Abstract:
The BCS to BEC evolution has been recently the focus of studies in superconductors and cold atomic gases. For a d-wave system, we show that a Lifshitz transition occurs at a critical particle density which separates two topologically distinct phases: a BCS-like gapless superconductor/superfluid in the higher density limit and a BEC-like fully gapped superconductor/superfluid in the lower density limit. This transition is second order according to Ehrenfest's classification, but it occurs without a change in the symmetry of the order parameter, and thus can not be classified under Landau's scheme. To illustrate the nature of the transition, we compute the compressibility and the superfluid density as functions of particle density.

Abstract:
We discuss the possibility of a quantum phase transition in ultracold spin polarized fermionic gases which exhibit a p-wave Feshbach resonace. We show that when fermionic atoms form a condensate that can be externally tuned between the BCS and BEC limits, the zero temperature compressibility and the spin susceptibility of the fermionic gas are non-analytic functions of the two-body bound state energy. This non-analyticity is due to a massive rearrangement of the momentum distribution in the ground state of the system. Furthermore, we show that the low temperature superfluid density is also non-analytic, and exibits a dramatic change in behavior when the critical value of the bound state energy is crossed.

Abstract:
We consider the possibility of quantum phase transitions in the ground state of triplet superconductors where particle density is the tunning parameter. For definiteness, we focus on the case of one band quasi-one-dimensional triplet superconductors but many of our conclusions regarding the nature of the transition are quite general. Within the functional integral formulation, we calculate the electronic compressibility and superfluid density tensor as a function of the particle density for various triplet order parameter symmetries and find that these quantities are non-analytic when a critical value of the particle density is reached.

Abstract:
We consider the possibility of topological quantum phase transitions of ultracold fermions in optical lattices, which can be studied as a function of interaction strength or atomic filling factor (density). The phase transitions are connected to the topology of the elementary excitation spectrum, and occur only for non-zero angular momentum pairing (p-wave, d-wave and f-wave), while they are absent for s-wave. We construct phase diagrams for the specific example of highly anisotropic optical lattices, where the proposed topological phase transitions are most pronounced. To characterize the existence of these topological transitions, we calculate several measurable quantities including momentum distribution, quasi-particle excitation spectrum, atomic compressibility, superfluid density, and sound velocities.

Abstract:
Current experimental results suggest that some organic quasi-one-dimensional superconductors exhibit triplet pairing symmetry. Thus, we discuss several potential triplet order parameters for the superconducting state of these systems within the functional integral formulation. We compare weak spin-orbit coupling $f_{xyz}$, $p_x$, $p_y$ and $p_z$ symmetries via several thermodynamic quantities. For each symmetry, we analyse the temperature dependences of the order parameter, condensation energy, specific heat, and superfluid density tensor.

Abstract:
We discuss the possibility of a quantum phase transition in ultra-cold spin-polarized Fermi gases which exhibit a p-wave Feshbach resonance. We show that when fermionic atoms form a condensate that can be externally tuned between the BCS and BEC limits, the zero temperature compressibility and the spin susceptibility of the fermionic gas are non-analytic functions of the two-body bound state energy. This non-analyticity is due to a massive rearrangement of the momentum distribution in the ground state of the system. Furthermore, we show that the low temperature superfluid density is also non-analytic, and exhibits a dramatic change in behavior when the critical value of the bound state energy is crossed.

Abstract:
We discuss ultra-cold Fermi gases in two dimensions, which could be realized in a strongly confining one-dimensional optical lattice. We obtain the temperature versus effective interaction phase diagram for an s-wave superfluid and show that, below a certain critical temperature T_c, spontaneous vortex-antivortex pairs appear for all coupling strengths. In addition, we show that the evolution from weak to strong coupling is smooth, and that the system forms a square vortex-antivortex lattice at a lower critical temperature T_M.

Abstract:
We describe quantum phase transitions in superconducting complex oxides which could be tuned by electrostatic charge transfer. Using a simple model for the superconductivity of a thin film or surface of a bulk copper oxide, we show that tuning the carrier density may allow the visitation of several superconducting phases with different pairing symmetries such as extended $s$- $(se)$, $d$- and $(se \pm id)$-wave. We construct a universal phase diagram for single-band superconductors with $se$- and d-wave components of the order parameter based on symmetry considerations alone. For a specific model with nearest neighbor attraction, we obtain the phase diagram in the interaction versus filling factor space showing the boundaries of the possible phases. Finally, we calculate the superfluid density and penetration depth as characteristic properties of each phase.

Abstract:
We consider the evolution of superfluid properties of a three dimensional p-wave Fermi gas from weak (BCS) to strong (BEC) coupling as a function of scattering volume. We analyse the order parameter, quasi-particle excitation spectrum, chemical potential, average Cooper pair size and the momentum distribution in the ground state ($T = 0$). We also discuss the critical temperature $T_{\rm c}$, chemical potential and number of unbound, scattering and bound fermions in the normal state ($T = T_{\rm c}$). Lastly, we derive the time-dependent Ginzburg-Landau equation for $T \approx T_{\rm c}$ and extract the Ginzburg-Landau coherence length.

Abstract:
We study decoherence effects in qubits coupled to environments that exhibit resonant frequencies in their spectral function. We model the coupling of the qubit to its environment via the Caldeira-Leggett formulation of quantum dissipation/coherence, and study the simplest example of decoherence effects in circuits with resonances such as a dc SQUID phase qubit in the presence of an isolation circuit, which is designed to enhance the coherence time. We emphasize that the spectral density of the environment is strongly dependent on the circuit design, and can be engineered to produce longer decoherence times. We begin with a general discussion of superconducting qubits such as the flux qubit, the Cooper pair box and the phase qubit and show that in these kinds of systems appropriate circuit design can greatly modify the spectral density of the environment and lead to enhancement of decoherence times. In the particular case of the phase qubit, for instance, we show that when the frequency of the qubit is at least two times larger than the resonance frequency of the environmental spectral density, the decoherence time of the qubit is a few orders of magnitude larger than that of the typical ohmic regime, where the frequency of the qubit is much smaller than the resonance frequency of the spectral density. In addition, we demonstrate that the environment does not only affect the decoherence time, but also the frequency of the transition itself, which is shifted from its environment-free value. Second, we show that when the qubit frequency is nearly the same as the resonant frequency of the environmental spectral density, an oscillatory non-Markovian decay emerges, as the qubit and its environment self-generate Rabi oscillations of characteristic time scales shorter than the decoherence time.