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arXiv:1007.3529 · Physical Review A

Measuring the equation of state of trapped ultracold bosonic systems in an optical lattice within situdensity imaging

Ping Nang Ma, Lode Pollet, Matthias Troyer

Published Sep 30, 2010• 17 citations
arXivHTMLPDF

Abstract

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 $10%$ accurate, but it is possible to reduce the variance by $2$ 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.

DOI
10.1103/physreva.82.033627

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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.
Beautiful experiments have shown that bosons in optical lattices provide a physical system with an unprecedented degree of control; that they can generate strongly correlated states; and that, as long-term goals, they may be useful for simulating other many-body systems, or realising a quantum computer. Recently a practical scheme for measuring temperature in an optical lattice has been demonstrated, which marks an important milestone necessary for the detailed study of the thermodynamics of this system. In this thesis, we consider both the translationally-invariant lattice, and the combined harmonic trap and optical lattice. We use an extended Bose-Hubbard Hamiltonian which goes beyond the usual Bose-Hubbard approach, and is valid for shallower lattices and higher temperatures, by allowing for beyond nearest-neighbour hopping and excited bands, and we have developed an approximation scheme for off-site interactions. We derive the equations of the Popov approximation to the Hartree-Fock-Bogoliubov method for our Hamiltonian, and diagonalise these equations in the local density approximation (LDA). We examine the density of states of the optical lattice in detail and in various limits. We derive new results on the structure of the density of states, and, in the ideal case, we compare the density of states from the full diagonalisation with our LDA calculation. We make an efficient numerical implementation of our theory and compare the results with the full diagonalisation (for the non-interacting case) and with the limited experimental results currently available. We consider the significance of beyond nearest-neighbour hopping and excited bands and illustrate the properties of our model. In contrast to the trapped gas with no lattice, few thermodynamic results for cold bosons in an optical lattice have been calculated. We analytically derive the first practical formula for the critical temperature in an optical lattice by using simplified shapes for the density of states. We derive corrections for the influence of excited bands, the finite-size effect and mean-field interactions. In all of these cases, we compare our results to full numerical calculations and show that the validity range of our method is complementary to that of the effective-mass approximation, so that the simple descriptions extend over a wide range.
Thermometry is essential for studying many-body physics with ultracold atoms. Accurately measuring low temperatures in these systems, however, remains a significant challenge due to the absence of a universal thermometer. Most widely applicable methods, such as fitting of in situ density profiles, are limited by the requirement of global thermal equilibrium and inapplicability to homogeneous systems. In this Letter, we introduce a novel in situ thermometry for quantum gases, leveraging single-atom-resolved measurements via quantum gas microscopy, and demonstrate it on an ideal Fermi gas. By analyzing number fluctuations in probe volumes containing as little as one atom on average, we extract both global and local temperatures over a broad dynamic range. Our method does not rely on the fluctuation-dissipation theorem and is based instead on the exact relationship between number fluctuations and density-density correlations. In the low-temperature regime, it allows us to observe significant deviations from fluctuation-dissipation predictions, uncovering subextensive fluctuations. Our method is applicable to systems with arbitrary trapping potentials, requiring neither precise trap calibration nor global thermal equilibrium. This thermometer for quantum gases overcomes key limitations of existing techniques, paving the way for more accurate and versatile temperature measurements in ultracold quantum systems.
In this article, we theoretically discuss how quantum simulators based on trapped cold bosons in optical lattices can explore the grand-canonical phase diagram of homogeneous lattice boson models, via control of the trapping potential independent of all other experimental parameters (trap squeezing). Based on quantum Monte Carlo, we establish the general scaling relation linking the global chemical potential to the Hamiltonian parameters for the Bose-Hubbard model in a parabolic trap, describing cold bosons in optical lattices; we find that this scaling relation is well captured by a modified Thomas-Fermi scaling behavior---corrected for quantum fluctuations---in the case of sufficiently high density and/or sufficiently weak interactions, and by a mean-field Gutzwiller ansatz over a much larger parameter range. This scaling relation allows us to control experimentally the chemical potential, independent of all other Hamiltonian parameters, via trap squeezing; given that the global chemical potential coincides with the local chemical potential in the trap center, measurements of the central density as a function of the chemical potential give access to the information on the bulk compressibility of the Bose-Hubbard model. Supplemented with time-of-flight measurements of the coherence properties, the measurement of compressibility enables one to discern among the various possible phases realized by bosons in an optical lattice with or without external (periodic or random) potentials---for example, superfluid, Mott insulator, band insulator, and Bose glass. We theoretically demonstrate the trap-squeezing investigation of these phases in the case of bosons in a one-dimensional optical lattice and in a one-dimensional incommensurate superlattice.
Strongly-correlated electron systems generate some of the richest phenomena and most challenging theoretical problems studied in physics. One approach to understanding these systems is with ultracold fermionic atoms in optical lattices, which can provide a level of control and ways of observing strongly-correlated fermionic systems that are not accessible with conventional materials. This thesis describes the development of an experimental technique where a quantum gas of fermionic 6Li atoms is prepared in a two-dimensional optical lattice and each atom can be frozen in place and imaged with single-site resolution. Combining a vacuum-compatible large numerical aperture microscope with Raman sideband cooling enables site-resolved fluorescence imaging with high fidelity. We observe several phases of the Hubbard model, including band and Mott insulators. The observed in-situ occupation distributions of atoms in the lattice are compared to theory with unprecedented detail and are used to determine the thermodynamic properties of the system. By combining site-resolved imaging with a spin-removal technique, we observe antiferromagnetic correlations in the Hubbard model with single-site resolution. We observe, for the first time in cold atom systems, beyond-nearest-neighbor magnetic correlations, which provide a direct measurement of the correlation length. We also present detailed measurements of the formation of correlations during lattice loading.
Correlation functions play an important role for the theoretical and experimental characterization of many-body systems. In solid-state systems, they are usually determined through scattering experiments, whereas in cold gases systems, time-of-flight, and in situ absorption imaging are the standard observation techniques. However, none of these methods allow the in situ detection of spatially resolved correlation functions at the single-particle level. Here, we give a more detailed account of recent advances in the detection of correlation functions using in situ fluorescence imaging of ultracold bosonic atoms in an optical lattice. This method yields single-site- and single-atom-resolved images of the lattice gas in a single experimental run, thus gaining direct access to fluctuations in the many-body system. As a consequence, the detection of correlation functions between an arbitrary set of lattice sites is possible. This enables not only the detection of two-site correlation functions but also the evaluation of non-local correlations, which originate from an extended region of the system and are used for the characterization of quantum phases that do not possess (quasi-)long-range order in the traditional sense.
We investigate the ground-state and finite-temperature properties of the Bose-Hubbard model using the standard basis operator method applied to the equation of motion for Green's functions. We show that by introducing the self-consistent approach, the theory goes significantly beyond the mean-field approximation, allowing more-precise description of quantum and thermal fluctuations. For optical lattice systems with ultracold bosons, we find relevant quantitative agreement with numerical and experimental data. In particular, the presented method reveals the importance of thermal fluctuations in time-of-flight experiments. For the most reliable fitting of the visibility analytical curve to the experimental data, the value of temperature is determined and proximity to the quantum regime is discussed. The problem of identification of the superfluid--Mott-insulator phase boundary from time-of-flight experiments is also considered.