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arXiv:0902.1017 · Physical Review A

Lattice thermodynamics for ultracold atoms

David C. McKay, Matthew White, Brian DeMarco

Published Jun 3, 2009• 49 citations
arXivHTMLPDF

Abstract

We measure the temperature of ultracold $^{87}\text{R}\text{b}$ gases transferred into an optical lattice and compare to noninteracting thermodynamics for a combined lattice-parabolic potential. Absolute temperature is determined at low temperature by fitting quasimomentum distributions obtained using band mapping, i.e., turning off the lattice potential slowly compared with the band gap. We show that distributions obtained at high temperature employing this technique are not quasimomentum distributions through numerical simulations. To overcome this limitation, we extract temperature using the in-trap size of the gas.

DOI
10.1103/physreva.79.063605

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We measure the temperature of ultracold 87Rb^{87}\text{R}\text{b}87Rb gases transferred into an optical lattice and compare to noninteracting thermodynamics for a combined lattice-parabolic potential. Absolute temperature is determined at low temperature by fitting quasimomentum distributions obtained using band mapping, i.e., turning off the lattice potential slowly compared with the band gap. We show that distributions obtained at high temperature employing this technique are not quasimomentum distributions through numerical simulations. To overcome this limitation, we extract temperature using the in-trap size of the gas.
We theoretically consider noninteracting fermions confined to optical lattices and apply a lattice amplitude modulation that we choose to be either homogeneous or of superlattice geometry. We study the atom excitation rate to higher Bloch bands which can be measured by adiabatic band mapping. We find that the atom excitation rate shows a clear signature of the temperature-dependent Fermi distribution in the lowest band of the equilibrium lattice as excitations are quasimomentum resolved. Based on typical experimental parameters and incorporating a trapping potential, we find that thermometry of one- and two-dimensional systems is within the reach of present-day experiments. Our scheme is valid down to temperatures of a few percent of the hopping amplitude comparable to the N\'eel temperature in interacting systems.
In the quest to reach lower temperatures of ultracold gases in optical-lattice experiments, nonadiabaticities during lattice loading represent one of the limiting factors that prevent the same low temperatures being reached as in experiments without lattices. Simulating the loading of a bosonic quantum gas into a one-dimensional optical lattice with and without a trap, we find that the redistribution of atomic density inside a global confining potential is by far the dominant source of heating. Based on these results we propose adjusting the trapping potential during loading to minimize changes to the density distribution. Our simulations confirm that a very simple linear interpolation of the trapping potential during loading already significantly decreases the heating of a quantum gas, and we discuss how loading protocols minimizing density redistributions can be designed.
Abstract. We describe a powerful method for determining the equation of state of an ultracold gas from in situ images. The method provides a measurement of the local pressure of an harmonically trapped gas and we give several applications to Bose and Fermi gases. We obtain the grand-canonical equation of state of a spin-balanced Fermi gas with resonant interactions as a function of temperature [1]. We compare our equation of state with an equation of state measured by the Tokyo group in [2], that reveals a significant difference in the high-temperature regime. The normal phase, at low temperature, is well described by a Landau Fermi liquid model, and we observe a clear thermodynamic signature of the superfluid transition. In a second part we apply the same procedure to Bose gases. From a single image of a quasi ideal Bose gas we determine the equation of state from the classical to the condensed regime. Finally the method is applied to a Bose gas in a 3D optical lattice in the Mott insulator regime. Our equation of state directly reveals the Mott insulator behavior and is suited to investigate finite-temperature effects. The equation of state of ultracold Bose and Fermi gases: a few examples 2
We discuss interacting and noninteracting one dimensional atomic systems trapped in an optical lattice plus a parabolic potential. We show that, in the tight-binding approximation, the noninteracting problem is exactly solvable in terms of Mathieu functions. We use the analytic solutions to study the collective oscillations of ideal bosonic and fermionic ensembles induced by small displacements of the parabolic potential. We treat the interacting boson problem by numerical diagonalization of the Bose-Hubbard Hamiltonian. From analysis of the dependence upon lattice depth of the low-energy excitation spectrum of the interacting system, we consider the problems of ``fermionization'' of a Bose gas, and the superfluid-Mott insulator transition. The spectrum of the noninteracting system turns out to provide a useful guide to understanding the collective oscillations of the interacting system, throughout a large and experimentally relevant parameter regime.
This thesis describes experiments focused on investigating out-of-equilibrium phenomena in the Bose-Hubbard Model and exploring novel cooling techniques for ultracold gases in optical lattices. In the first experiment, we study quenches across the Mott-insulator-to-superfluid quantum phase transition in the 3D Bose-Hubbard Model. The quench is accomplished by continuously tuning the ratio of the Hubbard energies. We observe that the degree of excitation is proportional to the fraction of atoms that cross the phase boundary, and that the amount of excitations and energy produced during the quench have a power-law dependence on the quench rate. These phenomena suggest an excitation process analogous to the mechanism for defect generation in non-equilibrium classical phase transitions. This experiment constitutes the first observation of the Kibble-Zurek mechanism in a quantum quench. We have reported our findings in Ref. [1]. In a second experiment, published in Ref. [2], we investigate dissipation as a method for cooling a strongly interacting gas. We introduce dissipation via a bosonic reservoir to a strongly interacting bosonic gas in the Mott-insulator regime of a 3D spin-dependent optical lattice. The lattice atoms are excited to a higher energy band using laser-induced Bragg transitions. A weakly interacting superfluid comprised of atoms in a state that does not experience the lattice potential acts as a dissipative bath that interacts with the lattice atoms through collisions. We measure the resulting bath-induced decay using the atomic quasimomentum distribution, and we compare the decay rate with predictions from a weakly interacting model with no free parameters. A competing intrinsic decay mechanism arising from collisions between lattice atoms is also investigated. The presence of intrinsic decay can not be accommodated within a non-interacting framework and signals that strong interactions may play a central role in the lattice-atom dynamics. The intrinsic decay process we observe may negatively impact the success of cooling via dissipation because a fraction of intrinsic decay events can deposit a large amount of energy into the lattice gas. In a third experiment, we develop and carry out the first demonstration of cooling an atomic quasimomentum distribution. Our scheme, applied in a proof-of-principle experiment to 3D Bose-Hubbard gas in the superfluid regime, involves quasimomentum-selective Raman transitions. This experiment is motivated by the search of new cooling techniques for lattice-trapped gases. Efficient cooling exceeding heating rates is achieved by iteratively removing high quasimomentum atoms from the lattice. Quasimomentum equilibration, which is necessary for cooling, is investigated by directly measuring rethermalization rates after bringing the quasimomentum distribution of the gas out of equilibrium. The measured relaxation rate is consistent at high lattice depths with a short-range, two-particle scattering model without free parameters, despite an apparent violation of the Mott-Ioffe-Regel bound. Our results may have implications for models of unusual transport phenomena in materials with strong interactions, such as heavy fermion materials and transition metal oxides. The cooling method we have developed is applicable to any species, including fermionic atoms. Our results are available in Ref. [3]
We analyze quantitatively how imaging techniques with single-site resolution allow to measure thermodynamical properties that cannot be inferred from time-of-flight images for the trapped Bose-Hubbard model. If the normal state extends over a sufficiently large range, the chemical potential and the temperature can be extracted from a single shot, provided the sample is in thermodynamic equilibrium. When the normal state is too narrow, temperature is low but can still be extracted using the fluctuation-dissipation theorem over the entire trap range as long as the local-density approximation remains valid, as was recently suggested by Q. Zhou and T.-L. Ho [arXiv:0908.3015]. However, for typical present-day experiments, the number of samples needed is on the order of 1000 in order to get the temperature at least 1010%10 accurate, but it is possible to reduce the variance by 222 orders of magnitude if the density-density correlation length is short, which is the case for the Bose-Hubbard model. Our results provide further evidence that cold gases in optical lattices can be viewed as quantum analog computers.