Quantum Chaos in Deformed Microcavities

- Abstract One of the behaviours to reach studying the quantum states is to consider them at the boundary of clasical mechanics whereas the applicability of clasical mechanics theory.
- For its nonclassical counterpart, the quantummechanics of chaotic systems, termed in short “quantum chaos”, the situation is completely different.
- The situation gradually changed in the middle of the eighties, since when numerous experiments have been performed.
- prevents a precise determination of the initial conditions.
- Quantum chaos is the study of no separable Schrodinger equations based on a knowledge of the underlying classical mechanics, which can be chaotic when the system is non-integrable..
- be the vector of the dynamical variables at the time t = 0.
- At any later time t we may write x(t) as a function of the initial conditions and the time as.
- The eigenvalues of the matrix.
- determine the stability properties of the trajectory.
- Stationary solutions of the Schrodinger equation are obtained by separating the time dependence,.
- (1.10) where c is the wave velocity, and if we separate again the time dependence by means of the ansatz (1.7).
- Integrable systems We have learnt that random matrix theory is perfectly able to explain the universal properties of the spectra of chaotic systems.
- Fortunately, random matrix theory is only one side of the coin.
- As we know from the correspondence principle, in the semiclassical limit of high quantum numbers.
- In there have two part : Trace formula, establishing a correspondence between the quantum mechanical spectrum and the periodic orbits of a system and a number of applications of the trace formula will be examined..
- Assume that in addition to the Hamiltonian H, let found another function K(x¯) of the phase space variables that is also conserved, i.e.
- The generalization of the above arguments to more than two degrees of freedom is that the trajectory will move on an N-dimensional torus in the 2N-dimensional phase space.
- However, the possibility of (classical) "beam splitting" at sharp corners implies that singularities of the above vector field can occur.
- The complexity of the problem is greatly in​creased in non-integrable systems, where it becomes impossible to reduce the wave equation to a collection of separate first-order differential equations.
- The ray dynamics analysis is facilitated by the axial symmetry of the droplets which implies (in the language of particle trajectories) that the z component of angular momentum, Lz is conserved.
- Thus a 3D specula reflection simply reverses the normal component of the 2D projected velocity.
- and reflections are also specula in the pro​jected coordinates.
- its apparent angle of incidence in the ρ — z-plane will be zero.
- The angle in the p — z-plane is then given by.
- At nonzero Lz certain regions of the SOS are forbidden due to the Lz angular momentum barrier (e.g.
- Before discussing ray escape in the deformed droplets it is important to note that as we proceed from higher to lower Lz in addition to the excluded regions of the SOS decreasing (because the angular momentum barrier becomes weaker) the degree of chaos grows rapidly.
- The reason for this is that high Lz trajectories are confined near the equator and a cross-section of the droplet at the equator is perfectly circular, i.e.
- high Lz orbits see an effective deformation which is much weaker than polar orbits (Lz = 0) which travel in the most deformed cross- section of the droplet.
- As just noted these high LZ modes are confined to orbits near the plane of the equator .
- Proceeding now to lower Lz, we see that the angular momentum barrier has weakened enough that the allowed region of the SOS passes through sinχc and rays with this value of Lz can escape.
- This situation persists all the way to Lz = 0 for deformations less than roughly 5% of the radius, so little Q-spoiling and approximately isotropic emission for smaller deformations than this..
- However for the 50% deformation used reducing Lz a little more causes the appearance of regions of chaos which extend from high sinχ across sinχc allowing classical Q-spoiling of the WG modes.
- But these low L~ modes are the only ones which can emit from the polar regions because of the angular momentum barrier for the high Lz modes.
- The prelates shape corresponds to a stretching of the droplet in the vertical direction and a compression in the equatorial plane.
- Because it is compressed in the equatorial plane there exists a large stable island at θ = π/2 corresponding to the two-bounce diametric orbit of the type we discussed in the 2D case.
- Ray trajectories The problems with the proper definition of the term “quantum chaos“ have their origin in the concept of the trajectory, which completely loses its significance in quantum mechanics.
- Typically the xN comprise all components of the positions and the moment of the particles.
- This straightforward generalization of arguments from the circle allows us to define the decay time as an average over an ensemble of trajectories on the adiabatic curve pm,p, of the time t needed by each orbit to escape.
- L of the ray up to this event is related to the escape time by L = ct/n, and the decay time is ( 1.17).
- This window is smeared out when the above wave effects are included.[21] The first of Poincare's integral invariants ( 1.18.
- As an implication of this argument, it is precisely the deviation of the trajectory from the adiabatic curve due to phase-space diffusion that determines the resonance lifetimes at high deformations.
- This does not constitute a contradiction to the validity of the semiclassical quantization provided the escape times due to classical diffusion are still long enough to permit the adiabatic curve to yield an accurate semiclassical quantization.
- In the limit of a clean resonator, τ is just the resonance lifetime..
- The latter is proportional to the number Ni of inverted atoms (or molecules) that interact with the mode, and to the intensity of the existing field.
- which is independent of the pump power.
- This is possible if the spatial overlap of the original mode and the new mode is incomplete, so that one has nodes where the other has antinodes.
- Its loss is larger than that of the first mode τ’< τ , corresponding to P’t > Pt.
- The same can be said in the presence of more than two lasing modes..
- In the absence of a suitable liquid for this purpose, a more immediate goal of an initial experiment is to test the universality of the emission directionality..
- In the absence of dynam​ical eclipsing, all that counts is that the tangent adiabatic curve be reached eventually, and the directionality is then prescribed.
- Far-field directionality for 5 different resonances of the quadruple at eccentric​ity e = 0.65 and refractive index n = 1.54, displaying the peak splitting due to dynamical eclipsing.[2] This was done by creating a cylin​drical stream of ethanol containing a lasing dye, which had an oval cross section due to the rectangular orifice at which it was produced.
- As in the ellipse, we still have escape predominantly from the minima of the invariant curve on which the ray moves.
- In particular, the directionality in the tunneling regime is correctly predicted by the pseudo classical model..
- Before that point, the emission looks similar to that of the billiard with n = 2.
- As shown in Fig.1.6 the four-peak structure has fully developed at e = 0.45, again well before chaotic diffusion becomes possible..
- Far-field directionality in the quadruple with increasing eccentricity e at n Emission directionality of quasi-bound states.
- then the corre​sponding solution of the time dependent wave equation decays at a rate γ since it has the form (1.24).
- in the past - but at that earlier time the field at the cavity was larger by a factor e-γΔt.
- in the far-field (r.
- Pulling out this common dependence, the field of the quasibound state factorizes into radial and angular functions, (1.26) This means that the directionality at large distances becomes independent of r, being con​tained solely in Ψ(Φ).
- Chosen r in this far-field region and plot the square of the electric field (which is proportional to the intensity) as a function of Φ to obtain the wave directionality..
- The directionality pattern in a scattering experiment will depend on the form of the incident wave both because of interference with the outgoing wave, and because.
- the incident wave may couple preferentially to different senses of circulation of the rays.
- If the interface can be made clean and smooth, the only leakage out of such a cavity stems from the fact that the surface has a finite curvature so that total internal reflection is violated, allowing a small fraction of the internal intensity to escape.
- the small violation of total internal reflection present even in the circle)..
- Directionality: Emission from a quasibound state is highly anisotropic at strong defor​mations, with intensity peaks in directions that are determined to high accuracy by the phase space structure of the classical ray dynamics inside the cavity.
- In the circle, we know that whispering-gallery WG resonances are narrow due to the low tunneling escape rate.
- The Husimi distribution is a Gaussian smoothed version of the Wigner function, representing the corresponding quantum mechanical probability distribution in phase space [4].
- The evident importance of the system’s underlying classical phase space for the behavior of the quantum or wave mechanical analogue (based on the analogy between Schr¨odinger and Helmholtz equation [1.
- To establish some experience with the phenomenology of the chaotic transition in billiards, it is instructive to discuss .
- Only convex deformations of the circle are of interest to us, because that is the requirement for the existence of whispering gallery orbits..
- To compare different shapes among each other, a measure of the deformation is required.
- For the display of the wave functions the stadium has been completed by a twofold reflection.
- This interaction comes from an absence of conserved quantities other than energy in the correspond​ing classical systems, and gives rise to avoided crossings when a parameter of the Hamiltonian is varied.
- These are the 289th and 290th states in the antisymmetric subspace of stadium billiard having the area π+4 [4] The relation between diabetic transformation and periodic orbits can be seen through Fourier transforma​tion of the level density (1.27) where k 2j is a value of the jth energy level..
- The study of classical periodic orbits can be a good starting point in fully chaotic systems because of the following three reasons..
- is the Maslov index, and a is deter​mined from the stability of the orbit.
- According to this formula, the Fourier transformation of the level density is expected to have peaks at each length of the periodic or​bit with a height corresponding to the stability of it [8]..
- The third reason is that, in fully chaotic systems, periodic orbits densely exist in phase space because of the ergodicity.
- Although they are isolated, they can play a role in coupling quantum states because of the finite h..
- Directional and Chaotic ray dynamic in some micro cavity.[16] Actually, we can control of the degree of deformation and thus we can follow a mode evolution continuously.
- All other deformations will have as their dominant multimode component this term, and we can therefore use the strength of the.
- quadruple part as a measure of the deformation that allows a comparison between different shapes.
- Adequacy of each approach depends on several factors, but most importantly on the size of micro cavity with respect to the wavelength of interest or the size parameter nkr, where n is the refractive index of the cavity medium, k - the wave vector with λ the wavelength and r the representative radius of the cavity..
- The Liquid jet micro cavity is the most suitable to study fully chaotic systems because of the following three reasons.
- The shape of the jet column can be approximated in the cylindrical coordinates by the following time independent equation.
- The bottom of the orifices are melted, narrowed and inclined, resembling the letter “V.
- in Fig.2.2 (a) of the amplitude oscillation, where the cavity boundary is given by Eq.2.2 The degree of possible three-dimensional (3D) effect due to a finite thickness of the region to be used in experiment around the cross- sectional plane, which is about 10µm, can be estimated in the following way.
- pressure of the jet at D2–D5 positions.[16].
- The pinching of the nozzle in one direction in the fabrication process has introduced different inner wall slopes, and these different wall slopes in turn induce different initial radial velocities, which makes it possible to tune the cavity deformation.
- It is noted that the oscillation of the quadruple deformation along the z direction is analogous to a damped harmonic oscillator with a nonzero launching velocity.
- In addition, the contribution due to the damping in the initial radial velocity is much smaller than that of the harmonic oscillation..
- Obtain the evolution of lasing directionality One of the behaviors to reach studying the quantum states is to consider them at the boundary of classical mechanics whereas the applicability of classical mechanics theory.
- Adequacy of each approach depends on several factors, but most importantly on the size of micro cavity with respect to the wavelength of interest or the size parameter nka, where n is the refractive index of the cavity medium, k= 2π/λ- the wave vector with the wavelength and a the representative radius of the cavity.
- A polarizer placed in front of the slit selects only the polarization component parallel to the QDM column.
- The emission spectrum is then measured for a fixed angle θ of the rotation stage (Figure 2.7).
- By background luminescence, mostly due to bulk fluorescence of the cavity medium, can be discriminated.
- The emission directionalities seen in the far-field distribution show the transition of mode dynamics from regularity (0%) to chaos (22%)..
- [6] M.V.Berry, in Chaotic Behavior of Deterministic Systems, Proceedings of the Les Huches Summer School, Session XXXVI edited by G..Iooss, R.H..G..Hellman [7] M.Wilkinson, J.
- Yang, T.T.A.Dao et al ” Observation of the far field evolution in a shape-variable microcavity”, TP-I1, OSK, Sokcho, July