DIPARTIMENTO INTERATENEO DI FISICA "M. MERLIN" Dottorato di ricerca in Fisica - Ciclo XXXII Settore Scientifico Disciplinare FIS/02 Bound states in the continuum in one-dimensional qubit arrays Dottorando: Domenico Pomarico Supervisori: Ch.mo Prof. Saverio Pascazio Dott. Francesco V. Pepe Coordinatore: Ch.mo Prof. Giuseppe Iaselli ESAME FINALE 2020 Contents Introduction1 I Friedrichs-Lee model simulators3 1 The Friedrichs-Lee model5 1.1 Unstable vacua............................6 1.2 Minimal coupling........................... 10 1.2.1 Canonical quantization................... 12 1.3 Dipole Hamiltonian and Rotating waves.............. 15 1.4 Waveguide QED in a nutshell.................... 19 1.4.1 Quasi-1D free field and interaction Hamiltonian..... 20 2 The resolvent formalism 23 2.1 Resolvent operator.......................... 23 2.2 Schrödinger equation......................... 25 2.3 Survival amplitude.......................... 26 2.4 Fermi golden rule........................... 28 2.5 Diagrammatics and Self-energy................... 30 2.6 Projection of the resolvent in a subspace.............. 33 3 Single emitter decay 35 3.1 Selection rules............................. 35 3.1.1 Approximations summary................. 37 3.2 Spontaneous photon emission.................... 39 3.2.1 Analytical continuation................... 42 3.2.2 Weisskopf-Wigner approximation............. 44 3.2.3 Plasmonic eigenstates.................... 46 iii iv Contents II Emitters pair: bound states and their dissociation 49 4 Entangled bound states generation 51 4.1 Resonant bound states in the continuum.............. 52 4.1.1 Entanglement by relaxation................. 55 4.1.2 Energy density........................ 57 4.1.3 Atomic population...................... 57 4.2 Time evolution and bound state stability.............. 59 4.3 Off-resonant bound states...................... 64 4.4 Extension to generic dispersion relations.............. 69 5 Correlated photon emission by two excited atoms in a waveguide 71 5.1 The two-excitation sector....................... 72 5.2 Self-energy and decay rate of double excitations......... 73 5.2.1 Fredholm integral equations................ 77 5.3 Two-photon amplitude........................ 79 5.4 Two-photon correlated emission.................. 81 5.4.1 Breit-Wigner self-energy expansion............ 86 III Many-body bound states 89 6 Bound states in the continuum for an array of quantum emitters 91 6.1 Physical system and Hamiltonian.................. 92 6.2 Bound states in the continuum................... 93 6.3 Eigenvalues and eigenstates..................... 96 6.3.1 Block-diagonal representation of the propagator..... 96 6.3.2 General properties of the eigenvalue equation...... 98 6.3.3 Large spacing approximation................ 101 6.3.4 Full form of the self-energy................. 103 6.3.5 Atomic populations..................... 110 6.4 Non-perturbative eigenstate pairs................. 117 6.4.1 Cut expansion......................... 120 6.4.2 Unstable states........................ 123 7 Spin waves and multimerization for many-body bound states 127 7.1 Self-energy as a Toeplitz-Hankel matrix.............. 128 7.2 Spin waves............................... 131 7.2.1 One-dimensional regular graph Laplacian matrix.... 132 7.2.2 Rank-one modification of the symmetric eigenproblem. 137 7.2.3 Nonlinear matrix pencils.................. 145 Contents v 7.3 Multimerization............................ 148 Conclusions and Outlook 153 A Waveguide modes 155 B Iterative methods 157 Newton’s method........................... 157 Halley’s method............................ 158 Bibliography 159 Acknowledgements 171 Introduction Things that have worked for a long time are preferable - they are more likely to have reached their ergodic states. At the worst, we don’t know how long they’ll last Nassim Taleb, "The black swan" Quantum systems display a wide variety of behaviors related in an es- sential way with mathematical structures. This characterization includes the complexity a physicist has to face with during experimental tests for a model: at this stage the implementation of approximations represents a key ingredient, which may affect predictions depending on the targeted coarse-grained level. In Chapter1 two example of the latter approach are encountered, consisting in the dipolar and rotating wave approximation used to simulate the Friedrichs- Lee model [42], defined in the restricted one-excitation sector of a Hamiltonian commuting with the total number of excitations, thus representing our family of commuting observables. Fundamental methods to characterize the eigensystem of the presented exactly solvable model are shown in Chapter2, where the Dyson-Neumann expansion is applied to the Hamiltonian resolvent, thus yielding the definition of self-energy. Our interest for this quantity resides in its predictions about the existence of bound states in the continuum according to a non-perturbative approach. This thesis is focused in overcoming the usual approach exploited in waveguide quantum electrodynamics (WQED), based on the linearization of the dispersion relation. If, on one hand, this approximation simplifies the theoretical framework for experimental feasibility, on the other hand it misses emerging phenomena such as plasmonic bound states, associated with non propagating photons, and degeneracy lifting for bound states in the continuum. In a more appropriate terminology these new features are included through a complete characterization of the interaction form factor, which determines the 1 2 Introduction analytical structure of the complex energy plane [40], as made in Chapter3 to test the model for a single two-level emitter. In Chapter4 the generation of entanglement between a pair of emitters, corresponding to a interatomic distance spanned by half-integer multiples of the emitted wavelength, appears as a benchmark for the relation linking the Friedrichs-Lee model and bound states associated with a purely point spectrum immersed in an absolutely continuous one. Renormalized propagator poles are described both numerically and according to an approximation which provides analytic expressions holding true in the weak coupling regime. Plasmonic off- resonance states are introduced, playing a key role also in Chapter5 because they emerge in stronger coupling regimes. Indeed, in the two-excitation sector, the double atomic excitation does not involve any bound state in the contin- uum, but just below the energetic cutoff for photons propagation, where the plasmonic type holds. A correlated emission of photons is characterized ac- cording to the variation of the interatomic distance. The focus on the one-excitation sector led to a complete characterization of the eigensystem for any number of two-level systems in proximity of eigenval- ues corresponding to the resonance condition between interatomic distance and emitted wavelength half-integers multiples, with distances sufficiently large to make the single mode coupling effective. In Chapter6 systems with three and four emitters are described, with a full treatment of the analytic structure imposed by the form factor, whose effect consists in removing eigenvalues de- generacy. Pole trajectories obtained in the variation of free parameters reveal a phenomenon, for small values of the distance, opposite to the splitting out of electromagnetic field in subsystems of the chain observed in the large distance regime: at some critical distances poles scatter along the real line, thus pro- viding the emergence of new eigenstates with the bosonic field covering wider parts of the one-dimensional lattice. In Chapter7 the complexity reduction implemented with the truncation of interactions up to nearest-neighbors char- acterizes the self-energy matrix as a Laplacian one associated with finite differ- ences on graphs, which provides an eigensystem composed by spin-waves, the aforementioned multimerization phenomenon for an odd number of emitters and the presence of super-radiant states as a typification of regular graphs. Part I Friedrichs-Lee model simulators 3 Chapter 1 The Friedrichs-Lee model It will be impossible to answer any one question completely without at the same time answering them all Paul Dirac, "The Quantum Theory of the Emission and Absorption of Radiation" During the 20th century the study of decay phenomena boosted an enormous amout of research in mathematics and physics [2, 25, 38, 85, 102, 120, 121, 136]: considering two major references per field we can mention E. Wigner in his foundation of random matrix theory [137], a topic involving usual tools for the analytical treatment as propagators resummations, E. Fermi with the first theoretical attempt in describing the coupling between vector currents associated with Dirac spinors [52]. The non-relativistic version of the latter approach corresponds to Lee’s model, where a neutral fermionic particle decays in another fermion of lower mass and a scalar relativistic boson, with key ingredients residing in renor- malization for physical quantities as mass, coupling constant and scattering amplitude. The mass of the decaying particle belongs to the free spectrum, while the complete Hamiltonian yields a dynamics not including the initial fermion as a steady state if the difference in energy associated with fermionic masses at rest is more than the scalar boson one. Viceversa, if the initial amount of energy is not sufficient, the fermionic particle remains stable with an eigen- value not included in the continuum [85].