The Quantum Dynamics Of Cold Atoms Under A Periodic Driving Action

Recently, the cold atom has been developed to be a quantum simulation system to understand the basic concepts in condensed matter physics and nonlinear physics theory, because its more parameters, such as atomic interaction, external potential, can be easily controlled by experimentalist. With experimentally realized of a circular external potential, the so called "quantum chaos" has been extensively studied. In the other hand, the cold atoms were expected to play an important role to develop quantum measurement since it was born in 1995. In this paper, we have investigated the quantum dynamics of generalized quantum kicked rotator model, especially in the effect of the nonlinearity on chaotic phenomena. The quantum dynamics of cold atom combined with harmonic trap and periodic periodical potential is explored. And as one application of its dynamics, one novel multi-mode atomic interferometry is proposed.In the first part, we presented our results of generalized quantum kicked rotator, which the effect of nonlinearity on the quantum dynamics are extensively studied. The ultra-cold Bose Gases trapped by cyclic potential can be described by nonlinear Schrodinger equation with periodic condition. We found a general analytical solution, and by which a relation between their chemical potential and the nonlinearity has been observed. The dynamic evolution is numerically observed by calculating the total energy, fidelity and distribution in momentum space, assuming the initial states are described by those analytical solutions. The numerical results show that the quantum beating does not depend on the initial states except the beating frequency in anti quantum resonance condition (The period of kicks is T= 2π) for weak nonlinearity. In quantum resonance condition (The period of kicks is T= 4π/3), the different rates of suppression depend on the initial states. It is interesting to note that the large nonlinearity induces an irregular motion of the system for quantum anti-resonance and quantum resonance conditions. The total energy does not increase with time, oscillates around a finite value. The distribution function in momentum space shows an exponential decay with the number of the occupied momentum states at the occupation boundary. This may reveal a kind of localization in momentum space.In the second part, we investigate the dynamics of cold atoms in harmonic trap (ω is the frequency) with periodic periodical potential. The periodic periodical potential creates a number of spatially addressable modes. At t= π/ω, the harmonic trap coherently recombines the wave packets back to the initial density configuration. Using the characteristics of dynamics, we presented a new theory of atom interferometer with high precision. The atom interferometer has the following four element steps.1. Beam-splitter: A Kaptiza-Dirac pulse (periodic periodical potential) is applied to the atom state at the special time tB. The Kaptiza-Dirac pulse creates a number of orthogonal, spatially addressable modes, which evolve along different paths under the harmonic confinement.2. Phase shift:Each spatial mode gains a phase shift θ with respect to its neighbor’s modes due to the action of an external potential.3. Beam splitter: When the harmonic trap has coherently recombined the wave packets back to the initial density configuration, one more Kaptiza-Dirac pulse is applied to again mix and separate the modes along different paths.4. Measurement:The external field is estimated by fitting the density profile or counting the number of atoms in each spatial mode at t?.The sequences 1-2 can be iterated an arbitrary number of times n before the final measurement 3-4. The expected sensitivity is rigorously calculated with the Fisher information and the Cramer-Rao lower bound. Applying the theory to the measuring acceleration of gravity, we calculate the sensitivity of atom interferometer analytically. With typical parameters of experiment, a temperature independent sensitivity is △g/g～10-9/n. If there is just one grating, the sensitivity can exceed current atomic interferometers by three orders. For initial state of finite temperature, wave packets of different modes begin to be overlap with increase of temperature. The sensitivity of atom interferometer decreases. In the last, we discuss the effect of two kinds of perturbation for the interferometer. For the perturbed harmonic trap, the numerical results show that it can reduce the sensitivity of atom interferometer. For perturbed Kapitza-Dirac pulses, the numerical results shows that it does not bring substantial effects to the sensitivity of atom interferometer.