SAR sent me a paper (Science, 320, p.356, 2008) which told a very intriguing story (if not conceptually new). Basically they can actually see the quantum chaos of Dirac billiard by the Coulomb blockade measurements. A relevant question would be: how to test the quantum chaos in our system?
As described here, we can make a strong analogue of our system, at the vicinity of wells, to swing spring plus a kinetic coupling. It would be interesting to study the swing spring system under magnetic field, and see how the levels quantize. As a started, there have been work done on the Diamagnetic hydrogen... Is it possible to employ similar approaches to actually identify the states in experiments?
Wednesday, April 30, 2008
A possible way to simplify (in steps) the potential surface
V = V(r,θ,φ), exact, useless
V = V(r) + V(θ,φ) + δV(r,θ,φ), separation + coupling, not very representative
V = V(θ,φ) + ½k[r-r0(θ,φ)]², closest to reality, note that k, if treated as a variable of (θ,φ), has a variance less than 6%.
A possible approach to the third potential: V = ½k[r-r0(θ,φ)]² + ε*V(θ,φ). The first part resembles a study of the Monodromy problem (of course it is much more complicated, because now we have 3 DOF, however locally we can approximate r0(θ,φ)≈r0(φ), for example, and from there we can further separate the phase space into tiles and continue the study.
V = V(r) + V(θ,φ) + δV(r,θ,φ), separation + coupling, not very representative
V = V(θ,φ) + ½k[r-r0(θ,φ)]², closest to reality, note that k, if treated as a variable of (θ,φ), has a variance less than 6%.
A possible approach to the third potential: V = ½k[r-r0(θ,φ)]² + ε*V(θ,φ). The first part resembles a study of the Monodromy problem (of course it is much more complicated, because now we have 3 DOF, however locally we can approximate r0(θ,φ)≈r0(φ), for example, and from there we can further separate the phase space into tiles and continue the study.
Friday, April 18, 2008
Notes on integrators
R-K 4th order: about 4.5 times of computation time of the same step V-V.
However, it has about the same conservation of energy (a gauge of accuracy) as V-V with 1/50 time step. Overall speaking, it's about 10 times more efficient. RK4 for E = 1000 cm-1, dt = 0.005 t0 for t=100 ps integration gives ~ 10-11 eV of total energy shift, for E = 500 cm-1, it is even better (~3×10-12 eV). Doubling dt (to 0.01 t0) leads to ~35 times higher Eshift.
We still ought to check the symplecticity of the integrator, i.e. whether it preserves the two-form:
ω2 = ∑ dpi ^ dqj. PS: as it turns out, the SIA4 integrator proposed by J. Candy in J. Comput. Phys, 92, 230 (1991) is a better (and symplectic) one than the plain RK4 method.
However, it has about the same conservation of energy (a gauge of accuracy) as V-V with 1/50 time step. Overall speaking, it's about 10 times more efficient. RK4 for E = 1000 cm-1, dt = 0.005 t0 for t=100 ps integration gives ~ 10-11 eV of total energy shift, for E = 500 cm-1, it is even better (~3×10-12 eV). Doubling dt (to 0.01 t0) leads to ~35 times higher Eshift.
We still ought to check the symplecticity of the integrator, i.e. whether it preserves the two-form:
ω2 = ∑ dpi ^ dqj. PS: as it turns out, the SIA4 integrator proposed by J. Candy in J. Comput. Phys, 92, 230 (1991) is a better (and symplectic) one than the plain RK4 method.
Monday, April 14, 2008
Quantum control, diffusion
It might be worthwhile to look into H=H0+μ·E. The driving term cause the diffusion perpendicular to energy surface, and H0 has diffusion by itself on the energy surface. If D∥»D⊥, it can be argued that the quasienergy (Floquet energy) eigenfunctions will be localized exponentially (see D.L. Shepelyansky, Physica D, 28, 103-114 (1987)). Also read Wigner distribution, Arnol'd diffusion.
Flashback: in the XCN isomerization studies (also between-tile hopping in the current study), everything might be broiled down to the comparison between diffusion rates along different directions (in the phase space). If D∥»D⊥, then the particle has a good chance to "isomerize" than to "dissociate".
For d>2 degrees of freedom systems, the invariant tori in phase space, being d-dimensional, can not slice the 2d-1 dimensional space of the surface of constant energy. This means for (d>2)-DOF systems the phase space is essentially "ergodic" (Arnol'd diffusion).
Flashback: in the XCN isomerization studies (also between-tile hopping in the current study), everything might be broiled down to the comparison between diffusion rates along different directions (in the phase space). If D∥»D⊥, then the particle has a good chance to "isomerize" than to "dissociate".
For d>2 degrees of freedom systems, the invariant tori in phase space, being d-dimensional, can not slice the 2d-1 dimensional space of the surface of constant energy. This means for (d>2)-DOF systems the phase space is essentially "ergodic" (Arnol'd diffusion).
Saturday, April 5, 2008
swing spring, 1:1:2 resonance, etc
It can be argued that although the radial mode is not exactly double the doubly degenerate peripheral mode frequency, a substantial coupling term (manifested by the strong overtone peak) could make an analog to the 1:1:2 resonance of swing spring valid. It would be interesting to actually observe the pulsation and precession effects described in Lynch's paper.
Regarding the polygon (Sinai) billiard analog - basically we treat the whole motion as motion in 'triangles' that tiled the icosahedron, with a reflection rule at the boundaries. However, since the triangle is actually on the S2 surface, special treatment is needed; for example, a rotation around the center of a pentagon should be mapped as a "bouncing" motion between adjacent two boundaries - which is quite ill-defined in the planar case. In fact, how much efforts are warranted to just carry out this simulation is unknown, as we can always do the whole simulation and mapped it back to the sub-region. The question is more interesting, only in analytical terms (since it reflects both the symmetry and/or interconnectivity).
Lastly, a measurement of 1/20 sub-region "revisit" frequency could be highly valuable. Say we have N trajectories originating from inside the boundaries, we want to calculate the number of trajectories that remain in the boundary as a function of time t. To check if this follows a polynomial or exponential decay law can be very enlightening on the stickiness of the phase space.
PS: assuming equal 'diffusion' toward the three adjacent tiles, in the long run, the decay should follow an exponential law of 0.95*exp(-0.255*r*t) + 0.05, in which r is the diffusion rate. (-0.254644 is the three leading exponents other than 0 of the eigenvalues of the diffusion matrix.) Update: preliminary simulations show a similar decay curve (the decay constant drops quickly) to an end value slightly above 0.05 and with a very fluctuating tail, even with 1000 trajectories. Despite being consistent with most of our speculations, the enormous fluctuation around the 0.05 tail is unaccounted for.
Regarding the polygon (Sinai) billiard analog - basically we treat the whole motion as motion in 'triangles' that tiled the icosahedron, with a reflection rule at the boundaries. However, since the triangle is actually on the S2 surface, special treatment is needed; for example, a rotation around the center of a pentagon should be mapped as a "bouncing" motion between adjacent two boundaries - which is quite ill-defined in the planar case. In fact, how much efforts are warranted to just carry out this simulation is unknown, as we can always do the whole simulation and mapped it back to the sub-region. The question is more interesting, only in analytical terms (since it reflects both the symmetry and/or interconnectivity).
Lastly, a measurement of 1/20 sub-region "revisit" frequency could be highly valuable. Say we have N trajectories originating from inside the boundaries, we want to calculate the number of trajectories that remain in the boundary as a function of time t. To check if this follows a polynomial or exponential decay law can be very enlightening on the stickiness of the phase space.
PS: assuming equal 'diffusion' toward the three adjacent tiles, in the long run, the decay should follow an exponential law of 0.95*exp(-0.255*r*t) + 0.05, in which r is the diffusion rate. (-0.254644 is the three leading exponents other than 0 of the eigenvalues of the diffusion matrix.) Update: preliminary simulations show a similar decay curve (the decay constant drops quickly) to an end value slightly above 0.05 and with a very fluctuating tail, even with 1000 trajectories. Despite being consistent with most of our speculations, the enormous fluctuation around the 0.05 tail is unaccounted for.
