Absolute temperature is usually bound to be positive. Under special conditions, however, negative temperatures-in which high-energy states are more occupied than low-energy states-are also possible. Such states have been demonstrated in localized systems with finite, discrete spectra. Here, we prepared a negative temperature state for motional degrees of freedom. By tailoring the Bose-Hubbard Hamiltonian, we created an attractively interacting ensemble of ultracold bosons at negative temperature that is stable against collapse for arbitrary atom numbers. The quasimomentum distribution develops sharp peaks at the upper band edge, revealing thermal equilibrium and bosonic coherence over several lattice sites. Negative temperatures imply negative pressures and open up new parameter regimes for cold atoms, enabling fundamentally new many-body states.
Absolute temperature is usually bound to be positive. Under special conditions, however, negative temperatures-in which high-energy states are more occupied than low-energy states-are also possible. Such states have been demonstrated in localized systems with finite, discrete spectra. Here, we prepared a negative temperature state for motional degrees of freedom. By tailoring the Bose-Hubbard Hamiltonian, we created an attractively interacting ensemble of ultracold bosons at negative temperature that is stable against collapse for arbitrary atom numbers. The quasimomentum distribution develops sharp peaks at the upper band edge, revealing thermal equilibrium and bosonic coherence over several lattice sites. Negative temperatures imply negative pressures and open up new parameter regimes for cold atoms, enabling fundamentally new many-body states.
As highly tunable interacting systems, cold atoms in optical lattices are ideal to realize and observe negative absolute temperatures, T<0. We show theoretically that, by reversing the confining potential, stable superfluid condensates at finite momentum and T<0 can be created with low entropy production for attractive bosons. They may serve as "smoking gun" signatures of equilibrated T<0. For fermions, we analyze the time scales needed to equilibrate to T<0. For moderate interactions, the equilibration time is proportional to the square of the radius of the cloud and grows with increasing interaction strengths as atoms and energy are transported by diffusive processes.
Motivated by the recent experimental observation of negative absolute temperature states in systems of ultracold atomic gases in optical lattices [Braun et al., Science 339, 52 (2013)], we investigate theoretically the formation of these states. More specifically, we consider the relaxation after a sudden inversion of the external parabolic confining potential in the one-dimensional inhomogeneous Bose-Hubbard model. First, we focus on the integrable hard-core boson limit, which allows us to treat large systems and arbitrarily long times, providing convincing numerical evidence for relaxation to a generalized Gibbs ensemble at negative temperature T<0, a notion we define in this context. Second, going beyond one dimension, we demonstrate that the emergence of negative temperature states can be understood in a dual way in terms of positive temperatures, which relies on a dynamic symmetry of the Hubbard model. We complement the study by exact diagonalization simulations at finite values of the on-site interaction.
We explore the zero-temperature behavior of an assembly of bosons interacting through a zero-range, attractive potential.Because the twobody interaction admits a bound state, the many-body model is best described by a Hamiltonian that includes the coupling between atomic and molecular components.Due to the positive scattering length, the low-density collection is expected to remain stable against collapse despite the attraction between particles.Although a variational many-body analysis indicates a collapsing solution with only a molecular component to its condensate at low density, the expected atomic condensate solution can be obtained if the chemical potential is allowed to be complex valued.In addition to revealing two discrete eigenfrequencies associated with the molecular case, an expansion in small oscillations quantifies the imaginary part of the chemical potential as proportional to a coherent decay rate of the atomic condensate into a continuum of collective phonon excitations about the collapsing lower state.
For systems of interacting, ultracold spin-zero neutral bosonic atoms, harmonically trapped and subject to an optical lattice potential, we derive an Extended Bose Hubbard (EBH) model by developing a systematic expansion for the Hamiltonian of the system in powers of the lattice parameters and of a scale parameter, the lattice attenuation factor. We identify the dominant terms that need to be retained in realistic experimental conditions, up to nearest-neighbor interactions and nearest-neighbor hoppings conditioned by the on-site occupation numbers. In the mean field approximation, we determine the free energy of the system and study the phase diagram both at zero and at finite temperature. At variance with the standard on site Bose Hubbard model, the zero-temperature phase diagram of the EBH model possesses a dual structure in the Mott insulating regime. Namely, for specific ranges of the lattice parameters, a density wave phase characterizes the system at integer fillings, with domains of alternating mean occupation numbers that are the atomic counterparts of the domains of staggered magnetizations in an antiferromagnetic phase. We show as well that in the EBH model, a zero-temperature quantum phase transition to pair superfluidity is, in principle, possible, but completely suppressed at the lowest order in the lattice attenuation factor. Finally, we determine the possible occurrence of the different phases as a function of the experimentally controllable lattice parameters.
We propose that negative absolute temperatures in ultracold atomic clouds in optical lattices can be used to simulate quantum systems in new regions of phase diagrams. First we discuss how the attractive SU(3) Hubbard model in three dimensions can be realized using repulsively interacting 173Yb atoms, then we consider how an antiferromagnetic S=1 spin chain could be simulated using spinor 87Rb or 23Na atoms. The general idea to achieve negative absolute temperatures is to reverse the sign of the external harmonic potential. Energy conservation in a deep optical lattice imposes a constraint on the dynamics of the cloud, which can relax toward a T<0 state. As the process is strongly nonadiabatic, we estimate the change of the entropy.
Transport properties and nonequilibrium dynamics in strongly correlated materials are \ntypically difficult to calculate. This holds true even for minimalistic model Hamiltonians \nof these systems, such as the fermionic Hubbard model. \nUltracold atoms in optical lattices enable an alternative realization of the Hubbard \nmodel and have the advantage of being free of additional complications such as phonons, \nlattice defects or impurities. This way, cold atoms can be used as quantum simulators of \nstrongly interacting materials. Being thermally isolated systems, however, we show that \ncold atoms in optical lattices can also behave very differently from solids and can show a \nplethora of novel dynamic effects. \nIn this thesis, several out-of equilibrium processes involving interacting fermionic atoms \nin optical lattices are presented. We first analyze the expansion dynamics of an initially \nconfined atomic cloud in the lowest band of an optical lattice. While non-interacting atoms \nexpand ballistically, the cloud expands with a dramatically reduced velocity in the presence \nof interactions. Most prominently, the expansion velocity is independent of the attractive \nor repulsive character of the interactions, highlighting a novel dynamic symmetry of the \nHubbard model. \nIn a second project, we discuss the possibility of realizing negative absolute temperatures in optical lattices. Negative absolute temperatures characterize equilibrium states \nwith an inverted occupation of energy levels. Here, we propose a dynamical process to realize equilibrated Fermions at negative temperatures and analyze the time scales of global \nrelaxation to equilibrium, which are associated with a redistribution of energy and particles \nby slow diffusive processes. \nWe show that energy conservation has a major impact on the dynamics of an interacting \ncloud in an optical lattice, which is exposed to an additional weak linear (gravitational) \npotential. Instead of ‘falling downwards‘, the cloud diffuses symmetrically upwards and \ndownwards in the gravitational potential. Furthermore, we show analytically that the \nradius R grows with the time t according to R ∼ t^1/3, consistent with numerical simulations \nof the Boltzmann equation. \nFinally, we analyze the damping of Bloch oscillations by interactions. For a homogeneous system, we discuss the possibility of mapping the dynamics of the particle current to \na classical damped harmonic oscillator equation, thereby giving an analytic explanation for \nthe transition from weakly damped to over-damped Bloch oscillations. We show that the \ndynamics of a strongly Bloch oscillating and weakly interacting atomic cloud can be discribed in terms of a novel effective “stroboscopic” diffusion equation. In this approximation, \nthe cloud’s radius R grows asymptotically in time t according to R ∼ t^1/5 .