Ultracold Atomic Gases in Optical Lattices: Mimicking Condensed Matter Physics and Beyond
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Advances in Physics Vol. 00, No. 00, Month-Month 200x, 1–135 Ultracold atomic gases in optical lattices: Mimicking condensed matter physics and beyond Maciej Lewenstein, ICREA and ICFO-Institut de Ci`encies Fot`oniques, E-08660 Castelldefels (Barcelona) Spain Anna Sanpera, ICREA and Grup de F´ısica Te`orica, Universitat Aut`onoma de Barcelona, E-08193 Bellaterra, Spain Veronica Ahufinger, ICREA and Grup d’Optica, Universitat Aut`onoma de Barcelona, E-08193 Bellaterra, Spain Bogdan Damski, Theory Division, Los Alamos National Laboratory, MS-B213, Los Alamos, NM 87545, USA Aditi Sen(De), and Ujjwal Sen ICFO-Institut de Ci`encies Fot`oniques, E-08660 Castelldefels (Barcelona) Spain (Received 00 Month 200x; In final form 00 Month 200x) We review recent developments in the physics of ultracold atomic and molecular gases in optical lattices. Such systems are nearly perfect realisations of various kinds of Hubbard models, and as such may very well serve to mimic condensed matter phenomena. We show how these systems may be employed as quantum simulators to answer some challenging open questions of condensed matter, and even high energy physics. After a short presentation of the models and the methods of treatment of such systems, we discuss in detail, which challenges of condensed matter physics can be addressed with (i) disordered ultracold lattice gases, (ii) frustrated ultracold gases, (iii) spinor lattice gases, (iv) lattice gases in “artificial” magnetic fields, and, last but not least, (v) quantum information processing in lattice gases. For completeness, also some recent progress related to the above topics with trapped cold gases will be discussed. 2 Contents 1. Introduction 4 1.1. Cold atoms from a historical perspective 4 1.2. Cold atoms and the challenges of condensed matter physics 6 1.3. Plan of the review 10 2. The Hubbard and spin models with ultracold lattice gases 11 2.1. Optical potentials 12 2.2. Hubbard models 13 2.3. Spin models 15 2.4. Control of parameters in cold atom systems 17 2.5. Superfluid - Mott insulator quantum phase transition in the Bose Hubbard model 19 3. The Hubbard model: Methods of treatment 22 3.1. Introduction 22 3.3. Weak interactions limit 23 3.4. Strong interactions limit 25 3.5. The Gutzwiller mean-field approach 26 3.6. Exact diagonalizations 28 3.7. Quantum Monte Carlo 29 3.8. Phase space methods 30 3.10. 1D methods 31 3.11. Bethe ansatz 33 3.12. A quantum information approach to strongly correlated systems 35 3.12.1. Vidal’s algorithm 36 3.12.2. Matrix product states 38 3.13. Fermi and Fermi-Bose Hubbard models 39 4. Disordered ultracold atomic gases 43 4.1. Introduction 43 4.2. Disordered interacting bosonic lattice models in condensed matter 45 4.3. Realization of disorder in ultracold atomic gases 47 4.4. Disordered ultracold atomic Bose gases in optical lattices 49 4.5. Experiments with weakly interacting trapped gases and Anderson localization 52 4.6 Disordered interacting fermionic systems 55 4.7 Disordered Bose-Fermi mixtures 57 4.8 Spin glasses 60 5. Frustrated models in cold atom systems 63 5.1. Introduction 63 5.2. Quantum antiferromagnets 64 5.2.1. The Heisenberg model 65 5.2.2. The J1 J2 model 66 5.3. Heisenberg antiferromagnets− and atomic Fermi-Fermi mixtures in kagom´elattices 68 5.3.1. Heisenberg kagom´eantiferromagnets 68 5.3.2. Realization of kagom´elattice by Fermi-Fermi mixtures 69 5.4. Interacting Fermi gas in a kagom´elattice: Quantum spin-liquid crystals 69 5.4.1. The quantum magnet Hamiltonian 69 5.4.2. Classical analysis 70 5.4.3. Quantum mechanical results 71 5.5. Realization of frustrated models in cold atom/ion systems 73 5.5.1. Simulators of spin systems with topological order 73 5.5.2. Frustrated models with polar molecules 74 5.5.3. Ion-based quantum simulators of spin systems 75 6. Ultracold spinor atomic gases 77 6.1. Introduction 77 6.2. Spinor interactions 78 6.3. F = 1 and F = 2 spinor gases: Mean field regime 79 6.3.1. F=1 gases in optical lattices 81 6.3.2. Bose-Hubbard model for spin 1 particles 82 6.3.3. F=2 gases in optical lattices 84 6.3.4. Bose-Hubbard model for F=2 particles 85 6.3.5. Spinor Fermi gases in optical gases 88 7. Ultracold atomic gases in “artificial” magnetic fields 89 3 7.1. Introduction – Rapidly rotating ultracold gases 89 7.2. Lattice gases in “artificial” Abelian magnetic fields 91 7.3. Ultracold gases and lattice gauge theories 98 8. Quantum information with ultracold gases 100 8.1. Introduction 100 8.2. Entanglement: A formal definition and some preliminaries 100 8.2.1. The partial transposition criterion for detecting entanglement 101 8.2.2. Entanglement measures 102 8.3. Entanglement and phase transitions 104 8.3.1. Scaling of entanglement in the reduced density matrix 105 8.3.2. Entanglement entropy: Scaling of spin block entanglement 106 8.3.3. Localizable entanglement and its scaling 107 8.3.4. Critical behaviour in the evolved state 108 8.4 Quantum computing with lattice gases 109 8.5. Generation of entanglement: The one-way quantum computer 111 8.5.1. The one-way quantum computer 111 8.5.2. Disordered lattice 113 9. Summary 114 Acknowledgements 114 Appendix A: Effective Hamiltonian to second order 115 Appendix B: Size of the occupation-reduced Hilbert space 117 References 117 4 Motto: There are more things in heaven and earth, Horatio, Than are dreamt of in your philosophy [1]. 1 Introduction 1.1 Cold atoms from a historical perspective Thirty years ago, atomic physics was a very well established and respectful, but ev- idently not a “hot” area of physics. On the theory side, even though one had to deal with complex problems of many electron systems, most of the methods and techniques were developed. The main questions concerned, how to optimize these methods, how to calculate more efficiently, etc. These questions were reflecting an evolutionary progress, rather than a revolutionary search for totally new phenomena. Quantum optics at this time was entering its Golden Age, but in the first place on the experimental side. Development of laser physics and nonlinear optics led in 1981 to the Nobel prize for A.L. Schawlow and N. Bloembergen “for their contribution to the development of laser spectroscopy”. Studies of quantum systems at the single particle level culminated in 1989 with the Nobel prize for H.G. Dehmelt and W. Paul “for the development of the ion trap technique”, shared with N.F. Ramsey “for the invention of the separated oscillatory fields method and its use in the hydrogen maser and other atomic clocks”. Theoretical quantum optics was born in the 60ties with the works on quantum coherence theory by the 2005 Noble prize winner, R.J. Glauber [2, 3], and with the development of the laser theory by M. Scully and W.E. Lamb (Nobel laureate of 1955) [4], and H. Haken [5]. In the 70ties and 80ties, however, theoretical quantum optics was not considered to be a separate, established area of theoretical physics. One of the reasons of this state of art, was that indeed the quantum optics of that time was primarily dealing with single particle problems. Most of the many body problems of quantum optics, such as laser theory, or more generally optical instabilities [6], could have been solved either using linear models, or employing relatively simple ver- sions of mean field approach. Perhaps the most sophisticated theoretical contributions concerned understanding of quantum fluctuations and quantum noise [6, 7]. This situation has drastically changed in the last ten – fifteen years, and there are several seminal discoveries that have triggered these changes: First of all, atomic physics and quantum optics have developed over the years • quite generally an unprecedented level of quantum engineering, i.e. preparation, manipulation, control and detection of quantum systems. Cooling and trapping methods of atoms, ions and molecules have reached regimes of • low temperatures (today down to nanoKelvin!) and precision, that 15 years ago were considered unbelievable. These developments have been recognised by the Nobel Foundation in 1997, who awarded the Prize to S. Chu [8], C. Cohen-Tannoudji [9] and W.D. Phillips [10] “for the development of methods to cool and trap atoms with laser light”. Laser cooling and mechanical manipulations of particles with 5 light [11] was essential for development of completely new areas of atomic physics and quantum optics, such as atom optics [12], and for reaching new territories of precision metrology and quantum engineering. Laser cooling combined with evaporation cooling technique allowed in 1995 for • experimental observation of Bose-Einstein condensation (BEC) [13,14], a phenom- enon predicted by S. Bose and A. Einstein more than 70 years earlier. The authors of these experiments, E.A. Cornell and C.E. Wieman [15] and W. Ketterle [16] re- ceived the Noble Prize in 2001, “for the achievement of Bose-Einstein condensation in dilute gases of alkali atoms, and for early fundamental studies of the properties of the condensates”. This was a breakthrough moment, in which “atomic physics and quantum optics has met condensed matter physics” [16]. Condensed matter community at this time remained, however, still reserved. After all, BEC was ob- served in weakly interacting dilute gases, where it is very well described by the mean field Bogoliubov-de Gennes theory [17]. Although the finite size of the sys- tems, and spatial inhomogeneity play there a crucial role, the basic theory of such systems was developed in the 50’ties.