures, the authors find that the resistance no longer scales linearly with the distance between the source and drain. The critical length at which nonlinear behavior starts to dominate is much larger than the electron mean free path, which suggests it is not the result of Anderson localization or a similar type of quantum interference. The phase coherence length is the same order of magnitude as the mean free path, and therefore much shorter than the critical length scale. The authors imply there is no satisfactory theoretical explanation of the observed behavior.
This was the first paper on nanotubes with a recommendation from the editors. It is well-written and reports an interesting result, so the endorsement is well-deserved.
Friday, May 11, 2007
Transfer of Spin Polarization
Spin Transfer from an Optically Pumped Alkali Vapor to a Solid
K. Ishikawa, B. Patton, Y.Y. Jau, and W. Happer
PRL 98, 183004 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e183004
This editor recommendation was a very well-written paper.
As the title indicates, the authors demonstrated the transfer of spin from optically pumped cesium vapor to cesium salt on the surface of a sealed glass cylinder. The authors deposited a layer of CsH salt on the walls of the container about 10 microns thick. Next, they put in pure cesium and nitrogen at various pressures and sealed the cylinders.
They used a laser tuned to a specific atomic transition in the cesium vapor to pump the system using both right and left handed circularly polarized light. To see the effects of pumping the vapor on the salt layers, the authors measured the free induction decay of the sample --- a common technique in NMR. They were able to discriminate between the vapor and the salt because the resonance of the vapor is at 52.39 MHz while the salt is at 53.15 MHz. They found that in optically pumped samples, the polarization of the salt was 4 times its value in the unpumped samples. They take this as demonstration of transfer of spin polarization from the vapor to the salt.
Charles Marcus spoke at Penn a couple months ago and discussed his work on one and two electron quantum dots. He makes artificial atoms on solid state chips and inserts electrons, makes them interact, and reads out their final states. It's one of the first approaches to quantum computing that I would consider feasible. Anyway, Marcus and his group observed that when they sent spin polarized electrons through the system, they could polarize the nuclear spins in the dot. I thought he called it a "nuclear zamboni," but I can't track down that reference right now. The idea is that you can use an electron beam to smooth out the nuclear environment and make it less likely that nuclear disorder will destroy your carefully prepared electron qbits.
The authors of the current paper discuss a similar phenomenon, and they give a nice theoretical description of the process. They argue that nuclear and electronic spins obey a set of coupled diffusion equations (and give very clear descriptions of where all the terms in the equation come from). They give approximate solutions to this set of equations, then show that these approximate and easy-to-understand solutions reproduce the major features of a much more detailed numerical model. Their simple model explains why the electron current scales inversely with pressure, and why the nuclear current is proportional to the pressure for small pressure, and constant at large pressures.
Despite the qualitative agreement between the diffusion model and experimental data, the authors imply the theory of spin transfer is not understood all that well. It's quite an interesting theoretical problem --- one I'm interested in myself. Usually in the analysis of collisions (as in particle physics calculations), one averages over the initial and final spin polarizations to compute the cross section. For a spin polarized sample, this is not the right approach.
In addition, atomic physics offers possibilities not allowed in particle physics. Suppose an atom in a singlet state scatters off a magnetic impurity. There is some probability for an interaction that would leave the atom in a triplet state and flip a spin in the magnetic impurity.
Another interesting situation is Coulomb drag. The Coulomb interaction does not allow for spin flips. This results in scattering processes that conserve charge current but reduce the spin current. I have an article on the phenomenon somewhere. I should write that up.
K. Ishikawa, B. Patton, Y.Y. Jau, and W. Happer
PRL 98, 183004 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e183004
This editor recommendation was a very well-written paper.
As the title indicates, the authors demonstrated the transfer of spin from optically pumped cesium vapor to cesium salt on the surface of a sealed glass cylinder. The authors deposited a layer of CsH salt on the walls of the container about 10 microns thick. Next, they put in pure cesium and nitrogen at various pressures and sealed the cylinders.
They used a laser tuned to a specific atomic transition in the cesium vapor to pump the system using both right and left handed circularly polarized light. To see the effects of pumping the vapor on the salt layers, the authors measured the free induction decay of the sample --- a common technique in NMR. They were able to discriminate between the vapor and the salt because the resonance of the vapor is at 52.39 MHz while the salt is at 53.15 MHz. They found that in optically pumped samples, the polarization of the salt was 4 times its value in the unpumped samples. They take this as demonstration of transfer of spin polarization from the vapor to the salt.
Charles Marcus spoke at Penn a couple months ago and discussed his work on one and two electron quantum dots. He makes artificial atoms on solid state chips and inserts electrons, makes them interact, and reads out their final states. It's one of the first approaches to quantum computing that I would consider feasible. Anyway, Marcus and his group observed that when they sent spin polarized electrons through the system, they could polarize the nuclear spins in the dot. I thought he called it a "nuclear zamboni," but I can't track down that reference right now. The idea is that you can use an electron beam to smooth out the nuclear environment and make it less likely that nuclear disorder will destroy your carefully prepared electron qbits.
The authors of the current paper discuss a similar phenomenon, and they give a nice theoretical description of the process. They argue that nuclear and electronic spins obey a set of coupled diffusion equations (and give very clear descriptions of where all the terms in the equation come from). They give approximate solutions to this set of equations, then show that these approximate and easy-to-understand solutions reproduce the major features of a much more detailed numerical model. Their simple model explains why the electron current scales inversely with pressure, and why the nuclear current is proportional to the pressure for small pressure, and constant at large pressures.
Despite the qualitative agreement between the diffusion model and experimental data, the authors imply the theory of spin transfer is not understood all that well. It's quite an interesting theoretical problem --- one I'm interested in myself. Usually in the analysis of collisions (as in particle physics calculations), one averages over the initial and final spin polarizations to compute the cross section. For a spin polarized sample, this is not the right approach.
In addition, atomic physics offers possibilities not allowed in particle physics. Suppose an atom in a singlet state scatters off a magnetic impurity. There is some probability for an interaction that would leave the atom in a triplet state and flip a spin in the magnetic impurity.
Another interesting situation is Coulomb drag. The Coulomb interaction does not allow for spin flips. This results in scattering processes that conserve charge current but reduce the spin current. I have an article on the phenomenon somewhere. I should write that up.
Thursday, May 10, 2007
(No) e/2 States in Graphene
Electron Fractionalization in Two-Dimensional Graphenelike Structures
C.Y. Hou, Claudio Channon, and Christopher Mudry
PRL 98, 186809 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e186809
This is an interesting paper that probably has no "practical application" whatsoever. The authors show how the order parameter of a Kekule phonon interaction in graphene can lead to fractionally charged states.
The Kekule texture, as the authors call it, is a periodic variation in the hopping amplitudes of the tight-binding model. One of my colleagues has done some work on this type of perturbation. It expands the unit cell in real space to include 4 lattice sites, and reduces the Brillouin zone accordingly. The order parameter that describes the Kekule phonons can be a complex number. However, the authors point out that a constant phase can be removed by an appropriate chiral transformation of the Hamiltonian.
Interesting things happen when the phase becomes a local parameter. Randy, one of my profs who taught a great course on liquid crystals, emphasized this point repeatedly. The central idea behind Goldstone modes is that gapless excitations arise when a global symmetry operation is applied locally. Phonons were the first example. In a lattice, the crystal is the same if the entire lattice is translated by any amount. However, if we make the translation a local operation --- i.e., each lattice site is translated by a different amount --- it leads to a gapless excitation in the system: acoustic phonons.
The work of the authors is similar. They introduce vortices in the Kekule parameter. The Hamiltonian can be solved to give a single-valued normalizable wave function for a zero mode --- a mode that occurs precisely in the middle of the band gap. (The Kekule parameter introduces a gap into the graphene energy surface.)
Next, they calculate the charge bound to a vertex by taking the difference of the local density of states when there is one vortex and no vortices. Conservation of the number of electrons says that 2 times the number of electrons associated with a vortex plus the integral of the zero mode --- i.e. 1 --- must vanish. Ergo, there is a charge of -e/2 associated with each vortex. This would seem to violate charge conservation --- adding one vortex changes the charge of the system by half an electron charge. The authors point out, however, that vortices come in pairs, so there is no contradiction.
The disappointed disclaimer of this work is that when you factor in the degeneracy associated with spin, the fractionally charged states disappear. Only scalar electrons permit fractionally charged excitations. Too bad. The authors go on to discuss how one might introduce local variations in the Kekule parameter, but the point seems moot.
The authors made a few statements I would like to know more about. They claim that the time-reversal symmetry of the tight-binding Hamiltonian follows from the fact that the hopping parameters are real. Moreover, they say that states come in pairs of energy +E and -E because there exists a gauge transformation that can flip the sign of the ladder operators on the A sublattice without affecting those on the B sublattices. This sounds like a 2 pi rotation of a spin-1/2 particle, but I don't understand how it works for a tight-binding model.
In a four page paper, the authors managed to explain quite a lot!
C.Y. Hou, Claudio Channon, and Christopher Mudry
PRL 98, 186809 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e186809
This is an interesting paper that probably has no "practical application" whatsoever. The authors show how the order parameter of a Kekule phonon interaction in graphene can lead to fractionally charged states.
The Kekule texture, as the authors call it, is a periodic variation in the hopping amplitudes of the tight-binding model. One of my colleagues has done some work on this type of perturbation. It expands the unit cell in real space to include 4 lattice sites, and reduces the Brillouin zone accordingly. The order parameter that describes the Kekule phonons can be a complex number. However, the authors point out that a constant phase can be removed by an appropriate chiral transformation of the Hamiltonian.
Interesting things happen when the phase becomes a local parameter. Randy, one of my profs who taught a great course on liquid crystals, emphasized this point repeatedly. The central idea behind Goldstone modes is that gapless excitations arise when a global symmetry operation is applied locally. Phonons were the first example. In a lattice, the crystal is the same if the entire lattice is translated by any amount. However, if we make the translation a local operation --- i.e., each lattice site is translated by a different amount --- it leads to a gapless excitation in the system: acoustic phonons.
The work of the authors is similar. They introduce vortices in the Kekule parameter. The Hamiltonian can be solved to give a single-valued normalizable wave function for a zero mode --- a mode that occurs precisely in the middle of the band gap. (The Kekule parameter introduces a gap into the graphene energy surface.)
Next, they calculate the charge bound to a vertex by taking the difference of the local density of states when there is one vortex and no vortices. Conservation of the number of electrons says that 2 times the number of electrons associated with a vortex plus the integral of the zero mode --- i.e. 1 --- must vanish. Ergo, there is a charge of -e/2 associated with each vortex. This would seem to violate charge conservation --- adding one vortex changes the charge of the system by half an electron charge. The authors point out, however, that vortices come in pairs, so there is no contradiction.
The disappointed disclaimer of this work is that when you factor in the degeneracy associated with spin, the fractionally charged states disappear. Only scalar electrons permit fractionally charged excitations. Too bad. The authors go on to discuss how one might introduce local variations in the Kekule parameter, but the point seems moot.
The authors made a few statements I would like to know more about. They claim that the time-reversal symmetry of the tight-binding Hamiltonian follows from the fact that the hopping parameters are real. Moreover, they say that states come in pairs of energy +E and -E because there exists a gauge transformation that can flip the sign of the ladder operators on the A sublattice without affecting those on the B sublattices. This sounds like a 2 pi rotation of a spin-1/2 particle, but I don't understand how it works for a tight-binding model.
In a four page paper, the authors managed to explain quite a lot!
Excitons in Boron Nitride Nanotubes
Excitons and Many-Electron Effects in the Optical Response of Single-Walled Boron Nitride Nanotubes
C.H. Park, C.D. Spataru, and Steve Louie
PRL 96, 126105 (2006)
URL: http://link.aps.org/abstract/PRL/v96/e126105
I took two main points from this paper.
First, the larger band gap in boron nitride nanotubes makes particle interactions even more important. The shift of the free particle band gap due to electron interactions is larger than in nanotubes, and the exciton binding energies are also larger. The authors do not present any scaling results as they only analyzed a single nanotube.
Second, the exciton is much more localized in boron nitride nanotubes than in carbon nanotubes. In a carbon nanotube, there is little variation in the magnitude of the exciton wave function around the tube circumference. In boron nitride, there is quite a lot of variation. The electron is localized around the hole, and the probability of finding it on the opposite side of the tube is very small. The amplitude is also strongly peaked on the boron sites.
In effect, the second result implies the excitons in boron nitride are qualitatively different than in carbon. They cannot be considered one-dimensional objects. The exciton envelope function in a carbon nanotube is a smooth localized function. That of a boron nitride nanotube must describe the variations in amplitdue between the two sublattices. The boron nitride exciton is a two-dimensional wave function that happens to live on a cylinder. It is not strongly influenced by the periodic boundary conditions. (At least not in the tube studied here. In a tube with a smaller radius, the amplitude might not decay to zero over half a circumference.)
An interesting question that has probably been answered: Are the excitons in a boron nitride nanotube much different from those of a boron nitride sheet? The sheet has 3-fold rotational symmetry, which the tube does not. However, the tightly-localized excitons in the tube might not "be able to tell the difference," so to speak. There are no excitons in graphene because it is a semi-metal with no band gap. The large gap of planar boron nitride should allow for excitons, and I would expect them to be quite similar to their nanotube analogs.
C.H. Park, C.D. Spataru, and Steve Louie
PRL 96, 126105 (2006)
URL: http://link.aps.org/abstract/PRL/v96/e126105
I took two main points from this paper.
First, the larger band gap in boron nitride nanotubes makes particle interactions even more important. The shift of the free particle band gap due to electron interactions is larger than in nanotubes, and the exciton binding energies are also larger. The authors do not present any scaling results as they only analyzed a single nanotube.
Second, the exciton is much more localized in boron nitride nanotubes than in carbon nanotubes. In a carbon nanotube, there is little variation in the magnitude of the exciton wave function around the tube circumference. In boron nitride, there is quite a lot of variation. The electron is localized around the hole, and the probability of finding it on the opposite side of the tube is very small. The amplitude is also strongly peaked on the boron sites.
In effect, the second result implies the excitons in boron nitride are qualitatively different than in carbon. They cannot be considered one-dimensional objects. The exciton envelope function in a carbon nanotube is a smooth localized function. That of a boron nitride nanotube must describe the variations in amplitdue between the two sublattices. The boron nitride exciton is a two-dimensional wave function that happens to live on a cylinder. It is not strongly influenced by the periodic boundary conditions. (At least not in the tube studied here. In a tube with a smaller radius, the amplitude might not decay to zero over half a circumference.)
An interesting question that has probably been answered: Are the excitons in a boron nitride nanotube much different from those of a boron nitride sheet? The sheet has 3-fold rotational symmetry, which the tube does not. However, the tightly-localized excitons in the tube might not "be able to tell the difference," so to speak. There are no excitons in graphene because it is a semi-metal with no band gap. The large gap of planar boron nitride should allow for excitons, and I would expect them to be quite similar to their nanotube analogs.
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