We study the problem of the attractive inverse square potential in quantum mechanics with a generalized uncertainty relation. Using the momentum representation, we show that this potential is regular in this framework. We solve analytically the $s$-wave bound states equation in terms of Heun’s functions. We discuss in detail the bound states spectrum for a specific form of the generalized uncertainty relation. The minimal length may be interpreted as characterizing the dimension of the system.

We have generalized the two-step approach to the solution of systems of $N$ coupled quantum anharmonic oscillators. By using the squeezed vacuum state of each individual oscillator, we construct the tensor product state, and obtain the optimal squeezed vacuum product state through energy minimization. We then employ this optimal state and its associated bosonic operators to define a basis set to construct the Heisenberg matrix. The diagonalization of the matrix enables us to obtain the energy eigenvalues of the coupled oscillators. In particular, we have applied our formalism to determine the eigenenergies of systems of two coupled quantum anharmonic oscillators perturbed by a general polynomial potential, as well as three and four coupled systems. Furthermore, by performing a first-order perturbation analysis about the optimal squeezed vacuum product state, we have also examined into the squeezing properties of two coupled oscillator systems.

We study experimentally a system comprised of linear chains of spin-1/2 nuclei that provides a test bed for multibody dynamics and quantum-information processing. This system is a paradigm for a class of quantum-information processing devices that can perform particular tasks even without universal control of the whole quantum system. We investigate the extent of control achievable on the system with current experimental apparatus and methods to gain information on the system state, when full tomography is not possible and in any case highly inefficient.

We present a dynamic interpretation of resonant x-ray Raman scattering where vibrationally selective excitation into molecular resonances has been employed in comparison with excitation into higher lying continuum states for condensed ethylene and benzene as molecular model systems. In order to describe the purely vibrational spectral loss features and coupled electronic and vibrational losses the one-step theory for resonant soft x-ray scattering is applied, taking multiple vibrational modes and vibronic coupling into account. The scattering profile is found to be strongly excitation energy dependent and to reflect the intermediate states dynamics of the scattering process. In particular, the purely vibrational loss features allow one to map the electronic ground state potential energy surface in light of the excited state dynamics. Our study of ethylene and benzene underlines the necessity of an explicit description of the coupled electronic and vibrational loss features for the assignment of spectral features observed in resonant x-ray Raman scattering at polyatomic systems, which can be done in both a time independent and a time dependent picture. The possibility to probe ground state vibrational properties opens a perspective to future applications of this photon-in-photon-out spectroscopy.

The resonance states of the systems consisting of three identical bosons and of three fermions interacting with an attractive $1∕r$ potential are investigated using highly correlated exponential basis functions. The method of complex-coordinate rotation is used to extract energies and widths for the resonances lying below the $n=2$ excited state threshold of the respective two-body subsystem. For the system consisting of three identical bosons, we have obtained a total of 12 $S$-wave resonances. It is concluded that nine of them belong to one series of Rydberg states. For the system consisting of three fermions, we have obtained a total of 23 $S$-wave resonances in total spin three-halfs (the ${}^{4}S$ state) and 17 $S$-wave resonances total spin one-half (the ${}^{2}S$ states). The resonances for the ${}^{4}S$ states belong to three different series, while those for the ${}^{2}S$ states belong to two Rydberg series. The resonance energies and widths of these states for the three-body self-gravitating bosons and fermions are reported.

We experimentally investigate one-dimensional (1D) expansion of a cold cloud of cesium atoms in orthogonal 2D configuration of near resonant laser beams by temporally modulating a pair of counterpropagating trapping beams of a magneto-optical trap (MOT). The cloud is observed to undergo ballistic expansion followed by superballistic explosive growth due to the fluctuations of the 2D radiation force. A model based on the theory of Brownian motion is developed and a comparison of experiments with theory is shown to provide a direct measure of the velocity diffusion coefficient. We also observe sudden contraction of the cloud immediately after switching off the pair of trapping beams, which provides direct evidence for the existence of the radiation trapping force in a MOT.

The effect of a rotation on a matter-wave-laser beam splitter is studied and modeled. This modeling is shown to give important nontrivial corrections to the Sagnac phase shift of matter-wave gyrometers when the duration of the laser action cannot be neglected with respect to the propagation time between beam splitters. This result is illustrated on a Mach-Zehnder atom gyrometer (rate gyroscope) with running laser beam splitters. A quasiclassical description of the interferometer arms inside the beam splitters is proposed to interpret the corrected Sagnac phase shift in terms of an effective interferometer area.

We report on measurements of an elementary surface mode in an ultracold, strongly interacting Fermi gas of ${}^6\mathrm{Li}$ atoms. The radial quadrupole mode allows us to probe hydrodynamic behavior in the crossover from Bose-Einstein condensation (BEC) to the Bardeen-Cooper-Schrieffer (BCS) regime without being influenced by changes in the equation of state. We examine the frequency and damping of this mode, along with its expansion dynamics. In the unitarity limit and on the BEC side of the resonance, the observed frequencies agree with standard hydrodynamic theory. However, on the BCS side of the crossover, a striking downshift of the oscillation frequency is observed in the hydrodynamic regime as a precursor to an abrupt transition to collisionless behavior; this indicates coupling of the oscillation to fermionic pairs.

We propose a transformation method that simplifies quantitative analysis in the $\Lambda$-type electromagnetically induced transparency (EIT) system containing degenerate Zeeman states. This transformation maps several coupled three-state systems into a simple three-state system. Moreover, we demonstrate that the transformation is an excellent approximation under adiabatic conditions and the calculation results from several coupled systems are in good agreement with those from the simple system. The transformation method is a powerful tool for relevant studies of EIT, slow light, and storage of light.

We consider a photorefractive crystal with an optically imprinted photonic lattice. Using a fully anisotropic model for the nonlinearity we numerically demonstrate that both focusing and defocusing gap vortex solitons exist in this system. We find that both may be dynamically stable under certain circumstances.

Using a single-channel active Raman gain medium, we show a $220\pm 20\mathrm{ns}$ advance time for an optical pulse of ${\tau }_{\mathit{FWHM}}=15.4\mu \mathrm{s}$ propagating through a $10\mathrm{cm}$ medium, a lead time that is comparable to what was reported previously using a two-mode pump field. In addition, we have verified experimentally all the features associated with this single-channel Raman gain system.

We present a necessary and sufficient condition for a finite-dimensional density matrix to be an extreme point of the convex set of density matrices with positive partial transpose with respect to a subsystem. We also give an algorithm for finding such extreme points and illustrate this by some examples.

We consider the dynamic escape of a small packet of ultracold atoms launched from within an optical dipole trap. Based on a theoretical analysis of the underlying nonlinear dynamics, we predict that fractal behavior can be seen in experimental escape data. These data can be collected by measuring the time-dependent escape rate for packets launched over a range of angles. This fractal pattern is particularly well resolved below the Bose-Einstein transition temperature—a direct result of the extreme phase-space localization of the condensate. We predict that several self-similar layers of this novel fractal should be measurable, and we explain how this fractal pattern can be predicted and analyzed with recently developed techniques in symbolic dynamics.

Experiments on the axial breathing mode in a rapidly rotating Bose-Einstein condensate are examined. Assuming a cold cloud without thermal component, we show that errors due to defocus of an imaging camera in addition to an inclination of the rotational axis can lead to a significant underestimate of the rotation rate in the fast rotation limit; within these uncertainties, our theoretical prediction agrees with the experimental data. We also show that, in the fast rotation regime, the Thomas-Fermi theory, which is inapplicable there, underestimates the rotation rate. Underestimation of the rotation rate due to these effects would also partly explain a discrepancy between theory and experiment for the Tkachenko mode frequency in the fast rotation regime.

We discuss the experimental signal for the Bose-Einstein condensation of cold atoms in an optical lattice at finite temperature. Instead of using the visibility of the interference pattern via time-of-flight imaging, we show that the momentum-space density profile in the first Brillouin zone, in particular its bimodal distribution, provides an unambiguous signal for the Bose-Einstein condensation. We confirm this point with detailed calculations of the change in the atomic momentum distribution across the condensation phase transition, taking into account both the global trapping potential and the atomic interaction effects.

The dipole-quadrupole and dipole-octopole polarizabilities $A$ and $E$ of $\text{Ge}{\left({\text{CH}}_3\right)}_4$ have been determined from collision-induced light-scattering experiments and ab initio calculations. Our experimental results are $\left|A\right|∕e^2{a}_0^3{E}_h^{-1}<143$ and $\left|E\right|∕e^2{a}_0^4{E}_h^{-1}<545$. Our best theoretical values are $A=45.48$ and $E=-389.9$, respectively. The calculated value for the dipole polarizability is $\alpha ∕e^2{a}_0^2{E}_h^{-1}=83.26$, in fine accord with our static experimental estimate of 83.2. We present a detailed discussion of the level of agreement between experiment and theory.

Calculations of photoionization cross sections for alkali-metal atoms are carried out in the framework of relativistic many-body perturbation theory (RMBPT) using quasicontinuum $B$-spline orbitals. All third-order terms are included, in contrast to previous calculations based on either random-phase approximation (RPA), Brueckner orbitals, or their combination. The particular advantage of quasicontinuum states is that high-order MBPT codes do not require modification for applications to the photoionization problem. The agreement with experiment is improved compared to RPA and Dirac-Hartree-Fock approximations. The results also exhibit close form invariance. The presented formalism can be extended to other photoionizing transitions.

We have calculated the energy loss of swift $\text{O}_2{}^{+}$, $\text{N}_2{}^{+}$, and $\text{C}_n{}^{+}$ $\left(n=2-60\right)$ molecular ions moving through an amorphous carbon target. The dielectric formalism is used to evaluate the vicinage effects in the energy loss of the atomic ions that form the molecular projectile, but we take into account that the charge state of these atomic ions is affected by their correlated motion through the target and by the screened Coulomb potential between them. When vicinage effects in the charge state are taken into account, the Coulomb repulsion is weakened, leading to a reduction in the interatomic separations ($\sim 3\%$ for $\text{N}_2{}^{+}$ and $\sim 9\%$ for $\text{C}_{60}{}^{+}$, both having similar velocities). These charge state effects can be neglected for diatomic molecular ions, but they give rise to a reduction of $\sim 8\%$ in the vicinage effects in the energy loss of larger molecular ions with cagelike geometrical structures, like $\text{C}_n{}^{+}$ $\left(n=20,60\right)$ projectiles.

By solving nonperturbatively the equations of Schrödinger and of Hamilton, we have studied the time-dependent multiphoton dissociation of a diatomic molecule induced by a moderate low-frequency laser field. The photodissociation probabilities are calculated and analyzed as a function of the laser frequency, the intensity, and the pulse shape. A well-established quantum and classical result is that for laser intensities of the order of ${10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$ the dissociation probability presents a maximum at the frequency ${\omega }_{\mathit{max}}\sim (0.80–0.90){\omega }_{01}$, where ${\omega }_{01}$ is the transition frequency from the ground to the first excited vibrational state (redshift phenomenon). In this work, we go further and explore the quantum and classical effects of radiation at the optimum frequency ${\omega }_{\mathit{max}}$ on the overall vibrational excitation and dissociation dynamics. First, it is shown that both quantum and classical results predict that ${\omega }_{\mathit{max}}$ continues to be the optimum frequency for photodissociation even at the order of $10\mathrm{TW}∕{\mathrm{cm}}^2$, i.e., low intensities. The quantum results show a multipeak structure versus laser frequency, which is attributed to resonant multiphoton transitions, while the classical results show a smooth curve with a broad maximum at ${\omega }_{\mathit{max}}$ which is explained by phase space arguments. Second, it is found that in both quantum and classical approaches ${\omega }_{\mathit{max}}$ marks a transition in the effects of turn-on time of the pulse shape on dissociation probability: for $\omega <{\omega }_{\mathit{max}}$ the gradual turn-on of the field leads to a noticeable reduction of the photodissociation probability, while for $\omega >{\omega }_{\mathit{max}}$ this effect is of minor importance. A classical interpretation of this finding is given, which is based on stickiness effects in phase space. Finally, the crucial role of ${\omega }_{\mathit{max}}$ is further demonstrated in the time dependence of bound-state occupation probabilities. The total survival probability decreases faster with time for $\omega <{\omega }_{\mathit{max}}$ rather than for $\omega >{\omega }_{\mathit{max}}$. Further, for $\omega >{\omega }_{\mathit{max}}$ the bound-state occupation probabilities exhibit multiphoton Rabi-type oscillations where more than two vibrational states are involved. These phenomena are predicted by both quantum and classical dynamics, although there are secondary differences which are revealed and discussed.

We propose a solvable generalization of the Tonks-Girardeau model that describes a coherent one-dimensional (1D) gas of cold two-level bosons which interact with two external fields in a Ramsey interferometer. They also interact among themselves by idealized, infinitely strong contact potentials, with interchange of momentum and internal state. We study the corresponding Ramsey fringes and the quantum projection noise which, essentially unaffected by the interactions, remains that for ideal bosons. The dual system of this gas, an ideal gas of two-level fermions coupled by the interaction with the separated fields, produces the same fringes and noise fluctuations. The cases of time-separated and spatially separated fields are studied. For spatially separated fields the fringes may be broadened slightly by increasing the number of particles, but only for large particle numbers far from present experiments with Tonks-Girardeau gases. The uncertainty in the determination of the atomic transition frequency diminishes, essentially with the inverse root of the particle number. The difficulties to implement the model experimentally and possible shortcomings of strongly interacting 1D gases for frequency standards and atomic clocks are discussed.