QUANTUM MECHANICS ADVANCED COURSE, HT 2017 FMFN01/FYSN17 PROJECTS

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1 QUANTUM MECHANICS ADVANCED COURSE, HT 2017 FMFN01/FYSN17 PROJECTS Within each project, the student group should together elaborate on the respective topic of choice, see below. The literature given should be studied and summarized, and you are encouraged to find additional material through your own literature search. The topic and your conclusions shall then be presented in the form of a short talk, in a group of up to four or five students, in one or two workshops to be held during the scheduled lecture/reporting hours in the last week of the course. Attendance at the workshops and active participation in the group presentation is compulsory! The projects will be distributed according to the s (sent to reimann@matfys.lth.se) we receive with your preference. Your MUST include your full name and your preference (ranked , highest to lowest) of project topics (include the project titles, not only their number!) The rule is first-come-first serve! The absolute deadline to submit your project wish is Friday Sept. 29 at 18:00. After this deadline no more project assignments will be handed out! 1. Electrons in Solids Provide a short summary for the concepts of energy bands and effective mass in a semiconductor. How does current flow in a solid? What is a quantum dot, and how can one in principle experimentally observe its energy levels? 1. Section 16.2 in Bransden-Joachain; 2. S. Tarucha, D. G. Austing, T. Honda, R. J. van der Hage and L. P. Kouwenhoven, Phys. Rev. Lett. 77, 3614 (1996); 3. A. Fuhrer, L.E. Fröberg, J.N. Pedersen, M.W. Larsson, A. Wacker, M.-E. Pistol, and L. Samuelson, Nano Letters 7, 243 (2007). 2. Quantum Dot Helium Discuss the eigenstates of a parabolic quantum dot confining two electrons. How (and why) does the spectrum differ from the single-particle case? 1. D. Pfannkuche, V. Gudmundsson and P.A. Maksym, Phys. Rev. 47, 2244 (1993). 2. T. Chakraborty, Quantum dots (Elsevier 1999) (will be provided)

2 3. Shell Structure in Quantum Dots Describe the general concept of shell structure. Relate this to the series of conductance peaks observed in circularly symmetric quantum dots by Tarucha et al. (see reference below). How does this shell structure change if you modify the geometry of the quantum dot, for example making it elliptic instead of circular? Can you describe this by a simple model? 1. S. Tarucha, D. G. Austing, T. Honda, R. J. van der Hage and L. P. Kouwenhoven, Phys. Rev. Lett. 77, 3614 (1996) 2. D. G. Austing, S. Sasaki, S. Tarucha, S. M. Reimann, M. Koskinen, and M. Manninen, Phys. Rev. B 60, (1999). 4. Wigner Localization in Quantum Dots and Quantum Wires What is a Wigner crystal? Does it occur in solids? In a harmonic trap, how could such a crystallized state form with just a few electrons? Has it been seen in experiments? 1. S.M. Reimann, M. Koskinen, and M.Manninen, Phys. Rev. B 62, 8108 (2000); 2. A. Ghosal, A. D. Guclu, C.J. Umrigar, D. Ullmo, and H.U. Baranger, Nature Physics 2, 337 (2006). 2. L.H. Kristinsdóttir, J.C. Cremon, H.A. Nilsson, HQ. Xu, L. Samuelsson, H. Linke, A. Wacker, and S.M. Reimann, Phys. Rev. B 83, (2011); 5. Quantum Rings How can one confine quantum particles in ring-like geometries? This has been done for electrons in semiconductor heterostructures; discuss the experimental setup. What makes ring geometries so special in quantum mechanics? Has this also been tried with ultra-cold quantum gases? T. Ihn, A. Fuhrer, L. Meier, M. Sigrist, and K. Ensslin, Europhysics News May/June 2005, p. 79, and references cited therein. S. Eckel, J.G. Lee, F. Jendrzejewski, N. Murray, C.W. Clark, C. J. Lobb, W. D. Phillips, M. Edwards, G. K. Campbell, Nature 506, (2014).

3 6. Quantum Dots in Strong Magnetic Fields: Fock-Darwin States What are the eigenstates and eigenenergies of a harmonic oscillator with an applied magnetic field in two dimensions? Show how the states interpolate between the oscillator states at B=0, and the Landau levels. How do interactions in a quantum dot at strong magnetic fields change the picture? 1. Section 2.1 of T. Chakraborty, Quantum dots (Elsevier 1999) 2. R.C. Ashoori, Nature 379, 414 (1996) 3. S. M. Reimann, M. Koskinen, M. Manninen, and B.R. Mottelson, Phys. Rev. Lett. 83, 3271 (1999). 7. Shell structure in large metallic clusters When metallic clusters are large, a strange beating pattern is observed in their abundacies. Can you relate the observations to what you know about the shell structure of a 3D sphere with hard walls? What happens if you soften the confinement? What determines the shape of the effective confinement of all the electrons in the cluster? 1. K. Hansen, H. Nishioka and B. R. Mottelson, Phys. Rev. B 42, 9377 (1990) 2. J. Pedersen, S. Bjornholm, J. Berggren, K. Hansen, T.P. Martin, and H.D. Rasmussen, Nature 353, 733 (1991) 8. Bose-Einstein condensation - theory What is a Bose-Einstein condensate? How does a system of bosonic particles confined in a harmonic oscillator potential differ from one with only fermions? 1. W. Ketterle, Nobel lecture: When atoms behave as waves: Bose-Einstein condensation and the atom laser*, Rev. Mod. Phys. 74, 1131 (2002) 2. C. Pethick and H. Smith, Bose-Einstein Condensation, Cambridge University Press (2002) 9. Bose-Einstein condensation - experiment Describe the early experiments where for the first time, a BEC was created. 1. W. Ketterle, Nobel lecture: When atoms behave as waves: Bose-Einstein condensation and the atom laser*, Rev. Mod. Phys. 74, 1131 (2002). 2. E. A. Cornell and C. E. Wieman, Rev. Mod. Phys. 74, 875 (2002). 3. C. Pethick and H. Smith, Bose-Einstein Condensation, Cambridge University Press (2002).

4 10. Vortices in Bose-Einstein Condensates What happens when you rotate a Bose-Einstein condensate? How could one model such a complicated interacting quantum system? 1. D.A. Butts and D.S. Rokhsar, Nature 397, 6717 (1999) 2. J. Cremon, G. Kavoulakis, B.R. Mottelson, and S.M. Reimann, Phys. Rev. A 87, (2013) 3. J. Cremon, A.D. Jackson, E.O. Karabulut, G.M. Kavoulakis, B.R. Mottelson and S.M. Reimann, Phys. Rev. A 91, (2015) 11. Atomic few-body systems More recently it has been shown that one can quantum-optically confine systems with only a few atoms, where one can control the atom number with high precision. Describe the experiment. Discuss possible impact of this work on future quantum technologies, as they are briefly mentioned in the article. Can you follow up the impact that this research has had in the field (for example, by checking some of its citations)? 1. F. Serwane, G. Zurn, T. Lompe, T.B. Ottenstein, A.N. Wenz and S. Jochim, Science 332, 336 (2011). 2. S. Murrmann, A. Bergschneider, V.M. Klinkhammer, G. Zurn, Th. Lompe and S. Jochim, Phys. Rev. Lett. 114, (2015). 12. Paired Fermions in a 2D Harmonic Trap (conceptionally more difficult) What happens if fermions attract each other? Discuss the structure of the possible quantum states for a system with attractive contact interactions between the particles. 1. M. Rontani, S.M. Reimann and S. Åberg, Phys. Rev. Lett. 102, (2009) 2. G. Zürn, A. N. Wenz, S. Murmann, T. Lompe, S. Jochim, Phys. Rev. Lett. 111, (2013) 3. J. Bjerlin, S.M. Reimann and G. Bruun, Phys. Rev. Lett. 116, (2016)

5 13. Quantum Szilard Engine (difficult but close to actual research) The famous Szilard engine is a thought experiment that shows how one in principle can extract work from a heat bath through information. The article below discusses how such a setup would perform in the quantum world. Could one really build such machines? Does quantum mechanics play a role here? 1. Sang Wook Kim, T. Sagawa, S. DeLiberato and M. Ueda, Phys. Rev. Lett. 106, (2011) 2. J.V. Koski, V. Maisi, J.P. Pekola, and D.V. Averin, PNAS 111, (2014) 3. J. Bengtsson et al., cond-mat arxiv: arxiv: (2017). 14. Self-bound dipolar droplets (difficult but close to actual research) New states of matter were found in ultra-cold atomic dipolar quantum gases, where stable droplets of an atomic dipolar liquid were experimentally discovered. Disucss the experiment and how one could possibly theoretically decribe the origin of these novel self-bound states. 1. M. Schmitt, M. Wenzel, F. Böttcher, I. Ferrier-Barbut and T. Pfau, Nature 539, 259 (2016). 2. I. Ferrier-Barbut, H. Kadau, M. Schmitt, M. Wenzel and T. Pfau, Phys. Rev. Lett. 116, (2016). 15. Quantum Transport with ultracold atoms (Overview article to read) It was more recently suggested that atomic ultra-cold quantum gases are in fact very similar, in many respects, to their solid-state counterparts. Researchers have even suggested to create atomtronic devices that could work with cold atoms instead of electrons. There has been a recent overview article on the subject by Krinner et al., from which you could choose some topics of interest, and discuss their possible feasibility. S. Krinner, T. Esslinger and J.P. Brantut, J. Phys.: Cond. Mat. 29, (2017).