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Lecture Notes in Physics 965 Jonas Larson Erik Sjưqvist Patrik Ưhberg Conical Intersections in Physics An Introduction to Synthetic Gauge Theories Lecture Notes in Physics Volume 965 Founding Editors Wolf Beiglböck, Heidelberg, Germany Jürgen Ehlers, Potsdam, Germany Klaus Hepp, Zürich, Switzerland Hans-Arwed Weidenmüller, Heidelberg, Germany Series Editors Matthias Bartelmann, Heidelberg, Germany Roberta Citro, Salerno, Italy Peter Hänggi, Augsburg, Germany Morten Hjorth-Jensen, Oslo, Norway Maciej Lewenstein, Barcelona, Spain Angel Rubio, Hamburg, Germany Manfred Salmhofer, Heidelberg, Germany Wolfgang Schleich, Ulm, Germany Stefan Theisen, Potsdam, Germany James D Wells, Ann Arbor, MI, USA Gary P Zank, Huntsville, AL, USA The Lecture Notes in Physics The series Lecture Notes in Physics (LNP), founded in 1969, reports new developments in physics research and teaching-quickly and informally, but with a high quality and the explicit aim to summarize and communicate current knowledge in an accessible way Books published in this series are conceived as bridging material between advanced graduate textbooks and the forefront of research and to serve three purposes: • to be a compact and modern up-to-date source of reference on a well-defined topic • to serve as an accessible introduction to the field to postgraduate students and nonspecialist researchers from related areas • to be a source of advanced teaching material for specialized seminars, courses and schools Both monographs and multi-author volumes will be considered for publication Edited volumes should, however, consist of a very limited number of contributions only Proceedings will not be considered for LNP Volumes published in LNP are disseminated both in print and in electronic formats, the electronic archive being available at springerlink.com The series content is indexed, abstracted and referenced by many abstracting and information services, bibliographic networks, subscription agencies, library networks, and consortia Proposals should be sent to a member of the Editorial Board, or directly to the managing editor at Springer: Dr Lisa Scalone Springer Nature Physics Editorial Department Tiergartenstraße 17 69121 Heidelberg, Germany lisa.scalone@springernature.com More information about this series at http://www.springer.com/series/5304 Jonas Larson • Erik Sjưqvist • Patrik Ưhberg Conical Intersections in Physics An Introduction to Synthetic Gauge Theories 123 Jonas Larson Department of Physics Stockholm University Stockholm, Sweden Erik Sjöqvist Department of Physics and Astronomy Uppsala University Uppsala, Sweden Patrik Öhberg IPaQS/EPS Heriot-Watt University Edinburgh, UK ISSN 0075-8450 ISSN 1616-6361 (electronic) Lecture Notes in Physics ISBN 978-3-030-34881-6 ISBN 978-3-030-34882-3 (eBook) https://doi.org/10.1007/978-3-030-34882-3 © Springer Nature Switzerland AG 2020 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland To the memory of Stig Stenholm Preface In the history of science we find many examples where the wheel has been reinvented It may be that some work, for some reason, fell into oblivion or it was simply never recognised by the community In Einstein’s two papers Quantentheorie des einatomigen idealen Gases from 1924 and 1925, he essentially predicted condensation Einstein realised that there is a critical temperature below which a single state becomes macroscopically populated; he calls it ‘condensation’ Interestingly, at that time the concept of an order parameter had not been introduced, but Einstein notices ‘One can assign a scalar wave field to such a gas’ What Einstein was hinting at was what today is known as the Gross–Pitaevskii equation It took, however, Gross and Pitaevskii more than 30 years to write down the equation for the classical field Another example is that of the Aharonov–Bohm effect presented in 1959 However, less known is that the effect was predicted already some 10 years earlier by Ehrenberg and Siday The above gives two examples where a result in its full glory has not been recognised (or it is simply not known) This often happens when essentially the same phenomenon is rediscovered in different communities The 1984 seminal work by Berry on ‘phase factors accompanying adiabatic changes’ had two precursors Already in 1956, Pancharatnam demonstrated how the interference of two polarised light beams depends on a geometric phase While in 1958, Longuet-Higgins et al showed that adiabatic following around a conical intersection between two electronic molecular potential surfaces resulted in a sign change of the wave function Berry’s work showed the generality of this phenomenon; whenever the wave function adiabatically encircles some sort of singularity in parameter space it acquires a geometric phase factor Such singularity appears at a conical intersection which is characterised by a point degeneracy of at least two energy surfaces in some parameter space Conical intersections are to be found in a range of different physical systems, and it seems that the importance of them has often been analysed independently in the different communities Thus, the physics of conical intersections can be regarded as yet another example for where the wheel has been reinvented Even if the concept of conical intersections as such is the same in the different communities, there are vii viii Preface still differences between conical intersections in say molecular and in condensed matter physics In this monograph we gather and discuss various systems where conical intersections have played an important role The similarities and differences are highlighted, as well as drawing attention to the origin of the physics With the ever ongoing experimental progress in the microworld, the boundaries between different fields of physics are becoming ever more diffuse The cooling and controlled manipulation of individual atoms/ions/molecules/photons have led to an avalanche of new physics, ranging from single particle quantum control to in situ explorations of quantum many-body systems The interaction among hundreds of ultracold atoms or ions can be monitored to such a degree that exotic phases of matter can be experimentally investigated and tailored dynamics of closed quantum many-body systems can for the first time be studied in the lab It is possible to construct on-chip electric circuits that realise novel photonic lattice models with an effective photon–photon interaction mediated by superconducting quantum dots This elimination of clear boundaries between the different subfields asks for a better and broader knowledge beyond ones expertise It is our hope that this book can help in bridging knowledge from different areas in physics Two of the main research directions in microscopic physics during the last couple of decades are quantum simulators and topological matter Geometric phases and synthetic gauge theories play central roles in both these fields It turns out that a correct picture of some novel states of matter is only emerging after introducing nonlocal quantities which are directly related to the geometric phase Many of the proposed quantum simulators consider noncharged particles (like atoms and photons), and in order to simulate, for example, charge particles in a magnetic field one needs to construct synthetic magnetic fields The physics of conical intersections serves as a very convenient introduction to synthetic gauge theories The sign change of the wave function upon adiabatically encircling a conical intersection is a manifestation of a geometric phase, and it may be envisioned as if a magnetic flux is penetrating the point of the conical intersection In this sense, it bares many similarities to the Aharonov–Bohm effect This book has been written with the hope that the reader must not be an expert in some particular field in order to grasp its content Ideally, it should be possible to understand the ideas in say molecular or condensed matter physics without having been particularly exposed to those fields Basic knowledge in quantum mechanics is, however, a prerequisite Chapter presents the general background to the topic Central is the concept of adiabaticity, which is essential in order to fully appreciate the following chapters However, it is not crucial to know Born–Oppenheimer theory to understand the content of Chap on condensed matter physics The following four chapters can be read individually, even if there exist a few cross-references between them in order to tie them together We have focused on three main fields in physics where conical intersections are found In Chap we discuss the basics of conical intersections in molecular physics It was also in this field that their importance was first discussed They mark the breakdown of the Born–Oppenheimer approximation and give rise to many observable effects The following Chap is devoted to conical intersections Preface ix in condensed matter physics A crucial difference between conical intersections in molecular and condensed matter physics is that in the former they appear in real space while in the latter they are found in (quasi) momentum space The example known for most people is probably that of Dirac cones in graphene, but we also discuss how they come about in spin–orbit coupled systems and how they generalise to higher dimensions, so-called Weyl points In Chap 5, conical intersections in systems of cold atoms are analyzed Like for molecules, the underlying theory here is that of Born–Oppenheimer But in comparison to molecules, in these systems there is a greater freedom to manipulate the actual model Hamiltonians which in certain cases allows for more clean experimental tests In the book we also include a chapter on other physical systems where conical intersections can emerge In particular, it is shown how Jahn–Teller models that appear in molecular physics are greatly related to the Jaynes–Cummings model which forms a backbone of cavity quantum electrodynamics and trapped ion physics One section of this chapter is assigned to open quantum systems Only within the last years we have seen an enormous interest in ‘non-Hermitian’ quantum mechanics and the importance of exceptional points Such exceptional points can be seen as conical intersections in the complex plane, and they typically show up in the theory for open quantum systems Acknowledgements Over the years, there have been many people who we have collaborated or discussed with, people whom, in one way or another, have had an influence on this book We are especially thankful to Alexander Altland, Emil Bergholtz, Jean Dalibard, Marie Ericsson, Barry Garraway, Gonzalo García de Polavieja, Nathan Goldman, Osvaldo Goscinski, Hans Hansson, Niklas Johansson, Gediminas Juzeli¯unas, Thomas Klein Kvorning, Åsa Larson, Maciej Lewenstein, Jani-Petri Martikainen, Luis Santos, Ian Spielman, Stig Stenholm, Robert Thomson, Dianmin Tong, Manuel Valiente, and Johan Åberg Stockholm, Sweden Uppsala, Sweden Edinburgh, UK September 2019 Jonas Larson Erik Sjưqvist Patrik Ưhberg Contents Introduction References Theory of Adiabatic Evolution 2.1 Introduction 2.2 Adiabatic Time-Evolution 2.2.1 Adiabatic Theorem 2.2.2 Adiabatic Approximation 2.2.3 The Marzlin–Sanders Paradox 2.2.4 The Importance of the Energy Gap: Local Adiabatic Quantum Search 2.3 Gauge Structure of Time-Dependent Adiabatic Systems 2.3.1 The Wilczek–Zee Holonomy 2.3.2 Adiabatic Evolution of a Tripod 2.3.3 Closing the Energy Gap: Abelian Magnetic Monopole in Adiabatic Evolution 2.4 Born–Oppenheimer Theory 2.4.1 Synthetic Gauge Structure of Born–Oppenheimer Theory 2.4.2 Adiabatic Versus Diabatic Representations 2.4.3 Born–Oppenheimer Approximation 2.4.4 Synthetic Gauge Structure of an Atom in an Inhomogeneous Magnetic Field References 5 6 Conical Intersections in Molecular Physics 3.1 Introduction 3.2 Where Electronic Adiabatic Potential Surfaces Cross: Intersection Points 3.2.1 The Existence of Intersections 3.2.2 Topological Tests for Intersections 11 13 13 17 20 23 23 25 27 28 31 33 33 35 35 38 xi 6.4 Open Quantum Systems 145 Fig 6.7 The real and imaginary parts of the APSs for the open E × JT model The losses appear either as adding an imaginary term −γ σˆ z to the Hamiltonian (6.42), (a) and (b), or adiabatically diagonalising the Liouvillian of the Lindblad master equation (6.46), (c) and (d) In the upper two plots, the potentials (6.43) are shown and we see a similar structure as in the Landau–Zener problem shown in Fig 6.6 For the Liouvillian, (c) and (d), there are three surfaces We see an evident resemblance between the two models (due to the construction of the master equation (6.46), (a) and (d), and (b) and (c) should be compared) However, one difference between the two models is the number of EPs; there are twice as many for the Lindblad master equation For this plot, we use dimensionless parameters, and to better visualise the structure of the EPs we take ω = with σˆ = (σˆ x , σˆ y , σˆ z ) By neglecting the kinetic energy term, (6.44) can be expressed in terms of the Bloch vector as ⎡ ⎡ ⎤ ⎤ Rx (t) Rx (t) ∂t ⎣ Ry (t) ⎦ = ML ⎣ Ry (t) ⎦ + bL Rz (t) Rz (t) 146 Conical Intersections in Other Physical Systems ⎡ ⎤⎡ ⎤ ⎡ ⎤ −γ /2 −2x Rx (t) ⎣ ⎦ ⎣ ⎣ ⎦ = 2x −γ /2 −2y Ry (t) + ⎦ 2y −γ Rz (t) 4γ (6.46) The matrix ML is the Liouvillian expressed in the Bloch representation The term bL is sort of a ‘pump’ that prohibits the null vector R = (0, 0, 0) to be a trivial steady state The corresponding adiabatic surfaces for the Lindblad master equation are the eigenvalues of ML , which we show in Fig 6.7c, d Instead of two surfaces, we now have three, and we see similarities between those of the Lindblad equation and those from the non-Hermitian Hamiltonian Note that the imaginary and real parts of the two sort of surfaces have been interchanged; the Bloch equations (6.46) are defined without the ‘i’ on the left- hand side Two additional EPs appear between the surfaces of the Lindblad equation Returning to the matrix ML of (6.46), we note that for the closed Hamiltonian system (γ = 0), the matrix iML is Hermitian (i.e ML is anti-symmetric) This is true for any dimension d—the Liouvillian matrix is anti-symmetric for a closed system The Bloch parametrisation used above can be used for any dimensions d by replacing the Pauli matrices with the generalised Gell-Mann matrices The length of the generalised Bloch vector will be d − 1, so that the number of adiabatic surfaces grows quadratically with the Hilbert space dimension References Scully, M.O., Zubairy, M.S.: Quantum Optics Cambridge University Press, Cambridge (1997) Larson, J.: Absence of vacuum induced Berry phases without the rotating wave approximation in cavity QED Phys Rev Lett 108, 033601 (2012) Niemczyk, T., Deppe, F., Huebl, H., Menzel, E.P., Hocke, F., Schwarz, M.J., Garcia-Ripoll, J.J., Zueco, D., Hümmer, T., Solano, E., Marx, A., Gross, R.: Circuit quantum electrodynamics in the ultrastrong-coupling regime Nature Phys 6, 772 (2010) Larson, J.: Analog of the spin-orbit-induced anomalous Hall effect with quantized radiation Phys Rev A 81, 051803(R) (2010) Schuster, D.I., Houck, A.A., Schreier, J.A., Wallraff, A., Gambetta, J.M., Blais, A., Frunzio, L., Majer, J., Johnson, B., Devoret, M.H., Girvin, S.M., Schoelkopf, R.J.: Resolving photon number states in a superconducting circuit Nature 445, 515 (2007) Leibfried, D., Blatt, R., Monroe, C., Wineland, D.: Quantum dynamics of single trapped ions Rev Mod Phys 75, 281 (2003) Porras, D., Ivanov, P.A., Schmidt-Kaler, F.: Quantum simulation of the cooperative Jahn-Teller transition in 1D ion crystals Phys Rev Lett 108, 235701 (2012) Davis, K.M., Miura, K., Sugimoto, N., Hirao, K.: Writing waveguides in glass with a femtosecond laser Opt Lett 21, 1729 (1996) Mukherjee, S., Spracklen, A., Choudhury, D., Goldman, N., Öhberg, P., Andersson, E., Thomson, R.R.: Observation of a localized flat-band state in a photonic lieb lattice Phys Rev Lett 114, 245504 (2015) 10 Gardiner, C., Zoller, P.: The Quantum World of Ultra-Cold Atoms and Light – Book I, Foundations and Quantum Optics Imperial College Press, London (2014) References 147 11 Lindblad, G On the generators of quantum dynamical semigroups Commun Math Phys 48, 119 (1976) 12 Walls, D.F., Milburn, G.J.: Effect of dissipation on quantum coherence Phys Rev A 31, 2403 (1985) 13 Heiss, W.D.: The physics of exceptional points J Phys A: Math Theor 45, 444016 (2012) 14 Landau, L.D.: Zur Theorie der Energieubertragung II Z Sowjetunion 2, 46 (1932) 15 Zener, C.: Non-adiabatic crossing of energy levels Proc R Soc Lond A 137, 696 (1932) 16 Zhen, B., Hsu, C.W., Igarashi, Y., Lu, L., Kaminer, I., Pick, A., Chua, S.-L., Joannopoulos, J.D., M Soljacic, M.: Spawning rings of exceptional points out of Dirac cones Nature 525, 354 (2015) Appendix A Identical Particles A.1 Second Quantisation The methods and ideas behind second quantisation can be found in numerous textbooks, see, e.g., [1] In this appendix, we follow a less standard, and also less rigorous, approach which still gives the general picture [2] When dealing with N indistinguishable particles, we have that the many-body wave function Ψ (r1 , , rN , t) (A.1) switches sign or remain intact upon swapping any two coordinates ri and rj This is determined by whether the particles are fermions or bosons, respectively Given the wave function, the swapping of coordinates leaves the particle density P (r1 , , rN , t) = |Ψ (r1 , , rN , t)|2 unchanged The wave function is a complex function living in a 3N-dimensional configuration space We are, however, more interested in knowing the particle density in the real three-dimensional space If we start by considering a single particle, we can expand the wave function in some time-independent orthonormal basis, Ψ (r, t) = an (t)φn (r) ⇔ |Ψ (t) = n an (t)|φn (A.2) n The Schrödinger equation in the given basis becomes i h¯ dan (t) = dt φn |Hˆ |φm am (t), (A.3) m © Springer Nature Switzerland AG 2020 J Larson et al., Conical Intersections in Physics, Lecture Notes in Physics 965, https://doi.org/10.1007/978-3-030-34882-3 149 150 A Identical Particles which alternatively can be written as ∂ Hˆ dan (t) = , dt ∂(i h¯ an∗ ) d(i h¯ an∗ (t)) ∂ Hˆ =− dt ∂an (A.4) with the energy expectation value Hˆ = ψ|Hˆ |ψ Equation (A.4) has the same form as Hamilton’s equations for the canonical variables x and p The physical interpretation of the amplitudes an is that |an |2 is the probability to find the particle in state |φn in a measurement If we would repeat the experiment N times, we have that the number of times we would find the system in the particular state is Nn ≡ N|an |2 Alternatively, if we had N non-interacting particles in a single experiment, we are expected to find N|an |2 of the particles in the same state |φn The idea of second quantisation is to replace the amplitudes an and an∗ by operators √ N an → aˆ n , √ ∗ N an → aˆ n† (A.5) and thus Nn → Nˆ n = aˆ n† aˆ n Given the canonical form of the equations of motion (A.4), we impose the commutation relations † ] = δnm [aˆ n , aˆ m (A.6) The algebra of the operators is that of a harmonic oscillator with aˆ n (aˆ n† ) annihilating (creating) a particle in the state |φn Furthermore, Nˆ n is the number operator of the nth state or mode, such that a Fock state with exactly Nn particles in that mode obeys Nˆ n |Nn = Nn |Nn The field operators Ψˆ (r) = aˆ n φn (r), n Ψˆ † (r) = aˆ n† φn∗ (r), (A.7) n respectively, annihilate and create a particle at position r Their commutation relation follows from (A.6), Ψˆ (r), Ψˆ † (r ) = δ(r − r ) (A.8) With these operators we can define the second quantised particle density n(r) ˆ = Ψˆ † (r)Ψˆ (r) that gives the number of particles at position r The second quantised Hamiltonian is analogously given by Hˆ = d rΨˆ † (r)Hˆ 1stΨˆ (r), (A.9) A Identical Particles 151 where Hˆ 1st denotes the first quantised Hamiltonian In fact, any first quantised operˆ has its second quantised companion given by the corresponding ator fˆ1st = f (ˆr, p) expression The Heisenberg equation for the field operator becomes i h¯ ∂ Ψˆ (r, t) = Ψˆ (r, t), Hˆ = Hˆ 1stΨˆ (r, t) ∂t (A.10) The interaction between two particles in the first quantisation form is often given in terms of the distance between the two particles, i.e., V1st (|r1 − r2 |) With the coordinates r1 and r2 for the two particles, we need to include both particles’ field operators Ψˆ (r1 ) and Ψˆ (r2 ) We not present the details here, but just state the result, V1st (|r1 − r2 |) → i
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