Discrete breathers and negative-temperature states

Abstract

We explore the statistical behaviour of the discrete nonlinear Schrödinger equation as a test bed for the observation of negative-temperature (i.e. above infinite temperature) states in Bose-Einstein condensates in optical lattices and arrays of optical waveguides. By monitoring the microcanonical temperature, we show that there exists a parameter region where the system evolves towards a state characterized by a finite density of discrete breathers and a negative temperature. Such a state persists over very long (astronomical) times since the convergence to equilibrium becomes increasingly slower as a consequence of a coarsening process. We also discuss two possible mechanisms for the generation of negative-temperature states in experimental setups, namely, the introduction of boundary dissipations and the free expansion of wavepackets initially in equilibrium at a positive temperature. Gesellschaft optics [1]. For instance, BEC in optical lattices are ideal benchmarks to investigate the role of nonlinearity and spatial discreteness in quantum transport phenomena [2,3]. The refined experimental techniques now available [4-8] enable investigations and applications of BEC in quantum coherence, quantum control, quantum information processing and the quantumclassical correspondence [9]. In addition, BEC in optical lattices can be considered as the 'atomic analogues' of light propagating in waveguide arrays [10].The discrete nonlinear Schrödinger equation (DNLSE) is a basic semiclassical model for the study of BEC in optical lattices [11] and light propagating in arrays of optical waveguides [10]. Numerical studies have revealed the important role played by localized solutions, notably discrete breathers [11,12]. Many aspects of the relationship between breathers and transport properties in BEC have recently been reviewed in [13]. The DNLSE has also been found to exhibit unusual thermodynamic features. A first statistical-mechanics study of the DNLSE identified a region in the parameter space characterized by the spontaneous formation of breathers and conjectured it to correspond to negative-temperature (NT) states [14]. NT states have attracted the curiosity of researchers since the pioneering work in systems of quantum nuclear spins [15]. From a thermodynamic point of view, the presence of NT states implies that the system's entropy is a decreasing function of the internal energy. In a series of recent papers [16][17][18][19], Rumpf provided a convincing theoretical argument that excludes the physical occurrence of NT equilibrium states in the DNLSE. In particular, he showed that even in the region of parameter space where breathers form spontaneously via a modulational instability, the DNSLE eventually reaches a maximum entropy (equilibrium) state formed by a background at infinite temperature superposed on a single breather that collects the 'excess' energy. It was also observed that the convergence to the equilibrium state predicted by Rumpf would need transients lasting over astronomical times [20]. Therefore, ...

A consistent description of fluctuations requires negative temperatures

“…It is interesting to notice that T B < 0 (and the corresponding clusterization) is not a peculiarity of the divergence of G(r) in r = 0, nor of the long range nature of the interaction: indeed, it can be obtained with any arbitrary G(r) having a maximum (even finite) in r = 0, and vanishing at large r, provided that the domain is bounded. The presence of spatial order at high values of energy, in the form of discrete breathers, has been observed also in the discrete non-linear Schrödinger equation and analogous systems [10,31]. In Section IV we introduce a different, in a way simpler, model which still exhibits spatial order at small negative temperatures.…”

“…3.D, the wrong concept of temperature (in non-isolated (sub)-systems) depending upon the energy of the microscopic configuration, see their Eq. (31), is used to claim the inconsistency of T B . Such confusion seems to be persistent, see [29] for a discussion of the topic of the (non existing) fluctuations of temperature.…”

On the dispute between Boltzmann and Gibbs entropy

“…Within the latter class for instance there are models usually employed for describing ultracold atoms. The possibility of observing negative temperature states in ultracold systems, has been theoretically predicted by some authors with different approaches [37,38,39] and, the experimental evidence of the existence of states for motional degrees of freedom of a bosonic gas at negative (Boltzmann) temperatures, have been achieved a few years ago by Braun et al [40]. The interpretation of such experimental results has been contested in [19].…”

Boundary-Induced Instabilities in Coupled Oscillators

“…Here below, we show that a similar scenario can be observed for the DNLS equation, iż n = −2|z n | 2 z n − z n+1 − z n−1 , where z n = (p n + iq n )/ √ 2 is a complex variable. The DNLS Hamiltonian has two conserved quantities, the mass/norm a and the energy density h [29,30], so that it is a natural candidate for describing coupled transport [10,31]. We have numerically studied a DNLS chain interacting with two Langevin thermostats at T = 0 and different chemical potentials µ 1 and µ N imposed at the boundaries (see Ref.…”