We demonstrate how the inverse scattering problem of a quantum star graph can be solved by means of diagonalization of the Hermitian unitary matrix when the vertex coupling is of the scale-invariant (or Fülöp-Tsutsui) form. This enables the construction of quantum graphs with desired properties in a tailor-made fashion. The procedure is illustrated on the example of quantum vertices with equal transmission probabilities.

Fractional-photon-assisted tunneling is investigated both analytically and numerically for few interacting ultracold atoms in the double wells of an optical superlattice. This can be realized experimentally by adding periodic shaking to an existing experimental setup [Cheinet et al., Phys. Rev. Lett. 101, 090404 (2008)]. Photon-assisted tunneling is visible in the particle transfer between the wells of the individual double wells. In order to understand the physics of the photon-assisted tunneling, an effective model based on the rotating-wave approximation is introduced. The validity of this effective approach is tested for wide parameter ranges that are accessible to experiments in double-well lattices. The effective model goes well beyond previous perturbation theory approaches and is useful for investigating in particular the fractional-photon-assisted tunneling resonances. Analytic results on the level of the experimentally realizable two-particle quantum dynamics show very good agreement with the numerical solution of the time-dependent Schrödinger equation. Far from being a small effect, both the one-half-photon and the one-third-photon resonances are shown to have large effects on the particle transfer.

We present a method utilizing the continuity equation for the condensate density to make predictions of the precessional frequency of single off-axis vortices and of vortex arrays in Bose-Einstein condensates at finite temperature. We also present an orthogonalized Hartree-Fock-Bogoliubov (HFB) formalism. We solve the continuity equation for the condensate density self-consistently with the orthogonalized HFB equations and find stationary solutions in the frame rotating at this frequency. As an example of the utility of this formalism we obtain time-independent solutions for quasi-two-dimensional rotating systems in the corotating frame. We compare these results with time-dependent predictions where we simulate stirring of the condensate.

We investigate the quantum fluctuations of a single atom in a weakly driven cavity where the center-of-mass motion of the atom is quantized in one dimension. We present analytic results for the second-order intensity correlation function $g^{(2)}(\tau )$ and the intensity-field correlation function $h_{\theta }(\tau )$ for both transmitted and fluorescent light for weak driving fields. We find that the coupling of the center-of-mass motion to the intracavity field mode can be deleterious to nonclassical effects in photon statistics, but less so for the intensity-field correlations.

The second-harmonic generation of cylindrical electromagnetic waves in a nonlinear nondispersive medium is investigated two separate ways. One method uses the exact solutions of the Maxwell equations, which describe propagation of cylindrical electromagnetic waves, and the other method uses the traditional coupled-wave equations, which are derived from the interaction between cylindrical electromagnetic waves and a nonlinear nondispersive medium. The results obtained by these two methods are concordant with each other. We also show that both methods are useful in dealing with the problem of cylindrical second-harmonic generation and each of them has particular advantages in several aspects. The method of using the exact solution can solve problems of second-harmonic generation under initial-value and boundary-value conditions and the approach of using coupled-wave equations can be directly extended to describe other nonlinear phenomena such as sum-frequency and four-wave mixing.

The theory of a quantum-limited phase-matching effect in a $\Lambda$-type lasing system is studied in detail based on the quantum Langevin approach. Two quasimonochromatic fields are directly generated based on two lasing transitions. We find that the coherence between two lasers can well exceed the linewidth of either laser field. This result denotes that two field phases match each other although either laser field has a high phase fluctuation. Unlike the phase-matching effect based on atomic absorption, the final coherence between two laser fields here is not limited by saturation broadening, and the higher laser intensities lead to a higher coherence. Additionally, based on a linear stability analysis, we find that the instability of the field steady state can substantially restrict the occurrence of this phase-matching effect in the bad-cavity limit for a high pump rate. We also discuss the spectrum of amplitude fluctuations of output fields, and the result shows that the squeezing of amplitude fluctuations at low frequencies for a single field oscillating inside the cavity is damaged in the case of two fields oscillating.

A Bethe-Salpeter equation for the three-particle correlation function is derived, representing a second-order response generalization of the usual Bethe-Salpeter equation. The equation can be solved formally, giving the three-particle correlation function in terms of the two-particle correlation function and many-body interaction kernels. The similarity to second-order response time-dependent density-functional theory can be used to improve the understanding of the higher-order exchange and correlation kernels of this theory. An exact expression for such a kernel is derived.

We present a scheme to prepare quantum correlated states of two mechanical systems based on the pouring of preavailable all-optical entanglement into the state of two micromirrors belonging to remote and noninteracting optomechanical cavities. We show that, under realistic experimental conditions, the protocol allows for the preparation of a genuine quantum state of a composite mesoscopic system whose nonclassical features extend beyond the occurrence of entanglement. We finally discuss a way to access such mechanical correlations.

We present a derivation of the Redfield formalism for treating the dissipative dynamics of a time-dependent quantum system coupled to a classical environment. We compare such a formalism with the master equation approach where the environments are treated quantum mechanically. Focusing on a time-dependent spin-$1/2$ system we demonstrate the equivalence between both approaches by showing that they lead to the same Bloch equations and, as a consequence, to the same characteristic times $T_1$ and $T_2$ (associated with the longitudinal and transverse relaxations, respectively). These characteristic times are shown to be related to the operator-sum representation and the equivalent phenomenological-operator approach. Finally, we present a protocol to circumvent the decoherence processes due to the loss of energy (and thus, associated with $T_1$). To this end, we simply associate the time dependence of the quantum system to an easily achieved modulated frequency. A possible implementation of the protocol is also proposed in the context of nuclear magnetic resonance.

A long-lived quantum memory was developed based on light-compensated cold ${}^{87}\mathrm{Rb}$ atoms in a dipole trap. The lifetime of the quantum memory was improved by 40 folds, from 0.67 to 28 ms with the help of a compensation laser beam. Oscillations of the memory efficiency due to the transverse mode breathing of the singly excited spin wave have been clearly observed and clarified with a Monte Carlo simulation procedure. With detailed analysis of the decoherence processes of the spin wave in cold atomic ensembles, this experiment provides a benchmark for the further development of high-quality quantum memories.

Multiphoton intrashell transitions in strongly driven Li ($n=25$) Rydberg atoms are studied experimentally. Orthogonal dc electric and magnetic fields lift the degeneracy of the $n$ shell and define the eccentricity $e$ of the initial coherent elliptic states, which are formed by laser excitation and subsequent adiabatic transformation. The intrashell transitions are driven by a time-harmonic electric field linearly polarized parallel to the major axis of the ellipse. $N$-photon resonances with $N=1-9$ are studied as a function of $e$. All resonances with $N⩾3$ are suppressed at certain $e$ values in between $0$ and $1$. A similar system was analyzed by Yabuzaki et al. [Phys. Rev. A 10, 1955 (1974)] who found a simple pattern of suppressions that applies also for the present experiments. The results of these experimentally confirm that each time $N$ is increased by two, an additional suppression is observed.

The momentum distribution is one of the most important quantities which provides information about interactions in many-body systems. At the same time it is a quantity that can easily be accessed in experiments on ultracold atoms. In this paper, we consider mixtures of light- and heavy-fermionic atoms in an optical lattice described effectively by the Falicov-Kimball model. Using a Monte Carlo method, we study how different ordered density-wave phases can be detected by measurement of the momentum distribution of the light atoms. We also demonstrate that ordered phases can be seen in Bragg scattering experiments. Our results indicate that the main factor that determines the momentum distribution of the light atoms is the trap confinement. On the other hand, the pattern formed by the heavy atoms seen in the Bragg scattering experiments is very sensitive to the temperature and possibly can be used in low-temperature thermometry.

We calculate several thermodynamic quantities for repulsively interacting one-dimensional fermions. We solve the Hubbard model at both zero and finite temperatures using the Bethe-ansatz method. For arbitrary values of the chemical potential, we calculate the particle number density, the double occupancy, various compressibilities, and the entropy as a function of temperature and interaction. We find that these thermodynamic quantities show a characteristic behavior so that measurements of these quantities can be used as a detection of temperature, the metal-insulator transition, and metallic and insulating phases in the trap environment. Further, we discuss an experimental scheme to extract these thermodynamic quantities from the column density profiles. The entropy and the compressibility of the entire trapped atomic cloud also reveal characteristic features indicating whether insulating and/or metallic phases coexist in the trap.

We investigate fast rotating quasi-two-dimensional dipolar Fermi gases in the quantum Hall regime. By tuning the direction of the dipole moments with respect to the $z$ axis, the dipole-dipole interaction becomes anisotropic in the $x$-$y$ plane. For a soft confining potential we find that, as we tilt the angle of the dipole moments, the system evolves from a $\nu =1/3$ Laughlin state with dipoles being polarized along the $z$ axis to a series of ground states characterized by distinct mean total angular momentum, and finally to an anisotropic integer quantum Hall state. During the transition from the fractional regime to the integer regime, we find that the density profile of the system exhibits crystal-like structures. We map out the ground states as a function of the tilt angle and the confining potential, revealing the competition of the isotropic confining potential and both the isotropic and anisotropic components of the dipole-dipole interaction.

We propose and theoretically demonstrate an infrared two-color polarization gating scheme for generating an intense isolated attosecond pulse (IAP) in the multicycle regime. Our simulations show that an IAP can be produced using a multicycle two-color driving pulse with a duration up to 60 fs. Moreover, the carrier-envelope phase (CEP) of the driving laser is not required to be stabilized, although the IAP intensity changes with the CEP slip. Such a gating scheme significantly relaxes the requirements for driving lasers and opens the door to easily create intense IAPs with a high-power conventional multicycle laser pulse.

We study the features of superfluid and Mott-insulator states in a two-dimensional array of cavities with a two-level atom embedded in each cavity which is strongly coupled to the cavity field and immersed in a bosonic bath. Employing a different quasiboson approach, we show analytically how the dissipation and decoherence influence the quantum phase transitions from two aspects and find a localized tendency. For the superfluid state, a dynamical instability will lead to a sweeping to a localized state of photons. For the Mott-insulator state, a dissipation-induced fluctuation will suppress the restoring of long-range phase coherence driven by interaction.

A single complex molecule with conformational changes (conformations 0 and 2) is considered. When such a molecule is irradiated by cw-laser light it can randomly change the intensity or polarization of its fluorescence due to jumps from one conformation to another. In fact, the molecule manifests itself either like the 0-type or the 2-type emitter. An expression for the matrix $s_{\alpha \beta }(t)$ called the start-stop correlator (waiting time distribution) in which $\alpha =0,2$ and $\beta =0,2$ is derived. An expression for the matrix $p_{\alpha \beta }(t)$ called the full correlator is derived as well. It determines the density of the probability of finding an event of $\alpha$ type and an event of $\beta$ type separated by time interval $t$. A relation between matrices $s_{\alpha \beta }(t)$ and $p_{\alpha \beta }(t)$ is found. A mathematical expression for the distribution $w_N(T)$ of events measured in time interval $T$ is derived. It is expressed solely via the matrix $s_{\alpha \beta }(t)$. Numerical calculations of the event distribution function for various rates of intra- and interconformational jumps are carried out with the help of the formula for $w_N(T)$ and by the Monte Carlo method. Both methods of the calculation yield identical distributions. Fluctuating fluorescence intensity $I$($t$) for a bin time of 5 ms is calculated for slow and fast interconformational jumps. A relation is found between the autocorrelation function $g^{(2)}(t)$ of fluorescence measurable in experiments and the matrix $p_{\alpha \beta }(t)$ calculated theoretically.

State mapping between atoms and photons, and photon-photon interactions play an important role in scalable quantum information processing. We consider the interaction of a two-level atom with a quantized propagating pulse in free space and study the probability $P_e(t)$ of finding the atom in the excited state at any time $t$. This probability is expected to depend on (i) the quantum state of the pulse field and (ii) the overlap between the pulse and the dipole pattern of the atomic spontaneous emission. We show that the second effect is captured by a single parameter $\Lambda \in [0,8\pi /3]$, obtained by weighting the dipole pattern with the numerical aperture. Then, $P_e(t)$ can be obtained by solving time-dependent Heisenberg-Langevin equations. We provide detailed solutions for both single-photon Fock state and coherent states and for various temporal shapes of the pulses.

We analyze the three-mode correlation properties of the electromagnetic field in a optical parametric oscillator below-threshold. We employ a perturbative expansion of the Itô equations derived from the positive-P representation of the density matrix. Using the generalized Cauchy-Schwarz inequality, we investigate the genuine quantum nature of the triple correlations between the interacting fields, since in this case continuous-variable entanglement is not detected by the van Loock-Furusawa criterion [Phys. Rev. A 67, 052315 (2003)]. Although not being a necessary condition, these triple correlations are sufficient evidence of tripartite entanglement. Of course, our characterization of the quantum correlations is applicable to non-Gaussian states, which we show to be the case of the optical parametric oscillator below-threshold, provided nonlinear quantum fluctuations are properly taken into account.

We study theoretically the dynamics of two-component Bose-Einstein condensates in two dimensions, which admit topological excitations related to the skyrmions of nuclear physics. We investigate a branch of uniformly propagating solitary waves, which, at high momentum, can be viewed as skyrmion-antiskyrmion pairs. We study these solitary waves for a range of interaction regimes and show that, for experimentally relevant cases, there is a transition to spatially extended spin-wave states at low momentum. We explain how this can be understood by analogy to the two-dimensional ferromagnet. Finally, we discuss how such solitary waves can be generated and studied in experiment.