TY - JOUR
AB - We prove that the critical temperature for the BCS gap equation is given by T c = μ ( 8\π e γ-2+ o(1)) e π/(2μa) in the low density limit μ→ 0, with γ denoting Euler's constant. The formula holds for a suitable class of interaction potentials with negative scattering length a in the absence of bound states.
AU - Hainzl, Christian
AU - Robert Seiringer
ID - 2377
IS - 2-3
JF - Letters in Mathematical Physics
TI - The BCS critical temperature for potentials with negative scattering length
VL - 84
ER -
TY - JOUR
AB - We derive a lower bound on the ground state energy of the Hubbard model for given value of the total spin. In combination with the upper bound derived previously by Giuliani (J. Math. Phys. 48:023302, [2007]), our result proves that in the low density limit the leading order correction compared to the ground state energy of a non-interacting lattice Fermi gas is given by 8πaσ uσ d , where σ u(d) denotes the density of the spin-up (down) particles, and a is the scattering length of the contact interaction potential. This result extends previous work on the corresponding continuum model to the lattice case.
AU - Robert Seiringer
AU - Yin, Jun
ID - 2378
IS - 6
JF - Journal of Statistical Physics
TI - Ground state energy of the low density hubbard model
VL - 131
ER -
TY - JOUR
AU - Frank, Rupert L
AU - Lieb, Élliott H
AU - Robert Seiringer
ID - 2379
IS - 4
JF - Journal of the American Mathematical Society
TI - Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators
VL - 21
ER -
TY - JOUR
AB - The Bardeen-Cooper-Schrieffer (BCS) functional has recently received renewed attention as a description of fermionic gases interacting with local pairwise interactions. We present here a rigorous analysis of the BCS functional for general pair interaction potentials. For both zero and positive temperature, we show that the existence of a non-trivial solution of the nonlinear BCS gap equation is equivalent to the existence of a negative eigenvalue of a certain linear operator. From this we conclude the existence of a critical temperature below which the BCS pairing wave function does not vanish identically. For attractive potentials, we prove that the critical temperature is non-zero and exponentially small in the strength of the potential.
AU - Hainzl, Christian
AU - Hamza, Eman
AU - Robert Seiringer
AU - Solovej, Jan P
ID - 2380
IS - 2
JF - Communications in Mathematical Physics
TI - The BCS functional for general pair interactions
VL - 281
ER -
TY - JOUR
AB - We determine the sharp constant in the Hardy inequality for fractional Sobolev spaces. To do so, we develop a non-linear and non-local version of the ground state representation, which even yields a remainder term. From the sharp Hardy inequality we deduce the sharp constant in a Sobolev embedding which is optimal in the Lorentz scale. In the appendix, we characterize the cases of equality in the rearrangement inequality in fractional Sobolev spaces.
AU - Frank, Rupert L
AU - Robert Seiringer
ID - 2381
IS - 12
JF - Journal of Functional Analysis
TI - Non-linear ground state representations and sharp Hardy inequalities
VL - 255
ER -
TY - JOUR
AB - We show that the Lieb-Liniger model for one-dimensional bosons with repulsive δ-function interaction can be rigorously derived via a scaling limit from a dilute three-dimensional Bose gas with arbitrary repulsive interaction potential of finite scattering length. For this purpose, we prove bounds on both the eigenvalues and corresponding eigenfunctions of three-dimensional bosons in strongly elongated traps and relate them to the corresponding quantities in the Lieb-Liniger model. In particular, if both the scattering length a and the radius r of the cylindrical trap go to zero, the Lieb-Liniger model with coupling constant g ∼ a/r 2 is derived. Our bounds are uniform in g in the whole parameter range 0 ≤ g ≤ ∞, and apply to the Hamiltonian for three-dimensional bosons in a spectral window of size ∼ r -2 above the ground state energy.
AU - Robert Seiringer
AU - Yin, Jun
ID - 2382
IS - 2
JF - Communications in Mathematical Physics
TI - The Lieb-Liniger model as a limit of dilute bosons in three dimensions
VL - 284
ER -
TY - JOUR
AB - We study the relativistic electron-positron field at positive temperature in the Hartree-Fock approximation. We consider both the case with and without exchange terms, and investigate the existence and properties of minimizers. Our approach is non-perturbative in the sense that the relevant electron subspace is determined in a self-consistent way. The present work is an extension of previous work by Hainzl, Lewin, Séré and Solovej where the case of zero temperature was considered.
AU - Hainzl, Christian
AU - Lewin, Mathieu
AU - Robert Seiringer
ID - 2383
IS - 10
JF - Reviews in Mathematical Physics
TI - A nonlinear model for relativistic electrons at positive temperature
VL - 20
ER -
TY - CONF
AB - We review our proof that in a scaling limit, the time evolution of a quantum particle in a static random environment leads to a diffusion equation. In particular, we discuss the role of Feynman graph expansions and of renormalization.
AU - László Erdös
AU - Salmhofer, Manfred
AU - Yau, Horng-Tzer
ID - 2702
TI - Feynman graphs and renormalization in quantum diffusion
ER -
TY - JOUR
AB - The field of cavity quantum electrodynamics (QED), traditionally studied in atomic systems, has gained new momentum by recent reports of quantum optical experiments with solid-state semiconducting and superconducting systems. In cavity QED, the observation of the vacuum Rabi mode splitting is used to investigate the nature of matter-light interaction at a quantum-mechanical level. However, this effect can, at least in principle, be explained classically as the normal mode splitting of two coupled linear oscillators. It has been suggested that an observation of the scaling of the resonant atom-photon coupling strength in the Jaynes-Cummings energy ladder with the square root of photon number n is sufficient to prove that the system is quantum mechanical in nature. Here we report a direct spectroscopic observation of this characteristic quantum nonlinearity. Measuring the photonic degree of freedom of the coupled system, our measurements provide unambiguous spectroscopic evidence for the quantum nature of the resonant atom-field interaction in cavity QED. We explore atom-photon superposition states involving up to two photons, using a spectroscopic pump and probe technique. The experiments have been performed in a circuit QED set-up, in which very strong coupling is realized by the large dipole coupling strength and the long coherence time of a superconducting qubit embedded in a high-quality on-chip microwave cavity. Circuit QED systems also provide a natural quantum interface between flying qubits (photons) and stationary qubits for applications in quantum information processing and communication.
AU - Johannes Fink
AU - Göppl, M
AU - Baur, Matthias P
AU - Bianchetti, R
AU - Leek, Peter J
AU - Blais, Alexandre
AU - Wallraff, Andreas
ID - 1763
IS - 7202
JF - Nature
TI - Climbing the Jaynes-Cummings ladder and observing its √n nonlinearity in a cavity QED system
VL - 454
ER -
TY - JOUR
AB - High quality on-chip microwave resonators have recently found prominent new applications in quantum optics and quantum information processing experiments with superconducting electronic circuits, a field now known as circuit quantum electrodynamics (QED). They are also used as single photon detectors and parametric amplifiers. Here we analyze the physical properties of coplanar waveguide resonators and their relation to the materials properties for use in circuit QED. We have designed and fabricated resonators with fundamental frequencies from 2 to 9 GHz and quality factors ranging from a few hundreds to a several hundred thousands controlled by appropriately designed input and output coupling capacitors. The microwave transmission spectra measured at temperatures of 20 mK are shown to be in good agreement with theoretical lumped element and distributed element transmission matrix models. In particular, the experimentally determined resonance frequencies, quality factors, and insertion losses are fully and consistently explained by the two models for all measured devices. The high level of control and flexibility in design renders these resonators ideal for storing and manipulating quantum electromagnetic fields in integrated superconducting electronic circuits.
AU - Göppl, M
AU - Fragner, A
AU - Baur, Matthias P
AU - Bianchetti, R
AU - Filipp, Stefan
AU - Johannes Fink
AU - Leek, Peter J
AU - Puebla, G
AU - Steffen, L. Kraig
AU - Wallraff, Andreas
ID - 1765
IS - 11
JF - Journal of Applied Physics
TI - Coplanar waveguide resonators for circuit quantum electrodynamics
VL - 104
ER -