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A quantum theory of cooling of a mechanical oscillator by radiation pressure-induced dynamical back-action is developed, which is analogous to sideband cooling of trapped ions. We find that final occupancies well below unity can be attained when the mechanical oscillation frequency is larger than the cavity linewidth. It is shown that the final average occupancy can be retrieved directly from the optical output spectrum.PACS numbers: 42.65. Sf ,42.65.Ky, 42.79.Gn Mesoscopic mechanical oscillators are currently attracting interest due to their potential to enhance the sensitivity of displacement measurements [1] and to probe the quantum to classical transition of a macroscopic degree of freedom [2,3]. A prerequisite for these applications is the capability of initializing an oscillator with a long phonon lifetime in its quantum ground state. So far this has not been demonstrated because the combination of sufficiently high mechanical frequencies (ω m /2π) and quality factors in the relevant regime ω m ≫ k B T has not been reached [3]. In contrast, in atomic physics laser cooling has enabled the preparation of motional ground states [4,5]. This has prompted researchers to study means of cooling a single mechanical resonator mode directly using laser radiation. Early work demonstrated cooling of a mechanical degree of freedom of a Fabry-Pérot mirror using a radiation pressure force controlled by an electronic feedback scheme [6,7], in analogy to stochastic cooling. In contrast, the radiation pressure induced coupling of an optical cavity mode to a mechanical oscillator [cf. Fig. 1(a)] can give rise to self-cooling via dynamical back-action [8]. In essence, the cavity delay induces correlations between the radiation pressure force and the thermal Brownian motion that lead to cooling or amplification, depending on the laser detuning. In a series of recent experiments, these effects have been used to cool a single mechanical mode [9,10,11]. While classical and semiclassical analysis of dynamical back-action have been developed [13,14], the question as to whether ground state cooling is possible has not been addressed.Here a quantum theory of cooling via dynamical backaction is presented. We find that final occupancies below unity can indeed be attained when the optical cavity's lifetime is comparable to or exceeds the mechanical oscillation period. Along these lines, an analogy between this mechanism and the sideband cooling of trapped ions in the Lamb-Dicke regime is elucidated [5]. In our setting the optical cavity mode plays the role of the ion's pseudospin mediating the frequency up-conversion underlying the cooling cycle. Finally, we discuss how the average phonon occupancy can be retrieved from the spectrum of the optical cavity output. We note that these results can be applied to a wide range of experimental realizations of cavity self-cooling [9,11,12].We treat the laser driven optical cavity mode coupled to the mechanical resonator mode as an open quantum system and adopt a rotating frame at the laser freq...

We present a self-consistent theory for the thermodynamics of the BCS-BEC crossover in the normal and superfluid phase which is both conserving and gapless. It is based on the variational many-body formalism developed by Luttinger and Ward and by DeDominicis and Martin. Truncating the exact functional for the entropy to that obtained within a ladder approximation, the resulting self-consistent integral equations for the normal and anomalous Green functions are solved numerically for arbitrary coupling. The critical temperature, the equation of state and the entropy are determined as a function of the dimensionless parameter 1/kF a, which controls the crossover from the BCS-regime of extended pairs to the BEC-regime of tightly bound molecules. The tightly bound pairs turn out to be described by a Popov-type approximation for a dilute, repulsive Bose gas. Even though our approximation does not capture the critical behaviour near the continuous superfluid transition, our results provide a consistent picture for the complete crossover thermodynamics which compare well with recent numerical and field-theoretic approaches at the unitarity point.

We discuss the superfluid to Mott-insulator transition of cold atoms in optical lattices recently observed by Greiner et.al. (Nature 415, 39 (2002)). The fundamental properties of both phases and their experimental signatures are discussed carefully, including the limitations of the standard Gutzwiller-approximation. It is shown that in a one-dimensional dilute Bose-gas with a strong transverse confinement (Tonks-gas), even an arbitrary weak optical lattice is able to induce a Mott like state with crystalline order, provided the dimensionless interaction parameter is larger than a critical value of order one. The superfluid-insulator transition of the Bose-Hubbard model in this case continuously evolves into a transition of the commensurateincommensurate type with decreasing strength of the external optical lattice.

We compute the shear viscosity of the unitary Fermi gas above the superfluid transition temperature, using a diagrammatic technique that starts from the exact Kubo formula. The formalism obeys a Ward identity associated with scale invariance which guarantees that the bulk viscosity vanishes identically. For the shear viscosity, vertex corrections and the associated Aslamazov-Larkin contributions are shown to be crucial to reproduce the full Boltzmann equation result in the high-temperature, low fugacity limit. The frequency dependent shear viscosity η(ω) exhibits a Drude-like transport peak and a power-law tail at large frequencies which is proportional to the Tan contact. The weight in the transport peak is given by the equilibrium pressure, in agreement with a sum rule due to Taylor and Randeria. Near the superfluid transition the peak width is of the order of 0.5 T F , thus invalidating a quasiparticle description. The ratio η/s between the static shear viscosity and the entropy density exhibits a minimum near the superfluid transition temperature whose value is larger than the string theory bound /(4πk B ) by a factor of about seven.

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We realize and study a strongly interacting two-component atomic Fermi gas confined to two dimensions in an optical lattice. Using radio-frequency spectroscopy we measure the interaction energy of the strongly interacting gas. We observe the confinement-induced Feshbach resonance on the attractive side of the 3D Feshbach resonance and find the existence of confinement-induced molecules in very good agreement with theoretical predictions. Two-dimensional Fermi gases play a pivotal role in quantum many-body physics. The restriction of particle motion to a plane profoundly increases the role of fluctuations and leads to qualitatively new effects in the interparticle interaction [1][2][3][4][5][6][7]. In the solid state context, strongly interacting two-dimensional Fermi gases are found in the cuprates, the two-dimensional electron gas in nanostructures, and in thin 3 He films. With the advent of ultracold atomic Fermi gases [8] and the ability to confine them to two-dimensional configurations [9][10][11][12][13], research has revived because tunable and ultraclean samples have become available. The direct experimental access to microscopic parameters, such as the particle interaction, promotes ultracold two-dimensional Fermi gases as quantum simulators of fundamental many-body effects.A quantum system is kinematically two-dimensional if the chemical potential and the thermal energy are smaller than the energy gap ω to the first excited state in the strongly confined direction. For harmonic confinement, the motion of particles is restricted to the quantum mechanical ground state with an extension l 0 = /mω, in which m is the mass of the particles. This new length scale l 0 competes with the three-dimensional s-wave scattering length a. As a result, two-dimensional gases display features not encountered in their three-dimensional counterparts. Beyond the absence of a true condensate at finite temperature there is also no scale invariant regime similar to the unitary gas at infinite scattering length. This has to do with the peculiar features of two-body scattering. Specifically, the amplitude of the outgoing cylindrical wave for low energy scattering with relative momentum q in two dimensions is of the form [3,4,7] f (q) = 4π ln(1/q 2 a 2 2D ) + iπ ,which defines the 2D scattering length a 2D . The logarithmic dependence on momentum shows that f (q) is never independent of energy and indicates that the dimensionless interaction strength 1/ ln(k F a 2D ) depends logarithmically on the Fermi wave vector [14,15]. In particular, the weak coupling limit of a Fermi gas, whose interactions can be described in a mean-field picture, is reached at high densities. Two-dimensional confinement also stabilizes a bound dimer state, the presence of which is both a necessary and also a sufficient criterion for the existence of an swave pairing instability in 2D [1]. For weak attractive interactions, with a negative scattering length and |a| < l 0 , the dimer binding energy is predicted to be [7] The associated size of the dimer is re...