It is shown here how the lifetimes and energies of resonance states can be calculated by applying the complex scaling transformation to the nonlinear Schrödinger equation. It is essential to first apply the complex scaling transformation to the full Hamiltonian and to subsequently derive from the result the correct complex scaled nonlinear Schrödinger equation. The latter equation is physically relevant and amenable to numerical calculations. To analyze the results obtained by solving this equation, it is necessary to realize the close association of resonance phenomena with fragmentation of the system. As an illustrative example, we apply this theory to the Gross-Pitaevskii nonlinear equation to calculate the tunneling lifetime of a condensate inside an external (either optical or magnetic) trap. We show that by varying the scattering length, the external potential acts like a “selective membrane” which controls the direction of the flux of the cold atoms through the barriers and, thereby, controls the size of the stable condensate inside the trap.

On the repulsive side of the Feshbach resonance fermionic atoms can bind into diatomic molecules which at low enough temperature condense into molecular Bose-Einstein condensate. Sufficiently close to the resonance the molecules are bound so weakly that a significant fraction of unbound fermionic atoms survives even below the temperature of condensation. In this paper we show that laser cooling of the unbound fermionic atoms can reduce temperature of mixture of atoms and molecules below the condensation temperature. Numerical simulations for reasonable experimental parameters show that it is possible to achieve molecular condensate after a few seconds of laser cooling.

We study quantum-noise squeezing and quantum correlations of gap solitons in a Bose-Einstein condensate loaded into a one-dimensional optical lattice. By employing a linearized quantum theory we find that quantum-noise squeezing of gap solitons, produced during their evolution, is enhanced compared with the atomic solitons in a lattice-free case due to intrasoliton structure of quantum correlations induced by the Bragg scattering in the periodic potential. We also show that nonlinear interaction of gap solitons in dynamically stable bound states can produce strong atom-number correlation between the soliton pairs.

A description of spontaneous parametric down-conversion in finite-length one-dimensional nonlinear photonic crystals is developed using semiclassical and quantum approaches. It is shown that if a suitable averaging is added to the semiclassical model, its results are in very good agreement with the quantum approach. We propose two structures made with $\mathrm{Ga}\mathrm{N}∕\mathrm{Al}\mathrm{N}$ that generate both degenerate and nondegenerate entangled photon pairs. Both structures are designed so as to achieve a high efficiency of the nonlinear process.

When a vapor absorbs laser light, the field’s intrinsic phase fluctuations (PM) induce variations in the atoms’ absorption cross section, which in turn yield fluctuations in the transmitted light intensity. In the Born approximation, the observed amplitude noise (AM) arises from a single scattering of the input field. Here, we show that the Born approximation breaks down for optically thick vapors and that the observed AM is influenced by scatterings of the “medium-perturbed” field. With importance for spectroscopy, we find that the multiple field-atom scatterings reduce the PM-to-AM conversion efficiency.

In the conventional quantum mechanics (i.e., Hermitian quantum mechanics) the adiabatic theorem for systems subjected to time-periodic fields holds only for bound systems and not for open ones (where ionization and dissociation take place) [D. W. Hone, R. Ketzmerik, and W. Kohn, Phys. Rev. A 56, 4045 (1997)]. Here with the help of the $(t,t^{\prime })$ formalism combined with the complex scaling method we derive an adiabatic theorem for open systems and provide an analytical criterion for the validity of the adiabatic limit. The use of the complex scaling transformation plays a key role in our derivation. As a numerical example we apply the adiabatic theorem we derived to a one-dimensional model Hamiltonian of Xe atom which interacts with strong, monochromatic sine-square laser pulses. We show that the generation of odd-order harmonics and the absence of hyper-Raman lines, even when the pulses are extremely short, can be explained with the help of the adiabatic theorem we derived.

We derive the optimal measurement for quantum-state discrimination without a priori probabilities, i.e., in a minimax strategy instead of the usually considered Bayesian one. We consider both minimal-error and unambiguous discrimination problems, and provide the relation between the optimal measurements according to the two schemes. We show that there are instances in which the minimum risk cannot be achieved by an orthogonal measurement, and this is a common feature of the minimax estimation strategy.

We show experimentally that quantum coincidence imaging with photon pairs generated by spontaneous parametric down conversion, in certain conditions, differs from the classical imaging in two aspects: the magnified image is produced by an “apodized” effective lens and its effective wavelength is the De Broglie wavelength of the photon pair $\lambda ∕2$, where $\lambda$ is the central wavelength of the single photon wave packet. Better resolution and apodization effects (reduced intensity of the diffraction fringes around the image) are observed in the detected images.

We demonstrate that the transition from plane-wave to focused pumping in type-II down-conversion is analogous to the transition from cw to pulsed pumping. We show experimentally that focused pumping leads to asymmetric broadening of both the ordinary and extraordinary light distribution. It hardly affects the entanglement quality if proper spatial filtering is applied.

We model the process of ultrabroadband light generation in which a pair of laser pulses separated by the Raman frequency drive a Raman transition. In contrast to the usual approach using separate field envelopes for the different frequency components, we treat the field as a single entity. This requires the inclusion of few-cycle corrections to the pulse propagation. Our single-field model makes fewer approximations and is mathematically (and hence computationally) simpler, although it does require greater computational resources to implement. The single-field theory reduces to the traditional multifield one using appropriate approximations.

We study a single incoherently pumped atom moving within an optical high-$Q$ resonator in the strong-coupling regime. Using a semiclassical description for the atom and field dynamics, we derive a closed system of differential equations to describe this coupled atom-field dynamics. For sufficiently strong pumping, the system starts lasing when the atom gets close to a field antinode, and the associated light forces provide for self-trapping of the atom. For a cavity mode blue detuned with respect to the atomic transition frequency, this is combined with cavity-induced motional cooling, allowing for long-term steady-state operation of such a laser. The analytical results for temperature and field statistics agree well with our earlier predictions based on quantum Monte Carlo simulations. We find sub-Doppler temperatures that decrease with gain and coupling strength, and can even go beyond the limit of passive cavity cooling. Besides demonstrating the importance of light forces in single-atom lasers, this result also gives strong evidence to enhance laser cooling through stimulated emission in resonators.

We study a quantum state transfer between two qubits interacting with the ends of a quantum wire consisting of linearly arranged spins coupled by an excitation conserving, time-independent Hamiltonian. We show that, if we control the coupling between the source and the destination qubits and the ends of the wire, the evolution of the system can lead to an almost perfect transfer even in the case in which all nearest-neighbour couplings between the internal spins of the wire are equal.

We discuss the detection of entanglement in interacting quantum spin systems. First, thermodynamic Hamiltonian-based witnesses are computed for a general class of one-dimensional spin-$1∕2$ models. Second, we introduce optimal bipartite entanglement observables. We show that a bipartite entanglement measure can generally be associated with a set of independent two-body spin observables whose expectation values can be used to witness entanglement. The number of necessary observables is ruled by the symmetries of the model. Illustrative examples are presented.

The dynamic scalar and tensor polarizabilities of Sc and Ti are computed with time-dependent density functional theory. These polarizabilities are used to compute the isotropic and anisotropic dispersion interactions in ScHe and TiHe. We find $C_{6,0}\left(\mathrm{ScHe}\right)=30.00$, $C_{6,2}\left(\mathrm{ScHe}\right)=-1.63$, $C_{6,0}\left(\mathrm{TiHe}\right)=28.40$, and $C_{6,2}\left(\mathrm{TiHe}\right)=-0.90$ $\left(E_h{a}_0^6\right)$. We show that an estimate of the dispersion anisotropy based on static polarizabilities results in an overestimation by a factor of 2.

The recent pioneering experiments of the [Nature 412, 52 (2001)] and [Science, 293, 274 (2001)] groups have demonstrated the dynamical tunneling of cold atoms interacting with standing electromagnetic waves. It has been shown [Phys. Rev. Lett. 89, 253201 (2002)], that the tunneling oscillations observed in these experiments correspondingly stems from two- and three-Floquet quantum state mechanism and can be controlled by varying the experimental parameters. The question where are the fingerprints of the classical chaotic dynamics in a quantum dynamical process which is controlled by 2 or 3 quantum states remains open. Our calculations show that although the effective $\hslash$ associated with the two experiments is large, and the quantum system is far from its semiclassical limit, the quantum Floquet-Bloch quasienergy states still can be classified as regular and chaotic states. In both experiments the quantum and the classical phase-space entropies are quite similar, although the classical phase space is a mixed regular-chaotic space. It is also shown that as the wave packet which is initially localized at one of the two inner regular islands oscillates between them through the chaotic sea, it accumulates a random phase which causes the decay of the amplitude of the oscillating mean momentum, $⟨P\left(t\right)⟩$, as measured in both experiments. The extremely high sensitivity of the rate of decay of the oscillations of $⟨P\left(t\right)⟩$ to the very small changes in the population of different Floquet-Bloch states, is another type of fingerprint of chaos in quantum dynamics that presumably was measured in the NIST and AUSTIN experiments for the first time.

We quantize a semiclassical system defined by the Hamiltonian obtained from the asymptotic self-similar solution of the Gross-Pitaevskii equation for a trapped Bose-Einstein condensate with a linear gain term. On the basis of a Schrödinger equation derived in a space of ellipsoidal parameters, we analytically calculate the quantum mechanical and thermal variance in the ellipsoidal parameters for Bose-Einstein condensates in various shapes of trap. We show that, except for temperatures close to zero, dimensionless dispersions do not depend on the frequencies of the trap and they have the same dependence on dimensionless temperatures.

The optical dipole trap scheme is generalized by considering the laser as a dynamical cavity field mode. For a large enough finesse, the far-detuned trapping field itself exerts a significant friction force on the atom. Unlike in free-space far-off resonance dipole traps, here the trapping is stable in the steady state. The scheme in principle applies to arbitrary linearly polarizable particles, e.g., molecules.