For geometries with the two leads facing each other, the contribution to the diagonal transmission amplitude from direct trajectories is given by Baranger, r. Lin, a generalized these results to the case in which the leads are not collinear.
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Direct trajectories are relevant in certain geometries, and their presence hampers the straightforward comparison between the semiclassical theory with numerical calculations or experimental data. The case of the square is rather special among integrable systems since the conserved quantities of the cavity are the same as in the leads.
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The families of trajectories can be found by going to an extended space spanned by copies of the original cavity, and treat them like direct trajectories. A continuous-fraction approach allows to identify the families of trajectories and calculate the semiclassical transmission amplitudes for the scattering through a rectangular billiard Pichaureau, a.
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Quantum mechanical calculations for a square cavity allowed to identify the peaks of the Fourier transform of the transmission amplitude with the families or bundles of classical trajectories contributing in the semiclassical expansion Wirtz, a. The inclusion of diffractive paths leads to a good quantitative agreement with the quantum calculations Wirtz, a.
The circular billiard is particularly interesting because it has been experimentally studied Marcus, a ; Berry, b-a ; Chang, a ; Persson, a ; Lee, a , and it is an integrable geometry where the semiclassical transmission amplitude 39 is applicable since the contributing trajectories are isolated. Also, the proliferation of trajectories with the number of bounces is much weaker than for the chaotic case, allowing for the explicit summation of Eq.
Lin and Jensen Lin, a undertook such a calculation considering trajectories up to bounces. The direct semiclassical sum yielded a coherent backscattering that is significantly reduced by off-diagonal contributions to the total reflection. The signature of classical trajectories in the numerically obtained quantum transmission amplitudes has been established for circular billiards Ishio, a ; Schwieters, a ; Schreier, a. In particular, the Fourier transform of the transmission amplitudes shows strong peaks for lengths corresponding to the classical trajectories contributing in the semiclassical expansion This is why in geometries with stable trajectories, like the circle, the Fourier peaks are more pronounced than for the stadium billiard.
Cavities with hyperbolic and regular classical dynamics are the most commonly studied cases of ballistic transport. However, the behavior with a mixed phase-space, containing both chaotic and regular regions, is the most generic situation for a dynamical system.
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It is also experimentally relevant since the microstructures do not have perfect hard-wall confining potentials and are not disorder free. The effect of isolated resonances Hufnagel, has been invoked to explain these numerical results, adding subtle issues to the description of quantum transport through cavities with generic mixed classical dynamics Takagaki, a ; Louis, a. Brouwer, t ; Beenakker, r ; Alhassid, r ; Mello, r ; Mello, b. Bohigas, r ; Guhr, r ; Haake, b ; Fyodorov, i. Random-matrix theory RMT has been applied to study the statistical properties in a variety of physical problems, ranging from Nuclear Physics to spectral distribution in small quantum systems and conductance fluctuations in disordered mesoscopic conductors.
The basic assumption of these approaches is that the matrix describing the problem at hand is the most random one among those verifying the required symmetries and constraints of the system under study. The Hamiltonian matrix is the relevant one for analyzing spectral statistics of complex systems disordered or classically chaotic. The mean-level spacing appears as the sole constraint in this case, and different ensembles are obtained according to the symmetries that may exist in addition to the Hermitian character of the Hamiltonian.
The transfer matrix, incorporating the constraint of the elastic mean-free-path, is the appropriate tool to study the conductance fluctuations in quasi-one dimensional disordered systems. The basic assumption of equal probability of all possible scattering process translates into a uniform distribution over the matrix ensemble. This is a suitable representation to obtain the distribution of the eigenphases, but it is not appropriate for the study of transport through the quantum dot. For instance, according to Eqs. Expressed in the coordinates of the polar decomposition 20 , the invariant measure of the COE ensemble can be written as Jalabert, a.
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Working Eq. A reduction factor of 2 is obtained for the variance of the conductance when a magnetic field is applied inducing the transition form the COE to the CUE. Figure 14 shows the weak-localization and the conductance fluctuations obtained from quantum numerical calculations for an asymmetric structure the "stomach" where direct paths and whispering gallery trajectories have been blocked. The distribution 68 has been extended to other situations. For instance, the joint probability density of reflection eigenvalues has been determined for the case of chaotic cavities with non ideal leads with ballistic or tunnel contacts and having an arbitrary number of propagating modes in the leads Jarosz, a.
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This is a remarkable result. Random-matrix theory yields universal results, equivalent to those of the loop-corrected semiclassical approximation, but in a considerable simpler fashion. It is important to remark that the universal values for the weak-localization correction and the conductance fluctuations obtained within RMT, or the loop-corrected SCA, rely on the ergodicity of the underlying classical dynamics, and therefore apply to a very restricted set of structures.
Obviously, random-matrix theory is not of any help when dealing with cavities with integrable or mixed dynamics, and semiclassics remains the preferred tool in these cases. Therefore, the numerical simulations of Figure 14 and Figure 15 were performed in cavities where the geometry were chosen to minimize the effect of direct and short trajectories. However, the simplicity and the usefulness of the random-matrix approach is diminished in this case.
The connection 28 between the Hamiltonian and scattering matrices allows to establish the relationships between the statistical distribution of both matrices. The random-matrix hypothesis for the Hamiltonian of a chaotic dot or the zero-dimensional nonlinear sigma model coupled to leads yields equivalent results to those of Eq. The weak-localization correction and the conductance fluctuations are different from those of Sec.
The SCA for the conductance of nearly symmetric quantum dots yields results equivalent to those of the RMT and allows to study the transition between the different symmetry classes Whitney, a. Random-matrix theory can be helpful for the study of decoherence process. This additional lead can be incorporated in a random-matrix theory approach Baranger, a ; Brouwer, a ; Brouwer, a. Westervelt, r ; Bird, r ; Bird, b. Marcus, r ; Marcus, r ; Bird, r ; Huibers, t ; Hackens, t.
Two geometrical shapes stadium and circle were lithographically patterned each in two different samples in order to achieve a steep-walled electrostatic confinement. The leads were oriented at right angles of each other aiming to reduce transmission via direct trajectories see insets in Figure These low numbers of modes place the experiment somehow at the limit of applicability of the semiclassical theory. The magnetoconductance was reported to be reproducible under thermal cycling, demonstrating that the fluctuations were a fingerprint of these samples. The traces corresponding to the stadium and the circle Figure 16 presented some likeness, but differed in the detail of the fluctuations.
Such a difference could be quantified by following the analysis of the power-spectrum of Sec. The power-spectrum of the stadium cavities showed a good agreement with Eq. Deviations for large and small areas were observed see Figure 17 and the inset in Figure The conductance fluctuations in the circular billiard appeared to be more structured more weight in the high harmonics of the power spectrum when compared with the case of the stadium.
The measurable difference in the transport through the two structures then appeared in the larger weight of the high harmonics of the magneto conductance for the case of the circle. The systematic study of conductance fluctuations requires a considerable amount of averaging. A given magnetoconductance curve offers only a limited interval for averaging, since once the cyclotron radius becomes comparable to the size of the structure, the nature of the classical dynamics may change.
The fact that thermal cycling could effectively produce different samples, by re-accommodation of impurities, hinted the importance of the smooth disorder in these microstructures that were ballistic, but not clean. The small amount of small-angle scattering affected the very long trajectories, without altering the statistical signature of chaotic trajectories Berry, a.
The power-spectrum of the conductance fluctuations in a circular cavity with a central bar "pacman billiard" was found to be well fitted by the theoretical prediction 56 , valid for a classically chaotic structure Berry, p-a. In the unpatterned circular cavity, the previous fit was considerably poorer, and the characteristic field was found to be smaller by a factor of 3 than in the case of the packman billiard.
Such difference follows from the fact that a barrier inside the cavity drives the spectrum of effective areas towards smaller values Schreier, a. The exponential decay of the power spectrum found in circular cavities was signed by steps at characteristic frequencies corresponding to integral fractions of flux quanta through the dot Persson, a.
Keller et al. However, the conductance fluctuations of the polygonal geometry did not show qualitative differences with those of the chaotic case. This departure from the theoretical prediction for clean integrable cavities was attributed to the effect of residual disorder. In the completely coherent picture of Sec. Keller, t ; Bird, r ; Chang, r. Separating these peaks from the background of conductance fluctuations requires some kind of average.
This observation is consistent with the lack of self-averaging of ballistic cavities. The weak-localization of the nominally chaotic billiards showed a good quantitative agreement with the numerical results of Sec. However, the polygon billiard failed to exhibit the weak-localization features expected for the integrable classical dynamics. The line-shape of the peak was found to be Lorentzian, in agreement with the semiclassical prediction Moreover, the field scales of the weak-localization and conductance fluctuations were found to be related by the factor of 2 that discussed in Sec.
Three different shapes were considered: stadium, circle, and rectangle. In each case, 48 cavities, nominally identical but actually slightly different due to uncontrollable shape distortions and residual disorder, were connected as 6 rows in a series of 8 in parallel. The resulting weak-localization peak was found to be Lorentzian for the stadium cavities and triangular for the circular ones see Figure 20 , in agreement with the semiclassical prediction and detailed numerical calculations Baranger, r. The lithographic shape of the cavity did not correspond to a chaotic geometry.
But the possible integrability of the dynamics was expected to be broken by the shape distortions employed for averaging. Moreover, in these relatively large structures, smooth disorder affected the long trajectories. Conductance was studied as a function of magnetic field and electrostatic shape distortion, allowing to gather very good statistics. The fluctuations as a function of magnetic field showed very good agreement with Eq. The shape-distortion fluctuations yielded an exponential power spectrum, in agreement with the calculations of Bruus and Stone Bruus, a showing that the semiclassical formalism of Sec.
A Lorentzian shape for the weak-localization peak was obtained, with a width related to the characteristic field of the conductance fluctuations, as predicted by semiclassical theory. The rich statistics that this type of structures allowed to gather was used to extract the moments of the conductance, as well as the whole conductance distribution in order to compare with the random-matrix theory predictions of Sec. Bird and collaborators Bird, a used the thermal smearing of the conductance fluctuations to measure the weak-localization correction in rectangular cavities.
The peak line-shape changed its profile from Lorentzian to triangular as the quantum point contacts at the entrance of the cavity were closed. Resistance measurements and numerical analysis Zozoulenko, a ; Zozoulenko, a on square cavities suggested that, depending on the geometry of the contacts, transport through the cavity could be mediated by just a few resonant levels or specific families of trajectories, illustrating the importance of the geometry and the injection conditions in the integrable case Ouchterlony, a. Different weak-localization profiles have been obtained in quantum numerical calculations within a fixed geometry by varying the election energy and the softness of the confining potential Akis, a.
Lee, Faini and Mailly used shape and energy averages to extract the weak-localization peak of chaotic stadium and stomach and integrable circular and rectangular cavities Lee, a. The former exhibited a Lorenztian line-shape, consistently with the theoretical prediction However, among the integrable cavities, only the rectangle showed the expected triangular shape, while the circle yielded a Lorentzian. Chang has proposed Chang, r that the discrepancy between the results of Refs.
Chang, a and Lee, a was due to the shorter physical cut-offs that were present in the latter experiment, hindering the long trajectories to exhibit the signatures of the integrable dynamics. These experimental results, that seemed to contradict the theoretical predictions for classically chaotic and integrable systems, were explained by invoking a mixed-dynamics in the case of the filled cavity and by the imperfections boundary roughness and small-angle scattering for the nominally regular system.
Experiments and numerical calculations have primarily focused on the differentiation between Lorentzian and linear line-shapes of the weak-localization peak according to the underlying classical dynamics. In this context it is important to recall that a complete semiclassical theory of weak-localization only exits for the case of a chaotic dynamics. Micolich, t ; Micolich, r ; Pilgrim, t.
A stadium and a Sinai billiard, which are paradigms of chaotic dynamics, become mixed systems when fabricated by lithographic methods that result in a soft-wall confinement.
These two geometries were patterned with purposely soft confining potentials and very wide leads 0. The resulting conductance fluctuations were claimed to have a fractal nature over two orders of magnitude in magnetic field Micolich, a ; Sachrajda, a. In square cavities, the soft confinement was found not to be effective in the smearing of the conductance fluctuations in comparison with the hard-wall scenario Ouchterlony, a. Marlow and collaborators Marlow, a performed a comprehensive comparison of magnetoconductance fluctuations in 30 devices spanning the ballistic, quasi-ballistic, and diffusive regimes, concluding that all of them exhibit identical fractal behavior.
The similar behavior, in terms of conductance fluctuations, of billiards with both kinds of confinement contradicts the claim Sachrajda, that the fractal conductance fluctuations are associated with the mixed dynamics resulting from a soft confinement. This comparative study concluded that the origin of the fractal conductance fluctuations found in all devices is the small-angle scattering that deflects electron trajectories away from straight paths Marlow, a. Such a picture was associated with the interpretation of scanning gate microscopy SGM studies in 2DEG Topinka, a and quantum dots Crook, a in terms of the drifting of electron trajectories due to small-angle scattering.
Concerning the link claimed with SGM studies, it is important to remark that the interpretation of SGM measurements as traces of the classical electron paths has been questioned for nanostructures surrounded by a 2DEG Jalabert, a , Gorini, a and QDs Kozikov, b-a. That is, considerably larger than the size of the quantum dots typically considered in quantum chaos studies. In order to further study the effects of disorder on conductance fluctuations, a comparison was established between two nominally identical geometries, where one of them was patterned on a standard modulation-doped heterojunction, and the other on an undoped gated heterostructure See, a.
Consistently with previous findings Berry, a , the magnetoconductance fingerprints of the modulation-doped sample changed under thermal cycling. While in the undoped case the magnetoconductance was reproducible. This different behavior demonstrated that the role of disorder was negligible in the undoped case.
However, the statistical analysis of the magnetoconductance presented similar features in both cases, with quantitative differences that could be attributed to the different electron density achieved with and without doping. It was then concluded the the presence of small-angle scattering in a modulation-doped structure is not capable of amplifying the fractal behavior of the magneto conductance fluctuations induced by a soft-confinement. The difficulties arising from the limited set of data, as well as the relevance of the characteristic lengths of the problem, were recognized in the early studies of quantum chaos in ballistic transport.
Thus, the analysis in terms of the power-spectrum of the conductance was proposed Jalabert, a , and later used to interpret the statistical properties of the measured magnetoconductance Marcus, a. As emphasized in Sec. In a closed cavity, the soft-wall confinement and the small-angle scattering generically drive nominally chaotic or integrable geometries into a mixed classical dynamics.
The effect of a weak smooth disorder in quantum dots has been analyzed in the context of orbital magnetism Richter, a ; Richter, p-a , and the small extra phase that an electron trajectory picks up was found not to be important in high-mobility samples. The global stability of a chaotic system is not altered by the small perturbation of a smooth disorder. Integrable geometries are more sensitive to weak disorder, but when the physical quantity under study is signed by contributions coming from short trajectories, the effect of disorder can be very small.
In an open system it is difficult to evaluate how much softness in the confinement and small-angle scattering is needed to drive the classical dynamics into a mixed one. This is due to the fact that the very long trajectories, which are the ones most affected by the perturbations, might contribute very little to the transport properties. Few studies exist to quantify the transition to mixed dynamics in an open system due to smooth confinement and weak small-anlge scattering. Among them, quantum calculations performed in order to describe the weak-localization experiments, showed that weak disorder does not necessarily mask the different behavior predicted between clean chaotic and integrable cavities Chang, a ; Baranger, r.
The non-perfect transmission of the quantum point contacts resulted in an additional source of noise that needed to be corrected for the comparison against the RMT prediction. Agam, a. In the opposite case, when transport is dominated by short trajectories, the deterministic character of the classical dynamics hiders the observation of shot noise.
In small cavities with large openings the suppression of the shot noise has been related with the existence of broad resonances that support direct process well described by deterministic classical dynamics Nazmitdinov, a. Kastner, r ; Kouwenhoven, r ; Kouwenhoven, r ; Aleiner, r.
In quantum dots that are sufficiently small and weakly connected to the leads, at low temperatures, the Coulomb repulsion of electrons cannot be ignored. The conditions for entering into the interaction-dominated regime are. Such a situation generically appears when tunnel barriers are imposed at the entrance and exit of the dot as sketched in Figure 24 , and thus the number of electrons within the dot is almost a good quantum number. When the condition 73 holds, the incoming electrons from the reservoirs do not have enough energy to overcome the gap in the tunneling density of states and cannot participate in transport Figure This suppression of the conductance is known as Coulomb blockade CB.
These are the so-called Coulomb blockade oscillations , presented in Figure 26 Meriav, a. The quasi-period depends on the geometrical dimensions of the dot which determine its capacitance. Such a simplification is appropriate to describe the Coulomb blockade features of dots with, roughly, more than hundred electrons. This so-called constant-interaction model CIM has the advantage of reducing a genuine many-body problem into an effective single-particle one.
Such a connection allows to apply the ideas of Quantum Chaos developed in the one-particle case. Leaving aside the case of very small dots, where Kondo physics emerges, and within the condition 73 for observing the discreteness of the charge, different regimes can be achieved:. The large fluctuations in the amplitude of adjacent peaks experimentally observed at low magnetic field see Figure 26 were proposed to arise from the chaotic nature of the eigenstates of irregular quantum dots Jalabert, a.
The statistical approach can be established from Eq. Mirlin, r ; Urbina, r. The maximum-entropy principle, expected to be applicable in classically chaotic systems, results in a Gaussian probability distribution of the wave-functions. According to the Voros-Berry conjecture, the Wigner function for a classically chaotic system is ergodically distributed on the energy manifold of phase-space Voros, a ; Berry, a , and the two-point correlation function is given by.
An important shortcoming of the Gaussian distribution 76 is that the wave-function normalization is only satisfied on average, and not by the individual realizations. In addition, a crucial limitation of the two-point correlation function 77 is the fact that it ignores boundary effects. Thus, according to 76 , their distribution should also be Gaussian. In presence of a magnetic field large enough to break the time-reversal symmetry in the dot, the peak-height distribution is Jalabert, a ; Prigodin, a.
Moreover, the departure from the random-matrix distributions obtained for integrable and nearly integrable geometries demonstrated that the distribution of level widths provides a tool for the Quantum Chaos task of differentiating quantum properties according to the underlying classical dynamics. Fire a machine gun at a wall with two slits. Look at the pattern bullets make that pass through. The pattern is the sum of two Gaussian curves. Classical physics makes sense. Repeat the exercise with electrons or photons.
Where there is constructive interference, the signal is strong. Where there is destructive interference, the signal can drop to zero. Waves behave that way all the time. Yet quantum interference works even if particles are fired at extremely low rates, such as once a day. The logic of quantum mechanics in experiments with two slits is not just strange: the explanation using quantum interference is logically inconsistent. Constructive interference has "this plus that" making the big signal.
Destructive interference has "this minus that" for the places with no signal. Yet one can use a super low intensity source, so low that "this never sees that". One cannot do the addition or subtraction. I accept the experimental results which have been confirmed time and time again. An explanation is a story we tell each other. The quantum interference story doesn't make sense. The number of slits is not relevant to this logical quandary. If there is only one slit, one sees quantum diffraction.
Use the same source with two slits and one sees quantum interference. For n slits, the result is a quantum diffraction grating. Here is what it looks like going from one to seven slits:. The locations of the light and dark bands is a function of the number of slits. What is not a function of the number of slits is that there are light and dark bands.
I deliberately neglected to use a vital word in my description: the source must be coherent. If one uses an incoherent source, no light and dark bands are seen, no matter how many slits are used.