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arXiv:1112.4299 · Physical Review A

Quantum simulators at negative absolute temperatures

Ákos Rapp

Published Apr 17, 2012• 10 citations
arXivHTMLPDF

Abstract

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 ${}^{173}$Yb atoms, then we consider how an antiferromagnetic $S=1$ spin chain could be simulated using spinor ${}^{87}$Rb or ${}^{23}$Na 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.

DOI
10.1103/physreva.85.043612

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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 173{}^{173}173Yb atoms, then we consider how an antiferromagnetic S=1S=1S=1 spin chain could be simulated using spinor 87{}^{87}87Rb or 23{}^{23}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<0T<0T<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 .
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.
Abstract Quantum antiferromagnets with geometrical frustration exhibit rich many-body physics but are hard to simulate by means of classical computers. Although quantum-simulation studies for analyzing such systems are thus desirable, they are still limited to high-temperature regions, where interesting quantum effects are smeared out. Here we propose a feasible protocol to perform analog quantum simulation of frustrated antiferromagnetism with strong quantum fluctuations by using ultracold Bose gases in optical lattices at negative absolute temperatures. Specifically, we show from numerical simulations that the time evolution of a negative-temperature state subjected to a slow sweep of the hopping energy simulates quantum phase transitions of a frustrated Bose–Hubbard model with sign-inverted hoppings. Moreover, we quantitatively predict the phase boundary between the frustrated superfluid and Mott-insulator phases for triangular lattices with hopping anisotropy, which serves as a benchmark for quantum simulation.
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<0T<0T<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.
Ultracold atomic gases may be the ultimate quantum simulator.These isolated systems have the lowest temperatures in the observable universe, and their properties and interactions can be precisely and accurately tuned across a full spectrum of behaviors, from few-body physics to highly-correlated many-body effects.The ability to impose potentials on and tune interactions within ultracold gases to mimic complex systems mean they could become a theorist's playground.One of their great strengths, however, is also one of the largest obstacles to this dream: isolation.This thesis touches on both of these themes.First, methods to characterize phases and quantum critical points, and to construct finite temperature phase diagrams using experimentally accessible observables in the Bose Hubbard model are discussed.Then, the transition from a weakly to a strongly interacting Bose-Fermi mixture in the continuum is analyzed using zero temperature numerical techniques.Real materials can be emulated by ultracold atomic gases loaded into optical lattice potentials.We discuss the characteristics of a single boson species trapped in an optical lattice (described by the Bose Hubbard model) and the hallmarks of the quantum critical region that separates the superfluid and the Mott insulator ground states.We propose a method to map the quantum critical region using the single, experimentally accessible, local quantity R, the ratio of compressibility to local number fluctuations.The procedure to map a phase diagram with R is easily generalized to inhomogeneous systems and generic many-body Hamiltonians.We illustrate it here using quantum Monte Carlo simulations of the 2D Bose Hubbard model.Secondly, we investigate the transition from a degenerate Fermi gas weakly coupled List of Tables Table Page 2.1 Techniques to Probe Atoms in Optical Lattices . . . . . .
Ultracold fermionic atoms in optical lattices offer pristine realizations of Hubbard models1, which are fundamental to modern condensed-matter physics2,3. Despite notable advancements4–6, the accessible temperatures in these optical lattice material analogues are still too high to address many open problems7–10. Here we demonstrate a several-fold reduction in temperature6,11–13, bringing large-scale quantum simulations of the Hubbard model into an entirely new regime. This is accomplished by transforming a low-entropy product state into strongly correlated states of interest via dynamic control of the model parameters14,15, which is extremely challenging to simulate classically10. At half-filling, the long-range antiferromagnetic order is close to saturation, leading to a temperature of T/t=0.05−0.05+0.06T/t=0.0{5}_{-0.05}^{+0.06}T/t=0.05−0.05+0.06​ based on comparisons with numerically exact simulations. Doped away from half-filling, it is exceedingly challenging to realize systematically accurate and predictive numerical simulations9. Importantly, we are able to use quantum simulation to identify a new pathway for achieving similarly low temperatures with doping. This is confirmed by comparing short-range spin correlations to state-of-the-art, but approximate, constrained-path auxiliary-field quantum Monte Carlo simulations16–18. Compared with the cuprates2,19,20, the reported temperatures correspond to a reduction from far above to below room temperature, at which physics such as the pseudogap and stripe phases may be expected3,19,21–24. Our work opens the door to quantum simulations that solve open questions in material science, develop synergies with numerical methods and theoretical studies, and lead to discoveries of new physics8,10. A several-fold reduction in temperature is accomplished using a neutral-atom Hubbard quantum simulator by transforming a low-entropy product state into strongly correlated states of interest via dynamic control of the model parameters.