This paper revisited quantum tunneling dynamics through a square double-barrier potential. We emphasized the similarity of tunneling dynamics through double-barrier and that of optical Fabry--P$\acute{e}$rot (FP) interferometer. Based on this similarity, we showed that the well-known resonant tunneling can also be interpreted as a result of matter multi-wave interference, analogous to that of FP interferometer. From this analogy, we also got an analytical finesse formula of double-barrier. Compared with that obtained numerically for a specific barrier configuration, we found that this formula works well for resonances at "deep tunneling region". Besides that, we also calculated standing wave spectrum inside the well of double barriers and phase time of double-barrier tunneling. The wave number spectrums of standing wave and phase time show another points of view on resonance. From semi-numerical calculations, we interpreted the peak of phase time at resonance as resonance life time, which coincides at least in order of magnitude with that obtained from uncertainty principle. Not to our surprise, phase time of double-barrier tunneling also saturates at long barrier length limit $l\rightarrow\infty$ as that of tunneling through a single barrier, and the limits are the same.
This paper revisited quantum tunneling dynamics through a square double-barrier potential. We emphasized the similarity of tunneling dynamics through double-barrier and that of optical Fabry--Peˊrot (FP) interferometer. Based on this similarity, we showed that the well-known resonant tunneling can also be interpreted as a result of matter multi-wave interference, analogous to that of FP interferometer. From this analogy, we also got an analytical finesse formula of double-barrier. Compared with that obtained numerically for a specific barrier configuration, we found that this formula works well for resonances at "deep tunneling region". Besides that, we also calculated standing wave spectrum inside the well of double barriers and phase time of double-barrier tunneling. The wave number spectrums of standing wave and phase time show another points of view on resonance. From semi-numerical calculations, we interpreted the peak of phase time at resonance as resonance life time, which coincides at least in order of magnitude with that obtained from uncertainty principle. Not to our surprise, phase time of double-barrier tunneling also saturates at long barrier length limit l→∞ as that of tunneling through a single barrier, and the limits are the same.
In close analogy with optical Fabry–Pérot (FP) interferometer, we rederive the transmission and reflection coefficients of tunneling through a rectangular double barrier (RDB). Based on the same analogy, we also get an analytic finesse formula for its filtering capability of matter waves, and with this formula, we reproduce the RDB transmission rate in exactly the same form as that of FP interferometer. Compared with the numerical results obtained from the original finesse definition, we find the formula works well. Next, we turn to the elusive time issue in tunneling, and show that the "generalized Hartman effect" can be regarded as an artifact of the opaque limit βl → ∞. In the thin barrier approximation, phase (or dwell) time does depend on the free inter-barrier distance d asymptotically. Further, the analysis of transmission rate in the neighborhood of resonance shows that, phase (or dwell) time could be a good estimate of the resonance lifetime. The numerical results from the uncertainty principle support this statement. This fact can be viewed as a support to the idea that, phase (or dwell) time is a measure of lifetime of energy stored beneath the barrier. To confirm this result, we shrink RDB to a double Dirac δ-barrier. The landscape of the phase (or dwell) time in k and d axes fits excellently well with the lifetime estimates near the resonance. As a supplementary check, we also apply phase (or dwell) time formula to the rectangular well, where no obstacle exists to the propagation of particle. However, due to the self-interference induced by the common cavity-like structure, phase (or dwell) time calculation leads to a counterintuitive "slowing down" effect, which can be explained appropriately by the lifetime assumptions.
The comprehension of the tunnel effect, the passage of matter waves and particles across a high potential barrier, was the most astounding of all discoveries in quantum mechanics. Research on tunnelling in semiconductors and superconductors, as well as the development of scanning tunnelling microscopy, garnered five Nobel prizes in physics. In this study, the resonant tunnelling effect of electron tunnelling through a one-dimensional double-potential barrier was found and verified by solving the time-independent Schrödinger equation and calculating the transmittance rate of the electron tunnelling through the two potential barriers by using the transmission matrix technique performed by MATLAB. According to the analysis, the transmittance rate was one under certain specific conditions (under some specific incident energy values or some specific distance between two potential barriers), which was called as resonant tunnelling. However, in this study, square potential wells were analyzed, which can be considered merely as ideal models in real life, and therefore all results are idealized. For further investigations, researchers can use more realistic parameters to set up potential wells and analyze real-life problems.
Tunneling phenomenon is one of the wondrous events that take place particularly in quantum mechanics and it has no counterpart in classical physics. We focus on its two important implications : tunneling time and tunneling probability. There has been no consensus on tunneling times among the tunneling time community for last seven decades. So far, most of the attempts to calculate tunneling times have apparently been centered on a simple rectangular barrier example. We implement the tunneling times with the most widely accepted approach, which is using physical Larmor clock, into a nonsymmetrical barrier and a double barrier. It results in different and unique insights into the usual understanding of tunneling time. Three characteristics tunneling time expressions are thoroughly studied in these quantum structures. The difference between dwell time and phase-delay time is once again approved. The unique characteristics of the phase-delay time, which was thought to be exactly similar to dwell time when we assign higher energies to the particles, is also revealed in the nonsymmetrical barrier that it should actually be regarded as a distinct feature from the dwell time. The singularity of reflection time in rectangular barrier is no longer valid in nonsymmetrical barrier. The finiteness of reflection time fairly shows that different quantum structures might bear some taboo breaking results. Since a double barrier involves much more complicated features and calculations, the calculation of tunneling times under this barrier is praiseworthy. The dwell time is the most crucial, troublesome and distinctive tunneling time element to look into for a double barrier. Besides the dwell and phase-delay time, we also calculate the three characteristic times as well as the reflection time for a double barrier. Another area that we study within the context of tunneling phenomenon is the tunneling probability. Recently, there has been ground shaking attempts in modifying the conventional WKB formulae. We follow one of the modified versions of WKB formulae in order to be able to calculate the tunneling probability through different quantum structures where conventional WKB formulae fail. These generalized WKB connection formulae are applied to some complex quantum potentials by seeking aid from the Green's function and the path integral approach. One of the biggest problems in quantum mechanics (visualization) is partially overcome through implementing the method of diagrammatic representations that make the life easier in understanding the events happening in the quantum world when we attempt to calculate the transmission and reflection probabilities. This part therefore has a pedagogical sense too in terms of interpreting the results in a diagrammatic way with the help of the path integral approach.
This thesis presents a theoretical investigation into the transmission and tunneling dynamics of a Bose-Einstein condensate (BEC) through single and double barriers. We show how quantum tunneling differs from transmission and provide a method for separating them out. We then focus on finding an experimentally-realisable parameter regime for making an atomic Fabry-Perot interferometer (FPI) using two repulsive barriers and study how interatomic interactions affect the dynamics of resonant transmission. Finally, we show that an atomic FPI can be used as an acceleration sensor and we compare its sensitivity to that of a Mach-Zehnder atom interferometer. We first investigate the transmission of an interacting BEC incident on a repulsive Gaussian barrier through numerical simulation. We study the dynamics associated with interatomic interactions across a broad parameter range not previously explored. Comparing effective one-dimensional Gross-Pitaevskii equation (GPE) simulations to classical Boltzmann-Vlasov equation (BVE) simulations, we isolate purely coherent matterwave effects. We then define quantum tunneling as the portion of the GPE transmission not described by the classical BVE. Transmission shows an exponential dependence on barrier height in the classical simulation, suggesting that observing such an exponential dependence is not a sufficient condition for quantum tunneling. Furthermore, we find that the transmission is predominately described by classical effects, although interatomic interactions modify the magnitude of the quantum tunneling. We show that the interactions also affect the amount of classical transmission, producing transmission in regions where the non-interacting equivalent has none. This theoretical investigation clarifies the contribution quantum tunneling makes to overall transmission in many-particle interacting systems, potentially informing future tunneling experiments with ultracold atoms. In the second part, we numerically demonstrate atomic Fabry-Perot resonances for a pulsed interacting BEC source transmitting through double Gaussian barriers. These resonances are observable for an experimentally-feasible parameter choice. By simulating an effective one-dimensional GPE, we investigate the effect of atom number, scattering length, and BEC momentum width on the resonant transmission peaks. For Rb-85 atomic sources with the current experimentally-achievable momentum width, we show that reasonably high contrast Fabry-Perot resonant transmission peaks can be observed using a) non-interacting BECs of 100000 atoms, b) interacting BECs of 100000 atoms with s-wave scattering lengths a_s= +-0.1a_0, and c) interacting BECs of 1000 atoms with a_s=+-1.0a_0. Our theoretical investigation impacts any future experimental realisation of an atomic Fabry-Perot interferometer with an ultracold atomic source. Finally, we explore the possibility of using an atomic FPI as an acceleration sensor. Using Fisher information, we quantify the acceleration sensitivity of the system. We study the dependence of sensitivity on various parameters of the system and determine the optimum parameter regime that gives the best sensitivity. For a BEC with infinitely narrow momentum width, the sensitivity can be improved without limit by increasing cavity length. However, when the atomic cloud has a finite momentum width, there exists an optimum cavity length corresponding to each momentum width. By comparing the sensitivity of the atomic FPI to that of a Mach-Zender interferometer with similar parameters, we show that the sensitivity of the atomic FPI overcomes that of the Mach-Zehnder interferometer in low device length and low momentum width regimes. In short, we demonstrate that an atomic FPI can be successfully used for sensing acceleration and the sensitivity of the device can be increased by reducing its length and by using a BEC with lower momentum width compared to the cavity linewidth.
The tunneling time of particle through given barrier is commonly defined in terms of "internal clocks" which effectively measure the interaction time with internal degrees of freedom of the barrier. It is known that this definition of the time scale for tunneling is not unique in the sense that it depends on the clock used to define it. For the case of resonance tunneling, a particular choice that in the limit of a high/broad square barrier yields the original result of Buttiker and Landauer (Phys. Rev. Lett. 1982, 49, 1739) is correlated to the lifetime of the resonance state. This is illustrated for analytically solvable one-dimensional double barrier models and for a realistic model of electron tunneling through a static water barrier. The latter calculation constitutes a novel application of this concept to a 3-dimensional model, and the observed structure in the energy dependence of the computed traversal time reflects the existence of transient tunneling resonances associated with instantaneous water structures. These models, characterized by the existence of shape resonances in the barrier, make it possible to examine different internal clocks that were proposed for measuring tunneling times in situations where a "clock independent" intrinsic time scale (the resonance life time) for the tunneling time exists. It is argued that this time may be used in order to estimate the relative importance of dynamical barrier processes that affect the tunneling probability.
An explicit expression is obtained for the phase-time corresponding to tunneling of a (non-relativistic) particle through two rectangular barriers, both in the case of resonant and in the case of non-resonant tunneling. It is shown that the behavior of the transmission coefficient and of the tunneling phase-time near a resonance is given by expressions with "Breit-Wigner type" denominators. By contrast, it is shown that, when the tunneling probability is low (but not negligible), the non-resonant tunneling time depends on the barrier width and on the distance between the barriers only in a very weak (exponentially decreasing) way: This can imply in various cases, as well known, the highly superluminal tunneling associated with the so-called "generalized Hartman Effect": but we are now able to improve and modify the mathematical description of such an effect, and to compare more in detail our results with the experimental data for non-resonant tunneling of photons . Finally, as a second example, the tunneling phase-time is calculated, and compared with the available experimental results, in the case of the quantum-mechanical tunneling of neutrons through two barrier-filters at the resonance energy of the set-up. Our analysis appears to explain satisfactorily both sets of experiments.