Transport Properties of Carbon Nanotube C_{60} Peapods
C.H.L. Quay, et al.
PRB 76, 073404 (2007)
URL: http://link.aps.org/abstract/PRB/v76/e073404
In this article, the authors present data that suggest room temperature transport properties of carbon nanotube peapods are basically the same as unfilled carbon nanotubes. The peapods are either semiconducting or metallic and there is a Coulomb blockade.
The authors are surprised by the similarities with unfilled nanotubes. Since the nanotubes tested were selected from an ensemble containing mostly --- but not all --- peapods, one possible explanation is that the authors happened to select 7 unfilled nanotubes from the ensemble. They present a Bayesian statistical analysis to show that the chance of that happening are small, and that the most likely number of peapods in the experiment is roughly 6 of 7.
The authors' conclusion is that the transport properties of fullerene peapods are not that different from unfilled nanotubes. Although their statistical analysis seems correct, they could bolster their claim with more samples. I share their initial surprise that a lattice of buckyballs inside a nanotube has virtually no effect on its transport properties.
If the authors are correct, there are no signatures of the buckyball lattice in the transport spectra of a nanotube. How else might one go about detecting them?
Sunday, August 12, 2007
Tuesday, July 10, 2007
Photons and the Aharonov Bohm Effect
Bound on the Photon Charge from the Phase Coherence of Extragalactic Radiation
Brett Altschul
PRL 98, 261801 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e261801
The conclusion of this paper will come as a surprise to very few people: the photon probably doesn't have a charge. Altschul, from Indiana University, has deduced an upper bound on the photon charge that is 32 orders of magnitude less than the electron charge (46 if photons have both a positive and a negative charge). It is his analysis rather than his conclusion that I found interesting.
Altshcul's analysis starts from the observation that we can use interferometry to study astrophysical objects. Basically, one collects light from the same source at two different receivers. By studying the interference between the signals at the two receivers, one can obtain information about the source object. For this to work, the light from the source must be coherent --- i.e., the phase difference between two photons traveling along the same path must be small compared to the phase difference they acquire due to the path difference between the two receivers.
Altschul points out a source of phase difference that does not immediately come to mind: the Aharonov-Bohm Effect. If photons have a charge, then photons at the two detectors of an interferometer will acquire a phase difference that depends on the magnetic flux through the triangle made up of the two detectors and the source. (The assumption here is that a charged photon would interact with an external electromagnetic field exactly the same way an electron does.) The fact that interferometry works means the Aharonov Bohm phase small. (Conservatively, Altschul interprets "small" as "less than one".)
Making order of magnitude estimates for the interstellar magnetic field and using the baseline of the Very Long Baseline Interferometry Space Observatory Program (VSOP) with a source distance of 1 Gpc (about 3 billion light years), Altschul places an upper bound of 10^{-32} on the ratio of the photon charge to the electron charge.
The small bound is possible because of the huge distances involved in astronomical observations. It's almost like running a lab experiment designed to probe the Aharonov Bohm effect for 3 billion years --- that's a lot of data!
If the photon can have both positive and negative charges (like electrons) or positive, negative, and neutral charges (like pions), then the bounds are even tighter. This is because the Aharonov Bohm phase is proportional to the charge of the particle. If a particle with positive charge and a particle with negative charge travel along the same path, they acquire equal and opposite phases. If photons have two charges, the fact that interferometry works places an upper limit of 10^{-46} on the ratio of photon and electron charges.
The major source of uncertainty in this analysis is the current lack of understanding regarding interstellar magnetic fields. Perhaps Altschul's study will inspire new methods of studying these fields using interferometry.
As I said, the fact that the one can place a very small upper bound on the photon charge is not surprising. The fact that it can be done by analyzing the Aharonov Bohm effect is.
Altschul mentions a couple interesting facts about the theory of photons. First, the problem of the photon mass has been studied much more than that of photon charge. He mentions three theories of photon mass (Proca, Higgs, and Stuckleberg). I've never heard of the third. He also points out that not much is known about the consequences of charged photons. This is surprising, given the large number of models studied in quantum field theory --- many of which have little relevance to the physical world as revealed by experiments. It sounds like the kind of problem one might find at the end of a chapter in Peskin and Schroeder.
Brett Altschul
PRL 98, 261801 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e261801
The conclusion of this paper will come as a surprise to very few people: the photon probably doesn't have a charge. Altschul, from Indiana University, has deduced an upper bound on the photon charge that is 32 orders of magnitude less than the electron charge (46 if photons have both a positive and a negative charge). It is his analysis rather than his conclusion that I found interesting.
Altshcul's analysis starts from the observation that we can use interferometry to study astrophysical objects. Basically, one collects light from the same source at two different receivers. By studying the interference between the signals at the two receivers, one can obtain information about the source object. For this to work, the light from the source must be coherent --- i.e., the phase difference between two photons traveling along the same path must be small compared to the phase difference they acquire due to the path difference between the two receivers.
Altschul points out a source of phase difference that does not immediately come to mind: the Aharonov-Bohm Effect. If photons have a charge, then photons at the two detectors of an interferometer will acquire a phase difference that depends on the magnetic flux through the triangle made up of the two detectors and the source. (The assumption here is that a charged photon would interact with an external electromagnetic field exactly the same way an electron does.) The fact that interferometry works means the Aharonov Bohm phase small. (Conservatively, Altschul interprets "small" as "less than one".)
Making order of magnitude estimates for the interstellar magnetic field and using the baseline of the Very Long Baseline Interferometry Space Observatory Program (VSOP) with a source distance of 1 Gpc (about 3 billion light years), Altschul places an upper bound of 10^{-32} on the ratio of the photon charge to the electron charge.
The small bound is possible because of the huge distances involved in astronomical observations. It's almost like running a lab experiment designed to probe the Aharonov Bohm effect for 3 billion years --- that's a lot of data!
If the photon can have both positive and negative charges (like electrons) or positive, negative, and neutral charges (like pions), then the bounds are even tighter. This is because the Aharonov Bohm phase is proportional to the charge of the particle. If a particle with positive charge and a particle with negative charge travel along the same path, they acquire equal and opposite phases. If photons have two charges, the fact that interferometry works places an upper limit of 10^{-46} on the ratio of photon and electron charges.
The major source of uncertainty in this analysis is the current lack of understanding regarding interstellar magnetic fields. Perhaps Altschul's study will inspire new methods of studying these fields using interferometry.
As I said, the fact that the one can place a very small upper bound on the photon charge is not surprising. The fact that it can be done by analyzing the Aharonov Bohm effect is.
Altschul mentions a couple interesting facts about the theory of photons. First, the problem of the photon mass has been studied much more than that of photon charge. He mentions three theories of photon mass (Proca, Higgs, and Stuckleberg). I've never heard of the third. He also points out that not much is known about the consequences of charged photons. This is surprising, given the large number of models studied in quantum field theory --- many of which have little relevance to the physical world as revealed by experiments. It sounds like the kind of problem one might find at the end of a chapter in Peskin and Schroeder.
Wednesday, July 4, 2007
Relativity on the Table-Top
Dirac Equation and Quantum Relativistic Effects in a Single Trapped Ion
L. Lamata, J. Leon, T. Schatz, and E. Solano
PRL 98, 253005 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e253005
In this article, the authors demonstrate the experimental possibility of simulating a Dirac Hamiltonian in an atomic system. In short, they propose a table-top experiment that would probe the effects of a relativistic system.
The requirements are relatively modest. To simulate the (1+1)-D or (2+1)-D Dirac equation, one needs a two-level atomic system; for the (3+1)-D Dirac equation, one needs a 4-level system. Three types of couplings are also required:
• Carrier Interaction -- a resonant coupling between two of the internal atomic states (such as a laser tuned to the transition frequency).
• Jaynes-Cummings Interaction -- couples two internal states with a vibrational mode of the center of mass. An upward transition between internal states is accompanied by the destruction of a phonon.
• Anti-Jaynes-Cummings Interaction -- also couples two internal states with a vibrational mode, but the frequency is tuned so that an upward transition between is accompanied instead by the creation of a phonon.
When the phases of the laser fields used to generate the above interactions are appropriately tuned, the effective Hamiltonian for the two-level or four-level system is identical in form with the free Dirac equation. By varying the couplings, one can tune the effective particle mass and velocity of light.
The authors dedicate a lot of space to a discussion of Zitterbewegung, which is a rapid oscillatory motion of an electron about its mean position. It was predicted in 1930 by Schrodinger, but has not yet been observed experimentally. For electrons, the amplitude of the oscillations is on the order of 10^{-13} m, and the frequency is on the order of 10^{21} Hz. In addition, Zitterbewegung is an effect of the single-particle Dirac equation. There is some controversy over whether or not the effect persists in QED.
The authors point out that by controlling the effective electron mass and the effective speed of light, one should be able to bring the amplitude and frequency of the oscillations into an experimentally accessible range. Another point they don't mention: even if the effect does not occur for real electrons, the dynamics of their atomic system are (in theory) accurately described by a single-particle Dirac equation. Whether or not real electrons jitter around, the trapped ions on their desks should.
The authors mention a few other relativistic effects that might be simulated: by controlling the particle mass, one could observe something akin to the Higgs mechanism for mass generation; one could simulate the Klein paradox, in which particle-hole pairs are created in a strong potential; one could simulate a (1+1)-D axial anomaly, the lower-dimensional counterpart of the chiral anomaly in (3+1)-D.
The possibility of tuning the parameters of the Dirac equation is interesting. However, such simulations of relativistic systems is a bit perplexing. A similar situation exists in the theory of graphene and carbon nanotubes, where the low-energy excitations are described by a Dirac equation. On a very fundamental level, the simulation can't be right. So how far can one push the analogy?
How similar are a pseudospinor and a spinor? Electron spin has to do with angular momentum, but pseudospin has nothing to do with it. The simulated system can't have pseudospin-orbit coupling, but the Dirac equation can.
How effective is an effective speed of light? In the table-top simulation, nothing prohibits the electron or the center of mass of the ion system from exceeding the effective speed of light. What then? Would the authors claim they were simulating tachyons?
In the Klein paradox, where are the holes going to come from? No corresponding anti-cesium ion is going to materialize in the trap.
It would be quite interesting to probe these unphysical effects in a trap and see exactly how the effective Dirac physics breaks down.
A final note: This was a well-written paper, but I must criticize the authors for a particular choice of phrase. On page 2, they refer to the "notorious analogy" between the Dirac equation and the effective Hamiltonian of the trapped ion. Jesse James was notorious. The three-body problem is notoriously difficult. An analogy between Hamiltonians is not "famous or well known, typically for some bad quality or deed."
L. Lamata, J. Leon, T. Schatz, and E. Solano
PRL 98, 253005 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e253005
In this article, the authors demonstrate the experimental possibility of simulating a Dirac Hamiltonian in an atomic system. In short, they propose a table-top experiment that would probe the effects of a relativistic system.
The requirements are relatively modest. To simulate the (1+1)-D or (2+1)-D Dirac equation, one needs a two-level atomic system; for the (3+1)-D Dirac equation, one needs a 4-level system. Three types of couplings are also required:
• Carrier Interaction -- a resonant coupling between two of the internal atomic states (such as a laser tuned to the transition frequency).
• Jaynes-Cummings Interaction -- couples two internal states with a vibrational mode of the center of mass. An upward transition between internal states is accompanied by the destruction of a phonon.
• Anti-Jaynes-Cummings Interaction -- also couples two internal states with a vibrational mode, but the frequency is tuned so that an upward transition between is accompanied instead by the creation of a phonon.
When the phases of the laser fields used to generate the above interactions are appropriately tuned, the effective Hamiltonian for the two-level or four-level system is identical in form with the free Dirac equation. By varying the couplings, one can tune the effective particle mass and velocity of light.
The authors dedicate a lot of space to a discussion of Zitterbewegung, which is a rapid oscillatory motion of an electron about its mean position. It was predicted in 1930 by Schrodinger, but has not yet been observed experimentally. For electrons, the amplitude of the oscillations is on the order of 10^{-13} m, and the frequency is on the order of 10^{21} Hz. In addition, Zitterbewegung is an effect of the single-particle Dirac equation. There is some controversy over whether or not the effect persists in QED.
The authors point out that by controlling the effective electron mass and the effective speed of light, one should be able to bring the amplitude and frequency of the oscillations into an experimentally accessible range. Another point they don't mention: even if the effect does not occur for real electrons, the dynamics of their atomic system are (in theory) accurately described by a single-particle Dirac equation. Whether or not real electrons jitter around, the trapped ions on their desks should.
The authors mention a few other relativistic effects that might be simulated: by controlling the particle mass, one could observe something akin to the Higgs mechanism for mass generation; one could simulate the Klein paradox, in which particle-hole pairs are created in a strong potential; one could simulate a (1+1)-D axial anomaly, the lower-dimensional counterpart of the chiral anomaly in (3+1)-D.
The possibility of tuning the parameters of the Dirac equation is interesting. However, such simulations of relativistic systems is a bit perplexing. A similar situation exists in the theory of graphene and carbon nanotubes, where the low-energy excitations are described by a Dirac equation. On a very fundamental level, the simulation can't be right. So how far can one push the analogy?
How similar are a pseudospinor and a spinor? Electron spin has to do with angular momentum, but pseudospin has nothing to do with it. The simulated system can't have pseudospin-orbit coupling, but the Dirac equation can.
How effective is an effective speed of light? In the table-top simulation, nothing prohibits the electron or the center of mass of the ion system from exceeding the effective speed of light. What then? Would the authors claim they were simulating tachyons?
In the Klein paradox, where are the holes going to come from? No corresponding anti-cesium ion is going to materialize in the trap.
It would be quite interesting to probe these unphysical effects in a trap and see exactly how the effective Dirac physics breaks down.
A final note: This was a well-written paper, but I must criticize the authors for a particular choice of phrase. On page 2, they refer to the "notorious analogy" between the Dirac equation and the effective Hamiltonian of the trapped ion. Jesse James was notorious. The three-body problem is notoriously difficult. An analogy between Hamiltonians is not "famous or well known, typically for some bad quality or deed."
Saturday, June 16, 2007
The Geometry of Phase Space
Geometry of Hamiltonian Chaos
L. Horwitz, et al.
PRL 98, 234302 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e234302
This group from Israel has recast classical Hamiltonian dynamics in the language of differential geometry. In doing so, they have discovered a new criterion for chaotic motion: the deviation of parallel geodesics. The geodesics of the system are classical trajectories. If nearby geodesics diverge, it indicates an instability in the system. This is a different measure than the Lyapunov exponent, although it’s possible there is a deeper connection between the two. I would speculate that the Lyapunov exponents could be derived from this geometric theory.
The general idea behind the method is to rewrite the Hamiltonian using a metric in the space of conjugate momenta. From this metric, one can derive the connection and curvature. From the curvature, one can calculate geodesic deviation. However, the authors find that the geodesic equation and the Hamiltonian equations of motion do not agree. It turns out that the geodesics in the dual manifold (defined by dx^{i} = g^{ij} dx_{j}) do indeed agree with the Hamiltonian equations of motion. The authors simply state this fact without discussion. There must be a deeper significance of this result. Or maybe the authors just started their calculation on the wrong manifold. Regardless of the connection, the geodesics on the dual manifold and the Hamiltonian equations of motion are mathematically equivalent; thus, one has a geometric way of thinking about phase space trajectories.
The new connection is anti-symmetric, so there is torsion. I don’t know much about torsion. In my course on general relativity, one of the postulates we used in deriving Einstein’s equations was the “no torsion condition.” It has something to do with twist of the geodesics. It is irrelevant in the derivation of the authors because the antisymmetric terms cancel from the geodesic equation, leaving only a symmetric connection.
In the end, the authors find that the curvature derived from this connection determines the stability of an orbit. If the curvature is negative, geodesics diverge and the system is chaotic. If curvature is positive, geodesics converge, and the system is stable. This seems like a very useful tool for analyzing unstable systems. The authors point out that this approach does not require one to use approximations of the true equations of motion, unlike other methods.
The idea behind the paper is quite interesting. My one complaint with the paper is that the phrase “a complete discussion will be given elsewhere” was used at least four times, but the authors don’t indicate where. It would be interesting to see these discussions, as well as a discussion of the connection between the curvature studied by the authors and the Lyapunov exponents.
L. Horwitz, et al.
PRL 98, 234302 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e234302
This group from Israel has recast classical Hamiltonian dynamics in the language of differential geometry. In doing so, they have discovered a new criterion for chaotic motion: the deviation of parallel geodesics. The geodesics of the system are classical trajectories. If nearby geodesics diverge, it indicates an instability in the system. This is a different measure than the Lyapunov exponent, although it’s possible there is a deeper connection between the two. I would speculate that the Lyapunov exponents could be derived from this geometric theory.
The general idea behind the method is to rewrite the Hamiltonian using a metric in the space of conjugate momenta. From this metric, one can derive the connection and curvature. From the curvature, one can calculate geodesic deviation. However, the authors find that the geodesic equation and the Hamiltonian equations of motion do not agree. It turns out that the geodesics in the dual manifold (defined by dx^{i} = g^{ij} dx_{j}) do indeed agree with the Hamiltonian equations of motion. The authors simply state this fact without discussion. There must be a deeper significance of this result. Or maybe the authors just started their calculation on the wrong manifold. Regardless of the connection, the geodesics on the dual manifold and the Hamiltonian equations of motion are mathematically equivalent; thus, one has a geometric way of thinking about phase space trajectories.
The new connection is anti-symmetric, so there is torsion. I don’t know much about torsion. In my course on general relativity, one of the postulates we used in deriving Einstein’s equations was the “no torsion condition.” It has something to do with twist of the geodesics. It is irrelevant in the derivation of the authors because the antisymmetric terms cancel from the geodesic equation, leaving only a symmetric connection.
In the end, the authors find that the curvature derived from this connection determines the stability of an orbit. If the curvature is negative, geodesics diverge and the system is chaotic. If curvature is positive, geodesics converge, and the system is stable. This seems like a very useful tool for analyzing unstable systems. The authors point out that this approach does not require one to use approximations of the true equations of motion, unlike other methods.
The idea behind the paper is quite interesting. My one complaint with the paper is that the phrase “a complete discussion will be given elsewhere” was used at least four times, but the authors don’t indicate where. It would be interesting to see these discussions, as well as a discussion of the connection between the curvature studied by the authors and the Lyapunov exponents.
Wednesday, June 13, 2007
Excitons Insensitive to Environment
Screening of Excitons in Single, Suspended Carbon Nanotubes
A.G. Walsh, et al.
Nanoletters 7, 1485--1488 (2007)
URL: http://pubs.acs.org/cgi-bin/abstract.cgi/nalefd/2007/7/i06/abs/nl070193p.html
This group from Boston University has investigated the binding energy of excitons in nanotubes as a function of the dielectric constant of the environment. They repeated photoluminescence measurements on nanotubes in dry air, humid air, and water. In general, they find that the binding energy is not very sensitive to changes in the dielectric environment. The binding energy only changes by a couple tens of meV between dry air and water.
The theoretical model they use is a 1D interaction of the form V / ( |z| + Z ). Surprisingly, they find that the parameter Z scales linearly with the dielectric constant! They claim this model was solved exactly in 1959. I’m going to track down that reference. With the exact solution and scaling relation for Z, they are able to reproduce the scaling of their data as well as a scaling relation reported by Perebeinos a couple years ago.
A.G. Walsh, et al.
Nanoletters 7, 1485--1488 (2007)
URL: http://pubs.acs.org/cgi-bin/abstract.cgi/nalefd/2007/7/i06/abs/nl070193p.html
This group from Boston University has investigated the binding energy of excitons in nanotubes as a function of the dielectric constant of the environment. They repeated photoluminescence measurements on nanotubes in dry air, humid air, and water. In general, they find that the binding energy is not very sensitive to changes in the dielectric environment. The binding energy only changes by a couple tens of meV between dry air and water.
The theoretical model they use is a 1D interaction of the form V / ( |z| + Z ). Surprisingly, they find that the parameter Z scales linearly with the dielectric constant! They claim this model was solved exactly in 1959. I’m going to track down that reference. With the exact solution and scaling relation for Z, they are able to reproduce the scaling of their data as well as a scaling relation reported by Perebeinos a couple years ago.
Tuesday, June 12, 2007
Exchange and Correlation in Graphene
Chirality and Correlations in Graphene
Y. Barlas, T. Pereg-Barnea, M. Polini, R. Asgari, and A.H. MacDonald
PRL 98, 236601 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e236601
This article was a pleasure to read. I've been reading about the use of techniques from field theory in condensed matter physics lately, and I got to see them in action here.
This group from Texas has evaluated the effects of exchange and correlation on the charge and spin susceptibility of graphene. They start from the massless Dirac equation that describes low energy excitations in graphene and calculate the interaction energy. The technique they used was new to me.
First, they write down an expression for the interaction energy that depends on the structure factor of the interacting system. The interaction energy appears to be calculated by adiabatically turning on the coupling between particles. The structure factor is calculated from the density-density response function, a relation based on the fluctuation dissipation theorem. The response function can be calculated in the random phase approximation using methods of quantum field theory. The authors combine a lot of neat tools to obtain the interaction energy. It's all condensed into one paragraph in this paper. I'll have to check out their reference, Giuliani and Vignale's "Quantum Theory of the Electron Liquid."
Though the techniques are well-established, they lead to surprising results in graphene. The authors find that the sign of the correections to the spin and charge susceptibilities is opposite that of the usual electron gas. The similarity of the response functions is also different from an electron gas. The exchange effect decreases the susceptibilities in graphene, while it enhances them in the normal 2DEG. The authors trace the difference in behavior to the renormalized fermi velocity, which increases with the interaction strength.
Once I am more comfortable with the techniques I'm learning from Abrikosov and Mattuck, I'm going to try and reproduce these results.
Y. Barlas, T. Pereg-Barnea, M. Polini, R. Asgari, and A.H. MacDonald
PRL 98, 236601 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e236601
This article was a pleasure to read. I've been reading about the use of techniques from field theory in condensed matter physics lately, and I got to see them in action here.
This group from Texas has evaluated the effects of exchange and correlation on the charge and spin susceptibility of graphene. They start from the massless Dirac equation that describes low energy excitations in graphene and calculate the interaction energy. The technique they used was new to me.
First, they write down an expression for the interaction energy that depends on the structure factor of the interacting system. The interaction energy appears to be calculated by adiabatically turning on the coupling between particles. The structure factor is calculated from the density-density response function, a relation based on the fluctuation dissipation theorem. The response function can be calculated in the random phase approximation using methods of quantum field theory. The authors combine a lot of neat tools to obtain the interaction energy. It's all condensed into one paragraph in this paper. I'll have to check out their reference, Giuliani and Vignale's "Quantum Theory of the Electron Liquid."
Though the techniques are well-established, they lead to surprising results in graphene. The authors find that the sign of the correections to the spin and charge susceptibilities is opposite that of the usual electron gas. The similarity of the response functions is also different from an electron gas. The exchange effect decreases the susceptibilities in graphene, while it enhances them in the normal 2DEG. The authors trace the difference in behavior to the renormalized fermi velocity, which increases with the interaction strength.
Once I am more comfortable with the techniques I'm learning from Abrikosov and Mattuck, I'm going to try and reproduce these results.
Wednesday, June 6, 2007
Ripping a Fluid Apart
Motion of a Viscoelastic Micellar Fluid around a Cylinder: Flow and Fracture
J.R. Gladden and A. Belmonte
PRL 98, 224501 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e224501
This duo from Penn state has experimentally demonstrated both the viscous and elastic regimes of a viscoelastic fluid. As with most fluid dynamics experiments, the photos are beautiful and fascinating.
Fluids flow. Solids deform or fracture. A viscoelastic fluid is a material that exhibits both types of response depending on how it is probed. In this Letter, Gladden and Belmonte have probed both types of response in a very simple way. The take a cylinder of diameter D and pull it through a layer of their fluid at constant velocity V. By varying the diameter and velocity, they observe three different types of response:
1) Flow: the fluid moves smoothly around the cylinder and recombines behind it.
2) Cut: the fluid still flows around the cylinder, but a cavity forms around it , and some air bubbles are trapped in its wake.
3) Tear: the cylinder rips through the fluid, leaving a trail of fin-shaped tears behind it, like a cylinder pulled through a thin plastic sheet.
These three types of response are displayed very effectively in Fig. 1. By plotting response as a function of V and D, the authors create a phase diagram for the fluid. There is a linear boundary between cut and flow, and a hyperbolic boundary between tear and the other two states. There is even a triple point.
The authors give an excellent discussion of their data. The boundary between flow and cut occurs when the time scale of fluid flow around the cylinder, D/V, exceeds the relaxation time of the fluid. The boundary between tear and the other two types of response has D*V constant. The authors use scaling arguments to determine the physical meaning of this constant, which is proportional to the tear strength of the fluid.
One more interesting observation is that when the cylinder is pulled fast enough, a crack forms in front of it. To analyze the stresses on the fluid, the authors used cross polarizers. They found a dipole pattern for the stress due to the moving cylinder, but the most fascinating picture from the paper is Fig. 6, the stress field due to a cubic probe. Amazing.
The pictures alone make this article worth reading. The authors analysis of the data is excellent.
J.R. Gladden and A. Belmonte
PRL 98, 224501 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e224501
This duo from Penn state has experimentally demonstrated both the viscous and elastic regimes of a viscoelastic fluid. As with most fluid dynamics experiments, the photos are beautiful and fascinating.
Fluids flow. Solids deform or fracture. A viscoelastic fluid is a material that exhibits both types of response depending on how it is probed. In this Letter, Gladden and Belmonte have probed both types of response in a very simple way. The take a cylinder of diameter D and pull it through a layer of their fluid at constant velocity V. By varying the diameter and velocity, they observe three different types of response:
1) Flow: the fluid moves smoothly around the cylinder and recombines behind it.
2) Cut: the fluid still flows around the cylinder, but a cavity forms around it , and some air bubbles are trapped in its wake.
3) Tear: the cylinder rips through the fluid, leaving a trail of fin-shaped tears behind it, like a cylinder pulled through a thin plastic sheet.
These three types of response are displayed very effectively in Fig. 1. By plotting response as a function of V and D, the authors create a phase diagram for the fluid. There is a linear boundary between cut and flow, and a hyperbolic boundary between tear and the other two states. There is even a triple point.
The authors give an excellent discussion of their data. The boundary between flow and cut occurs when the time scale of fluid flow around the cylinder, D/V, exceeds the relaxation time of the fluid. The boundary between tear and the other two types of response has D*V constant. The authors use scaling arguments to determine the physical meaning of this constant, which is proportional to the tear strength of the fluid.
One more interesting observation is that when the cylinder is pulled fast enough, a crack forms in front of it. To analyze the stresses on the fluid, the authors used cross polarizers. They found a dipole pattern for the stress due to the moving cylinder, but the most fascinating picture from the paper is Fig. 6, the stress field due to a cubic probe. Amazing.
The pictures alone make this article worth reading. The authors analysis of the data is excellent.
Bubble Dynamics
Role of Dimensionality and Axisymmetry in Fluid Pinch-Off and Coalescence
J.C. Burton and P. Taborek
PRL 98, 224502 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e224502
A very interesting paper. Figure 2 is amazing!
This pair of researchers from UC Irvine has investigated two phenomena that occure at fluid interfaces: pinch off and coalescence. As the authors point out, these are topological transitions which involve a conversion of interfacial energy into kinetic energy of fluid flow.
Pinch off is what happens when a droplet is pulled apart. At some point it breaks up into two or more pieces. Coalescence is the merging of two droplets into one. The authors have used high-speed, high-resolution video to study these two processes for both two-dimensional and three-dimensional alkane droplets.
What Burton and Taborek observed in 2D pinch off blows me away. Watch the video. When a 3D droplet is pulled apart, it transforms into two globules connected by a thin filament just before pinch off. At one of the filament-globule interfaces, the filament separates, leaving a flat surface and a cone whose opening angle is determined by the fluid properties of the droplet. For the 2D droplets in this paper, the filament starts to pinch off at both ends, so there are two large globules on the left and right connected to a small globule in the middle. As these filaments pinch off, they repeat the process on a smaller scale. The authors observe 5 generations of successive pinch offs, with each generation of droplets smaller than its parents by a factor of 3. The end result is that a single droplet has broken up into about 30 smaller droplets, the smallest of which are almost 1000 times smaller than the original droplet. If that's confusing, watch the video. It's amazing!
Burton and Taborek did not observe any striking differences in the coalescence of 2D and 3D droplets. As two spherical droplets start to merge, the radius of the connected region grows linearly in time. At longer times, the radius grows with the square root of time. The authors use the two scaling laws to determine the approximate size of the droplet at the crossover. The crossover length scale is 2 orders of magnitude larger than the natural length scale of the fluid system.
This is a very interesting paper, with beautiful images, clean data, and insightful explanations.
J.C. Burton and P. Taborek
PRL 98, 224502 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e224502
A very interesting paper. Figure 2 is amazing!
This pair of researchers from UC Irvine has investigated two phenomena that occure at fluid interfaces: pinch off and coalescence. As the authors point out, these are topological transitions which involve a conversion of interfacial energy into kinetic energy of fluid flow.
Pinch off is what happens when a droplet is pulled apart. At some point it breaks up into two or more pieces. Coalescence is the merging of two droplets into one. The authors have used high-speed, high-resolution video to study these two processes for both two-dimensional and three-dimensional alkane droplets.
What Burton and Taborek observed in 2D pinch off blows me away. Watch the video. When a 3D droplet is pulled apart, it transforms into two globules connected by a thin filament just before pinch off. At one of the filament-globule interfaces, the filament separates, leaving a flat surface and a cone whose opening angle is determined by the fluid properties of the droplet. For the 2D droplets in this paper, the filament starts to pinch off at both ends, so there are two large globules on the left and right connected to a small globule in the middle. As these filaments pinch off, they repeat the process on a smaller scale. The authors observe 5 generations of successive pinch offs, with each generation of droplets smaller than its parents by a factor of 3. The end result is that a single droplet has broken up into about 30 smaller droplets, the smallest of which are almost 1000 times smaller than the original droplet. If that's confusing, watch the video. It's amazing!
Burton and Taborek did not observe any striking differences in the coalescence of 2D and 3D droplets. As two spherical droplets start to merge, the radius of the connected region grows linearly in time. At longer times, the radius grows with the square root of time. The authors use the two scaling laws to determine the approximate size of the droplet at the crossover. The crossover length scale is 2 orders of magnitude larger than the natural length scale of the fluid system.
This is a very interesting paper, with beautiful images, clean data, and insightful explanations.
Friday, May 25, 2007
Three Body Topology
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.
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.
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.
Friday, April 27, 2007
Metamaterials and Cloaking
METAMATERIALS
David R. Smith was this year's Walter Selove lecturer at Penn. He gave two talks on his work in developing metamaterials and using them in a cloaking device. I did some background reading to understand the principles behind the cloaking device. There were two papers, published back to back in Science that I read through:
Optical Conformal Mapping
Ulf Leonhardt
Science 312, 1777--1780 (2006)
Controlling Electromagnetic Fields
J.B. Pendry, D. Schurig, and D.R. Smith
Science 312, 1780--1782
They make a lot more sense after hearing Dr. Smith's lectures and talking with him over lunch.
Question 1: What is a material?
Dr. Smith made a convincing argument that, from the perspective of electrodynamics, a material is something with a permittivity and a permeability. These are the only parameters that enter Maxwell's equations for describing electromagnetic fields in matter. This is already an effective theory --- fundamentally, QED would describe any electromagnetic phenomena. The permittivity and permeability represent a type of coarse-graining in which the atomic properties are averaged of distances that are large compared to the atomic scale (say the Bohr radius), but small compared with the wavelength of light in question.
Question 2: What is a metamaterial?
If the wavelength of light is large enough, we can imagine averaging the fields over scales large enough to be manipulated by intelligent beings. For visible light, one might consider nanoscale patterns etched on a wafer. For microwaves, objects as large as wires and loops could form the effective medium.
A metamaterial is a medium in which you control the atoms. Dr. Smith cited the work of John Pendry as demonstrating how one could build up a material with a negative index of refraction (negative permeability and permittivity). A negative value of the permittivity is not uncommon. It occurs in metals near a plasmon resonance. Dr. Smith was part of a team at UCSD that built an array of wire loops and posts with both a negative permittivity and a negative permeability.
Question 3: What does a metamaterial do?
Control over the permeability and permittivity allows for some interesting phenomena. Dr. Smith's group demonstrated a negative index of refraction by performing a simple Snell's law experiment. Light is bent in the opposite direction. Many other possibilities were discussed by a Soviet physicist named Veselago in 1968, such as a phase velocity opposite the direction of propagation, and lensing from a flat slab. More recently, Pendry proposed a "perfect lens" which would allow focusing of non-propagating modes --- the near fields that are always ignored in textbook problems.
Another phenomena is called cloaking. It's gotten a lot of media attention and Dr. Smith's work has made the covers of several scientific publications.
CLOAKING
Both the papers from Science address the possibility of cloaking. An interesting mathematical property of Maxwell's equations is that a coordinate transformation can be completely described by a transformation of the fields, the permittivity, and the permeability. This has practical consequences, as both papers illustrate.
Suppose you want to cloak a region of space --- i.e., you don't want any electromagnetic fields to penetrate the region, and you don't want any waves reflected or absorbed. Light rays follow geodesics (e.g., Fermat's principle). If the geodescis of spacetime were to travel around the region to be cloaked, this would be a neat solution of the problem. A coordinate transformation can implement this solution. That coordinate transfomration leads to field lines the curve around the cloaked region and their corresponding permittivity and permeability. This is where metamaterials come in. They allow one to engineer the permittivity and permeability as needed.
Dr. Smith's group at Duke perfomred numerical simulations to determine the properties their metamaterial would need to cloak a disc from microwave radiation. They built the required structure (rather, an approximation to the ideal structure), then demonstrated that waves pass right around the central disc for the most part, even when a strong scatterer is placed inside.
So has Dr. Smith ushered in the age of Klingon cloaking devices? Not yet. He is the first to point out the limitations of his devices. They have a very small bandwidth, meaning they only work for a very small range of wavelengths. In addition, it's hard to manipulate matter on very small scales, so cloaking in the optical range is still a technical challenge, even for a single frequency. To effective cloak a device, one would need to cover a large range of frequencies.
Cloaking was more a proof of principle than The Next Big Thing. More practical applications include antennas and lenses that can do things they don't tell you about in freshman physics.
The mechanism behind cloaking --- an effective warping of spacetime --- got me thinking about general relativity. Are there gravitational object that could actually warp spacetime in the same way, so that light would pass right around them? Black holes pull everything in. I suppose a very dense region of antigravity would be required to deflect light around an object. Still, if this were possible, the cloak would work at all frequencies, because the actual geodesics of spacetime would curve around the object. It would not be the result of an effective dielectric constant that depends on frequency. Light, particles, rocks, and anything else would travel along the same geodesics. It would be cloaked from everything, not just light!
David R. Smith was this year's Walter Selove lecturer at Penn. He gave two talks on his work in developing metamaterials and using them in a cloaking device. I did some background reading to understand the principles behind the cloaking device. There were two papers, published back to back in Science that I read through:
Optical Conformal Mapping
Ulf Leonhardt
Science 312, 1777--1780 (2006)
Controlling Electromagnetic Fields
J.B. Pendry, D. Schurig, and D.R. Smith
Science 312, 1780--1782
They make a lot more sense after hearing Dr. Smith's lectures and talking with him over lunch.
Question 1: What is a material?
Dr. Smith made a convincing argument that, from the perspective of electrodynamics, a material is something with a permittivity and a permeability. These are the only parameters that enter Maxwell's equations for describing electromagnetic fields in matter. This is already an effective theory --- fundamentally, QED would describe any electromagnetic phenomena. The permittivity and permeability represent a type of coarse-graining in which the atomic properties are averaged of distances that are large compared to the atomic scale (say the Bohr radius), but small compared with the wavelength of light in question.
Question 2: What is a metamaterial?
If the wavelength of light is large enough, we can imagine averaging the fields over scales large enough to be manipulated by intelligent beings. For visible light, one might consider nanoscale patterns etched on a wafer. For microwaves, objects as large as wires and loops could form the effective medium.
A metamaterial is a medium in which you control the atoms. Dr. Smith cited the work of John Pendry as demonstrating how one could build up a material with a negative index of refraction (negative permeability and permittivity). A negative value of the permittivity is not uncommon. It occurs in metals near a plasmon resonance. Dr. Smith was part of a team at UCSD that built an array of wire loops and posts with both a negative permittivity and a negative permeability.
Question 3: What does a metamaterial do?
Control over the permeability and permittivity allows for some interesting phenomena. Dr. Smith's group demonstrated a negative index of refraction by performing a simple Snell's law experiment. Light is bent in the opposite direction. Many other possibilities were discussed by a Soviet physicist named Veselago in 1968, such as a phase velocity opposite the direction of propagation, and lensing from a flat slab. More recently, Pendry proposed a "perfect lens" which would allow focusing of non-propagating modes --- the near fields that are always ignored in textbook problems.
Another phenomena is called cloaking. It's gotten a lot of media attention and Dr. Smith's work has made the covers of several scientific publications.
CLOAKING
Both the papers from Science address the possibility of cloaking. An interesting mathematical property of Maxwell's equations is that a coordinate transformation can be completely described by a transformation of the fields, the permittivity, and the permeability. This has practical consequences, as both papers illustrate.
Suppose you want to cloak a region of space --- i.e., you don't want any electromagnetic fields to penetrate the region, and you don't want any waves reflected or absorbed. Light rays follow geodesics (e.g., Fermat's principle). If the geodescis of spacetime were to travel around the region to be cloaked, this would be a neat solution of the problem. A coordinate transformation can implement this solution. That coordinate transfomration leads to field lines the curve around the cloaked region and their corresponding permittivity and permeability. This is where metamaterials come in. They allow one to engineer the permittivity and permeability as needed.
Dr. Smith's group at Duke perfomred numerical simulations to determine the properties their metamaterial would need to cloak a disc from microwave radiation. They built the required structure (rather, an approximation to the ideal structure), then demonstrated that waves pass right around the central disc for the most part, even when a strong scatterer is placed inside.
So has Dr. Smith ushered in the age of Klingon cloaking devices? Not yet. He is the first to point out the limitations of his devices. They have a very small bandwidth, meaning they only work for a very small range of wavelengths. In addition, it's hard to manipulate matter on very small scales, so cloaking in the optical range is still a technical challenge, even for a single frequency. To effective cloak a device, one would need to cover a large range of frequencies.
Cloaking was more a proof of principle than The Next Big Thing. More practical applications include antennas and lenses that can do things they don't tell you about in freshman physics.
The mechanism behind cloaking --- an effective warping of spacetime --- got me thinking about general relativity. Are there gravitational object that could actually warp spacetime in the same way, so that light would pass right around them? Black holes pull everything in. I suppose a very dense region of antigravity would be required to deflect light around an object. Still, if this were possible, the cloak would work at all frequencies, because the actual geodesics of spacetime would curve around the object. It would not be the result of an effective dielectric constant that depends on frequency. Light, particles, rocks, and anything else would travel along the same geodesics. It would be cloaked from everything, not just light!
Electroabsorption Spectroscopy of Carbon Nanotubes
Elucidation of the Electronic Structure of Semiconducting Single Walled Carbon Nanotubes by Electroabsoprtion Spectroscopy
Hongbo Zhao and Sumit Mazumdar
PRL 98, 166805 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e166805
That title is quite a mouthful!
The paper discusses how electroabsorption might be used to determine exciton binding energies and the free-particle excitation gap in carbon nanotubes.
The basic idea of electroabsorption spectroscopy is very simple. You make an absorption measurement on a nanotube sample, then you turn on an electric field and repeat the measurement. By subtracting off the free nanotube data, you can observe the effect of the electric field.
da(E,w) = a(E,w) - a(0,w)
Mazumdar and Zhao identify three noteworthy features in the nanotube spectrum, two of which can be used to determine the exciton binding energy. Surprisingly, it seems this quantity is not known. The splitting of different exciton peaks are usually studied in photoluminescence experiments, but it seems the free particle excitation gap is harder to probe. Without knowing the free particle gap, one cannot infer the exciton binding energy.
The most prominent feature in the graphs are the oscillations in the free-particle continuum. I believe these are the Franz-Keldysh oscillations studied by Perebeinos, et alii. I feel that Perebeinos description of the features and the physical mechanisms responsible are much clearer. That paper was published in Nanoletters, but I've only read the arXiv version so far.
After reading this letter, I finally understand what a Fano resonance is. It is a coupling between bound an continuum states. I don't know what its effects are, however. It seems like these would occur often in semiconductors for excitons in any band higher than the first. I don't see how it could occur for an atomic system, except in Rydberg systems where the ionization energy is very small.
Hongbo Zhao and Sumit Mazumdar
PRL 98, 166805 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e166805
That title is quite a mouthful!
The paper discusses how electroabsorption might be used to determine exciton binding energies and the free-particle excitation gap in carbon nanotubes.
The basic idea of electroabsorption spectroscopy is very simple. You make an absorption measurement on a nanotube sample, then you turn on an electric field and repeat the measurement. By subtracting off the free nanotube data, you can observe the effect of the electric field.
da(E,w) = a(E,w) - a(0,w)
Mazumdar and Zhao identify three noteworthy features in the nanotube spectrum, two of which can be used to determine the exciton binding energy. Surprisingly, it seems this quantity is not known. The splitting of different exciton peaks are usually studied in photoluminescence experiments, but it seems the free particle excitation gap is harder to probe. Without knowing the free particle gap, one cannot infer the exciton binding energy.
The most prominent feature in the graphs are the oscillations in the free-particle continuum. I believe these are the Franz-Keldysh oscillations studied by Perebeinos, et alii. I feel that Perebeinos description of the features and the physical mechanisms responsible are much clearer. That paper was published in Nanoletters, but I've only read the arXiv version so far.
After reading this letter, I finally understand what a Fano resonance is. It is a coupling between bound an continuum states. I don't know what its effects are, however. It seems like these would occur often in semiconductors for excitons in any band higher than the first. I don't see how it could occur for an atomic system, except in Rydberg systems where the ionization energy is very small.
Interband Excitons in Carbon Nanotubes
Polarized Photoluminescence Excitation Spectrum of Single-Walled Carbon Nanotubes
J. Lefebvre and P. Finnie
PRL 98, 167406 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e167406
The authors report photoluminescence measurements in carbon nanotubes. They present data for light polarized parallel to the nanotube axis, similar to previous experiments. However, this paper is the first I've seen to probe the photoluminescence spectrum of light polarized perpendicular to the nanotube axis.
Louie, et alii published a paper in 1995 --- a theoretical investigation of the polarizability of carbon nanotubes. They predict the response to light polarized along the nanotube axis to be an order of magnitude larger than for perpendicular polarization. This prediction seems to have been borne out in experiments, especially the one reported in this Letter.
The authors see familiar patterns of absorption at E22 and emission at E11, plus new data on the E12 peak (which is identical to the E21 peak, according to the authors). They claim to see dependence on the chiral angle that has not been included in an analytic theory yet.
Overall, I think the authors have done an excellent job collecting and presenting their data. However, I am confused by their analysis. They report two sidebands of the E11 peak, but it seems they have devoted far too much space to what these sidebands are not. After four paragraphs, I still don't know what the author believe the sidebands are.
This is definitely a good paper for my "Excitons: Experiments" folder. The data presented here might be relevant to the next phase of my research.
J. Lefebvre and P. Finnie
PRL 98, 167406 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e167406
The authors report photoluminescence measurements in carbon nanotubes. They present data for light polarized parallel to the nanotube axis, similar to previous experiments. However, this paper is the first I've seen to probe the photoluminescence spectrum of light polarized perpendicular to the nanotube axis.
Louie, et alii published a paper in 1995 --- a theoretical investigation of the polarizability of carbon nanotubes. They predict the response to light polarized along the nanotube axis to be an order of magnitude larger than for perpendicular polarization. This prediction seems to have been borne out in experiments, especially the one reported in this Letter.
The authors see familiar patterns of absorption at E22 and emission at E11, plus new data on the E12 peak (which is identical to the E21 peak, according to the authors). They claim to see dependence on the chiral angle that has not been included in an analytic theory yet.
Overall, I think the authors have done an excellent job collecting and presenting their data. However, I am confused by their analysis. They report two sidebands of the E11 peak, but it seems they have devoted far too much space to what these sidebands are not. After four paragraphs, I still don't know what the author believe the sidebands are.
This is definitely a good paper for my "Excitons: Experiments" folder. The data presented here might be relevant to the next phase of my research.
Quantum Chernoff Bound
Discriminating States: The Quantum Chernoff Bound
K.M.R. Audenaert, et al.
PRL 98, 160501 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e160501
As the title implies, this Letter demonstrates a quantum analogue of the classical Chernoff bound in information theory. Some of the mathematical formalisms in the paper were beyond me, but I did learn a few cool math tricks.
What is the Chernoff bound? Suppose you have a signal, and you know that the output comes from one of two sources. If you perform N measurements, what is the probability of attributing the data to the wrong source? In 1952, Chernoff showed that the probability of error decreases exponentially for a large number of measurements: P(N) -> exp(-kN). The largest possible k is known as the Chernoff bound (among other things, as the authors point out in footnote 2). It provides a concept of distance between probability distributions.
Until this Letter was published, there was apparently no quantum version of Chernoff's analysis. The authors analyze a two state system. They claim that the probability of error in determining the state also obeys P(N) -> exp(-kN). The quantum Chernoff bound is determined by minimizing the trace of the product of two density matrices. In the classical limit, the expression reduces to that of the classical Chernoff bound.
The one statement I could not verify was the one that follows Eq. (4). The authors say "the upper bound trivially follows" from a mathematical relation that is not at all obvious to me. Somehow, the authors are able to break the logarithm of the trace of a matrix product into a sum of two logarithms:
log [ Tr{ (A P^n)^s (B Q^n)^(1-s) } ] = log[ A^s B^(1-s) ] + n log[ Tr{ P^s Q^(1-s) } ]
This is stated without reference or justification. I don't see how the logarithm of a sum (the trace) can be rewritten as the sum of two logarithms. The authors are correct, however. If this relation is true, their expression for the Chernoff bound follows immediately.
The first paragraph of the "Proof" section contains two neat math tricks. One allows me to rewrite the power of a number (or positive definite matrix) as an integral, and the other allows me to rewrite the difference of two numbers as the integral of a single function. The second trick is similar to the Feynman trick we used in quantum field theory for combining denominators. It's interesting that it can be extended to matrices.
I didn't get much out of the rest of the paper. There are apparently a lot of subtle points and nice mathematical properties of the quantum Chernoff bound, but I lack the background to appreciate them.
The idea behind the Chernoff bound is interesting, and it seems relevant to a variety of fields. For instance, in cryptography, you might know a coded message uses one of two encryption schemes. Chernoff's theory says you only need to know the true value of a certain number of bits and perhaps something about the content of the message before you could distinguish between the two encryption schemes and focus your efforts.
Another area is in the analysis of experimental data. With a large data set, the probability of attributing your results to the wrong theory (distribution) becomes exponentially small. Of course, this assumes you know the distributions before you do the experiment.
In short, it's hard to make a wrong guess about the distribution if you have enough data.
K.M.R. Audenaert, et al.
PRL 98, 160501 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e160501
As the title implies, this Letter demonstrates a quantum analogue of the classical Chernoff bound in information theory. Some of the mathematical formalisms in the paper were beyond me, but I did learn a few cool math tricks.
What is the Chernoff bound? Suppose you have a signal, and you know that the output comes from one of two sources. If you perform N measurements, what is the probability of attributing the data to the wrong source? In 1952, Chernoff showed that the probability of error decreases exponentially for a large number of measurements: P(N) -> exp(-kN). The largest possible k is known as the Chernoff bound (among other things, as the authors point out in footnote 2). It provides a concept of distance between probability distributions.
Until this Letter was published, there was apparently no quantum version of Chernoff's analysis. The authors analyze a two state system. They claim that the probability of error in determining the state also obeys P(N) -> exp(-kN). The quantum Chernoff bound is determined by minimizing the trace of the product of two density matrices. In the classical limit, the expression reduces to that of the classical Chernoff bound.
The one statement I could not verify was the one that follows Eq. (4). The authors say "the upper bound trivially follows" from a mathematical relation that is not at all obvious to me. Somehow, the authors are able to break the logarithm of the trace of a matrix product into a sum of two logarithms:
log [ Tr{ (A P^n)^s (B Q^n)^(1-s) } ] = log[ A^s B^(1-s) ] + n log[ Tr{ P^s Q^(1-s) } ]
This is stated without reference or justification. I don't see how the logarithm of a sum (the trace) can be rewritten as the sum of two logarithms. The authors are correct, however. If this relation is true, their expression for the Chernoff bound follows immediately.
The first paragraph of the "Proof" section contains two neat math tricks. One allows me to rewrite the power of a number (or positive definite matrix) as an integral, and the other allows me to rewrite the difference of two numbers as the integral of a single function. The second trick is similar to the Feynman trick we used in quantum field theory for combining denominators. It's interesting that it can be extended to matrices.
I didn't get much out of the rest of the paper. There are apparently a lot of subtle points and nice mathematical properties of the quantum Chernoff bound, but I lack the background to appreciate them.
The idea behind the Chernoff bound is interesting, and it seems relevant to a variety of fields. For instance, in cryptography, you might know a coded message uses one of two encryption schemes. Chernoff's theory says you only need to know the true value of a certain number of bits and perhaps something about the content of the message before you could distinguish between the two encryption schemes and focus your efforts.
Another area is in the analysis of experimental data. With a large data set, the probability of attributing your results to the wrong theory (distribution) becomes exponentially small. Of course, this assumes you know the distributions before you do the experiment.
In short, it's hard to make a wrong guess about the distribution if you have enough data.
Friday, April 20, 2007
Macroscopic Laser Trapping
An All-Optical Trap for a Gram-Scale Mirror
Thomas Corbitt, et alii
PRL 98, 150802 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e150802
This experimental group has demonstrated laser cooling and trapping of a large object: a 1 gram mirror. (That doesn't sound large, but its 10^25 atoms --- a lot larger than the typical atomic and molecular clouds in these types of experiments.)
The authors were able to achieve an effective temperature of 0.8 K along the direction of the beams. (Temperature is a scalar quantity, so how can it have a direction? What the authors really measured were the mean-square fluctuations in the direction of the beam. The equipartition theorem allow one to turn this quantity into an effective temperature.)
It's not hard to point a laser at a mirror, so why hadn't someone done this before? The authors point out that there are two types of radiation pressure effects: damping forces and restoring forces. A damping force slows things down. It leads to effects like optical molasses. A restoring force keeps objects in a certain region, like in optical tweezers.
For a mirror in an optical cavity, both effects can be implemented by detuning a laser from the resonant frequency of the cavity, but not simultaneously. If the laser is above resonance, it will produce a restoring force, but also an anti-damping force. If the laser is below resonance, it gives rise to a damping force, but also an anti-restoring force. It is impossible to trap a mirror and damp its motion with a single laser. (Sounds like Heisenberg: you can't fix the momentum and position simultaneously. If you had a laser beam tuned to resonance, but with fluctuations above and below, could you achieve both effects with the same beam? Would the spread in momentum and position obey some uncertainty principle? Or would the whole system be totally unstable?)
The authors get around this difficulty by using two beams. One is tuned above resonance, the other below. The frequencies are chosen so that one gives a large restoring force with small anti-damping and the other a large damping force with small anti-restoring. The result is a very stable, rigid localizing force.
What caught my attention in this article was the rigidity. The authors imagine replacing the laser beam with a rigid rod of the same diameter. To achieve the same stiffness (spring constant) as their trap, you would need a material 20% stiffer than diamond!
The caveat of all this work is that the confinement is only to one dimension. The mirror still shows room temperature (or higher) fluctuations in the directions perpendicular to the beam. Perhaps a cubic mirror could be cooled and trapped just as effectively in all three directions.
Thomas Corbitt, et alii
PRL 98, 150802 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e150802
This experimental group has demonstrated laser cooling and trapping of a large object: a 1 gram mirror. (That doesn't sound large, but its 10^25 atoms --- a lot larger than the typical atomic and molecular clouds in these types of experiments.)
The authors were able to achieve an effective temperature of 0.8 K along the direction of the beams. (Temperature is a scalar quantity, so how can it have a direction? What the authors really measured were the mean-square fluctuations in the direction of the beam. The equipartition theorem allow one to turn this quantity into an effective temperature.)
It's not hard to point a laser at a mirror, so why hadn't someone done this before? The authors point out that there are two types of radiation pressure effects: damping forces and restoring forces. A damping force slows things down. It leads to effects like optical molasses. A restoring force keeps objects in a certain region, like in optical tweezers.
For a mirror in an optical cavity, both effects can be implemented by detuning a laser from the resonant frequency of the cavity, but not simultaneously. If the laser is above resonance, it will produce a restoring force, but also an anti-damping force. If the laser is below resonance, it gives rise to a damping force, but also an anti-restoring force. It is impossible to trap a mirror and damp its motion with a single laser. (Sounds like Heisenberg: you can't fix the momentum and position simultaneously. If you had a laser beam tuned to resonance, but with fluctuations above and below, could you achieve both effects with the same beam? Would the spread in momentum and position obey some uncertainty principle? Or would the whole system be totally unstable?)
The authors get around this difficulty by using two beams. One is tuned above resonance, the other below. The frequencies are chosen so that one gives a large restoring force with small anti-damping and the other a large damping force with small anti-restoring. The result is a very stable, rigid localizing force.
What caught my attention in this article was the rigidity. The authors imagine replacing the laser beam with a rigid rod of the same diameter. To achieve the same stiffness (spring constant) as their trap, you would need a material 20% stiffer than diamond!
The caveat of all this work is that the confinement is only to one dimension. The mirror still shows room temperature (or higher) fluctuations in the directions perpendicular to the beam. Perhaps a cubic mirror could be cooled and trapped just as effectively in all three directions.
Origami the Easy Way
Capillary Origami: Spontaneous Wrapping of a Droplet with an Elastic Sheet
Charlotte Py, Paul Reverdy, Lionel Doppler, Jose Bico, Benoit Roman, and Charles N. Baroud
PRL 98, 156103 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e156103
This article contains some of the pest pictures I've seen in a PRL article.
The authors place little sheets of polydimethylsiloxane (PDMS) on a hydrophobic surface and add a drop of water. As the water evaporates, the sheets fold up. The shape and thickness of the sheet determine what the final object will be. In this type of origami, Nature does the work for you. Like the chia pet, "Just add water!" (Of course, I am sure the authors invested a considerable amount of labor before they added the water and let Nature take over.)
The pictures on the first page show the folding of a square into a tube and a triangle into a tetrahedron. On the final page, the authors show of the expertise of their lab by folding a flower into a sphere, a cross into a cube, and a square with two rounded corners into a triangle with a tube at the bottom.
The theoretical explanation in this paper is excellent. If I had to summarize their entire theory of folding in one word, it would be "competition." Competition between bending energy and surface tension sets the fundamental length scale and determines the shape of the liquid-membrane interface. Competition between bending energy and stretching energy determines whether a sheet will bend or crumple. The authors explain these ideas and their model clearly.
This paper demonstrates a good balance between experiment and theory. Two-dimensional membranes are an interesting topic, because the systems are simple enough that analytic models can be derived and solved. However, new effects are reported frequently. Although the field has been around for hundreds of years, it continues to be a fertile area for research.
Charlotte Py, Paul Reverdy, Lionel Doppler, Jose Bico, Benoit Roman, and Charles N. Baroud
PRL 98, 156103 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e156103
This article contains some of the pest pictures I've seen in a PRL article.
The authors place little sheets of polydimethylsiloxane (PDMS) on a hydrophobic surface and add a drop of water. As the water evaporates, the sheets fold up. The shape and thickness of the sheet determine what the final object will be. In this type of origami, Nature does the work for you. Like the chia pet, "Just add water!" (Of course, I am sure the authors invested a considerable amount of labor before they added the water and let Nature take over.)
The pictures on the first page show the folding of a square into a tube and a triangle into a tetrahedron. On the final page, the authors show of the expertise of their lab by folding a flower into a sphere, a cross into a cube, and a square with two rounded corners into a triangle with a tube at the bottom.
The theoretical explanation in this paper is excellent. If I had to summarize their entire theory of folding in one word, it would be "competition." Competition between bending energy and surface tension sets the fundamental length scale and determines the shape of the liquid-membrane interface. Competition between bending energy and stretching energy determines whether a sheet will bend or crumple. The authors explain these ideas and their model clearly.
This paper demonstrates a good balance between experiment and theory. Two-dimensional membranes are an interesting topic, because the systems are simple enough that analytic models can be derived and solved. However, new effects are reported frequently. Although the field has been around for hundreds of years, it continues to be a fertile area for research.
Wednesday, April 18, 2007
General Adiabatic Theorem
Sufficiency Criterion for the Validity of the Adiabatic Approximation
D.M. Tong, K. Singh, L.C. Kwek, and C.H. Oh
PRL 98, 150402 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e150402
The authors take a close look at the adiabatic approximation. They find that the usual criterion presented in textbooks (for instance, Messiah) is valid for a restricted set of systems, and they show that two additional criteria are required for a general quantum system. This was a difficult read, but the result was worth the effort.
The adiabatic theorem sounds so sensible that I'm not surprised it was used without being proven for so many years: If you have a system that is in an eigenstate |n(0)>, then you modify the system very slowly, then it will remain in |n(t)>. For instance, if I have a magnetic dipole that points in the direction of an applied magnetic field, the adiabatic theorem says if I slowly rotate the magnetic field, the dipole continues to point along it. (For the quantum version, replace "magnetic dipole" with "spin.") Or, if I am in the ground state of a harmonic oscillator and I slowly change the spring constant, at any time, the system will be in the ground state of the current oscillator.
It turns out that there are cases where common sense is misleading. I read such a counterexample just a couple months ago. The system was designed to satisfy the requirement of the adiabatic approximation, yet violate its prediction. The authors of the current paper point out that the adiabatic theorem assumes the first time derivative is small and all higher derivatives are smaller. This is not the case in the counterexample.
Clearly the adiabatic theorem works a lot of the time. Otherwise, it wouldn't still be used and taught to graduate students. The question raised by the counterexample is, How can you tell if the adiabatic theorem will work or not? In a one and a half pages of straight-forward but tedious calculations, the authors derive three criteria that apply to any quantum system. (That's 1.5 PRL pages -- probably equivalent to 5 normal pages.) If these criteria are satisfied, then the system will be in |n(t)> at time t with a probability that approaches 1.
The first of these criteria is the industry standard. It applies to systems where the energy difference between states is a constant and the time evolution of the inner product of two states is a constant. The other two criteria involve integrals that can't be evaluated in general. However, an upper limit can be placed on the integrals by replacing the integrand with its largest value. This gives a product of the maximum value and the time interval that must be small, so it sets limits on how long you might expect the adiabatic approximation to be valid.
This could be quite useful. I've never seen a calculation that suggests how long you might expect the adiabatic theorem to hold. In my examples above, I used the term slowly. The authors have given theorists a way to quantify exactly what we mean by "slowly." They apply their criteria to a spin 1/2 system in a rotating magentic field and find that the adiabatic theorem will only be valid for a fixed number of periods. Even "slow evolution" isn't allowed to take forever!
D.M. Tong, K. Singh, L.C. Kwek, and C.H. Oh
PRL 98, 150402 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e150402
The authors take a close look at the adiabatic approximation. They find that the usual criterion presented in textbooks (for instance, Messiah) is valid for a restricted set of systems, and they show that two additional criteria are required for a general quantum system. This was a difficult read, but the result was worth the effort.
The adiabatic theorem sounds so sensible that I'm not surprised it was used without being proven for so many years: If you have a system that is in an eigenstate |n(0)>, then you modify the system very slowly, then it will remain in |n(t)>. For instance, if I have a magnetic dipole that points in the direction of an applied magnetic field, the adiabatic theorem says if I slowly rotate the magnetic field, the dipole continues to point along it. (For the quantum version, replace "magnetic dipole" with "spin.") Or, if I am in the ground state of a harmonic oscillator and I slowly change the spring constant, at any time, the system will be in the ground state of the current oscillator.
It turns out that there are cases where common sense is misleading. I read such a counterexample just a couple months ago. The system was designed to satisfy the requirement of the adiabatic approximation, yet violate its prediction. The authors of the current paper point out that the adiabatic theorem assumes the first time derivative is small and all higher derivatives are smaller. This is not the case in the counterexample.
Clearly the adiabatic theorem works a lot of the time. Otherwise, it wouldn't still be used and taught to graduate students. The question raised by the counterexample is, How can you tell if the adiabatic theorem will work or not? In a one and a half pages of straight-forward but tedious calculations, the authors derive three criteria that apply to any quantum system. (That's 1.5 PRL pages -- probably equivalent to 5 normal pages.) If these criteria are satisfied, then the system will be in |n(t)> at time t with a probability that approaches 1.
The first of these criteria is the industry standard. It applies to systems where the energy difference between states is a constant and the time evolution of the inner product of two states is a constant. The other two criteria involve integrals that can't be evaluated in general. However, an upper limit can be placed on the integrals by replacing the integrand with its largest value. This gives a product of the maximum value and the time interval that must be small, so it sets limits on how long you might expect the adiabatic approximation to be valid.
This could be quite useful. I've never seen a calculation that suggests how long you might expect the adiabatic theorem to hold. In my examples above, I used the term slowly. The authors have given theorists a way to quantify exactly what we mean by "slowly." They apply their criteria to a spin 1/2 system in a rotating magentic field and find that the adiabatic theorem will only be valid for a fixed number of periods. Even "slow evolution" isn't allowed to take forever!
Tuesday, April 17, 2007
Rack and Pinion a la Casimir
Noncontact Rack and Pinion Powered by the Lateral Casimir Force
Arahs Ashourvan, MirFaez Miri, and Ramin Golestanian
PRL 98, 140801 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e140801
The authors propose a nanoscale rack and pinion where the Casimir force (rather than contact between the cogs) allows the pinion to move the rack. They find two regimes. There is a contact regime, where the gears move as if they were in direct contact with one another, and a skipping regime where the teeth can slip by one another. The interesting aspect of this device is that the cogs are never in physical contact with one another.
The equation of motion for the system presented in Eq. (1) is rather simple. You could write it down without knowing much about racks, pinions, or vacuum fluctuations. It simply describes the motion of a pendulum with damping and a driving torque. It's a nonlinear equation that cannot be solved analytically, but it is a classical system.
The authors go on to study the system in four perturbative regimes. First, they study the case of no damping or torque. This system can be integrated, and shows crossover behavior between the contact and skipping regimes mentioned earlier. Depending on the velocity of the rack, the pinion can rotate in either direction. When a torque is applied, the same general behavior is observed. The main differenc is that the boundary between the two regions depends on the applied torque.
The case with dissipation cannot be solved exactly. In the case of weak dissipation, the authors treat the damping as a small perturbation using something called the Melnikov Method. This introduces an interesting regime where the pinion velocity is independent of the torque, and a load at which the velocity drops to zero --- the stall force.
In the case of strong dissipation, the authors discard the acceleration term (like a Langevin equation) and integrate the approximate equation of motion. There is again a stall force, and the behavior is similar to case of weak damping above the skipping velocity.
Finally, the authors investigate the actual form of the Casimir force in their system. As I said earlier, the above analysis is for a damped pendulum under constant torque. To say anything meaningful about the role of the Casimir force, the authors have to introduce it into this phenomenological model. The force decays exponentially with increasing separation between the rack and pinion. The skipping velocity is a power law at small separations and decays exponentially at large separations (like a gamma distribution, I suppose). This is quite useful in applications.
I kept this article in my files because it seems like the type of fundamental nanotechnology research needed to move the field forward. We can't just build little versions of big machines because effects like thermal fluctuations and friction can ruin everything on tiny scales. These authors have shown how you can take advantage of an effect that only happens at these small scales. The rack and pinion steering of an automobile will never utilize the Casimir force, but if you want to build a tiny ratchet out of nanotubes and buckyballs, then you might not have any other option. Well done guys.
Arahs Ashourvan, MirFaez Miri, and Ramin Golestanian
PRL 98, 140801 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e140801
The authors propose a nanoscale rack and pinion where the Casimir force (rather than contact between the cogs) allows the pinion to move the rack. They find two regimes. There is a contact regime, where the gears move as if they were in direct contact with one another, and a skipping regime where the teeth can slip by one another. The interesting aspect of this device is that the cogs are never in physical contact with one another.
The equation of motion for the system presented in Eq. (1) is rather simple. You could write it down without knowing much about racks, pinions, or vacuum fluctuations. It simply describes the motion of a pendulum with damping and a driving torque. It's a nonlinear equation that cannot be solved analytically, but it is a classical system.
The authors go on to study the system in four perturbative regimes. First, they study the case of no damping or torque. This system can be integrated, and shows crossover behavior between the contact and skipping regimes mentioned earlier. Depending on the velocity of the rack, the pinion can rotate in either direction. When a torque is applied, the same general behavior is observed. The main differenc is that the boundary between the two regions depends on the applied torque.
The case with dissipation cannot be solved exactly. In the case of weak dissipation, the authors treat the damping as a small perturbation using something called the Melnikov Method. This introduces an interesting regime where the pinion velocity is independent of the torque, and a load at which the velocity drops to zero --- the stall force.
In the case of strong dissipation, the authors discard the acceleration term (like a Langevin equation) and integrate the approximate equation of motion. There is again a stall force, and the behavior is similar to case of weak damping above the skipping velocity.
Finally, the authors investigate the actual form of the Casimir force in their system. As I said earlier, the above analysis is for a damped pendulum under constant torque. To say anything meaningful about the role of the Casimir force, the authors have to introduce it into this phenomenological model. The force decays exponentially with increasing separation between the rack and pinion. The skipping velocity is a power law at small separations and decays exponentially at large separations (like a gamma distribution, I suppose). This is quite useful in applications.
I kept this article in my files because it seems like the type of fundamental nanotechnology research needed to move the field forward. We can't just build little versions of big machines because effects like thermal fluctuations and friction can ruin everything on tiny scales. These authors have shown how you can take advantage of an effect that only happens at these small scales. The rack and pinion steering of an automobile will never utilize the Casimir force, but if you want to build a tiny ratchet out of nanotubes and buckyballs, then you might not have any other option. Well done guys.
High Powered Lasers
Laser physics: Extreme light
Ed Gernster
Nature, 446, 16--18 (2007)
URL: http://www.nature.com/news/2007/070226/full/446016a.html
Every so often, I come across an article that really excites me. It makes me feel like running to the library and checking out 3 or 4 textbooks so I can learn more about the subject. These books usually sit on my shelves for a couple months, until I come across another article. I feel guilty about having 8 library books out at a time, so I trade in my first set for a different one, a little disappointed that I don't have the knowledge I was so excited about learning a couple months ago. Well, I think it's about time for a trip to the library.
This article blew me away. Ed wrote about the advances in laser technology that have taken place in the last 50 years, and about the new facilities under construction.
The first tid-bit that caught my attention was that an electric field of 8 x 10^18 V/m will make the vacuum boil. It will rip apart the pairs of virtual particles that pop into and out of existence. This is called the Schwinger limit.
Sounds exciting, but all kinds of neat things are supposed to happen when you probe the Planck length too. The exciting thing about the Schwinger limit is that experimentalists are only 3 orders of magnitude away right now. (For comparison, I think the people at CERN will still be 15 orders of magnitude away from the Planck energy.)
You hear about lasers all the time. What's so great about them? Nonlinear optics. "In the 1960s, the fact that early lasers were powerful enough to change the refractive index of the medium through which they travelled opened up fresh vistas in nonlinear optics." Todays lasers can acclerate all the electrons around them to relativistic speeds. The next generation will be able to do the same for ions.
What a neat idea! I could shine a laser on a block of metal on my desk and be able to probe relativistic interactions between charged particles.
Another really neat idea is that these lasers could produce accelerations the same order of magnitude as the gravitational accelerations in black holes. Einstein said that gravity and acceleration are the same. Hawking said gravity can make radiation. In the 1970s, Unruh connected these ideas and said that an accelerated particle will see Hawking-like radiation, even if it's not in a gravitational field.
To quote one of Gernster sources, Bob Bingham, "The vacuum really doesn't care if it's an electric field, a magnetic field, a gravitational field, ... If you can packe enough energy i, you can excite particles out ofthe vacuum. ... Nothing generates fileds even close to those produced by an ultra-high intensity laser --- except perhaps a black hole."
Another sources points out the analog of ultrarelativistic lasers with the nonlinear phenomena of the 1960s through the present: "We're going to change the index of refraction of the vacuum." What a concept!
Gernster gives a very clear description of how lasers are able to do what they do. The take a lot of energy, but not a whole lot --- on the order of a kilowatt hour. But instead of spreading the energy out over an hour, the compress it down to a few femtoseconds. The energy is the same, but the difference in power is huge.
Excellent article.
Ed Gernster
Nature, 446, 16--18 (2007)
URL: http://www.nature.com/news/2007/070226/full/446016a.html
Every so often, I come across an article that really excites me. It makes me feel like running to the library and checking out 3 or 4 textbooks so I can learn more about the subject. These books usually sit on my shelves for a couple months, until I come across another article. I feel guilty about having 8 library books out at a time, so I trade in my first set for a different one, a little disappointed that I don't have the knowledge I was so excited about learning a couple months ago. Well, I think it's about time for a trip to the library.
This article blew me away. Ed wrote about the advances in laser technology that have taken place in the last 50 years, and about the new facilities under construction.
The first tid-bit that caught my attention was that an electric field of 8 x 10^18 V/m will make the vacuum boil. It will rip apart the pairs of virtual particles that pop into and out of existence. This is called the Schwinger limit.
Sounds exciting, but all kinds of neat things are supposed to happen when you probe the Planck length too. The exciting thing about the Schwinger limit is that experimentalists are only 3 orders of magnitude away right now. (For comparison, I think the people at CERN will still be 15 orders of magnitude away from the Planck energy.)
You hear about lasers all the time. What's so great about them? Nonlinear optics. "In the 1960s, the fact that early lasers were powerful enough to change the refractive index of the medium through which they travelled opened up fresh vistas in nonlinear optics." Todays lasers can acclerate all the electrons around them to relativistic speeds. The next generation will be able to do the same for ions.
What a neat idea! I could shine a laser on a block of metal on my desk and be able to probe relativistic interactions between charged particles.
Another really neat idea is that these lasers could produce accelerations the same order of magnitude as the gravitational accelerations in black holes. Einstein said that gravity and acceleration are the same. Hawking said gravity can make radiation. In the 1970s, Unruh connected these ideas and said that an accelerated particle will see Hawking-like radiation, even if it's not in a gravitational field.
To quote one of Gernster sources, Bob Bingham, "The vacuum really doesn't care if it's an electric field, a magnetic field, a gravitational field, ... If you can packe enough energy i, you can excite particles out ofthe vacuum. ... Nothing generates fileds even close to those produced by an ultra-high intensity laser --- except perhaps a black hole."
Another sources points out the analog of ultrarelativistic lasers with the nonlinear phenomena of the 1960s through the present: "We're going to change the index of refraction of the vacuum." What a concept!
Gernster gives a very clear description of how lasers are able to do what they do. The take a lot of energy, but not a whole lot --- on the order of a kilowatt hour. But instead of spreading the energy out over an hour, the compress it down to a few femtoseconds. The energy is the same, but the difference in power is huge.
Excellent article.
The Best Lorentz Frame for Calculations
Noninvariance of Space and Time Scale Ranges under a Lorentz Transformation and the Implications for the Study of Relativistic Interactions
J.L. Vay
PRL, 130405 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e130405
I don't think this was a particularly well-written article, but the ideas are quite interesting. The basic premise is that you can exploit time dilation and length contraction to find a frame that makes calculations simple.
In most experiments and numerical simulations, there is a hierarchy of length and time scales. For instance, in a carbon nanotube, the nanotube radius is probably the smallest important length scale; the largest might be the length of the nanotube, which can be thousands or millions of tube radii. If I had to carry out simulations that described all length scales, I would have a lot of grid points to worry about.
Vay says, "Well, Jesse, if you boosted to a frame that moves quickly enough, you could end up with a nanotube whose length is equal to or SMALLER than its radius. Einstein tells us that an experiment in this moving frame is just as good as one in the nanotube frame. Why not make it easy on yourself?"
Maybe it wouldn't help me out that much, but for simulations of relativistic beams of electrons, Vay shows that calculation times can be reduced by a factor of a thousand or more.
The first section of the paper is devoted to an example that shows the opposite: that the ratio of the longest to the shortest relevant length (and time) scales can be made extremely large depending on the Lorentz frame you choose for the calculation. Reading the paper a second time, I realized that the point was to demonstrate separation of scales, but since it contradicts the claim of the abstract, I was really confused the first time through.
The three physical examples in the second half of the paper clearly demonstrate the utility of choosing the right Lorentz frame.
To show that this approach is practical, Vay performed the same calculation of the passage of a relativistic beam of protons in a cylinder colliding with an electron gas. In the lab frame, the experiment spans a few kilometers, and the pipe radius is just a centimeter, so the length scales span 5 orders of magnitude.
The lab frame calculation took a week of supercomputer time. The boosted frame calculation took half an hour on the same computer. That's an amazing improvement, but I don't really know when this method will work. I wish Vay had devoted more time to explaining what types of calculations can be improved.
J.L. Vay
PRL, 130405 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e130405
I don't think this was a particularly well-written article, but the ideas are quite interesting. The basic premise is that you can exploit time dilation and length contraction to find a frame that makes calculations simple.
In most experiments and numerical simulations, there is a hierarchy of length and time scales. For instance, in a carbon nanotube, the nanotube radius is probably the smallest important length scale; the largest might be the length of the nanotube, which can be thousands or millions of tube radii. If I had to carry out simulations that described all length scales, I would have a lot of grid points to worry about.
Vay says, "Well, Jesse, if you boosted to a frame that moves quickly enough, you could end up with a nanotube whose length is equal to or SMALLER than its radius. Einstein tells us that an experiment in this moving frame is just as good as one in the nanotube frame. Why not make it easy on yourself?"
Maybe it wouldn't help me out that much, but for simulations of relativistic beams of electrons, Vay shows that calculation times can be reduced by a factor of a thousand or more.
The first section of the paper is devoted to an example that shows the opposite: that the ratio of the longest to the shortest relevant length (and time) scales can be made extremely large depending on the Lorentz frame you choose for the calculation. Reading the paper a second time, I realized that the point was to demonstrate separation of scales, but since it contradicts the claim of the abstract, I was really confused the first time through.
The three physical examples in the second half of the paper clearly demonstrate the utility of choosing the right Lorentz frame.
To show that this approach is practical, Vay performed the same calculation of the passage of a relativistic beam of protons in a cylinder colliding with an electron gas. In the lab frame, the experiment spans a few kilometers, and the pipe radius is just a centimeter, so the length scales span 5 orders of magnitude.
The lab frame calculation took a week of supercomputer time. The boosted frame calculation took half an hour on the same computer. That's an amazing improvement, but I don't really know when this method will work. I wish Vay had devoted more time to explaining what types of calculations can be improved.
Boron Nitride vs. Carbon Nanotubes
Theory of Graphitic Boron Nitride Nanotubes
Angel Rubio, Jennifer L. Corkill, and Marvin L. Cohen
PRB 49, 5081--5084 (1994)
URL: http://link.aps.org/abstract/PRB/v49/p5081
This paper came out just a year after the discovery of carbon nanotubes. It really was a recent discovery when this paper was written. The authors discuss what to expect from boron nitride nanotubes.
I didn't realize how different boron nitride was until I read this paper. It's definitely not graphene with a sublattice asymmetry! The lattice spacing is similar: 1.45 angstroms in BN, 1.42 in graphene. Apparently, the average on-site potential is similar as well. The authors say this can be inferred from the band withs of the parent crystals. However, the similarity ends there.
Boron nitride is a semiconductor with an INDIRECT gap on the order of 5 electron volts. All boron nitride nanotubes are semiconducting as well.
The most surprising result is the scaling of bandgap with tube radius. In carbon nanotubes, the well-known result is that the band gap is inversely proportional to the nanotube radius. In boron nitride, the band gap INCREASES with increasing radius until it approaches the free BN sheet band gap. The effect of curvature reduces the band gap of boron nitride.
This made no sense to me, with my background in nanotubes. The authors point out that the band gap of hexagonal boron nitride decreases with increasing pressure. Equating the strain of a curved tube with pressure, I can at least make sense of the effect.
It is interesting that only some of the boron nitride nanotubes are indirect gap semiconductors even though the parent BN sheet is. The (n,0) tubes are direct gap. All carbon nanotubes are direct gap semiconductors.
The authors use a tight-binding model with first and second nearest neighbor interactions. I'd like to know how many parameters are in their model. Surely more than the two parameters of the corresonding graphene model.
I'll have to keep these differences in mind as I continue my research. I would not have expected such disparity between these two similar structures.
Angel Rubio, Jennifer L. Corkill, and Marvin L. Cohen
PRB 49, 5081--5084 (1994)
URL: http://link.aps.org/abstract/PRB/v49/p5081
This paper came out just a year after the discovery of carbon nanotubes. It really was a recent discovery when this paper was written. The authors discuss what to expect from boron nitride nanotubes.
I didn't realize how different boron nitride was until I read this paper. It's definitely not graphene with a sublattice asymmetry! The lattice spacing is similar: 1.45 angstroms in BN, 1.42 in graphene. Apparently, the average on-site potential is similar as well. The authors say this can be inferred from the band withs of the parent crystals. However, the similarity ends there.
Boron nitride is a semiconductor with an INDIRECT gap on the order of 5 electron volts. All boron nitride nanotubes are semiconducting as well.
The most surprising result is the scaling of bandgap with tube radius. In carbon nanotubes, the well-known result is that the band gap is inversely proportional to the nanotube radius. In boron nitride, the band gap INCREASES with increasing radius until it approaches the free BN sheet band gap. The effect of curvature reduces the band gap of boron nitride.
This made no sense to me, with my background in nanotubes. The authors point out that the band gap of hexagonal boron nitride decreases with increasing pressure. Equating the strain of a curved tube with pressure, I can at least make sense of the effect.
It is interesting that only some of the boron nitride nanotubes are indirect gap semiconductors even though the parent BN sheet is. The (n,0) tubes are direct gap. All carbon nanotubes are direct gap semiconductors.
The authors use a tight-binding model with first and second nearest neighbor interactions. I'd like to know how many parameters are in their model. Surely more than the two parameters of the corresonding graphene model.
I'll have to keep these differences in mind as I continue my research. I would not have expected such disparity between these two similar structures.
Tuesday, April 10, 2007
Every Rock Cracks the Same Way
Scaling and Universality in Rock Fracture
Jorn Davidsen
PRL 98, 125502 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e125502
When a rock is squeezed, tiny cracks form inside. Each crack makes a sound. Davidsen recorded the waiting time between cracking sounds in a variety of rock samples and performed a statistical analysis. When the waiting times were divided by the mean waiting time for a given experiment, all the probability distributions collapsed onto a single curve. Even earthquake data falls onto the same curve when scaled by the mean waiting time between aftershocks.
The probability distribution for the scaled waiting time is a gamma distribution: a power law multiplied with an exponential. I suppose the name comes from the normalization constant. Davidsen showed that the distribution is independent of the sample, the mechanism used to crush the rock, and a cutoff intensity. (You get the same distribution even if you ignore the cracks you can't hear.) All of this suggests that the probability distribution is a universal feature of rock fracture.
A universal mechanism for cracking suggests that a detailed analysis of the molecular properties and bonding is unnecessary. Davidsen doesn't address this directly, but PhysicsWeb stressed the point. It could provide a useful check for numerical and analytic models of crack formation. If you use your favorite model to generate a series of crack, then analyze the scaled waiting time, your data should generate the same gamma distribution as real rocks and earthquakes. If not, then your model has failed to capture whatever mechanism is responsible for this universality.
Is it possible to work backwards with the renormalization group? I.e., knowing the universal scaling relation of the scaled waiting times, can one deduce something useful about the interactions on the microscopic level?
In concluding, Davidsen makes two interesting observations. First, although the waiting times for rock fracture and earthquakes have the same probability distribution, the correlation between waiting times do not. Second, the statistics of rare events often generates a Poisson distribution. The fact that the waiting times do not is important.
Jorn Davidsen
PRL 98, 125502 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e125502
When a rock is squeezed, tiny cracks form inside. Each crack makes a sound. Davidsen recorded the waiting time between cracking sounds in a variety of rock samples and performed a statistical analysis. When the waiting times were divided by the mean waiting time for a given experiment, all the probability distributions collapsed onto a single curve. Even earthquake data falls onto the same curve when scaled by the mean waiting time between aftershocks.
The probability distribution for the scaled waiting time is a gamma distribution: a power law multiplied with an exponential. I suppose the name comes from the normalization constant. Davidsen showed that the distribution is independent of the sample, the mechanism used to crush the rock, and a cutoff intensity. (You get the same distribution even if you ignore the cracks you can't hear.) All of this suggests that the probability distribution is a universal feature of rock fracture.
A universal mechanism for cracking suggests that a detailed analysis of the molecular properties and bonding is unnecessary. Davidsen doesn't address this directly, but PhysicsWeb stressed the point. It could provide a useful check for numerical and analytic models of crack formation. If you use your favorite model to generate a series of crack, then analyze the scaled waiting time, your data should generate the same gamma distribution as real rocks and earthquakes. If not, then your model has failed to capture whatever mechanism is responsible for this universality.
Is it possible to work backwards with the renormalization group? I.e., knowing the universal scaling relation of the scaled waiting times, can one deduce something useful about the interactions on the microscopic level?
In concluding, Davidsen makes two interesting observations. First, although the waiting times for rock fracture and earthquakes have the same probability distribution, the correlation between waiting times do not. Second, the statistics of rare events often generates a Poisson distribution. The fact that the waiting times do not is important.
Friday, April 6, 2007
Foolproof 3D Quasicrystals
Growing Perfect Decagonal Quasicrystals by Local Rules
Hyeong-Chai Jeong
PRL 98, 135501 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e135501
Perfect Penrose Tiling (PPT)
Penrose developed a set of rules which allow a perfect tiling of the 2D plane in a non repeating pattern. These are called quasicrystals. His rules allow such a tiling, but they do not guarantee it. At the edges, there are legal attachments that introduce defects and prevent you from tiling the entire plane. However, another guy called Onoda showed that if you start with a defect in the center, then local growth rules will fill the rest of the plane with a PPT. The only defect is the seed at the center.
To extend 2D tilings to 3D crystals, people thought the best you could do was stacking 2D planes on top of one another. This would introduce a line defect where all the decagons overlap. The authors of this letter developed an algorithm that starts with two defects, but their vertical and lateral attachment rules allow you to fill the rest of space --- "the bulk" --- with PPTs.
A decapod has 10 edges, and the way these growth rules work is to assign an arrow to each edge. Since the arrow at each edge can point in either of two directions, there are 1024 possible decagons. Symmetry under reflection and rotation reduces this number to 62. I'd like to see how that counting works, because 2x31 seems a difficult number to get from 2^10! Anyway, there are 62 unique decagons that will fill a plane with local growth rules. Of these, only one can be filled in: the cartwheel.
Using a special decagon at the bottom and a cartwheel on top of it, the authors were able to add a vertical growth rule that would overcome any dead zones. As a result, they are able to grow a 3D quasicrystal from a single point defect.
Who cares about local growth rules? Well, Nature for one. If I had a set of Penrose tiles (which I'd love to get my hands on), I could sit and meticulously place them one by one until I used up my bag, with a perfect tiling. However, Nature might not be as attentive as me. She'd probably take a tile and try to fit it at an edge. If it would stick (i.e. satisfied local growth rules), Nature would be happy and move on to the next tile. After enough of this, she'd eventually end up with nothing but dead zones and defects. Nature would have to be really lucky to make a quasicrystal as large as mine.
With a decagon in the middle, though, Nature couldn't go wrong. Every tile that fit on an edge would continue the pattern perfectly. Thus, a point defect at the center, the decagon seed, would allow a random growth algorithm to build a perfect quasicrystal except for the defect. The beauty of the author's work is that he showed you only need a point defect --- not an infinite line of them --- to do the same thing in three dimensions. He came up with a simple set of rules that make it impossible to mess up the tiling if you start with two special decagons stacked on top of each other. It's foolproof!
Hyeong-Chai Jeong
PRL 98, 135501 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e135501
Perfect Penrose Tiling (PPT)
Penrose developed a set of rules which allow a perfect tiling of the 2D plane in a non repeating pattern. These are called quasicrystals. His rules allow such a tiling, but they do not guarantee it. At the edges, there are legal attachments that introduce defects and prevent you from tiling the entire plane. However, another guy called Onoda showed that if you start with a defect in the center, then local growth rules will fill the rest of the plane with a PPT. The only defect is the seed at the center.
To extend 2D tilings to 3D crystals, people thought the best you could do was stacking 2D planes on top of one another. This would introduce a line defect where all the decagons overlap. The authors of this letter developed an algorithm that starts with two defects, but their vertical and lateral attachment rules allow you to fill the rest of space --- "the bulk" --- with PPTs.
A decapod has 10 edges, and the way these growth rules work is to assign an arrow to each edge. Since the arrow at each edge can point in either of two directions, there are 1024 possible decagons. Symmetry under reflection and rotation reduces this number to 62. I'd like to see how that counting works, because 2x31 seems a difficult number to get from 2^10! Anyway, there are 62 unique decagons that will fill a plane with local growth rules. Of these, only one can be filled in: the cartwheel.
Using a special decagon at the bottom and a cartwheel on top of it, the authors were able to add a vertical growth rule that would overcome any dead zones. As a result, they are able to grow a 3D quasicrystal from a single point defect.
Who cares about local growth rules? Well, Nature for one. If I had a set of Penrose tiles (which I'd love to get my hands on), I could sit and meticulously place them one by one until I used up my bag, with a perfect tiling. However, Nature might not be as attentive as me. She'd probably take a tile and try to fit it at an edge. If it would stick (i.e. satisfied local growth rules), Nature would be happy and move on to the next tile. After enough of this, she'd eventually end up with nothing but dead zones and defects. Nature would have to be really lucky to make a quasicrystal as large as mine.
With a decagon in the middle, though, Nature couldn't go wrong. Every tile that fit on an edge would continue the pattern perfectly. Thus, a point defect at the center, the decagon seed, would allow a random growth algorithm to build a perfect quasicrystal except for the defect. The beauty of the author's work is that he showed you only need a point defect --- not an infinite line of them --- to do the same thing in three dimensions. He came up with a simple set of rules that make it impossible to mess up the tiling if you start with two special decagons stacked on top of each other. It's foolproof!
Forked Fountains
Splitting of a Liquid Jet
Srinivas Paruchuri and Michael P. Brenner
PRL 98, 134502 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e134502
Interesting work --- hard to believe this problem had not been studied earlier! I guess that's the effect of a well-written paper: The ideas are presented so clearly they seem obvious. This was, in my opinion, a very well-written paper.
The authors analyzed the conditions under which a jet of fluid can split. They derive a Navier-Stokes equation for their model jet, then solve it numerically. The numerical results are used as the basis of an analyitc model that captures the essential features of the numerical work. The authors demonstrated a very nice interplay between numerical and analytic work.
The conclusions of the authors are that tangential stress on a jet can lead to splitting, but normal stress cannot. Tangential stress must overcome the surface tension of the fluid; thus, there is a critical stress, below which splitting cannot occur.
Doodling in my notepad, I made a simple model for the pinching and splitting of two surfaces. It is entirely mathematical and does not include any physical parameters. I compared my sketches with the authors results, and was surprised to see sharp cusps in their cross-sections. In my model, there is a linear crossing at one instant in time, but before and after, the surfaces are smooth. It seems to me that a smooth membrane would be vastly lower in energy than one with a cusp. Of course, when the membranes touch, it's got to lead to some kind of singularity in the differential equation, so a numerical routine or an analytic solution to the full equations might not "know what to do" after the membranes meet.
Srinivas Paruchuri and Michael P. Brenner
PRL 98, 134502 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e134502
Interesting work --- hard to believe this problem had not been studied earlier! I guess that's the effect of a well-written paper: The ideas are presented so clearly they seem obvious. This was, in my opinion, a very well-written paper.
The authors analyzed the conditions under which a jet of fluid can split. They derive a Navier-Stokes equation for their model jet, then solve it numerically. The numerical results are used as the basis of an analyitc model that captures the essential features of the numerical work. The authors demonstrated a very nice interplay between numerical and analytic work.
The conclusions of the authors are that tangential stress on a jet can lead to splitting, but normal stress cannot. Tangential stress must overcome the surface tension of the fluid; thus, there is a critical stress, below which splitting cannot occur.
Doodling in my notepad, I made a simple model for the pinching and splitting of two surfaces. It is entirely mathematical and does not include any physical parameters. I compared my sketches with the authors results, and was surprised to see sharp cusps in their cross-sections. In my model, there is a linear crossing at one instant in time, but before and after, the surfaces are smooth. It seems to me that a smooth membrane would be vastly lower in energy than one with a cusp. Of course, when the membranes touch, it's got to lead to some kind of singularity in the differential equation, so a numerical routine or an analytic solution to the full equations might not "know what to do" after the membranes meet.
Thursday, April 5, 2007
Efficiency of Non-Ideal Engines
Collective Working Regimes for Coupled Heat Engines
B. Jimenez de Cisneros and A. Calvo Hernandez
PRL 98, 130602 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e130602
The authors consider the efficiency of an array of coupled heat engines between two reservoirs at temperature t and T, with T > t. Long ago, Carnot showed that the maximum efficiency is
e = 1 - (t/T).
This efficiency can only be realized in an adiabatic, reversible process.
In the late 50s and mid 70s, physicists extended Carnot's analysis to finite-time endoreversible processes and found
e = 1 - sqrt(t/T).
This is called the Curzon-Ahlborn efficiency. (For T = 4t, this implies a reduction in efficiency from 50% to 25% -- significant!) The authors claim this efficiency provides a good approximation to the observed efficiency of several power plants, which suggests they are closer to maximum theoretical efficiency than one might have thought. If you want your power in finite time, you might have to settle for significantly less efficiency.
The authors analyze an array of coupled heat engines between two reservoirs. First, they show that the efficiency only depends on the endpoints and not the intermediate mechanisms. They derive an efficiency that depends on the heat fluxes at the ends, not the temperatures:
e = 1 - j/J.
They also calculate the rate of entropy production, which leads to an analysis of thermodynamic forces and Onsager coefficients, which I am not familiar with. The authors show that the Carnot efficiency is realized when the rate of entropy production is zero.
The rest of the paper is devoted to solutions of a Ricatti differential equation the authors derive for the Onsager coefficients. They show that the Carnot and Curzan-Ahlborn efficiencies are specific cases of their more general theory.
A surprising result is that global optimization of the total power of the system does not require that every element perform at its individual maximum power. I also infer from this analysis that the key to improving efficiency is reducing entropy production, or isolating the system from the environment.
A final note: I am nearly certain that Eq. (20) or (21) is incorrect. I can solve the differential equation, and the solution to Eq. (20) is not Eq. (21). I'm not sure where the error is, but something is amiss.
B. Jimenez de Cisneros and A. Calvo Hernandez
PRL 98, 130602 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e130602
The authors consider the efficiency of an array of coupled heat engines between two reservoirs at temperature t and T, with T > t. Long ago, Carnot showed that the maximum efficiency is
e = 1 - (t/T).
This efficiency can only be realized in an adiabatic, reversible process.
In the late 50s and mid 70s, physicists extended Carnot's analysis to finite-time endoreversible processes and found
e = 1 - sqrt(t/T).
This is called the Curzon-Ahlborn efficiency. (For T = 4t, this implies a reduction in efficiency from 50% to 25% -- significant!) The authors claim this efficiency provides a good approximation to the observed efficiency of several power plants, which suggests they are closer to maximum theoretical efficiency than one might have thought. If you want your power in finite time, you might have to settle for significantly less efficiency.
The authors analyze an array of coupled heat engines between two reservoirs. First, they show that the efficiency only depends on the endpoints and not the intermediate mechanisms. They derive an efficiency that depends on the heat fluxes at the ends, not the temperatures:
e = 1 - j/J.
They also calculate the rate of entropy production, which leads to an analysis of thermodynamic forces and Onsager coefficients, which I am not familiar with. The authors show that the Carnot efficiency is realized when the rate of entropy production is zero.
The rest of the paper is devoted to solutions of a Ricatti differential equation the authors derive for the Onsager coefficients. They show that the Carnot and Curzan-Ahlborn efficiencies are specific cases of their more general theory.
A surprising result is that global optimization of the total power of the system does not require that every element perform at its individual maximum power. I also infer from this analysis that the key to improving efficiency is reducing entropy production, or isolating the system from the environment.
A final note: I am nearly certain that Eq. (20) or (21) is incorrect. I can solve the differential equation, and the solution to Eq. (20) is not Eq. (21). I'm not sure where the error is, but something is amiss.
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