It is shown that the Greenberger-Horne-Zeilinger theorem can be generalized to the case with only two entangled particles. The reasoning makes use of two photons which are maximally entangled both in polarization and in spatial degrees of freedom. In contrast to Cabello’s argument of “all versus nothing” nonlocality with four photons [Phys. Rev. Lett. 87, 010403 (2001)], our proposal to test the theorem can be implemented with linear optics and thus is well within the reach of current experimental technology.

We calculate the geometric phase associated with the evolution of a system subjected to decoherence through a quantum-jump approach. The method is general and can be applied to many different physical systems. As examples, two main sources of decoherence are considered: dephasing and spontaneous decay. We show that the geometric phase is completely insensitive to the former, i.e., it is independent of the number of jumps determined by the dephasing operator.

We describe analytical and numerical methods for calculating forces between conductors due to variations of electrostatic surface potential across their surfaces. In the simple case where the spatial variation of surface potential gives rise to uniform power spectra, we show that the electrostatic force can be large in comparison with, and scale in approximately the same way with distance of closest approach as, the Casimir force. Patch potentials that are consistent with existing experimental data could give rise to forces with a magnitude of 4% of the Casimir force at separations of $0.1\mu \mathrm{m}$.

The difference of the thermal Casimir forces at different temperatures between real metals is shown to increase with a decrease of the separation distance. This opens new opportunities for the demonstration of the thermal dependence of the Casimir force. Both configurations of two parallel plates and a sphere above a plate are considered. Different approaches to the theoretical description of the thermal Casimir force are shown to lead to different measurable predictions.

We study the loss of spatial coherence in the extended wave function of fullerenes due to collisions with background gases. From the gradual suppression of quantum interference with increasing gas pressure we are able to support quantitatively both the predictions of decoherence theory and our picture of the interaction process. We thus explore the practical limits of matter wave interferometry at finite gas pressures and estimate the required experimental vacuum conditions for interferometry with even larger objects.

Over five years, we have compared the hyperfine frequencies of ${}^{133}\mathrm{C}\mathrm{s}$ and ${}^{87}\mathrm{R}\mathrm{b}$ atoms in their electronic ground state using several laser-cooled ${}^{133}\mathrm{C}\mathrm{s}$ and ${}^{87}\mathrm{R}\mathrm{b}$ atomic fountains with an accuracy of $\sim {10}^{-15}$. These measurements set a stringent upper bound to a possible fractional time variation of the ratio between the two frequencies: $\frac{d}{dt}\ln [({\nu }_{\mathrm{R}\mathrm{b}})/({\nu }_{\mathrm{C}\mathrm{s}})]=(0.2\pm 7.0)\times {10}^{-16}{\mathrm{y}\mathrm{r}}^{-1}$ ($1\sigma$ uncertainty). The same limit applies to a possible variation of the quantity $({\mu }_{\mathrm{R}\mathrm{b}}/{\mu }_{\mathrm{C}\mathrm{s}}){\alpha }^{-0.44}$, which involves the ratio of nuclear magnetic moments and the fine structure constant.

Over a two-year duration, we have compared the frequency of the ${}^{199}\mathrm{H}\mathrm{g}^{+}$ $5d^{10}6s{}^{2}S_{1/2}(F=0)\leftrightarrow 5d^{9}6s^2{}^{2}D_{5/2}(F=2)$ electric-quadrupole transition at 282 nm with the frequency of the ground-state hyperfine splitting in neutral ${}^{133}\mathrm{C}\mathrm{s}$. These measurements show that any fractional time variation of the ratio ${\nu }_{\mathrm{C}\mathrm{s}}/{\nu }_{\mathrm{H}\mathrm{g}}$ between the two frequencies is smaller than $\pm 7\times {10}^{-15}{\mathrm{y}\mathrm{r}}^{-1}$ ($1\sigma$ uncertainty). According to recent atomic structure calculations, this sets an upper limit to a possible fractional time variation of $g_{\mathrm{C}\mathrm{s}}(m_e/m_p){\alpha }^{6.0}$ at the same level.

We propose a scheme to observe strongly correlated fermionic phenomena with bosonic atoms in optical lattices. For different values of the sign and strength of the scattering lengths, it is possible to reach either a “superconducting” regime, where the system exhibits atomic pairing, or a Luttinger liquid behavior. We identify the range of parameters where these phenomena appear, illustrate our predictions with numerical calculations, and show how to detect the presence of pairing.

We propose a method to probe states in the Mott insulator regime produced from a condensate in an optical lattice. We consider a system in which we create time-dependent number fluctuations in a given site by turning off the atomic interactions and lowering the potential barriers on a nearly pure Mott state to allow the atoms to tunnel between sites. We calculate the expected interference pattern and number fluctuations from such a system and show that one can potentially observe a deviation from a pure Mott state. We also discuss a method in which to detect these number fluctuations using time-of-flight imaging.

We discuss the quantum phases and their diffusion in a spinor-1 atomic Bose-Einstein condensate. For ferromagnetic interactions, we obtain the exact ground state distribution of the phase fluctuations corresponding to the total atom number ($N$), the magnetization ($\mathcal{M}$), and the alignment (or hypercharge) ($Y$) of the system. The mean-field ground state is shown to be stable against these fluctuations, which dynamically recover the two continuous symmetries associated with the conservation of $N$ and $\mathcal{M}$ as in current experiments.

We study low-lying collective modes of an elongated ${}^{87}\mathrm{R}\mathrm{b}$ condensate produced in a 3D magnetic harmonic trap with the addition of a 1D periodic potential which is provided by a laser standing wave along the axial direction. While the transverse breathing mode remains unperturbed, quadrupole and dipole oscillations along the optical lattice are strongly modified. Precise measurements of the collective mode frequencies at different heights of the optical barriers provide a stringent test of the theoretical model recently introduced [M. Krämer et al., Phys. Rev. Lett. 88, 180404 (2002)].

We show that, in the Thomas-Fermi regime, the cores of vortices in rotating dilute Bose-Einstein condensates adjust in radius as the rotation velocity, $\Omega$, grows, thus precluding a phase transition associated with core overlap at high vortex density. In both a harmonic trap and a rotating hard-walled bucket, the core size approaches a limiting fraction of the intervortex spacing. At large rotation speeds, a system confined in a bucket develops, within Thomas-Fermi, a hole along the rotation axis, and eventually makes a transition to a giant vortex state with all the vorticity contained in the hole.

Coreless vortices were phase imprinted in a spinor Bose-Einstein condensate. The three-component order parameter of $F=1$ sodium condensates held in a Ioffe-Pritchard magnetic trap was manipulated by adiabatically reducing the magnetic bias field along the trap axis to zero. This distributed the condensate population across its three spin states and created a spin texture. Each spin state acquired a different phase winding which caused the spin components to separate radially.

We study the propagation of an atom laser beam through a spatial region with a magnetic field tuned around a Feshbach resonance. Magnetic fields below the resonance produce an effective focusing Kerr medium that causes a modulational instability of the atomic beam. Under appropriate circumstances, this results in beam breakup and filamentation seeded by quasiparticle fluctuations and in the generation of correlated atomic pairs.

We discuss the tunneling of phonon excitations across a potential barrier separating two condensates. It is shown that a strong barrier proves to be transparent for the excitations at low energy $\epsilon$. Moreover, the transmission is reduced with increasing $\epsilon$ in contrast to the standard dependence. This anomalous behavior is due to the existence of a quasiresonance interaction. The origin of this interaction is a result of the formation of a special well determined by the density distribution of the condensate in the vicinity of a high barrier.

An atomic gas subject to a commensurate periodic potential generated by an optical lattice undergoes a superfluid-Mott insulator transition. Confining a strongly interacting gas to one dimension generates an instability where an arbitrary weak potential is sufficient to pin the atoms into the Mott state; here, we derive the corresponding phase diagram. The commensurate pinned state may be detected via its finite excitation gap and the Bragg peaks in the static structure factor.

Cortical neurons in vivo show fluctuations in their membrane potential of the order of several millivolts. Using simple and biophysically realistic models of a single neuron we demonstrate that noise induced fluctuations can be used to adaptively optimize the sensitivity of the neuron’s output to ensembles of subthreshold inputs of different average strengths. Optimal information transfer is achieved by changing the strength of the noise such that the neuron’s average firing rate remains constant. Adaptation is fast, because only crude estimates of the output rate are required at any time.

We present a generalization of the classical Wang-Landau algorithm [Phys. Rev. Lett. 86, 2050 (2001)] to quantum systems. The algorithm proceeds by stochastically evaluating the coefficients of a high temperature series expansion or a finite temperature perturbation expansion to arbitrary order. Similar to their classical counterpart, the algorithms are efficient at thermal and quantum phase transitions, greatly reducing the tunneling problem at first order phase transitions, and allow the direct calculation of the free energy and entropy.

We introduce the concept of ring dark solitons in Bose-Einstein condensates. We show that relatively shallow rings are not subject to the snake instability, but a deeper ring splits into a robust ringlike cluster of vortex pairs, which performs oscillations in the radial and azimuthal directions, following the dynamics of the original ring soliton.

I introduce environment-assisted invariance or envariance—a symmetry exhibited by correlated quantum systems and related to causality—and describe how it can be used to understand the nature of ignorance and, hence, the origin and interpretation of Born’s rule for quantum probabilities.