Although originally developed for coherent paraxial scalar electromagnetic radiation in the visible-light regime, phase retrieval using the transport-of-intensity equation has been successfully applied to a range of paraxial radiation and matter-wave fields. Such applications include using electron wave fields to quantitatively image magnetic skyrmions and spin ices, propagation-based phase-contrast imaging using cold neutrons and hard x-rays, and visible-light refractive imaging of the projected column density of cold-atom clouds. Teague's method for phase retrieval using the transport-of-intensity equation, which renders the phase of a paraxial complex wave indirectly measurable via the existence of a conserved current, has been applied to a broad variety of situations which include all of the experiments described above. However, these applications have been undertaken without a thorough analysis of the underlying validity of the method. Here we derive sufficient conditions for the phase-retrieval solution provided by Teague's method to coincide with the true phase of the paraxial radiation or matter-wave field. We also present a sufficient condition guaranteeing that the discrepancy between the true phase function and that reconstructed using Teague's solution is small. These conditions demonstrate that, in most practical cases, for phase-amplitude retrieval using the transport-of-intensity equation, the Teague solution is very close to the exact solution. However, we also describe a counter example in the context of phase-amplitude retrieval using hard x-rays, in which the relative root-mean-square difference between the exact solution and that obtained using Teague's method is 9%. These findings clarify the foundations of one of the most widely applied methods for propagation-based phase retrieval of both paraxial matter and radiation wave fields and define a region for its applicability.

Motivated by possible applications of spectral singularities in optics, we develop a semiclassical method of computing spectral singularities. We use this method to examine the spectral singularities of a planar slab gain medium whose gain coefficient varies due to the exponential decay of the intensity of the pumping beam inside the medium. For both singly and doublypumped samples, we obtain universal upper bounds on the decay constant beyond which no lasing occurs. Furthermore, we show that the dependence of the wavelength of the spectral singularities on the value of the decay constant is extremely mild. This is an indication of the stability of optical spectral singularities.

We demonstrate the use of femtosecond laser pulse shaping for precise control of the interference between the coherent anti-Stokes Raman scattering (CARS) signal and the coherent nonresonant background generated within the same sample volume. Our technique is similar to heterodyne detection with the coherent background playing the role of the local oscillator field. In our experiment, we first apply two ultrashort (near-transform-limited) femtosecond pump and Stokes laser pulses to excite coherent molecular oscillations within a sample. After a short and controllable delay, we then apply a laser pulse that scatters off of these oscillations to produce the CARS signal. By making fine adjustments to the probe field spectral profile, we vary the relative phase between the Raman-resonant signal and the nonresonant background, and we observe a varying spectral interference pattern. These controlled variations of the measured pattern reveal the phase information within the Raman spectrum.

Here, we propose a solid-state quantum memory that does not require spectral holeburning, instead using strong rephasing pulses like traditional photon-echo techniques. The memory uses external broadening fields to reduce the optical depth and so switch off the collective atom-light interaction when desired. The proposed memory should allow operation with reasonable efficiency in a much broader range of material systems, for instance Er${}^{3+}$ doped crystals which have a transition at 1.5 $\mu \text{m}$. We present analytic theory supported by numerical calculations and initial experiments.

Matchgates are a family of two-qubit gates associated with noninteracting fermions. They are classically simulatable if acting only on nearest neighbors but become universal for quantum computation if we relax this restriction or use swap gates [Jozsa and Miyake, Proc. R. Soc. A 464, 3089 (2008)]. We generalize this result by proving that any nonmatchgate parity-preserving unitary is capable of extending the computational power of matchgates into universal quantum computation. We identify the single local invariant of parity-preserving unitaries responsible for this, and discuss related results in the context of fermionic systems.

Atomic inner-shell photoionization followed by double Auger decay is investigated. The focus of our study is the effect of post-collision interaction (PCI) on the photoelectron energy distribution. A semi-classical approach is employed to describe the PCI distortion of the photoelectron line shapes associated with both direct and cascade double Auger decays. This approach is shown to be valid at low photoelectron energies, whereas for large incident photon energies it reduces to the eikonal approximation. The theory is applied to the case of Ar $2p$ photoionization spectra and good agreement with available experimental data is achieved.

We present a method for efficient calculation of exchange correlation in a density-functional theory calculation with Gaussian basis functions. It seamlessly combines the treatments of smooth (low-energy cutoff) and nonsmooth parts of the electron density such that computational performance can be maximized without approximation. Benchmark calculations show that the scheme can speed up the calculation of the exchange-correlation by as much as ten times or more without error.

Bosonic dimer-dimer scattering is studied near the unitary limit using momentum-space equations for the four-particle transition operators. The impact of the Efimov effect on the dimer-dimer scattering observables is explored, and a number of universal relations is established with high accuracy. The rate for the creation of Efimov trimers via dimer-dimer collisions is calculated.

The dipole-dipole broadening of $ns-np$ microwave transitions of cold Rb Rydberg atoms in a magneto-optical trap has been recorded for $28⩽n⩽51$. Since the electric dipole transition matrix elements scale as $n^2$, a broadening rate scaling as $n^4$ is expected and a broadening rate of $8.2\times {10}^{-15}{n}^4$ MHz cm${}^3$ is observed. The observed broadening is smaller than expected from a classical picture due to the spin-orbit interaction in the $np$ atoms. The broadened resonances are asymmetric and cusp shaped, and their line shapes can be reproduced by a diatomic model which takes into account the dipole-dipole interaction, including the spin-orbit interaction, the strengths of the allowed microwave transitions, and the distribution of the atomic spacings in the trap.

An experimental study of the excitation of the ($2p^2$) ${}^{1}D$ and ($2s2p$) ${}^1{P}^o$ autoionizing states of helium by 250-eV electron impact is presented. The ejected-electron angular distributions and energy spectra are measured in coincidence with the corresponding scattered electrons for a scattering angle of $-{13}^{\circ }$ and for a range of ejected-electron angles in both the forward and backward directions. Resonance profiles are analyzed in terms of the Shore-Balashov parametrization to obtain the resonance asymmetry $A_{\mu }$ and yield $B_{\mu }$ parameters and the direct ionization cross section $f$. The spectra and their parameters are compared to the previous measurements of Lower and Weigold [J. Phys. B. 23, 2819 (1990)] and McDonald and Crowe [J. Phys. B 26, 2887 (1993)]. Comparison is also made with the recent theoretical triply differential cross-section calculations based on the first and second Born approximations. In general, good qualitative agreement is found between the experimental results. Some differences are found at the forward and backward directions. These differences in the shape and magnitude of the cross sections are attributed to the different incoming electron energies used in the experiments. The second Born approximation with inclusion of the three-body Coulomb interaction in the final state agrees reasonably well with experiments in the binary region. However, the ${}^1{P}^o$ resonance yield parameter $B_{\mu }$ is significantly overestimated at the recoil region, giving a relatively large recoil peak, in contradiction to the experiment. There is also a discrepancy between the two theories available for the ${}^{1}D$ resonance yield parameter $B_{\mu }$ in this region. Remaining discrepancies between theories and experiments are also discussed.

We present a method for measuring atomic parity violation in the absence of static external electric and magnetic fields. Such measurements can be achieved by observing the interference of parity-conserving and parity-violating two-photon transition amplitudes between energy eigenstates of zero electronic angular momentum. General expressions for induced two-photon transition amplitudes are derived. The signal-to-noise ratio of a two-photon scheme using the $6s{}^{21}{S}_0\to 6s6p{}^3{P}_0$ transition in ytterbium is estimated.

We investigate the quantum phase transition in an ultracold atom-molecule conversion system. It is found that the system undergoes a phase transition from a mixed atom-molecule phase to a pure molecule phase when the energy bias exceeds a critical value. By constructing a coherent state as variational state, we get a good approximation of the quantum ground state of the system. Using this variational state, we deduce the critical point analytically. We then discuss the scaling laws characterizing the transition and obtain the corresponding critical exponents. Furthermore, the Berry curvature signature of the transition is studied. In particular, we find that the derivatives of the Berry curvature with respect to total particle number intersect at the critical point. The underlying mechanism of this finding is discussed as well.

Motivated by recent experiments, we consider the dynamics of spin-1 spinor condensates after a quantum quench from the polar to ferromagnetic state from varying the quadratic Zeeman field $q$. We apply the truncated Wigner approximation (TWA) to the spinor system, including all spatial and spin degrees of freedom. For short times, we find full agreement with the linearized Bogoliubov analysis. For longer times, where the Bogoliubov theory fails, we find that the system reaches a quasisteady prethermalized state. We compute the Bogoliubov spectrum about the ferromagnetic state with general $q$ and show that the resulting finite-temperature correlation functions grossly disagree with the full TWA results, thus indicating that the system does not thermalize even over very long time scales. Finally, we show that the absence of thermalization over realistic time scales is consistent with calculations of Landau damping rates of excitations in the finite-temperature condensate.

A major obstacle for the experimental realization of a supersolid phase with cold atomic gases in an optical lattice is the weakness of the nearest-neighbor interactions achievable via magnetic dipole-dipole interactions. In this paper, we show that by using a large filling of atoms within each well, the characteristic energy scales are strongly enhanced. Within this regime, the system is well described by the rotor model, and the qualitative behavior of the phase diagram derives from mean-field theory. We find a stable supersolid phase for realistic parameters with chromium atoms.

The classical-field formalism has been widely applied in the calculation of normal correlation functions, and the characterization of condensation, in finite-temperature Bose gases. Here we discuss the extension of this method to the calculation of more general correlations, including the so-called anomalous correlations of the field, without recourse to symmetry-breaking assumptions. Our method is based on the introduction of $\mathrm{U}\left(1\right)$-symmetric classical-field variables analogous to the modified quantum ladder operators of number-conserving approaches to the degenerate Bose gas, and allows us to rigorously quantify the anomalous and non-Gaussian character of the field fluctuations. We compare our results for anomalous correlation functions with the predictions of mean-field theories, and demonstrate that the nonlinear classical-field dynamics incorporate a full description of many-body processes which modify the effective mean-field potentials experienced by condensate and noncondensate atoms. We discuss the role of these processes in shaping the condensate mode, and thereby demonstrate the consistency of the Penrose-Onsager definition of the condensate orbital in the classical-field equilibrium. We consider the contribution of various noncondensate-field correlations to the overall suppression of density fluctuations and interactions in the field, and demonstrate the distinct roles of phase and density fluctuations in the transition of the field to the normal phase.

A two-component Fermi gas with attractive $s$-wave interactions forms a superfluid at low temperatures. When this gas is confined in a rotating trap, fermions can unpair at the edges of the gas and vortices can arise beyond certain critical rotation frequencies. We compute these critical rotation frequencies and construct the phase diagram in the plane of scattering length and rotation frequency for different total numbers of particles. We work at zero temperature and consider a cylindrically symmetric harmonic trapping potential. The calculations are performed in the Hartree-Fock-Bogoliubov approximation which implies that our results are quantitatively reliable for weak interactions.

Equilibrium vortex formation in rotating binary Bose gases with a rotating frequency higher than the harmonic trapping frequency is investigated theoretically. We consider the system being evaporatively cooled to form condensates and a combined numerical scheme is applied to ensure the binary system being in an authentic equilibrium state. To keep the system stable against the large centrifugal force of ultrafast rotation, a quartic trapping potential is added to the existing harmonic part. Using the Thomas-Fermi approximation, a critical rotating frequency ${\Omega }_c$ is derived, which characterizes the structure with or without a central density hole. Vortex structures are studied in detail with rotation frequency both above and below ${\Omega }_c$ and with respect to the miscible, symmetrically separated, and asymmetrically separated phases in their nonrotating ground-state counterparts.

Investigations of the quantum interference in laser-assisted photoionization by an attosecond extreme ultraviolet (xuv) pulse shows an approximately constant value for the total photoionizations for different laser intensities. The square of the full width at half maximum of a photoelectron energy spectrum (PES) linearly depends on the laser intensity. By determining the laser-related phase of each streaked electron and using a transfer equation with linear corrections, an analytically quick method is proposed for precisely reconstructing the xuv pulse intensity (chirp) from one (two) measured PES(s) with a theoretical root-mean-square temporal (energy) difference of less than 1 attosecond (0.1 eV).

We calculate the radiative characteristics of emission from a system of entangled atoms which can have a relative distance larger than the emission wavelength. We develop a quantum multipath interference approach which explains both super- and subradiance though the entangled states have zero dipole moment. We derive a formula for the radiated intensity in terms of different interfering pathways. We further show how the interferences lead to directional emission from atoms prepared in symmetric $W$ states. As a byproduct of our work we show how Dicke’s classic result can be understood in terms of interfering pathways. In contrast to the previous works on ensembles of atoms, we focus on finite numbers of atoms prepared in well characterized states.

High-order harmonic generation by few-cycle 800 nm laser pulses in neon gas in the presence of a strong terahertz (THz) field is investigated numerically with propagation effects taken into account. Our calculations show that the combination of THz fields with up to 12 fs laser pulses can be an effective gating technique to generate single attosecond pulses. We show that in the presence of the strong THz field only a single attosecond burst can be phase matched, whereas radiation emitted during other half cycles disappears during propagation. The cutoff is extended and a wide supercontinuum appears in the near-field spectra, extending the available spectral width for isolated attosecond pulse generation from 23 to 93 eV. We demonstrate that phase-matching effects are responsible for the generation of isolated attosecond pulses, even in conditions when single-atom response yields an attosecond pulse train.