We construct an invisible quantum barrier which represents the phenomenon of quantum reflection using available data on atom-wall and Bose-Einstein-condensate--wall reflection. We use the Abel equation to invert the data. The resulting invisible quantum barrier is double valued in both axes. We study this invisible barrier in the case of atom and Bose-Einstein condensate (BEC) reflection from a solid silicon surface. A time-dependent, one-spatial-dimension Gross-Pitaevskii equation is solved for the BEC case. We found that the BEC behaves very similarly to the single atom except for size effects, which manifest themselves in a maximum in the reflectivity at small distances from the wall. The effect of the atom-atom interaction on the BEC reflection and correspondingly on the invisible barrier is found to be appreciable at low velocities and comparable to the finite-size effect. The trapping of an ultracold atom or BEC between two walls is discussed.
We construct an invisible quantum barrier which represents the phenomenon of quantum reflection using available data on atom-wall and Bose-Einstein-condensate--wall reflection. We use the Abel equation to invert the data. The resulting invisible quantum barrier is double valued in both axes. We study this invisible barrier in the case of atom and Bose-Einstein condensate (BEC) reflection from a solid silicon surface. A time-dependent, one-spatial-dimension Gross-Pitaevskii equation is solved for the BEC case. We found that the BEC behaves very similarly to the single atom except for size effects, which manifest themselves in a maximum in the reflectivity at small distances from the wall. The effect of the atom-atom interaction on the BEC reflection and correspondingly on the invisible barrier is found to be appreciable at low velocities and comparable to the finite-size effect. The trapping of an ultracold atom or BEC between two walls is discussed.
Recent developments in atom optics have brought Bose-Einstein condensates within 1 μm of solid surfaces where the atom-surface interactions can no longer be ignored. At longrange, the atom-surface interaction is described by the weakly attractive Casimir-Polder potential which is classically predicted to accelerate an incident atom toward the surface where it will interact strongly with the internal modes of the surface, lose energy, and land in a bound state of the surface. When the incident atom is very cold, on the order of a few nanokelvin, however, the acceleration of the atomic wavefunction is so abrupt that the atom may partially reflect from the attractive tail in a process known as quantum reflection. This work presents experimental evidence for quantum reflection from a solid surface at normal incidence. Using atoms from a 23Na BEC, cooled to a few nanokelvin in a recently demonstrated single-coil trap, controlled collisions were induced between atoms and solid silicon surface. A maximum reflection probability of ∼ 12% was observed for an incident velocity of 1 mm/s. Atoms confined against the surface at low density exhibited an enhanced lifetime due to quantum reflection. A surprising aspect of quantum reflection is that nano-structured surfaces are predicted to exhibit enhanced quantum reflection due to the reduction of the atom-surface interaction from reduced density surfaces. Using a pillared surface with an density reduced to 1% of bulk density, we observe an enhancement of the reflection probability to ∼ 60%. At velocities from 2-25 mm/s, predicted threshold dependence of the reflection probability was observed. At velocities below 2 mm/s, the reflection probability was observed to saturate. We develop a simple model which predicts the saturation as a result of mean-field interactions between atoms in the incident Bose-Einstein condensate. Thesis Supervisor: Wolfgang Ketterle Title: John D. MacAurthur Professor of Physics Thesis Supervisor: David E. Pritchard Title: Cecil and Ida Green Professor of Physics For the parents in my life,
Resonant reflection of a Bose–Einstein condensate by a double delta-function barrier has been considered analytically using the Gross–Pitaevskii approximation for nonlinearity. The reflecting coefficient has been derived taking into account a weak nonlinearity of the Schrodinger equation produced by the interaction between cold alkali atoms. A nonlinear term is given in the Hartree approximation for short-range interaction between atoms. The one-dimensional potential is approximated by two repulsive delta-function barriers. The analytical solution was obtained for the reflecting coefficient by a multiple-scale method in order to remove secular terms. The most interesting case corresponds to the condensate energies for which reflection is absent without a nonlinear term. Thus, reflection is determined only by the nonlinearity. The reflecting coefficient is derived in the first order on the nonlinearity parameter.
Resonant reflection of a Bose-Einstein condensate by a double delta-function barrier has been considered analytically using the Gross-Pitaevskii approximation for nonlinearity. The reflection coefficient has been derived taking into account a weak nonlinearity of the Schrödinger equation produced by the interaction between cold alkali atoms. Nonlinear term is given in the limit of asymptotically weak interaction and zero temperature. The one-dimensional potential is approximated by two repulsive delta-function barriers. The analytical solution was obtained for the reflection coefficient using a multiple-scale analysis in order to remove secular terms. The most interesting case corresponds to the condensate energies for which reflection is absent without nonlinear term. Thus, reflection is determined only by the nonlinearity. The reflection coefficient is derived in the first order on the nonlinearity parameter.
We applied the numerical combination of Runge-Kutta and Finite Difference (RKFD) scheme for a quantum reflection model of Bose-Einstein condensate (BEC) from a silicon surface. It is by the time-dependent Gross-Pitaevskii equation (GPE), a non-linear Schrödinger equation (NLSE) in the context of quantum mechanics. The role of cut-off potential δ and negative imaginary potential Vim is essential to estimating non-interacting BEC reflection models. Relying on these features, we performed a numerical simulation of the BEC quantum reflection model and calculated the effect of reflection probability R versus incident speed vx. The model is based on the three rapid potential variations: positive-step potential +Vstep, negative-step potential -Vstep, and Casimir-Polder potential VCP. As a result, the RKFP numerical scheme was successfully set up and applied to the quantum reflection model of BEC from the silicon surface. The numerical simulations results show that the reflection probability R decays exponentially to the incident speed vx.
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.
We observed quantum reflection of ultracold atoms from the attractive potential of a solid surface. Extremely dilute Bose-Einstein condensates of 23Na, with peak density 10(11)-10(12) atoms/cm(3), confined in a weak gravitomagnetic trap were normally incident on a silicon surface. Reflection probabilities of up to 20% were observed for incident velocities of 1-8 mm/s. The velocity dependence agrees qualitatively with the prediction for quantum reflection from the attractive Casimir-Polder potential. Atoms confined in a harmonic trap divided in half by a solid surface exhibited extended lifetime due to quantum reflection from the surface, implying a reflection probability above 50%.