On non-coercive mixed problems for parameter-dependent elliptic operators

We consider a (generally, non-coercive) mixed boundary value problem in a bounded domain $D$ of ${\mathbb R}^n$ for a second order parameter-dependent elliptic differential operator $A (x,\partial, \lambda)$ with complex-valued essentially bounded measured coefficients and complex parameter $\lambda$. The differential operator is assumed to be of divergent form in $D$, the boundary operator $B (x,\partial)$ is of Robin type with possible pseudo-differential components on $\partial D$. The boundary of $D$ is assumed to be a Lipschitz surface. Under these assumptions the pair $(A (x,\partial, \lambda),B)$ induces a holomorphic family of Fredholm operators $L(\lambda): H^+(D) \to H^- (D)$ in suitable Hilbert spaces $H^+(D)$ , $H^- (D)$ of Sobolev type. If the argument of the complex-valued multiplier of the parame\-ter in $A (x,\partial, \lambda)$ is continuous and the coefficients related to second order derivatives of the operator are smooth then we prove that the operators $L(\lambda)$ are conti\-nu\-ously invertible for all $\lambda$ with sufficiently large modulus $|\lambda|$ on each ray on the complex plane $\mathbb C$ where the differential operator $A (x,\partial, \lambda)$ is parameter-dependent elliptic. We also describe reasonable conditions for the system of root functions related to the family $L (\lambda)$ to be (doubly) complete in the spaces $H^+(D)$, $H^- (D)$ and the Lebesgue space $L^2 (D)$

Unconventional fermionic pairing states in a monochromatically tilted optical lattice

We study the one-dimensional attractive fermionic Hubbard model under the influence of periodic driving with the time-dependent density matrix renormalization group method. We show that the system can be driven into an unconventional pairing state characterized by a condensate made of Cooper pairs with a finite center-of-mass momentum similar to a Fulde-Ferrell state. We obtain results both in the laboratory and the rotating reference frames demonstrating that the momentum of the condensate can be finely tuned by changing the ratio between the amplitude and the frequency of the driving. In particular, by quenching this ratio to the value corresponding to suppression of the tunneling and the Coulomb interaction strength to zero, we are able to “freeze” the condensate. We finally study the effects of different initial conditions and compare our numerical results to those obtained from a time-independent Floquet theory in the large frequency regime. Our work offers the possibility of engineering and controlling unconventional pairing states in fermionic condensates.This work was conducted at the Center for Nanophase Materials Sciences, sponsored by the Scientific User Facilities Division (SUFD), Basic Energy Sciences (BES), U.S. Department of Energy (DOE), under contract with UT-Battelle. A.N. acknowledges support by the Center for Nanophase Materials Sciences and by the Early Career Research program, SUFD, BES, DOE. A.E.F. acknowledges the DOE, Office of Basic Energy Sciences, for support under Grant No. DE-SC0014407. A.P. was supported by NSF DMR-1506340, ARO W911NF1410540, and AFOSR FA9550-16-1-0334. (Scientific User Facilities Division (SUFD); Basic Energy Sciences (BES); U.S. Department of Energy (DOE); UT-Battelle; Center for Nanophase Materials Sciences; Early Career Research program; SUFD; BES; DOE; DE-SC0014407 - DOE, Office of Basic Energy Sciences; NSF DMR-1506340; ARO W911NF1410540; AFOSR FA9550-16-1-0334)Published versio

Breakdown of the adiabatic limit in low dimensional gapless systems

It is generally believed that a generic system can be reversibly transformed from one state into another by sufficiently slow change of parameters. A standard argument favoring this assertion is based on a possibility to expand the energy or the entropy of the system into the Taylor series in the ramp speed. Here we show that this argumentation is only valid in high enough dimensions and can break down in low-dimensional gapless systems. We identify three generic regimes of a system response to a slow ramp: (A) mean-field, (B) non-analytic, and (C) non-adiabatic. In the last regime the limits of the ramp speed going to zero and the system size going to infinity do not commute and the adiabatic process does not exist in the thermodynamic limit. We support our results by numerical simulations. Our findings can be relevant to condensed-matter, atomic physics, quantum computing, quantum optics, cosmology and others.Comment: 11 pages, 5 figures, to appear in Nature Physics (originally submitted version

Non-equilibrium coherence dynamics of a soft boson lattice

We study the non-equilibrium evolution of the phase coherence of a Bose-Einstein condensate (BEC) in a one dimensional optical lattice, as the lattice is suddenly quenched from an insulating to a superfluid state. We observe slowly damped phase coherence oscillations in the regime of large filling factor (~100 bosons per site) at a frequency proportional to the generalized Josephson frequency. The truncated Wigner approximation (TWA) predicts the frequency of the observed oscillations.Comment: 10 pages. 4 figure

Decay of super-currents in condensates in optical lattices

In this paper we discuss decay of superfluid currents in boson lattice systems due to quantum tunneling and thermal activation mechanisms. We derive asymptotic expressions for the decay rate near the critical current in two regimes, deep in the superfluid phase and close to the superfluid-Mott insulator transition. The broadening of the transition at the critical current due to these decay mechanisms is more pronounced at lower dimensions. We also find that the crossover temperature below which quantum decay dominates is experimentally accessible in most cases. Finally, we discuss the dynamics of the current decay and point out the difference between low and high currents.Comment: Contribution to the special issue of Journal of Superconductivity in honor of Michael Tinkham's 75th birthda

Effect of mesoscopic inhomogeneities on local tunnelling density of states

We carry out a theoretical analysis of the momentum dependence of the Fourier-transformed local density of states (LDOS) in the superconducting cuprates within a model considering the interference of quasiparticles scattering on quenched impurities. The impurities introduce an external scattering potential, which is either nearly local in space or it can acquire a substantial momentum dependence due to a possible strong momentum dependence of the electronic screening near a charge modulation instability. The key new effect that we introduce is an additional mesoscopic disorder aiming to reproduce the inhomogeneities experimentally observed in scanning tunnelling microscopy. The crucial effect of this mesoscopic disorder is to give rise to point-like spectroscopic features, to be contrasted with the curve-like shape of the spectra previously calculated within the interfering-quasiparticle schemes. It is also found that stripe-like charge modulations play a relevant role to correctly reproduce all the spectral features of the experiments.Comment: 11 pages and 5 figure

Dynamical Quantum Phase Transitions in the Transverse Field Ising Model

A phase transition indicates a sudden change in the properties of a large system. For temperature-driven phase transitions this is related to non-analytic behavior of the free energy density at the critical temperature: The knowledge of the free energy density in one phase is insufficient to predict the properties of the other phase. In this paper we show that a close analogue of this behavior can occur in the real time evolution of quantum systems, namely non-analytic behavior at a critical time. We denote such behavior a dynamical phase transition and explore its properties in the transverse field Ising model. Specifically, we show that the equilibrium quantum phase transition and the dynamical phase transition in this model are intimately related.Comment: 4+4 pages, 4 figures, Appendix adde

Dynamic Kosterlitz-Thouless transition in 2D Bose mixtures of ultra-cold atoms

We propose a realistic experiment to demonstrate a dynamic Kosterlitz-Thouless transition in ultra-cold atomic gases in two dimensions. With a numerical implementation of the Truncated Wigner Approximation we simulate the time evolution of several correlation functions, which can be measured via matter wave interference. We demonstrate that the relaxational dynamics is well-described by a real-time renormalization group approach, and argue that these experiments can guide the development of a theoretical framework for the understanding of critical dynamics.Comment: 5 pages, 6 figure

Superfluid-insulator transition in a moving system of interacting bosons

We analyze stability of superfluid currents in a system of strongly interacting ultra-cold atoms in an optical lattice. We show that such a system undergoes a dynamic, irreversible phase transition at a critical phase gradient that depends on the interaction strength between atoms. At commensurate filling, the phase boundary continuously interpolates between the classical modulation instability of a weakly interacting condensate and the equilibrium quantum phase transition into a Mott insulator state at which the critical current vanishes. We argue that quantum fluctuations smear the transition boundary in low dimensional systems. Finally we discuss the implications to realistic experiments.Comment: updated refernces and introduction, minor correction