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We investigate the classical and quantum dynamics of atoms moving in a phase-modulated standing light field. In both cases the width of the momentum distribution exhibits characteristic oscillations as a function of the modulation amplitude. We argue that at the maxima of these oscillations the system is chaotic, whereas in the valleys it is almost regular. Quantum localization appears only in the chaotic regime. We connect our analysis with a recent experiment [F. Moore et al. , Phys. Rev. Lett. 73, 2974(1994].PACS numbers: 05.45.+b, 42.50.Lc, 42.50.Vk Most recently the description of cold atoms [1] in the framework of atom optics [2] has opened [3 -5] a new avenue in the search for fingerprints of classical chaos in quantum systems: A strongly detuned atom in a standing light field moves like a particle in a spatially periodic potential [6,7]. An atom in a phase-modulated light field additionally experiences a time dependent force [3]. Classically, the resulting motion can be chaotic [8]. But how does classical chaos manifest itself in the quantum dynamics of the atom and how to reach the quantum domain'l The landmark experiment [4,5] by the Austin group answers these questions: The measured momentum transfer of atoms in a phase-modulated light field shows the characteristic signature of quantum chaos: dynamical localization. Moreover, the appropriate choice of the experimental parameters such as the wave number of the standing light field or the modulation frequency allows one to step from regimes which ask for a classical description to regimes which ask for a quantum description.In the present paper we predict a new effect in this playground of atom optics and quantum chaos: The width of the atomic momentum distribution shows characteristic oscillations as a function of the modulation depth A.These oscillations appear in the classical as well as in the quantum description. In the domains of A where maxima appear the classical system is chaotic, whereas in the valleys the motion is almost regular. In the quantum case the peaks are lower, that is the quantum mechanical momentum distribution is narrower than the classical one. Moreover, it is an exponential rather than a Gaussian a signature of quantum localization [9,10]. The motion of a strongly detuned atom of mass M, position x, and momentum p along a standing light field of wave number k, is described by the Hamiltonian H = P /2M -Vpcos(2kx), where Vp is the height of the periodic potential determined by the coupling of the atom to the light field [6]. The motion of the atom in a phase-modulated light field, as created, for example, by an oscillating mirror, is described [3] by the Hamiltonian H = P /2M -Vp cos[2kx -csin(tot)], where A denotes -+ p --a sin(x -Asinr) P(x, p;r) = 0 (2) l97 Bx Bp and periodic boundary conditions P(x + L, p; r) = P(x, p; r), where L is a multiple of 2~. The solution of Eq. (2) satisfying the initial condition P(xp, pp', r = 0) = Po(xp, pp) has the form P(x, p;r) = dxo dpp 6[x g(r; xp, pp)] X 6[p -7r(r;xp, pp)] && Po(xo, po)...

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We give a general introduction into polarized imaging and report on a Jones-pupil approach for a complete evaluation of the resulting optical performance. The Jones pupil assigns a Jones matrix to each point of the exit pupil describing the impact of both the global phase and the polarization on imaging. While we can learn already a lot about the optical system by taking a close look at the Jones pupil -and starting imaging simulations from it -a quantitative assessment is necessary for a complete evaluation of imaging. To do this, we generalize the concept of scalar Zernike aberrations to Jones-Zernike aberrations by expansion of the Jones pupil into vector polynomials. The resulting method is nonparaxial, i.e. the effect of the polarization dependent contrast loss for high numerical apertures is included. The aberrations of the Jones-matrix pupil are a suitable tool to identify the main drivers determining the polarization performance. Furthermore, they enable us to compare the polarized and the unpolarized performance of the such characterized lithographic system.