The Letter contains an incorrect statement. On p. 1 it says ''It turns out that this [damping] occurs if the initial state is a paired state with a small seed gap in 0 .'' The same statement is repeated on p. 4: ''if we start from the ground state with a small nonzero in 0 , the order parameter jtj asymptotes to a constant 1 .'' This statement corresponds to the following problem. Initially the system is in the BCS ground state with gap in . At t 0 the BCS coupling constant g is suddenly changed to a new value. The equilibrium gap for the new coupling is 0 .Actually, the statement that jtj asymptotes to a constant when in 0 is inconsistent with the classification of states developed in the Letter. It resulted from an erroneous assumption that for in 0 the square of the Lax vector L 2 u has only one pair of isolated roots. A detailed analysis of the isolated roots for this case was performed in Ref. [1]. It was shown that L 2 u has two pairs of isolated roots for in < e ÿ=2 0 . This means that as long as in < e ÿ=2 0 , the order parameter jtj exhibits periodic oscillations as it does when the original state is close to the Fermi state. The situation with one pair of roots takes place in the interval e ÿ=2 in = 0 < e =2 , while for in e =2 0 isolated roots are absent [1,2]. Therefore, the classification of initial states developed in the Letter implies that for in 0 the order parameter oscillates periodically at large times.As to Eq. (1) of the Letter, it was derived for the case in which there is a single pair of isolated roots (see the text). Therefore, it describes the long time behavior for e ÿ=2 < 0 = in e =2 and should not be applied outside of this interval of 0 = in . As soon as the ratio of the gaps 0 = in exceeds e =2 5 periodic oscillations occur [1], while for 0 = in < e ÿ=2 1=5 the order parameter exponentially decays to zero [1,2]. None of our other findings are affected.[1] R. A. Barankov and L. S. Levitov, cond-mat/0603317.[2] E. A. Yuzbashyan and M. Dzero, cond-mat/0603404.

We develop a theory of a pseudogap state appearing near the superconductor-insulator (SI) transition in strongly disordered metals with an attractive interaction. We show that such an interaction combined with the fractal nature of the single-particle wave functions near the mobility edge leads to an anomalously large single-particle gap in the superconducting state near SI transition that persists and even increases in the insulating state long after the superconductivity is destroyed. We give analytic expressions for the value of the pseudogap in terms of the inverse participation ratio of the corresponding localization problem.

We analyze the dynamics of a condensate of ultra-cold atomic fermions following an abrupt change of the pairing strength. At long times, the system goes to a non-stationary steady state, which we determine exactly. The superfluid order parameter asymptotes to a constant value. We show that the order parameter vanishes when the pairing strength is decreased below a certain critical value. In this case, the steady state of the system combines properties of normal and superfluid statesthe gap and the condensate fraction vanish, while the superfluid density is nonzero.Recently, several remarkable experiments have demonstrated Cooper pairing in cold atomic Fermi gases [1][2][3][4]. Key signatures of a paired state -condensation of Cooper pairs [1,2] and the pairing gap [4] have been observed. In addition, trapped gases provide a unique tool to explore aspects of fermion pairing normally inaccessible in superconductors. One of the most exciting prospects is a study of far from equilibrium coherent dynamics of fermionic condensates [5][6][7][8][9], made possible due to the precise experimental control over interactions between atoms [10,11]. The dynamics can be initiated by quickly changing the pairing strength with external magnetic field.In the present paper, we determine the time evolution of a fermionic condensate in response to a sudden change of interaction strength. Initially, the gas is in equilibrium at zero temperature on the BCS side of the Feshbach resonance with a coupling constant g i > 0. At t = 0 the coupling is suddenly changed to a smaller value g f > 0 on the same side of the resonance, g i → g f , Fig. 1 (inset). Ground states of the system at the old, g i , and new, g f , values of the coupling are characterized by corresponding BCS gaps, ∆ i and ∆ f , respectively. We consider the case ∆ i ≥ ∆ f . It has been shown previously that following the change of coupling, the time-dependent order parameter ∆(t) asymptotes to a constant value [9], |∆(t)| → ∆ ∞ on a timescale τ ∆ = 1/∆ i . Here we evaluate ∆ ∞ in terms of ∆ i and ∆ f .We show that when the coupling is decreased below a certain critical value, ∆ ∞ vanishes, Fig. 1. On a τ ∆ timescale the system goes to a steady non-stationary state that combines properties of normal and superfluid states in a peculiar way. For example, the gap vanishes, while the superfluid density remains finite. Provided the system is continuously cooled, the BCS ground state with a gap ∆ f is reached on the energy relaxation timescale τ ǫ , which is typically much larger than τ ∆ . Experimental signatures of the novel state include the absence of the gap in rf absorbtion spectrum and zero condensate fraction after a fast projection onto the Bose-Einstein Condensation (BEC) side (see below).At times t ≪ τ ǫ , dynamics of the condensate in the weak coupling regime can be described by theTime evolution of the BCS order parameter ∆(t) following an abrupt change of the coupling constant. The coupling is changed by varying the magnetic field on the BCS side of the Feshbach reso...

We obtain a complete solution for the mean-field dynamics of the BCS paired state with a large, but finite number of Cooper pairs in the nonadiabatic regime. We show that the problem reduces to a classical integrable Hamiltonian system and derive a complete set of its integrals of motion. The condensate exhibits irregular multi-frequency oscillations ergodically exploring the part of the phase-space allowed by the conservation laws. In the thermodynamic limit however the system can asymptotically reach a steady state.The study of the dynamics of the BCS superconductors has a long history [1]. Early attempts to describe nonstationary superconductivity were based on the timedependent Ginzburg-Landau (TDGL) equation [2,3,4], which reduces the problem to the time evolution of a single collective order parameter ∆(t). The TDGL approach is valid only provided the system quickly reaches an equilibrium with the instantaneous value of ∆(t), i.e. a local equilibrium is established faster than the time scale of the order parameter variation, τ ∆ ≃ 1/∆. This requirement limits the applicability of the TDGL to special situations where pair breaking dominates, e.g. due to a large concentration of magnetic impurities. An alternative to TDGL is the Boltzmann kinetic equation [5,6] for the quasiparticle distribution function coupled to a self-consistent equation for ∆(t). This approach is justified only when external parameters change slowly on the τ ∆ time scale, so that the system can be characterized by a quasiparticle distribution.Is it possible to describe theoretically the dynamics of a BCS paired state in the nonadiabatic regime when external parameters change substantially on the τ ∆ time scale? In particular, an important question is whether, following a sudden perturbation, the condensate reaches a steady state on a τ ∆ time scale or on a much longer quasiparticle energy relaxation time scale τ ǫ . In the nonadiabatic regime both TDGL and the Boltzmann kinetic equations fail and one has to deal with the coupled coherent dynamics of individual Cooper pairs. Recent studies [15,16,17,18] of this outstanding problem were motivated by experiments on fermionic pairing in cold atomic alkali gases [7,8]. The strength of pairing interactions in these systems can be fine tuned rapidly by a magnetic field, making it easier than in metals to access the nonadiabatic regime experimentally.The main result of the present paper is an explicit general solution for the dynamics of the BCS model, which describes a spatially homogenous condensate at times t ≪ τ ǫ . We employ the usual BCS mean-field approximation, which is accurate when the number of Cooper pairs is large [10,11]. It turns out that the mean-field BCS dynamics can be formulated as a nonlinear classical Hamiltonian problem. We obtain the exact solution for all initial conditions and a complete set of integrals of motion for the mean-field BCS dynamics.In this paper we assume that the number of Cooper pairs in the system is arbitrary large, but finite. In this case the typica...

We study the non-adiabatic dynamics of a 2D p+ip superfluid following an instantaneous quantum quench of the BCS coupling constant. The model describes a topological superconductor with a non-trivial BCS (trivial BEC) phase appearing at weak (strong) coupling strengths. We extract the exact long-time asymptotics of the order parameter ∆(t) by exploiting the integrability of the classical p-wave Hamiltonian, which we establish via a Lax construction. Three different types of asymptotic behavior can occur depending upon the strength and direction of the interaction quench. We refer to these as the non-equilibrium phases {I, II, III}, characterized as follows. In phase I, the order parameter asymptotes to zero due to dephasing. In phase II, ∆ → ∆∞, a nonzero constant. Phase III is characterized by persistent oscillations of ∆(t). For quenches within phases I and II, we determine the topological character of the asymptotic states. We show that two different formulations of the bulk topological winding number, although equivalent in the BCS or BEC ground states, must be regarded as independent out of equilibrium. The first winding number Q characterizes the Anderson pseudospin texture of the initial state; we show that Q is generically conserved. For Q = 0, this leads to the prediction of a "gapless topological" state when ∆ asymptotes to zero. The presence or absence of Majorana edge modes in a sample with a boundary is encoded in the second winding number W , which is formulated in terms of the retarded Green's function. We establish that W can change following a quench across the quantum critical point. When the order parameter asymptotes to a non-zero constant, the final value of W is well-defined and quantized. We discuss the implications for the (dis)appearance of Majorana edge modes. Finally, we show that the parity of zeros in the bulk out-of-equilibrium Cooper pair distribution function constitutes a Z2valued quantum number, which is non-zero whenever W = Q. The pair distribution can in principle be measured using RF spectroscopy in an ultracold atom realization, allowing direct experimental detection of the Z2 number. This has the following interesting implication: topological information that is experimentally inaccessible in the bulk ground state can be transferred to an observable distribution function when the system is driven far from equilibrium.